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NUMERICAL SIMULATION OF THE AIR AND FUEL
MIXING PROCESS IN FUEL-INJECTED ENGINES
USING DISCRETE VORTEX DYNAMICS

By

Asuquo Bassey Ebiana

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1990

(045-61996

ABSTRACT

NUMERICAL SIMULATION OF THE AIR AND FUEL
MIXING PROCESS IN FUEL-INJECTED ENGINES
USING DISCRETE VORTEX DYNAMICS

By

Asuquo Bassey Ebiana

The numerical modeling of turbulent combustion problems continues to invite
the application of novel analytical techniques. Contemporary numerical models seek
to utilize the observable eddy structures of the turbulent flow to simulate the physical
systems directly without recourse to conventional finite difference formulation.
Vortex methods represent such a class of numerical techniques. Vortex methods are
particularly appealing because they are suitable for use in high Reynolds number
flows without the problems of spatial resolution, stability and numerical viscosity

associated with finite difference schemes.

The discrete vortex method is used in this study to simulate the mixing
process prior to chemical reaction in the fuel-injected cylinder of a piston engine.
Mixing descriptions are restricted to a two—dimensional, incompressible, inviscid,
formulation of the bulk diffusion process. The analysis employs the fundamental

conservation equations of mass, momentum and scalar species with appropriate
simplifying assumptions. Although emphasis is limited to inviscid calculations, the
the theoretical and numerical considerations relating viscous and molecular diffusion
effects are fully addressed. The incorporation of viscous and molecular diffusion

effects will be the focus of a future study.

In the present study, the initial vorticity distribution inside the cylinder is
approximated by discrete fluid vortices each of quasi Gaussian-shaped distribution.
The initialization scheme makes use of prescribed velocity data and an imaging
system for the vortices found by modifying the circle theorem of Milne-Thomson.
Fuel injection is simulated using a discrete jet model analogous to Ashurst’s
scheme. At each time step, a grid—insensitive Vortex-In-Cell (VIC) solution
technique is used to approximate the velocity field inside the cylinder. Sub grid
accuracy is achieved through quadratic spline interpolation using Couet’s numerical
filter. Each vortex is advanced every time step using Heun’s second order accurate

time integration scheme.

Predictions of the numerical studies show excellent agreement with theoretical
results obtained using the method of images. Plausible, qualitative flow patterns
consistent with theory and intuition are illustrated. The primary conclusion of this
study is that optimum condition is reached for a 60° injection angle, swirl rate
proportional to 0.125 m2/s and 1.26 m/s injection speed. The artificial complete mixing
time for this optimum condition is estimated to be 0.159 sec. The implication that
there is an optimum swirl strength for minimum complete mixing time could be of

significance in engine design.

Dedicated to the memory of my father, Bassey Esang Ebiana,

who instilled in me the value of education.

ACKNOWLEGEMEN TS

I would first like to express my deepest gratitude to Professor Merle C.
Potter, my dissertation advisor, for the unselfish aid he has given me during the
research and writing of this dissertation. Without his invaluable insight, new ideas,
direction and guidance, much of the intricacies of this project would have been
difficulty to unravel. ' I am very much appreciative of the generous disposition of his
time. Thanks are also due Professors James V. Beck, Mahlon C. Smith and David H.
Y. Yen for serving on my committee. Their comments, corrections and suggestions

are well noted and appreciated.

I am deeply indebted to the Mathematics Department for its generous offer of
graduate assistantships without which I would not have been able to sustain myself
and family financially this far. Special appreciation is extended to Barbara Miller,
Kathy Trebilcott, Dr. Elizabeth Phillips, Professor Charles Seebeck and Professor
Jacob Plotkin for their sensitivity to my financial plight. I am also deeply appreciative
of the department’s recognition of my efforts by way of the Excellence-In-Teaching
citation award accorded me. I wish to thank my department, the Mechanical
Engineering Department, too for its financial support via laboratory teaching
opportunities. In particular, the Fluid Dynamics Laboratory teaching experience was
quite rewarding and I am grateful to Professor John F. Foss for refurbishing my
‘ undergraduate background with the fundamentals of experimental fluid dynamics and
for his guidance and direction.

Special thanks are extended to Dr. Kenneth Ebert of the Office for
International Students and Scholars for his kind disposition and professional
understanding of foreign students’ concerns. Appreciations are extended too to the
Stafi of the Case Center for Computer Aided Engineering who acquainted me with the
VAX/VMS, SUN and HP computer systems and helped me with system related
problems. In particular, Mike Mcpherson, Margaret Wilke, Fred Hall, Greg Fell, Dan
Mcknown and George Lupanoff deserve to be specially acknowledged. I would also
like to thank Mr. Tom Volkening, the chief librarian for his help in searching for
literature materials for me. Thanks too to my colleagues Dr. Bashar AbdulNour and

Dr. Syed Ali for their friendship throughout the years.

I wish to thank my mother, Mrs. Ukpong Bassey Ebiana for all she has done
for me. My parents-in-law Elder and Mrs. Ene Okon Iban and family have been
great sources of financial and emotional support and for this I am indeed very
sincerely thankful. My late sister-in-law Carol Okpok Ene Iban (may God bless her
soul) deserves special acknowledgement for her relentless efforts to encourage and
spur me on. My cousins Mr. Nweme Eyo Eyo, Mr. Asuquo Eyo Eyo and Mr. Abang
Okon Eyo and their families have also been emotionally very supportive and I thank
them for their support. My sincere thanks and deep appreciation to a very special
fiiend Dr. Abiodun Oyewale, for his encouragement and incredible financial sacrifice in
my behalf. I am especially and deeply indebted to my wife, Bassey and daughter
Victoria, for their encouragement in the face of great odds, emotional support,

sacrifice, patience and understanding throughout the duration of my study.

Above all I thank God Almighty for His care, protection and guidance always.

TABLE OF CONTENTS

Page

LIST OF TABLES ....................................................................................... viii

LIST OF FIGURES ....................................................................................... ix

NOMENCLATURE .................................................................................... x111

INTRODUCTION .......................................................................................... 1
CHAPTERS

1. PREVIOUS INVESTIGATIONS ............................................................ 8

1.1 Flow Field .................................................................................... 8

1.2 Mixing Models ........................................................................... 11

2. DESCRIPTION OF PROBLEM ......................................................... 14

3. THE DISCRETE VORTEX METHOD ............................................... 19

3.1 Introduction ................................................................................ 19

3.2 The Flow Field ........................................................................... 20

3.2.1 The interior region .......................................................... 23

3.2.2 Boundary layer region ..................................................... 28

3.2.3 Intake flow ...................................................................... 34

3.2.4 Summary ........................................................ 35

3.3 Exact Solution ............................................................................ 36

iv

3.4 Mixing ........................................................................................ 39
3.5 Choice of Numerical Parameters and Functions ........................ 46
NUMERICAL FILTERS ..................................................................... 47
4.1 Introduction ................................................................................ 47
4.2 Description of Filters ................................................................. 47
4.3 Analysis of Filter Performance .................................................. 51
4.4 Summary ..................................................................................... 54
NUMERICAL METHOD .................................................................... 56
5.1 Conformal Transformation .......................................................... 56
5.1.1 Grid transformation ......................................................... 56
5.1.2 Velocity transformation .................................................. 59
5.2 Point Vortex Initialization Scheme .............................................. 61
5.3 Vortex-In-Cell (VIC) Algorithm ................................................. 63
5.4 Vortex Sheet Algorithm ............................................................. 66
5.5 Advection Scheme ..................................................................... 68
5.6 Sorting Algorithm ....................................................................... 71
5.7 Mixing Algorithm ....................................................................... 72
NUMERICAL EXPERIMENTS
AND DISCUSSION OF RESULTS .................................................... 74
6.1 Introduction .................................................................................. 74
6.2 Optimization of Numerical Parameters ....................................... 75
6.2.1 Mesh size ......................................................................... 75
6.2.2 Injector nozzle diameter .................................................. 77
6.2.3 Vortex core radius ........................................................... 78

6.2.4 Time step .......................................................................... 81

6.3 Validation of Numerical Codes ................................................. 83

6.3.1 Conformal transformation ................................................. 83

6.3.1.1 Grid transformation ........................................... 83

6.3.1.2 Velocity transformation .................................... 84

6.3.1.3 Corner singularities ............................................ 85

6.3.2 Point vortex initialization .................................................. 86

6.3.3 Velocity field and vortex trajectories ............................... 87

6.3.3.1 Velocity field .................................................... 87

6.3.3.2 Vortex trajectories ............................................. 88

6.4 Numerical Simulations .............................................................. 91

6.4.1 Swirling motion with no jet flow ...................................... 95

6.4.2 Jet flow with no swirling motion ...................................... 96

6.4.3 Swirling motion with jet flow ........................................... 97

SUMMARY AND CONCLUSION .................................................... 99

RECOMMENDATIONS .................................................................... 103

TABLES ...................................................................................................... 106

FIGURES ...................................................................................................... 1 13
APPENDICES

Numerical Filters ............................................................................... 168

Al Derivation of Christiansen’s filter as an illustrative example 168

Functions ........................................................................................... 173

8.1

Smoothing function Gij for convergence method [39] ............ 173

vi

B.II Derivation of the streamfunction ‘1’ inside a

finite rectangular domain using method of images .................. 174

C. Coordinate Transformation ............................................................... 179
CI Square (Cm) -> Circle (x,y) ...................................................... 179

CH Circle (x,y) -> Square (Cm) ....................................................... 181

D. Evaluating the Strengths of the Point Vortices .................................. 187
E. List of Computer Programs ................................................................. 189
BIBLIOGRAPHY ........................................................................................ 286

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

LIST OF TABLES

One-dimensional filter models.
Dyer’s experimental data.

Comparison of the absolute value of the relative error (%) between the
modified exact and numerical tangential velocity values at grid points.

Error statistics for grid point coordinate and velocity transformations.

Velocity field initialization data.

L, 0- optimization data (swirl strength S0 = 12.5 cm-m/s).

Nc optimization data (swirl strength S9 = 12.5 cm°m/s).

Optimization data for swirl strength 39'

Figure 1.

Figure 2.

Rm 3.

Figure 4.

Figure 5.

Figure 6.

Figlue 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

LIST OF FIGURES

Three phases of heat release.
Sketch of intake flow.

Numerical filters for vorticity deposition on the grid and for
interpolation of local effects of the velocity field from the grid
to the Lagrangian points.

Vorticity distribution in a grid, with shading showing assignment
of vorticity to grid point for VIC method.

ISO-vorticity contours as represented by the various filters.
Large tick marks represent a half-grid length.

Fourier transform of the various filters.
Inverse Fourier transform of the various filters.

Error estimates between original filter and the inverse of
its transform.

Grid transformation.

Quadratic spline vorticity distribution of three typical
vortices on their nearby grid points.

Zone of influence of a vortex sheet.

Figme 12.

Figure 13.

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18.

Figure 19.

Figure 20.

Figure 21.

Figure 22.

Figure 23.

Vortex sorting.

Relative error between exact and numerical streamfunction values
in the square domain using a single vortex and a system of 105

vortices.

Relative error between the initial and final positions of a point
vortex after one complete revolution.

Radial distribution of tangential velocity component.

Geometric distribution of the system of 115 discrete vortices
(10:1m 1113).

Relative errors in Vr and V6 velocity components as a function
of time.

Trajectories of systems of discrete vortices.

Tangential velocity profiles at discrete time steps representative
of short and long time errors.

Radial velocity profiles at discrete time steps representative of
short and long time errors.

Uniform grid squares superimposed on circular domain.

Complete mixing time as a function of the number of cells
(Swirl strength 89:12.5 cm-m/s, Number of vortices per cell Nc=2).

Complete mixing time as a function of the inlet jet velocity
(Swirlstrength 89:12.5 cm-m/s, Number of vortices per cell N c=2).

Figure 24.

Figure 25.

Figure 26.

Figure 27.

Figure 28.

Figure 29.

Figure 30.

Figure 31.

Figure 32.

Figure 33.

Average complete mixing time as a function of the number of cells
(Swirl strength Se=12.5 cm°m/s. Number of vortices per cell Nc=2).

Complete mixing time as a function of thenumber of vortices per
cell (Swirl strength 80:12.5 cm-m/s. Number of cells L=32).

Complete mixing time as a function of inlet jet velocity (Swirl
strength 89:12.5 cm-m/s. Number of cells L=32).

Complete mixing time as a function of the swirl strength
(Number of cells L=32. Number of vortices per cell Nc=2).

Complete mixing time as a function of the inlet jet velocity
(Number of cells L=32. Number of vortices per cell Nc=2).

Geometric distribution of the system of 115 discrete vortices and
velocity vector representation of the induced swirling motion.

Fuel particle trajectory and vorticity profile at y=0 as a function time.
Injector at 60 degrees (counterclockwise from horizontal).
Um=210.0 cm/s

Streamlines and velocity profiles at y=0 as a function of time
(without swirl). Injector at 60 degrees (counterclockwise from
horizontal). Um=210.0 cm/s.

Air and Fuel particle trajectory and vorticity profile at y=0 as a
function of time. Injector at 60 degrees (counterclockwise from
horizontal). Um=210 cm/s.

Streamlines and velocity profiles at y=0 as a function of time
(swirl). Injector at 60 degrees (counterclockwise from
horizontal). Um=210.0 cm/s. ’

Figure 34.

Figure 35.

Figure 36.

Figure 37.

Figure 38.

Figure 39.

Figure 40.

Figure 41.

Figure 42.

Figure 43.

Fuel particle trajectory and vorticity profile at y=0 as a function time.
Injector at 90 degrees (counterclockwise from horizontal).
Um=210.0 cm/s.

Streamlines and velocity profiles at y=0 as a function of time
(without swirl). Injector at 90 degrees (counterclockwise from

horizontal). Uin=210°0 cm/s.

Air and Fuel particle trajectory and vorticity profile at y=0 as a
function of time. Injector at 90 degrees (counterclockwise from
horizontal). Um==210 cm/s.

Streamlines and velocity profiles at y=0 as a function of time
(swirl). Injector at 90 degrees (counterclockwise from
horizontal). Um=210.0 cm/s.

Fuel particle trajectory and vorticity profile at y=0 as a function time.
Injector at 120 degrees (counterclockwise from horizontal).
Uin=210.0 Cm/S.

Streamlines and velocity profiles at y=0 as a function of time
(without swirl). Injector at 120 degrees (counterclockwise from
horizontal). Um=210.0 cm/s.

Air and Fuel particle trajectory and vorticity profile at y=0 as a
function of time. Injector at 120 degrees (counterclockwise from
horizontal). Um=210 cm/s.

Streamlines and velocity profiles at y=0 as a function of time
(swirl). Injector at 120 degrees (counterclockwise from
horizontal). Um=210.0 cm/s.

Vorticity distribution in a grid, with shading showing assignment
of vorticity to grid point for VIC method.

Method of Images.

NOMENCLATURE

English Letter Symbols

a - Radius of circle in Milne-Thomson’s Circle Theorem.
C,CI,CZ - Constants.

c,s,d - Jacobian elliptic functions with real arguments (also c1,sl,d1).
cn,sn,dn - Jacobian elliptic functions with complex arguments.

D - Finite circular domain representing the combustion chamber.
1). - Total derivative.

Dt

d - Distance between mesh points.

dAiJ - Area element of vorticity assigned to the grid point (i,j).

dj - Injector nozzle diameter.

911,512 - Elements of a 2 x 2 matrix.

F - Matrix of geometric factors.

f(z) - Complex potential function.

f(¢,t) - Distribution function.

Gij - Green’s function or vortex core smoothing function

h - Boundary discretization length. Also, length of vortex sheet

segment.

xiii

Grid index in the x (or C) direction.
Grid index in the y (or 1]) direction.

The complete elliptic integral of the first kind. Also,
dimension of square domain.

Modulus. Also, wave number.

Number of square meshes or cells within the circular domain.
Initial number of vortex sheet segments created at wall test points.
Vortex sheet integer tags. Also number of dimensions.
Number of grid points in either C or 1] directions.
Number of vortex elements or fluid particles.

Number of vortex elements required per cell.

Number of indices.
Unit normal vector on S.
Averaged joint probability distribution function.

Pressure of fluid. Also, instantaneous joint probability
distribution function and parameter of interest.

Radius of combustion chamber.
Reynolds number.

Discrete vortex core radius.

Dimensionless coordinate r/R.
Matrix of discrete point vortex strengths.

Boundary of cireular domain.

‘ Swirl strength.

xiv

SI

W(z)
=(X.y)
(x,y)

z=x + iy

Symbols

Boundary of square domain.

Unit tangent vector on S.

Time.

Artificial time characterizing complete mixing.
Inlet jet velocity.

Velocity at the outer edge of the numerical shear layer
(i.e., velocity at"infinity").

Velocity component in the x-direction.
Velocity component in the y-dircction.
Velocity vector.

Radial velocity component

Tangential velocity component.

Average tangential velocity.

Matrix of tangential velocity components.

Complex potential function.
Position vector.

Cartesian coordinates in the circular domain.

Space coordinate in the complex circular plane.

Dummy variable.

Mixing fiequency. Also average distance between vortices.

XV

Length of time step.

Half-widths for the vorticity distribution in Men g/Thomson’s filter.
Numerical shear layer thickness.

Dirac delta function.

The del operator.

The Laplacian operator.

The coefficient of molecular diffusion.

Turbulent diffusion coefficient.
Discrete point vortex circulation.
Vortex sheet circulation.

Maximum allowable vortex circulation.

Vector with Gaussianly distributed random variables.

x—component random walk due to vortex j.

y-component random walk due to vortex j.

Of the order of.

Discrete point vortex strength.

Reduced wave number.
Mean.
Kinematic viscosity of fluid.

Angle of injection.

xvi

.5
a,

(C31)

Subscripts

Space coordinate in the complex square plane.

Set of o-dimensional scalars.

Velocity potential. Also, scalar function.
Filter function.
Density of fluid.

Number of independent scalar species. Also,
standard deviation (variance = oz).

Set of o-dimensional probability spaces corresponding to (D.

Dummy variable.
Stream function. Also probability space.
z-component vorticity.

Non-dimensional cartesian coordinates in the square domain.

Initial parameter.

For constants and functions.
Variable parameter, a = 1,2,...,o.
Valve lip region.

Valve seat region.

Index in the x(or C) direction.

Index in the y(or 11) direction. Also, jet.

xvii

p - Discrete point vortex.

pt. - Piston.

r - Radial component.

3 - Vortex sheet. Also, shear layer, and source.
t - Tln'bulent.

e — Tangential.

C - Vorticity.

Superscripts

n - Time step number.

6 - Number of scalar species.
t - Turbulent

— - Mean.

- Fluctuating component. Also, transformed boundary.

Abbreviations

CA. - Crank Angle.

CID - Coalescence and Dispersion.

CIC - Cloud-In-Cell.

CPU - Central Processor Unit.

ESCIMO - Engulfinent, Stretching, Coherent, Inter-diffusion, Moving, Observer.

xviii

FISHPAK

IMSL

NCAR

PDF

PLOTIT

TDC

2D

VAXIVMS

VIC

Fast Fourier Transform.
A portable elliptic boundary value problem solver.
Hewlett Packard.
International Mathematical and Statistical Libraries.
Laser-Doppler Velocimetry.
National Center for Atmospheric Research (Graphics package).
Probability Density Function.
Interactive Graphics and Statistics.
Top Dead Center.
Two dimensions.
Virtual Architecture Extended/Virtual Memory System.

Vortex-In-Cell.

INTRODUCTION

In recent years, growing concern over automobile emissions and fuel economy
has spurred great demand for further improvement of the internal combustion engine.
Since efficiency and emissions are influenced by the details of the combustion
process, it is logical that much of the recent engine research has focused on this
process. The important details of combustion include processes such as mixing,
chemical kinetics, vaporization, and so on. These processes are aided and abetted by
turbulent motions which produce large concentration gradients and large surface areas
of interface between fuel and air. To model these processes more accurately, it is
necessary to incorporate new ideas and approaches as our understanding of fluid
motion, turbulent structures, and their effects on combustion in an engine improves.
The current knowledge of the above processes is limited, and so a precise prediction

of these events in an engine has yet to be realized.

Turbulent motion of fluids pose problems of great mathematical and numerical
complexity. The theoretical analysis of turbulent motion has traditionally been by
statistical methods. Because the mechanism of turbulent mixing is intricately linked
to the dynamics of turbulent motion, the statistical theory of turbulent mixing has
been developed parallel to the statistical theory of the turbulent motion problem. In

either case, the choice of a statistical approach leads to a closure problem which

2
requires arbitrary postulates of the unknown terms. In turbulent motion, the closure

is a direct consequence of the non-linearity of the Navier-Stokcs equations.
However, in turbulent mixing it is caused by the non-linearity in the stochastic
variables even though the scalar conservation equation is essentially linear. Because
randomness and non-linearity combine to make the equations of turbulence nearly
intractable, turbulence theory suffers from the absence of sufficiently powerful
mathematical methods. In the absence of a completely formal mathematical theory,
experimental measurements and computer modeling of the actual behavior in a firing

engine have proven to be invaluable tools in the engine design process.

Description of experimental methods to measure and observe turbulent fluid
motion in an engine cylinder abound in the literature. These methods fall into either of
two broad classifications: direct measurements or Schlieren photography. Direct
measurements include the use of hot-wire anemometers for time-resolved
temperature distributions, continuous-wave Raman spectroscopy for fuel and air
concentrations, pulsed laser Raman spectroscopy for density fluctuations, Laser-
Doppler Velocimetry (LDV) for the velocity and turbulence intensity and more
conventional measuring devices for the cylinder pressure. The single point
measurement and directional ambiguity of the hot-wire anemometer, the large
pressure and temperature corrections that must be applied to hot-wire measurements
and the fact that hot wires do not survive the combustion process present significant
difficulties. The use of Laser-Doppler anemometry, which in principle has none of the
above disadvantages, evolved from the recognition that engine design must be firmly
based on the techniques of analysis and fundamental understanding of the in-cylinder

processes. Schlieren photographic methods have been used to provide a highly

3
informative qualitative assessment of the fluid motion inside the combustion

chamber. When coupled with other diagnostic efforts, the visualization allows the
detailed behavior at a point to be placed in context with other phenomena within the
chamber. Thus, the value of each diagnostic is enhanced by the application of the
other. For example, the understanding of the flow structure would alleviate the
directional ambiguity of the hot-wire anemometer and also enable investigators to

choose more strategic locations for point measurements.

A broad range of predictive models and mechanistic theories also abound in
the literatrue. ESCIMO, "population balance", "eddy breakup", coalescence/dispersion
and probability models are but a few of the prominent models proposed by
researchers in the field of turbulent combustion. The most widely applicable and
popular numerical implementation of the predictive models usually consists of finite
difference formulations of the appropriate equations. Although such techniques work
well in many cases, the need for a fine grid in the boundary layer where sharp
gradients exist places a computational upper bound on the size of the Reynolds
number that can be effectively modeled. Analysis implies that at least several grid

points must fall within the boundary layer whose thickness is O(Re'm) and one
always has a finite amount of space available on the computer. This problem is
especially important in cases dealing with flows in wakes or separated flows when
the initial boundary layer may not be visible to a coarse finite difference grid. Another
drawback of finite difi‘erences is that in areas near the boundaries sharp gradients
may give rise to large truncation errors which swamp the original approximation. This
problem is further complicated by the truncation errors which may cause a numerical

viscosity to form which is greater than the viscosity of interest, [32]. Also, strongly

4
interacting transient fuel sprays are difficult to model using finite difference

techniques. The difficulties stem from the fact that the spray contains a wide range of
droplet sizes, length scales and time scales. Furthermore, the inclusion of detailed
combustion chemistry in this approach is far beyond the capability of modern
computers, and cannot be seriously considered as a technique for modeling the

combustion processes in practical combustion devices.

The past two decades have brought enormous advances in computers and
computational techniques on the one hand, and measurements and data processing on
the other hand. The impact of such capabilities has led to a revolution both in the
understanding of the vortical structures which are important in the mechanics of
turbulence as well as in predictive methods for application in technology. The
existence of identifiable, large scale coherent structures (eddies) in turbulent flows,
as observed by Brown and Roshko [l2], Winant and Browand [72], Roshko [61] and
Dimotakis and Brown [27], has led to the growing belief that turbulence or vorticity
fluctuations are not so random as was commonly thought. This development has
prompted several researchers to propose turbulent mixing and combustion models
based on the creation, evolution, interaction and decay of these eddies. Vortex

methods represent a class of this new generation of numerical models.

The discrete vortex method was proposed by Chorin [16,17,18] for modeling
fluid flow. In Chorin’s scheme, the vorticity in the fluid and in the boundary layers is
subdivided into vorticity functions with finite cores (discrete vortices), and the
equations of motion are solved by following the discrete vortices throughout the fluid.

The fundamental advantage of this method is that it is suitable for use in high

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Reynolds number flows without the problems of spatial resolution, stability and

numerical viscosity associated with conventional finite difference schemes. Some
reliable results have been obtained using this scheme [2,16,64]. However, for
Green’s function formulation [45], the dynamics of the vorticity field translates into a

computational effort which scales as the square of the number of vortex elements.

This is due to the fact that if N discrete vortices are present in the fluid, then O(N2)
interactions must be computed per time step, since each vortex influences all the
others. Furthermore, the discrete vortices move according to two components, one of
which is a random displacement. This displacement leads to a partially random
distribution of vorticity which becomes more random as the number of discrete
vortices present in the fluid decreases. Hence to get accurate results, one must use

many vortices or average over ensembles.

In this study, air and fuel mixing processes are examined within the fuel-
injected cylinder of a piston engine. The mixing process is important because it
controls combustion which in turn controls the pollutant formation mechanisms. A
detailed understanding of the mixing phenomena allows advanced engine concepts to
be explored in search of higher efficiency, lower emissions and greater fuel flexibility.

The idea of mixing is best described perhaps by the combination of bulk, eddy
and molecular diffusion effects. Bulk diffusion includes diffusion other than eddy and
molecular, e.g., diffusion caused by the bulk effect of fluid motion. In nu'bulent flows,
the bulk motion of large groups of molecules (eddies) give rise to eddy diffusion.
Molecular diffusion is a product of relative molecular motion across the boundaries of

fluid elements. In any fluid system. molecular diffusion is always occurring and may

6
not always be neglected. Superimposed on this could be either bulk or eddy diffusion

or both. The attainment of sub-microsc0pic homogeneity, where molecules are
uniformly distributed over the field, is the ultimate measure of the effectiveness of
mixing in any mixing process. Chemical reaction occurs only where the fluids are

molecularly mixed.

Fuel-injected internal combustion engines typically employ injectors producing
finely atomized sprays with complex distribution patterns and time-varying injection
rates. The interaction of the injected fuel jet with the primary air motion inside the
engine cylinder compounds the complexity of the fluid motion. The ensuing chemical
kinetics of the combustion process makes the complexity of the fluid motion even
more staggering. Given this consideration, attention is limited to the mixing-
dominated phase prior to chemical reaction. Specifically, the bulk effect of the inviscid

fluid motion on the scalar field is numerically simulated.

The two—dimensional, inviscid calculations employ the discrete vortex method
developed by Chorin in conjunction with the Vortex-In-Cell (VIC) solution technique
to approximate the fuel intake and the mean flow generated by the primary air swirl
inside the combustion chamber. The VIC technique combines the best features of the
vortex tracking concept of the discrete vortices with the stream function-vorticity
finite-difference method. It employs a numerical filter to spread the vorticity onto a
grid for use by a fast Fourier transform routine. The VIC technique is conceptually
more efficient than Green’s function or Biot-Savart integration techniques, in part,
because the work load increases linearly with the number of vortex elements.

Although the numerical simulation of the viscous and eddy diffusion effects and

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7
molecular activity are neglected for simplicity, the theoretical fiamework relating

these effects are fully addressed in this study.

The basis for the simulation is discussed and tested against theoretical
results for the velocity field obtained by the method of images. The numerical
predictions of the effect of bulk diffusion on the scalar field are qualitatively assessed
and shown to be consistent with theory and intuition. Finally, a conclusion is drawn

with regard to the optimum swirl condition for minimum mixing time.

The plan of the rest of this dissertation is as follows: Chapter 1 provides a
brief historical perspective on the evolution of the discrete vortex method. The
direction of earlier efforts, difficulties encountered and the work of previous
investigators are outlined. A discussion of some of the mixing models consistent
with the ideas and techniques of the discrete vortex method is also presented.
Chapter 2 provides a clearer focus on the problem under investigation. The problem
parameters, assumptions and - governing equations are outlined. A more in-depth
discussion of the theoretical basis of the discrete vortex method and its numerical
implementation are presented in Chapters 3 and 5. Chapter 4 provides a description
of the available numerical filters in the literature for use in the VIC method and a
discussion of the considerations leading to the choice of an optimal filter. In Chapter
6 numerical experiments and discussion of results are presented. A summary and
concluding remarks regarding the study are provided in Chapter 7 and in Chapter 8

suggestions for future studies are outlined.

L1

CHAPTER 1

PREVIOUS INVESTIGATIONS

1.1 Flow Field

Calculations using vortices to simulate the gross features of an incompressible
two—dimensional fluid motion have a long history of evolution from Rosenhead’s [60]
hand calculations in 1931 using a few line vortices to recent computer calculations
involving hundreds of vortices. Rosenhead was the first to introduce a method of
analysis in which the interface between two uniform parallel streams of an inviscid,
incompressible fluid is approximated by an array of line vortices. Using similar
calculations, Abernathy and Kronauer [l] simulated the von Karman vortex street
and Westwater [71] studied the rolling-up of a vortex sheet of finite breadth. In this
classical line vortex method, it is assumed that the vorticity is concentrated at a
number of points. This corresponds to approximating the vorticity by a sum of delta-
functions. Each line vortex is moved by the velocity field induced by the other

vortices.

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9
With the advent of computers, Birkhoff and Fisher [10] reexamined and

refined Rosenhead’s calculation by following more line vortices with more accurate
integration time schemes. They showed that some of the vortices actually move
about irregularly in contrast to Rosenhead’s result. It was then apparent that the
velocity field becomes unbounded near a line vortex, and this leads to a spurious
interaction of neighboring vortices. This effect is not present in the original
calculations by Rosenhead and Westwater, possibly because of the small number of
vortices used or the limited accuracy of their calculations. Takami [69] and Moore

[52], using a large number of vortices, indicate that the classical line vortex method

is unreliable.

Most of the earlier efi‘orts were directed toward understanding the time
evolution of finite-area vorticity regions or initial break up of the shear layer in an
inviscid, incompressible fluid. Due to the difficulty with the velocity divergence near
the line vortex center, numerical solutions using line vortices have had some poor
convergence results. In the early part of the last two decades, Chorin and Bernard
[19] initiated a resurgence of interest in the use of vortex methods, for modeling real
high Reynolds number flows of practical significance, when they suggested the use of
discrete vortices (modified line vortices with finite cores) for smoothing the
singularity associated with line vortices. The finite core radius is analogous to the
introduction of a small viscosity which allows the vorticity in the line vortex to diffuse
only a small distance. The discrete vortices are assumed to retain the same shape for
all time. Chorin improved the convergence of flow past a circular cylinder [16] by an

appropriate choice of the core or "cutoff" radius. The inviscid component of the real

flow is solved via the discrete vortices while the viscous effects between vortices are

10
incorporated using a random walk for each vortex. Chorin [17] further improved the

slow rate of convergence near solid boundaries when he introduced a vortex sheet
algorithm to generate finite vortex sheets only as needed to . satisfy the no- slip
boundary conditions in the surrounding potential flow. The rate of convergence for the
various forms of the vortex core smoothing function have been discussed by various
authors including Hald and Del Prete [39], Milinazzo and Saffman [50] and Beal and
Madja [8]

Other investigators have performed discrete vortex calculations on a variety of
flow problems. Ashurst simulated turbulent mixing-layers [2,3], fluid motion in a
form-stroke piston cylinder [4] and various combustion problems [5]. Shestakov [64]
derived a hybrid method for the cavity flow problem. Men g and Thomson [49]
combined the discrete vortex method with the VIC technique to study a class of non-
linear hydrodynamic problems which includes the transport of aircraft trailing vortices
in a wind shear, and the gravity current. A literature search shows that the majority
of the calculations using discrete vortices have been performed for external flows.
The method has not been extensively applied to problems involving internal flows. A
comprehensive review of the general class of vortex methods which should serve as a
good entry point into the literature is given by Leonard [45]. Other notable reviews
appearing elsewhere include Clements and Mull [21] and Zabusky [74,75].

C31:

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1.2 Mixing Models

Contemporary mixing models which utilize the turbulent structure of the flow in
engines is perhaps best illustrated by Spalding [66] in his ESCIMO theory. The
acronym stands for Engulfment, Stretching, Coherent, Inter-diffusion, Moving
Observer. In this model, Spalding describes the engulfment process of the eddies as
well as the details of the internal processes within an eddy. Although the model
appears to conceptually embody much of the physics of turbulent mixing, validation of

the model is still at an early stage.

The Coalescence and Dispersion (CID) model, which admits finite rate
micromixing, was first suggested by Curl [26] when he proposed a rather complicated
integro-differential equation for which no analytical solution is known. Curl’s CID
model was originally intended to describe drop interactions in a two-liquid phase
chemical reactor. It was later extrapolated to gaseous turbulent systems and utilized
in the fields of chemical engineering and turbulent combustion [35,36,57]. The model
suggests a simple mechanism for the exchange of scalar concentration between
initially segregated fluid particles that are fed continuously into a perfectly-stirred
reactor. The feed rate was set to a predetermined rate so that a constant particle
population was maintained. The fluid particles are pictured as undergoing

independent pair collisions, representative of the molecular diffusion time scale.

Levenspiel and Spielman [46] first used the Monte Carlo numerical technique
to solve the integral term in Curl’s equation. The Monte Carlo method is a stochastic

fluid particle interaction model of the turbulent mixing process. It is conceptually

12
simple and especially suited for mixing in discrete systems. The flexibility, clarity and

practicality of this formulation has resulted in its subsequent widespread use. Flagan
and Appleton [36] give a lucid formulation of the Monte Carlo technique for the
solution of Curl’s equation. The works of Shreider [65] and Hammersley and
Handscomb [40] describe various applications of the Monte Carlo method.

The probability density function (PDF) model plays an increasingly significant
role in the theoretical treatment of turbulent flows and turbulent mixing of scalar
fields. Because transport of the PDF is accounted for in both position and probability
spaces, these equations give a more complete description of the system than their
moment counterpart and the physics of the system can be more exhaustively
explored. The PDF formalism is particularly attractive for reacting flows, since the
effects of reaction appear in closed form irrespective of the complexity and non-
linearity of the reaction scheme. These favorable attributes have led several authors
to base turbulence closures on transport equations for the PDF’s. Lundgren [47]
devised a method to derive a hierachy of transport equations for N-point probability
density functions for fluctuating variables in turbulent incompressible flow. The
Lundgren approach has been applied to turbulent flames by Pope [55] and J anicka, et.
al. [42]. Fox [37] and Dopazo and O’Brien [30,31] extended this method to

compressible and reacting flows.

The attraction of the PDF approach appears to diminish when solving the
transport equation. Analytical solutions have been obtained in a few simple cases,
but in general, numerical methods are required. Such numerical methods (e.g., the
Monte Carlo method) have to cope with the integro-differential nature of the

an 45‘

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13
equations and with the large dimensionality of the PDF when many species are

involved. An order of magnitude analysis [56] shows that the finite-difference
solution of the PDF transport equation is impractical for a number of species greater

than 3. The computational expense is found to rise exponentially with the number of

species.

CHAPTER 2

DESCRIPTION OF PROBLEM

The numerical simulation of air-swirl generated inside a flat disc-shaped
cylinder representing the compression volume of a piston en gine combustion chamber
is described in this chapter. Under the effect of a high pressure differential, a fuel jet
is introduced at an adjustable angle into the cylinder through a nozzle orifice located
on the cylinder wall and the problem of the evolution of the velocity and vorticity fields
due to bulk diffusion is addressed.

For liquid fuel injection into the cylinder of a piston engine, an aerodynamic
force will be encountered due to the swirling air in the cylinder. The fuel, internally
ttu'bulent and non-uniform at the exit plane of the nozzle, will exhibit a tendency to
disintegrate that ultimately will lead to the breakup of the fuel into droplets of
different sizes and concentrations in the spray. The distribution of droplet sizes will
depend on the relative magnitudes of the aerodynamic, viscous, inertial, and surface
forces present. The atomized fuel spray will be deflected by the air swirl and will
evaporate and mix with air at the fringe of the jet at a rate determined by the primary
air swirl, turbulent motion, and molecular viscosity. At some time during fuel

atomization and mixing, combustion would occur with the ensuing flame spreading

l4

15
across the chamber.

The time between injection and attainment of chemical reaction conditions is
the physical delay period. Lyn [48] observed that the combustion process can be
divided into three distinguishable heat release phases, as shown in Figure 1. The

first phase is associated with a period of rapid combustion and rapid cylinder pressure

rise and lasts for only about 3° .crank angle (CA). During this phase, fuel that has
evaporated and mixed with air during the physical delay period burns with a non-
luminous flame. The burning rate decays in the second phase which lasts about 40°
CA and the jet burns with a luminous, highly turbulent diffusion flame. In the final

phase, the remaining .fuel burns at a very low rate extending through the entire

expansion stroke.

The present study is concerned with the mixing—dominated physical delay
period prior to chemical reaction. The addition of combustion chemisn-y will be the

topic of a future study.

Experimental studies of the fluid flow in a firing engine cylinder prior to and
during injection, reveals a complex combination of fluid motions. The nature of the
motion in the engine cylinder can be broadly classified into three types : swirl,
turbulence and squish. Swirl denotes a large scale rotary motion of the charge in the
combustion chamber about the chamber axis, more or less. It is created during
induction and amplified during compression by the combustion chamber configuration.

Turbulence is the fluctuating velocity component superimposed on the mean velocity

16
of the fluid motion. Squish denotes the organized radial motion created by the piston

movement near t0p dead center (TDC), that is, near the end of compression. Because
it is anticipated that squish effects will significantly alter the flow patterns at the end
of compression, a right cylindrical disc-shaped cylinder with no squish zones is used
to model the combustion chamber near TDC. The formulation of a mixing model, which
neglects bulk fluid motion typified by the air swirl inside the cylinder cannot hope to

realistically predict design criteria.

The theoretical formulation will consider a system of equations composed of
the conservation of mass, momentum and scalar species. The problem of evolution
toward a uniform final state of the fuel undergoing inviscid bulk diffusion at constant
pressure and temperature will be numerically investigated using discrete vortex
dynamics. At high Reynolds number, the aerodynamics of the flow are dominated by
inertial forces except at the immediate vicinity of the solid surface where viscosity
plays a dominant role. In this study, the effects of molecular viscosity, turbulence and
molecular mixing are neglected for simplicity. The closure of the turbulent transport
term and a satisfactory model for the molecular mixing term in the scalar conservation
equation are particularly challenging. A future study will attempt to incorporate these
effects. A discrete jet model analogous to Ashurst’s scheme [4] and a Vortex-In-
Cell solution technique will be used to approximate the fuel injection and the fluid

motion inside the cylinder, respectively.

Because of the three-dimensional and turbulent nature of the phenomena
occurring within the engine combustion system, it is not advisable to solve the

fundamental equations even with the use of the largest computers. To keep the

l7
mathematical difficulties involved within reasonable bounds, assumptions must be

introduced to allow the simplification of the equations and the evolution of a tractable

mixing theory:

(i) It will be assumed that the controlling features of the problem
can be described by a two-dimensional, constant volume,
mean flow formulation. The flat, disc-shaped configuration of
the cylinder and the fact that combustion in an internal
combustion engine occurs near TDC where the volume does not

change appreciably, justify this assumption.

(ii) For two-dimensional flows, the stretching and rotation term
in the vorticity transport equation is zero. The only nonzero
component of the vorticity vector will be normal to both the

velocity vector and the direction of the piston motion.

(iii) Under engine operating conditions, the chamber conditions are
near or above the thermodynamic critical point of the fuel. Test
results by Shahed, et. al. [63] show that a major portion of the
spray appears in vapor form. Droplets appear only near the jet
nozzle exit within a few orifice diameters. This finding agrees
with the conclusions of several other workers [13,48,53]. For
simplicity, this study represents the fuel jet as air jet. The
consideration of a single phase eliminates complications caused

by fuel droplets.

18
(iv) The change in temperature and pressure of the vaporized

mixture in the physical delay period is minimal. This
assumption can be justified from energy conservation analysis.
Thus the kinematic viscosity is assumed constant since the

pressure is assumed constant prior to combustion.

(v) We will also assume that, to a first approximation, the
atomized fuel spray, idealized as discrete vortices, moves
with the entrained flow. This is a fair assumption because
parametric estimates [59] of the aerodynamic forces on typical
droplets in a fuel spray suggest that the relative velocities of
droplets in air rapidly decay so that droplets move with the

entrained air in the jet flow.

(vi) An incompressible fluid is assumed since there is no volume
source or sink in the flow situation considered. Volume
displacement effects are neglected since the volume occupied
by the fuel is small compared to the volume occupied by the
swirling air flow.

(vii) Gravity is not a factor. Centrifugal convection is due
primarily to the centrifugal body force which is accounted

for in the total derivative.

CHAPTER 3

THE DISCRETE VORTEX METHOD

3.1 Introduction

Modern experimental investigations identifying the existence of large scale
coherent structtues (eddies) in turbulent flows have rekindled much interest in the
numerical modeling of turbulent combustion problems. Contemporary numerical
models seek to utilize the observable eddy structures of the turbulent flow to
simulate the physical systems directly without recourse to conventional finite

difference formulation. Vortex methods represent such a class of numerical

techniques.

The central theme of Chorin’s discrete vortex method is based on the
realization that it is the interaction of the eddies that produces transport of mass,
momentum and energy. The basic concept of the method for flow simulation is to
raprescnt the observable eddy structures in the vorticity-containing regions of the
flow by discrete vortices and then track this discretization in a Lagrangian reference
frame. The essentially grid-flee formulation permits the simulation of the growth and
decay of eddies encountered in real fluids. The method is particularly appealing

19

20
because it is specifically designed to deal with problems of increased spatial

resolution, stability and numerical viscosity associated with finite difference

formulation.

3.2 The Flow Field

At any point in a non-reactive mixing flow system, the state of the fluid can be
characterized by the velocity V, the pressure p and a set of scalars <D=(¢1r¢2r-~-r¢o)
representing the species mass fi'actions; 0' is the number of independent species. As
a consequence of the idealizations discussed in the previous chapter, the

conservation of mass and the Navier-Stokcs equations govemin g the flow field can

be expressed in the following simple form :

Conservation of Mass : V°V = 0 (3.1)
Navier-Stoltes : g—" = - 3'52 + vVZV (3.2)
t

where V=(u,v) is the velocity vector, v, p and p are the kinematic viscosity, density
and pressure of the fluid, respectively. 8145+ VeV is the total derivative, V is the
del operator and v2 is the Laplacian operator. The flow field is satisfied by the
solution . of these equations in a finite circular domain D with boundary S (representing

the combustion chamber), subject to the no- slip boundary condition

V=0 on S. (3.3)

21
The pressure can be eliminated by introducing the vorticity

C: axv - Byu (3-4)

The curl of Eq. (3.2) gives the vorticity transport equation

2; _ 2
Dt -vV C (3.5)

The initial condition C(x,y,t=0) is assumed known in D. The coordinates x and
y are cartesian coordinates and t is time. The discrete vortex formulation makes use
of the simplified system of equations which now consists of Eqs. (3.1), (3.4) and
(3.5) to describe the flow.

A convenient solution device is the operator splitting technique, discussed
extensive by Yanenko [69], whereby the transport of vorticity (Eq. 3.5) can be
implemented in two fractional steps : advection and diffusion. The component
equations are

Advection : (3.6)

9.1.2
g

Diffusion: at: =vV2C (3.7)

22
This technique permits the application of the most efficient numerical algorithm

to compute each part independently and in succession. The combination of the
solutions of the advection and diffusion equations produces an approximation to the

solution for the full flow.

The Euler equation (Eq. 3.6) is equivalent to the statement that the conserved
vorticity field is advected by its own velocity field. The appearance of the vorticity
diffusion term in equation (3.7) has two ramifications : vorticity creation at the
boundary and diffusion of concentrations of vorticity in the flow field. Chorin
[16,17,18] has constructed a vorticity creation algorithm at the boundary using vortex
sheets and suggested that viscous diffusion in both the interior and boundary layer

regions can be accomplished in the Lagrangian framework by adding a random walk

component to each vortex at each time step, At.

The divergence free velocity field can be constructed from the vorticity
distribution through superposing the velocity field produced by each individual vortex.
Often times this is done in terms of the classical Green’s function or by use of the
Biot-Savart integration technique. A hybrid method utilizing Vortex-In-Cell
formulation which employs an efficient elliptic boundary value problem solver for the
interior region of the flow domain and Chorin’s [l7] vortex sheet algorithm near the

solid boundary can also be used to approximate the velocity field.

In Chorin’s vortex sheet algorithm, the random variables are to be drawn from

a Gaussian distribution with zero mean and a variance that grows with time as 2vAt.

This concept of stochastic modeling of diffusion is essentially based on Einstein’s

23
studies of Brownian motion [34] and the analysis by Courant, et. al. [25]. The idea is

that the effects of real viscosity are correctly reproduced in a statistical sense.

3.2.1 The interior region

Since the flow is incompressible and two-dimensional, the velocity

components 11 and v can be expressed in terms of the streamfunction \V as follows :

u =ayw ; v=- Bxul (3.8)

andfrom g=axv-ayu, ' VZW = J; ' (3.9)
with the normal boundary condition

Von= 0 (3-10)

where n is the unit normal vector on S.

An initial distribution of inotational discrete vortices is selected so that an
approximation to the vorticity field is obtained. Equation (3.9) is used to determine u!

up to an additive constant. In terms of the classical Green’s function, it is given by

N
V =j§10ij(zi-zj) 13 (3.11)

jrei

24

th

where N is the number of vortices, zi = (xi,yi) is the i point of interest in the flow

 

domain, zj = (xj,yj) is the position of the jth discrete vortex, Gij(zi-zj) is the
Green’s function and
(r )-
, _ P 1 (3.12)
K1 - 2 it

is the strength of vortex j. The parameter (yP)j is the circulation of vortex j. If the

normal boundary condition is to hold, then at the boundary, the streamfunction must

satisfy

w = constant. (3.13)

Each vortex is moved by the velocity field induced by all the other vortices. Thus
combining Eq. (3.9) with Eq. (3.11) we get

N
11 J = 1°21 ayGij(Zi‘Zj) Kj (3.14)
J¢i
N
vj = -j§laxGij(zi-zj) Kj (3.15)

J¢i

25

where uj and v- are the velocity components at the vortex position in the cartesian

J

coordinate system. To advance the vorticity to the next time step, the velocity is

calculated at the cm'rent position of each discrete vortex and using an appropriate

time integration scheme, the advection step is accomplished.

Typically, the Lagrangian coordinates of the vortices satisfy a non-linear

system of ordinary differenu'al equations :

dx- dy-
—! _ “'(X',Y’) 9 -_J - v (x'9y.)

This system can be solved by giving the time evolution of the coordinates:

t(r+An
n+1 n n
. = . + .
xJ x] f UJ dt
to
tot-At

n+1 n n
O 0 + 0 dt
yJ f vl

"<
e.
II

(3.16)

(3.17)

(3.18)

26
where to is the time at the beginning of integration, At is the time step and n

represents the step number. A higher-order time integration scheme is suggested by
Hald and Del Prete [39].

Viscous difl‘usion of vorticity in the interior region is modeled according to
Chorin’s random walk scheme [16]. If $011,112) is a vector whose components

are Gaussianly distributed random variables, with mean zero and variance 2vAt,

then Eq. (3.5) is approximated by

tot-At
x?“ = x31 + fugldt + (nipj (3.19)
to
to+At
n+1 n n n
to

where (nln)j and (112“)j are the Gaussianly distributed random variables at time

step number n.

It has long been known that the convergence of the rcsultin g flow fields to the
exact solutions of Equations (3.1) and (3.6) is rather poor; the reason is connected

with the singularity in the velocity field when zi=zj, i.e., at the vortex positions. An

obvious modification of the method is to pick Izi-zj I small but not zero, and ensure

27
that V remains smooth as zi—nj. The techniques for doing this are implicit in the

work of Hald and Del Prete [39], Millinazzo and Saffman [50] or Beal and Majda [8].
The form of the vortex core smoothing function or Green’s function Gij(zi'zj) given by

Hald and Del Prete is

 

 

IZ--Z-l lz--z-|
Gij=[cl(1' :33 )-C2(1- 12] )+10gr0] ; lzi-zj|gr0 (3.21)

where r0 is the radius or "cut-off" of the vortex, Cl=2l3 and Cz=3l2 . Outside of this

radius an irrotational motion is assumed and inside of this radius "rigid body" motion

is assumed. For details concerning the accuracy or consistency of Gij(zi‘zj) see

Hald and Del Prete [39].

The application of Green’s function formulation is clearly troubled by the
source singularities and by long running times since a large number of vortices are
required for reasonable accuracy. Recall that O(N2) interactions must be computed

per time step.

This study circumvents these problems by employing the Vortex-In-Cell
(VIC) formulation for the numerical approximation of the inviscid velocity field. In the
VIC formulation, the inversion of Poisson’s equation, (Eq. 3.9), is performed on a grid

superimposed over the vortices by using a Fast Fourier Transform (FFr) method

28
whose work load increases linearly with the number of vortex elements. A numerical

filter, or shape function is applied to each vortex element to temporarily create a mesh
record of the vorticity field. This eliminates the singularity associated with the vortex
position. A centered difference technique, which assures local conservation of mass,
is applied to the mesh record of the stream function resulting from Poisson inversion
to obtain local effects of the velocity field. The filter, again, provides the means for
interpolating the local effects of the velocity field fiom the grid to the vortex positions

which are now updated.

It turns out that interpolation to and fiom the grid are the most sensitive
procedures of the VIC algorithm. The choice of a good numerical filter is therefore
crucial in the successful implementation of the VIC algorithm. A discussion of

numerical filters is presented in chapter 4.

3.2.2 Boundary layer region

The boundary layer region acts as a transition zone from the still fluid layer at
the wall to the high speed flow in the interior. Since the viscous sublayer is thin
compared to the radius of the combustion chamber, the Prandtl boundary layer

equations for incompressible flow over a flat plate

2; _ 2
Dt — vayyC (332)
r; = - ayu (3.23)

29
provide a reasonable description of the propagation of vorticity in this region. Prandtl

boundary layer equations are approximations to the Navier-Stokes equations derived
from the assumptions that : (i) vorticity is mainly produced fi'om large gradients of u in

the y direction, hence C = axv - ayu s - ayu ; and, (ii) along the wall, the large gradients
in the y direction allow 32”; as 0. Here, x and y are directions parallel and normal to

the solid wall, respectively.

The computation of the flow field generated by point and discrete vortices near
solid boundaries has been shown by Chorin [17] to have a slow rate of convergence.
To improve the convergence rate in this region, Chorin introduced the vortex sheet
algorithm to simulate a solution of Prandtl’s boundary layer equations. Vortex sheets
are surfaces parallel to solid walls across which the fluid velocity has a continuous
normal component, but a discontinuous tangential component. The jump in the
tangential velocity is the strength of the vortex sheet. In addition to the improved
rate of convergence, the vortex sheet algorithm, like the VIC algorithm, avoids all
problems with singularities and offers substantial saving in the amount of

computational effort expended.
To describe the motion of vortex sheets, one begins with equations (3.1) and

(3.23). The vorticity in equation (3.23) can be integrated in the form

58
u(x.y) = U880!) + I got, ot)dot (3.24)
y

30
where 83 is the numerical shear layer thickness at the wall, U8:” is u(x,y) at y = 88,

that is, the velocity at "infinity" (the outer edge of the numerical shear layer) seen by
the layer. From Eq. (3.1) and the boundary condition v(x,0) = 0, we get using central

difference approximation

Y Y Y
v(x,y) = - axf u(x,a)doc = - %[Iu(x+h/2,a)da - fu(x-h/2,a)da] (3-25)
0 0 0

Equations (3.24) and (3.25) allow one to determine u and v if the vortex sheet

distribution C = C(x,y) is known.

Consider a collection of N vortex sheet segments each with circulation ys .

The equivalent discrete expression for V = (u,v) at the center r of the jth vortex
sheet segment due to the presence of other segments can be written as [17]
N
u(xj,yj) = U58(xj) + 1243], + zYsi d, (3.26)
i=1
i¢J
(I - 1_)
v(xjsj) = - -’T- (3.27)

31

where

di.,_ '_"i_"i_' ,oserr an yjry, (3,28)

 

h
h N i
= . - Y . -* 2
It U58[xJ:t: 2 ])’J + Z sid‘ yl (3 9)
i=1
#1
x. i .h. _ . i
dii=1- '3 I? X‘I; osdi31 (3.30)
yi”: min(yi.yj) (3.31)

and h is the length of the vortex sheet segment

Note that the total number of interactions between vortex sheets is small in
particular in comparison with what happens when discrete point vortex interactions
are taken into account; one can use sorting algorithms to minimize the number of

decisions involved in carrying out the several summations (see Article 5.6).

32
The position of the sheet is then given by

to+At ‘
x?” = x? + fugidt (3.32)
to
to+At
n+1 n n n .
to

The Gaussianly distributed random variable (112“)j appears only in the y-component

because Prandtl boundary layer equations take into account diffusion in the y direction
only. A first-order time integration scheme is used to advance the solution in this
case. Because the solution is computed in the (x,y) plane, without a change of
variables, it should be easy to match the computed boundary layer solution with an

inviscid solution outside the layer.

The coupling of the interior and boundary layer solutions is provided by U58(x).

This is the tangential velocity from the interior which is used as the velocity at infinity
for the boundary layer. This fact is evident in equations (3.26) and (3.27). This
allows the interior calculation to determine the production of vorticity within the
boundary layer. It should be noted that the number of vortex sheets will increase in
time and that discrete vortices appear in the interior domain only as a consequence of
the displacement of vortex sheets outside the boundary layer, thereby modeling the

mechanism of the generation of turbulence under actual flow conditions.

33
The combined flow fields of the two vortex algorithms must satisfy the no-slip

condition for the velocity. By construction, the normal boundary condition V - n = 0 on

S, is satisfied by both algorithms. However, the no-slip boundary condition V - s
= 0, s tangent on S, will require special attention. The vortex sheet algorithm was
specially designed to impose this condition by mimicking the actual vorticity
generation processes present on the wall and in so doing, provide a vehicle for the

introduction of vortex elements into the computational domain.

If at the wall the tangential velocity V0 it 0, the effect of viscosity will be to

create a thin boundary layer near the wall; the total circulation in the layer per unit
length of the wall is

interior interior a
fgdy = f (- -‘-‘)dy = - V9 (3.34)
o o 3’

Thus, one has to create a vortex sheet of circulation ‘Ys = -V6 per unit length of the

wall to satisfy the tangential no- slip condition.

Intuitively, the vortex sheets move about in ideal flow together with a random
y-component simulating viscous difl’usion. Vortex sheets newly created to
accommodate the boundary conditions diffuse out from the boundary by means of the
random component and then get swept by the main flow. This mechanism then has

most of the features a boundary layer should have.

34
3.2.3 Intake flow

The initial air-fuel composition ¢a(x,0), (a = l, 2) are established by an air
particle injection process simulating the intake fuel flow. The intake process of
internal combustion engines produces thin shear layers that dominate the interior flow
patterns. Since a finite-difference calculation would numerically smear these layers
[38], the dynamics of the injected air particles are analyzed using a discrete jet model

analogous to Ashurst’s scheme [4].

The injector nozzle valve is modeled as a single port with diameter dj equal to

the mesh size (1 (see Figure 2). At each time step, for the duration of the intake
stroke, the intake flow is simulated using two discrete vortices, one created at the
valve lip (region A) and the other created at the valve seat (region B) with the

appropriate sign and strength for the circulation given as [4]

(rp)A = - '2—At ; (7193 = + ——“-At (3.35)

where Uin is the inlet jet velocity, (719A is the source strength at the valve lip and

(7‘93 is the source strength at the valve seat. The source strength is assumed
constant in this study. The valve face is capable of closing the inlet flow area and the

angle of injection Oj can be varied between 0° and 180°. The particles are injected

nth 1

them

solid ll

I132. .

35
with initial velocity sufficient to clear the numerical boundary layer thickness 58. At

the end of the intake flow simulation, the valve face is closed and represented as a

solid wall.

Note that at each time step, for the duration of the injection process, the

number of vortices in the system increases by two and the velocity field has two parts

(3.36)

V(x,y) =V§ + VS

where Va is the velocity produced by all the vorticity that is already in the system and

V8 is the velocity produced by the source represented by the two discrete vortices at

the inlet flow area. The distribution of the jet vortices in the system at later times is

obtained by letting the vortices move with the swirling fluid.

3.2.4 Summary

A summary of the hybrid vortex algorithm for modeling the entire flow field is

as follows :

(1) Find the velocity V for each vortex in the interior flow domain.

(2) Find the value of the tangential velocity at the boundary, and use

this as the free stream velocity to compute V for each element of

33

36
the vortex sheet.

(3) Create vorticity as needed to satisfy the no-slip condition by adding

vortex sheets on the boundary.

(4) Move the interior discrete vortices according to Eqs. (3.19) and (3.20)
and the vortex sheets according to Eqs. (3.32) and (3.33), with the
stipulation that if a sheet moves into the interior, it becomes a discrete

vortex and vice versa.

(5) Steps (1) and (2) gives the velocity V at vortex locations in the
flow field. The velocity at any non- grid point in the flow field can be

found by interpolation.

3.3 Exact Solution

For purposes of comparison, the exact irrotational velocity field in the circular
domain and the streamfunction in the square domain are calculated via an imaging
system for the vortices. A modified circle theorem of Milne-Thomson [51] is

employed for the velocity field in the circular domain.

The circle theorem applies to an irrotational two-dimensional, incompressible,
inviscid flow and states in essence that a flow field’s complex potential f(z) will, upon

the introduction of a circular cylinder into the field of flow, be modified to

W(z) = f(z) +-f(a2/z) (3.37)

37
where the overbar signifies the complex conjugate of the complex function f and a is

the radius of the circular cross-section of the cylinder. Since all singularities of f(z)

are by hypothesis exterior to the circle, all the singularities of f(azlz) are interior to

the circle.

Consider the application of the theorem to the flow modeled by a single point

vortexj with complex potential given by

f(Z) = - i Kj 1n(Z'-Zj) (3.38)

By the circle theorem, the complex potential of the flow becomes

W(z) = - i 13 ln(z—zj) + i ls 1n(a2/z - 2]) (3.39)

which can be rearranged to give

W(z) = - i Kj [ ln(z - zj) - 1n(a2/z-j - z) + ln(zj/z) ] (3.40)

On the right hand side of Eq. (3.40), the first term represents the contribution
of the vortex outside of the circle, the second term is the image vortex located inside
the circle, and the final term, which can be neglected for internal flows [7], is a vortex

located at the origin to preserve the net circulation in the computational domain. With

38
this modification, the complex potential becomes

W(z) = - i Kj [1n(z - zj) - ln(a2/z_j - 2)] (3.41)

The velocity field produced by this complex potential can be expressed as

[u -iv ]j = gig?) .-. -in[ 1/(z - zj) - 1/(z-a2/ 23) 1 (3.42)

The singularities are smoothed using equation (3.21) as shown in Appendix B. The

velocity field due to a collection of vortices can be constructed through superposition :

N
[u-iv] = 2[u-iv]- (3.43)
i=1 1

A similar construction for the streamfunction in the square domain is

presented in Appendix B. The result is

Sinh ° 2

2 Sinh2(Cm) + Sinzmm)

where Cm = {if C-CO). Tim = fight-no) and Tip = ini(n+no)- The parameter

K is the dimension of the square domain as illustrated in Figure 43 in Appendix B.

39
3.4 Mixing

The scalar field ¢a(x,t) (a = 1,2,...,o) in an incompressible flow subject to a
forcing turbulent velocity V(x,t) and undergoing Fickian molecular diffusion is

governed by the scalar conservation equation [54] :

at» .% __a_ Bio.
”[73+vlaxi]'axi(raxi) (3.45)

where p(¢b) is the density, d) = (°l'°2"°"°a) is the set of o-dimensional scalars

th

representing the species mass fractions, Vi(x,t) is the i component of the

prescribed turbulent velocity vector field and I' is the coefficient of molecular diffusion.

Summation convention over repeated indices is implied.

Since averaged quantities are primarily of interest in turbulent flows, use can
be made of a formulation which models and solves the conservation equation for
P(‘P;x,t), the averaged joint probability density function (PDF) of the scalars. The
quantity ‘1' = (wl,vz,...vo) is the o-dimensional probability space corresponding to o.

Henceforth, the argument (‘1';x,t) is written (‘1’), the dependence upon x and t being
implied.

The formal derivation of the transport equation for POP) is given in [54]. The

resulting equation is

 

p[a_P(gL g agm] + an <Vr’ port) > = _a_(1. 833).) _ .02_ <p(‘P)F 331%
at . .

as ax] axi 3*. 3V2 axi
(3.46)
where vi’ is the fluctuating component of Vi; that is
Vi = <Vi> + vi' (3,47)

The averaged joint PDF PC?) can be defined as the ensemble average of the

instantaneous joint PDF p(\I'), a random function due to the dependence of the
distribution of 4’01 upon the initial conditions ca (x,0), and the random forcing

turbulence field V(x,t). Averaging over all possible realizations of initial conditions
and V fields yields

a a
PC?) = H < P(Va;x.t) > = H < 6 (Va - ¢a(xit)) > (3.48)

a=l a=1

where the angled brackets denote an ensemble averaged quantity and 8 is a Dirac
delta function. The justification for defining a probability density as the ensemble

average of a collection of Dirac spikes can be found in Stratonovich [67].

41
It is interesting to observe the correspondences between the physically

recognizable terms in equation (3.45) for the fluctuating scalar quantity ¢a and
equation (3.46) for the averaged joint PDF PCP). The convective term retains its
structure in the POP-equation but the turbulent convective contribution is added. The

molecular transport term appears in the PDF-equation twofold : as a transport term in
position space of the same structure as in the (pa-equation, and as a transport term in

probability space. Molecular transport in position space represents molecular
diffusion while molecular transport in probability space represents mixing by
molecular action. It is shown in [54] that the effect of the mixing term is to reduce the

variance of PC?) without affecting the mean.

Omitting the molecular diffusion terms, which are negligible at high Reynolds

number, equation (3.46) reduces to

[ML 4. <V->a—PSZL] =- 3p (vi p(‘I’) > - .32— <p(‘I’)F %-%-> (3.49)
p at 1 axi axi a\l’2 aXi aXi

The transport equation in this form is not immediately soluble, because, whereas the
term on the left is exact, the terms on the right, representing turbulent transport and

molecular mixing, are unknown and need to be modeled in order to close the transport

equation for POP).

42
The closure of the turbulent transport term is the fundamental stumbling block

to progress in probability formulation of turbulent scalar mixing. This is because
turbulent mass transport or eddy diffusion is a complex process that is dependent on
the scale and intensity of turbulence in the flow. Various models, which tend to make
the mathematics easier but at best are a crude representation of reality, have been

suggested in the literature [28,42,54] to describe the phenomenon. Pope’s [55]

 

simple gradient diffusion model,
at) <Vi' PCP) > _a_ 3P9?)
axi - - Bxi I‘t axi ’ (3°50)

introducing the turbulent diffusion coefficient I‘t(x,t), seems promising and better

behaved for inert scalars and for large departures from Gaussiarrity and homogeneity.

As with the turbulent transport term, a number of models for the molecular
mixing term have been proposed in the literature [26,31,42,55] but a completely
satisfactory model is yet to be demonstrated. The fluid particle coalescence and
dispersion model, proposed by Curl [26], is the only model applicable to an arbitrary
number of scalar quantities and can be expected to produce qualitatively correct
results. The model makes no attempt to describe the detailed dynamics of the
turbulent flow but, instead, relies on a probability density function of concentration,
itself dependent on empirically determined mixing frequency, to describe the non-
uniformities in the flow. Flagan and Appleton [36], Evangelista et. al., [35], and

Pratt [57] have modeled the mixing term in the form

43

- 3f? <p(‘11)r 3:???) = - BP(\P) + 2GB]. P(‘P+‘P')P(‘P-‘P')d‘l" (3.51)

where B(x,t) is the appropriate mixing frequency and ‘I" is a dummy variable. The
first term on the right-hand side models the disappearance of probability in the range
(‘PfiII't-d‘l’) by coalescence with other scalar ranges. The integral represents the

"generation" of probability in the range (\P,‘P+d‘¥) by coalescence from all

concentration ranges.

Given these modeling considerations, Eq. (3.49) can now be written as

pRPBl‘tl'L = Bax—i1": 3%? - pptr) + 2013f meow-mar (3.52)

Equation (3.52) is to be solved in a finite domain D with boundary S. The problem is
posed in the following terms : Given the initial joint PDF P(‘I';x,0), with two equally
probable states (zero and unity) for ¢a(x,t), a = 1,2, and the dynamic equation

(3.52), obtain P(‘i’) for t > 0.

A numerical solution for POP) can be attempted by treating the simultaneous
action of the advection, diffusion and mixing processes sequentially using the

technique of operator splitting mentioned in Section 3.2. The component equations are

Advection : ayatfli + (Vi)? = 0 (3.53)
Edd an 2 8P9?)

Diffleionz p at = 32;; (rt 88 ) (354)
miti‘g'i” p A1313: = - pptv) + 263 f P(‘P+‘P’)P(‘P-‘I”)d‘1” (3.55)

The advection of POP) by the mean flow (i.e., bulk diffusion) is obtained using
the solutions fi'om the analysis of the flow field discussed in Section 3.2. It is
assumed that, to a first approximation, the motion of the injected scalar particle is
identical to the motion of the Lagrangian point of fluid that would occupy the position
of the particle if it were not there. That is, the conserved, passive, scalar field is

advected by the fluid.

Turbulent difiusion of PCP) can be approximated by adding turbulent velocity
components to the droplet velocities. This requires that the turbulence intensity and
length scales be specified. This point remains a formidable closure problem. The use
of random walk in modeling turbulent diffusion was suggested by Chorin [15] as a

generalization of the random walk model used in his vortex sheet method for viscous

45
diffusion.

Molecular mixing of PC?) can be approximated using the Monte Carlo method,
mentioned in Chapter 1 for the simulation of the coalescence-dispersion term. An
essential feature of the Monte Carlo computional technique is that at any location in
the combustion chamber, the joint PDF is represented by an ensemble of fluid particles
which are identified in terms of their individual thermodynamic states. The turbulent
mixing process is described as a sequence of interactions between randomly chosen
groups of n representative fluid particles which mix (with frequency [3) completely
with one another to form it new particles having an identical thermodynamic state.
The simulation is allowed to continue until the ensemble statistics (mean and
variance of properties) are observed to stabilize. Finally, the ensemble average and
other statistics are obtained to represent the properties of interest. The Monte Carlo

method is insensitive to the dimensionality of the PDF and the computational expense

rises only linearly with 6, which is the best that can be achieved by any algorithm.

The combination of the solutions for the advection, diffusion and mixing
equations produces an approximation to the joint PDF solution for the turbulent mixing
problem. However, because of problems posed by closure approximations, the
present study focuses primary attention on the numerical solution of the advection or
bulk difi‘usion component equation. Eddy diffusion and molecular effects will be

incorporated in a future study.

45
3.5 Choice of Numerical Parameters and Functions

Specifying threshold values for the following nine control parameters :

(i) d, mesh size,

(ii) dj, injector nozzle diameter,

(iii) r0, discrete vortex core radius,
(iv) h, boundary discretization length,
(v) At, length of time step,

(vi) 58, numerical shear layer thickness,

(vii) 7m, maximum allowable vortex circulation.

(viii) 1, initial number of "smaller" vortex sheet segments
created at wall test points,

(ix) [3, mixing frequency,

and the proper choice of a vortex core smoothing function Gij and a numerical filter

function (p will produce a discrete vortex calculation which describes a real flow. If the
evolution of the important features of the real flow are to be predicted, then an

appreciation for the interplay between these control parameters must be developed.

Details of the numerical implementation of the foregoing theoretical analysis
are presented in Chapter 5. The numerical experiments germane to the present study

and discussion of results are presented in Chapter 6.

 

CHAPTER 4

NUMERICAL FILTERS

4.1 Introduction

q

Numerical filters serve a dual purpose in Vortex-In—Cell (VIC) formulation.
They are used for vorticity deposition on the grid and for interpolation of local effects
of the velocity field from the grid to the Lagrangian points. These interpolation
procedures are subject to errors. The extent to which these errors are minimized is
dependent upon the design of the filter. The choice of a carefully constructed filter for
use in the VIC algorithm is therefore a critical factor in obtaining consistent numerical
results. This choice is dictated by several considerations including accuracy, grid
independence, a reduction of undesirable grid effects or aliassing, Fourier

transformability and computational efficiency.

4.2 Description of Filters

A survey of the literature shows that there have been essentially five methods
used to derive the proper numerical filter. Some have assumed a "shape" for the
vortex core in real space while others have generated the shape in wave vector space

to obtain some desirable features during the Fast Fourier Transform (Fr-T). A sixth

47

 

 

48
filter developed by Bartholomew [7] is included in this description. An important

consideration is that the remainder of the fluid not sense that an interpolation has
occurred, hence, the vorticity carried by each vortex should be spread to as few mesh

points as possible.

The Cloud-In-Cell (CIC) method, first reported by Christiansen [20] and later
used by Baker [6], assumes each point vortex to possess a uniform or "top hat"
vorticity distribution within a square cell. The distribution involves the nearest two

grid points in a given coordinate direction.

Meng and Thomson [49] used a slightly modified "top hat" profile of

Christiansen. They assumed a vorticity distribution over a rectangle with dimensions

 

 

x and y of one cell according to
§(x.y) «4 1 - 1 (4.1)
3(x-x-) 2 3y-y-)
_.L _.L
[ 1 + Ax ] I: l + Ay :(2

where Ax/3 and Ay/3 are the half-widths characterizing the spreading and xj, yj are

th

the coordinates of the j vortex. Like Christiansen’s filter, the distribution involves

the nearest two grid points in a given coordinate direction.

To improve the numerics of the . CIC method, Hockney et. al. [41] proposed the

use of a truncated-Gaussian core having a larger cross-section (2d x 2d) for the

 

49
vorticity distribution function :

I

m .2
50%) .. exp {- EL } lxil s d
=1

2
2r0 (4.2)

=0 Ixi|>d

where 1 s i S m, r0 = 0.455d, d is the distance between mesh points, xi is the

coordinate in the ith dimension relative to the vortex and m is the number of
dimensions. This distribution involves the nearest three grid points in a given

coordinate direction. That is, the Gaussian distribution is truncated beyond 1.5 grid

lengths.

Couet [23,24] and Wang [70] developed their Gaussian distributions for the
vorticity in wave vector space and then transformed them into real space. The results
gave quadratic and cubic splines, respectively, and involve the nearest three and four

grid points, respectively, in a given coordinate direction.

Bartholomew [7] assumed a polynomial expression to describe the fraction of
the total circulation credited to the nearest three grid points in a given coordinate
direction.

These distribution functions can be put into a common format so that in two
dimensions the filter is the product of two one-dimensional filters. These one-

dimensional filter models, the scheme used in generating them, the number of nearest

50
grid points involved for each scheme and their authors are listed in Table 1. It is from

Bartholomew’s perspective of using the definition of circulation that the compact
forms of the filters, with the exception of Couet’s and Wang’s, were independently
derived. Profiles of the six filters are illustrated in Figure 3. It is observed that the
profile of Meng/Thomson’s filter is in fact a slight modification of Christiansen’s
triangular profile. The others appear to assume a quasi-Gaussian profile which is

brought smoothly to zero at the extent of each filter.

A generalization of the formulations listed in Table 1 can be made as follows :

A uniform rectangular grid system having coordinate directions x and y is first defined

as shown in Figure 4. Next, an assumed vorticity distribution function

§(x,y) = f(x,y). (4.3)

with the point vortex coordinates (xp,yp) taken to lie at the center of this distribution,

is normalized, C(Cm), where C=x/d and n=y/d, and a local grid point which serves as

the origin for the coordinates of the discrete vortex is chosen. The normalized filter

tp(C,t|) which determines the fiaction (71,)”; of the total circulation 71, that is credited

to the surrounding grid points of interest is now obtained by integrating the

normalized vorticity distribution over the overlapping area in the cell enclosing the

grid point :
<P(C.11) = C I €(C.Tl)dAi J- = <P(C)(P(Il) (4.4)

and (71,)” = (“0‘“an (4-5)

51
The quantity C is a constant, dA1,j is the area element of vorticity assigned to the grid

point (i,j) of interest and (p(C), (pm) are one dimensional filters. The grid point which
served as the origin is now generalized to give the compact expression for the filter.
This interpolation scheme has been sometimes referred to as "area weighting" in the

literature. An illustrative example of this procedure for Christiansen’s filter is

provided in Appendix A.

To reassemble the velocity information from the grid to the Lagrangian point
of interest, each grid point in the neighborhood of the Lagrangian point is assigned a
weighting value which shows the fraction that the Lagrangian point contributes to the

total velocity field estimate.

Because the expressions <p(C,n) for the filters are algebraically complicated,
they are numerically evaluated. Two-point Gaussian-Legendre quadrature has been
found to be sufficient for all the filters but Meng/Thomson’s. In this case, a ten-point

formulation is required for convergence.
4.3 Analysis of Filter Performance

Filter performance with regard to accuracy, grid independence, Fourier

transformability and computational efficiency are discussed in this section.

One possible measure of the formulation’s accuracy is its ability to satisfy

integral constraints of the fluid motion such as the conservation of circulation. How

 

52
well the local circulation is conserved in the Eulerian reference frame can be tested

analytically or numerically by varying the position of a single discrete vortex within a
square cell. The circulation deposited to the nearest grid points of interest is then
summed and checked to see if it reproduces the correct amount of circulation.
Analytical results show that the total circulation is conserved exactly for all of the
filters except Bartholomew’s. Numerical evaluation of Bartholomew’s filter gave an

error on the order of 0.01%.

Grid independence refers to the fact that the VIC formulation can resolve
length scales smaller than the grid scale used by FFI‘. For the formulation to be
essentially grid fiee, a necessary requirement is for the spreading filter to exhibit a
Spherical symmetry that is unaffected by the existence of the grid points. This point
is best understood from the following argument: consider a point in an infinite three-
dimensional space. Let the point diffuse for a short time interval, and then "turn-off"
the viscosity. The exact solution for this process, according to [14], shows that the

amount of vorticity at any location is proportional to

2 2
to...) - eagle) .. ”Mg-5;) ecu-3g) wage?) «to

where r is the distance from the point vortex to the spatial point of interest (x,y,z), v
is the kinematic viscosity and t is the time. This distribution is referred to as the "3D

filter" in Figure 5. To obtain the corresponding two-dimensional result, one must

distribute points along a line.

 

53
Figure 5 show iso-vorticity contours of the function (p(C,n) for five of the filters

as well as the 3D filter. The large tick marks on the figure represent a half grid
length. Hockney’s filter bears a striking similarity to Couet’s filter and has therefore
been omitted. It is evident that the 3D filter as well as Couet’s and Wang’s filters
are essentially spherically symmetric, implying that they exhibit considerable subgrid
resolution and therefore less numerical instability. The contours developed by
Christiansen’s, Meng/Thomson’s and Bartholomew’s filters tend to "square-off" into
rounded corners which translate into a positional dependence that is not grid

independent.

In addition to our concern for a spherically symmetric filter, one must be
concerned with the behavior of the filter in transform space where a mesh can resolve
only non-dimensional wave numbers less than 1.0. The Fourier transform of the
filters are shown in Figure 6. Any harmonic with a non-dimensional wave number
greater than 1.0 will be misinterpreted and thus will, in effect, add a mixture of
overtones to the approximation of the pure harmonic. This mesh-induced fluctuation

is known as "aliassing" and aliassing sets a limit at (klrt)max = 1.0 in transform

space. Christiansen’s and Meng/Ihomson’s filters produce the worst aliassing.
Wang’s filter transforms with virtually no "halos". However, even when the
transform smoothly approaches zero, there are still a significant number of
components outside a non-dimensional wave number of 1.0. This fact will prove

troublesome when the inverse transform is performed.

Figure 7 shows the inverse transform of the filters when a non-dimensional

wave number cutoff of 1.0 is used; with the exception of Wang’s filter which uses a

 

54
cutoff of 1.5. The "halos" in these functions are due to the fact that the transform

process can only resolve wave components through 7: modes. This problem would be
essentially non—existent if the transform could include modes through 27:. In practice,
the collection of vortices used in modeling a vortical region would result in a
"smoothing" of the solution and an additional reduction in the "halos". The result of
an error analysis between the original filter and the inverse of its transform for a wave
number cutoff of 1.0 is represented in Figure 8. Bartholomew’s, Couet’s and

Hockney’s filters are seen to be most recoverable with error in the neighborhood of

i6%.

A final criterion for choosing the optimal filter is based upon the central
processor unit (CPU) times for evaluating the filter and for using the filter to spread
vorticity to grid points. The results of these computational efforts are shown in Table
1. It is no surprise that from an execution speed point of view, Hockney’s filter is
least efficient since it requires 24 error function evaluations for spreading in 2D.
Christiansen’s filter evidently sacrifices accuracy for efficiency. On the average, the

CPU times for Meng/Ihomson’s and Couet’s filters compare quite well.

4.4 Summary

This investigation has highlighted the specific areas for consideration when
deciding on the choice of a numerical filter. A good filter should be grid independent,
conserve local circulation, minimize aliassing errors, be computationally accurate and

efficient, and lastly, certainly not the least, be in a convenient compact form for easy

implementation.

 

55
Based upon the preceeding discussion of spherical symmetry, Fourier

transformability, compact representation, accuracy and computational efficiency, the
filter developed by Couet represents, in this researcher’s best judgement, the best
choice of filter presently available in the literature. Couet’s filter was therefore
chosen for the VIC algorithm employed in this study to both spread the vorticity and

to reconstruct the velocity field.

CHAPTER 5

NUMERICAL METHOD

5.]. Conformal Transformation

The application of conformal transformation in potential theory provides the
flexibility for performing calculations in a simpler domain. In this study, conformal
Walnsformation of position and velocity is a necessary requirement imposed by the
chOice of the VIC formulation. The VIC algorithm is implemented on a uniform square
md and the resulting velocity field is transformed onto an equivalent non—uniform grid
system in the circular domain in order to advance the solution in time. Computational

economy dictates that only the mesh record of the position and velocity information be

traxlsformed. Information at off-grid locations is obtained via an interpolation scheme.

5 ‘ 1'- 1 Grid tramformation

The basis for the transformability of the grid systems shown in Figure 9 is
pr'b‘rided [11] by the equation

sn(c0)dn(00)

011(0)) . (5.1)

56

57
where z= x + iy and c0 = C + in; the coordinates (x,y) and (C,‘r]) are non-dimensional

grid point coordinates in the circular and square domains with boundaries S and 8’

respectively; the functions cn((l)), mm» and sn(a)) are J acobian elliptic functions with

complex arguments.

The transformation (5.1) will transform the square -Kf2 s C s K/2, -K/2 s n s M

onto the unit circle lz |=l The complete elliptic integral of the first kind K, defined as

do:

V1 - kzsinza

 

K = (5.2)

O'HNIFI

where the modulus k is a constant, is evaluated using Gauss-Legendre quadrature.
Note that the length of the square is equal to the value for K. The parameter or is a
dumIDy variable. The square domain is divided into M x M grid squares giving the
grid point coordinates (Cm). M is the desired number of grid points in the c and n
d‘il‘eCtions. The corresponding grid point coordinates (x,y) are determined by first
re<1‘1<:ing Eq.(5. 1) into a system of non-linear algebraic equations of the form

x = f(C.cl.d.d1.S.sl.k) = f(CJtrk) (5.3)

Y = g(csclrd9d1rssslrk) = g(Csnrk) (5.4)

58
where c = cn(C.k). c1 = cn(Tt,k’). d = dn(C.k). d1 = dn(n.k’). s = sn(§.k) and S1 =

sn(11.k’) are Jacobian elliptic functions with real arguments. The complementary
modulus k’ = W Equations (5.3) and (5.4) are derived in Appendix c. For a
square domain, it can be shown that k’2 = k2 = 0.5. Next, the elliptic functions are
numerically evaluated using a descending Landen transformation [11] and subtitution
in (5.3) and (5.4) gives the grid point coordinates (x,y). The resulting corner
singularities in the circular domain (see Figure 9) would prove to be a troublesome
factor with regard to interpolation procedures. This point is discussed further in

chapter 6.

The reverse transformation, (x,y) —> (Cm), involves an elaborate four—step

procedure:

(i) The functions s(C,k) and 31(11 ,k) are pretabulated.

(ii) Equation (5.1) is reduced to another system of non-linear,

algebraic equations of the form (see Appendix C)

f(k,s,sl,X) = O (5.5)

f(krssSPY) = 0 (5.6)

(iii) Equations (5.5) and (5.6) are then solved simultaneously for

s and SI, for every x and y input, using a quasi Newton-Raphson

numerical method.

59
(iv) The coordinates (Cm) are recovered floor the pretabulated functions

s(C,k) and sl(n,k) via linear interpolation.

5.1.2 Velocity transformation

Let a fluid motion inside the square region of the (u-plane be given by the

complex potential,

W(m) = q, + hp (5.7)

where e is the velocity potential and \y is the stream function. Then at corresponding
points z and 0), given by Eq. (5.1), W and therefore it and ‘1’ take the same values.
Now, 8’ is a boundary and so a streamline, and therefore 1]] = C, a constant, at all
points of 8’. Since S corresponds point by point with S’, u! = C at all points of S.
Therefore S is a streamline, in the motion given by (5.7) and (5.1) together, in the

z-plane.

The actual form of the complex potential in terms of 2 would be obtained by
eliminating (0 between (5.1) and (5.7), but it is often preferable to look on 2 as a
parameter and forgo the elimination. Thus, to find the fluid velocity at (x,y) in the z-

plane corresponding with (Cm) in the (D-plane, we have

d_W_ d_VY.d_w
dz ‘ d0) dz (5'8)

60
The equivalent matrix form of Eq. (5.8) can be written as

[3]

_ c11 e12 11]
(x,y) ‘ [W ”11 " (5'9)

(Cm)

where en, cm are elements of the 2 x 2 matrix. The velocity components in the z-

plane can be transformed to give velocity components in the cylindrical coordinate

system as :

V9 = vcosO - usin0 (5.10)

V = -vsin0 - ucosO (5.11)

Conversely, the fluid velocity at (Cm) in the (tr-plane corresponding with the
fluid velocity at (x,y) in the z-plane is given by

dW dW dz
m=arrs (m)
01’,
1°] [in “1
= 5.13
V (Cm) ’12 ell " (x,y) ( )

Notice that the matrix system in Eq. (5.13) is the inverse of that in Eq. (5.9).

 

61
Representation of the grid velocity field in the circular domain is done via

conformal transformation as discussed above. Off- grid point velocity calculation in
either domain can be accomplished using an interpolation routine relevant to each grid

system.

5.2 Point Vortex Initialization Scheme

The irrotational vorticity field at the initial time can be approximated using
discrete vortices in two ways. The first scheme involves analyzing the problem from
the point in time when the mechanism that is responsible for the production of
vorticity within the boundary layer first introduced vorticity into the interior domain to
the point where the flow field matches the desired initial conditions. This method is
computationally expensive. It would be computationally more advantageous to
develop a method that generates the initial distribution and strengths of the vortices
from a prescribed velocity field directly, or alternately, from the derivative of the

velocity field. In either case an iterative method is implied.

The initialization scheme employed in this study makes use of the tangential
velocity field prescribed by the experimental data of Dyer [33]; see Table 2. The data
is classified according to a time to which is the time from the closing of the shrouded
intake valve to the time of measurement. Dyer’s data were used to represent
different initial swirl conditions for this study. The strengths and optimum distribution

of the discrete vortices are determined as follows :

62
(i) Discrete vortices are arbitrarily but uniformly placed in

concentric circles about the origin of the circular domain.

(ii) The modified circle theorem of Milne-Thomson is applied
to the system of vortices to compute geometric factors and
using Dyer’s velocity data, a matrix inversion routine supplied

by IMSL is used to determine the strengths of the vortices.
(iii) The optimum distribution is found when the sum of the
circulation due to each vortex is equal to the global

circulation 21tRVe—, i.e.

N N
2 (719,- =21tR_V9_ or 2 K,- = Rig (5.14)
i=1 i=1

where N is the number of discrete point vortices. (7p)j is the circulation of vortex j, 1%

is the strength of vortex j, R is the radius of the combustion chamber and-VB is the

average tangential velocity or slip velocity obtained by numerically integrating Dyer’s

experimental velocity data.

To obtain the strengths of the discrete point vortices, consider an arbitrary

distribution of the vortices as noted above. The velocity at any point zk inside the

circular domain is the sum of the velocities due to all vortices located at 2]- given by

63

the modified circle theorem:
N
[u - i v ]k = 21 -itq [ 1/(zk - zj)-1/(zk - aZ/z‘jn (5.15)
1...

Equation (5.15) is composed of three factors : the desired velocity (u—iv)k, the

discrete point vortex strength Kj and a geometric factor relating the influence of vortex

j at point 21‘. This information can be represented in matrix form as

where (V e)k’ ij, Sj are matrices for the prescribed tangential velocities of the swirl
at the points zk, geometric factors and vortex strengths respectively. Solving for the

strengths by inverting the ij matrix, we get

_ -1
Sj — ij (Ve)k (5.17)
See the Appendix D for the development of the matrices.
5.3 Vortex-In-Cell (VIC) Algorithm

A VIC technique is essentially a formulation that combines the Lagrangian

vortex tracking concept with the stream function-vorticity finite difference

 

64
formulation. A distinguishing feature of the VIC method is the fact that vorticity

information is contained in the Lagrangian vortex element rather than on the Eulerian
mesh. The Eulerian mesh is introduced for fast Fourier transform and convenient
representation of results, but is not explicitly used to advance the solution in time as

would be done in the finite difference method.

As a first step towards solving Poisson’s equation (Eq. 3.8) on the square of
fixed mesh size d, we must create, at the beginning of each time step, a mesh record
of the vorticity field starting fi'om the Lagrangian representation. This is done by an

interpolation scheme using Couet’s filter

2
.thlnl] ; é—slnlsg-
(Pm) ={r} ~n2 ; In |s-2L} (5.18)
0 ; Otherwise

where 11 = y/d or x/d, to distribute the vorticity from each vortex over the nearest

three grid points in a given coordinate direction.

Figure 10 illustrates an application of the above one-dimensional model of
Couet’s filter. Three quadratic spline distributions of vorticity with different total
amplitudes are shown to have their centers located at positions a, b, and c. The three
nearest grid points to each of the centers will share the vorticity distribution according
to the spline function weighting on them. Grid points N-2 to N will share the vortex
"a" with grid points N-l receiving the largest weight. Similarly, points N—l to N+2

will share the vortices "b" and "c".

 

65
After the vorticity information is represented on the uniform grid, a grid stream

function can be generated by using FFI‘ techniques to invert Poisson’s equation (3.8).
The stream function is centrally difl‘erenced on the grid to produce a velocity field
which is conformally transformed onto an equivalent grid system in the circular
domain. The velocity at the vortex position is now obtained by interpolation. The
vortices are then convected to new positions and the procedure is repeated each time

step until the flow evolution is completed.

The convenience of FFI‘ is available only when boundary conditions are
imposed over the square domain. In order to satisfy the no slip normal boundary
condition, the value ‘1’ = 0 is specified on the grid boundary for this study. The grid
used is a modest 60 x 60 but extension to larger dimensions is feasible with the use
of mainframe computers. Swartztrauber, et. al. [68] have developed an elliptic
boundary value problem solver (FISHPAK) which allows for general boundary
conditions and is portable, i.e., it runs on a Cray, Cyber, IBM, or Vax computer with no
modifications. This software is used in this study to accomplish the Poisson

inversion.

In general, if the grid is M x M, this is an order leogzM calculation. In a
typical 2D calculation, the number of grid squares M x M is of the same order of the

total number of vortices N and thus FFI‘ requires on the order of NzlogzN operations

per time step. The interpolations take of the order N operations per time step. When

N is large, NzlogzN is considerably smaller than N2. Thus the VIC technique offers a

resolution of the problem associated with excessive computation time.

66
5.4 Vortex Sheet Algorithm

The velocity field in the boundary layer region can be constructed using vortex
sheets as described in Section 3.2.2. The u and v component velocities are given by

Eqs. (3.26) and (3.27). At the initial time, the tangential velocity V6 along the wall

is fixed by the potential flow solution of Eq. (3.8). Theoretically, the tangential no slip

boundary condition, Ve=0, can be satisfied by generating a continuous sheet of
vorticity along the wall with circulation 78 = — V9 per unit length of the wall. However,
it has been observed computationally [2,18] that 78 = —V9 will satisfy the tangential

no-slip boundary condition only in the limit as At —> 0.

In order to satisfy the boundary condition exactly (except possibly at a finite
set of values for t), Chorin [18], algorithmically, extended the velocity field across the
wall (y=0) at the beginning of each time step by anti-symmetry u(x,-y) = -u(x,y).

As a result C(x,-y) = C(x,y) for y at 0, and a continuous vortex sheet of circulation Vs

= -2 V0 per unit length of the wall appears at y = 0 if the tangential velocity V6 does

not vanish at the wall. This sheet is segmented into elements of finite length h,
centered at test points along the wall, specifying the spatial resolution in the x-

direction.

In order to have a reasonable approximation of the diffusion equation (3.7), the

resolution in the ydirection can be improved by replacing each vortex segment by 1

number of "smaller" segments with circulation (ys)j=ys/l such that |('Ys)j I S l¥nax'

67
where gnu is a predetermined value. Vortex elements from each segment h diffuse

normal to the wall via random walks and induce a velocity field which advects them

downstream At subsequent time steps, V9 is calculated using the obvious
specialization of Eq.(3.26). The correction term (ys)j/2 in Eq. (3.26) was added by

Chorin [18] to account for the effect of the image of the discrete point vortex the sheet

can generate (see Figure 12).

Unlike the "elliptic" flow modeled by point vortices, where the effect of each
point vortex extends throughout the field, the zone of influence of a vortex sheet is
restricted to the "shadow" below it, a consequence of Eq.(3.24). Thus, Eqs. (3.26)
and (3.27) imply that the sum is over all vortex sheets whose "shadow" on the wall

engulfs the sheet whose center is located at (xj,yj). This is illustrated in Figure 11.

The darker the shadow, the smaller u(x,y) becomes. Whatever fluid enters a shadow

region from the left and cannot leave on the right must leave upwards.

To reduce the total number of sheets retained each time step, the vortex sheet
algorithm can be implemented such that exactly half of them disappear. This is

accomplished as follows : at each wall test point an even number of sheets are
created and their intensity is adjusted such that if I (Yslil < IXnaxl no sheets are

created. A rejection technique [18] is then used to ensure that any two successive

112’s used at the wall will have differing signs. This rejection technique can be used

only at the wall, or else it would destroy the independence of the successive "I? ’s in

the interior and thus fail to describe the diffusion process correctly.

68
A tagging and variance reduction technique can be used to effect a substantial

reduction in the statistical error. Since diffusion takes place only in the y-direction,

the random numbers (n2)j used in (Eq. 3.33) need be independent of each other only

when they are used with vortex sheet segments whose centers lie in a narrow strip
perpendicular to the wall. As vortex sheets are created, they are assigned integer

th vortex sheet segment. All elements with

tags, mj being the tag assigned to the j
the same 112 are assigned the same tags. The effect of tagging is to piece together

the elements created at the several boundary points into coherent vortex sheets with

the elements of each sheet identified by a common tag.
5.5 Advection Scheme

Vortices can be advected in only two ways, by convection or by diffusion.
Once the local velocity is known, the convection step is accomplished by using an
appropriate time integration scheme. For vortices in the interior region, Heun’s time
integration method is used to evaluate the integrals in Eq.(3.l7) and Eq.(3.l8),

namely
n+1 _ n Atl n n
Xj — Xj + 2 11(Xj ,Yj ) + u(xj*’yj*) ] (5.19)

.n .n .n =1- *

t3?
I

°* = on on on 0* = on on on
where xJ xJ + AH: u(xJ ,yJ ) ] and yJ yJ + A2L[ V(xJ .YJ ) ] (5.21)

69
Heun’s technique is second order accurate in time. Such accuracy lessens the

tendency for discrete vortices to spread out in spinning motion as they come close
together. A first-order scheme merely takes the tangent at a given point and
constructs a straight line approximation to the trajectory, in effect spreading out the
vortices. Since Heun’s method requires two evaluations of the velocity field per time
step, this can prove expensive when using the direct O(N2) method of calculating

vortex interactions. The FFI‘ technique is advantageous in this respect.

Near the solid boundary, an explicit Euler integration which is first-order
accurate can be used to evaluate the integrals in Eq. (3.32) and (3.33) in order to

accomplish the convection step, namely

n+1 ___ on .n
xJ xJ + uJ At (5.22)
n+1 _ .n .n
yJ - yJ + vJ At (5.23)

The use of a second-order time integration scheme for the vortex sheets is not
necessary for two reasons. First, the convection field within the boundary layer is
essentially a one-dimensional flow in a direction parallel to the wall and thus poses
no great problem for a first-order scheme. The main component driving the sheets
away from the wall is the diffusion term. Second, the implementation of a second-
order scheme is a very complicated procedure, since sheets may turn into point

vortices during the predictor step and thus must be treated differently.

70
It can be assured that the vortices are not convected beyond one mesh width

(for discrete point vortices) or sheet length (for vortex sheets) as long as the Courant

condition

IVmax IA} 5 dorh (5.24)

is satisfied. The Courant condition sets an upper bound on At for numerical stability.

Diffusion as discussed by Chorin [18] is equivalent to a random walk which is
governed by statistical laws derived from the solutions of the diffusion equation
(3.7). The integration of this equation yields a Gaussian distribution for the flux of
vorticity [40]. Tschebysheff’s theorem [44] states that the Gaussianly distributed

random variable with mean zero will rarely be greater in magnitude than three

standard deviations. During a time step of duration At, vorticity will diffuse with a

standard deviation of \l 2vAt . Hence, in a time step, interior vortices will not likely

diffuse beyond one mesh width if 3% S d . In some physical sense, statistical
simulation of diffusion steps back from the concept of a material continuum, and
employs a sort of kinetic theory of giant and imaginary particles which wander
randomly through the region of interest carrying macroscopic elements of the diffusing
field. The outcome of a calculation is a list of particle positions and the elements of

the concentration field they are employed to transport.

7 1
5.6 Sorting Algorithm

Vortex sheets and discrete vortices are objects of a very similar nature; they
are both determined by position and circulation. A computational element (xj, yj, Cj)
can be treated as either a sheet or a discrete point vortex, depending on the
circumstances. A sheet of negative circulation casts a shadow which slows the fluid

under it; by the equation of continuity, this creates an upward flow to the left and a

downward flow to the right, just as if the sheet were a point vortex. Conservation of

circulation arguments show that a sheet segment of circulation Vs should transform

into a point vortex of circulation yp = ysh, where h is the length of the sheet.

These facts can be used to create a transition between the discrete vortices
and the vortex sheets. Following Chorin’s argument [l8], vorticity is transferred

from the boundary layer to the interior by allowing those vortex sheets located a

distance greater than 58 from the wall to become point vortices. Any vortex which
finds itself less than 58 from the solid boundary (inside or outside) becomes a sheet
or is reflected accordingly (see Figure 12 ). Reflection of any sheet which crosses the
wall back into the fluid imposes the antisymmetry condition. By taking 58 large
relative to the variance of the steps, it is unlikely that a discrete point vortex will
move further outside the domain than 53 in one time step; if this does occur, it is

removed.

72
5.7 Mixing Algorithm

The physics of the evolution of scalar field undergoing turbulent and molecular
mixing is embodied in the PDF transport equation (3.47). The Lagrangian view of
molecular mixing can be numerically simulated using the Monte Carlo coalescence

and dispersion (CID) scheme.

The initial bimodal joint PDF, P(\|!;X,0), can be characterized by the random
distribution inside the combustion chamber of N equal-mass, chemically-inert fluid
particles with values 0 designating the swirling air particles and 1 designating the
injected air particles. The fluid particles are idealized turbulent eddies (i.e., finite-
cored vortices) assumed to be larger than molecular dimensions, but sufficiently small
with respect to the turbulence microscale. The thermodynamic state of each particle
(i.e., pressure, mass fraction, temperature, etc.) is assumed to be uniform throughout
its volume at any instant of time, but the number of particles in a given state may vary

with time as a result of mixing between particles.

The Monte Carlo C/D scheme is a stochastic fluid particle interaction model.
Each fluid particle is assigned a number index, 1 S i S N, and a pseudo-random
number generator is used to generate it different indices, i1, i2, . . ., in (ij at ik) thereby
identifying the particles to be involved in the first interaction. In the simplest binary
C/D mixing model (n=2) proposed by Curl [26] pairs of fluid particles are allowed to
mix completely by adopting the average values of their properties and then separate.

The mixing interactions are computed sequentially; the first interaction representing a

73
forward step in time equal to

_ __1_
At(mix) - MN (5.25)

where [3(t) is the empirically determined, time dependent mixing frequency and N is

the fluid particle population in the combustion chamber.

A new group of pairs of fluid particles is next chosen at random, from the entire
ensemble of N fluid particles, to be involved in the next mixing interaction; this way
the calculation proceeds forward in time inside the combustion chamber. At any time
during the mixing process, the mean composition and other mean properties of the
fluid may be evaluated by taking an ensemble average over N fluid particles.
Furthermore, the shape of the evolving distribution function f(<b,t) may be determined
by plotting the ensemble population as a function of any one of the thermodynamic

state variables or any other local property.

 

CHAPTER 6

NUMERICAL EXPERIMENTS AND DISCUSSION OF RESULTS

6.1 Introduction

This chapter documents the numerical experiments conducted to evaluate the
Various aspects of the inert scalar mixing study. The results of the experiments are
discussed and compared with theoretical predictions. The experiments can be broadly
clas Sified into three sets :

(1) Optimization of Numerical Parameters
(ii) Validation of Numerical Codes

(iii) Numerical Simulation.

The computational analyses were conducted on a VMS/VAX 11/750 computer
system Several Fortran programs (see Appendix F) used to analyze the different
00111ponents of the study. A portable elliptic boundary value problem solver
msHPAK) which allows for general boundary conditions, developed by Swartztrauber

9‘- al. [68], IMSL routines, PLOTIT and NCAR graphics packages were the basic
74

75
computational tools used in these analyses. Some of the graphs were generated

using the FrameMaker utility on a Hewlett Packard workstation.
6.2 Optimization of Numerical Parameters

The relevant numerical parameters in the present study are the mesh size (1,

the jet nozzle diameter dj, the discrete vortex core radius r0 and the length of the

time step At. The relevant numerical functions are the vortex core smoothing function

or Green’s function G and the filter function (1). The accuracy of the exact and

ij’
numerical schemes presented in this study is dependent on these parameters, the

functions Gij and (p . The considerations leading to the choice of Hald and Del Prete’s

Vortex core smoothing function Gij and Couet’s filter function (p have been discussed
in chapters 3 and 4 respectively. In this section, error estimates guiding the choice of

‘1’ ‘15. to and At are considered.

6‘2- 1 Mesh size

Because the VIC formulation incorporates the streamfunction-vorticity finite-
difference scheme, the question of grid resolution must be addressed. The accuracy of
a finite difl'erence scheme depends on the scale of the grid spacing (mesh size)
relafive to the scale of the exact solution or features of the real flow which is being

approximated numerically. To obtain a meaningful solution, the mesh size must be
Small enough to resolve the flow adequately. Theoretically, convergence of the

n“merical solution with exact solution should occur as the mesh size tends to zero.

76
The choice of the mesh size (1 used in this study is based on a comparison of

£116 relative errors in the grid point stream function values between exact and
numerical solutions in the square domain. The percent relative error was computed

as afunctionofthemeshsizeby:

 

 

 

_ M .1/2
20V]; ' \I’N)2-d2
i=1
Relative error (%) = J“ M x 100 (6.1)
ZWEZ-dz
i=1
.. i=1

where M is the number of grid points in the C and 1] directions. The numerical stream

function, VIN=t|IN[C(id,jd)m(id,jd)], is computed via the VIC algorithm while the exact
Subaru function, VE=\|I5[C(id,jd)m(id,jd)], is obtained using equation (3.44). The

Parameters C(idjd) and n(id,jd) are the computed grid point positions.

Two tests were performed. The first compared the errors in the stream

furl<>tion field induced by a single discrete point vortex of strength K=21t, placed
cellh‘ally in the computational grid. The second examined the effect of a system of 105
ClisCI'ete point vortices (0.0<K<l.06) initialized in accordance with the initialization
s"'llerne discussed in section 5.2. Plots of the percent relative errors for the various
chOices of grid sizes are presented in Figure 13 . The plots confirm the theoretical
Medans that the numerical approximation will converge to the exact solution as
the mesh size tends to zero. This tendency is much more gradual for the second test
case, especially for grid sizes within the range 60 x 60 to 120 x 120. The choice of grid

Size anywhere in this range will produce comparable errors. From the practical

77
standpoint of computational time and storage, a modest 60 x 60 grid size ((1 = 0.0625

cm.) corresponding to 3721 computational grid points was decided upon. The
indicated relative error for this grid size is approximately 6% for the single vortex test
case and 0.3% for the system of 105 vortices. The peak-to—peak variation observed in
the second test suggests some interactions between neighboring vortices. These
interactions act to reduce the relative error by as much as a factor of 20 when
compared with the result for the single vortex test case. One should note that in
practice, computations will rarely deal with an isolated vortex. Instead, a collection of
vortices will be used to model a vortical region. Thus for the chosen mesh size,

convergence is assured to a reasonable degree.

6.2.2 Injector nozzle diameter

No particular emphasis was placed in regard to the details of the injector
nozzle design as these are not germane to the overall objective of this study. An
optimization scheme for the injector nozzle diameter was not feasible in the absence
of an exact jet flow solution. It is known however that for appropriate resolution of
the jet feattues, the mesh size should be comparable to or smaller than the injector

nozzle diameter. An injector nozzle diameter d] equivalent to the mesh size (1 was

deemed satisfactory for this study.

78
6.2.3 Vortex core radius

Turbulent flows can be approximated by the vortex method with many
computational vortices used per physical vortex (see Ashurst [2]) or the amount of
labor can be reduced by choosing computational vortices which agree in structure and
spacing with the physical vortices. In either case, an assumption is made concerning
the local vorticity distribution within the computational vortex element and this leads

to a discussion of the vortex core cut-off, to. There is considerable difference of

opinion in the literature as to the optimal smoothing parameter or cut-off. Chorin and
Bernard [19] have observed that the results of the computations are quite insensitive
to the exact details of the smoothing. Shestakov [64] and Chorin [18] use fairly
large values proportional to the length of the segments into which the boundary was

divided (r0 ~ h). This selection is connected with the formation of new vortices at a

grid wall in order to satisfy the tangential no-slip boundary condition. On the other
hand, Millinazzo and Saffman [50] believe that the cut-off r0 should be as small as

possible and use 8/50, where B is the average distance between vortices. Saffman
questions the relevance of Chorin’s criterion for the diffusion and motion of the
vortices away from walls. Hald and Del Prete [39] have proved some interesting
convergence results. In particular, they have shown that the optimum cut-off depends

upon the smoothness of the flow under consideration, and that for smooth flows, as
the number of vortices increases, r0 should be of order 132/3. This implies that r0 tends
to zero more slowly than B; that is, there must be an ever increasing overlap of the
circular vortices in order to maintain high accuracy in approximating a continuum of

vorticity. This is nattual, since as the vortices become packed closer together, the

79
effect of the singularity in the fundamental solution of the Poisson equation [Eqn.

(3.8)] becomes more pronounced and requires ever greater smoothing.

In light of the disagreement in the literature over what constitutes an

acceptable value for the vortex core cut-off r0, an empirical procedure has been
devised in this study to optimize r0. Since the vortex core cut—off is needed to

evaluate Green’s function [Eq.(3.21)] in the modified exact solution scheme and the

VIC numerical solution scheme is independent of r0, the procedure seeks to vary r0
and compare at a fixed initial time, the exact and numerical grid point solutions for the
tangential component of velocity induced by a single discrete point vortex of strength

tc=2n in the circular domain . The choice of vortex strength is arbitrary. A value for r0
that yielded the smallest error was considered the optimal choice for to for use in this

study.

The numerical solution was initiated in the square domain followed by the
transformation equation (5.11) which provides the corresponding solution in the
circular domain. A test grid 50 x 50 was chosen, within the 60 x 60 computational
square domain, so the influence of the boundary would not be felt. This was
necessary so that the entire quasi Gaussian distribution of each vortex lies within the

square domain. For each arbitrarily chosen to, the absolute value of the percent

relative error at each grid point was measured as

 

”9"”- ' ”9)” x 100 (6.2)
(VO)E

Relative error (%) =

80
and the minimum and maximum values noted. The parameters (V9)E=(V9)E[x(id,jd)]

and (V0)N---(Ve )N[x(id,jd)] are the exact and numerical tangential velocities respectively
at the computed grid points x(id,jd). Note that the numerical solution is independent

of the cut-off radius to

Two tests cases were considered. In the first test, the vortex was located at
the grid point in the middle of the square domain for symmetry. In the second test,
the vortex was located at the center of one of the four cells surrounding the centrally
located grid point. In each case, an equivalent vortex location in the circular domain
was used for computing the exact solution. The results of the calculations for all grid
points and for all grid points sufficiently removed fiom the vortex singularity are
shown in Table 3. Note that the criterion "sufficiently removed from vortex
singularity" is assured for grid points greater than two mesh lengths from vortex
position at a grid point and for grid points greater than one and a half mesh lengths
fiom vortex position at a cell center. When all grid points are considered the maximum

error was observed to be minimum at r0: 0.036 for grid point location and ro=0.048 for

cell center location. For grid points sufficiently removed from the vortex location, the

trend in this case suggests that convergence would occur in the limit as re tends to
zero. For equivalent r0, the maximum errors were lower than for all grid points by

less than factors of 4 and 6 for grid point and cell center locations, respectively. This

was anticipated. It would seem reasonable to choose an average value of ro=0.042.
However, r0=0.05 was considered convenient for this study. The associated errors

were comparable and within acceptable range. The minimum relative error was zero

for each consideration in both test cases.

VOI
Ihe

line

8 1
6.2.4 Time step

A general guideline for choosing a value for At is not discussed in the
literature. Instead, an upper bound for At is given by the Courant condition (Eq.
5.24). In this study, the value of the time step At is optimized by observing the
uajectory of a single discrete point vortex in the circular domain. Theoretically, it is
expected that the vortex, whose motion is induced by its own velocity field, would
uace a circle in one complete revolution. An error results if the initial radial position

ri of the vortex differs fiom the final radial position rf after a complete revolution.

Several tests were performed using a vortex of strength lt=2rt. For each chosen

radial location, At is varied and the relative error after one complete revolution is

calculated as

Relative error (%) = Eff—r1 x 100 (6.3)
I

The results of these tests are shown in Figure 14. It is observed that the

vortex motion in the central portion of the domain (r < 0.6) is relatively insensitive to
the time step and that convergence would occm' as At tends to zero, which is again in

line with theory. The time step At=0.001 seconds was chosen sufficiently small, so
that a decrease in the time step would not significantly affect the results.

82
To summarize, the threshold values for the control parameters and the

numerical functions employed in this study are listed below :

Grid size, M x M = 60 x 60
Mesh size, (I x d = 0.0625 cm. x 0.0625 cm.
Injector nozzle diameter, dj = 0.0625 cm.
Vortex core cut-off, r0 = 0.05 cm.
Time step, At = 0.001 sec.
Couet’s filter :
.1. 3.- 2 . .3.
2[2 Inl] .%SIn|s2
$01) ={% 412 ; |n |s-2L (6.4)
0 ; Otherwise
Vortex core smoothing function :
lz z I 3 I I2
i‘ ' Zr 2'
Gij =[CI(1 - Tl )- 02(1- 71' )+ logro] ; Izi'zj ISTO (6.5)
0 0

83
6.3 Validation of Numerical Codes

The numerical codes consist of the conformal transformation, point vortex
initialization, velocity and vortex trajectory algorithms. Each component was tested

for my by comparison with exact data.

6.3.1 Conformal transformation

The validity of the conformal grid point coordinate and velocity transformation
schemes discussed in chapter 5 was tested using the reverse transformation
procedure for the grid point coordinates and by comparing exact with numerical
solutions for the velocities in the circular domain. The resulting error statistics are
listed in Table 4. The large maximum relative errors noted for the velocity
components result from the exact values being zero at the test point. Thus to avoid

dividing by zero a small number (1.0E-06) is added to the denominator.

6.3.1.1 Grid transformation

The distribution of the non-uniform grid points in the circular domain is
induced by the coordinate transformation equation (5.1) starting with the uniform grid
points in the square domain (see Figure 9). The reverse coordinate transformation
seeks to recover the uniform grid system from the non-uniform grid system. As
stated in chapter 5, this involves the use of pretabulated J acobian elliptic functions,
intensive algebraic manipulations, a quasi Newton-Raphson numerical method and

linear interpolation.

84
A smaller error is observed (see Table 4) in the results for the n-coordinate

than for the C-coordinate. This asymmetry in error could be attributed to at least two
factors. First, the construction, by hand, of the algebraic expressions for the system
of equations (5.3) through (5.6) is quite a substantial challenge (see Appendix C).
The algebraic complexities offer many Opportunities for mistakes both in their
derivations and in their coding for Fortran programming. Secondly, the use of Newton-
Raphson’s method brings with it certain difficulties, e.g., the lack of convergence
because of poor initial guesses. The hope is that the initial guesses are close enough
to the actual solutions and that the J acobian matrix of the system of equations (5.5)

and (5.6) is continuous and invertible at the solution points.

6.3.1.2 Velocity transformation

The errors resulting fi'om the velocity transformation scheme were calculated
by comparing the exact solutions obtained using the modified circle theorem for the
system of 105 vortices with numerical solutions resulting from the transformation
equation (5.11). Because of algebraic complexities, the elements of the matrix
system in equation (5.11) were obtained by numerically differentiating equation
(5.1). The resulting error is smaller for the v-component velocity than for the u-
component velocity and the values in general are more symmetric than for the
coordinate transformation scheme. This is perhaps attributable to the fact that the
algebraic manipulations for the velocity transformation scheme is less prone to

mistakes.

85
In line with expectation, the result of the error calculation shows

comparatively poor convergence at the corners of the domains because of the
singularities inherent in the coordinate transformation scheme. Outside of these
singular regions excellent convergence was obtained, the constraints of algebraic
complexities and initial guesses implicit in the Newton-Raphson numerical method

nothwithstanding.

6.3. 1.3 Corner singularities

The corner singularities precluded velocity vector and streamline
representations of the grid point results in the circular domain. Clearly, interpolation
schemes appropriate to the non-uniform grid system and implicit in the associated
NCAR graphics routines would not recognize the singular points. Procedures to
circumvent these restrictions were devised. For plots of velocity vectors and
streamlines, the circular domain was superimposed with a uniform grid (mesh width <
d) and velocity values from the non-uniform grid system were interpolated onto the
superimposed uniform grid points lying within the circle. Velocity interpolation at
vortex locations was performed in the square domain, using Couet’s filter, by first
transforming vortex coordinates in the circular domain to equivalent positions in the
square domain and then transferring grid velocity values floor the circular domain to
corresponding grid points in the square domain. The resulting off grid velocity values
were then used to adth the vortices in the circular domain. The justification for
doing this is implicit in the conformal transformation concept and the invariance of the
complex potential and hence the streamfunction at corresponding transformable

points. The validity of these schemes was tested by observing that the qualitative

86
features of the velocity vector and streamline p10ts and the locus of a discrete vortex

after a complete revolution were correct and consistent with theory.

6.3.2 Point vortex initialization

The results from the vorticity initialization program described in Section 5.2

are presented in Table 5. Table 5(a) illustrates the velocity field initialization
iteration data for 10:100 ms. The optimum distribution of vortices satisfying the
requirement of equation (5.14) was obtained at trial number 13. This illustration is

representative of similar iterations performed for 100 S to S 500 ms. The initialization

results for the different times are presented in Table 5(b). The mean value It and the
standard deviation 0’ of the strengths of the vortices are observed to decrease and
increase, respectively, with time. Naturally, the swirling motion looses its intensity

in the course of time if no external sustaining mechanism for the motion is in place.

The experimental data of Dyer (see Table 2) are plotted in Figure 15 at
corresponding positions in the circular domain used in this study. Curves show the
decay of velocity with time. Radial position is normalized by the chamber radius, R =
1.0 cm. The geometric distribution of the vortices for to=100 ms is plotted in Figure

16. The geometric disuibutions at other times are similar except for the difference in
the number of vortices initialized. The initialization scheme successfully reproduced
the indicated velocity profiles. This is not surprising since the scheme makes use of
the prescribed velocity information to initialize the positions and strengths of the

vortices. It should be noted that the iteration procedure is obviously open ended and

87
time consuming and that the number of vortices and geometric distribution for each to

are not unique. The iterations were performed with a view toward minimizing the

number of vortices since the computational work load is 0(N2) for the exact solution

scheme.
6.3.3 Velocity field and vortex trajectories

To estimate the accuracy of the numerical approximation for Equations (3.8)-
(3.10) with assumed initial condition C(x,y,0) in D, computed velocity components
(tangential and radial) and vortex trajectories in the circular domain are compared
with exact results obtained via the modified circle theorem of Milne-Thomson. The
velocity field induced by the system of 105 discrete vortices and the trajectories of

vortex systems containing 1, 2, 13 and 37 discrete vortices are compared.
6.3.3.1 Velocity field

The system of 105 discrete vortices used to generate the velocity field in the
circular domain were initialized in accordance with the initialization scheme discussed
in Section 5.2. The relative errors in both the tangential and radial component of

velocity are measured at each time step as

P

M " 1/2
20’s ' V102
i=1

Relative error (%) = P‘ x 100 (6.4)

 

 

 

88
where VE=VE[x(id,jd),t] is the exact velocity and VN=VN[x(id,jd),t] is the numerical

velocity at the computed grid points x(id,jd) and time t. The errors are computed until
t=l.1 seconds. Figure 17 shows the plot of the percent relative error versus time.
Shortly after the onset of calculation, the error is observed to grow almost linearly
with time for about 0.20 sec. Beyond t=0.25 sec., the error is observed to grow at a
rate which is relatively independent of time, i.e. the method is essentially stable.
Comparable errors in the component velocities noted earlier in the velocity
transformation scheme are again realized. The velocity errors give reliable indication
of the accuracy of the numerical prescription. In general, the results indicate that the
method is convergent to within 20% relative error and essentially stable. In light of
the susceptibility to errors of the various components involved in the numerical
evaluation of the velocity field, this result seems acceptable. Fm'ther more vortex
methods are generally low to medium accuracy methods. Plots of the tangential and
radial velocity profiles at selected times corresponding to short and long time errors

are illustrated in Figures 19 and 20 respectively.

6.3.3.2 Vortex trajectories

Figure 18 illustrates the trajectories for the systems of discrete vortices. In
each system, the initial conditions for each vortex are determined by position,
strength and sign of circulation. The initial coordinates for the single vortex is
(0.6,0.0) (Frgme 18a); for the 2-vortex systems, the vortex coordinates are (06,00)
and (0.0,0.6) (Figure 18b and c) and for the 13- and 37-vortex systems, the vortices

are uniformly initialized on concentric circles with a vortex placed in the center of the

domain (Figure 18d, e, f and g). The single vortex is assigned a strength 1c=27t and

89
positive circulation. For each of the systems with more than one vortex, two cases

with the same initial vortex positions were considered. In the first case (Figure 18b,
d and f), the vortices were assigned the same strengths (x=21t) and positive
circulations. In the second case (Figure 18c, e and g), the vortices assumed different
circulations and the same or different strengths. Specifically, the vortices in the 2-

vortex system (Figure 18c), maintained the same strengths (lt=21t) but were
assigned opposite circulations; counter clockwise for vortex 1 and clockwise for
vortex 2 . In the 13- and 37-vortex systems (Figure 18e and g), the vortices

assumed randomly assigned strengths and signs of circulation .

The trajectories after one complete cycle are shown for the 1- and 2-vortex
systems. The vortices in the 13- and 37-vortex systems were allowed to proceed for
1/4th and 1/8 th of a cycle, respectively, enough to show a clear trend for comparison.
A satisfactory agreement is observed in each case between the exact and numerical
solutions validating the numerical procedure. Differences that are apparent are due to
numerical errors resulting from interpolation procedures implicit in the numerical

scheme and finite core effects implicit in the exact solution scheme.

The motions of the constituents in the 2-vortex systems are noteworthy . The
mutually induced velocities of the two vortices gives the pair qualities reminiscent of
a classical particle. The vortices with the same strength and same sign of circulation
(Figure 18b) are beginning to wrap around one another. After several periods, it is
expected that a single vortex will develop from the interaction of the two vortices.
Rossow [62] introduced the term "convective merging" to describe this process which

is available within the fiamework of a two-dimensional Euler equation, i.e., it does

not require any viscous effects. ggnvective merging survives as an important
mechanism in vortex pairing [72] and it plays an essential role in our understanding of
the dynamics of vortex structures in a shear layer. Additionally, the phenomenon of
merging illustrates very clearly the generation of large scales from smaller scales
(two small vortices fuse to one larger vortex), which in two-dimensional turbulence is
responsible for the inverse energy cascade [43, 58]. In contrast, as the constituents
in the pair of vortices with opposite sign and same strength (Figure 18c) come close
together the vortices deviate from their original paths, pass one another and continue
moving in opposite directions. The exact form of the paths followed depends on the
relative strength of the vortices. This type of interaction between vortices of opposite

sign thus never involves any pairing behavior of the vortices.

The realization that deterministic dynamic systems with very few degrees of
freedom can display chaodc behavior has had a profound influence in our
understanding of transition to turbulent flow. As the number of vortices is increased
(Figure 18d, e, f and g), individual uajectories become of less concern and attention is
shifted to collective modes exhibited by groups of many vortices. For simple initial
conditions as described above, these collective modes take the form of clusters or
regions of vorticity that take on an identity of their own and become the basic building
blocks of the flow. We now have chaotic dynamics with many degrees of freedom or a
turbulent flow situation (Figure 18g). But the surprise is that this regime docs nor
correspond to a completely disorganized distribution of vorticity. Rather, we find
flows with considerable structural features. As mentioned in the Introduction, the
stimulus for studying the two-dimensional case has come from the experimental

discoveries of so-called coherent vortex structures in certain laboratory and

9 l
geophysical shear flows.

6.4 Numerical Simulations

The Fortran computer code for this study has the capability to track the jet
vortices from the near wall injection point through their evolution in the interior of the
cylinder. The jet algorithm was intended not as a rigorous model of the real jet
behavior but more to explore by a numerical method how a jet flow might be expected
to interact with a swirling cross flow. Numerical experiments were conducted that

systematically allowed different parameters to influence the mixing action. The
parameters involved are the angle of injection Gj, inlet jet velocity Um, swirl strength
So (~ 21%) and time t. The angle of injection Oj is set counter clockwise from the
positive x-axis. The inlet jet velocity Um was determined by calculating, for each 0]-
orientation, the distance per unit time step At covered by the vortex pair exiting the
plane of the injector nozzle into the cylinder interior and just clearing the numerical
shear layer thickness 53' To simulate different initial swirl conditions, vortices were
initialized as shown on Table 5 to match Dyer’s data (see Table 2) at the different
times. The ultimate objective was to identify fiom the available data, the optimum
conditions for minimum complete mixing time t' defined as the time required for the jet

vortices to "completely fill" the cylinder. The parameter t' is therefore artificial. The
criterion for "completely fill” was established by superimposing a square grid on the

circular domain as shown in Figure 21 and ensuring that at least Nc vortices reside in

each of the L square meshes or cells whose centers lie on the boundary or within the

circular domain.

92
The effects of varying the grid size (and therefore the number of cells L), the

number of vortices per cell NC, the angle of injection 0j, the swirl strength Se and the
inlet jet velocity Um on the mixing time t' were first investigated to enable an

appropriate choice of values for these parameters. Three tests were performed each

with the inlet jet velocity Um as a variable parameter. The first test sought to
optimize L and Oj for fixed values of Nc and 50' The second sought to optimize Nc for

the chosen values of L, Oj, 50' In the final test, the swirl strength S9 was varied with

L, Oj 311d NC fixed.

The considerations leading to the choice of the optimum grid size (or L) and
angle of injection Gj began with an investigation of three grid sizes 6 x 6, 8 x 8 and 10
x 10 illustrated in Figure 21. For Nc=2, 89:12.5 cmm/s and L, 0j and Uin as variable
parameters, t' data are recorded as shown in Table 6. These results are plotted in

Figure 22.. As expected, t', is observed to increase with L for any combination of OJ-

and Uin' When t. is viewed from the standpoint of its dependence on Uin’ results for

L=32 and L=80 (0j=30°) seem relatively well behaved as illustrated in Figure 23.
The case of L=32, 0j=60° best illustrates the expected exponentially decreasing

function. However, for the cases L=52 and L=80 (01:60" and 90°) t' appears

essentially insensitive to variations in the injection speed. These observations
suggest that the optimum grid size may have been exceeded at L=52. Given this

relative independence on Uin, the average of the t' values, t'avg, in each column of

Table 6 is plotted in Figure 24 as a function of the number of cells L with the angle of

93
injection Gj as a variable parameter. Although the lowest injection angle 9j=30°

appears slightly advantageous, the advantage is not significant, especially in

comparison with 0j=60°. Given the observed trends, it would seem most appropriate

to choose L=32 and 0j=60°.

Having chosen the grid size 6x6 (L=32) and the angle of injection 01:60", the
optimum number of vortices required per cell was investigated next. With L=32,

0j=60°, Se=12.5 cm-m/s and NC, Um as variable parameters t," data are recorded as

shown on Table 7. The results are plotted in Figure 25. As with the case of t*(L),
t*(Nc), in general, exhibited the expected trend; that is, t' increases with increase in

the number of vortices per cell. The notable exception is the case for Um=112.3 cm/s

which appears to be insensitive to N . In observing the trends for t*(U- ), more
C in

clearly illustrated in Figure 26, it becomes evident that Nc=2 is the best candidate
since t' decreases with increase in Um throughout the range of values for Urn in a -

much smoother fashion. The t' (Um) trends for Nc=1 and Nc=3 are not as smooth.

With the parameters L, Nc and 0]- established, the final test examined the

effect of varying the swirl strength Se on the complete mixing time t', for a given inlet
jet velocity Um. The data recorded for L=32, Nc=2, and 0j=60° is shown on Table 8

and plotted in Figure 27. With the exception of the result for Um=125.7 cm/s the

general trend suggests that t'" is essentially independent of the swirl strength.

94
Perhaps the absence of viscosity may in part explain the indicated trend. It should be

noted that randomness is inherently characteristic of vortex methods and that the
observed outcome is a manifestation of the interactions and changing dynamics of the
inviscid vortices in the flow domain. Bulk diffusion results from the interactions of the
swirl and jet vortices. If the force field of a vortex or cluster of neighboring vortices
modeling the swirling flow is stronger than that of a neighboring jet vortex or cluster,
mixing is inhibited. The higher swirl strength will only aid the transport of the jet
vortices to the cylinder wall due to the centrifugal force acting on the jet vortices. On
the other hand, mixing is enhanced when a stronger jet overcomes the centrifugal
force field and penetrates the flow. These contending phenomena are dictated by the
relative strengths of neighboring vortices, and it is the net effect of these vortex
interactions over time that results in the "completely filled" situation. It should be
mentioned too that each swirl condition for these tests was obtained from a single
realization of the vortex initialization scheme (see Table 5). Randomness would be
reduced if each swirl condition is obtained by averaging over several possible

realizations. That is, ensemble averaging would give a more realistic result. From

the point of view of the variation of t' with Uin’ for fixed 89’ the choice of 80:12'5 cm.

m/s and Um=l25.7 cm/s are preferred as can be deduced from Figure 28.
In summary, the arguments fi'om these investigations point to the choice of the
following parameters for the mixing simulation: 6 x 6 grid size (L=32 cells), Nc=2

vortices per cell, 0j=60°, 89:12.5 cm-m/s and Um=125.7 cm/s.

To help illustrate the nature of the flow in the engine cylinder, the results of

95
the numerical experiments are presented for (i) swirling motion with no jet flow, (ii)

jet flow with no swirling motion and (iii) swirling motion with jet flow. Streamline,
velocity, vortex trajectory and vorticity results for incremental time steps of 10 ms up
to when the cylinder wall is hit and including the time needed for jet vortices to fill the
entire domain is filled with jet vortices are illustrated. The velocity and vorticity

measurements serve to quantify the simulation process.

6.4.1 Swirling motion with no jet flow

Figure 29 shows the geometric distribution for the system of 115 discrete
vortices and velocity vector representation of the swirling motion induced by these
vortices as a function of time. Results are presented for selected times corresponding
to short and long time errors illustrated in Figure 18. Because of interpolation
problems associated with the comer singularities in the non-uniform grid system in
the circular domain, the velocity vector field is displayed on a uniform 60 x 60 grid
superimposed on the circular flow domain. Vector information at every other grid point
is illustrated. The magnitude of the vector at each grid point denotes the relative
speed of the flow. The swirling fluid motion is characterized by a much slower moving,
small central core region and a nearly constant tangential velocity for the bulk of the
faster moving fluid. At the initial time, a high degree of swirling air-flow symmetry is
achieved. In the course of time, the center-of-rotation of the swirling flow is offset
from the cylinder axis resulting in the oscillatory motions most apparent near the
center of the cylinder. The precession of the off-set center-of-rotation is gradually
dissipated by the bulk of the fluid, the dissipation rate stabilizing somewhat beyond

t=200 ms. It was observed that at times less than t=50 ms, corresponding to the

96
relatively short, horizontal, lower left portion of Figure 18, the vortex distribution was

still relatively organized. But at times greater than t=50 ms, the organized point
distribution was lost as is evident from Figure 29. The larger relative error
associated with the latter is essentially due to the critical closeness of the vortex
elements, which allows the motion to be dominated by the effect of the singularity.
Hald and Del Prete [39] suggest a larger smoothing parameter in order to ensure

better convergence results in the disorganized state.

6.4.2 Jet flow with no swirling motion

Figures 30 and 31 show stages in the computed development of the jet

vortices issuing at Um=210.0 cm/s from an injector nozzle oriented at 0j=60° into a

quiescent environment. It is observed from Figure 30 that the dorminant structure in
the first few diameters of the jet after the onset of injection consists of an array of
vortex pairs. A non-linear region followed where instabilities give rise to small
clumps, or eddies which interact with each other and evolve into larger eddies as they
move across the cylinder toward the wall. The vorticity coalescence results in the
eddies loosing their identity. The occurrence and abruptness of the coalescence
process depend on the irregularity in lateral spacing of the structures. A jet vortex
first hits the wall at t=29 ms and its vorticity theoretically is destroyed when it is
combined with its image vortex implying no flow through the wall. Algorithmically,
this is simulated by bouncing back onto the wall vortices which translate beyond the
boundary of the cylinder. In Figure 31, the streamlines begin to distort as the vortices
coalesce and the "completely filled" situation at t=271 ms translates into the flow

becoming more random, approaching a "turbulent" state. The closeness of

97
streamlines are indicative of a high fluid velocity in that region. The gradual evolution

of the velocity and vorticity fields across the cylinder at y=0 is observed to change
drastically as the jet spread gets close to the wall. The highest velocity and vorticity
values are recorded when the cylinder is filled with vortices. Note that at the onset of
injection, the radial velocity is comparable to the tangential velocity, but as the

vortices get closer to the wall, the radial velocity begins to diminish.

6.4.3 Swirling motion with jet flow

A global view of the mixing process is provided by Figures 32 and 33. The jet

nozzle is set at 0j=60°. Jet fluid at Um=210.0 cm/s issues into a swirling fluid of

strength 12.5 cm°m/s. As the jet enters the cylinder, it is deflected by the swirling
flow and moves with it. The core of the jet is not significantly deflected by the air
swirl as the jet passes across the cylinder toward the wall, implying that the jet
vortices are stronger at those locations and are able to penetrate the centrifugal force
field of the swirling flow. Soon after the jet vortices interact with the swirling flow,
the organized point distribution in the immediate vicinity of contact is lost. Because
of the absence of turbulent motion and molecular viscosity, the distribution of jet
vortices (i.e., mixing via bulk diffusion) is determined primarily by the air swirl. The
streamline patterns, illustrated in Figure 33, provide a qualitative appreciation of the
mixing process. The regions of great distortions of the streamline patterns are
indicative of intense vortex interactions. The radial motion of the vortex particles is
observed to be a small fraction of the tangential velocity due to centrifugal forces

acting on them. This accounts for the jet vortices reaching the vicinity of the cylinder

.i

 

98
wall earlier (t=10 ms) than for the no swirl case (t=20 ms). Note that the fact that a jet

vortex first hits the wall (t=29 ms) for the no swirl case than for the swirl case (t=32 ms) in
no way diminishes the transport of jet fluid to the wall by the swirling flow. The algorithm
was set up to identify the time when a vortex first hits the wall and not when the bulk of the

fluid first hits the wall. The fact that swirling flow does indeed aid the mixing process is seen
in the faster complete mixing time (t'=198 ms) registered by this situation than for the no

swirl case (t'=271 ms).

The above results are representative of results for 9j=90° and 0j=120° presented in
Figures 34 through 41. At 0j=120° the injector is opposing the swirling flow. Some

symmetry is observed between the results for 0j=60° and 0j=120° at corresponding times.

CHAPTER 7

SUMMARY AND CONCLUSION

A major challenge in engine research is to produce an engine that runs
efficiently and smoothly on lean mixtures. To do this an engine must have the ability
to burn very lean mixtures quickly and completely. Cylinder charge motion is thought
to be the most practical means of reaching this goal. To use the mixture motion
effectively, it is desirable that the lean-bum engine designer be aware of how the
flame behaves in turbulence and swirl. The effectiveness of turbulence in reducing
combustion times has received and is still receiving attention. But there appears to

be little information available on how a flame behaves under the effect of swirl.

A flexible research program has been initiated in this study to address the
mixing effect of swirl on an inert scalar field. Mixing descriptions based on the
concepts of discrete vortex dynamics are restricted to a two-dimensional, inviscid,
Lagrangian formulation. The general structure and flexibility of the numerical schemes
employed in this study make the inclusion of viscosity, turbulence and chemical

kinetics of the combustion process a viable future objective.

99

100
In meeting the present goal, several topics were explored. An empirical

scheme was developed to determine the vortex cut-ofi' radius, the associated
controversy in the literature notwithstanding. A foundation was laid for a technique
to initialize the vortex element distribution from prescribed velocity data. Useful
insights have been provided on conformal transformation techniques and real space

application of numerical filters.

Although the model provided herein is an idealized representation of a real
complex problem, the results and physical insights gained from this study can be
extended qualitatively to actual engines. Comparison with exact solutions shows the
validity of this method of approximation. Streamfunction values compare to within 6%
for values induced by a single vortex and 0.3% for values induced by a system of 105
vortices. An estimate of the uncertainty in a given velocity measurement is about
20%. This seems an acceptable result considering the many variables involved in the
velocity computation. The predictions of theory have been validated to the extent that

convergence would occur as the mesh size d and time step At tend to zero. The

method is relatively stable and d and At are constrained only by the accuracy
requirement of the Courant condition. The swirling air motion and the features of a
real jet are well simulated, at least qualitatively, by the motion of a system of discrete
point vortices, adding confidence to the numerical description. Some general features

of mixing in swirling flow have emerged from this work :
(i) Swirl enhances the transport process.

(ii) The strengths of the jet vortices scale with flow velocities
at the nozzle valve. ‘ ‘

101
(iii) The stronger the jet kinetic energy, the greater the centripetal

movement of fuel particles. This implies greater mixing. It
can be inferred from this that the combustion time will be
shorter and the leaner the mixture, the more obvious this

trend.

(iv) There is an optimal swirl-rate for minimum mixing time, a

conclusion which could be of significance in engine design.

The advantages gained from this study are derived from the use of an Eulerian
and Lagrangian formulation. The Eulerian grid in the VIC method allows the
application in real space of an efficient numerical filter capable of subgrid resolution
and leaving invariant the total circulation within each cell. The method generates a
solution independent of the core radius of the discrete vortex elements thereby
avoiding the singularity inherent in the Green’s function method. The Lagrangian
treatment eliminates the need to explicity treat convective derivatives. Unlike other
methods, the discrete vortex methods are not limited at high Reynolds number of
practical significance by the difficulty in distinguishing real from numerical diffusion.
The number of mesh points as well as the amount of computational labor required to
obtain a solution would be prohibitive, in particular for the finite element and finite
difference methods which can handle only a bounded range of frequencies. The VIC
technique ofl'ers a resolution of the problem of excessive computation time. The fast
Fourier uansform allows for efficient solution of the Poisson’s equation for the stream
function in every time step of the simulation. Computational economy is achieved
because only O(N) computer memory is required per time step, where N is the

number of vortices, and the number of operations per time step in the FFI‘ increases

102
linearly with the number of vortex elements and mesh points. Additional benefits

include an increased understanding of vortex interactions at a level of sophistication

consistent with the present needs of the design engineer.

Vortex methods are limited, however, like other methods, by the fact that
effects not resolved cannot be seen, i.e., if there are not enough computational
elements to represent a phenomenon, that phenomenon will not be observed. The
dependence of the method on an assumed non-deformable structure of the vortex
makes analysis difficult since in reality vortices of finite size deform.
Misrepresentation of the interaction of neighboring vortices can lead to a significant
statistical error in any one realization of the solution. This obviously is the price that
must be paid for the removal of numerically induced viscosity. In the VIC method, the
vortex core cutoff is replaced, in effect, by the cell size which has an effect on the
computations, e.g., the finite mode Fourier transform error, also known as aliassing
error. It is necessary to eliminate such errors generated by the numerical procedure
in order to obtain a smooth solution. Finally, proper application requires a

mathematical effort which is relatively substantial.

It should be noted that mixing experiments remain one of the most important
tools in combustion research since chemical reactions can only occur when the
reactants have been mixed to the appropriate composition. When the airflow process
in an engine cylinder is generally understood and we can model the non-combustion
mixing flow effectively, we will have a much better model of the combustion flow and

thus equip the engine designer with the necessary facts for predicting design trends.

CHAPTER 8

RECOMMENDATIONS

Vortex methods have certainly not reached the reliability of, say, finite
element methods for elliptic problems. They are useful and important because they
apply to problems where other methods are not necessarily applicable. It is hoped
that the further development of vortex methods will lead to ever increasing reliability.
The brunt of future work should be focused on the fluid mechanics which is an integral

and essential aspect of the mixing problem.

The distribution of the computational vortex elements in the vorticity
initialization technique will usually be dictated by the geometry of the system as well
as accuracy considerations of the integration formula implicit in the scheme. It is
recommended that a more systemmatic approach employing an appropriately derived
integration formula for the initialization procedme be investigated instead of the use
of the arbitrary, non-unique distribution employed in this study. For circular
domains, it is expected that a radially symmetric distribution, with the distribution
weighted more heavily toward the center, will produce smaller errors than a

distribution on a rectangular grid. The imaging system which makes use of

103

104
experimental information to produce the desired vorticity and velocity fields provides

a good foundation upon which to build. Averaging over all possible realizations of the
initial conditions and velocity fields would lessen the characteristic random outcome

of a vortex method and result in a smoother solution.

The extent to which chaotic motion is damped by finite core effects is an
important issue that should be addressed more thoroughly in the future especially
considering the controversy in the literature concerning the core radius. Boundary
effects (neglected in this study) should also be included in the optimization scheme

for the cut-off radius.

Future studies should investigate the inter-dependence of mesh size and jet
nozzle diameter on the resolution of the features of the jet flow. The jet nozzle

diameter was chosen to be equal to the mesh size in this study.

It is recommended that alternative uansformation schemes, free of
singularities be chosen. The exact form of the transformation is not important.
However, it is essential that the transformation function be smooth. Some computing
centers now offer programs for the symbolic manipulations of expressions and
functions. Such programs could be of tremendous help in the construction of the
system of equations used for the grid transformation. This should eliminate the rather
lengthy and tedious algebra employed in this study.

No particular efi‘ort has been directed toward computational efficiency in this
study since the emphasis of this study has been the development of a model rather
than a computer program. Future studies could focus some attention on computer

time usage.

105
It is believed that if these improvements are made, a versatile discrete vortex

approximation scheme would emerge that would provide a firm and promising
foundation for a more accurate two-dimensional inviscid flow simulation and
ultimately a more reliable framework for the incorporation of viscous effects and

combustion chemistry.

106

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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107

Table 2. Dyer’s experimental data.

V6 = Mean tangential velocity, (m/s)

r = Radial distance (cm)

R = Chamber radius (cm)

1:0 = Time from the closing of the shrouded

intake valve to time of measurement.

 

 

 

 

 

 

 

 

 

 

 

Ve (m/s)
to (ms)
IR

100 150 200 300 500
0.125 12.00 8.60 7.30 5.30 3.50
0.200 24.50 14.70 12.60 10.00 4.90
0.300 33.10 21.60 16.60 11.40 5.50
0.400 37.20 25.10 17.00 10.30 5.20
0.500 39.00 23.90 16.30 9.60 4.90
0.600 37.60 22.20 15.10 9.00 4.70
0.700 33.70 20.40 14.00 8.40 4.50
0.800 30.00 18.80 13.10 7.80 4.30
0.900 28.20 17.35 12.40 7.60 4.30
0.950 27.80 17.00 12.10 7.50 4.30
A3? , 27.80 17.50 12.50 7.90 4.10

 

 

Table 3. Comparison of the absolute value of the relative emor (%) estimates between

the modified exact and numerical tangential velocity values at grid points.

POINT VORTEX AT GRID POM

For All Grid Points

in

'108

 

Maximum

ooooooooooooo
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
mvmmavmavmamm
FQNHFMHOP¢OWN
§fi¢Nmthhmmhomm1
mm¢mmmmmmvhah
mommflhmmmmmmm
Nmm¢wbma¢bovm
NNNNNNNmmmwvv

0000000000000

 

For All Grid Points > 2.0d

intmum

M

ooooooooooooo
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
ooooooooooooo
.ooooooooooooo.
Efiooooooooooooo,
ooooooooooooo
ooooooooooooo
ooooooooooooo~
ooooooooooooo

0000000000000

 

 

 

Maximum

HHHHHHHHHHHHH
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
ommmNthomumm
mnmmmoabmmmmo
EfiNmMNNmHmmvmom:
MMHHOm¢MMFPPm
omvmovmrmoamm
HHommHmHmNmNo
MMMNNNHNNMMVW

mmmmmmmmmmmmm

l

 

intmum

M

 

0000000000000
0000000000000

ooooooooooooo
ooooooooooooo
sfiooooooooooooo:fl
ooooooooooooo
ooooooooooooo
ooooooooooooo
ooooooooooooo

0000000000000

 

 

Cutoff Value

 

oamm¢mmnomoam
.Hmmmmmmmmmm¢¢”.u
;Hooooooooooooognl

0000000000000

4

 

 

 

Maximum

ooooooooooooo
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
mammmflm¢mvmmo
mommHHmnNmovH
Efiflmmammaaommmm:
MOOmemHWNmHH
mmmmm¢mmmmwmm
hmammmmmamabv
mmmmmmhbmmmmo

Q'VV‘Q‘V‘VQ‘Q‘Q‘Q‘Q‘Q‘ID

 

For All Grid Points > l.5d

intmum

M

ooooooooooooo
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
ooooooooooooo
ooooooooooooo
Efiooooooooooooo:
ooooooooooooo
ooooooooooooo
ooooooooooooo
ooooooooooooo

0000000000000

 

 

 

POINT VORTEX AT CELL CENTER

For All Grid Points

Maximum

HHHHHHHHHHHHH
ooooooooooooo
+++++++++++++
mmmmmmmmmmmmm
mm¢mmmombmmma
mmmmmmmmmmbwm
ffimmmmmmmmmmavv:
NH¢mommh¢mmmm
havommmhvmwmo
mmmmmv¢ormwoa
mbmmvvvmwmavm

NNNNNNNNNNMMM

A

 

intmum

M

 

0000000000000
0000000000000
+++++++++++++
mmmmmmmmmmmmm
0000000000000
0000000000000
gfiooooooooooooogH
0000000000000
0000000000000
0000000000000
0000000000000

0000000000000

 

 

Cutoff Value

 

mmwmmbmmoamm¢
vvvvvc¢vmmmmm _
gfioooooooooooooih

0000000000000

A

 

 

109

Table 4. Error statistics for grid point coordinate and velocity transformations.

 

Coordinate Transformation

Velocity Transformation

 

 

 

 

 

 

 

C 11 u v
Avg. absolute error 2.0397014507 1.4282386507 1.5206969501 1.4264177501
your. absolute error 1.3470206505 6.3128371506 3.4926014E+00 4.0869408E+00
lMax. relative. error (76)) 1.9993558502 1.0198522502 4.09980205+07 9.25844255+05
Root mean square error 6.8885259E-O7 35780748307 3.3185267E-01 3.0976245E-01
Grid point location (2,1), (2,61), (1,3), (159), (2,1), (2,61), (1,3), (159),
°f m‘ mu” m (60,1), (60,61) (61,59), (61,3) (60,1), (60,61) (61,59), (61,3)

 

Note :

Average absolute error =

Relative error (%)

Root mean square error

 

 

In

M [‘2
[2(1’5 ' PN)2/M2T
i=1

j=1

where p is the parameter of interest.

(6.6)

(6.7)

(6.8)

 

110

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Table 6. L, 6. optimization data (Swirl strength S6 = 12.5 cm-m/s).

111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nc = 2 ej = 30°
Grid Size
U“ 6 x 6 8 x 8 10 x 10
(cm/s) (L=32) (L=52) (L=80)
t’ (1118)
112.3 173 185 344
125.7 149 229 327
152.6 112 213 319
210.0 124 184 250
NC = 2 Bi = 60°
Grid Size
Uin 6 x 6 8 x 8 10 x 10
(cm/s) (L=32) (L=52) (L=80)
t' (m8)
112.3 185 207 343
125.7 159 198 307
152.6 130 213 304
210.0 99 198 343
Nc = 2 Bi = 900
Grid Size
Uin 6 x 6 I 8 x 8 I 10 x 10
(cm/s) (L=32) (L=52) (L=80)
6" (ms)
112.3 215 267 356
125.7 191 236 371
152.6 138 268 373
210.0 108 167 330

 

 

 

 

 

 

112

Table 7. Ne optimization data (Swirl strength S9 = 12.5 cm-m/s).

 

 

 

 

 

 

 

 

 

 

 

 

 

L = 32 9j = 60°
N
Uin c
1 2 3
(cm/s)
1‘ (ms)

112.3 179 185 185

125.7 128 159 173

152.6 113 130 167

210.0 82 99 152

 

 

 

Table 8. Optimization data for swirl strength Se.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N,= 2
Swirl strength 89(cm-m/s)
Uin

4.1 7.9 12.5 17.5 . 27.8

(cm/s)

t‘ (1118)

112.3 175 178 185 152 146
125.7 210 135 159 150 128
152.6 141 164 130 123 149
210.0 116 88 99 100 118

 

 

 

 

 

Rate of Heat Release

 

 

113

 

 

Figure l.

Crank Angle

Three phases of heat release.

114

>X

 

. Fuel

Oj Angle of injection

Figure 2. Sketch of intake flow

115

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o
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1 I 0.0.0.
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Figure 4. Vorticity distribution in a grid, with shading showing
assignment of vorticity to grid point for VIC method.
is assigned to grid point (i,j)
@ is assigned to grid point (i+1.j)
111111 is assigned to end point (i+1.j+1)
g is assigned to grid point (i,j+1)

117

 

©

 

 

 

 

 

 

WW (POLYNOINL) WSW/BAKER (AREA wacumc)

 

 

 

 

 

 

 

COUEY (OUIDRATIC SPURS) RENO/THOMSON (MODIFIED AREA WGHHNG)

 

CZ3©

 

 

 

 

 

WANG (WC W5) 30 SPREADING flLTER

Figure 5. Iso-vorticity contours as represented by the various filters.
Large tick marks represent a half-grid length.

118

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mg... 10. Quadratic spline vorticity distribution of three
typical vortices on their nearby grid points.

123

 

 

 

1"‘11211—1‘1 hdilh

t —l Hat-1 ,

Figure 11. Zone of influence of a vortex sheet.

124

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126

 

 

 

 

 

40.0
q r‘-o.o
30.0-
20.0-
10.0-
r‘-o.a
r‘-o.7
o.o- , . I .
0.0 4.0 8.0 12.0 16.0

At x 1.0E—03 sec.

Figure 14. Relative error between the initial and final positions

of a point vortex after one complete revolutl'

 

1011.

 

20.0

127

 

40.0

30.0 -

Vs (m/S)

 

300 mg.

. b 500 me.

 

 

 

 

 

 

  

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Ct

>00 0 oooooo oooooooo
00° 00

-o.sa

 

 

 

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-1.0 -O.5 0.0 0.5 1.0

 

 

Figure 16. Geometric distribution of the system of 105 discrete vortices (10:100 ms.)

128

 

 

 

 

30.0
— Vg—component
. --- V,-component
25.01
" ‘1 \ \s
f I V V \ \ 1‘ \ ' ' A I J
g? 20-0- P ‘ “i ' 1"(i J» t I‘\ ’\
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t.
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' T ' I ' I T I ' I ' l ' 1 ' I ' l ' T '
0.0 0.1 0.2 0.3 0.4- 0.5 0.6 0.7 0.8 0.9 1.0 1.1
Time (sec)

Figure 17. Relative errors in V1. and V9 component velocities
as a function of time.

EXACT

129

NUMERICAL

.0

 

 

 

-11

1.0

fi1!1 .1

T
Y rfir r—1
1

r r V r

r
0.8
i
0.5
r
0.5

0.0

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Y rf‘rrr'r r
2

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at;

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1
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c

m n... M w M..- u M m .4... u u «m.

Figure 18. Trajectories of systems of discrete vortices

EXACT

130

NUMERKIAL

 

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3
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Figure 19. Tangential velocity profiles at discrete time steps
representative of short and long time errors.

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Figure 20. Radial velocity profiles at discrete time steps
representative of short and long time errors.

134

 

 

 

 

 

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Figure 21. Uniform grid squares superimposed on circular domain.

1. (ms)

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Figure 22. Complete mixing time as a function of the number of cells.
(Swirl strength 89:12.5 cm-m/s. Number of vortices per cell Nc=2)

1. (ms)

1‘ (ms)

t. (m a)

136

 

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Inlet jet velocity. ”in (cm/s)

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Figure 23. Complete mixing time as a function of the inlet jet velocity.
(Swirl strength 89:12.5 cm-m/s. Number of vortices per cell Nc=2)

137

 

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W 0":30‘3
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400.01

 

 

 

 

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20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0
Number of cells, L

Figure 24. Average complete mixing time asa function of the number of cells.
(Swirl strength 89:12.5 cmom/s. Number of vortices per cell Nc=2)

138

 

 

 

 

 

 

 

 

 

 

300.0
8-8 U1n=112.3 cm/s
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Figure 25. Complete mixing time as a function of the number of vortices per cell
(Swirl strength Se=12.5 cm-m/s. Number of cells L=32).

1* (ms)

139

 

 

 

 

 

 

 

 

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Inlet jet velocity, Uin (cm/s)

Figure 26. Complete mixing time as a function of the inlet jet velocity
(Swirl strength 80:12.5 cm-m/s. Number of cells L=32).

250.0

140

 

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Swirl strength 56 (cm m/s)

Figure 27. Complete mixing time as a function of the swirl strength.
(Number of cells L=32. Number of vortices per cell Nc=2)

50 .0

t* (ms)

141

 

 

 

 

 

 

 

 

300.0
8-8 59: 4.1 cm. m/s
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- H 89-175 cm. m/a
H Sa‘27.8 cm. m[s
200.0-
100.0d
090 r I r I ' I
50.0 100.0 150.0 200.0

Inlet jet velocity. Uin (cm/S)

Figure 28. Complete mixing time as a function of the inlet jet velocity.

(Number of cells L=32. Number of vortices per cell Nc=2)

 

250.0

142

      

      

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Figure 30. Fuel particle trajectory and vorticity profile at y=0 as a function of time.
Injector at 60 degrees (cormterclockwise from haimntal). Uin’ 210.0 m/s.

 

v/R

 

145

1311.0

-mu

Vodidly (1 [609)

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Vorticity (t/uc.)

 

 

 

Vodiaty (t/uc.)

 

 

 

Figure 30 (con’ul).

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(without swirl).
210 0 m/s.

time
Uin=

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Figure 31.

 

 

 

 

t=29tns.

 

 

 

 

 

 

 

 

 

t a 271 ms.

Figure 31 (con’td).

148

 

 

 

 

 

 

 

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Vorticity (l/sec )

thuculy (l/Icc.)

Vorticity (l/soc )

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Injecurat 60mm“

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149

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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150

 

 

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8

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x/R

 

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Figure 33. Streamlines and velocity profiles at y=0 as a function of time (with swirl).
Injector at 60 degrees (counterclockwise from horizontal). Uin= 210.0 m/s.

151

 

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Figure 33 (con’td).

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1.3
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Figure 34. Fuel particle trajectory and vorticity profile at y=Q as a function of time.
Injector at 90 degrees (counterclockwise from horizontal). Um: 210.0 m/s.

153

 

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Velocity (tn/e)

 

 

 

 

Velocity (tn/e)

 

 

 

 

 

t=20tn8.

Figure 35. Streamlines and veloci profiles at y=0 a a function of time (without swirl).
Injector at 90 degrees counterclockwbe from haizontal). Uin" 210.0 m/s.

 

Velocity (m/e)

 

 

 

 

Velocity (tn/e)

 

 

 

 

Velocity (In/e)

 

 

 

t-256ms.

Figure 35 (MM).

156

 

tom:

 

   

  
     

 

Vorticity (I/uc.)

 

-‘m‘ r v v
-10 -0.5 0-0 0.5 10

 

 

 

 

 

 

 

 

Vorticity (t/uc.)

 

 

 

 

 

 

 

 

 

m4

1/8

 

Vorticity (l/eec.)

 

 

 

 

 

 

 

l/R

Figure 36. AirandFuelpmicleuajectoryandvaficitypmfileatyaOasafimctionoffime.
Injecttr at 90 degrees (counterclockwise from horizontal). Uin’ 210.0 m/s.

0.5

0.0

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2

-1.: .

157

 

 

 

Vorticity (I /eoc)

 

 

 

Vorticity (I /uc.)

 

Vorticity (i/eoc.)

 

 

 

1M3

 

 

 

 

is

010

 

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«mom

 

 

 

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Figure 36 (con’td).

43.3

 

 

Velocity (m/e)

1

 

 

 

 

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t-lOms.

 

 

 

 

 

Figure 37. Streemlinee and velocity profiles at y=0 as a function of time (with swirl).
Injector at 90 degrees (counterclockwise from haizonml). Um: 210.0 m/s.

 

1?
\
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v
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Veloeky (In/e)

 

 

 

 

 

Velocity (tn/e)

 

 

 

 

 

t-257ms.

Figure 37 (con’td).

 

0.54

00-4

v/8

 

 

 

 

 

fl.

 

 

 

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Figure38.

 

 

Vorticity (l/eoc )

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Vertlclty (t leee.)

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thlculy (l/eoc)

 

104

0.0‘

 

 

 

 

 

 

 

 

 

 

 

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Fuel particle trajectory and vorticity profile at y=0 as a function of time.
Injector at 120 degrees (countacloclrwise from horizontal).U m: 210.0 mls

161

 

 

 

m

1m
‘

A.uvu\ .v 3.093)

0.04

 

flu:

 

0.0

0.0

'03

 

 

 

 

x/n

x/I

t=30tns.

l.0

 

 

ob

 

 

m4-

Ade.\.v .93...)

w

004

 

4w:

 

 

 

 

 

l.0

 

r/R

-6.e

 

 

an:

t-36ms

4 w J1 m
-

..uee\.v been.)

 

-l.0

 

x/R

 

 

 

tszfium.

figmnw(umm&

Velocity (m/s)

4.54

 

 

 

 

 

 

 

 

 

1:
E
V
3.

 

m 010 ole .o
x/I
‘ 1
W
A
i ‘ /"‘\
é ./
3 '/
4 V/
-W
.9; .0 _a‘. 0.. 0:0 “

 

 

 

 

 

t-20ms.

Figure 39. Streamlines and velocity profiles at y=0 as a function of time (without swirl).
Injector at 120 deuce: (counterclockwise

from horizontal). um: 210.0 m/s.

163

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 39 (con’td).

164

 

10m;

 

scum

 

0
$0.04 0 e: eee eeeeeeee 091
.

 

Vorthlty (l /sec.)

 

 

 

 

 

 

 

 

 

seem

 

Vorticity (l/eoc.)

 

 

 

4”:

 

 

 

 

 

 

v/R
Vorticity (l/eec.)

 

 

 

 

 

 

 

x/R

Figure 40. Air and Fuel Barrie le trajectory and vaticity profile at y=0 as a function of time.
lnjccttr at l deucee (counterclockwise from horizontal). Uin’ 210.0 m/s.

165

 

1m;

 

”.04

00‘

 

Vorfiaty (I lac.)

40mm

 

 

 

 

d”;

 

 

 

 

 

r/R

 

Vorticity (t/eec.)
l

 

 

 

 

 

 

 

an

 

 

Vorticity (I [sec]

«mead

 

 

 

 

 

- «can: . , . m
i " .0 ‘05 0.0 0.5
Lo ~05 ‘/R

“an 40 (con’td).

166

   

Velocity (m/e)

Velocity (m/e)

 

t-20rns.

Velody (tn/e)

 

 

 

 

 

”its:
tan-4
0.:
45.04
J
- 4.0 43.6 on 0;: 1.0
n/R

 

 

 

 

 

 

olu'l

 

 

 

 

-1.o «is

03 ole 1.0

Figure 41. Streamlines and velocity profiles at yaO as a function of tune (with swirl).
Injector at 120 degrees (counterclockwise from horizontal). ”in“ 210.0 m/s.

167

 

 

Velocity (tn/e)

 

 

 

0.5 1.0

 

1W
3‘ .
\
5
g 0.0-
>

 

45.01

 

 

 

-l .e 4'» is of u

 

Vcbdty (tn/a)

 

 

 

 

 

t=332rns.

Figure 41 (con'td).

APPENDICES

APPENDIX A

APPENDIX A

NUMERICAL FILTERS

AI. Derivation of Christiansen’s filter as an illustrative example

 

 

 

 

 

 

 

 

 

 

 

 

 

YA
r l r
| I l
—fl-——+-——P-
I I l
= 1 : #1:
I I I
d--| .———I——
I l
J " l
: i+l,j '
I I l
-—|—-—+-—-|—-d
I l I I
3 1 l >X

Figure 42. Vorticity distribution in a grid, with shading showing
assignment of vorticity to grid point for VIC method

is assigned to grid point (i,j)
is assigned to grid point (i+l.j)
is assigned to grid point (i+1J+l)

W§Q§

is assigned to grid point (i,j+1)

168

Notes :

(1)

(2)

(3)

(4)

(5)

(6)

169

Each point vortex is assumed to possess uniform vorticity
within a square cell with dimensions (d x d).

Unit circulation (yp=l.0) is assumed for each vortex.

The point vortex coordinates (xp,yp) lie at the center of the

square constant-vorticity core and must fall between four
grid points.

The origin for (xpyp) is the lower left hand grid point location.
The filter which credits the circulation to the surrounding four

grid points is obtained through a bilinear interpolation scheme
(area weighting).

AL)" (719i J, are the area and vortex circulation assigned to the

grid point (i,j) respectively; §(x,y) is the vorticity and (xg,yg)
are the grid point coordinates.

§A=1.0

f §(x.y)dA

Wit = f anymu = EAIJ

Dividing equation (A.2) by (A.1) :

We

(7pm i
'Yp A

Aid.

(A.1)

(A2)

(A3)

(A.4)

:45?

Ai+1,j

 

170
[(xg+dl2)-(xp-d/2)l-[(yg+d/2Hyp-d/2)]

[xg+d12-xp+d/2]°[yg+d/2-yp+d/2]

[xs'xpfifl'ws'ypm] (A.5)

[xg'xpw] . U8- Yp+dl
(1 ° (1

[(xg-xp/d+1]-[(yg-yp)/d+1]
[1-(xp-xg)/d]-[l-(yp-ygvd]

[l-Tl(X)]°[1-11(y)] (A6)

[(Xp+dl2)-(Xg+d/2)]°[(Yg‘I-d/Z)-(Yp-df2)]
[xp+d/2-xg-d/2]°[yg+d/2-yp+d/2]

pr-ng-Iyg-ypwl (A7)

[xp-xgl-[yg-YPM]
d- d

 

[(xp-xgydlttyg-ypxdnl
[(xp-xgydI-Il-(yp-ygydl

[Tl(X)]°[1-Tl(y)] (A8)

171
Am” = [(xp+d/2>-<xg+d/2>I-[<y,,+d/2)-(yg+d/2)1

= [xp+d/2-xg-d/2]°[yp+d/2-yg-d/2]

= [xp-xgl-[yp-yg] (A.9)

Ai+l,J+l [xp'ngUp‘yS]
A d° d

 

= [(xp-xgydI-uyp-ygydI

= 1100-110) (A.10)

Am = [cram-(gala)-I(yp+d/2)-(yg+d/2)1
= ngm-xme-[ypm/z-yga/zl

= [xg-xpwl-Up-yg] (A. 1 1)

Ai.J+l [_xg’xpl'dl'b'p’ygL
A d-d

= [(xg-xPI/d+II-I(yp-yg)/d1
= [1-(xp-xg)/d]°[(yp-Yg)/d]

= [l-fl(X)]'[Tl(y)] (A-12)

172

We can generalize equations (A.6), (A.8), (A.10) and (A.12) by letting

are.
1-|n(x)| ; In(x)| 51.0
q’iJIMXH = {
O ; otherwise.
1- mm ; |n(y)| s 1.0
mid-[110)] = {
0 ; otherwise.
‘PiJ = ‘Pi J[11(X),TI(Y)] = ‘P1 J[11(X)]'<Pi 3310)]

and from equation (A.4) : (79)“ = (Pinp

(xg,yg) be the coordinates of either grid point (i,j) or (i+1,j). The results

(A. 13)

(A. 14)

(A. 15)

(A. 16)

(A. 17)

(A. 18)

APPENDIX B

APPENDIX B

FUNCTIONS

BI. Smoothing function Gi j for convergence of vortex method [39]

 

 

W(Z) = -inGij (B.1)
lz-z- I3 Iz-z- I2
=ns [C1(1'—§L )- (12(1- ——2J— )+logro] ; lz-zjlsl'o
r0 r0
Iz-z. '2 Cl IZ-Zj I
= 'in I: —r—21— (C2 - r0 ) + (Cl-CZ + log r0)] (B-2)
0
[u-iv]j = %)- (B3)
Iz-z- I2 C (z-z-) Iz-z- I 2(z-z-)
=-i._l_(__1_J_+C2,Cl_J_ _J_
'3 r02 r0 Iz i’ zjl 1'0 r02

 

 

 

= -in [ 202(z-z!-) - 3C1(z'zj) Iz-zj I J

 

. 202 3C1 Iz-z- I
= ""5 I71)—2 ' _rTJ—Ia'zj’ ; '“j ' Sr0 (13.4)

174
where C1=2l3 and 02:3/‘2.

BI]. Derivation of the stream function, w, inside a finite rectangular domain
using method of images

 

 

 

 

 

 

—————a

I |

l I

L__QIZJ

I O #4 I

I |

I I

I"""---‘I

I I

I J #2 |
___.Sn—3 #1
TI K
o

>C

I I

L__@.f3__I

l J #5 I

I I

L.______I

I l

I I

I ~ #7 I

I---Q---I

Figure 43. Method of Images

175

Notes : (i) Sign Convention :
Counter clockwise is positive, clockwise is negative.
(ii) to = §+in ; 0) = C-i'n (B-5)
«)0 = co+ino . «00 = co-ino (13.6)

Consider the ‘n-direction,

#1 #2 #3
W((D) = 'iK{1n[m-(Co+iflo)I-1n[(0-I§0+i(2K-flo)II-IIIIOJ-(CO-iflo)]+

#4 #5
1n[a)-{ §0+i(2K+110) ] ]+ln[to-{I;O-i(2K—n0) H-

#6 #7
ln[o)-{§0+i(4K-no)}]—1n[co-{§O-i(2K-I-n0)}]+... }

= -iIc{ln(u)—coo)-ln(to—EO-iZIQ-In(a)-E'O)+ln(co-mo-i2K)+

1n(o)—coo+i2K)-ln(0)-coo-i4K)-ln(0)-'cb‘0-i2K)+...}

= -iI<{[ln((o—mo)+ln(0)—mo-i2K)+ln(tn—coo+i2K)+...]-

[ln((I)—(To)-ln(co-50-i2K)-ln(0)-50+i2K)+...] }

nz-oo

= -iK {Eln[m—mo+i2nK]-2In[w—6fo+i2nK]}
n=-oo

= f(w) + f(_m) (13.7)

176

where f(to) = 41:23, [Manx] (B.8)
[1.1-co

f(co) = ixifh [to—50+i2nK] (13.9)
n=-oo

Now, consider f((o) :

f(m) = - iKln{(to-coo)[(to—mo)+i2K)][(m—wo)-i2K][(m—mo)+i4K]+...}

= - iK1n[(m—coo)fi£(0)-too)2+(2nK)2]]
n =

= “II "—‘§’§°’- ”n‘ nll+r(2£—n'£301121(2nK)2]

= -ix{ln[m: HIM-m nK)}2]] + m[2,th:ll(2 nK)2]I

I
= -ix[1n[3inh{“(—2HK01}] + 1n[2K “321111)le
= -ixln[smh{"—(“;'—I;°i1] + Cl (13.10)
where Shun—(92%} = [fi—ZmI—(ioflfigl+r—(%mfl}2]] (3.11)
(13.12)

D
II
5'
l—'—I
”I;
5
.__Z

177
Similarly, it can be shown that :

E25) = ixm[smh{’i(—°;-_me)—I] — C1 (B.13)

Substituting (3.10) and (B.13) in (B.7) :

2K )II}

“((03) = -iK{1n[Sinh{n—(%_iw-Q)-}] — 1n[3inh{7£(_‘L“—b_

 

' (“(20.1%)
- -ncln 1t( —)
Sinh{ 2K }

. SinhICm+inml 1
= -1K1n —,—-—-,——
Smh[§m+111p] )

, Sinhtcm)Cos( nm)+iCosh<t;m)8in< 11m)
""1"{ Sinh(§m)Cos( up)+iCosh(§m)Sin( 11p) ) 03-14)

where Cm = flag—go), Tlm = 'ZIIL((n-no) and TIP = infi(n+no).

Similarly,

(B.15)

W(TO') - 1K { sinh(t;m)Cos( np)-iCosh(§m)Sin( up) I

From

and

178

W(co) ¢ + iv

i\I’

II
'9'
I

We»

W(m) - We

.16
2i (B )

 

Substituting (3.14) and (B.15) in (B.16) :

-€
I

 

. [Sinh(Cm)C08( nm)+iCosh(§m)Sin( TIm)
-u<1n —,———= _
Smh(§m)Cos( np)+1Cosh(§m)Sm( 11p)

Sinh(§m)Cos( np)-iCosh(I;m)Sin( 11p)

 

2i

 

__E 1n[Sinh2(§m)Cos2( nm)+Cosh2<cm)8in2( nm)]
2 Sinh2(§m)Cos2( nP)+Cosh2(§m)Sin2( up)

.3 1n[_S____inI:2<t;.,.>+Sin2< _____n_m)]
2 Sinh2(§m)+Sin2( 11p)

12 1n[Sinh2(§m)+Sin2( no)
2 Sinh2(§m)+Sin2( nm)

 

(B.17)

APPENDIX C

CI.

where

APPENDIX C

COORDINATE TRANSFORMATION

Square (Cm) -) Circle (x,y)

We begin with the transformation (5.1) (See Bowman [11]):

z = sn(co)dn(to)

 

cn(0))
z =x+iy
a) = §+in

sn(0)) = [(sd1+icdslc1)/(1-d2512)]
dn(o)) = [(dcldl-ikzscs1)/(1—dzslz)]

cn((o) = [(ccl-isdsldl)/(l-dzslz)]

c = cn(c.k)
cl = cn(n.k’)
d = dn(c.k)
d1 = dn(n,k’)
s = sn(C.k)
81 = sn(11.k’)
k’ = m

179

(CI)

(C2)

(C3)
(C4)
(C5)
(C6)
(C7)
(C.8)
(C9)

(C10)

(CH)
(C. 12)

(C. 13)

180
Substituting equations (C2), (C4), (C5) and (CG) in (CI) we get

+. _ [(sd1+icdslc1)/(l-d2812)] [(dcldl-ikzscsl)/(142312)] (C14)
x 1" " [(ccl-isdsldlyu-dzslzn '

 

(sdl-I-icdslclxdcldl-ikzscsl)(cc1+isdsldl)
(ccl-isdsldlxl-dzslzch1+isdsld1)

 

[sdlzdcl-ikzszdlcsl+icdzslc12dl+k2c2dslzcls)(ccl+isdsldl)

 

[clds(d12+k2c2512)+icdlsl(clzdz-k232)](ccl+isdsldl)
(02c12+d2d1252312X1-dzslz)

 

_ (cclzds+ic1d2d1szsl)(d12+k2czslz)+(ic2cldlsl-cdlzdSSIZXClzdz-kzsz)
_ (czc12+d2dlzszslz)(l-d2312)

 

ic ldl sl [d252(d12+k2czs 12)+c2(c 12d2,k252)]

 

= cds[c12d12(1-d2812)+k2312(c2c124-d1252)] +
icldls,[d2(d12s2+c2c12)+k2c2s2(d2s1-1)]
(c2c12+d2d12s2512)(l-dzslz)

 

 

_ [cds[c12d12(1-d2s12)+k2512(c2012+d1232)] ]
(c2c,2+d2d12s2s,2)(1-d2s12)

 

. I:cldlsl[d2(d1252+c2c12)+k2c252(d251-1)] ]
l (C2C12+d2d1252512)(1412512)

18 1
Thus,

 

 

x = = f(Canvk)
(c2c12+d2d1252812X1—dzslz)
c d s [d2(d 282+c2c 2)+k2c252(dzs -1)]
=[111 izziz 22l :I=g(c’“’k)
(czcl +d (11 s 51 )(l-d 51 )

II. Circle (x,y) --> Square (Cm)

X-Coordinate

Squaring both sides of equation (C. 15) we get

 

2 [cds[c12d12(1-d2312)+k2312(c2c12+d1252)] T
x =
(c2c12w2d1232312)(1-d2812)

(C. 15)

(C. 16)

(C. 17)

Now, c2, d2 can be expressed in terms of 32; and of, d12 in terms of 512 (see [11]) as

c2 =l-s2
d2=1-k232

012:1 ' $12

(112: 1 ‘ k’zslz

(C. 18)

(C. 19)

(C.20)

(C.21)

182
Substituting equations (C.18)—-9 (C21) in equation (C.22) gives

(1-s2)(1-k2s2)s2[(1-s12)(1- 2312){1-(l-k232)312]+
x2 = 1:2.o.12{(1-s.2)(1-s12)+(1.1831232I]2 (C22)
[ (1—s2)(1-512)+(l-k252)(1-k’2512)s2312]2[1-(1-k252)512]2

an252+an4s4+an656+an858+an10510

 

(C .23)
P0+P282+P4s4+P686+P888+P1Os10+P12812

where an2’ an4, an6' an8' an10' P0, P2, P4, P6, P8, P10, P12 are polynomial functions

of 51:
Pm = [1-2(3-2k2)s12+(15-18k2+4k4)s144(5-3k2)(1-k2)s16+ (c 24)
(15-13k2)(1-k2)s13-6(1-k2)2s110+(1-k2)2s112] '
Pm = -[(1+k2)-2{3+2(1-k2)k2}812+(15+7k2-22k4+4k6)sl4-
4(5+7k2-5k4)(1-k2)s15+(15+22k2-29k4)(1-k2)313- (c.25)
2(3+8k2)2(1-k2)25110+(1+3k2)2(1-k2)2s112]
ans = [l-2(4—k2)s12+(5+k2)(5-4k2)s14-4(5-k2)(2+k2)(l-k2)s16+
(7+5k2)(5-4k2)(l-k2)s18-2(8+7k2)(1-k2)s110+ (C.26)
3(1+k2)2(1-k2)23112]k2
ans = -[-2812+(11-7k2)sl4-4(6-k2)(l-k2)315+2(13-7k2-2k4)(1-k2)318

‘2(7+2k2)( 1-k2)281 10+(3+k2)(1-k2)281 12] [(4 (C.27)

an10= [$14-4(1-k2)s15+2(3-2k2)(1-k2)s18-4(1-k2)s110+(1-k2)23112]k6 (C.28)

183
[1-4812+6Sl4'4516+818]

-2[l-(5+k2)slz+2(5+k2)sl4-10s15+(5-2kz)s18-(1-k2)23110]

1-6(1+k2)812+(15+22kz-k4)s14-4(5+7k2-2k4)s15+
3{5+4(1-k2)k2)}$13-2(3+2kz)(1-k2)sl10+(1,k2)2sl12

-4[-s12+(5+k2)sl4-(10+k2-k4)sl6+2(10-3k2-2k4)318-
5(1-k2)2s11°+(1-k2)23112]k2

[5314-4(6-k2)s16+(36-20k2-k4)s13-4(6.1;2)(”(2)8110+
6(l-k2)23112]k4

-2[-2315+3(2-k2)s18-(6-k2)(1-k2)sl1°+2(1-k2)251121165

[SIS-2(1-k2)sl10+(1-k2)5112]k3

Equation (C.23) can be written as

where

2 4 6 8 10 12
PXO+PX28 +Px4s +PX68 +Px88 +leos +PX128 = O
P x0 = ~P0x2
sz - P -P x2

" nx2 2

P14 = an4'P4x2

(C29)

(C30)

(C31)

(C32)

(C33)

(C34)

(C35)

(C36)

(C37)

(C38)

(C39)

184

Px6 = an5‘P5X2 (C40)

ng = ngxz-ng2 (C -41)

leo = anlO’Plox2 (C42)

P1112 = -P12x2 (c.43)
Y-Coordinate

Squaring both sides of equation (C.16) we get

 

d d2 d 2s2+c2c 2 +k2c2 2 d2 -1
y2::[31 151I ( 1 1 ) S ( 81 I] T (CA4)

(c2c12+d2d1252312X1- 2312)
Substituting equations (C.18)—> (C.21) in equation (C.44) gives

(l-slz)(l-k’2812)312[(1-k252){(1-k’2512)52+(1-52)(l-512)}+

y2 = k2(1-s2)s2{(1-k2s2)s12-1)12 (c.45)

[ (l-sz)(1-slz)+(l-k2s2)(1-k'2312)52312]2[1-(l-k252)512]2

2 4 6 8 10 12
= PnyO+Pny28 +Pny4s +Pny68 +Pny88 +Pnylos +Pny128 (C 46)
P0+P282+P4s4+P636+P858+Plos104-P128” '

Where Pnyo, Pnyz, Pny4’ PHYO’ Pays, PHYIO’ Pny12, P0, P2, P4, P6, P8, P10, P12 are p01yn0mial
functions of 31 viz:

ny6 =

ny8 =

Pny10=

Pny12 =

185
512-(4—k2)sl4+3(2-k2)s16-(4—3k2)s18+1-k2)sl10

—2[2s12-(9-2k2)sl4+(15-7k2)315-(11-8k2)518+3(1-k2)sllo]k2

[2(1+2k2)812-2(4+1 1k2-2k4)sl4+(12+43k2-20k4)516_
(8+36k2-29k4)s13+(2+13k2)(1-k2)s 110119

-4[812-(5+k2)s14+(3-k2)(3+2kz)s15-(7+3k2-5k4)s18+
(l-k2)(2+3k2)s11°]k4

[$12-(4+7k2)s14+(6+23k2-4k4)s16-(4+25k2-10k4-4k6)318+
(1+10k2+4k4)(1-k2)sllo]k4

-2[-sl4+(3+k2)sl6-{3+2k2(1-k2)]518+(1+2k2)(1-k2)5110]k5

[316-(2-k2)sls+(1-k2)sl10]k3

Equation (C.46) can be written as

where

2 4 6 8 10 12
Py0+Py28 +Py4s +Py68 +Py85 +Py10$ "I'Py125 = O
P .. p -P 2
, Yo - nyO 0y

2

(C47)

(C48)

(C49)

(C30)

(C31)

(C32)

(C33)

(C34)

(C35)

(C36)

186

‘34 = Pny4-P4y2 (C57)
Pys = Pnyfi'P6yz (c158)
Pys = Pny8'P8y2 (C39)
Ple = Pnle'PlOy2 (C60)
Pyl2 = Pny12'P12y2 (C61)

Equations (C36) and (C34) are, respectively, of the form

f(k,s,s 1X)

II
C

(C.62)

f(k9598 1),)

II
C

(C.63)

Equations (C62) and (C.63) are solved simultaneously for s and $1, for every x and y
input, using a quasi N ewton-Raphson numerical method.

APPENDIX D

APPENDIX D

EVALUATING THE STRENGTHS OF THE POINT VORTICES

We separate
N

[u -iv]k = 2 -in[1/(zk- zj)'1/(Zk' a2/'z‘j)1 (13.1)
'=l _
I...
into velocity components to get :
A02 A .
u = fi[(@% - 'g—EIEI‘YO—gn'] = f(vortex strength, geometnc factors) (D.2)

A02 .
v = Kifi - (%J = g(vortex strength, geometnc factors) (D3)

where Kj = Vortex strength
A02 = sz + yj2 a = Radius of circle
AY = yk(A02) - azyj AX = xk(A02) - azxj
YMYO =yk-yj XMXO =xk-xj

(m2 + (A102

DENOMI = (XMXO)2 + (YMYO)2 DENOM2
j,k = indices for vortex location and point of interest respectively

Now,

(V9)k = vCosek - uSinGk

' I
, xmxo (A02)(_A_X) _ (A02)(A1_Q YMYO .
I303312110111 ' DENOM2 Icosek [DENOMZ ' DENOMI IsmekI

187

188

m0 _ (AozxA_X) ] _ [(AozxAx) _ YMYO ] - .
“’hm fkj = [DENOMI DENOM2 C0591: DENOM2 DENOMI smek (D 5)

From Equation (D.4) we express the tangential velocity at k points of interest due to
j vortices as

(V9)1 = Klf“ + Iczfu + K3f13 + + xjflj
(V9)2 = K1f21 + K2f22 + x3f23 + + 1(5ij

(V0)3 = Klf31 + K2f32 + K3f33 + ... + fif3j

(Ve)k = Klfkl ‘I' 'szu + K3fB ‘I' ... + Kjfkj

Or, in matrix form we get

 

 

 

 

 

 

"(V191 ' ' f11 f12 f13 f1)“ "‘1 '
(V192 f21 f22 f23 f2j K2
. K
(V193 = f31 f32 f33 f3) 3. or (Ve)k=ij 5. (D6)
.. (V9)k I .. fkl sz fB fkja - IS -

Solving for Kj there results

' f11 f21 f31 fkl‘ "(V191 ‘ “‘1 ‘

f12 f22 f32 sz (V192 "2

f13 f23 f33 fk3 (V193 = "3 or Flgjl (V9)k=sj (D?)
L f11 f21 f31 fkj~ -(Ve)k . “‘5 ~

 

 

 

 

 

 

APPENDIX E

APPENDIX E

LIST OF COMPUTER PROGRAMS

Page
Flow Chart ................................................................................................. 192
Program Nomenclature for some Arrays and Control Parameters ............. 193
Program Swirl .......................................................................................... 197
DVM Library ............................................................................................. 199
- Subroutine INITP ..................................................................................... 199
- Subroutine INITPO ................................................................................... 201
- Subroutine IPV ......................................................................................... 203
- Subroutine GFCT OR ................................................................................ 204
- Subroutine JETMIX ................................................................................. 206
- Subroutine DVM ...................................................................................... 208
- Subroutine JET ......................................................................................... 209
- Subroutine FULL ...................................................................................... 210
- Subroutine VIC ......................................................................................... 212
— Subroutine INITPV ................................................................................... 213

189

190

- Subroutine BORDER ............................................................................... 215
- Subroutine GRIDVZ ................................................................................. 215
- Subroutine GRDV21 ................................................................................ 217
- Subroutine EXACT .................................................................................. 217
- Subroutine VORT ..................................................................................... 219
- Subroutine LAYGRD ............................................................................... 221
- Subroutine PNTV 2 ................................................................................... 224
- Subroutine PRNT ..................................................................................... 226
- Subroutine ADVECT ............................................................................... 228
- Subroutine UPDATE ................................................................................ 231
- Function PHI ........................................................................................ 231
DMAP Library ........................................................................................... 233
- Subroutine SSNl ...................................................................................... 233
- Subroutine XYUV .................................................................................... 234
- Subroutine NASZ ...................................................................................... 234
- Subroutine NASNWR .............................................................................. 236
- Subroutine NASJME ................................................................................ 240
- Subroutine NASFNB ................................................................................ 241
- Function NASEPS ................................................................................ 242
- Subroutine NASABX ............................................................................... 243
- Subroutine DECOM ................................................................................. 243

- Subroutine SUBST ................................................................................... 244

191

- Subroutine MOVEDD .............................................................................. 245
- Subroutine NORMSD .............................................................................. 245
- Subroutine EXCHSD ................................................................................ 246
- Subroutine VIPASD ................................................................................. 246
- Subroutine F ............................................................................................. 247
- Subroutine INTPLU ................................................................................. 249
- Subroutine CNDNSN ............................................................................... 250
- Function CN .......................................................................................... 251
- Function DN ........................................................................................... 251
- Function SN ........................................................................................... 251
- Function FUN CTN ................................................................................ 252
- Function GAUSS ................................................................................... 252
POISSON Library ..................................................................................... 254
- Subroutine COSGEN ................................................................................ 254
- Subroutine GENBUN ............................................................................... 255
- Subroutine HWSCRT ............................................................................... 256
- Subroutine MERGE .................................................................................. 262
- Subroutine POISD2 .................................................................................. 263
- Subroutine POISN2 .................................................................................. 269
- Subroutine POISP2 ................................................................................... 279
— Subroutine TRIX ...................................................................................... 282

- Subroutine TRI3 ....................................................................................... 283

192

 

 

 

 

 

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193

 

 

C***

C***********************************************************************
C*** PROGRAM NOMENCLATURE FOR SOME ARRAYS AND CONTROL PARAMETERS **
C***********************************************************************
C*** PROGRAM **
C***********************************************************************
C*** it
C*** SWIRL : **
C*** **
C*** **
C*** Parameters : **
C*** IDIMAX - Maximum number of gridpoints (for PET) **
C*** - 4*NCOUNT + [13+INTGER(LN(NCOUNT))]*NCOUNT **
C*** JERT = 1/0: Injector on/off **
C*** MAXPV - Maximum.number of vortex elements **
C*** NCOUNT - 181: Subgrid resolution for interpolation purpose **
C*** NCASE - 1/2: Circle/Square domain **
C*** NCOUNT - Number of grid points in the x and y directions **
C*** NDATA - Total no. of grid points(NDATA=NCOUNT**2 for a square) **
C*** NPRNT - 1/0: Print data : yes/no **
C*** NWALL - Number of wall test points **
C*** SWRL - 1.0/0.0: Swirling motion : yes/no **
C*** TOL - Tolerance **
C*** **

C***********************************************************************

C*** SUBROUTINES **
C***********************************************************************

 

 

 

 

 

C*** **
C*** INITP : **
C*** **
C*** **
C*** Parameters : **
C*** LCODE - Flag for monitoring array size **
C*** MBDCND - 1 Prescribed BC's on the left and right boundary **
C*** NBDCND - 1 Prescribed BC's on the lower and upper boundary **
C*** SGMA - Vortex cut-off radius **
C*** **
C*** **
C*** INITPO : **
C*** **
C*** **
C*** Arrays : **
C*** P - Interpolation points **
C*** SN - Jacobian elliptic function (S or $1) **
C*** UVDATA - (u,v) coordinates in square domain **
C*** XYDATA - (x,y) coordinates in circular domain **
C*** **
C*** Parameters : **
C*** RMODZ - Modulus. One of the arguments of the elliptic fctns. **
C*** RK - Complete elliptic integral of the first kind **
C*** RW - Radius of chamber - 1.0 cm. **
C*** **
C*** ****

C*** IPV: ****

194

 

 

 

 

 

C*** **
C*** **
C*** Arrays : **
C*** GPNT - Vortex strength **
C*** RC - Radial locations of concentric circles **
C*** RPNT - Radial-coordinates of point vortices **
C*** S - Geometric shape factors **
C*** SINV - Inverted geometric shape factors **
C*** SRD - Dyer's velocity test points **
C*** SVD - Dyer's velocity data at test points **
C*** TPNT - Tangential-coordinates of point vortices **
C*** XPNT - x-coordinates of point vortices **
C*** YPNT - y-coordinates of point vortices **
C*** **
C*** Parameters : **
C*** GPVMN - Minimum vortex strength **
C*** GPVMX - Maxmum vortex strength **
C*** MXPV - Maximum allowable size for arrays NPVCI), RC() **
C*** NPV - Number of point vortices **
C*** NCCl - Number of concentric circles inside of circular do **
C*** NCC2 - Number of reference points for Dyer’s data **
C*** NPVC - Number of point vortices per concentric circle **
C*** **
C*** **
C*** JETMIX : **
C*** **
C*** **
C*** Parameters : **
C*** DSPM - Dsplcmt from injector lip to inner edge of shear layer **
C*** JETON = Turn injector on **
C*** JETOFF - Turn injector off **
C*** NXDIR - Number of grid cells in the x-direction **
C*** NYDIR - Number of grid cells in the y-direction **
C*** VIN - Inlet jet velocity **
C*** XA - x-location of injector lip **
C*** XAB — Horizontal distance between injector lip and seat **
C*** XB - x-location of injector seat **
C*** XPA — x-distance between vortex particle and lip **
C*** XPB - x-distance between vortex particle and seat **
C*** YAB - Vertical distance between injector lip and seat **
C*** YB - y-location of injector seat **
C*** YPA - y-distance between vortex particle and lip **
C*** YPB - y-distance between vortex particle and seat **
C*** **
C*** **
C*** JET **
C*** **
C*** **
C*** Arrays : **
C*** FTAG - Tags for jet vortices **
C*** **
C*** Parameters : **
C*** GPA - Strength of vortex issuing from valve lip **
C*** GPB - Strength of vortex issuing from valve seat **

C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
c***
c***
C***
c***
C***
C***
C***
C***
C***
C***

195

 

 

 

 

 

 

XLA - x-location of vortex particle issuing from valve lip
XLB - x-location of vortex particle issuing from valve seat
YLA - y-location of vortex particle issuing from valve lip
YLB - y—location of vortex particle issuing from valve seat
XYUV

Parameters :

61,62 - Initial guesses for the iteration process NAS routine
UPNT - x-coordinate of vortex in square domain

VPNT - y-coordinate of vortex in square domain

HWSCRT :

Arrays -

BDA,BDB,BDC,BDD - Boundary data

F - Streamfunction

W - Work space

Parameters :

IDIMF - Number of grid points in the x-direction

IDIMW - 2*NCOUNT*NCOUNT

IDIMWP - IDIMW+2*NCOUNT

JDIMF - Number of grid points in the y-direction

GRDV21 :

Arrays :

UGl - u-component velocity in the circular domain

UGlS - u-comp. vel. in the circular domain (single precision)
UGZ - u—component velocity in the square domain

VGl - v-component velocity in the circular domain

VGlS - v-comp. vel. in the circular domain (single precision)
VGZ - v-component velocity in the square domain

Parameters :
E11,E12- Matrices for velocity transformation

 

EXACT

 

Arrays :

UGlE - u-component velocity in the circular domain
VGlE - v-component velocity in the circular domain
Parameters :

NAP - Non-analytical grid points

**
**

'k*
**
**
*‘k
**

*‘k
*‘k
*‘k
**
*‘k
**
**
*‘k
**
**
*‘k
**
**
*‘k
*‘k
**
**
**
**
**
**
**
*‘k
*‘k
**
**
*‘k
*‘k
**
**
*‘k
*‘k
**
*‘k
*‘k
**
**
*‘k
**
*‘k
**
**
**
**
*‘k

C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***

196

 

 

 

 

 

**
**
**
**
**
**
**
**
**
**
**
**
**
**

**
**

**

VORT

Arrays :

Q - Vorticity

LAYGRD :

Arrays :

IBOX - Domain tags

NG - Singularity tags

UGN - u-component velocity on uniform grid in circular domain**
VGN — v-component velocity on uniform grid in circular domain**
XY - uniform (x,y) coordinates in circular domain

PNTV2

 

Arrays

U
V

u-component velocity at vortex location
v-component velocity at vortex location

**
**
**
**
**
**

**

C******************************************t****************************

C***

FUNCTIONS

**

C***********************************************************************

C***
C***
C***
C***
C***
c***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
c***
C***
C***

 

PHI

 

PHI

DHU,DHV-

Interpolation function (Filter)

Weight values

 

CN,DN,SN

 

CN, DN, SN - Jacobian elliptic functions

 

GAUSS

 

Gauss-Legendre quadrature for integration purposes

**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**
**

C***********************************************************************

C***

**

C***

197

C***********************************************************************

PROGRAM SWIRL

c***********************************************************************

C***
C***
C***
C***
C***
C***
C***
C***
C***
C***
C***

THIS PROGRAM EMPLOYS DISCRETE VORTICES TO SIMULATE THE MIXING
PROCESS PRIOR TO CHEMICAL REACTION IN THE FUEL-INJECTED CYLINDER
MIXING DESCRIPTIONS ARE RESTRICTED TO A TWO

OF A PISTON ENGINE.
DIMENSIONAL, INCOMPRESSIBLE, INVISCID FORMULATION OF THE BULK
DIFFUSION PROCESS. THE NUMERICAL SIMULATION IS BASED ON A
VORTEX-IN-CELL FORMULATION. POISSON INVERSION IS ACCOMPLISHED
USING THE FAST FOURIER TRANSFORM PACKAGE "FISHPAK" FROM THE
NATIONAL CENTER FOR ATMOSPHERIC RESEARCH (NCAR), BOULDER,
COLORADO. PLOTIT AND NCAR GRAPHICS PACKAGES ARE USED.

**
**
**
**
**
**
*a
**
**
**
**

C***********************************************************************

C****

C***

c***

C***

DOUBLE PRECISION G1,GZ,P,SN,SSl,RK,RMOD2,XX,YY

DOUBLE PRECISION UGl,UGlE,UVDATA,VGI,VG1E,XYDATA
PARAMETER(JERT-1,MCOUNT=181,NPRNT-l,SWRL-l.0)
PARAMETER (NCASE'1,NCOUNT=61,G1=0.3719D0,G2=0.3719D0)
PARAMETER(IDIMAX=1300,MAXPV-12000,NDATA-NCOUNT*NCOUNT)

COMMON/ENDARI/MBDCND,NBDCND
COMMON/GRIDl/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/GRIDZ/ULNGTH,UMN,UMX,VLNGTH,VMN,VMX
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/GRIDB/DX,DXINVT,DY,DYINVT,XMN,XMX,YMN,YMX
COMMON/JETPAR/DJ,PMSIGN,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/MIXPAR/JETOFF,JETON,NXDIR,NYDIR
COMMON/PTPARI/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,HZ,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
COMMON/WORKSP/RWKSP(2727472)

VIRTUAL UVDATA(2,NDATA),XYDATA(2,NDATA)

VIRTUAL P(MCOUNT),SN(MCOUNT),SSl(2,NDATA)

VIRTUAL XPNT(MAXPV),YPNT(MAXPV)

VIRTUAL GPNT(MAXPV),RPNT(MAXPV),TPNT(MAXPV)

VIRTUAL UGlE(NCOUNT,NCOUNT),VG1E(NCOUNT,NCOUNTI

VIRTUAL UPNT(MAXPV),VPNT(MAXPV)

DIMENSION F(NCOUNT,NCOUNT),W(IDIMAX)

VIRTUAL BDA(NCOUNT),BDB(NCOUNT),BDC(NCOUNT),BDD(NCOUNT)
VIRTUAL UGZ(NCOUNT,NCOUNT),VG2(NCOUNT,NCOUNT)

VIRTUAL Q(NCOUNT,NCOUNT)

VIRTUAL UN(MAXPV),VN(MAXPV),XPN(MAXPV),YPN(MAXPV)

VIRTUAL UG1(NCOUNT,NCOUNT),VG1(NCOUNT,NCOUNT)

VIRTUAL UGlS(NCOUNT,NCOUNT),VG1$(NCOUNT,NCOUNT)

VIRTUAL UPV1(MAXPV),VPV1(MAXPV)

VIRTUAL IBOX(NCOUNT,NCOUNT),NG(NCOUNT,NCOUNT),XY(2,NDATA)
VIRTUAL FTAG(2,MAXPV),UGN(NCOUNT,NCOUNT),VGN(NCOUNT,NCOUNT)

198

OPEN(UNIT‘1,FILE"SWIRLS.CLNS',TYPEa’NEW')
OPEN(UNIT'2,FILE-'SWIRLS.CLNS’,TYPE-'NEW')
OPEN(UNIT'3,FILE-'SWIRLS.CLNS',TYPEI'NEW’)
OPEN(UNIT=4,FILE-'SWIRLS.CLNS',TYPE-'NEW’)
OPEN(UNIT-5,STATUs-'OLD',FILE-’SYS$OUTPUT')
OPEN(UNIT'6,FILE-’SWIRLS.CLNS’,TYPEI'NEW')
OPEN(UNIT-7,FILE"SWIRLS.CLNS',TYPE='NEW')
OPEN(UNIT-8,FILE-'SWIRLS.CLNS',TYPEa’NEW')

C***
C***

 

C*** Initialize parameters :
c***

 

C***

CALL IWKIN(2727472)
CALL INITP(GPNT,JERT,MAXPV,MCOUNT,NCOUNT,P,RPNT,SN,SSl,

*SWRL,TPNT,UVDATA,XPNT,XYDATA,YPNT)
C***

 

C***

C*** Perform.discrete vortex vortex calculations :

 

c***

C***
DO 1 NTIME - NSTART,NSTOP

C***

CALL DVM(BDA,BDB,BDC,BDD,F,FTAG,G1,G2,GPNT,IBOX,JERT,
*MAXPV,MCOUNT,NCASE,NCOUNT,NG,NPRNT,P,Q,RPNT,SN,SSl,SWRL,
*TPNT,UGN,UGI,UGlE,UGIS,UG2,UN,UPNT,UPV1,UVDATA,VGN,VG1,
*VGlE,VGIS,VGZ,VN,VPNT,VPV1,VR,W,XPN,XPNT,XY,XYDATA,YPN,YPNT)

C***

IF(LCODE.EQ.0.0R.NTIME.EQ.NSTOP) GO TO 2

 

 

C***
1 CONTINUE
C***
C***
C*** Close the debugging file
C***
C***

2 CLOSE(UNIT-1)
CLOSE(UNIT-2)
CLOSE(UNIT-3)
CLOSE(UNIT-4)
CLOSE(UNIT-6)
CLOSE(UNIT-7)
CLOSE(UNIT-8)

c***

STOP 'NORMAL TERMINATION OF PROGRAM SWIRLS'

END
C***

199

***
g***********************************************************************
C*** DVM LIBRARY CONTAINS THE FOLLOWING SUBROUTINES/FUNTIONS : ***
C*** **
C*** SUBROUTINES : **
C*** 1. INITP - Inititializes parameters **
C*** -INITPO - Initializes grid and elliptic parameters **
C*** -IPV - Initializes coords. & strengths of the vortices **
C*** -GFCTOR - Computes the geometric shape factors **
C*** -LSGRR - Inverts the geometric shape factors [IMSL] **
C*** -JETMIX - Initializes jet and mixing parameters **
C*** 2. DVM - Makes calls vortex routines : **
C*** -JET - Simulates fluid injection **
C*** -FULL - Checks if cylinder fluid is "completely mixed" **
C*** -VIC - Calls Vortex-In—Cell formulation routines **
C*** -XYUV - Maps coordinates from circle to square [Dmaplib] **
C*** -INITPV- Initializes vorticity on square grid **
C*** -BORDER- Borders active points according to bndary conds. **
C*** -HWSCRT- Does the Fast Fourier Transform.[Poissonlib] **
C*** -GRIDV2- Computes velocities on a square grid **
C*** -GRDV21- Maps grid velocities : square -> circle **
C*** -EXACT- Computes exact velocities for non-analytical pts. **
C*** -VORT - Computes vorticity at grid points in uv-plane **
C*** -LAYGRD- Computes velocities on imposed uniform grid **
C*** -PNTV2 - Interpolates velocities at off grid locations **
C*** -PRNT - Prints data **
C*** -ADVECT - Advects vortices via Heun's time integrtn.scheme **
C*** -UPDATE - Updates parameters/array size **
C****** *****
C*** FUNCTIONS : **
C*** **
C*** 1. PHI - Evaluates the interpolation functions **
C*** **
C*** SUBROUTINE(S)/FUNCTIONS CALLED BUT NOT IN THIS LIBRARY : **
C*** SUBROUTINES : **
C*** 1. CNDNSN - Evaluates Jacobian elliptic functions CN,DN and SN **
C*** 2. SSNl - Evaluates Jacobian elliptic function SN **
C*** 3. XYUV - (See above) **
C*** 4. HWSCRT - (See above) **
C*** 5. PRNT - (See above) **
C****** **
C*** FUCTIONS : **
C*** 2. FUNCTN - Evaluates integrand of elliptic integral K(or K') **
C*** [1/sqrt(l-k**23in**2(phi)]/[1/sqrt(l-k'**23in**2(phi)]**
C*** 3. GAUSS - Performs Gaussian integration of elliptic integrals **
c*** **

C*********************i************************************************t

SUBROUTINE INITP(GPNT,JERT,MAXPV,MCOUNT,NCOUNT,P,RPNT,SN,SSl,

*SWRL,TPNT,UVDATA,XPNT,XYDATA,YPNT)
C***********************************************************************

c*** **
C*** THIS SUBROUTINE INITIALIZES PARAMETERS ' **
C*** **

C***************************************************************t*******

C***

200

DOUBLE PRECISION P,RK,RMOD2,SN,SSl,UVDATA,XX,XYDATA,YY
COMMON/BNDARY/MBDCND,NBDCND
COMMON/GRIDl/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/GRIDZ/ULNGTH,UMN,UMX,VLNGTH,VMN,VMX
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/GRIDB/DX,DXINVT,DY,DYINVT,XMN,XMX,YMN,YMX
COMMON/JETPAR/DJ,PMSIGN,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/MIXPAR/JETOFF,JETON,NXDIR,NYDIR
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
EXTERNAL GAUSS,FUNCTN

C***********************************************************************

c***
C***
C***
C***

C***
C***
c***
C***
C***

C***
C***
c***
C***
C***

C***
C***
C***
C***
c***

 

 

 

 

 

 

 

 

Grid, elliptic and interpolation parameters/arrays :
CALL INITPO(DU,DV,DX,DY,MCOUNT,NCOUNT,P,PI,RW,SN,SSl,TWOPI,
*UVDATA,XYDATA)

Time step parameters :
NFULL - 0

NSTART - 1

NSTOP - 1001

DTIME - 0.001

Vortex parameters :

LCODE - l

NPV - 0

GPVMN - 1.0E+06

GPVMX - -GPVMN

CNSTl - 2.0/3.0

H2 - 3.0

SGMAO - 1.0E-06

SGMA - 0.05

SGMAZI - 1.0/(SGMA*SGMA)
TOL - 10.*AMACH(4)
Square domain parameters :
MBDCND - 1

NBDCND - 1

DUINVT - 1 O/DU

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***

C***
C***

201

DXINVT = 1.0/DX
DVINVT - 1.0/DV
DYINVT - 1.0/DY
HDUINV = 0.5*DUINVT
HDVINV 8 0.5*DVINVT
UMN = -0.5*SNGL(RK)
UMX - 0.5*SNGL(RK)
ULNGTH - SNGL(RK)
VLNGTH - ULNGTH

VMN - UMN

VMX - UMX

XMN - -RW

XMX - RW

YMN - XMN

YMX = XMX

 

Initialize coordinates and strengths of vortices

 

IF(SWRL.EQ.1.0) CALL IPV(GPNT,RPNT,TPNT,XPNT,YPNT)

 

Initialize jet and mixing parameters :

 

CALL JETMIX(JERT,NCOUNT)

RETURN
END

C***********************************************************************
SUBROUTINE INITPO(DU,DV,DX,DY,MCOUNT,NCOUNT,P,PI,RW,SN,SSl,TWOPI,

*UVDATA,XYDATA)

C***********************************************************************

C***
C***
C***
C***
C***

THIS SUBROUTINE INITIALIZES GRID AND ELLIPTIC PARAMETERS
(i) S(U) : U in P() and S in SN()
(ii) Grid point coordinates:(U,V) in UVDATA(); (X,Y) in XYDATA()

**
**
**
**
**

C***********************************************************************

C***

C***

IMPLICIT DOUBLE PRECISION (A-H, O-Z)
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
EXTERNAL GAUSS,FUNCTN

VIRTUAL P(*),SN(*),SSl(2,*),UVDATA(2,*),XYDATA(2,*)

C***********************************************************************

C***
c***
c***
C***

 

Grid parameters :

 

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

202

IMIN - 1

IMAX - NCOUNT

IMID - (IMAX+IMIN)/2
ISIZE - IMAX-l

JMID - IMID

JMIN - IMIN

JMAX - IMAX

JSIZE - ISIZE

 

Elliptic parameters

 

RMODZ ' 0.5D0

PI - DACOS(-1.0D0)

TWOPI - PI+PI

RK - GAUSS( 0.0D0,0.5D0*PI,15,FUNCTN,RMOD2 )
DU - RK/DFLOAT(ISIZE)

DV - DU

 

Pretabulate S elliptic function :

 

DUU - RK/DFLOAT(MCOUNT-1)

DO 1 I - 1,MCOUNT

P(I) - DUU*DFLOAT(I-1) - 0.5D0*RK
CALL SSN1( P(I), RMOD2, SN(I) )
CONTINUE

 

Initialize grid point coordinates (U,V) and (X,Y)

 

DO 2 J ' JMIN,JMAX

V - DV*DFLOAT(J-1) - 0.5D0*RK
CALL CNDNSN( V, RMOD2, C1, D1, 31 )

C12 - C1*C1

012 - D1*D1

$12 - 51*31

DO 2 I ' IMIN,IMAX

K - I+(J-JMIN)*JMAX

U - DU*DFLOAT(I-1) - 0.5D0*RK
CALL CNDNSN( U, RMOD2, C, D, S )

C2 - C*C

D2 - D*D

$2 - S*S

DENOM-(C2*C12+D2*D12*SZ*812)*(1.0DO-D2*SIZ)
XNUMPC*D*S*(C12*D12*(1.0D0-D2*512)+RMOD2*512*(C2*C12+D12*52))
YNUMPC1*D1*S1*(D2*(D12*SZ+C2*C12)+RMOD2*C2*SZ*(D2*512-1.0D0))
UVDATA(1,K) - U

UVDATA(2,K) - V

XYDATA(1,K) - XNUM/DENOM

XYDATA(2,K) - YNUM/DENOM

C***

C***
C***
C***
C***
C***

C***

C***

C***
C***

203

IF(I.EQ.IMAX.AND.J.EQ.JMAX) THEN

RW - DSQRT(XYDATA(1,K)**2+XYDATA(2,K)**2)
DX ' 2.0D0*RW/DFLOAT(ISIZE)

DY - DX

END IF

 

Store non-analytical points :

 

IF(DABS(U).EQ.DABS(V)) THEN
SSl(1,K) = $1

SSl(2,K) = S

END IF

CONTINUE

RETURN
END

C***********************************************************************

SUBROUTINE IPV(GPNT,RPNT,TPNT,XPNT,YPNT)

C****************************************************t******************

C***
c***
C***
C***
C***

THIS SUBROUTINE INITIALIZES THE COORDINATES AND STRENGTHS
OF THE POINT VORTICES USING DYER'S VELOCITY DATA [33] AND
MATRIX INVERSION ROUTINE FROM IMSL ROUTINES.

**
**
**
**
**

C***********************************************************************

C***

C***
c***
C***

C
C

C
C

C***
C***
c***

C***
c***
C***

c***
C***
C***

PARAMETER(MXPV-200,NCC1-8,NCC2=10)

Number of vortices per concentric circle

DATA NPVC/4,6,6,6,28,32,15,8/ !@100 ms.
DATA NPVC/4,6,6,8,31,32,15,8/ !@150 ms.
DATA NPVC/4,6,6,13,31,32,15,8/ !@200 ms.
DATA NPVC/4,6,6,16,34,34,18,8/ !@300 ms.
DATA NPVC/4,6,6,6,26,30,16,8/ !@500 ms.

Radial location of concentric circles

DATA RC/0.1,0.2,0.30,0.40,0.50,0.60,0.70,0.80/

Velocity test points :

DATA SRD/0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90/

Dyer’s velocity data at test points :

DATA SVD/0.00,7.80,24.SO,33.10,37.20,39.00,37.60,33.70,!@100 ms.

*30.00,28.20/

204

DATA SVD/0.00,6.20,14.70,21.60,25.10,23.9o,22.2o,20.4o,!@150 ms.
*18.80,17.40/

DATA SVD/0.00,S.30,12.60,16.60,17.00,16.30,15.10,14.00, !@2oo ms.
*13.10,12.40/
C DATA SVD/0.00,3.20,10.oo,11.40,10.3o,9.6o,9.00,8.40, !@300 ms.
c *7.80,7.60/
C DATA SVD/0.00,2.80,4.90,5.50,5.20,4.90,4.70,4.50, !@500 ms.
C *4.30,4.30/

00

COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNSTl,H2,RW,SGMAZI,TOL

VIRTUAL S(NCC2,MXPV),SINV(MXPV,NCC2)

VIRTUAL NPVC(NCC1),RC(NCC1),SVD(NCC2),SRD(NCC2)

C***
C***********************************************************************
C***

 

C*** Determine the total number of point vortices

 

C***
C***
DO 20 N - 1,NCC1
NPV - va + NPVC(N)
20 CONTINUE
C***
C***

 

C*** Compute the geometric factors, perform matrix inversion

C*** and then compute the strengths of the point vortices
C***

 

C***
CALL GFCTOR(GPNT,NCC2,NPV,NCC1,NCC2,NPV,NPVC,NCC2,
*RC,S,SINV,SRD,SVD,RPNT,TPNT,XPNT,YPNT)
c***
RETURN
END
C***
c***

C***********************************************************************

SUBROUTINE GFCTOR(GPNT,LS,LSINV,NCC1,NCC2,NCS,NPVC,NRS,RC,S,SINV,

*SRD,SVD,RPNT,TPNT,XPNT,YPNT)
C***********************************************************************

C*** **

C*** THIS SUBROUTINE COMPUTES THE GEOMETRIC FACTORS, PERFORMS MATRIX **

C*** INVERSION AND THEN COMPUTES THE STRENGHTS OF THE POINT VORTICES **
C*** *t

C***********************************************************************
C***

COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/PTPARllGPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL

VIRTUAL GPNT(*),RPNT(*),TPNT(*),XPNT(*),YPNT(*)

VIRTUAL NPVC(*),RC(*),S(LS,*),SINV(LSINV,*),SRD(*),SVD(*)
EXTERNAL LSGRR

C***
Ct****************t*t****************************t**********************

205

 

C***
C*** Compute the geometric shape factor relative to the reference point

 

C***

 

 

C***
DO 30 I - 1,NCC2
J - 0
T - 0.0
x - SRD(I)
Y - 0.0
C***
DO 20 N - 1,NCC1
DELTO ' TWOPI/NPVC(N)
DO 10 K - 1,NPVC(N)
J = J + 1
T0 - DELTO*FLOAT(K)
x0 - RC(N)*COS(T0)
Y0 - RC(N)*SIN(T0)
IF(I.EQ.1) THEN
RPNT(J) - SQRT(X0*X0+Y0*Y0)
TPNT(J) - T0
XPNT(J) ' X0
YPNT(J) = Y0
END IF
C***
C***
C*** Initialize parameters for the shape factors
C***
C***
XMXO - X-XO
YMYO - Y-YO
A02 - X0*X0+Y0*Y0
AX - A02*X-(RW*RW)*XO
AY - A02*Y-(RW*RW)*Y0
AX2 - AX*AX
AY2 = AY*AY
XMXOZ - XMXO*XMXO
YMY02 - YMYO*YMYO
DENOMl - XMX02+YMY02
DENOMZ - AX2+AY2
IF(DENOM2.EQ.0.0) DENOMZ - SGMAO
SQRTD - SQRT(DENOM1)
H1 - (3.0*CNST1*SQRTD)/SGMA
C***
IF(SQRTD.LE.SGMA) THEN
SU ' (A02*AY/DENOM2)-SGMA2I*(H2-H1)*YMYO
SV - -(A02*AX/DENOM2)+SGMAZI*(H2-H1)*XMXO
ELSE
SU - (A02*AY/DENOM2)-(YMYO/DENOM1)
SV 8 -(A02*AX/DENOM2)+(XMXO/DENOMl)
END IF
C***
' S(I,J) - SV*COS(T)-SU*SIN(T)
C***

10 CONTINUE

20 CONTINUE
30 CONTINUE

206

 

 

 

 

C***
C***
C*** Invert the 3 matrix :
C***
C***

CALL LSGRR(NRS,NCS,S,LS,TOL,IRANK,SINV,LSINV)
C***
C***
C*** (i) Compute the strengths of the point vortices.
C*** (ii) Check for the minimum and maximum strengths.
C***
C***

DO 50 J = 1,NPV

GPNT(J) - 0.0

DC 40 I - 1,NCC2

GPNT(J) - GPNT(J) + SVD(I)*SINV(J,I)

40 CONTINUE
GPVMN - AMIN1(GPNT(J),GPVMN)
GPVMX - AMAX1(GPNT(J),GPVMX)
50 CONTINUE

C***

RETURN

END
C***

C***********************************************************************

SUBROUTINE

JETMIX(JERT,NCOUNT)

C***********************************************************************

C***
C***
C***
C***
C***
C***

THIS SUBROUTINE INITIALIZES JET AND MIXING SIMULATION PARAMETERS

JDIR - 0/1 - Same/opposite direction of flow

SIV

- 0.0/1.0 - Different/same initial velocity.

**
**
**
ti
**
**

C***********************************************************************

c***

c***

PARAMETER(JDIR-1,SIv-1.0)

COMMON/GRID

1/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV

COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/JETPAR/DJ,PMSIGN,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/MIXPAR/JETOFF,JETON,NXDIR,NYDIR
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME

C***********************************************************************

C***
C***
C***
C***

 

Initialize

jet parameters

 

IF(JERT.EQ.
JETON - NS
JETOFF - NS
END IF

1) THEN
TART
TOP

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***

C***

C***
C***
C***
C***
c***
c***

C***
C***
C***
C***
c***

c***
c***
C***
C***

207

 

Specify injector orifice diameter :

 

DJ = DU

 

Specify injector angle. CCW from horizontal :

 

TJ = PI/3.0
IF(JDIR.EQ.1) THEN

XB = -0.5*DJ*SIN(TJ)
ELSE

XB 3 0.5*DJ*SIN(TJ)
END IF

IF(XB.GE.0.0) PMSIGN = -1.0
IF(XB.LE.0.0) PMSIGN - 1.0

YB = 0.5*DJ*COS(TJ)
YAB - DJ*COS(TJ)
XAB = DJ*SIN(TJ)
XA = PMSIGN*XAB+XB

 

Compute the initial minimum allowable
displacement of jet particles from injector orifice

 

IF(TJ.EQ.(PI/2.0)) DSPM - 3.633021*DJ
IF(TJ.EQ.(PI/2.4)) DSPM - 4.067413*DJ
IF(TJ.EQ.(PI/3.0)) DSPM - 4.938183*DJ
IF(TJ.EQ.(PI/4.0)) DSPM - 6.794676*DJ
IF(TJ.EQ.(PI/6.0)) DSPM - 16.39000*DJ
XPB = DSPM*COS(TJ)

YPB - DSPM*SIN(TJ)

pr - xps

YPA = YPB

 

Initialize jet velocity :

 

IF(SIV.EQ.1.0) THEN

VIN - 6.794676*DJ/DTIME !If velocity at 90 deg. is desired
ELSE
VIN = DSPM/DTIME
END IF

 

Initialize mixing parameters :

 

208

C***
NXDIR I 8
NYDIR I 8

C***

RETURN
END
c***
C***

c***********************************************************************

SUBROUTINE DVM(BDA,BDB,BDC,BDD,F,FTAG,G1,G2,GPNT,IBOX,JERT,
*MAXPV,MCOUNT,NCASE,NCOUNT,NG,NPRNT,P,Q,RPNT,SN,SSl,SWRL,
*TPNT,UGN,UG1,UG1E,UGIS,UGZ,UN,UPNT,UPV1,UVDATA,VGN,VG1,VGlE,VG1$,

*VG2,VN,VPNT,VPV1,VR,W,XPN,XPNT,XY,XYDATA,YPN,YPNT)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE MAKES CALLS TO DISCRETE VORTEX ALGORITHMS **
C*** **

C***********************************************************************

 

 

 

 

C***
DOUBLE PRECISION G1,GZ,P,SN,SSl,UG1,UGlE,UVDATA,VG1,VGlE,XYDATA
COMMON/MIXPAR/JETOFF,JETON,NXDIR,NYDIR
COMMON/PTPARI/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME

C***

C***

C*** Turn injector on/Off

C***

C***
IF(JERT.EQ.1) THEN
IF(NTIME.GE.JETON.AND.NTIME.LE.JETOFF) THEN
CALL JET(FTAG,GPNT,MAXPV,RPNT,TPNT,XPNT,YPNT)
IF(LCODE.EQ.0) GO TO 3
END IF

C***

C***

C*** Check if cylinder fluid is “completely mixed"

C***

C***
CALL FULL(IBOX,NCOUNT,RPNT,XPNT,YPNT)

c***

c***

 

C*** Compute advection velocities

 

 

 

C***
C***

CALL VIC(BDA,BDB,BDC,BDD,F,GI,GZ,GPNT,IBOX,1,JERT,MCOUNT,NCASE,
*NCOUNT,NG,P,Q,SN,SSl,UGN,UGl,UGlE,UGlS,UGZ,UPNT,UPV1,UVDATA,VGN,
*VGl,VGlE,VG1$,VGZ,VPNT,VPV1,W,XPNT,XY,XYDATA,YPNT)

C***
C***
C*** Check if time to print data :
C***
C***

IF(NPRNT.EQ.1) THEN
CALL PRNT(F,FTAG,NCASE,NCOUNT,Q,RPNT,UGlS,UGZ,UGN,UVDATA,VGlS,

209

*VG2,VGN,XPNT,XY,XYDATA,YPNT)

END IF
C***

C***

C*** Initialize vortex elements for the next time step :
C***

 

 

 

 

C***

CALL ADVECT(BDA,BDB,BDC,BDD,F,FTAG,Gl,GZ,GPNT,IBOX,JERT,MCOUNT,
*NCASE,NCOUNT,NG,P,Q,RPNT,SN,SSl,TPNT,UGl,UGlE,UGlS,UGZ,UGN,UN,
*UPV1,UPNT,UVDATA,VGl,VGlE,VG1$,VGZ,VGN,VN,VPV1,VPNT,XPN,XPNT,YPN,
*YPNT,W,XY,XYDATA)

C***
C***
C*** Updata parameters and monitor array size :
C***
C***
CALL UPDATE(MAXPV)
C***
RETURN
END
C***

C***********************************************************************

SUBROUTINE JET(FTAG,GPNT,MAXPV,RPNT,TPNT,XPNT,YPNT)
c***********************************************************************

C*** **
C*** THIS SUBROUTINE SIMULATES THE FLUID INJECTION PROCESS **
C*** Ref.: **

C*** "Piston-Cylinder Fluid Motion Via Vortex Dynamics", W.T. Ashurst **

C*** Fluid Mechanics of Combustion Systems, ed. by T. Morel et. a1. **
C*** **

C***********************************************************************
C***

COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/JETPAR/DJ,PMSIGN,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,HZ,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME

VIRTUAL GPNT(*),FTAG(2,*),RPNT(*),TPNT(*),XPNT(*),YPNT(*)
C***

C***********************************************************************
C*** SEAT CALCULATIONS (B) **
C***********************************************************************
C***

 

C*** Compute location of particle issuing from valve seat (region b) :
C***

 

c***
XLB I XB+PMSIGN*XPB
YLB I (YB+YPB)-RW
GPB I PMSIGN*0.5*(VIN*VIN*DTIME)
GPB I GPB/TWOPI
C***
c***

 

C*** Update parameters and store values in point vortex arrays :
C***

 

210

C***

NPV I NPV + 1

IF(NPV.GT.MAXPV) GO TO 4

XPNT(NPV) I XLB

YPNT(NPV) I YLB

GPNT(NPV) I GPB

RPNT(NPV) I SQRT(XLB*XLB+YLB*YLB)

TPNT(NPV) I ATAN2(YLB,XLB)

NJP I NPV
c***
Ct**********************************************************************
C*** LIP CALCULATIONS (A) **

C**********‘k‘k***********************************************************

 

C***

C*** Compute location of particle issuing from valve lip (region a) :
C***

 

C***
XLA - XA+PMSIGN*XPA
YLA - YLB-YAB

C***

C***

 

C*** Update parameters and store values in point vortex arrays :
C***

 

 

 

C***
NPV I NPV + 1
IF(NPV.GT.MAXPV) GO TO 4
XPNT(NPV) I XLA
YPNT(NPV) I YLA
GPNT(NPV) I IGPB
RPNT(NPV) I SQRT(XLA*XLA+YLA*YLA)
TPNT(NPV) I ATAN2(YLA,XLA)
C***
C***
C*** Tag the jet vortices as fuel particles
C***
C***

2 DO 3 N I NJP,NPV
FTAG(1,N) I 1.0
3 CONTINUE
C***
4 IF(NPV.GT.MAXPV) LCODE I 0
RETURN

END
c***

C***
C***********************************************************************

SUBROUTINE FULL(FTAG,IBOX,NCOUNT,RPNT,XPNT,YPNT)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE DETERMINES IF DOMAIN IS FILLED WITH VORTICES **
c*** **

Ctttt***§***************************************************************
c***

PARAMETER(NSKMI1)

211

COMMON/MIXPAR/JETOFF,JETON,NXDIR,NYDIR
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
VIRTUAL FTAG(2,*),IBOX(NCOUNT,*),RPNT(*),XPNT(*),YPNT(*)

 

 

C***
C***********************************************************************
c***
C*** Initialize parameters :
C***
C***
NCELLS I 0
R0 I 2.0*RW/FLOAT(NCIRS)
DO 7 J I 1,NRAYS
DO 7 I I 1,NCIRS
IBOX(I,J) I 0
7 CONTINUE
C***
C***

 

C*** Tag grid cells within Circle

 

 

 

C***
C***
DO 1 J I 1,NRAYS
RY I IRW+R0*FLOAT(J-l)
RYP I RY+R0
HRYP I RY+(0.5*R0)
DO 1 I I 1,NCIRS
RX I -RW+R0*FLOAT(I-l)
RXP I RX+R0
HRXP I RX+(0.5*R0)
RCC I SQRT(HRXP*HRXP+HRYP*HRYP)
C***
IF(RCC.LE.RW) THEN
IBOX(I,J) I 1
NCELLS I NCELLS + 1
END IF
C***
1 CONTINUE
C***
C***
C*** Identify cell
C***
C***
DO 3 J I 1,NRAYS
RY I -RW+R0*FLOAT(J-1)
RYP I RY+R0
D0 3 I I 1,NCIRS
NYES I 0
RX I IRW+RO*FLOAT(I-1)
RXP I RX+R0
C***
C***

 

C*** Tag cell with at least NO vortex/vortices
c***

 

212

C***
NO I 2
IF(IBOX(I,J).EQ.1) THEN
DO 2 N I 1,NPV
IF(FTAG(1,N).EQ.1.0) THEN
IF((XPNT(N).GE.RX.AND.XPNT(N).LE.RXP).AND.
*(YPNT(N).GE.RY.AND.YPNT(N).LE.RYP)) THEN
NYES I NYES + 1
IF(NYES.EQ.NO) THEN
IBOX(I,J) I 2
GO TO 3
END IF
END IF
END IF
2 CONTINUE
END IF
3 CONTINUE
C***
C***

 

C*** Check if each cell contains at least NO vortex/vortices
C***

 

C***
NYES I 0
DO 4 J I 1,NRAYS
DO 4 I I 1,NCIRS
IF(IBOX(I,J).EQ.1) GO TO 70
IF(IBOX(I,J).EQ.2) NYES I NYES + 1
4 CONTINUE
C***
5 IF(NYES.EQ.NCELLS) THEN
NFULL I 1
END IF
C***
70 RETURN
END
C***
c***

C***********************************************************************

SUBROUTINE VIC(BDA,BDB,BDC,BDD,F,G1,G2,GPNT,IBOX,IFLG,JERT,
*MCOUNT,NCASE,NCOUNT,NG,P,Q,SN,SSl,UGN,UG1,UG1E,UGlS,UGZ,UPNT,U,
*UVDATA,VGN,VG1,VG1E,VG1$,VG2,VPNT,V,W,XPNT,XY,XYDATA,YPNT)

C***********************************************************************
C*** **

C*** THIS SUBROUTINE COMPUTES GRID VELOCITIES IN SQUARE DOMAIN USING **
C*** VORTEX-IN-CELL (VIC) FORMULATION. OFF-GRID VELOCITIES CALCULATED **

C*** USING QUADRATIC SPLINE INTERPOLATION TECHNIQUE (COUET’S FILTER). **
c*** **

c***********************************************************************
C***

PARAMETER(NAPI1)
DOUBLE PRECISION Gl,G2,P,SN,SSl,UGl,UGlE,UVDATA,VGl,VG1E,XYDATA

COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
C***

c***********************************************************************

213

C***

IF(NPV.EQ.0) RETURN
C***
IF(NCASE.EQ.2.AND.NTIME.GT.NSTART) GO TO 1
CALL XYUV(G1,GZ,1,MCOUNT,0,P,SN,SSl,UPNT,UVDATA,VPNT,XPNT,
*XYDATA,YPNT)
IF(NAP.EQ.0) GO TO 2
1 CALL INITPV(F,GPNT,NCOUNT,UPNT,VPNT)
CALL BORDER(F,NCOUNT)
CALL HWSCRT(BDA,BDB,BDC,BDD,0.0,F,NCOUNT,PERTRB,IERROR,W)
CALL GRIDV2(F,NCOUNT,UG2,VG2)
C***
2 IF(NCASE.EQ.1) THEN
CALL GRDV21(GPNT,NAP,NCOUNT,UG1,UGlE,UGlS,UG2,VG1,VG1E,VG1$,VG2,
*XPNT,XYDATA,YPNT)
C***
IF(IFLG.EQ.1) THEN
IF(NAP.EQ.0) GO TO 3
CALL VORT(NCOUNT,UG2,VG2,Q)
3 CALL LAYGRD (IBOX, NCOUNT, NG, UGlS , UGN, XY, VGl S , VGN, XYDATA)

END IF
C***
CALL PNTV2(1,NCOUNT,0,UG15,UPNT,U,VG1$,VPNT,V)
ELSE
CALL PNTV2(1,NCOUNT,0,UG2,UPNT,U,VGZ,VPNT,V)
END IF
C***
RETURN
END
c***
c***

C***********************************************************************

SUBROUTINE INITPV(F,GPNT,NCOUNT,UPNT,VPNT)
C****t******************************************************************

 

 

C*** **
C*** THIS SUBROUTINE SPREADS VORTICITY TO GRID POINTS **
C*** it
C***********************************************************************
c***
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/GRIDl/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/GRIDZ/ULNGTH,UMN,UMX,VLNGTH,VMN,VMX
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/PTPARI/GPVMN,GPVMX,LCODE,NO,NPV
DIMENSION F(NCOUNT,*)
VIRTUAL GPNT(*),UPNT(*),VPNT(*)
EXTERNAL PHI
C***
C***********************************************************************
C***
C*** Initialize the charge array :
C***
C***

DO 1 J = JMIN,JMAX

214

DO 1 I I IMIN,IMAX
F(I,J) I 0.0

 

 

1 CONTINUE
C***
C***
C*** Locate the point vortices :
C***
C***
FACTR I ITWOPI*DUINVT*DVINVT
DO 60 N I 1,NPV
GPOINT I FACTR*GPNT(N)
HUP I (UPNT(N)-UMN)*DUINVT+1.
HVP I (VPNT(N)-VMN)*DVINVT+1.
ICOL I IFIX(HUP)
JROW I IFIX(HVP)
DCOL I HUP-FLOAT(ICOL)
DROW I HVP-FLOAT(JROW)
C***
IPV I IFIX(HUP+.5)-2
JPV I IFIX(HVP+.5)-2
C***

DO 50 II I 1,3
IDELTA I IPV + II
HUC I FLOAT(IDELTA)

DHU I ABS(HUP-HUC)
C***

C***

 

C*** Check if outside the computational domain in the u-direction :
c***

 

C***
IF(IDELTA.LT.IMIN) IDELTA I IMIN
IF(IDELTA.GT.IMAX) IDELTA I IMAX

c***

DO 40 JJ I 1,3
JDELTA I JPV + JJ
HVR I FLOAT(JDELTA)
DHV I ABS(HVP-HVR)
c***
c***

 

C*** Check if outside the computational domain v-direcction :
C***

 

c***

IF(JDELTA.LT.JMIN) JDELTA I JMIN
IF(JDELTA.GT.JMAX) JDELTA I JMAX
C***
F(IDELTA,JDELTA) I F(IDELTA,JDELTA) + GPOINT*PHI(DHU,DHV)
40 CONTINUE
50 CONTINUE
60 CONTINUE
C***
RETURN

END
C***

C***

215

C***********************************************************************

SUBROUTINE BORDER(F,NCOUNT)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE SPECIFIES THE BOUNDARY CONDITION **
C*** **

C***********************************************************************

 

 

C***
COMMON/NSIZE /IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
DIMENSION F(NCOUNT,*)
C***
C***********************************************************************
C***
C*** Initialize the border of the E-array :
C***
C***

DO 1 I I IMIN,IMAX,ISIZE
DO 1 J I JMIN,JMAX
F(I,J) I 0.0

1 CONTINUE
DO 2 J I JMIN,JMAX,JSIZE
DO 2 I I IMIN+1,ISIZE
F(I,J) I 0.0

2 CONTINUE
C***
RETURN
END
C***
c***

C***********************************************************************

SUBROUTINE GRIDV2(F,NCOUNT,UGZ,VG2)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES VELOCITIES AT GRID POINTS **
C*** IN THE UV-PLANE USING A CENTERED DIFFERENCE TECHNIQUE **
C*** WHICH ASSURES LOCAL CONSERVATION OF MASS **
c*** **

C***********************************************************************

C***
COMMON/GRIDl/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
DIMENSION F(NCOUNT,*)

VIRTUAL UG2 (NCOUNT , *) , VG2 (NCOUNT, *)

C***

C***********************************************************************

 

C***—
C*** Initialize arrays

 

C***

C***
DO 50 J - JMIN,JMAX
DO 50 I - IMIN,IMAX
UG2(I,J) - 0.0
VG2(I,J) a 0.0

50 CONTINUE

216

 

 

 

 

 

 

 

 

C***
C***
C*** Interior grid points
C***
C***
DO 1 J I JMIN+1,JSIZE
DO 1 I I IMIN+1,ISIZE
UG2(I,J) I UGZ(I,J)+(F(I,J+1)-F(I,J-1))*HDVINV
VG2(I,J) I VGZ(I,J)+(F(I-1,J)-F(I+1,J))*HDUINV
1 CONTINUE
C***
C***
C*** Lower boundary grid points
C***
C***
DO 3 I -IMIN,IMAX .
UGZ(I,JMIN) I UGZ(I,JMIN)+2.0*F(I,JMIN+1)*HDVINV
IF(I.EQ.IMIN) VG2(I,JMIN)IVGZ(I,JMIN)-2.0*F(I+1,JMIN)*HDUINV
IF(I.EQ.IMAX) VG2(I,JMIN)IVGZ(I,JMIN)+2.0*F(I-1,JMIN)*HDUINV
IF(I.GT.IMIN.AND.I.LT.IMAX) VGZ(I,JMIN) I VGZ(I,JMIN) +
* (F(I-1,JMIN)-F(I+1,JMIN))*HDUINV
C***
3 CONTINUE
C***
C***
C*** Upper boundary grid points :
C***
C***
DO 4 I I IMIN,IMAX
UGZ(I,JMAX) I UG2(I,JMAX)-2.0*F(I,JMAXI1)*HDVINV
IF(I.EQ.IMIN) VGZ(I,JMAX)IVGZ(I,JMAX)-2.0*F(I+1,JMAX)*HDUINV
IF(I.EQ.IMAX) VG2(I,JMAX)IVGZ(I,JMAX)+2.0*F(I-1,JMAX)*HDUINV
IF(I.GT.IMIN.AND.I.LT.IMAX) VGZ(I,JMAX) I VGZ(I,JMAX) +
* (F(I-1,JMAX)-F(I+1,JMAX))*HDUINV
4 CONTINUE
C***
C***
C*** Left boundary grid points :
C***
C***
DO 5 J I JMIN+1,JSIZE
UGZ(IMIN,J) I UGZ(IMIN,J)+(F(IMIN,J+1)-F(IMIN,J-1))*HDVINV
VGZ(IMIN,J) I VG2(IMIN,J)-2.0*F(IMIN+1,J)*HDUINV
5 CONTINUE
C***
C***

 

C*** Right boundary grid points

C***

 

C***

DO 6 J I JMIN+1,JSIZE
UG2 (IMAX, J) I UG2 (IMAX, J) + (F (IMAX, J+1) IF (IMAX, J-1))*HDVINV
VG2(IMAX,J) I VG2(IMAX,J)+2.0*F(IMAX-1,J)*HDUINV

6 CONTINUE

C***

217

RETURN

END
C***

C***
C***********************************************************************

SUBROUTINE GRDV21(GPNT,NAP,NCOUNT,UG1,UG1E,UGlS,UG2,VG1,VG1E,

*VG1$,VG2,XPNT,XYDATA,YPNT)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES GRID VELOCITIES IN THE CIRCLE DOMAIN **
C*** **

C***********************************************************************

C***
DOUBLE PRECISION E11,E12,UG1,UG1E,VG1,VG1E,XYDATA
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
VIRTUAL UG2 (NCOUNT, *) ,VGZ (NCOUNT, *)

VIRTUAL UGl (NCOUNT, *) , UG1E(NCOUNT, *) , UGlS (NCOUNT, *)
VIRTUAL VGl (NCOUNT, *) ,VG1E(NCOUNT, *) , VGIS (NCOUNT, *)
C***

C***********************************************************************

 

 

C***
C*** Compute exact solution
C***
C***
CALL EXACT(GPNT,NCOUNT,NAP,UGlE,UGlS,VG1E,VGlS,XPNT,XYDATA,YPNT)
IF(NAP.EQ.0) RETURN
C***

OPEN(UNITI3,FILEI'MATRCES.CLNS’,TYPEI'OLD’)
DO 2 J I JMIN,JMAX
DO 2 I I IMIN,IMAX
READ(3,1) E11,E12
1 FORMAT(' ',T2,2(2X,1PD15.7))
UGl(I,J) I E11*DBLE(UG2(I,J))+E12*DBLE(VG2(I,J))
VG1(I,J) I IE12*DBLE(UG2(I,J))+E11*DBLE(VG2(I,J))
IF(NAP.EQ.1) THEN
IF(I.EQ.IMIN.OR.I.EQ.IMAX) THEN
IF(J.EQ.JMIN.OR.J.EQ.JMAX) THEN
UG1(I,J) I UG1E(I,J)
VG1(I,J) I VGlE(I,J)
END IF
END IF
END IF
UGIS(I,J) I SNGL(UG1(I,J))
VG1$(I,J) I SNGL(VG1(I,J))

2 CONTINUE
CLOSE(UNITI3)
C***
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE EXACT(GPNT,NCOUNT,NAP,UG1E,UGIS,VGlE,VG1$,XPNT,XYDATA,
*YPNT)

218

C***********************************************************************

C*** it
C*** THIS SUBROUTINE COMPUTES EXACT VELOCITIES **
C*** USING A MODIFIED MILNE-THOMSON CIRCLE THEOREM **
C*** **

c***********************************************************************
c***

DOUBLE PRECISION UGlE,VG1E,XYDATA
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/PTPAR1/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL

VIRTUAL GPNT(*),XYDATA(2,*)

VIRTUAL UG1$(NCOUNT,*),VGlS(NCOUNT,*)

VIRTUAL UGlE (NCOUNT, *) ,VG1E(NCOUNT, *) , XPNT (*) , YPNT (*)
C***

C***********************************************************************

 

 

 

 

C***
C*** Initialize the arrays :
C***
C***
DO 1 J I JMIN,JMAX
DO 1 I I IMIN,IMAX
UG1E(I,J) I 0.0D0
VGlE(I,J) I 0.0D0
UGIS(I,J) I 0.0
VGlS(I,J) I 0.0
1 CONTINUE
C***
IF(NAP.EQ.0) THEN
JNC I 1
INC I 1
ELSE
JNC I JSIZE
INC I ISIZE
END IF
C***
DO 3 N I 1, NPV
A02 I XPNT(N)*XPNT(N)+YPNT(N)*YPNT(N)
C***
DO 2 J I JMIN,JMAX,JNC
DO 2 I I IMIN,IMAX,INC
K I I + (J-JMIN)*JMAX
C***
C***
C*** Initialize parameters for u,v velocities :
C***
C***
AX I A02*SNGL(XYDATA(1,K))-(RW*RW)*XPNT(N)
AY I A02*SNGL(XYDATA(2,K))I(RW*RW)*YPNT(N)
AX2 I AX*AX
AY2 I AY*AY
XMXO I SNGL(XYDATA(1,K))-XPNT(N)
YMYO I SNGL(XYDATA(2,K))IYPNT(N)

 

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***

C***

C***

C***
C***

219

XMXOZ I XMXO*XMXO

YMYOZ I YMYO*YMYO

DENOMl I XMX02+YMY02

DENOM2 I AX2+AY2
IF(DENOM2.EQ.0.0) DENOM2 I SGMAO
SQRTD I SQRT(DENOM1)

H1 I (3.0*CNST1*SQRTD)/SGMA

 

Remove singularity according to Bald & Del Prete’s formulation

 

IF(SQRTD.LE.SGMA) THEN

UTMP I GPNT(N)*( (A02*AY/DENOM2)-SGMA2I*(H2-H1)*YMYO)
VTMP I GPNT(N)*(-(A02*AX/DENOM2)+SGMA2I*(H2-H1)*XMXO)
ELSE

UTMP I GPNT(N)*( (A02*AY/DENOM2)-(YMYO/DENOM1))

VTMP I GPNT(N)*(-(A02*AX/DENOM2)+(XMXO/DENOMl))

END IF

 

Compute the velocity components :

 

N

UG1E(I,J) I UGlE(I,J)+DBLE(UTMP)
VG1E(I,J) I VG1E(I,J)+DBLE(VTMP)

IF(NAP.EQ.0) THEN
UG1$(I,J) I SNGL(UG1E(I,J))
VGlS(I,J) I SNGL(VG1E(I,J))
END IF

CONTINUE
CONTINUE

RETURN

END

C***********************************************************************

SUBROUTINE VORT(NCOUNT,UGZ,VG2,Q)
C***********************************************************************

C***
C***
C***
C***
C***

THIS SUBROUTINE COMPUTES VORTICITY AT GRID POINTS IN
THE UV-PLANE USING A CENTERED DIFFERENCE TECHNIQUE
WHICH ASSURES LOCAL CONSERVATION OF VORTICITY

**
**
**
**

**

C***********************************************************************

C***

C***

COMMON/GRIDI/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE

VIRTUAL Q(NCOUNT,*),UG2(NCOUNT,*),VGZ(NCOUNT,*)

C***t*******************************************************************

220

C***

 

C*** Initialize array

 

C***
C***

DO 50 J I JMIN,JMAX
DO 50 I I IMIN,IMAX
Q(I,J) - 0.0

50 CONTINUE

C***

 

C***

C*** Interior grid points :

 

 

 

 

 

 

C***
C***
DO 1 J I JMIN+1,JSIZE
DO 1 I I IMIN+1,ISIZE
DUDY I (UGZ(I,J+1)-UGZ(I,J-1))*HDVINV
DVDX I (VGZ(I+1,J)-VGZ(I-1,J))*HDUINV
Q(I,J) I DVDX-DUDY
1 CONTINUE
C***
C***
C*** Lower boundary grid points
C***
C***
DO 3 I I IMIN,IMAX
DUDY I 2.0*UGZ(I,JMIN+1)*HDVINV
IF(I.EQ.IMIN) DVDX I 2.0*VGZ(I+1,JMIN)*HDUINV
IF(I.EQ.IMAX) DVDX I I2.0*VGZ(I-1,JMIN)*HDUINV
IF(I.GT.IMIN.AND.I.LT.IMAX) DVDX I (VGZ(I+1,JMIN)-VGZ(I-1,JMIN))*
*HDUINV
Q(I,JMIN) I DVDX-DUDY
C***
3 CONTINUE
C***
C***
C*** Upper boundary grid points :
C***
C***
DO 4 I I IMIN,IMAX
DUDY I I2.0*UGZ(I,JMAX-1)*HDVINV
IF(I.EQ.IMIN) DVDX I 2.0*VGZ(I+1,JMAX)*HDUINV
IF(I.EQ.IMAX) DVDX I I2.0*VGZ(I-1,JMAX)*HDUINV
IF(I.GT.IMIN.AND.I.LT.IMAX) DVDX I (VGZ(I+1,JMAX)-VGZ(I-1,JMAX))*
*HDUINV
Q(I,JMAX) I DVDX-DUDY
4 CONTINUE
C***
C***
C*** Left boundary grid points :
C***

 

c***
' DO 5 J - JMIN+1,JSIZE
DUDY - (UG2(IMIN,J+1)IUG2(IMIN,J-1))*HDVINV
DVDX - 2.0*VGZ(IMIN+1,J)*HDUINV

221
Q(IMIN,J) = DVDX-DUDY

 

 

5 CONTINUE
C***
C***
C*** Right boundary grid points :
C***
C***
DO 6 J I JMIN+1,JSIZE
DUDY I (UG2(IMAX,J+1)-UG2(IMAX,J-1))*HDVINV
DVDX I I2.0*VG2(IMAX-1,J)*HDUINV
Q(IMAX,J) I DVDX-DUDY
6 CONTINUE
C***
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE LAYGRD(IBOX,NCOUNT,NG,UGlS,UGN,XY,VG1$,VGN,XYDATA)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES VELOCITIES ON UNIFORM **
C*** GRID POINTS SUPERIMPOSED ON THE CIRCULAR DOMAIN **
C*** **

C***********************************************************************
C***

DOUBLE PRECISION XYDATA
COMMON/GRID3/DX,DXINVT,DY,DYINVT,XMN,XMX,YMN,YMX
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
EXTERNAL PHI

VIRTUAL UG1$(NCOUNT,*),UGN(NCOUNT,*),XY(2,*)

VIRTUAL VGlS(NCOUNT,*),VGN(NCOUNT,*),XYDATA(2,*)
VIRTUAL IBOX(NCOUNT,*),NG(NCOUNT,*)

C***
C***********************************************************************

 

 

C***

C*** Initialize the arrays/parameters

C***

C***
DO 1 J I JMIN,JMAX
DO 1 I I IMIN,IMAX
UGN(I,J) I 0.0
VGN(I,J) I 0.0
NG(IIJ) ' 0

1 CONTINUE
C***
C***

 

C*** Assign radial location of singularity (RS)

C*** and velocity reduction factor (RF)
C***

C***

 

RS

ll
c3<>

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***

c***
c***
c***
C***
C***

C***

C***
C***
C***
C***
C***

C***
C***

222

 

Tag grid points outside circle

 

IF(NTIME.EQ.NSTART) THEN

DO 2 J I JMIN,JMAX

Y I DY*FLOAT(J-JMIN)+YMN
DO 2 I I IMIN,IMAX

X I DX*FLOAT(I-IMIN)+XMN
K I I + (J-JMIN)*JMAX
IBOX(I,J) I 0

XY(1,K) I X

XY(21K) " Y

R I SQRT(X*X+Y*Y)
IF(R.GT.XMX) IBOX(I,J) I 1
CONTINUE

END IF

 

Interpolate velocities to grid points

 

DO 70 NS I 1,2

DO 60 J I JMIN,JMAX
DO 60 I I IMIN,IMAX
K I I + (JIJMIN)*JMAX

 

Locate vortex position

 

HXP I (SNGL(XYDATA(1,K))-XMN)*DXINVT+1.
HYP I (SNGL(XYDATA(2,K))IYMN)*DYINVT+1.
ICOL I IFIX(HXP)

JROW I IFIX(HYP)

DCOL I HXP-FLOAT(ICOL)

DROW I HYP-FLOAT(JROW)

IPV I IFIX(HXP+.S)-2

JPV I IFIX(HYP+.5)-2

 

Determine argument, dx, for the weighting function :

 

DO 50 II I 1,3

IDELTA I IPV + II
IF(NS.EQ.2) GO TO 3

HXC I FLOAT(IDELTA)
DHX I ABS(HXP-HXC)

 

C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
c***
C***
C***

C***
C***
C***
C***
C***

C***

C***
C***
C***
C***
C***

C***

C***
C***
C***
C***
C***
c***
c***
c***
C***

C***
c***
C***
c***

223

Check if outside the computational domain

 

IF(IDELTA.LT.IMIN) IDELTA I IMIN
IF(IDELTA.GT.IMAX) IDELTA I IMAX

 

Determine argument, dy, for the weighting function :

 

DO 40 JJ I 1,3

JDELTA I JPV + JJ
IF(NS.EQ.2) GO TO 4

HYR I FLOAT(JDELTA)
DHY I ABS(HYP-HYR)

 

Check if outside the computational domain

 

r—

IF(JDELTA.LT.JMIN) JDELTA I JMIN
IF(JDELTA.GT.JMAX) JDELTA I JMAX

 

Check if grid point is within circle

 

IF(IBOX(IDELTA,JDELTA).EQ.1) GO TO 40

IF(NS.EQ.2) GO TO 5

 

Interpolate velocities to grid points

 

UGN(IDELTA,JDELTA)IUGN(IDELTA,JDELTA)+UG18(I,J)*PHI(DHX,DHY)
VGN(IDELTA,JDELTA)IVGN(IDELTA,JDELTA)+VGIS(I,J)*PHI(DHX,DHY)

IF(NS.EQ.2) THEN

 

Modify velocities within singularity zone

 

 

Check (x,y) coordinates of grid points

 

X I (IDELTA-1)*DX + XMN
Y I (JDELTA-1)*DY + YMN

 

Check radial location of grid points

 

224

 

 

 

 

 

 

C***
R I SQRT(X*X+Y*Y)
C***
IF(R.GE.RS) THEN
C***
C***
C*** Check angular location of grid points
C***
C***
T I ATAN2D(Y,X)
IF(Y.LT.0.0) T I T + 360.0
C***

IF((T.GE.35.0.AND.T.LE.55.0) .OR.
*(T.GE.125.0.AND.T.LE.145.0) .OR.
*(T.GE.215.0.AND.T.LE.235.0) .OR.
*(T.GE.305.0.AND.T.LE.325.0)) THEN

C***

C***

C*** Check if duplicating. If not tag grid point

C***

C***
IF(NG(IDELTA,JDELTA).EQ.1) GO TO 40
NG(IDELTA,JDELTA) I 1

C***

C***

C*** Modify grid velocity :

C***

C***
UGN(IDELTA,JDELTA) I RF*UGN(IDELTA,JDELTA)
VGN(IDELTA,JDELTA) I RF*VGN(IDELTA,JDELTA)

C***

END IF

END IF

END IF

C***

40 CONTINUE
50 CONTINUE
60 CONTINUE
70 CONTINUE
C***
RETURN

END
C***

c***
C***********************************************************************
SUBROUTINE PNTV2(IFLG,NCOUNT,NTP,UGC,UPT,U,VGC,VPT,V)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE CALCULATES VELOCITIES AT OFF GRID LOCATIONS **
c*** **

C***********************************************************************
C***

COMMON/GRIDl/DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/GRIDZ/ULNGTH,UMN,UMX,VLNGTH,VMN,VMX

225

COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV

VIRTUAL UPT(*),VPT(*)

VIRTUAL UGC(NCOUNT,*),U(*),VGC(NCOUNT,*),V(*)

EXTERNAL PHI

 

 

 

 

C***
C***********************************************************************
C***
IF(IFLG.EQ.1) NP I NPV
IF(IFLG.EQ.2) NP I NTP
DO 90 N I 1,NP
U(N) I 0.
V(N) I 0.
UTEMP I 0.
VTEMP I 0.
HUP I (UPT(N)-UMN)*DUINVT+1.
HVP I (VPT(N)IVMN)*DVINVT+1.
C***
ICOL I IFIX(HUP)
JROW I IFIX(HVP)
DCOL I HUP-FLOAT(ICOL)
DROW I HVP-FLOAT(JROW)
C***
IPV I IFIX(HUP+.5)-2
JPV I IFIX(HVP+.5)-2
C***
DO 80 II I 1,3
IDELTA I IPV + II
HUC I FLOAT(IDELTA)
DHU I ABS(HUP-HUC)
C***
C***
C*** Check if outside the computational domain :
C***
C***
IF(IDELTA.LT.IMIN) IDELTA I IMIN
IF(IDELTA.GT.IMAX) IDELTA I IMAX
C***
DO 70 JJ I 1,3
JDELTA I JPV + JJ
HVR I FLOAT(JDELTA)
DHV I ABS(HVP-HVR)
C***
C***
C*** Check if outside the computational domain :
C***
C***
IF(JDELTA.LT.JMIN) JDELTA I JMIN
IF(JDELTA.GT.JMAX) JDELTA I JMAX
C***
UTEMP I UTEMP + UGC(IDELTA,JDELTA)*PHI(DHU,DHV)
VTEMP I VTEMP + VGC(IDELTA,JDELTA)*PHI(DHU,DHV)
70 CONTINUE

80 CONTINUE

226

 

 

C***
C***
C*** Store the velocity
C***
C***
U(N) - UTEMP
V(N) - VTEMP
90 CONTINUE
C***
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE PRNT(F,FTAG,NCASE,NCOUNT,Q,RPNT,UGlS,UG2,UGN,

*UVDATA,VGlS,VG2,VGN,XPNT,XY,XYDATA,YPNT)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE PRINTS RESULTS OF VORTEX CALCULATION **
C*** **

C***********************************************************************

c***

DOUBLE PRECISION RK,RMOD2,UVDATA,XX,YY,XYDATA
PARAMETER(NGRD-0,NPos-o,TANG-1.0,XDIR-1.0)
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/JETPAR/DJ,PMSIGN,STOPER,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
COMMON/PTPARI/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNSTI,H2,RW,SGMAZI,TOL
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
DIMENSION F(NCOUNT,*)

VIRTUAL UVDATA(2,*),XY(2,*),XYDATA(2,*)

VIRTUAL FTAG(2,*),XPNT(*),YPNT(*),RPNT(*)

VIRTUAL Q(NCOUNT,*),UG1$(NCOUNT,*),VGlS(NCOUNT,*)

VIRTUAL UGN(NCOUNT,*),UG2(NCOUNT,*),VGN(NCOUNT,*),VG2(NCOUNT,*)
C***

C***********************************************************************

 

 

C***
IF(NPV.EQ.0) GO TO 300
C***
C***
C*** NON-GRID POINT DATA :
C***
C***
IF(NPOS.EQ.1) THEN
IF(NCASE.EQ.1) THEN
WRITE(1,108) NTIME
WRITE(2,108) NTIME
DO 1 N - 1,NPV
D IF(FTAG(1,N).EQ.1.0) THEN
D WRITE(1,110) XPNT(N),YPNT(N),NTIME*DTIME,RPNT(N)
D ELSE
D WRITE(2,110) XPNT(N),YPNT(N),NTIME*DTIME,RPNT(N)

227

D END IF
D l CONTINUE
D END IF
D END IF

108 FORMAT(' ’,T2,’NTIME I',T10,I5,/)
D 110 FORMAT(' ',T2,1PE15.7,T19,1PE15.7,T37,1PE15.7,T54,1PE15.7)

 

 

C***
C*** GRID POINT VELOCITY DATA :
C***
c***
IF(NGRD.EQ.1.0) THEN
C***
DO 2 NLOOP I 1,2
IF(NLOOP.EQ.1) THEN
J1 I JMID
J2 I JMID
J3 I 1
13 I 10
ELSE
J1 I JMAX
J2 I JMIN
J3 I -2
I3 I 2
END IF
C***
DO 131 J I J1,J2,J3
DO 131 I I IMIN,IMAX,I3
K I I + (J-JMIN)*JMAX
IF(NLOOP.EQ.1) THEN
IF(XDIR.EQ.0.0) K I IMID + (I-JMIN)*JMAX
IF(TANG.EQ.1.0) THEN
IF(SNGL(XYDATA(1,K)).EQ.0.0.AND.SNGL(XYDATA(2,K)).EQ.0.0) THEN
T I ATAN2(SNGL(XYDATA(2,K))ySGMAO)
ELSE
T I ATAN2(SNGL(XYDATA(2,K))ISNGL(XYDATA(1,K)))
END IF
VTC I ABS(VGlS(I,J))*COS(T)-ABS(UG1$(I,J))*SIN(T)
VRC I IABS(VG1$(I,J))*SIN(T)-ABS(UG1$(I,J))*COS(T)
c***
ELSE
C***
VTC I UGIS(I,J)
VRC I VGlS(I,J)
END IF
C***
IF(I.EQ.IMIN.AND.J.EQ.J1) THEN
IF(NCASE.EQ.1) WRITE(3,3) NTIME
IF(NCASE.EQ.2) WRITE(7,3) NTIME
END IF
C***

IF(XDIR.EQ.1.0) THEN
IF(NCASE.EQ.1) THEN
WRITE(3,4) SNGL(XTDATA(1,K)),SNGL(XXDATA(2,K)),P(I,J),

228

*VTC,VRC,Q(I,J)

ELSE

WRITE(7,4) SNGL(UVDATA(1,K)/RK),SNGL(UVDATA(2,K)/RK),
*F(I,J).VG2(I,J).UGZ(I,J),Q(I,J)

END IF
C***
ELSE
C***
IF(NCASE.EQ.1) THEN
WRITE(3,4) SNGL<XYDATA(1,K)),SNGL(XYDATA(2,K)),P(I,J),
*VTC,VRC,Q(I:J)
ELSE
WRITE(7,4) SNGL(UVDATA(1,K)/RK),SNGL(UVDATA(2,K)/RK),
*F(IIJ)IUGZ(IIJ)(VG2(IIJ)IQ(IIJ)
END IF
END IF
C***
ELSE
C***
IF(I.EQ.IMIN.AND.J.EQ.J1) THEN
IF(NCASE.EQ.1) THEN
WRITE(4,3) NTIME
WRITE(6,3) NTIME
ELSE
WRITE(8,3) NTIME
END IF
END IF
C***
IF(NCASE.EQ.1) THEN
c WRITE(4,4) SNGL(XYDATA(1,K)),SNGL(XYDATA(2,K)),P(I,J),
C *UGlS(I,J),VG1$(I,J),Q(I,J)
WRITE(6,4) XY(1,K),XY(2,K),F(I,J),UGN(I,J),VGN(I,J),
*Q(I,J)
ELSE
WRITE(8,4) SNGL(UVDATA(1,K)),SNGL(UVDATA(2,K)),
*F(I:J):UGZ(I:J):VGZ(I:J)[Q(I,J)
END IF
END IF
C***
131 CONTINUE
2 CONTINUE

3 FORMAT(' ',/,T2,'NTIME I’, I5,/)
4 FORMAT(' ',T2,1PE15.7,T20,1PE15.7,T38,1PE15.7,T56,1PE15.7,T74,
*1PE15.7,T92,1PE15.7,T110,1PE15.7)

C***
END IF
c***
300 RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE ADVECT(BDA,BDB,BDC,BDD,F,FTAG,Gl,G2,GPNT,IBOX,JERT,
*MCOUNT,NCASE,NCOUNT,NG,P,Q,RPNT,SN,SSl,TPNT,UG1,UG1E,UG1$,UG2,

229
*UGN,UN,UO,UPNT,UVDATA,VG1,VGlE,VGlS,VGZ,VGN,VN,VO,VPNT,XPN,XPO,

 

 

 

 

*YPN,YPO,W,XY,XYDATA)
C******************************************************-*****************
C*** **
C*** THIS SUBROUTINE ADVECTS THE VORTICES **
C*** USING HEUN’S METHOD WHICH IS SECOND-ORDER ACCURATE IN TIME, **
C*** AS OPPOSED TO EULER'S METHOD WHICH IS FIRST-ORDER ACCURATE **
C*** (Ref. ”Turbulent Combustion in Open and Closed Vessels", by **
C***James Sethian, Journal of Computational Physics 54, 425-456 (1984)**
C*** **
C***********************************************************************
C***

DOUBLE PRECISION GI,GZ,P,SN,SSl,UGl,UGlE,UVDATA,VGl,VG1E,XYDATA
COMMON/CONSNT/PI,SGMA,SGMAO,TWOPI
COMMON/JETPAR/DJ,PMSIGN,TJ,VIN,XA,XB,XPA,XPB,YAB,YB,YPB
COMMON/GRIDZ/ULNGTH,UMN,UMX,VLNGTH,VMN,VMX
COMMON/TIMER/DTIME,NFULL,NSTART,NSTOP,NTIME,TIME
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
COMMON/PTPARZ/CNST1,H2,RW,SGMAZI,TOL
VIRTUAL RPNT(*),TPNT(*),UPNT(*),VPNT(*),FTAG(2,*)
VIRTUAL UN(*),UO(*),VN(*),VO(*),XPN(*),XPO(*),YPN(*),YPO(*)
C***
C***********************************************************************
C***
C*** Advect the point vortices :
C***
C***
IF(NPV.EQ.0) GO TO 3
DO 1 N I 1,NPV
IF(NCASE.EQ.1) THEN
XPN(N) I XPO(N) + UO(N)*DTIME
YPN(N) I YPO(N) + VO(N)*DTIME
C***
C***
C*** Check if outside the computational domain :
C***
C***
IF(XPN(N).EQ.0.0.AND.YPN(N).EQ.0.0) THEN
TPNT(N) I ATAN2(YPN(N),SGMAO)
ELSE
TPNT(N) I ATAN2(YPN(N),XPN(N))
END IF
RPNT(N) I SQRT(XPN(N)*XPN(N)+YPN(N)*YPN(N))
C***
IF(RPNT(N).GT.RW) THEN
RPNT(N) I RW
XPN(N) I RPNT(N)*COS(TPNT(N))
YPN(N) I RPNT(N)*SIN(TPNT(N))
END IF
C***
ELSE
c***
XPO(N) I UPNT(N)
YPO(N) I VPNT(N)

230

UPNT(N) I UPNT(N) + UO(N)*DTIME
VPNT(N) I VPNT(N) + VO(N)*DTIME

 

 

 

 

 

 

 

 

 

C***
C***

C*** Check if outside the computational domain :

C***

C***

IF(UPNT(N).LT.UMN) UPNT(N) I UMN

IF(UPNT(N).GT.UMX) UPNT(N) I UMX

IF(VPNT(N).LT.VMN) VPNT(N) I VMN

IF(VPNT(N).GT.VMX) VPNT(N) I VMX

END IF

C***
1 CONTINUE
C***
C***
C*** Compute velocities at new vortex positions
C***
C***

CALL VIC(BDA,BDB,BDC,BDD,F,Gl,GZ,GPNT,IBOX,2,JERT,MCOUNT,
*NCASE,NCOUNT,NG,P,Q,SN,SSl,UGN,UGl,UG1E,UGlS,UGZ,UPNT,UN,UVDATA,
*VGN,VGl,VG1E,VGIS,VGZ,VPNT,VN,W,XPN,XY,XYDATA,YPN)

C***
C***
C*** Advect point vortices with average velocities
C***
C***

DO 2 N I 1,NPV

UO (N) I 0 . 5* (U0 (N) +UN (N) )

VO(N) I 0.5*(VO(N)+VN(N))

XPO(N) I XPO(N)+UO(N)*DTIME

YPO(N) I YPO(N)+VO(N)*DTIME

C***
C***
C*** Check if outside the computational domain :
C***
C***

IF(NCASE.EQ.1) THEN

IF(XPO(N).EQ.0.0.AND.YPO(N).EQ.0.0) THEN

TPNT(N) I ATAN2(YPO(N),SGMAO)

ELSE

TPNT(N) I ATAN2(YPO(N),XPO(N))

END IF

RPNT(N) I SQRT(XPO(N)*XPO(N)+YPO(N)*YPO(N))

C***

IF(RPNT(N).GT.RW) THEN

RPNT(N) I RW

XPO(N) I RPNT(N)*COS(TPNT(N))

YPO(N) I RPNT(N)*SIN(TPNT(N))

END IF

C***
C***

 

C*** Compute the tangential & radial velocities of the point vortices
C***

 

231

C***
VT(N) I VO(N)*COS(TPNT(N))-UO(N)*SIN(TPNT(N))
VR(N) I IVO(N)*SIN(TPNT(N))IUO(N)*COS(TPNT(N))
C***
ELSE
C***
IF(XPO(N).LT.UMN) XPO(N) I UMN
IF(XPO(N).GT.UMX) XPO(N) I UMX
IF(YPO(N).LT.VMN) YPO(N) I VMN
IF(YPO(N).GT.VMX) YPO(N) I VMX
UPNT(N) I XPO(N)
VPNT(N) I YPO(N)
END IF
2 CONTINUE
C***
3 RETURN
END
C***
C***

c***********************************************************************

SUBROUTINE UPDATE(MAXPV)

c***********************************************************************

c*** **
C*** THIS SUBROUTINE UPDATES PARAMETERS/ARRAY SIZE **
C*** **

c***********************************************************************
C***

COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
C***
C***********************************************************************
C***

 

C*** Monitor array size :

 

C***
C***
IF(NPV.GT.MAXPV) THEN
LCODE I 0
WRITE(5,1) NPV,MAXPV
1 FORMAT(' ',T12,'TOO MANY VORTICES CREATED :',/,T12,
*’NPV I ',I5,' (MAXPV I ',IS,')',/,T12,
*’PLEASE ADJUST ARRAYS AND RESTART CALCULATION')
END IF
C***
RETURN
END
C***
C***

C***********************************************************************

FUNCTION PHI( DHU, DHV)

C**************************************t********************************

c*** **
C*** THIS FUNCTION ROUTINE EVALUATES THE INTERPOLATION FUNCTIONS **
c*** **

C***********************************************************************
C***

C***

C***
C***
C***
C***
C***

C***
C***
C***
C***
c***

c***

DATA

DATA
DATA
DATA

DHU2
DHV2

232

A, B, c, D, E / 1.763668E-03, -3.395062E-02, 2.573854E-01,
-7.229663E-01, 6.651786E-01/

ALPHA / 1.55408084 /

CTAN / 0.50875376 /

CONST/ 0.51438377 /

I DHU*DHU
I DHV*DHV

 

Filter developed by Couet ( Quadratic Spline Interpolation )

 

IF(
IF(
IF(
IF(
IF(
PHI

DHU2.GE.
DHU2.LT.
DHU2.GE.
DHV2.LT.
DHV2

.25.0R.DHV2.GE.2.25 )

.25 ) PHIU 0.75 - DHU2
.25 ) PHIU 0.5*((1.5-ABS(DHU))**2)
.25 ) PHIV 0.75 - DHV2

.GE.0.25 ) PHIV 0.5*((1.5-ABS(DHV))**2)
I PHIU*PHIV

GO TO 7

OOON

RETURN

 

Zero

value for the filter

 

PHI I 0.0
RETURN

END

233

C***
C***********************************************************************

C****** DMAP LIBRARY CONTAINS THE FOLLOWING SUBROUTINES/FUNCTIONS : **

C****** **
C*** SUBROUTINES : **
C*** 1. SSNl - Evaluates Jacobian elliptic functions SN **
C*** 2. XYUV - Performs coord. transformation from (x,y) to (u,v) **
C*** 3. NASZ I Prepares input parameters for NASNWR **
C*** NASNWR - Solves a system of nonlinear algebraic equations **
C*** based on the Newton-Raphson method **
C*** F - Provides the system.of nonlinear algebraic eqns. **
C*** NASJME - Evaluates the function Jacobi matrix **
C*** NASFNE - Evaluates the function residuals and their norm, **
C*** NASABX _ Solves A*XIB by LU decomp. & backwards substitution **
C*** DECOM - Does the decomposition **
C*** SUBST - Does the substitution **
C*** MOVEDD - Moves elements of a vector to another vector **
C*** NORMSD - Computes the Euclidean norm of a vector **
C*** EXCHSD - Swaps elements of two vectors **
C*** VIPASD - Calc. a vector inner product & adds to given value **

C*** 4. INTPLU - Performs linear interpolation for (u,v) coordinates **
C*** 5. CNDNSN - Evaluates Jacobian elliptic functions CN,DN and SN **

C****** *****
C*** FUNCTIONS : **
C*** **
C*** 1. CN,DN,SN Jacobian elliptic functions **
C*** 2. FUNCTN - Evaluates integrand of elliptic integral K (or K') **
C*** [1/sqrt(1-k**23in**2(phi)]/[1/sqrt(1-k’**23in**2(phi)]**
C*** 3. GAUSS - Performs Gaussian integration of elliptic integrals **
C*** **

C***********************************************************************

SUBROUTINE SSN1( V0, CMO, SNl )

C***********************************************************************

C*** **
C*** THIS SUBROUTINE EVALUATES JACOBIAN ELLIPTIC FUNCTIONS **
C*** USING A DESCENDING LANDEN TRANSFORMATION WITH THE REDUCED **
C*** FUNCTIONS APPROXIMATED BY CIRCULAR (TRIG) FUNCTIONS **

C***(See page 573, "Handbook Of Mathematical Functions" by Abramowitz)**
C****************************************************t******************

C***
IMPLICIT DOUBLE PRECISION ( A-H, O-z )

C***

C***********************************************************************

C***

 

C*** Perform three applications of reduction to reduce the parameter :
c***

 

C***
CMI - 1.0D0 — CMO
RMUl - ( ( 1.0Do-DSORT(CM1) ) / ( 1.0D0+DSORT(CM1) ) )**2
v1 - vo / ( 1.0D0+DSORT( RMUl ) )

C***
CM2 I 1.000 - RMUl
RMU2 - ( ( 1.0D0-DSQRT(CM2) ) / ( 1.0D0+DSORT(CM2) ) )**2
v2 - V1 / ( 1.0Do+DsORT( RMU2 ) )

234

 

C***
CM3 - 1.0D0 - RMU2
RMU3 - ( ( 1.0D0IDSQRT(CM3) ) / ( 1.0D0+DSQRT(CM3) ) )**2
v3 - v2 / ( 1.0D0+DSORT( RMU3 ) )
C***
3N3 - (1.0D0+DSORT(RMU3))*SN(V3,RMU3)
8N3 - SN3/( 1.0Do + DSQRT(RMU3)*SN(V3,RMU3)**2 )
SN2 - (1.0D0+DSQRT(RMU2))*SN3
SN2 - SN2/( 1.0D0 + DSQRT(RMU2)*SN3**2 )
SNl - (1.0D0+DSQRT(RMU1))*SN2
SNl - SN1/( 1.0D0 + DSQRT(RMU1)*SN2**2 )
C***
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE XYUV(Gl,GZ,IFLG,MCOUNT,NL,P,SN,SSl,UPT,UVDATA,

*VPT,XPT,XYDATA,YPT)
C***********************************************************************

 

C*** **
C*** THIS SUBROUTINE PERFORMS COORDINATE TRANSFORMATION : **
C*** CIRCLE (x,y) -> SQUARE (u,v) **
C*** **

C***********************************************************************
C***

DOUBLE PRECISION G1,GZ,P,SN,SSl,UVDATA,XYDATA
DOUBLE PRECISION RK,RMOD2,X,XX,YY
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/PTPARl/GPVMN,GPVMX,LCODE,NO,NPV
DIMENSION X(2)

VIRTUAL UPT(*),VPT(*),XPT(*),YPT(*)
C***

C***********************************************************************
C***

IF(IFLG.EQ.1) NP I NPV
IF(IFLG.EQ.2) NP I NL
DO 1 I I 1,NP
XX I DBLE(XPT(I))
YY I DBLE(YPT(I))
CALL NASZ(G1,GZ,SSl,UVDATA,X,XYDATA)
CALL INTPLU(MCOUNT,P,UPT(I),SN,X(2))
CALL INTPLU(MCOUNT,P,VPT(I),SN,X(1))
1 CONTINUE
C***
RETURN

END
C***

C***
Ct**********************************************************************

VSUBROUTINE NASZ(G1,G2,SSl,UVDATA,X,XYDATA)
C***********************************************************************
C***

C*** THIS SUBROUTINE PREPARES INPUT PARAMETERS FOR SUBROUTINE NASNWR **

235

C***
***
C***********************************************************************

C***
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
EXTERNAL F
INTEGER N, MAXN, MAXIT, IRSTAT
PARAMETER (MAXNI42)
LOGICAL AGIVEN
DIMENSION X(2), R(2), A(MAXN,MAXN), WK(MAXN,MAXN+3)
VIRTUAL SSl(2,*),UVDATA(2,*),XYDATA(2,*)
COMMON/MAP/RK,RMOD2,XX,YY
COMMON/NSIZE/IMAX,IMID,IMIN,ISIZE,JMAX,JMID,JMIN,JSIZE
C***

C***********************************************************************

 

 

C***
C*** Determine the sign of S and $1 :
C***
C***
AGIVEN I .FALSE.
IF(XX.GE.0.0DO) THEN
IF(YY.GE.0.0D0) THEN
X(l) I G1
X(2) I 62
ELSE
X(l) I -61
X(2) I GZ
END IF
ELSE
IF(YY.GE.0.0D0) THEN
X(l) I 61
X(2) I IGZ
ELSE
X(l) I -G1
X(2) I -G2
END IF
END IF
C***
TOL I 1.0D-6
MAXIT I 400
N I 2
C***
C***

 

C*** Call the subroutine that would solve

C*** F(x)-0 using a quasi Newton method :
c***

 

c***

CALL NASNWR(F,X,N,R,A,MAXN,TOL,WK,MAXIT,AGIVEN,IRSTAT)
C***

 

C***

C*** Limiting cases :
C***

 

C***

IF(XX.EQ.0.0DO) X(2) I 0.0D0

 

IF(YY.EQ.
DO 22 J I
DO 22 I I
K -

236

0.0D0) X(1) I 0.0D0
JMIN,JMAX
IMIN,IMAX
I + (J-JMIN)*JMAX

IF(XX.EQ.XYDATA(1,K).AND.YY.EQ.XYDATA(2,K)) THEN
IF(DABS(UVDATA(1,K)).EQ.DABS(UVDATA(2,K))) THEN

X(1) I
X(2) I
END IF
END IF
22 CONTINUE
C***
RETURN
END
C***
C***

SSl(1,K)
SSl(2,K)

C***************t*******************************************************

SUBROUTINE NASNWR(F,X,N,R,A,MAXN,TOL,WK,MAXIT,AGIVEN,IRSTAT)
C***********************************************************************

c***
C***
C***
c***

THIS SUBROUTINE SOLVES A SYSTEM OF NONLINEAR ALGEBRAIC
EQUATIONS BASED ON THE NEWTON-RAPHSON METHOD

**
**
**
**

C***********************************************************************

C***

IMPLICIT DOUBLE PRECISION (A-H,O-Z)

EXTERNAL
INTEGER
LOGICAL
DIMENSION
INTRINSIC
EXTERNAL
EXTERNAL
INTEGER
INTEGER
INTEGER
INTEGER
LOGICAL
PARAMETER
PARAMETER

C***

F

N, MAXN,
AGIVEN
X(1),
DABS,
NASABX,
MOVEDD,
MAXJME,
IT,
NPLUSl,
J77, J88,
EVALJM, XFOUND
(MAXJMEIIO)
(MAXJMSIZ)

R(l),
DMINl,
NASJME,
VIPASD
MAXJMS
IT1,
NPLUSZ,

MAXIT, IRSTAT

A(MAXN,1), WK(MAXN,1)
DMAXl, DSQRT

NASFNE

JMS, J10

NPLUS3

J99, IDUMMY

C***********************************************************************

C***
NPLUSI I N+1
NPLUSZ I NPLU51+1
NPLUS3 I NPLU82+1
EVALJM I .NOT.AGIVEN

C***

IT I 0

ITl I 0

JMS I 0

DFRAC I .5000
C***

C***

 

C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
c***
C***
c***

C***
C***
C***
C***
C***

188

237
Investigate the initial guess :

 

XNORMZ I 0.D00

CALL VIPASD(X,1,X,1,N,XNORM2)
IF (XNORM2.EQ.0.D00) THEN
IRSTAT I -101

RETURN

ENDIF

CALL NASFNE(F,X,N,R,RNORM)
CALL MOVEDD(N,R,WK(1,NPLU51))

 

Reevaluate the function Jacobi matrix if necessary :

 

CONTINUE
IF (EVALJM) THEN

CALL NASJME(F,X,N,R,A,MAXN)
ENDIF

 

Iterate with an estimate of the function Jacobi matrix

 

CONTINUE

DO 399 J99I1,MAXJME
XFOUND I RNORM.LT.TOL
IF (XFOUND) THEN

 

The residual norm is within given tolerance. All done :

 

IRSTAT I 0
RETURN
ELSE

 

Next iteration is necessary :

 

TNORM I RNORM

 

Compute the nominal iteration step :

 

DO 188 J88I1,N
WK(J88,NPLUS3) I -R(J88)
WK(J88,NPLU82) I WK(J88,NPLUSI)
CALL MOVEDD(N,A(1,J88),WK(1,J88))
CONTINUE
CALL NASABX(WK,MAXN,N,WK(1,NPLU83),WK(1,NPLUSl),R,IDUMMY)

 

C***

238

IF (IDUMMY.EQ.0) THEN
JMS - JMS+1
IF (JMS.GT.MAXJMS) THEN
CALL NASFNE(F,X,N,R,RNORM)
CALL NASJME(F,X,N,R,A,MAXN)
IRSTAT - -10
RETURN
ELSE
CALL MOVEDD(N,WK(1,NPLUSZ),WK(1,NPLUSI))
ENDIF
ELSE
JMS I 0
ENDIF
DNORM2 I 0.D00
CALL VIPASD(WK(1,NPLU81),1,WK(1,NPLUSI),1,N,DNORM2)
IF (DNORM2.EQ.0D00) THEN
IRSTAT - -102
RETURN
ENDIF
CORR - DMINl(DFRAC/DSQRT(DNORM2/XNORM2),1.D00)
X8 I CORR
J10 - o

 

C***
C***
C***

Evaluate the function for new arguments :

 

C***

200

288

C***
C***

CONTINUE
DO 288 J88I1,N
WK(J88,NPLUSl) I CORR*WK(J88,NPLUSl)
WK(J88,1) I X(J88)+WK(J88,NPLU31)
CONTINUE
CALL NASFNE (F, WK, N, R, RNORM)
IF(RNORM.GE.TNORM) GOTO 4
IT1 I IT1+1
IF(IT1.EQ.1) GOTO 5
IT1 I 0
DFRAC I DMIN1(DFRAC+.2D00,2.D00)
GOTO 5
CORR I .ZDOO
X8 I X8*.2D00
IT1 I 0
DFRAC I DMAX1(DFRAC-.05D00,.1D00)
J10 I J10+1
IF(J10.LE.2) GOTO 200
X8 I X8*5.D00
CONTINUE

 

C***
C***

Extrapolate the function Jacobi matrix :

 

C***

TEMPl - 1.D00/(DNORM2*X8**2)
DNORM2 - 1.D00-X8

239

D0 388 J88I1,N
X(J88) I WK(J88,1)
TEMP2 I (R(J88)+WK(J88,NPLUS3)*DNORMZ)*TEMPl
DO 377 J77I1,N
A(J88,J77) I A(J88,J77)+TEMP2*WK(J77,NPLU31)
377 CONTINUE
388 CONTINUE
ENDIF
399 CONTINUE
XFOUND I RNORM.LT.TOL
IF (XFOUND) THEN

C***

 

C***

C*** The residual norm is within given tolerance, fortunatelly :

 

C***

 

C***
IRSTAT I 0
RETURN
ELSE
C***
C***

C*** Next iteration is necessary :

 

 

 

C***
C***
IT - IT+MAXJME
IF (IT.GE.MAXIT) THEN
C***
C***
C*** Iteration limit reached :
C***
C***
IRSTAT I I1
RETURN
ELSEIF (RNORM.LE..5D00*TNORM) THEN
C***
C***

 

C*** The function Jacobi matrix need not be updated :

 

C***

C***
GO TO 2
ELSE
c***
C***

 

C*** The function Jacobi matrix must be updated :

 

C***
C***
EVALJM I .TRUE.
GO TO 1
ENDIF
ENDIF
c***
RETURN
END

c***

240

C***
C***********************************************************************

SUBROUTINE NASJME(F,X,N,R,A,MAXN)

C***********************************************************************

c*** **
C*** THIS SUBROUTINE EVALUATES THE FUNCTION JACOBI MATRIX **
C*** **

c*****************************************************************************

 

 

 

 

C***
EXTERNAL F
INTEGER N, MAXN
DOUBLE PRECISION X, R, A
DIMENSION X(1), R(1), A(MAXN,1)
INTRINSIC DABS
EXTERNAL NASEPS
DOUBLE PRECISION DABS, NASEPS
DOUBLE PRECISION DIFFR, XINCR, XPERT, XSAVE
INTEGER ZEROS
INTEGER J88, J99
LOGICAL FIRST, MODIFY
SAVE DIFFR, FIRST
DATA FIRST/.TRUE./
C***
c***********************************************************************
C***
C*** Deternime the nominal difference-to-argument ratio :
C***
C***
IF (FIRST) THEN
DIFFR I DSQRT(NASEPS(1))
FIRST I .FALSE.
ENDIF
C***
C***
C*** Compute the Jacobi matrix (by columns) :
C***
C***

DO 199 J99-1,N
MODIFY I .FALSE.

100 CONTINUE
c***

 

C***

C*** Compute perturbed argument :
C***

 

C***

XSAVE I X(J99)

IF (XSAVE.EQ..0D00.0R.MODIFY) THEN
XINCR I DIFFR

ELSE
XINCR I XSAVE*DIFFR

ENDIF

XPERT I XSAVE+XINCR

X(J99) I XPERT
c***

C***
C***
C***
C***

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***

188

C***
c***
C***
C***
C***
c***

241

 

Evaluate the function residuals

 

CALL F(X,N,A(1,J99))

 

Restore the argument :

 

X(J99) I XSAVE

 

Compute an estimated partial derivative :

 

ZEROS I 0
DO 188 J88I1,N

A(J88,J99) I (A(J88,J99)-R(J88))/XINCR
IF (A(J88,J99).EQ..0D00) THEN

ZEROS I ZEROS+1

ENDIF
CONTINUE

 

Test for possibly misleading results. Deside whether to
modify the finite difference and recompute the column :

 

IF (ZEROS.EQ.N.AND..NOT.MODIFY) THEN

MODIFY I .TRUE.
GO TO 100
ENDIF

199 CONTINUE

C***

C***
C***

RETURN
END

C***********************************************************************

SUBROUTINE NASFNE(F,X,N,R,NORM)

Ci**********************************************************************

C***

**

C*** THIS SUBROUTINE EVALUATES THE FUNCTION RESIDUALS AND THEIR NORM **

C***

**

C***********************************************************************

c***

C***

EXTERNAL F
INTEGER N
DOUBLE PRECISION X,
DIMENSION X(1),
INTRINSIC DSQRT
DOUBLE PRECISION DSQRT

R:
R(l)

NORM

242

C***********************************************************************

 

 

 

 

C***
C*** Evaluate residuals first
C***
C***
CALL F(X,N,R)
C***
C***
C*** Evaluate the norm of R :
C***
C***
CALL NORMSD(R,1,N,NORM)
C***
RETURN
END
C***
C***

C***********************************************************************

DOUBLE PRECISION FUNCTION NASEPS(IDUMMY)

C***********************************************************************

C*** **
C*** THIS FUNCTION COMPUTES THE HOST ARITHMETIC UNIT ACCURACY **
C*** **

c***********************************************************************

C***
INTEGER IDUMMY
DOUBLE PRECISION EPS, BUFFER
INTEGER NR
INTEGER J99
LOGICAL FIRST
PARAMETER (NRI32)
DIMENSION BUFFER(NR)
SAVE EPS, FIRST
DATA EPS/l.D00/,FIRST/.TRUE./
C***

C***********************************************************************
C***

IF (FIRST) THEN
10 CONTINUE
EPS - EPS/2.D00
EUFFER(1) - 1.DOO+EPS
Do 199 J99-1,NR—1
BUFFER(J99+1) - BUFFER(J99)
199 CONTINUE
DO 299 J99-1,NR
IF (BUFFER(J99).EQ.1.DOO) THEN
FIRST - .FALSE.
EPS - EPS*2.D00
NASEPS - EPS
RETURN
ENDIF
299 CONTINUE
Go To 10
ELSE

a—v

243
NASEPS - EPS

ENDIF
C***

RETURN

END
C***
C***

c***********************************************************************

SUBROUTINE NASABX(A,NA,N,B,X,MP,IDET)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE SOLVES A*XIB BY LU DECOMPOSITION **
C*** AND BACKWARDS SUBSTITUTION **
C*** **

C***********************************************************************

C***

INTEGER NA, N, IDET, MP

DOUBLE PRECISION A, B, x

DIMENSION A(NA,1),B(1), X(1), MP(1)
C***
C***********************************************************************
C***

CALL DECOM(A,NA,N,X,MP,IDET)

IF (IDET.EQ.O) RETURN

CALL SUBST(A,NA,N,B,x,MP)

RETURN

END
C***
C***

C***********************************************************************

SUBROUTINE DECOM(A,NA,N,VS,MP,IDET)
C***********************************************************************

C***
IMPLICIT DOUBLE PRECISION(A-H,O-Z)
DIMENSION A(NA,1),VS(1),MP(1)
DOUBLE PRECISION NASEPS

C***

C***********************************************************************
C***

EPS - NASEPS(1)*2**MIN(4,N/2)
DO 100 J00I1,N
CALL NORMSD(A(J00,1),NA,N,TEMP)
IF (TEMP.EQ..0D0) GO TO 4
VS(J00) I .lDl/TEMP
MP(J00) I J00
100 CONTINUE
IDET - 1
DO 200 J00I1,N
IS I J00
X I .000
I00 I JOO-l
Do 110 J10-J00,N
TEMP I -A(J10,J00)
IF (Ioo.GT.0) CALL VIPASD(A(J10,1),NA,A(1,J00),1,Ioo,TEMP)

 

244

A(J10,J00) I ITEMP
TEMP I DABS(TEMP*VS (J10))
IF (X.GE.TEMP) GO TO 110
X I TEMP
IS I J10

110 CONTINUE
IF (IS.EQ.J00) GO TO 1
IDET I -IDET
CALL EXCHSD(A(IS,1),NA,A(J00,1),NA,N)
TEMP I VS(J00)
VS(J00) I VS(IS)
VS(IS) I TEMP
JS I MP(J00)
MP(J00) I MP(IS)
MP(IS) I JS

1 CONTINUE

IF (X.LE.EPS) GO TO 4
X I -.1D1/A(J00,J00)
K00 I J00+1
IF (K00.GT.N) GO TO 200
D0 120 J20IK00,N
TEMP I -A(J00,J20)
IF (IOO.GT.0) CALL VIPASD(A(J00,1),NA,A(1,J20),1,I00,TEMP)
A(J00,J20) I X*TEMP

120 CONTINUE

200 CONTINUE

 

RETURN
4 IDET I 0
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE SUBST(A,NA,N,B,VS,MP)
c***********************************************************************
C***

IMPLICIT DOUBLE PRECISION(A-H,OIZ)

DIMENSION A(NA,1),B(1),VS(1),MP(1)
C***

C***********************************************************************
c***

DO 100 J00I1,N

VS(J00) I B(MP(J00))
100 CONTINUE

VS(l) I IVS(1)/A(1:1)

IF (N.EQ.1) GO TO 1

DO 200 J00I2,N

TEMP I VS(J00)

CALL VIPASD(A(J00,1),NA,VS(1),1,J00-1,TEMP)

VS(J00) I -TEMP/A(J00,J00)
200 CONTINUE

1 M00 I N
DO 300 J00I1,N
100 I J00I1

245

TEMP I VS(M00)
IF (I00.GT.0) CALL VIPASD(A(M00,M00+1),NA,VS(M00+1),1,I00,TEMP)
VS(M00) I ITEMP
M00 I MOO-1
300 CONTINUE
RETURN

END
C***

C***
C***********************************************************************
SUBROUTINE MOVEDD(N,X,Y)

C***********************************************************************

c*** **
C*** THIS SUBROUTINE MOVES ELEMENTS OF A VECTOR TO ANOTHER VECTOR **
C*** **

C***********************************************************************

C***
INTEGER N
DOUBLE PRECISION X, Y
DIMENSION X(1), Y(l)
INTEGER J99

C***

C***********************************************************************
C***
DO 199 J99-1,N
Y(J99) I X(J99)
199 CONTINUE

C***
RETURN
END

c***

c***

C***********************************************************************

SUBROUTINE NORMSD(X,INCX,N,NORM)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES THE EUCLIDEAN NORM OF A VECTOR **
c*** **
C***********************************************************************
c***

INTEGER INCX, N

DOUBLE PRECISION X, NORM

DIMENSION X(1)

INTEGER J99
cr**

C***********************************************************************
C***

NORM I .0000
DO 199 J99I1,INCX*N,INCX
NORM I NORM+X(J99)**2
199 CONTINUE
NORM I DSQRT(NORM)
C***

RETURN

246

END
C***

C***
C***********************************************************************

SUBROUTINE EXCHSD(X,INCX,Y,INCY,N)

c***********************************************************************

C*** **
C*** THIS SUBROUTINE SWAPS ELEMENTS OF TWO VECTORS **
C*** **
C***********************************************************************
C***

INTEGER INCX, INCY, N

DOUBLE PRECISION X, Y

DIMENSION X(1), Y(l)

INTEGER I99, J99

DOUBLE PRECISION TEMP
C***

C***********************************************************************
C***
I99 I 1
DO 199 J99-1,INCX*N,INCX
TEMP I X(J99)
X(J99) I Y(I99)
X(199) I TEMP
I99 I I99+INCY
199 CONTINUE

C***
RETURN
END

C***

C***

C***********************************************************************

SUBROUTINE VIPASD(X,INCX,Y,INCY,N,VIP)
c***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES A VECTOR INNER PRODUCT **
C*** AND ADDS IT TO A GIVEN VALUE **
C*** **

C***********************************************************************

C***
INTEGER INCX, INCY, N
DOUBLE PRECISION X, Y, VIP
DIMENSION X(1), Y(1)
INTEGER I99, J99

c***

c***********************************************************************
C***
I99 I 1
DO 199 J99I1,INCX*N,INCX
VIPI VIP+X(J99)*Y(I99)
I99 I I99+INCY
199 CONTINUE

C***

RETURN

247

END
c***

C***
C***********************************************************************

SUBROUTINE F(X,N,R)
Ci*********************************************************************t

 

 

 

C*** **
C*** THIS SUBROUTINE PROVIDES THE SYSTEM OF NONLINEAR **
C*** ALGEBRAIC EQUATIONS FOR USE IN SUBROUTINE NASNWR **
C*** **
C***********************************************************************
C***

IMPLICIT DOUBLE PRECISION (A-H,O-Z)

INTEGER N

DIMENSION X(1), R(l)

COMMON/MAP/RK,RMOD2,XX,YY
C***
C***********************************************************************
C***
C*** Initialize parameters :
C***
C***
C***

C*** K parameter :

 

 

 

 

 

 

 

C***
C***
RKZ I RMODZ
RK4 I RK2*RK2
RK6 I RK4*RK2
RKB I RK6*RK2
C***
C***
C*** K prime parameter :
C***
C***
RKPZ I RMODZ
RKP4 I RKP2*RKP2
C***
C***
C*** S parameter :
C***
C***
32 I X(2)*X(2)
$4 I 82*82
56 I 84*52
$8 I 86*82
$10 I 88*82
$12 I 510*82
C***
C***
C*** 51 parameter :
C***
C***

802 I X(1)*X(1)

C***
C***
C***
C***
C***

C***
C***
C***
C***
C***
C***
C***
C***
C***

C***

C***
c***
C***

248

 

 

 

 

 

 

 

814 I 502*802

$16 I 814*802

$18 I 516*802

$110 I 518*802

$112 I 3110*302

X,Y parameters :

X2 I XX*XX

Y2 I YY*YY

Compute the 81 polynomials :

Numerator :

PNX2 I 1.-2.*(3.-2.*RK2)*SOZ+(15.-18.*RK2+4.*RK4)*814-

& 4.*(S.-3.*RK2)*RKP2*Sl6+(15.-13.*RK2)*RKP2*818-

& 6.*RKP4*5110+RKP4*8112

PNX4 I -((1.+RK2)-2.*(3.+2.*RK2*RKP2)*302+(15.+7.*RK2-22.
& *RK4+4.*RK6)*Sl4-4.*(5.+7.*RK2-5.*RK4)*RKP2*516+

& (15.+22.*RK2-29.*RK4)*RKP2*818-2.*(3.+8.*RK2)*

& RKP4*8110+(1.+3.*RK2)*RKP4*3112)

PNXG I (l.-2.*(4.-RK2)*802+(25.-15.*RK2-4.*RK4)*Sl4-4.*

& (lO.+RK2-2.*RK4)*RKP2*816+(35.-3.*RK2-20.*RK4)*

& RKP2*818-2.*(8.+7.*RK2)*RKP4*5110+3.*(1.+RK2)*

& RKP4*5112)*RK2

PNX8 I -(-2.*802+(11.-7.*RK2)*Sl4-4.*(6.-RK2)*RKP2*516+

& 2.*(13.-7.*RK2-2.*RK4)*RKP2*818-2.*(7.+2.*RK2)*

& RKP4*8110+(3.+RK2)*RKP4*8112)*RK4

PNXlOI (Sl4-4.*RKP2*816+2.*(3.—2.*RK2)*RKP2*318-4.*RKP4

& *8110+RKP4*3112)*RK6

PNYO I $02-(4.-RK2)*814+3.*(2.-RK2)*816-(4.-3*RK2)*818+RKP2*8110
PNY2 I -2.*(2.*302-(9.-2.*RK2)*Sl4+(15.-7.*RK2)*816-

& (11.-8.*RK2)*818+3.*RKP2*8110)*RK2

PNY4 I (2.*(1.+2.*RK2)*502-2.*(4.+11.*RK2-2.*RK4)*Sl4+

& (12.+43.*RK2-20.*RK4)*316-(8.+36.*RK2-29.*RK4)*318
& +(2;+13*RK2)*RKP2*5110)*RK2

PNY6 I -4.*($02-(S.+RK2)*Sl4+(3.-RK2)*(3.+2.*RK2)*816-(7.+
& 3.*RK2-5.*RK4)*818+(2.+3.*RK2)*RKP2*5110)*RK4

PNY8 I (SOZ-(4.+7.*RK2)*Sl4+(6.+23.*RK2-4.*RK4)*Sl6-(4.+25.*RK2
& I10.*RK4-4.*RK6)*818+(1.+10.*RK2+4.*RK4)*RKP2*3110)*RK4
PNYlOI I2.*(ISl4+(3.+RK2)*SlS-(3.+2.*RK2*RKPZ)*SlB+

& (1.+2.*RK2)*RKP2*8110)*RKG

PNY12I ($16-(2.-RK2)*818+RKP2*5110)*RK8

Denominator :

249

 

 

 

 

 

C***
C***
PDO I 1.-4.*602+6.*Sl4-4.*Sl6+818
PD2 I I2.*(1.-(5.+RK2)*502+2.*(5.+RK2)*Sl4-10.*816+
& (5.-2.*RK2)*SlB-RKP2*3110)
P04 I 1.-6.*(1.+RK2)*802+(15.+22.*RK2-RK4)*314-
& 4.*(5.+7.*RK2-2.*RK4)*Sl6+3.*(5.+4.*RK2*RKP2)*
& $18-2.*(3.+2.*RK2)*RKP2*8110+RKP4*8112
PD6 I -4.*(-SOZ+(5.+RK2)*Sl4-(10.+RK2-RK4)*Sl6+(lO.-3.*
& RK2-2.*RK4)*818-5.*RKP2*5110+RKP4*5112)*RKZ
PD8 I (6.*Sl4-4.*(6.-RK2)*Sl6+(36.-20.*RK2-RK4)*318-
& 4.*(6.—RK2)*RKP2*5110+6.*RKP4*8112)*RK4
PDlO I -2.*(I2.*316+3.*(2.-RK2)*818-(6.-7.*RK2+RK4)*SllO
& +2.*RKP4*3112)*RK6
P012 I (818-2.*RKP2*3110+RKP4*3112)*RKB
C***
C***
C*** Compute the coefficients of the S polynomial
C***
C***
PXO I -PDO*X2
PXZ I PNXZ-PD2*X2
PX4 = PNX4-PD4*X2
PX6 I PNX6-PD6*X2
PX8 I PNXB-PD8*X2
PX1O I PNXlO-PD10*X2
PX12 I -PD12*X2
C***
PYO I PNYO-PDO*Y2
PYZ I PNYZ-PD2*Y2
PY4 I PNY4-PD4*Y2
PY6 I PNYG-PD6*Y2
PY8 I PNYB-PD8*Y2
PYlO I PNYlO-PD10*Y2
PY12 I PNYlZ-PD12*Y2
C***
C***
C*** Compute the S polynomials
C***
C***
R(l) I PYO+PY2*SZ+PY4*S4+PY6*SG+PY8*88+PY10*SlO+PY12*SlZ
R(Z) I PXO+PX2*SZ+PX4*S4+PX6*$6+PX8*38+PX10*810+PX12*812
C***
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE INTPLU(MCOUNT,P,P0,SN,SO)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE PERFORMS LINEAR INTERPOLATION FOR U OR V **
C*** **

C***********************************************************************

250

C***

DOUBLE PRECISION P,PD,SO,SN,SNQ

VIRTUAL P(*),SN(*)
C***

C***********************************************************************

 

C***

C*** Check the bounds on $0 : S/Sl(1) < SO < S/Sl(NPOINT) :
c***

 

 

C***
IF(SO.LT.SN(1)) So - SN(I)
IF(SO.GT.SN(MCOUNT)) so - SN(MCOUNT)
C***
C***

C*** Determine the relative location of 80
C*** and interpolate for the value of U or V

 

C***
C***
DO 1 I I 1,MCOUNT
IF(SO.GT.SN(I)) GO TO 1
SNQ I (SO-SN(I))/(SN(I-1)-SN(I))
PD I P(I-1)IP(I)
P0 I SNGL(SNQ*PD + P(I))
GO TO 2
1 CONTINUE
2 RETURN
END
C***
C***

C***********************************fi***********************************

SUBROUTINE CNDNSN( V0, CMO, CN1, DN1, SNl)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE EVALUATES JACOBIAN ELLIPTIC FUNCTIONS **
C*** USING A DESCENDING LANDEN TRANSFORMATION WITH THE REDUCED **
C*** FUNCTIONS APPROXIMATED BY CIRCULAR (TRIG) FUNCTIONS **
C***(See page 573, "Handbook of Mathematical Functions" by Abramowitz)**
c*** **
C***********************************************************************
c***
IMPLICIT DOUBLE PRECISION ( A-H, O-z )

C***

c***********************************************************************

 

C***

C*** Perform three applications of reduction to reduce the parameter
c***

 

C***
CMl I 1.000 - CMO
RMU1 - ( ( 1.0Do-DSQRT(CM1) ) / ( 1.0D0+DSQRT(CM1) ) )**2
v1 - v0 / ( 1.0D0+DSQRT( RMU1 ) )

C***
CM2 - 1.0D0 - RMUl _ .
RMU2 - ( ( 1.0D0-DSQRT(CM2) ) / ( 1.0D0+DSQRT(CM2) ) )**2
v2 - v1 / ( 1.0D0+DSQRT( RMU2 ) )

C***

251

CM3 - 1.0D0 - RMU2
RMU3 - ( ( 1.0Do-DSQRT(CM3) ) / ( 1.0D0+DSQRT(CM3) ) )**2
v3 a v2 / ( 1.0D0+DSQRT( RMU3 ) )
C***
DN3 - DN(V3,RMU3)**2 + DSQRT(RMU3) - 1.0D0
DN3 - DN3/( 1.0Do + DSQRT(RMU3) - DN(V3,RMU3)**2 )
8N3 - (1.0D0+DSQRT(RMU3))*SN(V3,RMU3)
8N3 - SN3/( 1.0D0 + DSQRT(RMUB)*SN(V3,RMU3)**2 )
CN3 - CN(V3,RMU3)*DN(V3,RMU3)
CN3 - CN3/( 1.0Do + DSQRT(RMU3)*SN(V3,RMU3)**2 )
C***
DN2 - DN3**2 + DSQRT(RMUZ) - 1.0D0
DNz - DN2/( 1.0Do + DSQRT(RMU2) - DN3**2 )
SN2 - (1.0D0+DSQRT(RMU2))*SN3
SN2 - SN2/( 1.0D0 + DSQRT(RMU2)*SN3**2 )
CN2 - CN3*DN3
CN2 - CN2/( 1.0Do + DSQRT(RMU2)*SN3**2 )
C***
DN1 - DN2**2 + DSQRT(RMUl) - 1.0Do
DN1 - DN1/( 1.0D0 + DSQRT(RMUl) - DN2**2 )
SN1 - (1.0D0+DSQRT(RMU1))*SN2
SN1 - SNl/( 1.0Do + DSQRT(RMU1)*SN2**2 )
CN1 - CN2*DN2
CN1 - CN1/( 1.0D0 + DSQRT(RMU1)*SN2**2 )
C***
RETURN
END
C***
C***

C***********************************************************************

FUNCTION CN( X, RM )
C***********************************************************************

C***
IMPLICIT DOUBLE PRECISION ( A-H, OIZ )
CN I DCOS(X) + 0.25D0*RM*(X-DSIN(X)*DCOS(X))*DSIN(X)
RETURN
END
C***
C***

C***********************************************************************

FUNCTION DN( X, RM )

C***********************************************************************

C***
IMPLICIT DOUBLE PRECISION ( A-H, O-Z )
DN I 1.0D0 - 0.SDO*RM*DSIN(X)**2
RETURN
END

C***

C***

ctttttttttt*********************************************t***************

FUNCTION SN( X, RM )
C***ttt*ttt*******************t*****************************************
C***

IMPLICIT DOUBLE PRECISION ( A-H, O-Z )

 

252

SN I DSIN(X) - 0.25D0*RM*(X-DSIN(X)*DCOS(X))*DCOS(X)
RETURN

END
C***

C***
C***********************************************************************

FUNCTION FUNCTN( DX, CMO )

C***********************************************************************

C*** **
C*** THIS FUNCTION EVALUATES THE VALUE OF THE INTEGRAND **
C*** [1/sqrt(1-k**23in**2(phi)] OR [l/sqrt(1—k'**23in**2(phi)] **
C*** IN THE COMPLETE ELLIPTIC INTEGRAL (K) OR ITS COMPLEMENTARY (K') **
C*** RESPECTIVELY **
C*** **

c***********************************************************************

C***
IMPLICIT DOUBLE PRECISION (A-H, O-z )
C***
C***********************************************************************
C***
FUNCTN - ( 1.0D0 - CMO*DSIN(DX)**2 )
FUNCTN - DSQRT( FUNCTN )
FUNCTN - 1.0D0/FUNCTN
RETURN
END
C***
C***

C***********************************************************************

FUNCTION GAUSS( A, B, M, FUNCTN, CMO )
C***********************************************************************
C***

IMPLICIT DOUBLE PRECISION ( A-H, O-z )

DIMENSION NPOINT(7), KEY(8), 2(24), WEIGHT(24)

DATA NPOINT / 2, 3, 4, S, 6, 10, 15 /

DATA KEY / 1, 2, 4, 6, 9, 12, 17, 25 /

DATA 2 / 0.577350269, 0.0, 0.774596669, 0.339981044, 0.861136312,

.0, 0.538469310, 0.906179846, 0.238619186, 0.661209387,

.932469514, 0.148874339, 0.433395394, 0.679409568,

.865063367, 0.973906529, 0.0, 0.201194094, 0.394151347,

.570972173, 0.724417731, 0.848206583, 0.937273392,

0.987992518 /

DATA WEIGHT / 1.0, 0.888888889, 0.555555556, 0.652145155,
0.347854845, 0.568888889, 0.478628671, 0.236926885,
0.467913935, 0.360761573, .171324493, 0.295524225,
0.269266719, 0.219086363, .149451349, 0.066671344,
0.202578242, 0.198431485, .186161000, 0.166269206,
0.139570678, 0.107159221, .070366047, 0.030753242 /

m m m m m
CDCDCDCD

m b m b m
C)C>C)O

C***
C***********************************************************************
C***

C*** Find the subscripts of the first 2 and weight values
C***

 

 

C***

DO 1 II1,7

 

 

 

253

 

 

IF( M.EQ.NPOINT(I) ) GO TO 2
1 CONTINUE
C***
C*** INVALID M USED
C***
GAUSS I 0.0D0
RETURN
C***
C***
C*** Setup initial parameters :
C***
C***

2 JFIRST I KEY(I)
JLAST I KEY(I+1) - 1
C I 0.5D0*(B-A)
D I 0.5D0*(B+A)

 

 

 

 

 

C***
C***
C*** Accumulate the sum in the M-POINT formula
C***
C***
SUM I 0.0D0
DO 3 JIJFIRST, JLAST
IF( Z(J).EQ.0.0D0 ) SUM I SUM + WEIGHT(J)*FUNCTN(D,CMO)
IF( Z(J).NE.0.0D0 ) SUM I SUM + WEIGHT(J)*(FUNCTN(Z(J)*C+D,CMO)
* + FUNCTN(-Z(J)*C+D,CMO))
3 CONTINUE
C***
C***
C*** Make the interval correction and the return
C***
C***
GAUSS I SUM*C
RETURN
END
C***

254

C***
C***********************************************************************
c********* POISSON LIBRARY CONTAINS THE FOLLOWING SUBROUTINES : **
C****** **
C*** 1. COSGEN - Computes requiredcosine values in ascending order **
C*** 2. GENBUN - Reorders unknowns, reverses columns and returns **
C*** 3. HWSCRT - Does the Fast Fourier Transform [In Poisson/Lib] **
C*** 4. MERGE - Merges two ascending strings of nos. in array TCOS **
C*** S. POISD2 - Solves Pooisson’s eqn. for Dirichlet bndry conds. **
C*** 6. POISN2 - Solves Pooisson's eqn. for Neumann bndry conds. **
C*** 7. POISP2 - Solves Pooisson's eqn. for Periodic bndry conds. **
C*** 8. TRIX - Solves a system of linear equations where the **
C*** coefficient matrix is a rational fctn. given by **
C*** Tridiagonal (..., A(I), B(I), C(I), ...) **
C*** 9. TRIB - Solves three linear systems whose common coeffs. **
C*** matrix is a rational fctn.in the matrix given by **
C*** Tridiagonal (..., A(I), 8(1), C(I), ...) **
C*** **

C***********************************************************************

SUBROUTINE COSGEN (N,IJUMP,FNUM,FDEN,A)

C***********************************************************************

C*** **
C*** THIS SUBROUTINE COMPUTES REQUIRED COSINE VALUES IN ASCENDING **
C*** ORDER. WHEN IJUMP .GT. 1 THE ROUTINE COMPUTES VALUES **
C*** **
C*** 2*COS(J*PI/L) , JI1,2,...,L AND J .NE. 0(MOD N/IJUMP+1) **
C*** WHERE L - IJUMP*(N/IJUMP+1). **
C*** WHEN IJUMP - 1 IT COMPUTES **
C*** 2*COS((J-FNUM)*PI/(N+FDEN)) , J=1, 2, ... ,N **
C*** WHERE **
C*** FNUM - 0.5, EDEN - 0.0, FOR REGULAR REDUCTION VALUES **
C*** FNUM - 0.0, EDEN - 1.0, FOR B-R AND C-R WHEN ISTAG - 1 **
C*** FNUM I 0.0, EDEN I 0.5, FOR B-R AND C-R WHEN ISTAG - 2 **
C*** FNUM - 0.5, EDEN - 0.5, FOR B-R AND C-R WHEN ISTAG - 2 **
C*** IN POISN2 ONLY. **
C*** **

C***********************************************************************
C***

DIMENSION A(l)
C***
C***********************************************************************
C***

PI I 3.14159265358979
IE (N .EQ. 0) GO To 105
IE (IJUMP .E0. 1) GO TO 103
K3 - N/IJUMP+1
K4 I K3-l
PIBYN - PI/ELOAT(N+IJUMP)
Do 102 KI1,IJUMP
K1 I (K-1)*K3
K5 - (H-1)*K4
DO 101 II1,K4
x - K1+I
K2 - KS+I

 

 

255

A(K2) I I2.*COS(X*PIBYN)
101 CONTINUE
102 CONTINUE
GO TO 105
103 CONTINUE
NPl I N+1
Y I PI/(FLOAT(N)+FDEN)
DO 104 II1,N
X I FLOAT(NPl-I)IFNUM
A(I) I 2.*COS(X*Y)
104 CONTINUE
105 CONTINUE
RETURN

END
C***

C***
C***********************************************************************
SUBROUTINE GENBUN (NPEROD,N,MPEROD,M,A,B,C,IDIMY,Y,IERROR,W)
c***********************************************************************

C***

DIMENSION Y(IDIMY,1)

DIMENSION W(1) ,B(1) ,A(l) ,C(1)
C***
C***********************************************************************

C***

IERROR I 0
IF (M .LE. 2) IERROR I 1
IF (N .LE. 2) IERROR I 2
IF (IDIMY .LT. M) IERROR I 3
IF (NPEROD.LT.0 .OR. NPEROD.GT.4) IERROR I 4
IF (MPEROD.LT.0 .OR. MPEROD.GT.1) IERROR I 5
IF (MPEROD .EQ. 1) GO TO 102
DO 101 II2,M
IF (A(I) .NE. C(1)) GO TO 103
IF (C(I) .NE. C(1)) GO TO 103
IF (B(I) .NE. 8(1)) GO TO 103
101 CONTINUE
GO TO 104
102 IF (A(l).NE.0. .OR. C(M).NE.0.) IERROR I 7
GO TO 104
103 IERROR I 6
104 IF (IERROR .NE. 0) RETURN

MP1 I M+1
IWBA I MP1
IWBB I IWBA+M
IWBC I IWBB+M
IWBZ I IWBC+M
IWB3 I IWBZ+M
IWW1 I IWB3+M
IWWZ I IWW1+M
IWW3 I IWW2+M

IWD I IWW3+M
IWTCOS I IWD+M
IWP I IWTCOS+4*N

256

D0 106 II1,M
K I IWBA+II1
W(K) I -A(I)
K I IWBC+II1
W(K) I -C(I)
K I IWBB+II1
W(K) I 2.IB(I)
DO 105 JI1,N
Y(I,J) - 'Y‘IIJ)
105 CONTINUE
106 CONTINUE
MP I MPEROD+1
NP I NPEROD+1
GO TO (114,107),MP
107 GO TO (108,109,110,111,123),NP
108 CALL POISP2 (M,N,W(IWBA),W(IWBB),W(IWBC),Y,IDIMY,W,W(IWB2),

1 W(IWB3),W(IWW1),W(IWW2),W(IWW3),W(IWD),W(IWTCOS),
2 W(IWP))
GO To 112
109 CALL POISD2 (M,N,1,W(IWBA),W(IWBB),W(IWBC),Y,IDIMY,W,W(IWW1),
1 W(IWD),W(IWTCOS),W(IWP))
GO To 112
110 CALL POISN2 (M,N,1,2,W(IWBA),W(IWBB),W(IWBC),Y,IDIMY,W,W(IWBZ),
1 W(IWB3),W(IWW1),W(IWW2),W(IWW3),W(IWD),W(IWTCOS),
2 W(IWP))
GO To 112
111 CALL POISN2 (M,N,1,1,W(IWBA),W(IWBB),W(IWBC),Y,IDIMY,W,W(IWBZ),
1 W(IWBB),W(IWW1),W(IWW2),W(Iww3),W(IWD),W(IWTCOS),
2 W(IWP))
112 IPSTOR - W(IWWl)
IREv - 2

IF (NPEROD .EQ. 4) GO TO 124
113 GO TO (127,133),MP

114 CONTINUE
C***

 

c***
C*** Reorder unknowns when MPIO

 

C***
C***

MH I (M+1)/2
MHMl I MH-l
MODD I 1
IF (MH*2 .EQ. M) MODD I 2
DO 119 JI1,N
DO 115 II1,MHM1
MHPI I MH+I
MHMI I MH-I
W(I) I Y(MHMI,J)-Y(MHPI,J)
W(MHPI) I Y(MHMI,J)+Y(MHPI,J)
11$ CONTINUE
W(MH) I 2.*Y(MH,J)
GO TO (117,116),MODD
116 W(M) I 2.*Y(M,J)
117 CONTINUE

118
119

120

121
122

C***

257

D0 118 II1,M
Y(I,J) I W(I)
CONTINUE
CONTINUE
K I IWBC+MHM1~1
I I IWBA+MHM1
W(K) I 0.
W(I) I 0.
W(K+1) I 2.*W(K+1)
GO TO (120,121),MODD
CONTINUE
K I IWBB+MHM1I1
W(K) I W(K) -W(I-1)
W(IWBCIl) I W(IWBC-1)+W(IWBB-l)
GO TO 122
W(IWBB-l) I W(K+1)
CONTINUE
GO TO 107

 

C***
C***

Reverse columns when NPERODI4

 

C***
C***

123

124

125
126

127

128

129
130

131
132
133

C***
C***

IREV I 1
NBY2 I N/2
DO 126 JI1,NBY2
MSKIP I N+1-J
DO 125 II1,M
A1 I Y(I,J)
Y(I,J) I Y(I,MSKIP)
Y(I,MSKIP) I A1
CONTINUE
CONTINUE
GO TO (110,113),IREV
CONTINUE
DO 132 JI1,N
DO 128 II1,MHM1
MHMI I MH-I
MHPI I MH+I
W(MHMI) I .5*(Y(MHPI,J)+Y(I,J))
W(MHPI) I .5*(Y(MHPI,J)-Y(I,J))
CONTINUE
W(MH) I .S*Y(MH,J)
GO TO (130,129),MODD
W(M) I .5*Y(M,J)
CONTINUE
DO 131 II1,M
Y(I,J) I W(I)
CONTINUE
CONTINUE
CONTINUE

 

C***

Return storage requirements for W array

 

 

258

 

C***

C***
W(l) - IPSTOR+IWP-1
RETURN
END

C***

C***

C***********************************************************************
SUBROUTINE HWSCRT(BDA,BDB,BDC,BDD,ELMBDA,F,IDIMF,PERTRB,IERROR,W)

C*************************************i*it******************************

C*** **

C*** W:A ONE-DIMENSIONAL ARRAY THAT MUST BE PROVIDED BY THE USER FOR **
C*** WORK SPACE. W MAY REQUIRE UP TO 4*(N+1)+(13+INT(LOG2(N+1)))*(M+1)**
C*** LOCATIONS. THE ACTUAL NUMBER OF LOCATIONS USED ID COMPUTED BY **
C*** HWSCRT AND IS RETURNED IN LOCATION W(l). **

C*** **
C*********************************t*************************************

 

 

C***
COMMON/BNDARY/MBDCND,NBDCND
COMMON/GRID1 /DU,DUINVT,DV,DVINVT,HDUINV,HDVINV
COMMON/GRIDZ /ULNGTH,A,B,VLNGTH,C,D
COMMON/NSIZE /MP1,IMID,IMIN,M,NP1,JMID,JMIN,N
DIMENSION F(IDIMF,1), W(I)
VIRTUAL BDA(*), BDB(*), BDC(*), BDD(*)
C***
C***********************************************************************
C***
C*** Check for invalid parameters :
C***
C***

IERROR - 0

IF (A .GE. B) IERROR - 1

IF (MBDCND.LT.0 .OR. MBDCND.GT.4) IERROR - 2

IE (C .GE. D) IERROR - 3

IF (N .LE. 3) IERROR - 4

IF (NBDCND.LT.0 .OR. NBDCND.GT.4) IERROR - 5

IE (IDIMF .LT. M+1) IERROR - 7

IE (M .LE. 3) IERROR - 8

IF (IERROR .NE. 0) RETURN

NPEROD - NBDCND

MPEROD - 0

IF (MBDCND .GT. 0) MPEROD - 1

TWDELx - 2./DU

DELXSQ - 1./DU**2

TWDELY - 2./Dv

DELYSQ I 1./DV**2

NP - NBDCND+1

MP - MBDCND+1

NSTART - 1

NSTOP - N

NSKIP - 1

GO To (104,101,102,103,104),NP
101 NSTART - 2 .

GO To 104

102
103

104

C***
C***

259

NSTART I 2

NSTOP I NP1

NSKIP I 2

NUNK I NSTOP-NSTART+1

 

C***
C***

Enter boundary data for X-boundaries :

 

C***

105

106

107
108
109
110
111
112
113
114

115

116
117

C***
C***

MSTART I 1
MSTOP I M
MSKIP I 1
GO TO (117,105,106,109,110),MP
MSTART I 2
GO TO 107
MSTART I 2
MSTOP I MP1
MSKIP I 2
DO 108 JINSTART,NSTOP
F(2,J) I F(2,J)-F(1,J)*DELXSQ
CONTINUE
GO TO 112
MSTOP I MP1
MSKIP I 2
DO 111 JINSTART,NSTOP
F(1,J) I F(1,J)+BDA(J)*TWDELX
CONTINUE
GO TO (113,115),MSKIP
DO 114 JINSTART,NSTOP
F(M,J) I F(M,J)-F(MP1,J)*DELXSQ
CONTINUE
GO TO 117
DO 116 JINSTART,NSTOP
F(MP1,J) I F(MP1,J)-BDB(J)*TWDELX
CONTINUE
MUNK I MSTOP-MSTART+1

 

C***

Enter boundary data for Y-boundaries :

 

C***
C***

118
119
120
121
122
123

124

GO TO (127,118,118,120,120),NP
DO 119 IIMSTART,MSTOP

F(I,2) I F(I,2)-F(I,1)*DELYSQ
CONTINUE
GO TO 122
D0 121 IIMSTART,MSTOP

F(I,1) I F(I,1)+BDC(I)*TWDELY
CONTINUE
GO TO (123,125),NSKIP
DO 124 IIMSTART,MSTOP

F(I,N) I F(I,N)-F(I,NP1)*DELYSQ
CONTINUE
GO TO 127

260

125 D0 126 IIMSTART,MSTOP
F(I,NPI) I F(I,NP1)-BDD(I)*TWDELY

126 CONTINUE
C***

 

C***

C*** Multiply right side by DV**2 :
C***

 

C***

127 DELYSQ I DV*DV
DO 129 IIMSTART,MSTOP
DO 128 JINSTART,NSTOP
F(I,J) I F(I,J)*DELYSQ
128 CONTINUE

129 CONTINUE
C***

C***

C*** Define the A,B,C coefficients in W-array :
C***

 

 

 

C***
ID2 I MUNK
ID3 I ID2+MUNK
ID4 I ID3+MUNK
S I DELYSQ*DELXSQ
5T2 I 2.*S
DO 130 II1,MUNK
W(I) I S
J I ID2+I
W(J) I 'ST2+ELMBDA*DELYSQ
J I ID3+I
W(J) I S
130 CONTINUE
IF (MP .EQ. 1) GO TO 131
W(I) I 0.
W(ID4) I 0.
131 CONTINUE
GO TO (135,135,132,133,134),MP
132 W(IDZ) I 8T2
GO TO 135
133 W(IDZ) I 5T2
134 W(ID3+1) I 3T2
135 CONTINUE
PERTRB I 0.
IF (ELMBDA) 144,137,136
136 IERROR I 6

GO TO 144
137 IF ((NBDCND.EQ.0 .OR. NBDCND.EQ.3) .AND.
1 (MBDCND.EQ.0 .OR. MBDCND.EQ.3)) GO TO 138
GO TO 144
C***
C***

 

C*** For singular problems must adjust data

C*** to insure that a solution will exist
C***

C***

 

 

 

261

138 A1 I 1.
A2 I 1.
IF (NBDCND .EQ. 3) A2 = 2.
IF (MBDCND .EQ. 3) A1 I 2.
$1 I 0.
MSPl I MSTART+1
MSTMl I MSTOP-1
NSPl I NSTART+1
NSTMl I NSTOP-1
DO 140 JINSP1,NSTM1
S I 0.
DO 139 IIMSP1,MSTM1
S I S+F(I,J)
139 CONTINUE
$1 I Sl+S*A1+F(MSTART,J)+F(MSTOP,J)
140 CONTINUE
$1 I A2*Sl
S I 0.
DO 141 IIMSP1,MSTM1
S I S+F(I,NSTART)+F(I,NSTOP)
141 CONTINUE
$1 I Sl+S*A1+F(MSTART,NSTART)+F(MSTART,NSTOP)+F(MSTOP,NSTART)+
1 F(MSTOP,NSTOP)
S I (2.+FLOAT(NUNK-2)*A2)*(2.+FLOAT(MUNK-2)*Al)
PERTRB I 81/3
DO 143 JINSTART,NSTOP
DO 142 IIMSTART,MSTOP
F(I,J) I F(I,J)-PERTRB
142 CONTINUE
143 CONTINUE
PERTRB I PERTRB/DELYSQ

 

C***
C***

 

C*** Solve the equation :

 

 

C***
c***
144 CALL GENBUN (NPEROD,NUNK,MPEROD,MUNK,W(1),W(ID2+1),W(ID3+1),
1 IDIMF,F(MSTART,NSTART),IERR1,W(ID4+1))
W(l) I W(ID4+1)+3.*FLOAT(MUNK)
C***
c***

C*** Fill in identical values when have periodic boundary conditions
c***

 

C***

IF (NBDCND .NE. 0) GO TO 146
DO 145 IIMSTART,MSTOP
F(IrNPl) I F(I,1)
145 CONTINUE
146 IF (MBDCND .NE. 0) GO TO 148
DO 147 JINSTART,NSTOP
F(MplrJ) - F(I,J)
147 CONTINUE
IF (NBDCND .EQ. 0) F(MP1,NP1) I F(I,NPI)
148 CONTINUE

262

RETURN

END
C***

C***
c***********************************************************************

SUBROUTINE MERGE (TCOS,I1,M1,I2,M2,I3)
C***********************************************************************

C*** **
C*** THIS SUBROUTINE MERGES TWO ASCENDING STRINGS OF NUMBERS IN THE **
C*** ARRAY TCOS. THE FIRST STRING IS OF LENGTH M1 AND STARTS AT **
C*** TCOS(I1+1). THE SECOND STRING IS OF LENGTH M2 AND STARTS AT **
C*** TCOS(I2+1). THE MERGED STRING GOES INTO TCOS(I3+1). **
C*** **

C***********************************************************************
C***

DIMENSION TCOS(1)

c***

 

C***********************************************************************

C***
J1 I 1
J2 I 1
J I I3
IF (M1 .EQ. 0) GO TO 107
IF (M2 .EQ. 0) GO TO 104
101 J I J+1
L I J1+Il
X I TCOS(L)
L I J2+I2
Y I TCOS(L)

IF (X-Y) 102,102,103
102 TCOS(J) - x
J1 - J1+1
IE (J1 .GT. M1) Go To 106
Go TO 101
103 TCOS(J) I Y
J2 I J2+1
IF (J2 .LE. M2) GO TO 101
IF (J1 .GT. M1) GO TO 109
104 K - J-J1+1
DO 105 J-J1,M1
M - K+J
L - J+Il
TCOS(M) - TCOS(L)
105 CONTINUE
GO TO 109
106 CONTINUE
IF (J2 .GT. M2) GO TO 109
107 K - J-J2+1
DO 108 J-J2,M2
M - K+J
L - J+I2 J
TCOS(M) - TCOS(L)
108 CONTINUE
109 CONTINUE

263

RETURN

END
C***

C***
C***********************************************************************

SUBROUTINE POISD2 (MR,NR,ISTAG,BA,BB,BC,Q,IDIMQ,B,W,D,TCOS,P)
C***********************************************************************

C*** **
C*** SUBROUTINE TO SOLVE POISSON'S EQUATION FOR DIRICHLET BOUNDARY **
C*** CONDITIONS. **
C*** ISTAG I 1 IF THE LAST DIAGONAL BLOCK IS THE MATRIX A. **
C*** ISTAG I 2 IF THE LAST DIAGONAL BLOCK IS THE MATRIX A+I. **
C*** **

C***********************************************************************

C***
DIMENSION Q(IDIMQ,1) ,BA(1) ,BB(1) ,BC(1) ,
1 TCOS(1) ,B(1) ,D(1) ,W(1) ,
2 P(l)

C***

C***********************************************************************
C***
M I MR
N I NR
JSH I 0
PI I 1./FLOAT(ISTAG)
IP I IN
IPSTOR I 0
GO TO (101,102),ISTAG
101 KR I 0
IRREG I 1
IF (N .GT. 1) GO TO 106
TCOS(1) I 0.
GO TO 103
102 KR I 1
JSTSAV I 1
IRREG I 2
IF (N .GT. 1) GO TO 106
TCOS(1) I I1.
103 DO 104 II1,M
B(I) I Q(I,1)
104 CONTINUE
CALL TRIX (1,0,M,EA,BB,EC,E,TCOS,D,W)
DO 105 II1,M
Q(I,1) I 3(1)
105 CONTINUE
GO TO 183
106 LR I 0
DO 107 II1,M
P(I) I 0.
107 CONTINUE
NUN I N
JST I 1

JSP I N
C***

C***

264

 

C***
C***

IRREG-1 when no irregularities have occurred, otherwise it is 2

 

C***

108

C***
C***

L I 2*JST
NODD I 2-2*((NUN+1)/2)+NUN

 

C***
C***

NODD-1 when NUN is odd, otherwise it is 2

 

C***

109

110

111

C***
C***

GO TO (110,109),NODD

JSP I JSP-L

GO TO 111

JSP I JSP-JST

IF (IRREG .NE. 1) J5? I JSP-L
CONTINUE

 

C***
C***

Regular reduction :

 

C***

112

113

114
115

116
117

C***

CALL COSGEN (JST,1,0.5,0.0,TCOS)
IF (L .GT. JSP) GO TO 118
DO 117 JIL,JSP,L

JMl I J-JSH
JPl I J+JSH
JMZ I J-JST
JP2 I J+JST

JM3 I JMZ-JSH

JP3 I JP2+JSH

IF (JST .NE. 1) GO TO 113

DO 112 II1,M
3(1) I 20*Q(IIJ)
Q(IIJ) - Q(IIJM2)+Q(IIJP2)

CONTINUE

GO TO 115

DO 114 II1,M
T I Q(I,J)-Q(I,JMl)-Q(I,JP1)+Q(I,JM2)+Q(I,JP2)
B(I) I T+Q(I'J)-Q(IIJM3).Q(IIJP3)
Q(IIJ) ' T

CONTINUE

CONTINUE

CALL TRIX (JST,0,M,BA,BB,BC,B,TCOS,D,W)

‘DO 116 II1,M
Q(IIJ) - Q(IIJ)+B(I)

CONTINUE

CONTINUE

 

c***
c***
C***

Reduction for last unknown :

 

C***

118

GO TO (119,135),NODD

 

 

 

119

C***
C***

265
GO To (152,120),IRREC

 

C***
c***

Odd number of unknowns :

 

C***

120

121

122

123

124

125

126

127
128

129
130

131

132

133

134

JSP - JSP+L
J - JSP
JMl - J-JSH
JPl - J+JSH
JMZ - J-JST
JP2 - J+JST
JM3 I JM2-JSH
GO To (123,121),ISTAG
CONTINUE
IF (JST .NE. 1) GO To 123
D0 122 II1,M
s(I) - Q(I,J)
Q(I,J) - o.
CONTINUE
GO TO 130
GO To (124,126),NODDPR
DO 125 II1,M
IPl - IP+I
3(1) I .5*(Q(IpJM2)-Q(I:JM1)-Q(I'JM3))+P(IP1)+Q(I,J)
CONTINUE
GO To 128
DC 127 II1,M
8(1) I .5*(Q(I'JMZ)-Q(I:JM1)-Q(I,JM3))+Q(I,JP2)-Q(I,JP1)+Q(I,J)
CONTINUE
DO 129 II1,M
Q(IIJ) I 05*(Q(IIJ)-Q(IIJM1).Q(IIJP1))
CONTINUE
CALL TRIX (JST,0,M,BA,BB,BC,B,TCOS,D,W)
IP - IP+M
IPSTOR - MAXO(IPSTOR,IP+M)
DO 131 II1,M
1P1 - IP+I
P(IPl) - Q(I,J)+B(I)
3(1) I Q(I,JP2)+P(IPI)
CONTINUE
IF (LR .NE. 0) GO TO 133
D0 132 I-1,JST
KRPI - KR+I
'TCOS(KRPI) - TCOS(1)
CONTINUE
GO To 134
CONTINUE
CALL COSGEN (LR,JSTSAV,0.,FI,TCOS(JST+1))
CALL MERGE (TCOS,0,JST,JST,LR,KR)
CONTINUE
CALL COSGEN (KR,JSTSAV,0.0,FI,TCOS)
CALL TRIX (KR,KR,M,BA,BB,BC,B,TCOS,D,W)
DO 135 II1,M

 

135

C***
C***

266

IPl - IP+I
Q(I,J) - Q(I,JM2)+B(I)+P(IP1)
CONTINUE
LR - KR
KR - KR+L
GO To 152

 

C***

Even number of unknowns :

 

C***
C***

136

137

138

139

140

141

142

143

144

145

146

147

148

JSP I JSP+L
J I JSP
JMl J-JSH
JPl J+JSH
JMZ J-JST
JPZ J+JST
JM3 JM2-JSH
GO TO (137,138),IRREG
CONTINUE
JSTSAV I JST
IDEG I JST
KR I L
GO TO 139
CALL COSGEN (KR,JSTSAV,0.0,FI,TCOS)
CALL COSGEN (LR,JSTSAV,0.0,FI,TCOS(KR+1))
IDEG I KR
KR I KR+JST
IF (JST .NE. 1) GO TO 141
IRREG I 2
DO 140 II1,M
8(1) I Q(I,J)
Q(I,J) I Q(I,JMZ)
CONTINUE
GO TO 150
D0 142 II1,M
8(1) I Q(I,J)+.5*(Q(I,JM2)-Q(I,JM1)-Q(I,JM3))
CONTINUE
GO TO (143,145),IRREG
DO 144 II1,M
Q(I,J) I Q(I,JM2)+.5*(Q(I,J)-Q(I,JMl)-Q(I,JP1))
CONTINUE
IRREG I 2
GO TO 150
CONTINUE
GO TO (146,148),NODDPR
DO 147 II1,M
IPl I IP+I
Q(I,J) I Q(I'JM2)+P(IP1)
CONTINUE
IP I IP-M
GO TO 150
DO 149 II1,M
Q(I,J) I Q(I,JM2)+Q(I,J)-Q(I,JM1)

 

149
150

151
152

C***
C***

267

CONTINUE
CALL TRIX (IDEG,LR,M,BA,BB,BC,B,TCOS,D,W)
DO 151 II1,M
Q(I,J) I Q(IrJ)+B(I)
CONTINUE
NUN - NUN/2
NODDPR - NODD
an - JST
JST - 2*JST
IF (NUN .GE. 2) GO TO 108

 

C***
C***

Start solution :

 

C***

153

154

155

156

157

158

159

160

161

162

163
164

c***
C***

J I JSP
DO 153 II1,M
3(1) I Q(I,J)
CONTINUE
GO TO (154,155),IRREG
CONTINUE
CALL COSGEN (JST,1,0.5,0.0,TCOS)
IDEG I JST
GO TO 156
KR I LR+JST
CALL COSGEN (KR,JSTSAV,0.0,FI,TCOS)
CALL COSGEN (LR,JSTSAV,0.0,FI,TCOS(KR+1))
IDEG I KR
CONTINUE
CALL TRIX (IDEG,LR,M,BA,BB,BC,B,TCOS,D,W)
JMl I J-JSH
JP1 I J+JSH
GO TO (157,159),IRREG
DO 158 II1,M
Q(IrJ) I .S*(Q(I,J)-Q(I,JM1)-Q(I,JP1))+B(I)
CONTINUE
GO TO 164
GO TO (160,162),NODDPR
DO 161 II1,M
IPl I IP+I
Q(I,J) I P(IP1)+B(I)
CONTINUE
IP I IP-M
GO TO 164
D0 163 II1,M
Q(I,J) I Q(I,J)-Q(I,JM1)+B(I)
CONTINUE
CONTINUE

 

C***

Start back substitution :

 

C***
C***

JST I JST/2

 

268

JSH I JST/2
NUN I 2*NUN
IF (NUN .GT. N) GO TO 183
D0 182 JIJST,N,L
JMl I J-JSH
JP1 I J+JSH
JM2 I J-JST
JP2 I J+JST
IF (J .GT. JST) GO TO 166
D0 165 II1,M
3(1) I Q(I,J)+Q(I'JP2)
165 CONTINUE
GO TO 170
166 IF (JP2 .LE. N) GO TO 168
D0 167 II1,M
8(1) I Q(I,J)+Q(I'JM2)
167 CONTINUE
IF (JST .LT. JSTSAV) IRREG I 1
GO TO (170,171),IRREG
168 D0 169 II1,M
8(1) I Q(LJ)+Q(I'JM2)+Q(I,JP2)
169 CONTINUE
170 CONTINUE
CALL COSGEN (JST,1,0.5,0.0,TCOS)
IDEG I JST
JDEG I 0
GO TO 172
171 IF (J+L .GT. N) LR I LR-JST
KR I JST+LR
CALL COSGEN (KR,JSTSAV,0.0,FI,TCOS)
CALL COSGEN (LR,JSTSAV,0.0,FI,TCOS(KR+1))
IDEG I KR
JDEG I LR
172 CONTINUE
CALL TRIX (IDEG,JDEG,M,BA,BB,BC,B,TCOS,D,W)
IF (JST .GT. 1) GO TO 174
DO 173 II1,M
Q(I,J) I B(I)
173 CONTINUE
GO TO 182
174 IF (JP2 .GT. N) GO TO 177
175 DO 176 II1,M
Q(LJ) I .5*(Q(I,J)-Q(I,JM1)-Q(I,JPI))+B(I)
176 CONTINUE
GO TO 182
177 GO TO (175,178),IRREG
178 IF (J+JSH .GT. N) GO TO 180
D0 179 II1,M
IPl I IP+I
Q(I,J) I B(I)+P(IP1)
179 CONTINUE
IP I IP-M
GO TO 182
180 DO 181 II1,M

 

 

269

Q(I,J) I B(I)+Q(IIJ)-Q(IIJM1)
181 CONTINUE
182 CONTINUE

 

 

 

L - L/Z
GO TO 164
183 CONTINUE
C***
C***
C*** Return storage requirements for P vectors :
C***
C***
W(l) I IPSTOR
RETURN
END
C***
C***

C*****************************************************************************

SUBROUTINE POISN2 (M,N,ISTAG,MIXBND,A,BB,C,Q,IDIMQ,B,B2,B3,W,W2,
1 W3,D,TCOS,P)

C*****************************************************************************

 

C*** **
C*** SUBROUTINE TO SOLVE POISSON’S EQUATION WITH NEUMANN BOUNDARY **
C*** CONDITIONS. **
C*** **
C*** ISTAG I 1 IF THE LAST DIAGONAL BLOCK IS A. **
C*** ISTAG I 2 IF THE LAST DIAGONAL BLOCK IS A-I. **
C*** MIXBNDI 1 IF HAVE NEUMANN BOUNDARY CONDITIONS AT BOTH BOUNDARIES.**
C*** MIXBNDI 2 IF HAVE NEUMANN BOUNDARY CONDITIONS AT BOTTOM AND **
C*** DIRICHLET CONDITION AT TOP.(FOR THIS CASE, MUST HAVE ISTAG I 1.) **
C*** **

C***********************************************************************

C***
DIMENSION A(l) ,BB(1) ,C(1) ,Q(IDIMQ,1) ,
1 3(1) 132(1) 133(1) :W(1) r
2 W2(1) ,W3(1) ,D(1) ,TCOS(1) ,
3 K(4) :P(1)
EQUIVALENCE (K(1),K1) :(K(2),K2) ,(K(3),K3) :(K(4),K4)
C***

c***********************************************************************
c***

FISTAG I 3-ISTAG

FNUM I 1./FLOAT(ISTAG)

FDEN I 0.5*FLOAT(ISTAG-1)

MR I M

IP I IMR

IPSTOR I 0

I2R I 1

JR I 2

NR I N

NLAST I N

KR I 1

LR I 0

GO TO (101,103),ISTAG
101 CONTINUE

 

102

103
104

105

106

107

108

C***

270

D0 102 II1,MR
Q(IIN) I .5*Q(I,N)
CONTINUE
GO TO (103,104),MIXBND
IF (N .LE. 3) GO TO 155
CONTINUE
JR I 2*IZR
NROD I 1
IF ((NR/2)*2 .EQ. NR) NROD I 0
GO TO (105,106),MIXBND
JSTART I 1
GO TO 107
JSTART I JR
NROD I 1-NROD
CONTINUE
JSTOP I NLAST-JR
IF (NROD .EQ. 0) JSTOP I JSTOP-IZR
CALL COSGEN (I2R,1,0.5,0.0,TCOS)
IZRBYZ I I2R/2
IF (JSTOP .GE. JSTART) GO TO 108
J I JR
GO TO 116
CONTINUE

 

C***
C***

Regular reduction :

 

C***
C***

109

110

111

112
113

DO 115 JIJSTART,JSTOP,JR
JPI I J+I2RBY2
JP2 I J+I2R
JP3 I JP2+IZRBY2
I JIIZRBYZ
JM2 I J-I2R
I JM2-I2RBY2
IF (J .NE. 1) GO TO 109
JMl I JP1
JM2 I JPZ
JM3 I JP3
CONTINUE
IF (IZR .NE. 1) GO TO 111
IF (J .80. 1) JM2 I JP2
DO 110 II1,MR
3(1) I 2-*Q(IoJ)
Q(I'J) ' Q(Ilm)+Q(IIJP2)
CONTINUE
GO TO 113
CONTINUE
DO 112 II1,MR
FI I Q(I,J)
Q(I,J) I Q(I,J)-Q(I,JMl)-Q(I,JP1)+Q(I,JM2)+Q(I,JP2)
3(1) I FI+Q(I:J)“Q(I,JM3)-Q(I:JP3)
CONTINUE
CONTINUE

 

114

C***
c***

271

CALL TRIX (I2R,0,MR,A,BB,C,B,TCOS,D,W)
DO 114 II1,MR

Q(IuJ) I Q(I,J)+B(I)
CONTINUE

 

C***
C***

End of reduction for regular unknowns

 

C***

115
C***
C***

CONTINUE

 

C***

Begin special reduction for last unknown :

 

C***
C***

116

C***
C***

J I JSTOP+JR

NLAST I J

JMl I J-IZRBYZ

JM2 I J-IZR

JM3 I JM2-IZRBY2

IF (NROD .EQ. 0) GO TO 128

 

C***
C***

Odd number of unknowns :

 

C***

117

118

119

120

121

122
123

124

125
126

IF (12R .NE. 1) GO To 118
D0 117 II1,MR
3(1) - FISTAG*Q(I,J)
Q(I,J) I Q(I,JMZ)
CONTINUE
GO TO 126
D0 119 II1,MR
3(1) I Q(IIJ)+05*(Q(IIJM2)-Q(IIJM1)-Q(IIJM3))
CONTINUE
IF (NRODPR .NE. 0) GO To 121
DC 120 II1,MR
II - IP+I
Q(I,J) I Q(I,JM2)+P(II)
CONTINUE
IP - IP-MR
GO To 123
CONTINUE
DO 122 II1,MR
Q(IoJ) I Q(I,J)-Q(I'JM1)+Q(I,JM2)
CONTINUE
IF (LR .EQ. 0) GO TO 124
CALL COSGEN (LR,1,0.5,FDEN,TCOS(KR+1))
GO To 126
CONTINUE
DO 125 II1,MR
B(I) - FISTAG*B(I)
CONTINUE
CONTINUE

 

 

 

127

128

C***
C***

272

CALL COSGEN (KR, 1 , 0 . 5, FDEN, TCOS)
CALL TRIX (KR,LR,MR,A,BB,C,B,TCOS,D,W)
DO 127 II1,MR
Q(IIJ) - Q(IIJ)+B(I)
CONTINUE
KR I KR+I2R
GO TO 151
CONTINUE

 

C***
C***

Even number of unknowns :

 

C***

129

130

131

132

133

134

135

136

137

138

139

JPl I J+IZRBY2
JP2 I J+I2R
IF (I2R .NE. 1) GO TO 135
D0 129 IIl,MR
B(I) I Q(I,J)
CONTINUE
CALL TRIX (1,0,MR,A,BB,C,B,TCOS,D,W)
IP I 0
IPSTOR I MR
GO TO (133,130),ISTAG
DO 131 II1,MR
P(I) I B(I)
B(I) I B(I)+Q(I,N)
CONTINUE
TCOS(1) I 1.
TCOS(Z) I 0.
CALL TRIX (1,1,MR,A,BB,C,B,TCOS,D,W)
DO 132 II1,MR
Q(I,J) I Q(I,JM2)+P(I)+B(I)
CONTINUE
GO TO 150
CONTINUE
DO 134 II1,MR
P(I) I B(I)
Q(I,J) I Q(I,JM2)+2.*Q(I,JP2)+3.*B(I)
CONTINUE
GO TO 150
CONTINUE
DO 136 II1,MR
B(I) I Q(I,J)+.5*(Q(I,JM2)-Q(I,JM1)-Q(I,JM3))
CONTINUE
IF (NRODPR .NE. 0) GO TO 138
D0 137 II1,MR
II I IP+I
B(I) I 3(I)+P(II)
CONTINUE
GO TO 140
CONTINUE
DO 139 II1,MR
B(I) I B(I)+Q(I,JP2)-Q(I,JP1)
CONTINUE

140

141

142

143
144

145

146

147
148

149
150

151

152

153

154

155

C***

273

CONTINUE
CALL TRIX (12R,o,MR,A,EE,C,E,TCOS,D,W)
IP - IP+MR
IPSTOR - MAXO(IPSTOR,IP+MR)
DO 141 II1,MR
II - IP+I
P(II) - B(I)+.5*(Q(I,J)-Q(I,JMI)-Q(I.JP1))
B(I) - P(II)+Q(I,JPZ)
CONTINUE
IE (LR .EQ. 0) GO To 142
CALL COSGEN (LR,1,0.5,FDEN,TCOS(IZR+1))
CALL MERGE (TCOS,0,12R,IZR,LR,KR)
GO To 144
DO 143 I-1,I2R
II - KR+I
TCOS(II) - TCOS(I)
CONTINUE
CALL COSGEN (KR,1,0.5,FDEN,TCOS)
IE (LR .NE. 0) GO To 145
GO To (146,145),I$TAG
CONTINUE
CALL TRIX (KR,KR,MR,A,33,C,3,TCOS,D,W)
GO To 148
CONTINUE
DO 147 II1,MR
B(I) - FISTAG*3(I)
CONTINUE
CONTINUE
DO 149 II1,MR
II - IP+I
Q(I,J) - Q(I,JM2)+P(II)+B(I)
CONTINUE
CONTINUE
LR - KR
KR - KR+JR
CONTINUE
GO To (152,153),MIXBND
NR - (NLAST-1)/JR+1
E (NR .LE. 3) GO To 155
GO To 154
NR - NLAST/JR
IE (NR .LE. 1) GO TO 192
I2R - JR
NRODPR - NROD
GO To 104
CONTINUE

H

 

c***
c***

Begin solution :

 

C***
C***

J I 1+JR
JMl I J-I2R
JPl I J+I2R

 

C***
C***

274

JM2 I NLAST-12R

IF (NR .EQ. 2) GO TO 184
IF (LR .NE. 0) GO TO 170
IF (N .NE. 3) GO TO 161

 

C***
C***

Case N-3

 

C***

156

157

158

159

160

C***

GO TO (156,168),ISTAG
CONTINUE
DO 157 II1,MR
B(I) I Q(IIZ)
CONTINUE
TCOS(1) I 0.
CALL TRIX (1,0,MR,A,BB,C,B,TCOS,D,W)
DO 158 II1,MR
Q(IIZ) I B(I)
B(I) I 4.*B(I)+Q(I,1)+2.*Q(I,3)
CONTINUE
TCOS(1) I -2.
TCOS(Z) I 2.
I1 I 2
I2 I 0
CALL TRIX (I1,I2,MR,A,BB,C,B,TCOS,D,W)
DO 159 II1,MR
Q(IIZ) ' Q(II2)+B(I)
B(I) I Q(I'1)+2.*Q(I:2)
CONTINUE
TCOS(1) I 0.
CALL TRIX (I,0,MR,A,BB,C,B,TCOS,D,W)
DO 160 II1,MR
Q(Ipl) I B(I)
CONTINUE
JR I 1
IZR I 0
GO TO 194

 

C***
c***
c***

Case N - 2**P+1

 

C***

161

162

163

164

CONTINUE
GO TO (162,170),ISTAG
CONTINUE
DO 163 II1,MR
3(1) I Q(IIJ)+-5*Q(II1)_Q(IIJM1)+Q(IINLAST)-Q(IIJM2)
CONTINUE
CALL COSGEN (JR,1,0.5,0.0,TCOS)
CALL TRIX (JR,0,MR,A,BB,C,B,TCOS,D,W)
DO 164 II1,MR
Q(IIJ) ' 05*(Q(IIJ)-Q(IIJM1)-Q(IIJP1))+B(I)
B(I) I Q(I,I)+2.*Q(I,NLAST)+4.*Q(I,J)
CONTINUE

 

 

 

165

166

167

C***
C***

275

JR2 I 2*JR
CALL COSGEN (JR, 1, 0.0, 0.0,TCOS)
DO 165 II1,JR
11 I JR+I
12 I JR+1II
TCOS(II) I -TCOS(I2)
CONTINUE
CALL TRIX (JR2,0,MR,A,BB,C,B,TCOS,D,W)
DO 166 II1,MR
Q(I,J) I Q(I,J)+B(I)
B(I) I Q(I,1)+2.*Q(I,J)
CONTINUE
CALL COSGEN (JR,1,0.5,0.0,TCOS)
CALL TRIX (JR,0,MR,A,BB,C,B,TCOS,D,W)
DO 167 II1,MR
Q(I,1) I .5*Q(I,1)-Q(I,JM1)+B(I)
CONTINUE
GO TO 194

 

C***
C***

Case of general N with NR I 3

 

C***

168

169

170

171

172

173

174
175

176
177

DO 169 II1,MR
B(I) I Q(IIZ)
Q(IIZ) - 0.
32(I) I Q(I,3)
33(I) I Q(Irl)
CONTINUE
JR I 1
I2R I 0
J I 2
GO TO 177
CONTINUE
DO 171 II1,MR
B(I) I 05*Q(Ill)-Q(IIJM1)+Q(IIJ)
CONTINUE
IF (NROD .NE. 0) GO TO 173
DO 172 II1,MR
II I IP+I
B(I) I B(I)+P(II)
CONTINUE
GO TO 175
D0 174 II1,MR
B(I) I B(I)+Q(I,NLAST)‘Q(I,JM2)
CONTINUE
CONTINUE
DO 176 II1,MR
T I .5*(Q(IIJ)-Q(IIJM1)“Q(I,JPl))
Q(I,J) - T
82(I) I Q(I,NLAST)+T
B3(I) I Q(I,1)+2.*T
CONTINUE
CONTINUE

 

 

178

179

180

181

182

183

184

C***

K1 I
K2 I

276

KR+2*JR-l
KR+JR

TCOS(K1+1) I I2.

K4 I
CALL
K4 I
CALL
CALL
K3 I
CALL
K4 I
CALL
CALL

IF (LR .EQ.

CALL
CALL
CALL
K3 I
K4 I
CALL

K1+3-ISTAG

COSGEN (K2+ISTAG-2,1,0.0,ENUM,TCOS(K4))
K1+K2+1

COSGEN (JR-1,1,0.0,1.0,TCOS(K4))

MERGE (TCOS,K1,K2,K1+K2,JR-1,0)
K1+K2+LR

COSGEN (JR,1,0.5,0.0,TCOS(K3+1))
K3+JR+1

COSGEN (KR,1,0.5,FDEN,TCOS(K4))

MERGE (TCOS,K3,JR,K3+JR,KR,K1)

0) GO TO 178

COSGEN (LR,1,0.S,EDEN,TCOS(K4))

MERGE (TCOS,K3,JR,K3+JR,LR,K3-LR)
COSGEN (KR,1,0.5,FDEN,TCOS(K4))

KR

KR

TRI3 (MR,A,BB,C,K,3,32,33,TCOS,D,W,W2,W3)

DO 179 II1,MR

B(I) I B(I)+32(I)+B3(I)
CONTINUE
TCOS(1) I 2.

CALL

TRIX (1,0,MR,A,BB,C,B,TCOS,D,W)

DO 180 II1,MR
Q(I,J) - Q(I,J)+B(I)
B(I) - Q(I,1)+2.*Q(I.J)
CONTINUE

CALL
CALL

IF (JR .NE.

COSGEN (JR,1,0.5,0.0,TCOS)
TRIX (JR, 0,MR,A,BB,C, B, TCOS,D,W)
1) GO TO 182

DO 181 II1,MR
Q(I,1) I B(I)

CONTINUE

GO TO 194

CONTINUE

DO 183 II1,MR

Q(Ill) I

.5*Q(I,1)-Q(I,JM1)+B(I)

CONTINUE
GO TO 194
CONTINUE
IF (N .NE. 2) GO TO 188

 

C***
C***
C***

Case

N I 2

 

C***

185

D0 185 II1,MR
3(I) I Q(I,1)

CONTINUE

TCOS(1) I 0.

CALL

TRIX (1,0,MR,A,BB,C,B,TCOS,D,W)

DO 186 II1,MR

186

187

188

c***
C***

277

Q(Icl) I B(I)
B(I) I 2.*(Q(I,2)+B(I))*FISTAG
CONTINUE
TCOS(1) I IFISTAG
TCOS(Z) I 2.
CALL TRIX (2,0,MR,A,BB,C,B,TCOS,D,W)
DO 187 II1,MR
Q(Ill) " Q(I11)+B(I)
CONTINUE
JR I 1
IZR I 0
GO TO 194
CONTINUE

 

C***
C***

Case of general N and NR I 2

 

C***

189

190

191

192

193

194

C***

DO 189 II1,MR
II I IP+I
33(1) I 0.
B(I) I Q(I,1)+2.*P(II)
Q(II 1) a .5*Q(II1)-Q(IIJM1)
32(I) I 2.*(Q(I,1)+Q(I,NLAST))
CONTINUE
K1 I KR+JRI1
TCOS(K1+1) I I2.
K4 I K1+3IISTAG
CALL COSGEN (KR+ISTAG-2,1,0.0,FNUM,TCOS(K4))
K4 - K1+KR+1
CALL COSGEN (JR-1,1,0.0,1.0,TCOS(K4))
CALL MERGE (TCOS,K1,KR,K1+KR,JR-1,0)
CALL COSGEN (KR,1,0.5,FDEN,TCOS(K1+1))
K2 I KR
K4 I K1+K2+1
CALL COSGEN (LR,1,0.5,FDEN,TCOS(K4))
K3 I LR
K4 I 0
CALL TRI3 (MR,A,BB,C,K,3,32,B3,TCOS,D,W,W2,W3)
DO 190 II1,MR
B(I) I B(I)+32(I)
CONTINUE
TCOS(1) I 2.
CALL TRIX (1,0,MR,A,BB,C,B,TCOS,D,W)
DO 191 II1,MR
Q(I,1) I Q(I'1)+B(I)
CONTINUE
GO TO 194
DO 193 II1,MR
B(I) I Q(I,NLAST)
CONTINUE
GO TO 196
CONTINUE

 

 

 

C***

278

 

C***
C***

Start back substitution :

 

C***

195
196

197

198

199

200

201
202

203
204

205

206

207

208
209

J I NLAST-JR
DO 195 II1,MR
B(I) I Q(I,NLAST)+Q(I,J)
CONTINUE
JM2 I NLAST-IZR
IF (JR .NE. 1) GO TO 198
DO 197 II1,MR
Q(I,NLAST) I 0.
CONTINUE
GO TO 202
CONTINUE
IF (NROD .NE. 0) GO TO 200
D0 199 II1,MR
II I IP+I
Q(I,NLAST) I P(II)
CONTINUE
IP I IP-MR
GO TO 202
DO 201 II1,MR
Q(I,NLAST) I Q(I,NLAST)IQ(I,JM2)
CONTINUE
CONTINUE
CALL COSGEN (KR,1,0.5,FDEN,TCOS)
CALL COSGEN (LR,1,0.5,FDEN,TCOS(KR+1))
IF (LR .NE. 0) GO TO 204
DO 203 II1,MR
B(I) I FISTAG*B(I)
CONTINUE
CONTINUE
CALL TRIX (KR,LR,MR,A,BB,C,B,TCOS,D,W)
DO 205 II1,MR
Q(I,NLAST) I Q(I,NLAST)+B(I)
CONTINUE
NLASTP I NLAST
CONTINUE
JSTEP I JR
JR I I2R
I2R I I2R/2
IF (JR .EQ. 0) GO TO 222
GO TO (207,208),MIXBND
JSTART I 1+JR
GO TO 209
JSTART I JR
CONTINUE
KR I KR-JR
IF (NLAST+JR .GT. N) GO TO 210
KR I KR-JR
NLAST I NLAST+JR
JSTOP I NLAST-JSTEP
GO TO 211

210

211

212

213

214
215

216

217

218
219

220
221

222

C***
C***

279

CONTINUE
JSTOP I NLAST-JR
CONTINUE
LR I KRIJR
CALL COSGEN (JR,1,0.5,0.0,TCOS)
DO 221 JIJSTART,JSTOP,JSTEP
JM2 I J-JR
JPZ I J+JR
IF (J .NE. JR) GO TO 213
DO 212 III,MR
B(I) I Q(IIJ)+Q(IIJP2)
CONTINUE
GO TO 215
CONTINUE
DO 214 II1,MR
3(1) I Q(IIJ)+Q(IIJM2)+Q(IIJP2)
CONTINUE
CONTINUE
IF (JR .NE. 1) GO TO 217
DO 216 II1,MR
Q(I,J) I 0.
CONTINUE
GO TO 219
CONTINUE
JMI I JIIZR
JPI I J+I2R
DO 218 II1,MR
Q(IIJ) I .5*(Q(I,J)-Q(I,JMl)-Q(I,JP1))
CONTINUE
CONTINUE
CALL TRIX (JR,0,MR,A,BB,C,B,TCOS,D,W)
DO 220 II1,MR
Q(IIJ) I Q(IIJ)+B(I)
CONTINUE
CONTINUE
NROD I 1
IF (NLAST+I2R .LE. N) NROD I 0
IF (NLAST? .NE. NLAST) GO TO 194
GO TO 206
CONTINUE

 

C***
C***

Return storage requirements for P vectors

 

C***

C***
C***

W(I) I IPSTOR
RETURN
END

C***********************************************************************

SUBROUTINE POISP2 (M,N,A,BB,C,Q,IDIMQ,B,B2,B3,W,W2,W3,D,TCOS,P)

C***********************************************************************

C***

**

280

C*** SUBROUTINE TO SOLVE POISSON EQUATION WITH PERIODIC BOUNDARY **
C*** CONDITIONS. **
C*** **

C***********************************************************************

C***
DIMENSION A(1) ,33(1) ,C(1) ,Q(IDIMQ,1) ,
1 8(1) ,BZ(1) .83<1) .W(1) ,
2 w2(1) ,w3(1) ,D(1) ,TCOS(1) ,
3 9(1)

C***

C***********************************************************************

 

 

C***

MR I M

NR - (N+1)/2

NRMl I NR-l

IF (2*NR .NE. N) GO To 107
C***
C***
C*** Even number of unknowns :
C***
C***

DO 102 JI1,NRM1
NRMJ I NRIJ
NRPJ I NR+J
DO 101 II1,MR
5 I Q(IINRMJ)-Q(IINRPJ)
T I Q(I,NRMJ)+Q(I,NRPJ)
Q(I,NRMJ) I S
Q(I,NRPJ) I T
101 CONTINUE
102 CONTINUE
DO 103 II1,MR
Q(IINR) I 2.*Q(I:NR)
Q(I,N) I 2.*Q(I,N)
103 CONTINUE
CALL POISD2 (MR,NRM1,1,A,BB,C,Q,IDIMQ,B,W,D,TCOS,P)
IPSTOR I W(l)
CALL POISN2 (MR,NR+1,1,1,A,BB,C,Q(1,NR),IDIMQ,B,BZ,B3,W,W2,W3,D,
l TCOS,P)
IPSTOR I MAXO(IPSTOR,INT(W(1)))
DO 105 JI1,NRM1
NRMJ I NRIJ
NRPJ I NR+J
DO 104 II1,MR
S I .5*(Q(I,NRPJ)+Q(I,NRMJ))
T I .5* (Q(I,NRPJ)'Q(I,NRMJ))
Q(IINRMJ) - S
Q(IINRPJ) I T
104 CONTINUE
105 CONTINUE
DO 106 II1,MR
Q(I,NR) I .5*Q(I,NR)
Q(IpN) I 05*Q(IIN)
105 CONTINUE

281

 

 

GO TO 118
107 CONTINUE
C***
C***
C*** Odd number Of unknowns :
C***
C***
DO 109 J-1,NRM1
NRPJ - N+1-J
DO 108 I-1,MR
S - Q(I,J)-Q(I,NRPJ)
T I Q(I,J)+Q(I,NRPJ)
Q(IIJ) ' S
108 CONTINUE

109 CONTINUE
DO 110 I-1,MR
Q(I.NR) = 2.*Q(I,NR)
110 CONTINUE
LH - NRMl/Z
DO 112 J-1,LH
NRMJ - NR-J
DO 111 I-1,MR
S ‘ Q(IIJ)
Q(I,J) - Q(IpNRMJ)
Q(IINRMJ) a S
111 CONTINUE
112 CONTINUE
CALL POISDZ (MR,NRM1,2,A,BB,C,Q,IDIMQ,B,W,D,TCOS,P)
IPSTOR - W(l)
CALL POISN2 (MR,NR,2,1,A,BB,C,Q(1,NR),IDIMQ,B,82,BB,W,W2,W3,D,
1 TCOS,P)
IPSTOR - MAXO(IPSTOR,INT(W(1)))
DO 114 J-1,NRM1
NRPJ - NR+J
DO 113 I-1,MR
S - .S*(Q(I,NRPJ)+Q(I,J))
T - .5*(Q(I,NRPJ)-Q(I.J))
Q(I,NRPJ) - T
Q(IIJ) ' S
113 CONTINUE
114 CONTINUE
DO 115 I-1,MR
Q(I,NR) - .5*Q(I,NR)
115 CONTINUE
DO 117 J-1,LH
NRMJ - NR-J
DO 116 I-1,MR
S - Q(I,J)
Q(I,J) - Q(IINRMJ)
Q(IINRMJ) ' S
116 CONTINUE
117 CONTINUE
118 CONTINUE

282

 

 

C***
C***
C*** Return storage requirements for P vectors :
C***
C***
W(l) I IPSTOR
RETURN
END
C***
C***

C***********************************************************************

SUBROUTINE TRIX(IDEGBR,IDEGCR,M,A,B,C,Y,TCOS,D,W)

C***********************************************************************

C*** **
C*** SUBROUTINE TO SOLVE A SYSTEM OF LINEAR EQUATIONS WHERE THE **
C*** COEFFICIENT MATRIX IS A RATIONAL FUNCTION IN THE MATRIX GIVEN BY **
C*** TRIDIAGONAL ( . . . , A(I), B(I), C(I), . . . ). **
C*** **

C***********************************************************************
C***

DIMENSION A(1) ,B(1) ,C(1) ,Y(l) ,

l TCOS(1) ,D(1) ,W(1)

C***
C***********************************************************************
C***
MMl I M-l
EB I IDEGBR+1
PC I IDEGCR+1
L I FB/FC
LINT I 1
DO 108 KI1,IDEGBR
X I TCOS(K)
IF (K .NE. L) GO TO 102
I I IDEGBR+LINT
XX I X-TCOS(I)
DO 101 II1,M
W(I) I Y(I)
Y(I) I XX*Y(I)
101 CONTINUE
102 CONTINUE
Z I 1./(B(1)-X)
0(1) I C(1)*Z
Y(1) I Y(1)*Z
DO 103 II2,MM1
Z I 1./(B(I)-X-A(I)*D(I-l))
9(1) I C(I)*Z
'Y(I) I (Y(I)-A(I)*Y(I-l))*Z
103 CONTINUE
z I B(M)—X-A(M)*D(MM1)
IF (2 .NE. 0.) GO TO 104
Y(M) I 0.
GO TO 105
104 Y(M) I (Y(M)-A(M)*Y(MM1))/Z
105 CONTINUE

283
DC 106 IP=1,MM1

I I M-IP
Y(I) I Y(I)-D(I)*Y(I+1)
106 CONTINUE

IF (K .NE. L) GO To 108
D0 107 II1,M
Y(I) - Y(I)+W(I)

107 CONTINUE

LINT - LINT+1

L - (FLOAT(LINT)*FB)/FC
108 CONTINUE

RETURN

END
C***

C***
C***********************************************************************
SUBROUTINE TRI3 (M,A,B,C,K,Y1,Y2,Y3,TCOS,D,W1,W2,W3)

C***********************************************************************

C*** **
C*** SUBROUTINE TO SOLVE THREE LINEAR SYSTEMS WHOSE COMMON COEFFICIENT**
C*** MATRIX IS A RATIONAL FUNCTION IN THE MATRIX GIVEN BY **
C*** TRIDIAGONAL (...,A(I),B(I),C(I),...) **
C*** **

C***********************************************************************

C***
DIMENSION A(1) ,B(l) ,C(1) ,K(4) ,
1 TCOS(1) ,Y1(l) ,Y2(l) ,Y3(l) ,
2 D(l) ,W1(1) ,W2(1) ,w3(1)

C***

C***********************************************************************

C***
MMl I M-l
K1 I X(1)
K2 I X(2)
K3 I K(3)
K4 I K(4)
F1 I Kl+1
F2 I K2+1
F3 I K3+1
F4 I K4+1

K2K3K4 I K2+K3+K4
IF (K2K3K4 .EQ. 0) GO TO 101

L1 I Fl/FZ

L2 I F1/F3

L3 I F1/F4

LINTl I 1

LINTZ I 1

LINT3 I 1

KINTl I K1

KINTZ I KINT1+K2
KINT3 I KINT2+K3

101 CONTINUE
DO 115 NI1,K1
X I TCOS(N)

102
103

104
105

106
107

108

109

110

111

112

113

284

IF (K2K3K4 .EQ. 0) GO TO 107
IF (N .NE. L1) GO To 103
D0 102 1=1,M
w1(1) = 21(1)
CONTINUE
IF (N .NE. L2) GO TO 105
DO 104 1:1,M
w2(1) a 22(1)
CONTINUE
1E (N .NE. L3) GO TO 107
D0 106 II1,M
W3(I) s 23(1)
CONTINUE
CONTINUE
z - 1./(B(1)-X)
0(1) - C(1)*Z
21(1) - 21(1)*z
22(1) = 22(1)*z
23(1) = 23(1)*z
DO 108 I=2,M
z - 1./(B(I)-X-A(I)*D(I-l))
0(1) = C(I)*Z
21(1) = (Y1(I)-A(I)*Y1(I-1))*Z
22(1) = (Y2(I)-A(I)*Y2(I-l))*z
23(1) - (Y3(I)-A(I)*Y3(I-1))*Z
CONTINUE
DO 109 IPI1,MM1
I - M-IP
21(1) = 21(1)-D(I)*21(1+1)
22(1) - 22(1)-D(1)*22(1+1)
23(1) - Y3(I)-D(I)*Y3(I+l)
CONTINUE
IF (K2K3K4 .EQ. 0) GO TO 115
IF (N .NE. L1) GO TO 111
1 - LINT1+KINT1
xx - x-TCOS(1)
DO 110 II1,M
21(1) - xx*21(1)+w1(1)
CONTINUE
L1NT1 - L1NT1+1
L1 - (FLOAT(LINT1)*F1)/F2
IF (N .NE. L2) GO TO 113
1 - LINT2+KINT2
xx - X-TCOS(I)
DO 112 1-1,M
22(1) - xx*22(1)+w2(1)
CONTINUE
LINTZ - LINT2+1
L2 - (FLOAT(LINT2)*F1)/F3
IF (N .NE. L3) GO TO 115
1 - LINT3+KINT3
xx - x-TCOS(1)
DO 114 1=1,M
23(1) - xx*23(1)+w3(1)

 

 

 

285

114 CONTINUE
LINT3 I LINT3+1
L3 I (FLOAT(LINT3)*F1)/F4
115 CONTINUE
RETURN
END

 

 

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