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LAMINAR FLOW SEPARATION
IN
A CONSTRICTED CHANNEL

BY

Najdat Nashat Abdulla

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1987

ABSTRACT

LAMINAR FLOW SEPARATION IN A CONSTRICTED CHANNEL

By
Najdat Nashat Abdulla

The two-dimensional, incompressible, laminar flow in the
enxtrance region of a channel with and without constrictions (in the
form of forward, backward and finite steps) has been analyzed
runnerically for various step-to-channel height ratios, step lengths

axui step positions for Reynolds numbers up to 2000, based on the

Channel height.

A stream function-vorticity formulation is used in conjunction

xvith a finite-difference, over-relaxation method utilizing

accelerating parameters to solve the full Navier-Stokes equations

'which.describe the steady flow. The power of the method is contained

in the structure of the finite-difference equations, which, for all

Reynolds numbers, yields a diagonally dominant system of linear,

algebraic equations. This avoids the numerical instability of the

finite-difference equations at high Reynolds numbers.

The stream function, vorticity, streamwise velocity and

pressure are reported at each grid point. The inviscid-core region

and profile-development region, which form the entrance length, are
identified for various Reynolds numbers and inlet velocity profiles.
In addition, separation and reattachment points are obtained for

various step-to-channel height ratios, step lengths and positions for

‘flm constricted channel. Furthermore, the convergence domain for the

mmcessive over-relaxation method and the optimum values of

u-

1..

 

 

v...

hi

 

 

accelerating parameters, which minimize the computing time, are
obtained.

The centerline velocity and entrance length for the channel
without a constriction are compared with the results obtained by
approximate techniques. Also, the separation and reattachment points

for a constricted channel are compared with both numerical and

experimental results.

A DEDICATION

This thesis is dedicated to my wife,
Najah, who has displayed great
patience and tolerance during the
preparation of this study and without
her encouragement and confidence this
work would not have been completed.
It is also dedicated to my son,

Zayd and daughter, Noora.

ii

.n

 

c .
.‘I
n.

 

 

 

ACKNOWLEDGMENT

The author wishes to express his sincere thanks to his major
professor Dr. Merle C. Potter for his valuable guidance throughout
this study. The author's graduate work has been made possible through
his patience, understanding and personal interest.

Special thanks to Dr. Richard W. Bartholomew for his
constructive comments about numerical technique used in this study.

Thankful acknowledgment is extended to Dr. John Lloyd,
chairman, Mechanical Engineering Department and to other members of
the Guidance Committee: Dr. Craig Somerton and Dr. Robert H.
Wasserman.

Thanks are also extended to Carol Bishop for her accuracy and

neatness in typing this dissertation.

iii

.1.

 

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . vi

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

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter

1. INTRODUCTION 1

1.1 Background . . . l

1.2 Entrance Region of a Channel . . . 2

1.3 Methods of Solving the Entrance Flow Problem . 2

1. 3.1 The Integral Method . . . . 2

1. 3. 2 Axially Patched Solutions . 3

1.3.3 Linearization of the Momentum Equation 3

1. 3. 4 Finite Difference Methods . 4

1.4 Separated Flows . 6

1.5 Methods of Solving the Constricted Flow Problem 7

1. 5.1 Matched Asymptotic Expansions . . 7

1.5.2 Numerical Methods . . . . 9

1.5.3 Experimental Methods 13

1.6 Description of the Present Work 14

2. MATHEMATICAL FORMULATION 17

2.1 Primary Flow . . . . . . . . . . . . . . . . . . . . 17

2.2 Governing Equations . . . . . . . . . . . . . . . . 18

2.3 Boundary Conditions . . . . 20

2. 3.1 The Channel Entrance with no Constriction . . 20

2.3.2 The Channel Entrance with a Constriction . . 23

3. NUMERICAL METHODS . . . . . . . . . . . . . . . . . . . 25

3.1 Introduction . . . . . . . . . . . . . . . . . . 25

3.1.1 Direct Methods . . . . . . . . . . . . . . . 26

3.1.1.1 Cramer' 8 Rule . . . . . . . . . . . . . . 26

3.1.1. 2 Gaussion Elimination . . . . . . . . . . . 26

3.1.2 Iterative Methods . . . . . . . . . 27

3.1.2.1 Gauss- Seidel Iteration . . . 28

3.1. 2. 2 Successive Over- Relaxation Methods 29

3.2 Numerical Solutions . . . . . . . . . . . . . . . 30

iv

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v...

ba~

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L11

 

4. CONVERGENCE CRITERIA AND NAVIER-STOKES EQUATIONS .

4
4.
I4

LBNH

4.4

Introduction .

The Problem Under Consideration .

Convergence Conditions .

4.3.1 Sufficient Condition for Convergence of
the Successive Over- Relaxation Method

Acceleration Parameters .

5. RESULTS, DISCUSSION AND CONCLUSIONS

5.1
2

5.3

mm
U'IL‘

TABLES .

FIGURES

Background
Results for the Channel Entrance with no
Constriction

Results for the Channel Entrance with Constriction.

5.3.1 Forward Step .

5. 3. 2 Backward Step

5.3.3 Finite Step . .
Optimum Over- Relaxation and Weighting Factors
Conclusions . . . . . . . .

BIBLIOGRAPHY .

APPENDICES .

Appendix A:
Appendix B:
Appendix C:
Appendix D:

Computer Program for the Entrance Region
Computer Program for the Forward Step
Computer Program for the Backward Step
Computer Program for the Finite Step

Page
38

38
39
40
41
41
43

46
46

48
50
50
52
55
58
61
66

97
131
135
135
142

149
155

LIST OF TABLES

Table

1. Actual velocity profile values

2. Summary of entrance flow problems studied

3. Entrance length LE

4. Comparison of 2LE/H and LE/(HRe) with other researchers

5. The inviscid-core length, the profile-development length
and the entrance length for various Reynolds numbers

6. Vorticity values in the entrance of a straight channel,
Re-200

7. Summary of cases studied for a forward step

8. The Y-values for selected streamlines for the flow past

a forward step located in the profile—development
region, (Re)H-20

9. The Y-values for selected streamlines for the flow past
a forward step located in the profile-development
region, (Re)H-200

10. The Y-values for selected streamlines for the flow past
a forward step located in the profile-development
region, (Re)H-2000

11. Summary of cases studied for a backward step

12. The Y-values for selected streamlines for the flow past
a backward step located in the profile-development
region, (Re)H-20 . . .

13. The Y-values for selected streamlines for the flow past
a backward step located in the profile-development
region, (Re)H-200

14. The Y-values for selected streamlines for the flow past
a backward step located in the profile-development
region, (Re)H-2000 . . . . . . . .

15. Summary of cases studied for a finite step

vi

Page
66
67
68

69

70

71

72

73

74

75

. 76-77

78

80

. 81-82

 
 

Table Page

16. Effect of downstream length on the downstream
separation streamline Y- coordinate for the finite
step of height 0. 3H located in the fully- developed
region, (Re)H-20 . . . . . . . . . . . . . . . 83

17. Effect of downstream length on the downstream
separation streamline Y-coordinate for the finite
step of height 0.3H located in the inviscid-core
region, (Re)H-200 . . . . . . . . . . . . . . . . . . . . 83

18. Effect of step length on downstream separation
streamline Y-coordinate for the finite
step of height 0.3H located in the inviscid-core
region, (Re)H-20 . . . . . . . . . . . . . . . . . . . . 84

19. Effect of step position on downstream separation
streamline Y-coordinate for the finite
step of height 0.3H, (Re)H-20 . . . . . . . . . . . . . . 84

20. Effect of step position on downstream separation
streamline coordinates for the finite step of
height 0.3H, (Re)H-200 . . . . . . . . . . . . . . . . . 85

21. The Y-values for selected streamlines for the flow
past a finite step located in the profile-development
region, (Re) H—20 . . . . . . . . . . . . . . . . . . . . 86

22. The Y—values for selected streamlines for the flow
past a finite step located in the profile-development
region, (Re)H-200 . . . . . . . . . . . . . . . . . . . . 87

23. The Y-values for selected streamlines for the flow
past a finite step located in the profile-development
region, (Re)H-1300 . . . . . . . . . . . . . . . . . . . 88

24. Summary of cases studied for a single step . . . . . . . 89
25. The Y-values for selected streamlines for the flow

past a single step located in the inviscid-core

region, (Re)H-20 . . . . . . . . . . . . . . . . . . . . 90

26. The Y-values for selected streamlines for the flow
past a single step located in the inviscid-core

region, (Re)H-200 . . . . . . . . . . . . . . . . . . . . 91
27. Relaxation factor (FS) vs. computing time, h-l/15 . . . . 92
28. Relaxation factor (FS) vs. computing time, h-1/20 . . . . 92
29. Relaxation factor (FV) vs. computing time, h-l/lS . . . . 93
30. Relaxation factor (FV) vs. computing time, h-1/20 . . . . 93
31. Weighting factor (KS) vs. computing time, h=1/15 . . . . 94

vii

i.

(J.

(to

Table

32.
33.
34.

35.

Weighting factor (KS) vs. computing time, h-l/20
Weighting factor (KV) vs. computing time, h-1/15
Weighting factor (KV) vs. computing time, h-1/20

Factors of reduction in the computing time

viii

Page
94
95
95

96

“L-.!
v

‘(

IJ

Figure

10.

11.

12.

13.

14.

15.

LIST OF FIGURES

The two-dimensional channel

Forward and backward steps with notations

Finite step with notations

Finite difference representation of basic equations
Flow development in the inlet region of a straight
channel with uniform inlet velocity profile,
(Re)H-20 . . . .

Flow development in the inlet region of a straight
channel with uniform inlet velocity profile,
(Re)H-200 . . . . . .

Flow development in the inlet region of a straight
channel with uniform inlet velocity profile,
(Re)H-500 .

Flow development in the inlet region of a straight
channel with uniform inlet velocity profile,
(Re)H-2000 . . . . .
Comparison of centerline velocities

The streamwise pressure gradient, (Re)H-20

The streamwise pressure gradient, (Re)H-200

Flow development in the inlet region of a straight
channel with an actual inlet velocity profile,
(Re)H-20

Flow development in the inlet region of a straight
channel with an actual inlet velocity profile,
(Re)H-500 . .
Streamlines in the vicinity of the forward step
located in the profile-development region,
(Re)H-20 . . .

Streamlines in the vicinity of the forward step
located in the profile-development region, (Re)H-200

ix

Page
97
97
98

98

99

99

100

100

101

102

102

103

103

104

104

(ll

‘11

 

Figure

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

Streamlines in the vicinity of the forward step
located in the profile-development region,
(Re)H-2000 . . . . . . . . . .

Effect of forward step position on separation region,
(Re)H-200

Effect of forward step position on separation region,
(Re)H -2000 . .

Effect of forward step height on separation region,
(Re)H -200 . . . . . . .
Effect of forward step height on separation region,
(Re)H -1000 . . . . . . .

Effect of forward step height on
reattachment points, (Re)H-200

separation and

Effect of forward step height on
reattachment points, (Re)H-1000

separation and

Separation and reattachment points of a forward step
for various Reynolds numbers

Comparison of separation point for a forward step with
numerical results .

Comparison of reattachment point for a forward step with
numerical results

Effect
a-0.2H

of backward step position on separation region,

and (Re) “-20

Effect
a-O.3H

of backward step position on separation region,

and (Re) H—20

Effect
a-0.4H

of backward step position
and (Re)H-20

on separation region,

Effect
a-0.4H

of backward step position on separation region,

and (Re) H-200

Effect
a-O.ZH

of backward step position
and (Re)H-2000

on separation region,

Streamlines in the vicinity of the
in the profile-development region,

backward step located
(Re)H-20

Streamlines in the vicinity of the
in the profile-development region,

backward step located
(Re)H-200

Page

104

105

105

106

106

107

107

108

108

109

109

110

110

111

111

112

112

 

 

Figure

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

Streamlines in the vicinity of the backward step located
in the profile-development region, (Re)H-2000

Effect of backward step height on separation region,
(Re)H -20 . . . . . . . . . . . . . . . .
Effect of backward step height on separation region,

(Re) H-200 . . . . . . . . . . . . . . . .

Effect of backward step height on separation region,
(Re)H-500 . . . . . . . . . .

Separation point vs. backward step height for different
Reynolds numbers . . . . . . . . . . . . .

Reattachment point vs. backward step height for different
Reynolds numbers . . . . . . . . . .

Separation and reattachment points of backward step for
various Reynolds numbers, a-0.3H

Separation and reattachment points of backward step for
various Reynolds numbers, a-0.2H

Variation of separation point for backward step with
Reynolds number .

Comparison of reattachment point for backward step with
theoretical results for Reynolds numbers in the
range of (4-100) . . .

Comparison of reattachment point for backward step with
experimental results for Reynolds numbers in the
range of (100-400)

Effect of finite step length on downstream separation
region, (Re)H-200 . . . . . .

Effect of step position on downstream separation region,
(Re)H-500

Streamlines in the vicinity of the finite step located
in the profile-development region, (Re)H-20

Streamlines in the vicinity of the finite step located
in the profile-development region, (Re)H-200

Streamlines in the vicinity of the finite step located
in the profile-development region, (Re)H-1300

Effect of step height on downstream separation region,
(Re)H-20

Effect of step height on downstream separation region,
(Re)H-200 . . .
xi

Page

112

113

113

114

114

115

115

116

116

117

117

118

118

119

119

119

120

120

.\-

Figure Page

51. Downstream separation point vs. step height for various

Reynolds numbers . . . . . . . . . . . . . . . . . . . 121
52. Downstream reattachment point vs. step height for various

Reynolds numbers . . . . . . . . . . . . . . . . . . . 121
53. Effect of Reynolds number on downstream separation region,

a-0.3H . . . . . . . . . . . . . . . . . . . . . . . . 122
54. Downstream reattachment point vs. Reynolds number . . . 122
55. Effect of single step position on separation region,

(Re)H-20 . . . . . . . . . . . . . . . . . . . . . . . . 123
56. Effect of single step position on downstream separation

region, (Re)H-200 . . . . . . . . . . . . . . . . . . . 123
57. Streamlines in the vicinity of the single step located

in the inviscid-core region, (Re)H-20 . . . . . . . . . 124
58. Streamlines in the vicinity of the single step located

in the inviscid-core region, (Re)H-200 . . . . . . . . . 124
59. Effect of single step height on downstream separation

region, (Re)H-200 . . . . . . . . . . . . . . . . . . . 125
60. Effect of Reynolds number on downstream separation

region, a-0.3H . . . . . . . . . . . . . . . . . . . . . 125
61. Effect of single step height on downstream reattachment

point, (Re)H-200 . . . . . . . . . . . . . . . . . . . . 126
62. Downstream reattachment point of a single step vs. Reynolds

number . . . . . . 126
63. Comparison of downstream reattachment points for a

finite and a single step with numerical results . . . . 127
64. Comparison of downstream separation points for different

steps . . . . . . . . . . . . . . . . . . . . . . . . . 127
65. Comparison of downstream reattachment points for different

steps . . . . . . . . . . . . . . . . . . . . . . . . . 128
66. Optimum accelerating parameters vs. Reynolds numbers,

h-1/15 . . . . . . . . . . . . . . . . . . . . . . . . . 129
67. Optimum accelerating parameters vs. Reynolds numbers,

h-1/20 . . . . . . . . . . . . . . . . . . . . . . . . . 129
68. Optimum over-relaxation factors vs. Reynolds numbers . . 130
69. Optimum weighting factors vs. Reynolds numbers . . . . . 130

xii

.0\.

u. '\ ‘

ll

“
s IH

.-
\-

FS

Xs

NOMENCLATURE

Step height

Dimensionless vertical coordinate

Optimum value of the over-relaxation factor for stream
function

Optimum value of the over-relaxation factor for vorticity
Gravity

Mesh size equal in both X- and Y-directions
Channel height

Number of iterations

Weighting factor for the stream function
Weighting factor for the vorticity

Step position

Pressure

Dimensionless kinematic pressure

Reynolds number based on the channel height
Reynolds number based on the step height
Dimensionless streamwise velocity component
Dimensionless centerline velocity

Average uniform velocity at channel inlet
Dimensionless normal velocity component
Streamwise coordinate

Dimensionless streamwise coordinate (X/H)

Upstream separation point

xiii

I‘.
.c n

Xr

Ys

Yr

Downstream reattachment point

Normal coordinate

Dimensionless normal coordinate (Y/H)

Downstream separation point

Upstream reattachment point

Dimensionless stream function

Vorticity

Fluid density

Fluid kinematic viscosity

Difference between two values of stream function or

vorticity

xiv

 

v-.

.3;

 

 

CHAPTER 1

INTRODUCTION

1.1 W

The study of incompressible fluid flow through an entrance
region of a pipe or a duct and through constricted channels is of
considerable practical significance. The applications of such flows
are quite numerous; they include fluid flows found in physiology (flow
through blood vessels and lung airways, flow separation due to build-
up of deposits on artery walls, and measurements of blood pressure
using a cuff on the arm), and in machinery (flow in the vicinity of
junctions and valves).

The Navier-Stokes equations, which are considered to describe
the fluid motion of interest, are nonlinear. Because of this
nonlinearity, some difficulties have arisen in numerical as well as in
analytical studies. One of the greatest difficulties with the
numerical studies is the problem of divergence of the iterative
methods at high Reynolds numbers. Since an analytical solution of the
actual problem is extremely difficult, if not impossible, a number of
assumptions together with a numerical solution may be employed to
obtain approximate results.

Since the pioneering work of Prandtl early in this century,
boundary-layer theory has provided the principal basis for the

theoretical analysis of laminar flow phenomena near solid boundaries.

 

 

 

 

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-

 

 

 

2
It is now possible to conduct a more rigorous analysis of laminar
flow; the development of high-speed computers and sophisticated
numerical techniques permit the solution of the complete set of field

equations describing a particular fluid motion.

1.2 Ent ance e ion 0 a Channe

In the entrance region of a channel, our primary concern is
with changes in the streamwise velocity component. The entrance
region extends a considerable distance downstream and may be quite
significant in high Reynolds number flows. It may take up to 100 gap
widths before a fully developed flow is produced. So, in any study of
a high Reynolds number channel flow, the assumption of a fully
developed velocity profile implicitly assumes a substantial length of
entrance flow that must be accounted for. Many channels, ducts, or
pipes are not sufficiently long to allow developed flow to occur. A
variety of methods have been employed for the determination of the
flow characteristics in the entrance region as reported in the large

number of references in the literature.

1.3 Methods of Solving the Entrance Flow Problem

In general, four different methods have been applied to solve

the entrance flow. These methods will be outlined in this section.

1.3.1 The Integral Method

An early analysis of the entrance region in a tube was
presented by Schiller [l]. The entrance region was considered to be
composed of two zones: a boundary layer developing on the wall and an
inviscid core. The core flow terminated as the boundary layers merged

resulting in a fully-developed parabolic profile. Subsequent

 

 

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s
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3
modifications to this integral method have been presented elsewhere
[2,3].

Mohanty and Asthano [4] investigated the flow in the entrance
region of a pipe. They solved the boundary-layer equations in the
inlet region and the Navier-Stokes equations, with order—of-magnitude
analysis, in the "filled region" using a fourth-degree velocity
profile. This work was the first to recognize that the core region
terminated with a non-parabolic profile; a "filled region" separated

the core region from the developed flow region.

1.3.2 Axially Patched Solutions
In this method, initially used by Schlichting [5,6], the

entrance region is divided into two regions. Near the entrance a
boundary layer model is used and an approximate solution is obtained
in terms of a perturbation of the Blasius boundary layer solution. In
the region where the flow is nearly fully developed, the velocity
profile is approximated in terms of a small perturbation to the fully
developed parabolic profile. The two solutions are then matched at
some approximate streamwise location.

Van Dyke [7] improved Schlichting's solution near the entrance
by an upstream expansion whose first approximation is the leading edge
solution for a semi-infinite plate, which had been presented by Davis
[8]. The displacement effect of the boundary layer on the inviscid

core is accounted for in this higher order approximation.

1.3.3 Linearigation 9f the Momentum Equation

The nonlinear inertia terms in the x-component Navier-Stokes
equations were linearized and the solution to the resulting equation

found in a method proposed by Langhaar [9]. Sparrow, et.al. [10], who

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followed the linearization method of Langhaar, solved both channel and
circular pipe flow.

Lundgren [ll] employed the linearized equations of motion to
predict the incremental pressure drop due to the entrance region for
ducts of arbitrary cross-section.

Morihara and Cheng [12] investigated the entrance flow in a
channel between semi-infinite parallel plates using a quasi-

linearization method.

Recent work by Du Plessis [13], who followed the linearization

method of Lungren [ll], solved a channel flow with an arbitrary inlet

velocity profile.

1.3.4 Finite Difference Methods

The Navier-Stokes equations have been solved by finite
difference methods for flow inside circular pipes and for parallel
plate channels. In these solutions, the assumptions inherent in
boundary-layer theory have been used; that is, both the streamwise
velocity derivative azu/ax2 and the pressure gradient 6p/6y normal to
the plate have been neglected.

In a pipe flow, Christiansen and Lemmon [14] numerically
studied the flow in the entrance region of a circular pipe with a
‘uniform inlet velocity profile. They solved boundary layer equations
‘near the entrance and restricted the equations of motion in
cylindrical coordinates to conditions such that the flow is
iruiependent of time, the radial component of the equations of motion
is negligible, any angular motion is negligible, and the flow is
irudependent of any existing body force field far from the entrance.

Robert W. Hornbeck [15] analyzed the laminar flow of an

incompressible fluid in the inletof a pipe up to Reynolds number of

 

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5
0.9. He solved an approximate form of the governing differential
equations by neglecting the axial molecular transport of momentum.
This is accomplished numerically by means of a finite-difference
marching procedure in which the velocities and pressure at any axial
position in the pipe are determined by using values upstream from the
point.

In a channel flow, Hwang and Fan [16] investigated a laminar
magneto-hydrodynamic flow in the entrance region of a flat rectangular
duct. They assumed that the duct walls are electrically non-
conducting, with a uniform magnetic field imposed perpendicular to the
duct walls. They employed a finite—difference method to solve the
usual boundary-layer equations.

Bodia and Osterle [17] investigated the flow in the inlet
region of a straight channel. They used finite-difference techniques
to solve an approximate form of the governing differential equation by
neglecting the axial diffusion of vorticity.

Several publications have described the use of finite
difference methods to solve the full Navier-Stokes equations,
maintaining the axial transport of vorticity terms as well as the
pressure gradient terms in the radial direction; these however, have
‘been.limited to relatively low Reynolds numbers.

Vrentas, Duda and Bargeron [18] analyzed the development of
the steady, laminar flow of an incompressible Newtonian fluid in the
eurtrance of a circular tube at a Reynold number of 250. The circular
conduit is considered to be infinite in extent with a fully developed
parabolic velocity profile existing far downstream from the entrance.

13me3r numerically studied the effect of axial diffusion of vorticity on

flow development in circular conduits, by solving the boundary-layer

equations and the complete equations of motion.

:9" .
unifies EC:

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6
Friedman, Gillis, and Liron [19] solved the complete Navier-
Stokes equations for the steady-state, axisymmetric flow in the inlet
region of a straight circular pipe at low and moderate Reynolds

numbers.

Wang and Longwell [20] studied laminar flow in the inlet
section of parallel plates at a Reynolds number of 300. They solved
the complete Navier-Stokes equations. A transformation from x to a
new independent variable 0 to make the boundaries finite and an
exponential solution are used for a numerical treatment of the

problem.

1.4 Se arated ow

There have been numerous computational studies made of the
Navier-Stokes equations for laminar flow involving separation. These
have been two-dimensional or axisymmetric steady flows in both

external and internal flow situations at Reynolds numbers such that
laminar flow exists. For constricted flows, constriction was always
placed in the fully-developed flow region with an initial parabolic
velocity profile upstream of the constriction.

In external flows, the most classical type of such problems
concerns the fluid motion past a bluff body. For incompressible
fluids, numerical solutions for the flow around bluff bodies have been
(flatained by many authors, over various ranges of Reynolds numbers.

In constricted flows, such as flow through a channel, the
near-field motions due to a constriction or an enlargement of the
channel resulting in a separated flow are of particular interest.

An understanding of laminar separation in a channel or pipe
flow is incomplete at this time. Approximations have resulted in

significant errors in the predicted flows.

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7
1.5 Me d o v n the Constricted Flow Problem
Generally, three methods have been utilized to obtain
predictions for the separated streamline shape and for the separation
and reattachment points. Each method is discussed in the following

sections.

1.5.1 Matched Asymptotic Expansions

Using the method of matched asymptotic expansions (MAX), two
limits of the solutions to the Navier-Stokes equations may be
considered as the Reynolds number becomes large while still remaining
laminar. The outer solution describes the inviscid core flow, while
the inner solution satisfies the surface boundary conditions and is
valid near the wall. These two solutions are then matched in an
intermediate range.

Using MAX, Smith [21] studied the influence of the uniform
entrance conditions on a steady, laminar flow through a constricted
tube for large Reynolds numbers. A linearized asymptotic solution of
the governing equations of the inviscid core flow and the two viscous
boundary layers is used to determine the influence of the size and
position of the constriction. Effects of the constriction's position
on the boundary layers were described.

In another study, Smith [22] constructed a triple-deck
structure in the vicinity of the separation point for a laminar flow
(Jf an incompressible fluid streaming past a smooth surface. A finite-
<iifference approach was used to solve the boundary layer equations
saith an elliptic relation between the unknown pressure and streamline
displacement within the triple-deck structure. Comparisons with the
separating fluid motion in a similarly constricted flow, determined

numerically from an approximate form of the Navier-Stokes equations by

 

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8
Dennis and Chang [23], and obtained experimentally by Dimopoulos and
Hanratty [24], give some support to the triple-deck description.

In still another study, Smith [25] described the effects on
the otherwise unidirectional flow field within a long, straight,
rigid-wall channel suffering a severe asymmetric constriction at some
downstream station. The flow is considered to be laminar, steady, and
fully-developed with a large Reynolds number. A numerical approach,
similar to that used by Smith and Stewartson [26], is adopted for the
boundary layer equations which describe the nonlinear flow of the
lower and upper viscous zones. Free streamline theory is used for the
inviscid portion of the flow. Both upstream and downstream separation
regions were found. The upstream separation point was found to move
further upstream as the upstream slope of the constriction was
increased.

In a fourth study, Smith [27], using free-stream theory,
located the separation point in the axisymmetric flow of an
incompressible fluid through a pipe suffering a severe constriction,
with incoming Poiseuille flow. The upstream viscous separation, the
downstream eddy, and the drag on the constriction were considered for
the very severe constriction. The limiting solution in the upstream
region was found using a numerical solution of Euler's equations of
Inotion. Qualitatively, all the flow patterns given by the approximate
solutions and the experimental data tend to support the limiting
solution.

Smith and Duck [28] extended the study of laminar flow in a
ccnastricted channel by utilizing free streamline theory to describe

separation and reattachment of a steady, plane flow at high Reynolds
numbers through a channel suffering a severe non-symmetric

constriction. A numerical approach is used to solve the non-linear

 

 

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9
equations, using a Runge-Kutta scheme, for the positions of the
separation and reattachment points. The upstream separation takes
place asymptotically far ahead of the constriction. The separation on
the constriction is described by Smith's [22] triple-deck structure.
The first reattachment, described by an inviscid process near the
constriction surface, induces only small reversed velocities; the
second reattachment takes place at a large distance downstream of the
constriction. Discrepancies between these predictions and the

measurements of Blowers [29] are noted.

1.5.2 Numerieal Methods

On the numerical side, the Navier-Stokes equations have been
used to describe the constricted flow, and have been solved
approximately using numerical methods. Dennis and Smith [30] solved
the Navier-Stokes equations numerically for the flow of a two-
dimensional, laminar flow through a channel suffering an asymmetric
abrupt decrease of its width in the form of a semi-infinite step, for
Reynolds numbers up to 2000. Poiseuille flow is assumed far upstream
and far downstream of the step. In the numerical technique, the
Navier-Stokes equations are separated into two equations; by suitable
exponential expansions, an approximate value for stream function and
'vorticity is obtained at internal grid points. Good agreement was
(Retained for the upstream separation and the wall vorticity with the
analytical solutions of Smith [27].

Hung and Macagno [31] have obtained a finite-difference
sciLution of the Navier-Stokes equations for flow in a channel with a
symmetric sudden expansion. The velocity profile upstream of the
expansion is taken to be parabolic. They found that the point of

reattachment and the distance of the center of the eddy measured from

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10
the expansion are linear with Reynolds number. Results obtained by
Morihara [32] for the above problem also show a linear trend.

Hurd and Peters [33] numerically studied the motion through a
right-angle bend in a channel. They assumed an approximate solution
for the Navier-Stokes equations.

A numerical investigation of separated flow in a channel with
a backstep, or with single or multiple obstructions, has been carried
out by Nallasamy [34].

Ralph [35] solved the Navier-Stokes equations using the
finite-element method for a two-dimensional fluid flow in a straight
channel for various contraction ratios up to Reynolds number of 100.
An unsteady, quasi-linear approach is used to circumvent the

difficulties associated with the nonlinearity of the governing
equations. A steady-state solution is assumed when the time-dependent
solution becomes convergent. The flow patterns and separation regions
are detailed for a wide range of Reynolds numbers. Reasonable
agreement with the results of Lee and Fung [36], who used a method
which combines conformal mapping with a finite-difference technique,
was obtained.

Deshpande, Giddens and Mabon [37] numerically solved the case
of steady flow through a localized axisymmetric constriction in a
‘rigid tube for Reynolds numbers up to 200. The continuity and Navier-
Stokes equations, in cylindrical coordinates, are taken as the
governing relations. The numerical scheme employed closely follows
tflmat of Gosman [38] with modifications to treat the curved boundaries;
a (monstriction similar to that employed in the experimental study by
‘Youung and Tsai [39] is used. Separation regions were detailed up to a

Reynolds number of 100. The results agreed well with those determined

experimentally by Young and Tsai [39] for the separation location and

ll
pressure drop across the constriction. The reattachment prediction
was not in good agreement.

A more accurate solution of the Navier-Stokes equations is
given by Greenspan [40] who numerically investigated a two-
dimensional, laminar flow through a channel with a constriction formed
by a finite step on one wall. Boundary conditions consisted of a
parabolic velocity profile upstream, and a horizontal flow and
constant pressure downstream. Upstream and downstream separation
regions were detailed up to a Reynolds number of 1000. In this study,
a coarse mesh size is used and consequently an upstream vortex for
Reynolds numbers less than 200 was not observed. The problem is
modified by eliminating the downstream step to reduce computing time.

Friedman [41] studied the same problem of Greenspan [40] but
with a corrected sign in the downstream boundary condition. A
numerical approach, similar to that developed by Greenspan [40], is
used for the small Reynolds number analysis, and a linearized
numerical technique for moderate and high Reynolds numbers reduces the
computing time needed for convergence. A fine mesh size is used and
the upstream vortex is detailed up to a Reynolds number of 500.

Andreas and Mark [42] examined the sudden expansion
(symmetric) of a laminar flow in a two-dimensional channel in the
limit of large Reynolds number. Boundary layer equations are solved
‘numerically using a finite-difference technique for selected values of
A, the ratio of the upstream channel half-width to the step height.
‘Velocity profiles, the streamline pattern and the wake length are
ftnxnd for values of A in the range of (0.3-19) when the inlet velocity
profile is parabolic.

Taylor and Ndefo [43] studied the viscous incompressible flow

it! a two-dimensional channel with a backstep (asymmetric) for Reynolds

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12
numbers up to 100 using a splitting method. In this technique, the
two-dimensional, unsteady Navier-Stokes equations are reduced to a
coupled set of one-dimensional, unsteady flow equations. Velocity
profiles, streamline patterns, the pressure gradient and separation
and reattachment points are obtained for the range of Reynolds numbers
considered. The results show that separation occurs at about 2/3 the
step height.

Roache and Mueller [44] obtained solutions for both
incompressible and compressible laminar separated flows using time-
dependent finite-difference equations. These include backstep flow
with and without splitter plates, and flow over square cavities up to
a Reynolds number of 100. The results indicate that the separation
point moves down from the conjectured limit position at the sharp
corner toward a Stokes flow limit as the Reynolds number is decreased.

Kitchens [45] solved the steady-state, Navier-Stokes equations
and described the flow field near a square, two-dimensional
protuberance immersed in a plane Couette flow. Numerical results have
been obtained for Reynolds numbers between 1 and 200 based on plate
velocity and protuberance height. The downstream separation region is
detailed up to a Reynolds number of 200. The reattachment length and
the distance of the center of the eddy from the protuberance vary
linearly with the Reynolds number.

Recently, Frank and Andreas [46] studied steady, laminar flow
past a sudden channel expansion at large Reynolds number. A global
Newton's method was used to obtain accurate finite-difference
scilutions for uniform inflow to several sudden expansion geometries.
Eddy shapes and length, the pressure gradient and streamline contours
were obtained. The results suggest that for uniform inflows and

snualler values of the expansion ratio, the eddy length will no longer

13
increase linearly with Reynolds number when the latter is sufficiently
large.

Kwon and Pletcher [47] studied the laminar and turbulent
incompressible flow in a two-dimensional channel with a sudden
expansion by using viscous-inviscid iteration techniques. The viscous
flow solutions are obtained by solving the boundary-layer equations
using a finite-difference scheme; the inviscid flow is computed by
numerically solving the Laplace equation for the stream function using
an Alternating-Direction Implicit Method (ADI). The viscous and
inviscid solutions are matched interatively along displacement
surfaces. The flow fields were detailed up to a Reynolds number of

500 and for a ratio of step height to channel inlet height of 0.0664.

1.5.3 Experiments; Methode

On the experimental side, various experimental investigations
were performed in a wind tunnel to obtain a better understanding of
fluid flow following separation including reattachment and the
redevelopment of the flow following reattachment in this regime.
Several experimental investigations for the laminar separating flow
over back steps have been reported.

Mueller and O'Leary [48] have done an experimental as well as
a numerical study in a channel with a back step up to a Reynolds
‘number of 200. Their results show that for Reynolds numbers in the
range of 50 to 200, the reattachment length and the distance of the
center of the eddy from the step vary linearly with the Reynolds
ruxmber, confirming the numerical results presented earlier.

Goldstein, et.a1., [49] investigated the flow over a back step
with a laminar free shear layer until reattachment. The experiments

iru31ude visual observations of smoke filaments. Velocity profiles,

 

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14
reattachment points, and the momentum thickness are reported over a
range of ratio of the step height to the wind tunnel height of 0.023
to 0.0625.

Honji [50] investigated the incompressible starting flow past
a back step at Reynolds numbers less than 500 by means of flow
visualization techniques. The distance between the step and the point
of reattachment on the downstream wall was found to increase linearly
with time at intermediate stages of the flow development.

Leal and Crivos [51] investigated the effect of base bleed on
a recirculating wake behind a bluff body under conditions of a laminar
shear layer at the separation point and a steady flow field. The
streamline pattern in the wake region was observed photographically by
means of a bubble-tracer technique; in addition, a number of
quantities, such as the physical dimensions of the wake region, were
measured.

Sinha, et.a1., [52] investigated the incompressible laminar
flow over back steps by flow visualization over a range of step to
channel height ratios of 0.02 to 0.08. The flow fields were detailed
up to a Reynolds number of 1000. The experimental results indicate
that the reattachment length increases linearly with Reynolds number

as long as the reattachment is laminar.

1.6 so i on of the Present Work

The influence of a non-parabolic upstream velocity profile on
time fluid motion in the vicinity of a step may be considerable in high
Reynolds number motion. In many cases of high Reynolds number channel
or: pipe flow, the upstream assumption of a fully-developed velocity
profile demands a substantial length and may render the study

inapplicable in practice. In seeking to understand realistic

 

15
situations there arose a need, therefore, for an investigation of the
effects that the entrance conditions may have on the flow in the
vicinity of a constriction.

In the entrance region the derivative 82u/8x2 is small
relative to azu/ay2 but it may influence the solution; the pressure
gradient in the y-direction is also small but the y-component momentum
equation may not be negligible. The solution of the complete set of
the Navier-Stokes equations, without any simplifying assumption is
desirable in the solution of the entrance flow problem.

The purposes of the present work are first, to numerically
investigate the steady, two-dimensional, Newtonian, incompressible
laminar flow in the entrance region of a channel using the full
Navier-Stokes equations. The two regions making up the entrance
region are to be quantified; these include the inviscid-core region,
in which a viscous layer is assumed to exist on the wall, and the
profile-geveiopmene :egien, in which viscous effects completely
dominate the channel, as shown in Figure 1. Second, a constriction in
the form of a step, will be positioned in the inviscid core, in the
profile-development region, and in the fully developed region of the
channel flow, as shown in Figures 2 & 3. Both a finite step and a
semi-infinite step will be considered. The resulting flow will be
investigated numerically to identify the separation and reattachment
points using the full Navier-Stokes equations. Various degrees of
severity will be considered by using different heights for the step.

A wide range of Reynolds numbers will be considered. The full Navier-
Stokes equations will be expressed in terms of a stream function and
the vorticity and solved by a finite-difference scheme. A numerical
solution will be utilized which does not invoke the boundary-layer

assumptions and therefore will represent an exact solution in the

(J‘:
D
f“

 

 

 

 

l6
sense that no terms in the x- and y-momentum equations will be
neglected. Also, no approximations or exponential solutions for the
governing equations are assumed. A vorticity-stream function scheme,
which utilizes a finite-difference approximation to the Navier-Stokes
equations and has second-order accuracy in the whole flow field,
possessing conservative and transportive properties and utilizing
upwind differencing for advection terms, avoids the numerical
instability of an iterative solution at high Reynolds numbers. Third,
optimum over-relaxation values and the weighting factors for stream
function and vorticity values will be determined; these factors
minimize the spectral radius of the over-relaxation iteration matrix

and thereby maximize the rate of convergence of the method.

CHAPTER 2

MATHEMATICAL FORMULATION

2.1 r ma ow

For duct flows, which are completely bounded by solid
surfaces, the flow is assumed to be uniform at the duct entrance with

average velocity U0. Because of the no-slip condition, the velocity

at the wall must be zero along the entire length of the duct. A
boundary layer develops along the walls of the channel due to the
retarding shear force of the solid surface on the flow; thus, the
speed of the fluid in the neighborhood of the surface is reduced. At
successive sections along the pipe the viscous effects of the solid
surface diffuse farther and farther out into the flow.

Eventually, the viscous effects dominate the entire flow
thereby terminating the inviscid core region. Viscous effects finally
result in a fully developed velocity profile: a parabolic velocity
profile in a pipe or a wide channel. This defines the end of the
entrance region.

The velocity profiles of a laminar flow in a channel entrance,
0!: in a channel with a constriction in the form of a step, undergo a
change from an assumed uniform profile at the inlet to that of a
fiillyudeveloped, parabolic profile at a location far downstream. Both
a.:fi111te step (sudden contraction followed by a sudden expansion) and
a semi-infinite step (sudden contraction or a sudden expansion) are

17

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18
considered. The straight two-dimensional channel, and a channel with
semi-infinite and finite steps, are shown in Figures 1, 2 and 3.

In the case of a sudden contraction, the flow separates
upstream for sufficiently large Reynolds numbers. Flow separation
occurs in the case of a sudden enlargement on the downstream side,
while in the case of a finite step it may occur upstream as well as
downstream. Basically, the size of the separation region depends on

the Reynolds number Re, step height a, length w and position L.

2.2 Gover n uations

For the entrance configuration described above we consider the
steady state, two-dimensional, incompressible, laminar flow of a
Newtonian fluid with constant physical properties.

The dimensionless streamwise and normal velocity components
(u,v) are referenced to the average velocity Uo at the inlet, pk is
the dimensionless kinetic pressure referenced to pUg/Z, where p is the
fluid density, and the streamwise and normal dimensionless coordinates
(x,y) are normalized with H, the height of the channel. The

dimensionless Navier-Stokes equations are

8p 2 2
ufl+v§i1___k+1_i_u+u (2.2.1)
6x 6y 6x Re a 2 2
x 8y
6p 2 2
uiY+VQY___1S+L§_l+M (2.2.2)
ax 6y 8y Re ax2 ay2

where the Reynolds number is represented by

0

Re - “‘ (2.2.3)

19

In addition, the pressure term includes the body force term; that is

(2.2.4)

where b is a dimensionless vertical coordinate. The quantity (gH/Ug)
will not play a role in this problem since there are no pressure

boundary conditions that would demand the imposition of Eq. (2.2.4).

The continuity equation is

5; 8y (2.2.5)
Since it proves to be more convenient to work in terms of a stream

function and vorticity, the dimensionless stream function ¢(x,y) is

introduced in the usual manner:

u - 6y (2.2.6)
v - - ax (2.2.7)
It is evident from Eqs. (2.2.6) and (2.2.7) that the stream function

satisfies the continuity equation identically. Furthermore, for this

plane flow field, the only non-zero component of the vorticity is

w - 6x - (2.2.8)

Combining the definition of vorticity and the velocity components in
terms of the stream function, and cross-differentiating the Navier-

Stokes equations to reduce the number of equations and eliminate the

20
pressure terms, a new set of equations is obtained with independent

variables p and w:

32! 52!
+ - - w (2.2.9)
2 2
6x 6y
sz Q29 fie aw 6Q Qw
+ + Re - ] - 0 (2.2.10)
ax2 ay2 6x 6y 8y 8x

Equation (2.2.9) is a Poisson equation, an elliptic, partial
differential equation. Equation (2.2.10), which represents the steady
Navier-Stokes equations, is also an elliptic partial differential
equation in terms of w if the stream function terms are assumed to be
known coefficients. The numerical solution technique selected treats
the equations such that the stream function derivatives in Eq.
(2.2.10) are known; hence, this equation will be considered to be
elliptic in the vorticity w. These two equations are to be solved in
a given region subject to the condition that the values of the stream
function and the vorticity, or their derivatives, are prescribed on

the boundary of the domain.

2.3 Bounda Cond tions

2.3.1 The Qhehhei Ehtgance with no Constriction
The boundary condition for the two-dimensional channel, shown

in Figure l, are stated in the following:

1. The no-slip condition is applicable at the walls:
lower wall AB: u(x,0) - 0, v(x,0) - 0 (2.3.1.1)

upper wall CD: u(x,l) e 0, v(x,l) - 0 (2.3.1.2)

21
2. Two cases for the entrance velocity distribution are considered:

a) Uniform velocity profile:
u(0,y) - l, v(0,y) - 0 (2.3.1.3)

b) Actual velocity profile:
u(0,y) - f(y). V(0.y) - 0 (2.3.1.4)

where f(y) is specified from actual data, as given in

Table l.

3. The flow approaches the fully-developed parabolic channel flow far

downstream (x>LE):

u(x,y) - 6y - 6y2, v(x,y) = 0 (2.3.1.5)

For the problem under consideration to be completely
specified, the stream function and the vorticity must now be specified
on all boundaries. For the channel with and without a constriction, a
vorticity condition at the solid boundaries (AB, CD in Figure l and
AB, BC, CD, DE, EF and GH in Figures 2 and 3) is determined by using a
method presented by Thom [53]: if the subscript "0" represents a mesh
point on a boundary and "1" represents a neighboring mesh point on the

inward normal to "0" we expand in a Taylor series as

22
21 1 2 33g 1 3 gig
¢ - p + h + - h + r h (2.3.1.6)
l 0 6y 0 2 a 2 6 a 3
y o y o
neglecting terms of higher order. But,

ii -
[3y 0 - u 0 (2.3.1.7)

According to Eq. (2.2.9),

2
6.12 Q. fill:
y o

Differentiating once again results in
3 6w w - w
0
[2_§ _ _ 5;. : - l 1 h 0] (2.3.1.9)
8y 0

It follows from Eqs. (2.3.1.6), (2.3.1.8), and (2.3.1.9) that at the

boundary, the vorticity is related to the stream function by

301’ - 1b ) w
0 l l

 

where h is the mesh size equal in both the x- and y-directions.
In terms of the stream function and vorticity, the boundary
conditions used to solve Eqs. (2.2.9) and (2.2.10) are:

entrance AD: gf(o,y) - o, Qf(0.y) - 1. u(0,y) - y.

w(0,y) — o (2.3.1.11)

23

lower wall AB: %£(x,0) - o, fi$(x,0) - o, ¢(x,0) - o,

 

2

 

w(x,0) - 2 - (2.3.1.12)
h
upper wall on: %£(x,1) — o, fi$(x,l) — o, u(x,1) — 1,
3(¢ - ¢ ) w
w(x,l) - 0 2 1 - 2; (2.3.1.13)

h

exit BC: ¢(x,y) - 3y2 - 2y3, w(x,y) - 12y

6 for x 2 LE (2.3.1.14)
2.3.2. Ihe Channel Entrance With a Constriction

The boundary conditions, in terms of the stream function and
vorticity for the two-dimensional channel with a constriction in the
form of both a finite step (sudden contraction and expansion) and a
semi-finite step as shown in Figures 2 and 3, which are used to solve

Eqs. (2.2.9) and (2.2.10) are:

entrance AG:

uniform flow ¢(O,y) — y
w(0,y) - 0 (2.3.2.1)
2 3
parabolic flow u(0,y) - 3y - 2y

w(0.y)

ll

12y - 6 (2.3.2.2)

7 5
n‘l.

24

 

 

lower walls AB, BC, CF, %f(x,0) - 0
FE, EF, GH:
a1
8y(x’0) - 0 (2.3.2.3)
v(x.0> = 0
w(x,0) - - “—
h2 2
upper wall GH: %£(x,l) - 0
%$(x,l) - 0 (2.3.2.4)
¢(X.1) - 1
3<¢0 - 11) ”1
w(x,l) - - ‘-
h2 2
. 2 3
exit FG. u(x,y) - 3y - 2y for x 2 LE
(2.3.2.5)

w(x,y) - 12y - 6 for x 2 LE

CHAPTER 3

NUMERICAL METHODS

3-1 lnEIQQBEELQQ

Numerical methods have been developed to handle problems
involving nonlinearities in the describing equations, or complex
geometries involving complicated boundary conditions. A finite-
difference method is commonly used to solve either ordinary or partial
differential equations. The describing differential equations and the
necessary boundary conditions form a boundary value problem.

Any finite-difference method, used to solve a boundary value
problem, leads to a system of simultaneous algebraic, difference
equations. Their number, however, depends on the number of nodal
points which is generally very large and, for this reason, the
solution becomes a major problem.

The matrices associated with the difference equations,
approximating the partial differential equations, are either banded or
not banded. Banded matrices (the coefficient matrix is dense) are
Inatrices with non-zero elements lying between two sub-diagonals
parallel to the main diagonal. Non-banded matrices (the coefficient
lnatxix is sparse) are matrices in which the number of zero elements in
tflme matrix is much greater than the number of non-zero elements.

The two commonly used methods of solving simultaneous
algebraic equations include the direct method, that makes use of the

25

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In

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nOgva
lubed

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26

Gauss elimination or Gauss-Jordan elimination procedure, and the
iterative method, that makes use of the Gauss-Seidel iteration or a
successive over-relaxation procedure to solve the equations. These

two methods will now be discussed in some detail.

3.1.1 Direct methods

Direct methods are used to solve the system of equations in a
known number of arithmetic Operations. The most elementary methods of
solving simultaneous linear algebraic equations are Cramer's rule and

the various forms of Gaussian elimination.

3.1.1.1 Crameg's Rule

This is one of the most elementary methods. Unfortunately the
algorithm is immensely time consuming, the number of operations being
approximately proportional to (N+1)!, where N is the number of
unknowns. A number of horror stories have been told about the large
computation time required to solve systems of equations by Cramer's

rule. Even if time were available, the accuracy would be unacceptable

due to round-off error.

3.1.1.2 Gaussian Elimination

This method is a very efficient tool for solving many systems
of algebraic equations, particularly for the special case of a
tridiagonal system of equations. However, the method is not as fast
as: some others to be considered for more general systems of algebraic
equations. Approximately N3 multiplications are required in solving N
equations. Also, round-off errors, which can accumulate through the

lnangr algebraic operations, sometimes cause deterioration of accuracy

vflueri DJ is large. Actually the accuracy of a method depends on the

27
specific system of equations and the matter is too complex to resolve
by a simple general statement.

Rearranging the equations to the extent possible, in order to
locate the coefficients which are largest in magnitude on the main
diagonal, will tend to improve accuracy; this is known as "pivoting".
For a matrix that is not banded, standard Gaussian elimination is
inefficient in that the band is filled with non-zero numbers that have
to be stored in the computer and used at subsequent stages of the

elimination process.

3.1.2 itegative Methods

When large sets of equations with sparse, non-banded
coefficient matrices are to be solved and if computer storage is
critical, it is desirable to use a method that does not require a
large storage capacity. An iterative method is suitable for such
purposes. In this method an initial guess at the solution is improved
with a second approximation, which in turn is improved with a third
approximation, and so on. The iterative procedure is said to be
convergent when the differences between the successive approximations
tend to zero as the number of iterations increase. In general, the
exact solution is never obtained in a finite number of steps, but this
does not matter. What is important is that the successive iterations
«conwerge fairly rapidly to values that are within specified accuracy.

With iterative methods, however, no manipulations are
associated with zero coefficients so considerably fewer numbers have

to be stored in the computer memory. As a consequence, they can be
used to solve systems of equations that require matrices which are too
large when direct methods are used. Programming and data handling are

also much simpler using iterative methods than when using direct

28

methods, especially in the solution of sets of nonlinear equations.
The efficient use of iterative methods is very dependent, however,
upon the direct calculation or estimation of the value (or values) of
some numerical parameter called an acceleration parameter, and upon
the coefficient matrix being well-conditioned; otherwise, convergence
will be slow and the volume of computations enormous. With optimum
acceleration parameters the volume of computations, when using an
iterative method with large sets of equations, may actually be less
than the computations involved when using a direct method. Iterative
methods need or require approximately N2 operations. In addition, the
coefficient matrix of the system which results from the finite
difference approximation has many strategically placed zeroes.
However, no special account of these zeroes is taken in most direct
methods. It is reasonable to expect that a particular method,
designed in accordance with the general structure of the coefficient
matrix, could further reduce the number of operations. Many such
special iteration schemes have been devised and conditions on the
coefficient matrix have been established, which are sufficient to
insure the convergence to an acceptable solution. However, there is
no general procedure available to determine which of the many possible
methods is "best” in a given case.

The most frequently used iterative method is the Gauss-Seidel
iteration. One difficulty with the Gauss-Seidel method is that
(convergence is relatively slow. Convergence is improved when a

successive over-relaxation method is used.

3. 1.2.1 us -Seide Iteratio
Although many different iterative methods have been suggested

over the years, Gauss-Seidel iteration (often called Liebmann

 

 

 

 

29
iteration when applied to the algebraic equations that results from
the differencing of an elliptic, partial differential equation) is one
of the most efficient and useful point-iterative procedures for large
systems of equations. The method is extremely simple but converges
only under certain conditions related to ”diagonal dominance" of the
matrix of coefficients. The method makes explicit use of the

sparseness of the coefficient matrix.

3.1.2.2 §hccessive Over-Relaxation Method

Successive over-relaxation (SOR) is a technique which can be
used in an attempt to accelerate any iterative procedure. Often, the
number of iterations required to reduce the error, of an initial
estimate of the solution of a system of equations, by a predetermined
factor can be substantially reduced by a process of extrapolation from
previous iterations of the Gauss-Seidel method. Actually, the
solution of a system of simultaneous algebraic equations by Gauss-
Seidel iteration requires numerous recalculation, or iterations before
convergence to an acceptable solution is achieved. During this
process there are changes in the values of the unknowns at each mesh
point between two successive iterations; a correction of the values in
the anticipated direction before the next iteration is necessary to
accelerate convergence. The parameter which is used to accelerate the
convergence is known as a relaxation factor. If the Optimum
relaxation factor is found, it is apparently possible to reduce the
«commutation time in some problems by a factor of up to 30. It is
(flyviously very important to find this optimum factor. Occasionally,
Stuzcessive over-relaxation may not be of much help in accelerating

convergence, but it should be considered and evaluated. The potential

for savings in computation time is simply too great to ignore.

 

 

 

##A“
—_——____.—————__—fi__—w———_———.——w_—

 

30

3-2 W

Because of the simplicity and effectiveness of an iterative
technique in solving large sets of equations with sparse coefficient
matrices, which result from the finite-difference approximations of
the governing equations, an over-relaxation technique is used to solve
the full Navier-Stokes equations which describe the steady flow.

In terms of the stream function p and the vorticity w, the two

dimensional, steady state Navier-Stokes equations are

2 2
L15 + L? - - (0 (3.2.1)
6x a

y
1352,53,], [flig-fli@]-o 322
2 2 eaxay ayax (..)

It will be convenient to approximate these coupled equations by
linear, elliptic difference equations; the numerical solution of such
equations is well understood.

A square computational grid of size Ax - Ay - h is selected,
with a grid lines parallel to the x and y axes such that the grid fits
exactly the geometry of the channel with and without a constriction.
Around a typical internal grid point (x,y) we adopt the convention
that quantities at (x,y), (x+h,y), (x,y+h), (x-h,y) and (x,y-h) are
[denoted by the subscripts l, 2, 3, 4, 5, respectively, as shown in
jFigmre 4.

Equation (3.2.1) which is an elliptic, partial differential
equation is to be solved simultaneously with the nonlinear, partial

differential equation (3.2.2) in a rectangular region subject to the
condition that the values of the stream function and vorticity are

prescribed on the boundary of that domain.

31
Eq. (3.2.1) can be approximated using central-difference at

the representative interior point (x,y) by

 

2
M 1.
2 - 2 [$2 + $4 - 2¢1] (3.2.3)
ax h
' 351 L.
1 6y h

with an error 0(h2). Thus, Eq. (3.2.1) can be written for the square

mesh as

 

¢1--}; [¢2+¢3+¢4+¢5] +%h2wl (3.2.5)

We could also use a central—difference formulation for Eq. (3.2.2),

but we anticipate that the problem will need to be solved for

] reasonably high values of Reynolds number; it is known that such a
formulation may not be satisfactory owing to the loss of diagonal
dominance in the sets of difference equations, with resulting
difficulties in convergence when using an iterative procedure.

Eq. (3.2.2) can be approximated by a difference equation,
using central-differences for the second derivatives and a forward-
difference for the first derivatives; there results

+ w + w + w

(-4w + w

1 2 3 a 5)

¢ - ¢ w - w ¢ - u w - w
+ Re [ 2h . 2h - 2h . 2h ] - 0 (3'2'6)

or , equivalently ,

32
.13.. k
'4“1 + [1 ' 48 ($3 ‘ *5’] “2 + [1 + 4 ($2 ' $4)] “3

+ [1 + %9 (¢3 — ¢5)] wa + [1 - %§ (¢2 - ¢4)] wS = 0 (3.2.7)

Eqs. (3.2.5) and (3.2.7) with the appropriate boundary
conditions can be solved using the Gauss-Seidel scheme. The numerical
solution for the stream function and the vorticity will be denoted by

¢k+1 and wk+1, where k is the number of iterations. This solution

 

£ works relatively well but diverges for Reynolds numbers greater than
250 and h - 1/40. The reason for this divergence is that, for high
Reynolds numbers, the terms Re(¢3 - ¢S)/4 and Re(¢2 - ¢4)/4 in Eq.
(3.2.7) become so large that the matrix of the resulting system loses
its diagonal dominance.

A forward-backward technique can be introduced to maintain the
diagonal dominance coefficient of ml in Eq. (3.2.2) which determines
the main diagonal elements of the resulting linear system; this

technique is outlined as follows:

Set

a - $2 - $4 (3.2.8)
3 _ $3 - ¢5 (3.2.9)
Innen approximate Eq. (3.2.2) by

_ 2 s_@_fl_§_2=
4w1 + w + w + w + w + h Re [ 6y 2h 8x] 0 (3.2.10)

2 3 4 5 2h

lflonr , if

 

 

33

ee : ”3 ' “1
a a 0, 6y h
(0'0)
0 < o, g? 2 1 h 5 (3.2.11)
If
em : “2 ' “4
fl 2 0’ 8x h
‘0‘“)
p < o, 3% 2 4 h 2 (3.2.12)

To assure the diagonal dominance of the coefficient matrix for ml,
which depends on the sign of a and 6, Eq. (3.2.2) is expressed in the

following difference forms:

+ [1 + 9%9] w3 + [1 - QBE] wh + ws - o (a 2 o, a z 0) (3.2.13)

2
['4 ’ 222 + £32] ”1 + [1 ' £32] ”2
+ [1 + 9%£} w3 + wa + ws - o (a z o, a < 0) (3.2.14)
['4 ’ g2g ' £37] “1 + “2 + “3
4- [1 + 5%2] wh + [1 - 959] ws - o (a < o, 5 a 0) (3.2.15)

34

+ + w + [1 - 9%g] ws - o (a < o, 5 < 0) (3.2.16)

”3 a
It has been found that the new equations result in convergence
for o 5 Re 5 105, h - 1/15 but diverge for h < 1/15.

Weighted averages can be introduced to avoid any possible
divergence. A smoothing formula results which corrects the value of
the vorticity in the interior region; that is, the vorticity is
assumed to be

w* - xv wk + (1 - KV) wk+1 (3.2.17)

and the stream function

¢* - KS wk + (1 - KS) ¢k+1 (3.2.18)

where wk+1

and ¢k+1 are the calculated vorticity and stream function.
The values of the weighted averages KS and KV are within the range of
0 to l and their determination will be discussed in more detail in the
next chapter.

An over-relaxation technique can be applied to accelerate the

convergence of Eqs. (3.2.5) and (3.2.13-3.2.l6); the expressions are

used in this technique presented in the following:

For'Poisson's equation

¢§+1 = (1 - F8) wf

£3 2
+4 (¢2+¢3+¢4+¢5+hw1) (3.2.19)

35

For the vorticity equations

wllfil - (1 - FV)w1{+ FV{[w2 + [l + (1112 - 1,1119%] (03

+[1+(¢3'¢5)§—e]w4+w5]/[AI-($2-104)?

+ (153 - 16912351} (or Z 0, fl 2 0) (3.2.20)
w§+1 - <1 - FV) w: + Fv {[11 - (v3 - (5) %21 w,

+ [1 + ($2 - o4) §91 w3 + ma + ws] /
is; 3.9.
[4 + (¢2 - ‘54) 2 + “)3 - m5) 2 ]} a _>_ 0, fl < 0 (3.2.21)

wllc+1-(l-FV)w1{+FV{[w2-+w3+[1+(¢3-¢5)%‘§]w4

+ [1 - (w, - $4) §§1 “5] / [a - (e, - ¢,) %Q

+ (¢3 - ¢S)%§]} (a < o, 5 2 0) (3.2.22)
”1+1 ' (1 ' FV) ”1 + FV {[[l ' ($3 ' $5) 37] “2 + ”3 + “a

+ [1 - <¢2 - 14) fig] w5] / [a - <¢2 - v4) %Q

+ (e3 - ¢5)%§]}- (a < o, s < 0) (3.2.23)

In the above equations FS and FV are the relaxation factors for the

stream function and vorticity, respectively. The values of these

 

 

 

36
relaxation factors are in the range of 0 to 2 and the determination of
their optimum values will be discussed in the next chapter.
A numerical solution of Eqs. (3.2.19-3.2.23) is carried out be

an iterative procedure according to the following steps:

(1) Initial values of wi j and $1 j are assumed at all mesh
points. Here w. represents the vorticity at x = ih and

1,j
y - jh.

(2) Calculate the values of the vorticity on the boundary

using Eq. (2.3.1.10).

(3) Successively calculate for every mesh point:
a. the values of the stream function and vorticity
from Eqs. (3.2.19) and (3.2.20-3.2.23).
b. corrected the values pi’j using Eq. (3.2.18).
c. corrected the values w. using Eq. (3.2.17).

1.]

(4) Except for the points where w - 0, continue the

1.1“ $1.1

iteration until the following error criterion is

satisfied:
fk+1 _ f1; -6
I-171k:1—_‘i| s 10 (3.2.24)
f
1.
Here fi,j represents either wi,j or ¢i,j' When the above

is satisfied the iteration is terminated, k being the

number of iterations. If this relation is not satisfied

 

 

*

 

 

37
for some preselected maximum number of iterations, then

return to step (2) and repeat the process.

The computer time is significantly reduced by using the optimal over-

relaxation factors.

 

 

CHAPTER 4

CONVERGENCE CRITERIA AND THE NAVIER-STOKES EQUATIONS

4.1 ih§;oductiom

The numerical solution of boundary value problems for partial
differential equations usually requires the solution of large systems
of linear algebraic equations. The order N of such systems is
generally equal to the number of mesh points in the domain under
consideration. Since direct inversion procedures require the order of
N3 operations they are not practical, even when using high speed
digital computers, for reasonable mesh size in two dimensions. Thus,
iterative methods for solving linear systems are of interest as they
usually require an order of N2 operations. In addition, the
coefficient matrix of the system, which results from the finite
difference approximations, has many strategically placed zeroes.
However, no special account of these zeroes is taken in most direct
inversions. It is reasonable to expect that a particular method,
(designed in accordance with the general structure of the coefficient
nuatrix, could further reduce the number of operations. Because of
simnplicity and effectiveness, the successive over-relaxation method
has been the most popular of the iterative methods for solving a large

system of linear algebraic equations possessing a sparse, non—banded

coefficient matrix .

38

 

 

 

 

39
In this chapter, our main objective is to determine optimum
values of the over-relaxation and weighting factors that maximize the

rate of convergence of the successive over-relaxation method.

4.2 The Pgoblem Under Censideration

The two-dimensional, incompressible, laminar flow in the
entrance region of a channel is investigated numerically. The non-
dimensional Navier-Stokes equations in terms of a stream function u

and vorticity w as the governing equations are

2 2
fi-i + fl-i - - w (4.2.1)
6 2 2

x 6y

2 2
M+M+Re MQ‘B_@§£BO (422)
ax2 ay2 ax 6y 8y 8x ' '

In a finite-difference form, using over-relaxation factors, the

equations take the forms

¢E+1 - (1-FS) ¢§ + gfi (d2 + $3 + $4 + ms + hzwi) (4.2.3)

w§+1 - (1-FV) w: + Z— {[1 - %9 (¢3 - ¢S)] w2 +
[1+%§(¢2'¢4)] w3+[1+%§(¢3'¢5)] (04+
1 BE
[ _ 4 (¢2 - ¢4)] “5} (4.2.4)

Using smoothing formulas they become

1* - KS ek + (1 - KS) ekil o 5 KS 5 1 (4.2.5)

 

fr H_<4 4

 

 

40

w* - xv wk + (1 - KV) wk+l (o s xv

IA

1) (4.2.6)

where FS and FV are optimum over-relaxation factors for Poisson's and
the vorticity equations, respectively; KS and KV are weighted averages
for the stream function and vorticity, respectively.

With rectangular field boundaries represented by i - 0, I + l,
and j - 0, J + 1, each of these difference Eqs. (4.2.3) and (4.2.4)
represents a set of J x J equations, so that there are 2I x J

algebraic equations to be solved simultaneously.

4.3 Qomvergence Conditions

The question of stability and convergence of any iterative
procedure can only be answered completely by a consideration of Eqs.
(4.2.3) and (4.2.4), one of which is nonlinear. However, Poisson's
equation is known to have excellent convergence properties when solved
along. Therefore, it is reasonable to assume that the convergence of
the simultaneous solution of the nonlinear vorticity Eq. (4.2.4) and
the Poisson Eq. (4.2.3), for the stream function will be most affected
by the convergence properties of the nonlinear equation. Since
equations (4.2.3) and (4.2.4) are coupled in e and w, the accelerating
parameters, which are optimum for the Poisson's Eq. (4.2.3) when
solved alone with m constant during the iteration, may not accelerate
the convergence of the simultaneous solution of Eqs. (4.2.3) and
(4.2.4).

For the general solution of the simultaneous Eqs. (4.2.3) and
(4.2.4), the iteration will be continued until the relative error

criterion

 

 

 

 

is satisfied. Here, f1 j represents either wi j or mi j and k will be

the number of iterations.

4.3.1 Smfficient Conditions for Convergence of the Succeeeive Over-
Relamation Method

The general linear algebraic system of N equations in the N

unknowns $1, $2,..., ¢N or wl, w2,..., wN can be written in the form
a11¢1 + a12¢2 + a13¢3 + '°' + a1N"’N ' b1
a214’1 + a22¢2 + a23¢3 + °'° + a2N¢N ‘ b2
aNl¢l + aN2¢2 + aN3¢3 + ... + aNNwN - bN (4.3.1.1)

If the matrices m, b and A are defined by

 

 

 

 

 

 

r a r 1 r \
$1 b1 a11 a12 8111
m b a a ... a
w - .2 , b - .2 , A - .21 .22 .2” (4.3.1.2)
LIN, LPN] [8N1 aN2 aNN,
Then
A¢ - b (4.3.1.3)

jLet: us assume that A is nonsingular so that for a given A and b, m

eximsts and is unique. In order to provide a compact notation, we will

 

 

 

 

42
order the equations, if possible, so that the coefficient largest in
magnitude in each row is on the diagonal. Then if the system is
irreducible (cannot be arranged so that some of the unknowns can be

determined by solving less than N equations) and if

N

Iaiil a} Iaijl (4.3.1.4)
j-l
j¢i

for all i and if for at least one i,

N

Iaiil >§ Iaijl (4.3.1.5)
j-l
jei

then the over-relaxation iteration will converge. This is a
sufficient condition which means that convergence may sometimes be
observed when the above condition is not met. A necessary condition
can be stated but it is impractical to evaluate.

The sufficient condition can be interpreted as requiring for
each equation that the magnitude of the coefficient on the diagonal be
greater than or equal to the sum of the magnitudes of the other
coefficients in the equation with the "greater than" holding for at
least one (usually corresponding to a point near a boundary for a
physical problem) equation. The matrix which satisfies this condition
is called a diagonal dominant matrix. Therefore, for convergence, the
lnatrix of the resulting system must be diagonally dominant.

Perhaps we should relate the above iterative convergence

crdjzeria to the system of equations which results from a finite-

difference approximation of Poisson's and the vorticity equations.

 

 

w— ———— r‘;‘ AF44-

 

 

43
Consider that at any point in our iteration our intermediate
values of stream function p's and vorticity w's are the exact solution
+ e;

plus some tolerance e, i.e., $1 - ($1) + e and wl - (wl)

exact exact

then our condition of diagonal dominance is forcing the e's to become
smaller and smaller as the iteration is repeated cyclically.

For a general system of equations, the multiplications per
iteration could be as great as N2 but could be much less if the matrix

is sparse. This is the case in our system of equations.

4.4 Accelerating Parametere

For different flow situations (i.e., different Reynolds
numbers) and mesh size h, the values of FS, FV, KS and KV in Eqs.
(4.2.3-4.2.6) have a significant effect on the convergence of the
solution as well as the computing time. These parameters are called
accelerating factors and they play an important role in the solution.

The successive over-relaxation (SOR) method can be used in an
attempt to accelerate any iterative procedure but we will propose it
here primarily as a refinement to the Gauss-Seidel method
(unaccelerated method). With the determination of optimized
accelerating parameters, it is possible to reduce the required number
of overall iterations in the solution by more than an order of
magnitude from that required by Gauss-Seidel iteration; in addition,
‘we may remove the restriction placed on the maximum size of the space
step imposed by the Gauss-Seidel technique. The general idea of
.accelerating the solution is well known; however, the determination of
the optimum acceleration parameter for, and the application to, this

nonlinear set of simultaneous equations has not heretofore been given.

Therefore, a search must be made for the optimum acceleration

parameters .

 

. ._ .....~_ _.

 

 

44
The optimum value of the over-relaxation factor F8 for the
Poisson equation depends on the mesh size, the shape of domain, and
the boundary conditions. For the problem in a rectangular domain of
size (I-1)Ax by (J-1)Ay with constant Ax and Ay, it has been shown

[54] that

FS - gift”: (4.4.1)
‘7
with 1 - cos(«/M) + cos(n/N), where M and N are, respectively, the
total number of increments into which the horizontal and vertical
sides of the rectangular region are divided.

The optimum value of the over-relaxation factor FV for the
vorticity equation depends on the Reynolds number, which identifies
the coefficient of the matrix which results from the finite-difference
form of the governing equations; the mesh size also plays a role.

In addition, the values of the weighting factor KS for the
stream function and RV or the vorticity are determined by
experimentation; the values that fall within the range of 0 to 1 will
accelerate the convergence of the solution; this results due to the
different percent of the old and the new values of stream function and
vorticity used during the matrix iteration.

The main idea behind the convergence of the solution is that
the matrix that results from the finite-difference equations must be
diagonally dominant; this is the case for low Reynolds number flows.
.For'high Reynolds number flows, the matrix of the resulting system
loses its diagonal dominance. A forward-backward technique can be

introduced to maintain the diagonal dominance and, consequently,

(cornvergence will also be maintained. Actually, the optimum value of

 

 

 

45
FV minimizes the spectral radius (i.e., results in the smallest
magnitude of the maximum eigenvalue of matrix A, Eq. (4.3.1.3)) of the
over-relaxation iteration matrix and thereby maximizes the rate of

convergence of the method.

 

 

 

 

 

 

CHAPTER 5

RESULTS, DISCUSSION AND CONCLUSIONS

5.1 Mackgremhe

The numerical solution of the full Navier-Stokes equations,
for the entrance flow and constricted flow problems, has been obtained
using a successive over-relaxation technique. In the development of a
numerical scheme, one is never sure of the accuracy of the numerical
solution obtained. At times, convergence in an iterative procedure
may not mean that the solution is convergent to the solution of the
differential equations. Comparing the numerical results with a known
analytical solution is one possibility, but on the other hand,
analytical solutions that are available use either simplified Navier-
Stokes equations or an assumption is made concerning the approximate
form of the solution. Alternatives are to compare the results with
other numerical or experimental studies and to perform a grid-
independency test for confidence in the numerical results.

Numerous solutions to the entrance flow problem have been
reported in the literature. All of those available, both analytical

and.numerical, report methods that solve boundary-layer equations or

Simplified versions of the Navier-Stokes equations (for Reynolds
numbers up to 2000), or full Navier-Stokes equations with a
trarjsformation and an exponential solution for numerical treatment

(fkrr Reynolds numbers up to 300). Solutions to the full Navier-Stokes

46

47
equations in the entrance region, with and without constrictions for
Reynolds numbers based on the channel height up to 2000, have not been
presented in the literature.

The laminar incompressible flow in the entrance region of a
high aspect-ratio, plane channel with and without constriction in the
form of a step (forward, backward, and finite) has been analyzed using
the full Navier-Stokes equations.

The stream function, vorticity, and streamwise velocity are
reported at each grid point for Reynolds numbers up to 2000 for
various step-to-channel height ratios and step lengths for the
constricted channel. In addition, separation and reattachment points
are obtained by fitting a polynomial to the separated streamline
coordinates. An actual profile, obtained by fitting a polynomial near
the wall to a uniform central section, using velocity measurements
from a hot wire annometer, as shown in Table 1, is also used.

The convergence domain for the successive over-relaxation
method and the optimum values of over-relaxation and weighting
factors, often referred to as accelerating parameters, required by the
numerical scheme, are utilized to maximize the rate of convergence
thereby minimizing the computing time.

The first case solved is for a Reynolds number of 20, based on
channel height, with a mesh size of 0.091 by 0.091, eleven elements
normal to the flow and a sufficient number of elements in the flow
direction to allow a fully-developed flow to occur. Several other
Reynolds numbers are used up to 2000, the limit for laminar flows of
interest. In order to improve the accuracy of the solution and to
.avoid excessive computing time, a mesh size of 0.05 by 0.05 is used
for subsequent cases. Using the 0.05 mesh size, the majority of the

calculations are performed by the VAX-ll/750 VMS 4 computer. In

 

48
addition, however, mesh sizes in the range of 0.02 to 0.1 are used at
selected Reynolds numbers to check the validity of the numerical work,

i.e., that the solution is independent of the mesh size.

 

5.2 Beem1§s fie; ehe Qhannei Entrance with no Constriction

Table 2 summarizes the cases considered providing each
Reynolds number, inlet condition, mesh size, number of iterations, and
the time needed for convergence.

The velocity profiles for the cases Re-20, 200, 500, and 2000
are shown in Figures 5-8, assuming a uniform velocity inlet profile.
It is noted that for only very small X, in fact, at only the first X-
step, the velocity profiles include a minimum on the axis and
symmetrically located maxima on either side of the centerline, where
the maximum velocity is 0.05% higher than that at the centerline.
This contradicts the results obtained by other authors [12, 55, 56],
in which these local maxima are much more pronounced over significant
downstream distances.

The centerline velocity for Reynolds numbers of 200 and 2000
is shown in Figure 9 along with those obtained by other researchers
[6, 12, 56, 63]. Their values are generally smaller than those
obtained in the present study; however the velocity distributions are
similar in shape.

Near the entrance where the velocity gradients are large near
the wall, large viscous stresses develop. Therefore, the streamwise
pressure gradient dp/dx is largest near the entrance. Also, the
thermal pressure gradient dp/dy, neglected in other studies, is quite
significant near the wall for small X. These characteristics are more
pronounced for the high Reynolds number cases. A typical normalized

pressure gradient, for Reynolds number 20 and 200, is plotted versus

49
X/Re in Figures 10 and 11, respectively. It approaches unity
asymptotically as X/Re become large.

The streamwise pressure gradient along the wall and centerline
is always negative and [-(dp/dx)(Re/12)] is large near the entrance
and decreases asymptotically to unity. This contradicts the result
obtained by Morihara and Cheng [12], in which a localized adverse
pressure gradient resulted due to the maximas in the velocity profile.
This is undoubtedly due to the approximate form of the governing
equations used in the solution.

The entrance length LE, which is defined as the distance from
the inlet to the point where the centerline velocity reaches 99% of
the parabolic centerline velocity, is calculated using both the
uniform profile and the actual profile. The velocity profiles that
develop from an actual inlet profile are shown in Figures 12 and 13.
The entrance length is found to be insensitive to the inlet velocity
distribution, as shown in Table 3. It is noted that the entrance
length increases slightly as the inlet velocity gradient at the wall
decreases.

The entrance length calculated in this study is compared with
that of other researchers [6, 12, 16, 17, 55, 57] in the Table 4. No
significant difference is noted.

The entrance region in a channel is analyzed suggesting the
existence of two distinct regions: the inviscid-core region and the
profile-development region. The lengths of these regions and their
ratios are obtained for various Reynolds numbers as presented in Table
5.

The end of the inviscid-core region occurs when the boundary
layer thickness becomes equal to half of the channel height. This is

determined numerically to occur when the velocity at the centerline

50
exceeded the velocity at the first node above the centerline. The
inviscid-core region is observed to be approximately one-fifth of the
entrance length, a much shorter length than has been reported by
Mohanty and Asthana [58] for a pipe flow.
Finally, the vorticity distribution in the entrance region of
a straight channel at different locations X/Re for Reynolds number

200, is given in Table 6.

5.3 Resulte for she Channei Entrance with e Conetriction

5.3.1 Eogmagd Step

Solutions of the finite-difference equations are obtained for
flow through a channel whose width is altered sharply, asymmetrically
and by a finite amount of 0.4 of the channel height (a forward step).
This step is positioned at various locations in the entrance region,
and Reynolds numbers based on the channel height up to 2000 are
considered. Table 7 summarizes the cases considered and their
Reynolds number, step height and position, computational domain,
purpose and computing time. Streamlines in the vicinity of the step
are shown in Figures l4-16 for Reynolds numbers (Re)H-20, 200, and
2000, for a step located in the profile-development region. Also, the
Y-values for selected streamlines for the entire flow field are given
in Tables 8-10 for (Re)H-20, 200, and 2000, respectively. The
streamline plots give a qualitative picture of the flow solutions;
quantitative information is presented in Figures 17-23. The numerical
results show that there is a detectable eddy of recirculating fluid
upstream of the step for Re-50; as the Reynolds number increases the
size of the eddy increases. The step's position has no observable

effect on the reattachment point, as shown in Figures 17 and 18.

51
Increasing the height of the step forces the separation point further
upstream and reattachment point upward in a linear nature as shown in
Figures 19 and 20 for (Re)H-200 and 1000, respectively. The
streamline separates ahead of the step at a distance approximated by

0.0215 (Re)'4319 and reattaches to the vertical face of the step at a

height of 0.0282 (Re)'2572. This separation of the streamline is
observed at Re-50, while, in Greenspan's study [40], it is observed at
Re-2000, due to a coarse mesh size used. Figures 21-23 show the
separation and reattachment points as a function of step height and
Reynolds number.

The separation point, which is predicted as Xs-0.0215
(Re)'2572 in this study, is compared in Figure 24 with that found in
Smith's asymptotic theory [27] for a channel with an asymmetric
constriction in the form of a semi-infinite step and Dennis and
Smith's numerical solution [30] for a channel with a symmetric
contraction. The trend of the results is consistent with Dennis and
Smith [30] for low Reynolds numbers and with Smith [27] for high
Reynolds numbers. Also, the reattachment point is compared with
Dennis and Smith [30] in Figure 25. Their values are relatively
higher than the present work, however, the trends are the same.

On the qualitative side, asymmetric or symmetric constrictions
produce a sizeable upstream adjustment of the flow when the Reynolds
number is large. As the Reynolds number increases, the size of the
separation region grows. Also, the dividing streamline upstream,
reported by Greenspan [40] and Smith [27], exhibits the concave-
upwards behavior for all values of Reynolds numbers, in agreement with

the results displayed in Figure 17 for low Reynolds number and Figure

18 for high Reynolds number.

52
5.3.2 Eachmerg Etep

In this section, the numerical results for incompressible flow
past a backward step will be presented and discussed. The ratio of
step height to channel downstream height is 0.2. The step is
positioned at various locations in the entrance region and Reynolds
numbers (Re)H, based on the downstream channel height, up to 2000 are
considered. The presence of the step is observed to induce a
noticeable acceleration in the flow near the step. The general
features of the flow are separation of a shear layer from the vertical
face of the step and its reattachment to the surface of the downstream
lower wall, resulting in the formation of a separation region
immediately behind the step.

The cases are summarized in Table 11; their Reynolds numbers,
step height, position, computational domain, purpose and computing
time are listed.

The numerical results show that the step's position has little
effect on the separation point; it is more pronounced for a high step—
to-channel ratio and high Reynolds number, as shown in Figures 26-30.
Although, the reattachment points are different for the step in the
different positions, the trend of the results is the same. Therefore,
the streamline patterns are shown for selected positions. In Figures
31-33 the streamline patterns are shown for Reynolds numbers (Re)H-20,
200 and 2000, for flow in the vicinity of a backward step located in
the profile-development region. Also, the Y-values are given for
selected streamlines for the entire flow field, in Tables 12-14.

Figures 34-36 show the effect of step height on the separation
region for (Re)H-20, 200 and 500 for a step located in the inviscid-
core region. The results show both separation and reattachment points

are sensitive to the step height. The separation point moves upward

'— fv—rfw ty— 7

53
as the step height increases in a nonlinear manner and approaches the
top of the step for high step height and Reynolds number as shown in
Figure 37. In addition, the step height has significant effect on the
reattachment point; as the step height increases, the reattachment
point moves further downstream in a linear fashion, as shown in Figure
38, for Reynolds numbers 20, 200 and 500.

Increasing Reynolds number (Re)H forces the separation point
further upward, as shown in Figure 39 for the step height 0.3H and
Reynolds numbers up to 500, and in Figure 40 for the step height of
0.2M and Reynolds numbers up to 2000, for the step in the inviscid-
core region.

The location of the separation point, Ys/a, for the step
height 0.2H, is plotted versus Reynolds numbers in Figure 41. The
separation point does not show the linear variation with Reynolds
numbers as found by Kawaguti [59] and by Macagno and Hung [60] in
channels with a sudden expansion. It approaches asymptotically to
unity (i.e., the top of the step) as Reynolds number becomes large.
The present nonlinear trend is probably due to the influence of the
upper wall. A similar nonlinear trend is found by Roache and Mueller
[44] for a backward step.

To compare the numerical results of this study with the
theoretical and experimental results of others, the Reynolds number
(Re)ais based on the step height "a" rather than the channel
downstream height.

The numerical results indicate separation occurring at about
2/3 the step height for low step height and Reynolds number. This is
consistent with the numerical results obtained by Taylor [43] for low
Reynolds number [(Re)a-4]; it is not a constant value for Reynolds

ruunfl5ers higher than 4, as Taylor [43] claimed. For Reynolds numbers

54
in the range of 20 to 40, separation occurs in the range of 90% to 95%
of the step height, which is found to be consistent with the numerical
results of Roache and Mueller [44], Kitchens [45] and Mueller and
O'Leary [48]. For high Reynolds numbers (Re)a of 100 to 400 the
streamline separates at the top of the step. A similar trend is
clearly seen for large Reynolds numbers in an experimental study by
Honji [50] for the backward step, and in a numerical study by Kummar
and Yajnih [61] for a sudden expansion in a channel flow.

The numerical results also show that, for low Reynolds numbers
[(Re)a below 100], the reattachment point is a nonlinear function of
Reynolds number, while, for high Reynolds numbers [(Re)a 100 to 400],
it is a linearly increasing function of Reynolds number.

Convergence could not be obtained using the iterative
procedure as reported by Kitchens [45] for Reynolds number higher than
200. This is probably caused by the local mesh size and related to
the numerical stability problems encountered by Macagno and Hung [60].
Nonconvergence is also noted by Mueller and O'Leary [48] and Roache
and Mueller [44] for Reynolds number higher than 100 (based on step
height and free stream velocity). In this work, convergence is
obtained for mesh sizes of 0.05 and 0.07 for all Reynolds numbers (up
to 400); this is because upwind differencing is used for the advection
terms in the Navier-Stokes equations. This avoids the numerical
instability of an iterative solution at high Reynolds numbers; in
addition, optimum accelerating parameters are used to accelerate the
solution.

For low Reynolds number range, the reattachment points compare
:faNorably with the theoretical results obtained by Roache and Mueller
[44] and Mueller and O'Leary [48], as Shown in Figure 42. Also, the

reattachment points for high Reynolds numbers in the range of 100 to

55
400, are in good agreement with experimental data obtained by Sinha,
et.al. [52], Leal and Acrivos [51], and Goldstein [49] as shown in
Figure 43. Furthermore, the trend of the results is consistent with
the numerical results of Kumar and Yajnih [61], Andreas and Mark [42]
and Schrader [62] for flow through a sudden expansion at large

Reynolds numbers.

5.3-3 MED

The numerical results of the steady-state, Navier-Stokes
equations for the flow field near a finite step immersed in a two-
dimensional channel entrance region are described in this section.
The qualitative features of the separation phenomena induced by the
finite step are expected to be similar to those found in the forward
and backward step cases. The cases are summarized in Table 15 by
listing the Reynolds number, step height, length and position,
computational domain, purpose and computing time. Numerical results
are obtained for Reynolds numbers between 20 and 1300 based on the
channel height and average velocity.

The numerical solutions for a finite step immersed in the
channel entrance flow show a very small separated flow region upstream
of the step, with separation region length and height almost
independent of Reynolds number, for (Re)H between 200 and 1300.
Recall that for the same range of Reynolds numbers, a significant
Iapstream separation region is found for the forward step case. A
simdlar upstream influence of the finite step is reported by Kitchens
[115] and Greenspan [40]. Obviously, the downstream region is
significantly influencing the upstream separation region.

On the other hand, the numerical results show that the

downstream separation region introduced by the backward step, extends

 

56
further downstream than that associated with the finite step; this
effect increases as the Reynolds number increases.

The numerical results show that channel length downstream of
the step, the step length, and the position of the step have an
insignificant effect on the downstream separation region for the cases
considered, as given in Tables 16-20 and shown in Figures 44 and 45.
Therefore, the streamline patterns are shown for a selected position
of the finite step.

Streamlines in the vicinity of the finite step are shown in
Figures 46-48 for (Re)H of 20, 200 and 1300, for a step in the
profile-development region. The Y-values for selected streamlines for
the entire flow field are given in Tables 21-23 for various Reynolds
numbers.

The effect of the step height on the downstream separation
region is shown in Figures 49 and 50. As in a backward step, the
downstream separation point moves upward toward the top of the step as
the step height increases, in a nonlinear manner, and approaches the
top of the step for high step height and Reynolds numbers as shown in
Figure 51. Also, as the step height increases, the downstream
reattachment point moves further downstream in a linear fashion, as
shown in Figure 52, for (Re)H-20 and 200. The locations of the
downstream separation point and reattachment point with (Re)H are
shown in Figure 53 for the step located in the inviscid-core region,
with height of 0.3H and (Re)H up to 1300. The linear relationship of
the downstream reattachment point as a function of Reynolds numbers
laased on the step height, in.the range of 60 up to 390 is shown in
Figure 54. A similar trend was obtained numerically by Kitchens [45]

‘for flow past square protuberance.

57

As a special case, a finite step with one mesh size (0.05) of
length (called a single step) is investigated up to (Re)H-SOO. The
cases are summarized in Table 24. As in a finite step, a single step
causes a very small separated flow region upstream of the step for
high Reynolds number. The numerical results show that for low (Re)H
up to 20, the step position has a significant effect on the downstream
separation region due to the short distance of the inviscid-core
region, as shown in Figure 55. This effect is diminished as the
Reynolds number increases, as shown in Figure 56.

Streamlines in the vicinity of the single step, located in the
inviscid-core region, are shown in Figures 57 and 58 for (Re)-20 and
200. The Y-values for selected streamlines for the entire flow field
are given in Tables 25 and 26 for (Re)H-20 and 200, respectively. For
(Re)H greater than 20, the streamlines separate from the top of the
single step, as shown in Figures 59 and 60. The step height, for
(Re)H greater than 20, has no effect on the downstream separation
points; however, it does effect the downstream reattachment point
which increases linearly with step height, as shown in Figure 61.
Furthermore, as (Re)H increases, the downstream reattachment point
moves further downstream and increases almost linearly with Reynolds
number, as shown in Figure 62. A similar trend is also shown for flow
past a square protuberance in a Couette flow studied by Kitchens [45].

The downstream reattachment points for the flow past a finite
and a single step, are compared with the numerical results of Kitchens
[45] for a square protuberance in Figure 63. His values are
relatively higher than the present work. However, the trends are
similar. In addition, the comparison of downstream separation and
reattachment points for different steps located in the inviscid-core

region, and for the Reynolds numbers considered, are shown in Figures

 

58
64 and 65. The trend of the downstream separation and reattachment

points is similar for the different steps.

5.4 Qpeimmm Qger-Relamatioh end Weighting Factors

The starting point of the numerical analysis is the
consideration of the full Navier-Stokes equations and Poisson equation
in the entrance region of the unconstricted channel. The rate of
convergence to the solution of the above equations, using a finite-
difference scheme, can be significantly increased by using the optimum
values of the over-relaxation factors (FV) for the Navier-Stokes
equations and (F8) for the Poisson equation, and the optimum values of
the weighting factors (KS) for the stream function and (KV) for the
vorticity.

The purpose of this section is to report the optimum values of
the over-relaxation and weighting factors, often referred to as
accelerating parameters. These parameters depend on the mesh size and
the Reynolds number and significantly minimize the computing time for
the simultaneous solution of Eqs. (4.2.3) and (4.2.4). They are
determined primarily by computer experimentation.

A large number of combinations for Reynolds numbers and mesh
sizes are attempted using different values for the accelerating
parameters (1 to 1.9 for F8 and FV and 0 to l for KS and KV).
Convergent results are obtained with a relative error criterion of
e-10'6, for (Re)H-20, 50, 100, 200, 500, 1000 and 2000 with a mesh
size of h-l/lS and h-l/20. Tables 27 and 28 give the computing time
required for convergence for the range of the Reynolds numbers and
different mesh sizes considered over a range of values of F8. The
mannerical results show that the optimum value of FS depend on the mesh

size (assumed equal in the X- and Y-direction) of the computational

 

 

 

59
domain. It is noted that the computing time decreases as FS increases
until a minimum computing time is achieved; further increases in the
value of FS result in an increase in the computing time. At the
minimum computing time, FS represents the optimum value. The
variation in computing time is more pronounced for high Reynolds
numbers, as shown in Tables 27 and 28.

Reduction in computing time, at least by factor of 2, is
obtained by using the optimum value of the over-relaxation factor FS.
Reducing FS below unity significantly increases the computing time.

It is also found that the results converge most rapidly when FS is
given by Eq. (4.4.1).

Tables 29 and 30 show the effect of the relaxation factor FV
on the computing time for various Reynolds numbers and mesh sizes. It
is noted that as FV increases for a certain value of Reynolds number,
the computing time decreases until a minimum computing time is reached
at the optimum FV value; further increases in the FV value cause the
computing time to increase for low Reynolds numbers of 100 or below.
However, the value of FV is equal to unity for Reynolds numbers of 200
or higher; the numerical solution does not converge for values of FV
slightly greater than unity.

The influence of the weighting factors KS and KV, defined in
Eqs. (4.2.5) and (4.2.6), respectively, which allow for a different
percent of the old and the new values of the stream function and
'vorticity during the matrix iteration, are given in Tables 31-34 for
'various Reynolds numbers and mesh sizes. It is noted that the
computing time decreases as KS decreases to zero for low Reynolds

rnnmbers (Re-20, 50 and 100). It has a value in the range of 0.1 to
0.13 for Reynolds numbers in the range of 200 to 2000 for minimum

computing time. Also, as the value of RV increases, the computing

60
time decreases until a minimum computing time is obtained at an
optimum value of KV; further increases in the value of KV beyond this
optimum value result in an increase in the computing time, as shown in
Tables 33 and 34. A reduction in computing time by a factor of 2 to 4
is possible using the optimum values of the weighting factors KS and
KV.

Finally, for each Reynolds number and mesh size there is an
optimum combination of the values for F8, FV, KS and KV to minimize
computing time. The optimum values, as a function of Reynolds number,
are shown in Figures 66 and 67 for the two different mesh sizes
considered. It may be noted that the optimum value of FS increases as
Reynolds number increases up to 50 for h-l/20 and 500 for h-l/lS; for
higher Reynolds numbers, it approaches a constant value of 1.8, as
shown in Figure 68. On the other hand, the optimum value of FV is
large at low Reynolds number and decreases asymptotically to unity for
high Reynolds numbers, for both mesh sizes, as shown. The optimum
values of KS is nearly zero for Reynolds numbers up to 100 and
increases as Re increases.

The Figures also show that the optimum value of the weighting
factor KV increases as the Reynolds number increases for the range of
the Reynolds numbers considered. For the range of the mesh size
considered, it is noted that for low Reynolds numbers, the over-
relaxation factors FS and FV are a function of Re, while they approach
(constant values of 1.8 and 1, respectively, for high Re as shown in
iFigure 68. Furthermore, the weighting factors KS and KV have
relatively high values for the smaller mesh grid than larger grid as

shown in Figure 69. This Figure also shows that KV has a constant

‘vaJJae for high Re, while KS has a constant value for IOW'Re.

61

For a channel with a constriction in the form of a step
(forward, backward and finite) several runs are also performed using
different values of the accelerating parameters. The numerical
results show that the optimum values of the accelerating parameters,
which are used to reduce the computing time for the channel flow
without a constriction, also represent the optimum values for the
channel flow with a constriction in the form of a step.

In summary, a reduction in computing time, by factors of 1.5
to 4 for mesh size h-1/15 and factors of 2 to 6.6 for h-l/20, is
obtained.by using the optimum values of the accelerating parameters
FS, FV, KS and KV as compared with the unaccelerated case (FS-FV-l and

KS-KV-O); this is shown in Table 35.

5.5 Qanclusiess

A successive over-relaxation method, utilizing optimum
accelerating parameters, is numerically stable for all Reynolds
numbers, step-to-channel ratios and mesh sizes considered. The
entrance region in a rectangular channel with and without a
constriction has been studied using a grid size of 0.05 by 0.05. The

following conclusions are based on the results presented earlier.

Ch e an e Re on

1. By solving the full Navier-Stokes equations, it is found that the
local maxima in the velocity profiles are essentially non-
existent; they are apparently the result of solving modified
Navier-Stokes equations with certain terms neglected or they are

a manifestation of the numerical algorithms.

62
The results show that an actual inlet profile with velocity
gradients near the two walls does not influence the flow in the

entrance region significantly.

The inviscid-core region for the channel flow is approximately
one-fifth of the entrance length, substantially shorter than that
reported for pipe flow. The profile-development region makes up

the remaining four-fifths of the entrance region.

Forwerd Shep

For the downstream region of the step, at least 0.55 of the
channel height is needed using the selected algorithm to satisfy
the fully-developed flow downstream boundary conditions for the
stream function and vorticity. This is true for all Reynolds
numbers considered. Therefore, a step height of 0.4 is used for

the analysis of the flow.

Separation occurs for Reynolds numbers greater than 20; no
separation occurs for a step height of 0.2 of the channel height

for the range of Reynolds numbers considered.

No separation of the fluid downstream of the step is observed at
any Reynolds number; use of very fine grids would be necessary to
obtain this separation and recover the true flow situation in the

region immediately downstream of the step.

63

l. The location of the separation point from the vertical face is a
nonlinear function of step height. The location of the
reattachment point on the lower surface is a linear function of

the step height.

2. The step location has negligible effect on the separation point,
however, it does effect the reattachment point and is more

pronounced for high step-to-channel ratios and Reynolds numbers.

 

For example, for Re-400 (based on step height), the reattachment
point for the step located in the profile-development region is
further downstream than the reattachment point for the step in the
inviscid-core region by 42.5% and 8% further downstream than the

reattachment point for the step in the fully-developed region.

3. The separation point approaches the top of the step for high

Reynolds numbers.

4. The reattachment point is a nonlinear function of Reynolds numbers
(based on the step height) up to 100 and a linear function for

high Reynolds number of 100 to 400.

F te te

1. Both the finite and single step possess a very small upstream

separated region, with length and height almost independent of

Reynolds number, quite unlike the forward step.

64
2. The finite step position and length, and the length of the channel
downstream of the step have negligible effect on the downstream

separation region.

3. The location of the downstream separation point is a nonlinear
function of the finite step height and a constant value for the
single step. The downstream reattachment point is a linear

function of step height for both finite and single steps.

4. For a single step, the streamlines separate from the top of the
step for all Reynolds numbers (based on the downstream channel
height) greater than 20. This is not the case for backward and

finite steps.

5. The location of the downstream reattachment point is a nonlinear
function of Reynolds numbers, based on the step height, up to 60
for a finite step and 15 for a single step, and a linear function

of Reynolds number for higher values.

6. Generally, the upstream and downstream separation regions
introduced by the finite step are smaller than those associated

with the forward and the backward step cases.

Optimmm Accelerating Parameters

1. Generally, for a uniform grid size in a rectangular domain, the
iterated results converge most rapidly when FS is defined by Eq.

(4.4.1) for the range of Reynolds numbers considered. For fine

rfffv—f 4.—

65
mesh, or high Reynolds number, FS is constant and equal to 1.808

as predicted by Eq. (4.4.1).

The fastest rate of convergence of the Navier-Stokes equations is
obtained when FV-l, for high Reynolds number (200 or greater), and

in the range of 1.1 to 1.5 for low Reynolds number.

The values of weighting factors KS and KV for mesh size h-l/20 are

slightly greater than for mesh size h-l/15.

Using optimum values of the accelerating parameters, the maximum
reduction in computing time is a factor of 4 for h-l/lS and a

factor of 6.6 for h-l/20.

The optimum values of the accelerating parameters FS, FV, KS and
KV, which are found in this study for the channel flow without a
constriction, are also applicable for a channel flow with a

constriction in the form of a step.

The optimum values of the four accelerating parameters should
serve as a guide to reduce the computing time for other flow

situations which use this system of equations.

TABLES

66

Table 1. Actual inlet velocity profile

 

 

 

 

Normal distance Velocity Stream function

Y u/UO m

0.00 0.000 0.0000
0.05 0.725 0.0300
0.10 0.875 0.0725
0.15 0.975 0.1175
0.20 1.075 0.1700
0.25 1.100 0.2250
0.30 1.100 0.2800
0.35 1.100 0.3350
0.40 1.100 0.3900
0.45 1.100 0.4450
0.50 1.100 0.5000
0.55 1.100 0.5550
0.66 1.100 0.6100
0.65 1.100 0.6650
0.70 1.100 0.7200
0.75 1.100 0.7750
0.80 1.075 0.8300
0.85 0.975 0.8825
0.90 0.875 0.9275
0.95 0.725 0.9700
1.00 0.000 1.0000

 

 

 

 

67

 

 

 

 

 

 

 

 

 

Table 2. Summary of entrance flow problems studied

(Re)H Inlet velocity Mesh No. of CPU

profile size iterations time
5 Uniform velocity 0.0833 44 0 00:02:35.19
20 Uniform velocity 0.0500 75 0 00:06:10.17
20 Actual velocity 0.0500 64 0 00:05:42.03
50 Uniform velocity 0.0500 96 0 00:20:03.55
50 Actual velocity 0.0500 94 0 00:19:48.42
100 Uniform velocity 0.0500 131 0 00:27:10.24
100 Actual velocity 0.0500 118 0 00:22:03.35
200 Uniform velocity 0.0500 143 0 01:20:13.69
200 Actual velocity 0.0500 136 0 01:13:43.36
500 Uniform velocity 0.0500 210 0 04:46:31.47
500 Actual velocity 0.0500 190 0 04:04:14.62
1000 Uniform velocity 0.0500 233 0 10:50:31.09
2000 Uniform velocity 0.0500 592 l 05:54:53.35

 

 

68

Table 3. Entrance length LE

 

 

 

 

 

 

 

 

 

(Re)H Inlet velocity profile LE LE/H Re
5 Uniform velocity 0.83 0.1666

20 Uniform velocity 0.90 0.0450
20 Actual velocity 1.05 0.0525
50 Uniform velocity 2.20 0.0440
50 Actual velocity 2.40 0.0480
100 Uniform velocity 4.40 0.0440
100 Actual velocity 4.60 0.0460
200 Uniform velocity 8.85 0.0442
200 Actual velocity 9.10 0.0455
500 Uniform velocity 22.15 0.0443
500 Actual velocity 22.55 0.0451
1000 Uniform velocity 44.25 0.0442
2000 Uniform velocity 88.55 0.0443

 

L is the distance at which the velocity at the centerline
reaches 99 percent of the fully developed value.

 

69

Table 4. Comparison of 2LE/H and LE/H Re with other researchers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2LE/H
(Re)H Morihara Schlichting Gillis, et a1. Present work
& Cheng

5 0.33

20 2.24 1.60 2.26 1.80

50 4.40

100 8.80

200 18.06 16.00 18.23 17.70

500 44.30

1000 88.50

2000 171.60 160.00 177.10

LE/H Re

5 0.1666

20 0.0559 0.0400 0.0565 0.0450

50 0.0400 0.0440

100 0.0400 0.0440

200 0.0452 0.0400 0.0456 0.0442

500 0.0400 0.0443

1000 0.0400 0.0442

2000 0.0429 0.0400 0.0443

At large Re limit

Researcher LE/H Re
Present work 0.0443
Schlichting 0.0400
Hwang and Fan 0.0422
Morihara and Cheng 0.0423
Bodoia and Osterle 0.0440
Gillis, et a1. 0.0442
Roidt and Cess 0.0454

70

Table 5. The inviscid-core length, the profile-development length,
and the entrance length for various Reynolds numbers

 

 

 

 

 

 

 

 

Re Li Ld LE LE/H Re Li/H Re Ld/H Re Li/LE
20 0.18 0.72 0.9 0.0450 0.0090 0.0360 0.200
50 0.44 1.76 2.2 0.0440 0.0088 0.0352 0.1999

100 0.88 3.52 4.4 0.0440 0.0088 0.0352 0.2000
200 1.75 7.1 8.85 0 0442 0.0087 0.0355 0.1977
500 4.43 17.72 22.15 0.0443 0.0088 0.0354 0 2000
1000 8.8 35.45 44.25 0.0442 0.0088 0 0354 0.1990
2000 17.5 71.05 88.55 0.0447 0.0087 0 0355 0.1978

 

L - Inviscid-core length
Ld - Profile-development length

LE - Entrance length

3

71

Table 6. Vorticity values in the entrance region of a straight
channel, Re-200

 

 

\ lee

Y \ 0.00025 0.005 ’ 0.00875 0.02 0.05

0.00 -43.814 -13.967 -13.316 -11.633 -11.470
0.05 - 9.629 - 6.574 - 6.233 - 5.507 - 5.435
0.10 - 2.635 - 5.608 - 5.295 - 4.861 - 4.820
0.15 - 0.683 - 4.336 - 4.263 - 4.207 - 4.202
0.20 - 0.164 - 2.903 - 3.172 - 3.550 - 3.583
0.25 - 0.036 - 1.664 - 2.137 - 2.896 - 2.965
0.30 - 0.007 - 0.819 - 1.288 - 2.259 - 2.354
0.35 - 0.001 - 0.348 - 0.690 - 1.650 - 1.751
0.40 - 0.000 - 0.127 - 0.325 - 1.074 - 1.159
0.45 0.000 - 0.038 - 0.122 - 0.528 - 0.576
0.50 0 000 0 000 0.000 0 000 0.000
0.55 0.000 0.038 0.122 0.528 0.576
0.60 0.000 0 127 0.325 1.074 1.159
0.65 0.001 0.348 0 690 1.650 1.751
0.70 0 007 0 819 1.288 2.259 2.354
0.75 0.036 1.664 2.137 2.896 2.965
0.80 0.164 2.903 3.172 3.550 3.583
0.85 0.683 4.336 4.263 4.207 4.202
0.90 2.635 5 608 5.295 4.861 4.820
0.95 9.629 6.574 6.233 5.507 5.435
1.00 43.814 13.967 13.316 11.633 11.470

 

I

 

72

 

 

Table 7. Summary of cases studied for forward step

Step Step Computational CPU

(Re)H height position domain purpose time
20 0.40H Inviscid-core 1.5H 0 00:01:40 27
20 0.40H Profile-dev 2.0H 0 00:02:55.10
20 0.40H Fully-dev 2.5H Effect 0 00:05:37.22
50 0.40H Inviscid-core 3.0H of O 00:12:57.57
50 0.40H Profile-dev 4.0H step 0 00:15:26.18
50 0.40H Fully-dev 6.0H position 0 00:14:17.52
200 0.40H Inviscid-core 10.5H on the 0 00:13:16.79
200 0.40H Profile-dev 15.0H separation 0 00:30:21.31
200 0.40H Fully-dev 12.0H region 0 01:09:14.29
2000 0.40H Inviscid 110.0H 0 08:25:49.09
2000 0.40H Profile-dev 130.0H 0 10:32:07.14
2000 0.40H Fully-dev 110.0H l 07:03:24.28
200 0.45H Inviscid-core 10.5H 0 01:52:25.10
200 0.40H Inviscid-core 10.5H 0 00:13:16.79
200 0.35H Inviscid-core 10.5H 0 00:14:06.21
200 0.30H Inviscid-core 10.5H Effect 0 00:14:42.46
200 0.25H Inviscid-core 10.5H of 0 00:15:10.48
200 0.20H Inviscid-core 10.5H step 0 00:15:58.90
1000 0.45H Inviscid-core 60.0H height 0 15:20:42.20
1000 0.40H Inviscid-core 60.0H on the 0 05:01:07.95
1000 0.35H Inviscid-core 60.0H separation 0 04:59:12.72
1000 0.30H Inviscid-core 60.0H region 0 04:51:41.50
1000 0.25H Inviscid-core 60.0H 0 04:20:55.28
1000 0.20H Inviscid-core 60.0H 0 04:01 11.37
50 0.40H Inviscid-core 3.0H 0 00:12:57.57
100 0.40H Inviscid-core 6.5H 0 00:06:55.01
200 0.40H Inviscid-core 10.5H 0 00:13:16.79
500 0.40H Inviscid-core 32.5H 0 01:42:27.66
1000 0.40H Inviscid-core 60.0H 0 05:01:07.95
2000 0.40H Inviscid-core 110.0H 0 08:25:49.09
50 0.40H Profile-dev 4.0H Effect 0 00:15:26.18
100 0.40H Profile-dev 8.0H of O 00:11:44.22
200 0.40H Profile-dev 15.0H Reynolds 0 00:30:21.31
500 0.40H Profile-dev 38.5H number 0 02:08:35.41
1000 0.40H Profile-dev 70.0H on 0 08:58:23.21
2000 0.40H Profile-dev 130.0H separation 0 10:32:07.14
50 0.40H Fully-dev 6.0H region 0 00:14:17.52
100 0.40H Fully-dev 12.0H 0 00:53:59.04
200 0.40H Fully-dev 20.0H 0 01:09:14.29
500 0.40H Fully-dev 50.0H 0 05:27:02.47
1000 0.40H Fully-dev 100.0H l 01:25:49.72
2000 0.40H Fully-dev 110.0H l 07:03:24.28

 

73

Table 8. The Y-values for selected streamlines for the flow past a
forward step located in the profile-development region,
(Re)H-20

 

X W - 0.005 W - 0.1 u - 0.2 p - 0.5 W - 0.7 w - 0.8

 

0.05 .0090 .1320 .2370 .5312 .7106 .8000
0.10 .0160 .1643 .2754 .5585 .7196 .8000
0.15 .0280 .1986 .3158 .5824 .7280 .8000
0.20 .0470 .2365 .3584 .6040 .7363 .8010
0.25 .0591 .2821 .4030 .6233 .7446 .8056
0.30 .0807 .3385 .4467 .6408 .7528 .8106
0.35 .1208 .4016 .4843 .6564 .7607 .8157
0.40 .2130 .4650 .5153 .6698 .7680 .8206
0.45 .4050 .4811 .5380 .6809 .7744 .8250
0.50 .4080 .4996 .5535 .6898 .7794 .8317
0.55 .4111 .5090 .5635 .6966 .7838 .8340
0.60 .4130 .5153 .5703 .7018 .7870 .8355
0.65 .4142 .5190 .5748 .7054 .7893 .8360
0.70 .4147 .5212 .5776 .7079 .7903 .8365
0.75 .4148 .5223 .5792 .7093 .7916 .837
0.80 .4147 .5227 .5798 .7100 .7918 .8371
0.85 .4146 .5226 .5800 .7101 .7918 .8369
0.90 .4144 .5222 .5796 .7099 .7914 .8364
0.95 .4142 .5217 .5790 .7093 .7908 .8359
1.00 .4140 .5210 .5783 .7086 .7901 .8352
1.10 .4138 .5203 .5775 .7077 .7893 .8345
1.15 .4136 .5197 .5767 .7068 .7885 .8338
1.20 .4135 .5191 .5759 .7060 .7877 .8324
1.25 .4132 .5180 .5745 .7043 .7864 .8319
1.30 .4131 .5176 .5739 .7036 .7857 .8315
1.35 .4130 .5171 .5733 .7029 .7851 .8311
1.40 .4128 .5167 .5728 .7024 .7847 .8308
1.45 .4127 .5163 .5722 .7018 .7843 .8305
1.50 .4124 .5158 .5716 .7014 .7841 .8302
1.55 .4122 .5151 .5710 .7010 .7840 .8302
1.60 .4120 .5144 .5704 .7000 .7839 .8302
1.65 .4120 .5142 .5703 .7000 .7839 .8302
l

.70 .4120 .5142 .5703 .7000 .7839 .8302

 

l

74

 

 

Table 9. The Y-values for selected streamlines for the flow past a
forward step located in the profile-development region,
(Re)H- 200
X m - 0.005 8 - 0.1 w - 0.2 u - 0.5 w - 0.6 W - 0.8
0.05 .0070 .1169 .2110 .5000 .5960 .7880
0.20 .0190 .1486 .2363 .5000 .5888 .7637
0.30 .0240 .1591 .2461 .5000 .5852 .7539
0.50 .0270 .1691 .2570 .5000 .5802 .7430
0.75 .0280 .1742 .2618 .5000 .5785 .7374
1.00 .0280 .1767 .2660 .5000 .5770 .7342
1.50 .0290 .1803 .2703 .5000 .5754 .7303
2.00 .0302 .1830 .2740 .5000 .5742 .7270
2.50 .0316 .1866 .2770 .5020 .5740 .7262
3.00 .0330 .1907 .2820 .5059 .5760 .7264
3.20 .0341 .1930 .2846 .5070 .5779 .7277
3.40 .0357 .1964 .2882 .5102 .5807 .7299
3.60 .0384 .2008 .2932 .5149 .5850 .7333
3.80 .0417 .2065 .3005 .5218 .5915 .7385
4.00 .0480 .2150 .3104 .5320 .6010 .7460
4.20 .0527 .2283 .3256 .5476 .6158 .7583
4.40 .0595 .2511 .3502 .5705 .6371 .7751
4.60 .0670 .2660 .3893 .6040 .6512 .7850
4.80 .0130 .3660 .4000 .6480 .7040 .8212
4.90 .2240 .4410 .5090 .6710 .7230 .8322
4.95 .4041 .4730 .5300 .6812 .7312 .8366
5.05 .4105 .5040 .5570 .6975 .7445 .8360
5.15 .4166 .5174 .5714 .7084 .7535 .8490
5.25 .4190 .5240 .5790 .7150 .7590 .8510
5.35 .4190 .5274 .5835 .7185 .7619 .8527
5.45 .4180 .5280 .5855 .7190 .7630 .8520
5.55 .4169 .5276 .5819 .7195 .7622 .8508
5.65 .4159 .5260 .5840 .7180 .7614 .8490
5.75 .4151 .5248 .5826 .7171 .7593 .8472
5.85 .4142 .5230 .5812 .7150 .7572 .8450
5.95 .4140 .5219 .5798 .7139 .7558 .8434
6.05 .4136 .5210 .5785 .7120 .7540 .8420
6.25 .4132 .5190 .5764 .7099 .7514 .8391
6.55 .4129 .5175 .5745 .7062 .7485 .8363
6.75 .4128 .5169 .5738 .7054 .7472 .8351
6.95 .4127 .5167 .5732 .7050 .7453 .8342
7.25 .4127 .5167 .5730 .7050 .7434 .8332
7.75 .4127 .5167 .5730 .7050 .7435 .8310
8.05 .4127 .5167 .5730 .7030 .7435 .8310
9.05 .4127 .5167 .5730 .7030 .7435 .8310
111.05 .4127 .5167 .5730 .7030 .7435 .8310
11.05 .4127 .5167 .5730 .7030 .7435 .8310
12.05 .4127 .5167 .5730 .7030 .7435 .8310
13.05 .4127 .5167 .5730 .7030 .7435 .8310
14.05 .4127 .5167 .5730 .7030 .7435 .8310
14. 95 .4127 .5167 .5730 .7030 .7435 .8310

 

75

 

 

Table 10 . The Y-values for selected streamlines for the flow past a
forward step located in the profile-development region,
(Re)H - 2000
X ¢-0.005 w—0.l ¢-0.2 w—0.5 w—0.6 u-O.8
0.05 .0230 .1600 .2482 .5000 .5831 .7520
2.00 .0242 .1639 .2518 .5000 .5826 .7482
4.00 .0247 .1672 .2557 .5000 .5811 .7443
6.00 .0256 .1702 .2590 .5000 .5798 .7410
8.00 .0264 .1727 .2618 .5000 .5786 .7382
10.00 .0271 .1748 .2642 .5000 .5776 .7358
12.00 .0277 .1767 .2664 .5000 .5767 .7336
14.00 .0283 .1783 .2681 .5000 .5759 .7319
16.00 .0288 .1798 .2698 .5000 .5751 .7302
18.00 .0292 .1811 .2713 .5000 .5745 .7288
20.00 .0298 .1826 .2729 .5004 .5743 .7279
22.00 .0324 .1868 .2773 .5038 .5770 .7298
23.00 .0399 .1967 .2875 .5134 .5861 .7378
23.25 .0448 .2017 .2929 .5186 .5899 .7421
23.50 .0507 .2082 .3006 .5260 .5981 .7482
23.75 .0544 .2156 .3086 .5341 .6083 .7572
24.00 .0606 .2324 .3265 .5517 .6227 .7697
24.20 .0700 .2506 .3449 .5688 .6388 .7829
24.40 .0906 .2739 .3703 .5923 .6604 .7990
24.60 .1187 .3122 .4093 .6243 .6890 .8205
24.70 .1470 .3407 .4367 .6439 .7058 .8317
24.80 .1787 .3815 .4724 .6651 .7235 .8427
24.90 .2473 .4450 .5149 .6861 .7405 .8531
24.95 .4042 .4762 .5354 .6957 .7483 .8579
25.05 .4115 .5073 .5625 .7118 .7616 .8659
25.15 .4225 .5229 .5787 .7236 .7716 .8719
25.25 .4347 .5329 .5891 .7316 .7785 .8760
25.35 .4449 .5394 .5958 .7368 .7829 .8785
25.45 .4500 .5432 .5999 .7400 .7856 .8799
25.55 .4470 .5451 .6017 .7416 .7869 .8805
25.75 .4388 .5456 .6022 .7414 .7865 .8796
25.95 .4325 .5423 .5994 .7388 .7839 .8776
26.25 .4257 .5367 .5934 .7340 .7796 .8740
26.55 .4220 .5313 .5884 .7363 .7764 .8718
26.75 .4205 .5290 .5865 .7289 .7751 .8709
27.05 .4192 .5275 .5853 .7279 .7741 .8700
27.55 .4181 .5268 .5848 .7271 .7733 .8689
28.05 .4172 .5263 .5844 .7266 .7726 .8679
29.05 .4161 .5254 .5836 .7252 .7709 .8658
i30.05 .4149 .5213 .5788 .7211 .7670 .8622
4().00 .4087 .5005 .5559 .7028 .7500 .8618
5C).00 .4080 .5000 .5500 .7000 .7500 .8616
6C).00 .4077 .5000 .5500 .7000 .7500 .8612
70. 00 .4072 .5000 .5500 .7000 . 7500 .8608
90 . 00 . 4050 . 5000 . 5500 . 7000 . 7500 . 8600
1 10 . 00 . 4050 . 5000 . 5500 . 7000 . 7500 . 8600
130 . 00 . 4050 . 5000 . 5500 . 7000 . 7500 . 8600

 

.1

Table I

f9
C)

C.)

J

A» be.) Pm» r ,
c.) c. C-) k

r .. 7 I“. 't
(._,‘I ‘s. '

(1

.)

:1»
J

i ‘1‘,

.5)

.5
5

La",
‘10:]
.V

:’]A
...:U'

o’
VJ

76

 

 

Table 11. Summary of cases studied for backward step
Step Step Computational CPU
(Re)H height position domain purpose time
20 0.2H Inviscid-core 2.2H 0 00:04:43.21
20 0.2H Profile-dev 2.6H 0 00:10:08.95
20 0.2H Fully—dev 2.2H 0 00:08:10.08
20 0.3H Inviscid-core 2.2H Effect 0 00:07:45.44
20 0.3H Profile-dev 2.6H of 0 00:18:20.34
20 0.3H Fully-dev 2.2H step 0 00:18:19.98
20 0.4H Inviscid-core 2.2H position 0 00:09:07.74
20 0.4H Profile-dev 2.6H on 0 00:14:48.01
20 0.4H Fully-dev 2.2H separation 0 00:12:50.17
200 0.4H Inviscid-core 11.0H region 0 02:40:35.81
200 0.4H Profile-dev 14.0H 0 04:59:37.56
200 0.4H Fully-dev 12.0H 0 03:12:28.17
2000 0.2H Inviscid-core 110.0H l 20:21:31.61
2000 0.2H Profile-dev 130.0H 4 06:51:40.12
2000 0.2H Fully-dev 110.0H 2 23:32:51.82
20 0.2H Inviscid-core 2.2H 0 00:04:43.21
20 0.3H Inviscid-core 2.2H 0 00:07:45.44
20 0.4H Inviscid-core 2.2H Effect 0 00:09:07.74
20 0.5H Inviscid-core 2.2H of 0 00:10:47.43
200 0.2H Inviscid-core 11.0H step 0 00:49:04.14
200 0.3H Inviscid-core 11.0H height 0 01:15:36.18
200 0.4H Inviscid-core 11.0H on 0 02:40:35.81
200 0.5H Inviscid-core ll.0H separation 0 04:10:14.78
500 0.2H Inviscid-core 32.5H region 0 06:26:52.48
500 0.3H Inviscid-core 32.5H 0 10:47:03.67
500 0.4H Inviscid-core 32.5H l 03:50:45.79
20 0.3H Fully-dev 2.2H Effect of 0 00:18:19.98
50 0.3H Fully-dev 4.0H Reynolds 0 00:44:33.64
100 0.3H Fully-dev 8.0H number on 0 00:24:51.98
200 0.3H Fully-dev 12.0H separation 0 02:08:35.05
500 0.3H Fully-dev 32.5H region 0 10:47:03.67
20 0.2H Inviscid-core 2.2H 0 00:04:43.21
50 0.2H Inviscid-core 3.4H 0 00:11:00.81
100 0.2H Inviscid-core 6.5H 0 00:31:52.23
200 0.2H Inviscid-core 11.0H Effect 0 00:49:04.14
500 0.2H Inviscid-core 32.5H of 0 06:26:52.48
1000 0.2H Inviscid-core 55.0H Reynolds 0 12:52:31.50
2000 0.2H Inviscid-core 110.0H number on 1 20:21:31.61
20 0.2H Profile-dev 2.2H separation 0 00:08:10.08
50 0.2H Profile-dev 6.0H and 0 00:35:28.24
100 0.2H Profile-dev 8.5H reattachment 0 01:01:18.85
200 0.2H Profile-dev 14.0H points 0 02:02:19.48
500 0.2H Profile-dev 38.5H 0 09:35:33.49
1000 0.2H Profile-dev 70.0H 1 01:05:31.14
2000 0.2H Profile-dev 130.0H 4 06:51:40.12

 

 

Table 11 (cont'd.)

20
50
100
200
500
1000
2000

OOOOOOO

.2H
.2H
.2H
.2H
.2H
.2H
.2H

Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev

12.
32.
55.
110.

77

.2H
.0H
.5H

0H
5H
0H
0H

NOOOOOO

00:
00:
00:
01:
06:
14:
23:

08

29

24:
36:

32

:10.
13: .
52:20.
:09.
38.
38.

:51.

 

78

Table 12. The Y-values for selected streamlines for the flow past a
backward step in the profile-development region, (Re)H-20

 

 

X ¢-0.05 ¢-0.l ¢-0.2 ¢-0.5 ¢-0.7 ¢-0.8
0.00 .2050 .3000 .3750 .5985 .7475 .8220
0.05 .2066 .3011 .3760 .5976 .7463 .8210
0.10 .2100 .3156 .3882 .5954 .7350 .8058
0.15 .2130 .3254 .3969 .5934 .7258 .7940
0.20 .2154 .3317 .4027 .5914 .7184 .7850
0.25 .2166 .3355 .4060 .5892 .7123 .7779
0.30 .2169 .3371 .4074 .5869 .7072 .7723
0.35 .2163 .3366 .4071 .5843 .7027 .7675
0.40 .2143 .3340 .4053 .5814 .6986 .7634
0.45 .2111 .3291 .4019 .5782 .6949 .7597
0.50 .1812 .3217 .3969 .5746 .6914 .7562
0.55 .1529 .3125 .3903 .5706 .6879 .7528
0.60 .1206 .3025 .3828 .5663 .6843 .7500
0.65 .1017 .2900 .3746 .5617 .6808 .7466
0.70 .0795 .2776 .3663 .5569 .6771 .7438
0.75 .0672 .2664 .3581 .5520 .6734 .7409
0.80 .0649 .2565 .3502 .5471 .6699 .7381
0.85 .0558 .2473 .3417 .5422 .6664 .7353
0.90 .0528 .2381 .3340 .5375 .6629 .7326
0.95 .0508 .2304 .3271 .5329 .6596 .7300
1.00 .0479 .2241 .3209 .5287 .6564 .7276
1.05 .0447 .2188 .3155 .5247 .6534 .7253
1.10 .0423 .2144 .3108 .5210 .6506 .7232
1.15 .0404 .2108 .3067 .5176 .6482 .7213
1.20 .0389 .2077 .3031 .5145 .6460 .7195
1.25 .0378 .2052 .3001 .5118 .6440 .7179
1.30 .0369 .2031 .2972 .5093 .6423 .7165
1.35 .0361 .2014 .2947 .5072 .6408 .7153
1.40 .0356 .2000 .2926 .5054 .6394 .7143
1.45 .0351 .1985 .2909 .5037 .6383 .7134
1.50 .0348 .1974 .2895 .5024 .6373 .7126
1.55 .0345 .1964 .2883 .5012 .6365 .7120
1.60 .0343 .1957 .2874 .5002 .6359 .7116
1.65 .0342 .1951 .2866 .5000 .6354 .7112
1.70 .0340 .1947 .2860 .4988 .6350 .7109
1.75 .0340 .1943 .2856 .4984 .6347 .7108
1.80 .0340 .1941 .2853 .4980 .6346 .7107
1.85 .0340 .1939 .2850 .4978 .6345 .7107
1.90 .0340 .1938 .2848 .4976 .6344 .7107
1.95 .0340 .1938 .2848 .4975 .6344 .7107
2.00 .0340 .1938 .2848 .4975 .6344 .7107
2.05 .0340 .1938 .2848 .4975 .6344 .7107
2.11) .0340 .1938 .2848 .4975 .6344 .7107
2.315 .0340 .1938 .2848 .4975 .6344 .7107
2.20 .0340 .1938 .2848 .4975 .6344 .7107
2. 25 .0340 .1938 .2848 .4975 .6344 .7107

2.30 .0340 .1938 .2848 .4975 .6344 .7107

2.135 .0340 .1938 .2848 .4975 .6344 .7107

 

 

 

i

 

 

 

79

 

 

Table 13. The Y—values for selected streamlines for the flow past a
backward step in the inviscid core region, (Re)H-200

X ¢'0-005 w—0.l ¢-0.2 ¢—0.5 ¢-0.7 ¢-0.8
0.05 .2119 .3222 .3930 .6000 .7380 .8065
0.20 .2139 .3232 .3940 .6000 .7378 .8059
0.40 .2178 .3357 .4063 .6000 .7286 .7936
0.80 .2184 .3420 .4133 .6000 .7230 .7866
1.20 .2189 .3449 .4164 .6000 .7203 .7835
1.80 .2196 .3480 .4196 .5998 .7173 .7800
2.40 .2201 .3500 .4216 .5994 .7146 .7773
3.00 .2201 .3503 .4219 .5979 .7119 .7742
3.50 .2191 .3477 .4190 .5933 .7049 .7692
3.60 .2187 .3464 .4177 .5916 .7033 .7669
3.70 .2180 .3445 .4158 .5894 .7017 .7646
3.80 .2171 .3419 .4133 .5867 .7001 .7623
3.90 .2152 .3380 .4098 .5834 .6968 .7599
3.95 .2134 .3353 .4077 .5815 .6951 .7582
4.00 .2085 .3320 .4052 .5794 .6932 .7564
4.05 .2039 .3285 .4025 .5771 .6913 .7545
4.10 .2000 .3245 .3994 .5747 .6892 .7525
4.15 .1867 .3204 .3958 .5721 .6870 .7503
4.20 .1771 .3160 .3920 .5713 .6877 .7482
4.25 .1683 .3116 .3881 .5666 .6823 .7461
4.30 .1603 .3071 .3840 .5637 .6799 .7441
4.35 .1529 .3025 .3800 .5607 .6773 .7420
4.40 .1403 .2974 .3759 .5577 .6748 .7398
4.45 .1268 .2917 .3718 .5547 .6722 .7375
4.50 .1165 .2862 .3677 .5516 .6696 .7354
4.60 .1017 .2758 .3598 .5456 .6644 .7311
4.70 .0814 .2665 .3524 .5397 .6593 .7269
4.80 .0693 .2581 .3447 .5341 .6545 .7227
4.90 .0623 .2507 .3375 .5289 .6500 .7185
5.00 .0577 .2427 .3311 .5242 .6463 .7153
5.20 .0524 .2299 .3205 .5161 .6395 .7090
5.40 .0489 .2210 .3125 .5098 .6355 .7050
5.60 .0443 .2148 .3066 .5056 .6315 .7023
5.80 .0417 .2105 .3024 .5023 .6293 .7009
6.00 .0402 .2076 .2995 .5001 .6277 .6993
6.40 .0386 .2044 .2952 .4988 .6273 .6998
6.80 .0378 .2029 .2940 .4985 .6283 .7016
7.40 .0375 .2018 .2932 .5000 .6305 .7040
7.60 .0371 .2012 .2929 .5000 .6317 .7053
8.20 .0368 .2006 .2921 .5000 .6326 .7066
8.60 .0364 .2001 .2913 .5000 .6334 .7080
9.20 .0360 .1992 .2903 .5000 .6342 .7096
10.00 .0357 .1982 .2894 .5000 .6350 .7107
1!).60 .0357 .1976 .2892 .5000 .6356 .7113
311.20 .0357 .1976 .2892 .5000 .6350 .7116
12 . 20 .0357 .1976 .2892 .5000 .6350 .7121
12 . 80 .0357 .1976 .2892 .5000 .6350 .7121
13 .40 .0357 .1976 .2892 .5000 .6350 .7121
14. 00 .0357 .1976 .2892 .5000 .6350 .7121

 

.1

.. 6!... a; «but \. o A rs All. Ikv .1: I.“ .....IU alt ...HtHJ p§a any... .s - .. ... .\.J ..t.» .54
. it a, .1 .. \J ' ;. v. n ‘ . .... ... a . ...: .. \ .. n v. r, L. o s N
. a . a - Q . a . a s Q « ~ . a u a u

.\J .3 a) all. 9 0U Ad. Ad. AIJ Ahlv AU “.14 films a“. n. 5.1. «..U fixJ ....IU «...... A) «UN. uald
u. f. 0.3 .

n

0

 

 

 

 

Table 14. The Y-values for selected streamlines for the flow past a
backward step in the profile-development region, (Re)H-2OOO
X ¢-0.005 ¢-0.1 ¢-0.2 ¢'0-5 ¢-0.7 w-O.8
0.00 .2220 .3310 .4000 .6000 .7330 .7991
0.55 .2200 .3320 .4019 .6000 .7320 .7981
1.05 .2173 .3324 .4030 .6000 .7310 .7970
1.55 .2162 .3326 .4034 .6000 .7308 .7967
2.05 .2162 .3329 .4040 .6000 .7302 .7959
4.05 .2167 .3360 .4074 .6000 .7276 .7925
6.05 .2173 .3385 .4100 .5998 .7253 .7897
8.05 .2174 .3399 .4114 .5989 .7225 .7865
9.05 .2168 .3388 .4103 .5955 .7193 .7833
9.55 .2157 .3365 .4080 .5933 .7156 .7796
9.65 .2154 .3356 .4072 .5922 .7145 .7785
9.75 .2150 .3346 .4062 .5910 .7132 .7773
9.85 .2145 .3334 .4050 .5896 .7117 .7758
9.95 .2137 .3318 .4035 .5879 .7100 .7742
10.00 .2133 .3309 .4026 .5870 .7091 .7733
10.10 .2122 .3287 .4006 .5850 .7070 .7714
10.20 .2096 .3260 .3982 .5827 .7049 .7692
10.30 .2064 .3230 .3953 .5801 .7022 .7668
10.40 .2032 .3197 .3921 .5773 .6995 .7642
10.50 .2005 .3160 .3881 .5742 .6966 .7614
10.60 .1921 .3122 .3849 .5710 .6935 .7584
10.80 .1768 .3039 .3769 .5639 .6869 .7518
11.20 .1528 .2833 .3597 .5484 .6725 .7385
11.40 .1330 .2730 .3509 .5405 .6651 .7316
11.60 .1161 .2633 .3416 .5327 .6578 .7249
11.80 .1040 .2544 .3329 .5254 .6508 .7185
12.00 .0859 .2455 .3251 .5186 .6445 .7125
12.50 .0604 .2270 .3099 .5051 .6322 .7004
13.00 .0528 .2166 .3008 .4969 .6277 .6938
13.50 .0502 .2117 .2961 .4932 .6216 .6912
14.00 .0492 .2100 .2948 .4927 .6216 .6916
14.50 .0494 .2102 .2953 .4950 .6232 .6935
15.00 .0497 .2105 .2964 .4955 .6252 .6958
15.50 .0497 .2110 .2974 .4970 .6268 .6978
16.00 .0492 .2116 .2980 .4982 .6285 .6995
16.50 .0483 .2109 .2980 .4987 .6292 .7005
17.00 .0472 .2105 .2981 .4993 .6300 .7016
18.00 .0451 .2092 .2973 .4994 .6306 .7026
19.00 .0433 .2079 .2963 .4996 .6308 .7032
20.00 .0422 .2067 .2955 .4998 .6310 .7038
25.00 .0380 .2030 .2925 .5000 .6320 .7058
.30.00 .0370 .2007 .2910 .5000 .6320 .7078
40.00 .0360 .1984 .2895 .5000 .6320 .7090
50.00 .0350 .1975 .2881 .5000 .6320 .7100
60.00 .0350 .1965 .2878 .5000 .6320 .7100
70.00 .0348 .1962 .2876 .5000 .6320 .7100
8C).00 .0347 .1959 .2872 .5000 .6320 .7100
9C).00 .0346 .1959 .2868 .5000 .6320 .7100

 

 

81

 

 

Table 15. Summary of cases studied for a finite step
Step Step Step Computational CPU
(Re)H height position length domain purpose time
20 0.38 Fully-dev 1.8 2.98 Effect of 0 00:32:27.79
20 0.38 Fully-dev 1.8 4.48 downstream 0 01:04:01.78
200 0.38 Inviscid-core 1.8 15.58 length on 0 02:39.36.21
200 0.38 Inviscid-core 1.8 18.08 separation 0 03:25:55.59
region
20 0.38 Fully-dev 2.8 3.98 Effect of 0 00:27:14.46
20 0.38 Fully-dev 1.8 2.98 step 0 00:22:45.18
200 0.38 Inviscid-core 1.8 15.58 length on 0 02:39:36.21
200 0.38 Inviscid-core 2.8 16.58 separation 0 02:47'54.68
200 0.38 Inviscid-core 4.8 18.58 region 0 03:40 00 39
20 0.38 Inviscid-core 1.8 2.68 0 00:16:34.08
20 0.38 Profile-dev 1.8 3.18 0 00:22:45.18
20 0.38 Fully-dev 1.8 2.98 Effect of 0 00:32:27.79
200 0.38 Inviscid-core 1.8 15.58 step 0 02:39:36.21
200 0.38 Profile-dev 1.8 17.28 position 0 03:08:48.26
200 0.38 Fully-dev 1.8 16.28 on 0 04:02:22 66
500 0.38 Inviscid-core 1.8 36.58 separation 0 11:36:29 60
500 0.38 Profile-dev 1.8 45.08 region 0 13:17:28.62
500 0.38 Fully-dev 1.8 39.08 0 18 30:30.93
20 0.28 Inviscid-core 1.8 2.68 O 00:15:33.99
20 0.38 Inviscid-core 1.8 2.68 0 00:16:34.08
20 0.48 Inviscid-core 1.8 2.68 0 00:18:41.81
20 0.58 Inviscid-core 1.8 2.68 Effect of 0 00:23:12.63
200 0.28 Inviscid-core 1.8 15.58 step 0 03:43:51.59
200 0.38 Inviscid-core 1.8 15.58 height 0 02:39:36.21
200 0.48 Inviscid-core 1.8 15.58 on 0 08:51:26 13
200 0.58 Inviscid-core 1.8 15.58 separation 0 14:42:32.75
500 0.28 Inviscid-core 1.8 36.58 region 0 17:22:51.43
500 0.38 Inviscid-core 1.8 36.58 0 11:36:29.60
500 0.48 Inviscid-core 1.8 36.58 1 06:12:04.45
20 0.38 Inviscid-core 1.8 2.68 0 00:16:34.08
50 0.38 Inviscid-core 1.8 4.88 0 00:53:41.08
100 0.38 Inviscid-core 1.8 8.48 O 02:23:27.22
200 0.38 Inviscid-core 1.8 15.58 0 02:39:36.21
500 0.38 Inviscid-core 1.8 36.58 0 11:36:29.60
1000 0.38 Inviscid-core 1.8 72.58 Effect of 2 01:37:59.41
1300 0.38 Inviscid-core 1.8 88.58 Reynolds 3 02:49:26 68
20 0.38 Profile-dev 1.8 2.98 number 0 00:22:45.18
50 0.38 Profile-dev 1.8 5.88 on 0 01:06:13.55
100 0.38 Profile-dev 1.8 10.18 separation 0 02:44:28.51
200 0.38 Profile-dev 1.8 17.28 region 0 03:08:48.26
500 0.38 Profile-dev 1.8 45.08 0 13:17:28.62
1000 0.38 Profile-dev 1.8 87.28 2 11:06:04.09
1300 0.38 Profile-dev 1.8 103.58 3 14:56:18.67

 

Table 15 (cont'd )

20
50
100
200
500
1000
1300

0006600

.38
.38
.38
.38
.38
.38
.38

Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev
Fully-dev

HHHHHI—‘H
2222222

82

16

77

.98
.38
.68
.28
39.

08

.28
88.

90000000

 

83

 

 

 

 

 

Table 16. Effect of downstream length on the downstream separation

streamline Y-coordinate for the finite step of 0.38 located
in the fully-developed region, (Re)H - 20

X Length-1.358 Length=2.858

0.00 0.2200 (Ys) 0.2200 (Ys)

0.05 0.2190 0.2190

0.10 0.2021 0.2020

0.15 0.1665 0.1664

0.20 0.1284 0.1283

0.25 0.0774 0.0771

0.26 (Xr) 0.0000 0.0000

Table 17. Effect of downstream length on the downstream separation

streamline Y-coordinate for the finite step of 0.38 located
in the inviscid-core region, (Re)H - 200

X length-13.258 length-17.758

Y Y

0.00 0.2788 (Ys) 0.2788 (Ys)

0.05 0.2785 0.2785

0.15 0.2769 0.2768

0.25 0.2692 0.2692

0.35 0.2594 0.2593

0.45 0.2471 0.2471

0.55 0.2257 0.2257

0.65 0.2078 0.2077

0.75 0.1825 0.1825

0.85 0.1575 0.1574

0.95 0.1212 0.1211

1.05 0.0813 0.0812

1.21 (Xr) 0.0000 0.0000

 

84

Table 18. Effect of step length on the downstream separation
streamline Y-coordinate for the finite step of height of
0.38 located in the inviscid-core region, (Re)H-20

 

 

Step length-28 Step lengthan
X Y Y
0.00 0.2205 (Ys) 0.2202 (Ys)
0.05 0.2203 0.2190
0.10 0.2029 0.2019
0.15 0.1685 0.1656
0.20 0.1325 0.1284
0.25 0.0844 0.0791
0.26 (Xr) 0.0000 0.0000

 

Table 19. Effect of step position on the downstream separation
streamline Y-coordinate for the finite step of height 0.3H,

 

(Re)H-20
Inviscid—core Profile-development Fully-developed
X Y Y Y
0.00 0.2200 (Ys) 0.2200 (Ys) 0.2200 (Ys)
0.05 0.2190 0.2192 0.2191
0.10 0.2019 0.2020 0.2019
0.15 0.1665 0.1666 0.1665
0.20 0.1284 0.1285 0.1284
0.25 0.0771 0.0772 0.0771
0.26 (Xr) 0.0000 0.0000 0.0000

 

 

85

Table 20. Effect of step position on the downstream separation
streamlines Y-coordinate for the finite step of height 0.3H,

 

 

(Re)H-200
X Inviscid-core Profile-development Fully-developed
0.00 0.2800 (Ys) 0.2805 (Ys) 0.2810 (Ys)
0.05 0.2794 0.2797 0.2794
0.15 0.2780 0.2789 0.2781
0.25 0.2705 0.2707 0.2706
0.35 0.2608 0.2610 0.2609
0.45 0.2500 0.2508 0.2502
0.55 0.2286 0.2291 0.2287
0.65 0.2100 0.2109 0.2105
0.75 0.1872 0.1880 0.1878
0.85 0.1611 0.1613 0.1612
0.95 0.1279 0.1284 0.1282
1.05 0.1001 0.1010 0.1006
1.15 0.0508 0.0520 0.0516
1.30 X(r) 0.0000 0.0000 0.0000

 

QDL‘ .71

16 LL

all“

J

v‘d
\‘J

NJ 0 PJ Rave
filWJ nIV .\... .3 {H.-
nld. .thub. - I .v. t. It r u

. ‘J A ..-
f: V! 1:
killhv. l . In

.\J 0 S nan...
Aliii

‘wll

n... ..i

.J

-. ..IU

B:

-

.Jl

{v.1

..IrU ab
. . L {1‘
u 4 ...I.

86

Table 21. The Y-values for selected streamlines for the flow past a
finite step located in the profile-development region,

 

 

(Re)H-20

}( $—0.005 ¢-0.1 ¢-0.2 ¢-0.5 w-0.7 ¢-0.8
().05 .0101 .1627 .2411 .5387 .7008 .7801
().10 .0159 .1800 .2847 .5594 .7054 .7820
(3.20 .0413 .2182 .3278 .5706 .7124 .7833
().25 .0552 .2532 .3597 .5843 .7169 .7843
().30 .0708 .2919 .3905 .5971 .7218 .7863
0.35 .1067 .3336 .4186 .6086 .7268 .7890
0.40 .1766 .3700 .4430 .6189 .7318 .7919
0.45 .3061 .3971 .4619 .6279 .7364 .7948
().50 .3099 .4126 .4757 .6355 .7406 .7975
0.55 .3130 .4214 .4848 .6414 .7438 .7987
0.60 .3162 .4298 .4935 .6467 .7470 .8020
0.65 .3173 .4341 .4982 .6504 .7488 .8032
(){70 .3185 .4377 .5025 .6534 .7510 .8049
(3.75 .3187 .4391 .5044 .6551 .7518 .8057
(3.80 .3189 .4407 .5060 .6565 .7528 .8060
().85 .3188 .4409 .5062 .6569 .7537 .8059
(3.90 .3187 .4410 .5066 .6571 .7538 .8057
().95 .3185 .4405 .5061 .6564 .7524 .8051
1.00 .3183 .4398 .5056 .6560 .7514 .8043
1. 20 .3168 .4348 .5004 .6500 .7459 .7990
1.4L) .3137 .4248 .4886 .6390 .7370 .7915
1.50 .3009 .4134 .4778 .6308 .7304 .7862
11.55 .2718 .4044 .4703 .6259 .7266 .7831
11.60 .2429 .3915 .4616 .6203 .7224 .7797
11.65 .2112 .3763 .4518 .6142 .7179 .7761
1. 70 .1807 .3613 .4400 .6077 .7130 .7722
21.75 .1538 .3459 .4277 .6007 .7078 .7680
1. 80 .1221 .3283 .4155 .5934 .7024 .7637
l. 85 .1029 .3127 .4037 .5859 .6970 .7592
1 . 90 .0828 .2986 .3910 .5785 .6918 .7546
1. 95 .0702 .2830 .3787 .5711 .6866 .7501
2.00 .0627 .2699 .3675 .5638 .6814 .7461
2.11) .0545 .2500 .3478 .5501 .6715 .7386
2.20 .0503 .2313 .3303 .5377 .6626 .7317
2 . 30 .0440 .2190 .3170 .5273 .0548 .7258
2.40 .0399 .2106 .3072 .5188 .6485 .7210
4. 50 .0374 .2048 . 3000 .5142 .6438 .7205
2 . 60 .0358 .2024 .2950 .5120 .6412 .7190
2 . 80 .0356 .2000 .2900 .5100 .6400 .7180
3 . 00 .0356 .2000 .2900 .5100 .6400 .7180

 

 

 

87

 

 

 

Table 22. The Y-values for selected streamlines for the flow past a

finite step located in the profile-development region,
(Re) -200
8

X ¢-0.05 ¢-0.1 $~0.2 t—O.5 ¢-0.7 ¢-0.8
0.05 .0677 .1170 .2116 .5001 .6931 .7885
0.25 .1048 .1549 .2422 .5006 .6735 .7587
0.55 .1172 .1738 .2596 .5013 .6618 .7427
0.75 .1198 .1756 .2644 .5020 .6590 .7391
1.05 .1225 .1798 .2695 .5037 .6575 .7368
1.25 .1245 .1825 .2727 .5054 .6576 .7365
1.55 .1283 .1876 .2787 .5094 .6594 .7375
1.75 .1320 .1925 .2841 .5138 .6623 .7397
2.05 .1411 .2032 .2966 .5248 .6705 .7465
2.55 .1768 .2459 .3438 .5664 .7028 .7740
2.75 .2207 .2921 .3876 .5961 .7237 .7904
2.85 .2706 .3352 .4197 .6126 .7344 .7985
2.95 .3509 .3875 .4531 .6281 .7427 .8061
3.00 .3655 .4036 .4656 .6349 .7485 .8095
3.05 .3753 .4139 .4755 .6409 .7524 .8124
3.15 .3883 .4272 .4894 .6503 .7586 .8169
3.25 .3957 .4348 .4979 .6565 .7627 .8196
3.35 .3990 .4387 .5024 .6601 .7647 .8208
3.45 .3997 .4401 .5043 .6616 .7652 .8205
3.55 .3985 .4397 .5045 .6615 .7643 .8192
3.65 .3963 .4382 .5034 .6601 .7623 .8170
3.75 .3934 .4356 .5013 .6577 .7595 .8140
3.85 .3898 .4323 .4980 .6543 .7558 .8105
3.95 .3849 .4277 .4934 .6501 .7515 .8062
4.05 .3772 .4213 .4875 .6450 .7466 .8013
4.15 .3680 .4136 .4805 .6391 .7412 .7963
4.35 .3481 .3948 .4639 .6254 .7289 .7853
4.55 .3203 .3710 .4447 .6098 .7150 .7729
4.65 .3075 .3594 .4338 .6015 .7076 .7663
4.85 .2773 .3336 .4124 .5845 .6928 .7525
5.05 .2507 .3091 .4022 .5681 .6789 .7402
5.15 .2352 .2981 .3913 .5603 .6722 .7345
5.45 .2019 .2658 .3808 .5395 .6543 .7192
5.75 .1758 .2437 .3547 .5238 .6414 .7076
6.25 .1562 .2209 .3337 .5084 .6297 .6978
6.75 .1489 .2120 .3133 .5025 .6265 .6965
7.05 .1465 .2095 .3045 .5015 .6269 .6977
8.05 .1429 .2060 .2999 .5021 .6311 .7039
9.05 .1403 .2034 .2956 .5024 .6338 .7080
10 . 05 . 1382 . 2012 . 2930 . 5020 . 6350 . 7099
11.05 .1367 .1996 .2911 .5015 .6357 .7111
12.05 .1357 .1983 .2987 .5011 .6362 .7118
13.05 .1350 .1976 .2888 .5009 .6365 .7123
14. 05 .1346 .1970 .2881 .5006 .6366 .7127
15.05 .1340 .1963 .2877 .5000 .6367 .7129
16. 05 .1339 .1960 .2874 .5000 .6368 .7130
17. 25 .1338 .1955 .2868 .5000 .6369 .7132

 

88

Table 23. The Y-values for selected streamlines for the flow past a
finite step located in the profile-development region,
(Re)H-l300

 

X H.05 xiv-0.1 \6-0.2 $0.5 $0.7 \b-0.8

 

0.05 .0658 .1146 .2095 .5000 .6943 .7904
0.55 .1117 .1620 .2485 .5000 .6679 .7579
1.05 .1126 - .1646 .2520 .5000 .6651 .7480
1.55 .1123 .1652 .2531 .5000 .6643 .7470
2.05 .1128 .1663 .2545 .5001 .6634 .7457
4.05 .1165 .1717 .2607 .5013 .6607 .7417
6.05 .1319 .1904 .2809 .5176 .6723 .7512
7.05 .1945 .2585 .3539 .5815 .7232 .7951
7.15 .2114 .2765 .3720 .5950 .7328 .8029
7.25 .2363 .3032 .3961 .6100 .7429 .8112
7.35 .2780 .3406 .4250 .6255 .7530 .8190
7.45 .3509 .3884 .4561 .6402 .7620 .8261
7.50 .3666 .4023 .4683 .6467 .7666 .8292
7.60 .3840 .4249 .4861 .6575 .7735 .8342
7.70 .3964 .4341 .4976 .6652 .7783 .8376
7.80 .4026 .4411 .5045 .6702 .7812 .8392
8.00 .4057 .4463 .5095 .6737 .7822 .8385
8.20 .4033 .4441 .5082 .6717 .7788 .8343
8.40 .3977 .4385 .5037 .6669 .7733 .8284
8.50 .3932 .4350 .5008 .6638 .7700 .8252
8.55 .3908 .4334 .4991 .6622 .7683 .8235
8.65 .3860 .4292 .4954 .6587 .7646 .8200
8.75 .3810 .4250 .4914 .6550 .7610 .8164
8.85 .3760 .4206 .4871 .6511 .7572 .8126
8.95 .3709 .4160 .4827 .6471 .7532 .8088
9.05 .3658 .4114 .4782 .6430 .7492 .8049
9.15 .3608 .4067 .4736 .6387 .7451 .8008
9.25 .3558 .4019 .4688 .6343 .7410 .7969
9.45 .3438 .3900 .4592 .6254 .7326 .7892
9.65 .3300 .3780 .4506 .6161 .7239 .7812
9.85 .3175 .3665 .4380 .6066 .7150 .7729
10.05 .3057 .3552 .4273 .5969 .7058 .7643
10.55 .2723 .3250 .3995 .5722 .6829 .7426
11.15 .2307 .2850 .3665 .5443 .6570 .7189
12.05 .1882 .2497 .3316 .5147 .6302 .6935
13.05 .1693 .2316 .3173 .5035 .6209 .6860
14.05 .1676 .2306 .3176 .5061 .6250 .6911
15.05 .1678 .2312 .3197 .5104 .6239 .6973
16 . 05 . 1659 . 2302 . 3194 . 5119 . 6230 . 7005
17.05 .1626 .2269 .3171 .5112 .6224 .7015
18.05 .1594 .2235 .3144 .5099 .6219 .7017
20.05 .1549 .2185 .3103 .5081 .6212 .7026
30 . 05 . 1434 . 2066 . 2995 . 5044 . 6192 . 7076
40.05 .1386 .2017 .2937 .5026 .6180 .7103
50.05 .1363 .2008 .2906 .5016 .6171 . 7117
60.05 .1351 .2000 .2889 .5010 .6165 . 7125
70.05 .1350 .2000 .2889 .5010 .6165 . 7124

78.45 .1350 .2000 .2889 .5010 .6165 .7124

 

 

 

89

Table 24. Summary of cases studied for a single step

 

 

Step Step Step Computational CPU

(Re)H height position length domain purpose time
20 0.38 Inviscid-core 0.058 1.78 Effect 0 00:11:

20 0.38 Profile-dev 0.058 2.08 of step 0 00:13:

20 0.38 Fully-dev 0.058 3.08 position on 0 00:30:

200 0.38 Invisicid-core 0.058 14.68 separation 0 02:45:
200 0.38 Profile-dev 0.058 15.68 region 0 05:16:
200 0.38 Fully-dev 0.058 14.68 0 04:51:
200 0.18 Inviscid-core 0.058 14.68 Effect 0 01:07:
200 0.28 Inviscid-core 0.058 14.68 of step 0 01:24:
200 0.38 Inviscid-core 0.058 14.68 height on 0 02:45:
200 0.48 Inviscid-core 0.058 14.68 separation 0 05:45:
200 0.58 Inviscid-core 0.058 14.68 region 0 09:54:
20 0.38 Inviscid-core 0.058 1.78 Effect 0 00:11:

50 0.38 Inviscid-core 0.058 3.98 of Reynolds 0 00:39:

100 0.38 Inviscid-core 0.058 7.58 number on 0 02:06:
200 0.38 Inviscid-core 0.058 14.68 separation 0 02:45:
500 0.38 Inviscid-core 0.058 36.38 region 0 20:00:

 

Table 23.

.‘4 fl r\J A, J. {J 0 .\J 0 5 0U. I .5.
3..» .JN AIJ 1 ..Q ,, .4 .7 .‘J 5 It: 7.. Va .. g 6.3.. c: a... .v . 4
. . . .. I 1 . n. q i 2 .. ...;- n. ..... . I h I 6.. I e 1e \ c v. \\\\\\\

A Mi

90

Table 25. The Y-values for selected streamlines for the flow past a
single step located in the inviscid core region, (Re)H-20

 

X $—0.05 ¢-0.l w—0.2 ¢-0.5 ¢-0.7 ¢-0.8

 

0.05 .0988 .1802 .3139 .5637 .7231 .8046
0.10 .2430 .3206 .4032 .6036 .7385 .8084
0.15 .3497 .3867 .4531 .6299 .7492 .8119
0.20 .3757 .4177 .4828 .6477 .7570 .8151
0.25 .3975 .4377 .5026 .6596 .7625 .8176
0.30 .4079 .4509 .5141 .6673 .7660 .8192
0.35 .4127 .4562 .5204 .6716 .7679 .8199
0.40 .4128 .4573 .5224 .6733 .7682 .8196
0.45 .4089 .4547 .5211 .6728 .7671 .8184
0.50 .4020 .4489 .5170 .6704 .7649 .8164
0.55 .3896 .4389 .5107 .6665 .7616 .8133
0.60 .3748 .4271 .5027 .6613 .7573 .8096
0.65 .3594 .4142 .4921 .6549 .7523 .8052
0.70 .3412 .4006 .4801 .6477 .7467 .8002
0.75 .3206 .3831 .4677 .6396 .7408 .7954
0.80 .3020 .3664 .4550 .6310 .7344 .7904
0.85 .2695 .3406 .4310 .6220 .7278 .7850
0.90 .2501 .3170 .4100 .6080 .7209 .7795
1.00 .2237 .3000 .4002 .5943 .7067 .7680
1.10 .1973 .2716 .3737 .5761 .6930 .7565
1.20 .1754 .2503 .3521 .5593 .6804 .7460
1.30 .1613 .2316 .3327 .5440 .6689 .7370
1.40 .1518 .2183 .3174 .5307 .6587 .7291
1.50 .1437 .2086 .3055 .5192 .6498 .7225
1.60 .1377 .2015 .2954 .5092 .6429 .7172
1.80 .1370 .2013 .2951 .5090 .6423 .7169
2

.00 .1370 .2013 .2951 .5090 .6423 .7169

 

‘

233.6

VA

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.-H‘ ....4‘ .54 o.A - ‘1 x. ALI... 0.. a”... Q
. s .

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A

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0.05 .44 ....‘ .‘J
< .s n a s s . x

\

91

Table 26. The Y-values for selected streamlines for the flow past a
single step located in the inviscid-core region, (Re)H=200

 

 

X ¢-0.05 ¢-0.l ¢-0.2 ¢=0.5 ¢—0.7 ¢=0.8
0.05 .0688 .1189 .2144 .5036 .6963 .7913
0.20 .1058 .1574 .2479 .5137 .6891 .7745
0.40 .1286 .1855 .2772 .5273 .6893 .7690
0.60 .1382 .1979 .3020 .5439 .6974 .7737
0.80 .1690 .2201 .3317 .5662 .7111 .7836
1.00 .2140 .2830 .3789 .5955 .7288 .7963
1.10 .2644 .3279 .4025 .6112 .7379 .8028
1.15 .3081 .3570 .4299 .6187 .7421 .8058
1.20 .3452 .3809 .4457 .6256 .7460 .8087
1.25 .3579 .3948 .4575 .6317 .7494 .8111
1.30 .3657 .4046 .4664 .6369 .7524 .8131
1.40 .3766 .4161 .4785 .6446 .7568 .8159
1.50 .3826 .4220 .4849 .6489 .7589 .8169
1.60 .3839 .4236 .4872 .6500 .7588 .8162
1.70 .3811 .4218 .4862 .6492 .7569 .8139
1.80 .3753 .4172 .4824 .6460 .7533 .8102
1.90 .3671 .4103 .4765 .6412 .7484 .8053
2.00 .3574 .4018 .4689 .6350 .7426 .7995
2.20 .3298 .3774 .4503 .6196 .7286 .7869
2.40 .3006 .3518 .4268 .6019 .7127 .7727
2.60 .2657 .3223 .4038 .5834 .6962 .7578
2.80 .2342 .2967 .3806 .5658 .6811 .7438
3.00 .2079 .2719 .3608 .5500 .6675 .7321
3.50 .1654 .2167 .3249 .5217 .6432 .7111
4.00 .1509 .2146 .3079 .5083 .6329 .7024
4.50 .1448 .2082 .3012 .5036 .6305 .7015
5.00 .1426 .2058 .2986 .5034 .6315 .7038
6.00 .1405 .2036 .2960 .5033 .6345 .7084
7.00 .1385 .2016 .2936 .5030 .6359 .7107
8.00 .1369 .2002 .2918 .5023 .6364 .7117
9.00 .1359 .1989 .2901 .5017 .6367 .7123

10.00 .1352 .1976 .2894 .5013 .6368 .7127
11.00 .1347 .1972 .2884 .5009 .6369 .7129
12.00 .1343 .1962 .2875 .5007 .6370 .7131
13.00 .1341 .1962 .2866 .5005 .6370 .7130
14.00 .1339 .1960 .2857 .5000 .6370 .7130

15.00 .1339 .1960 .2857 .5000 .6370 .7130

 

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fin. q-.. 1 a 11 «.1 a1 «.1 1 Vi. I ~25 l 1-.. I x. .6 Al. Via .1. alt I . A TIL 71 1 alt l l 1“ a4

92

 

 

 

 

 

Table 27. Relaxation factor (PS) vs. computing time*, h-l/15
Relaxation Reynolds number
factor 20 50 100 200 500 1000 2000
0.90 2.20 16.90 16.90 49.10 98.40 + +
1.00 1.90 13.71 14.30 40.72 78.00 + +
1.10 1.75 11.23 13.19 33.81 75.00 + +
1.20 1.64 10.55 12.17 29.20 72.00 + +
1.30 1.60 9.92 10.78 27.50 69.00 333.60 +
1.40 1.51 9.43 9.62 24.90 64.80 283.80 +
1.50 1.44 8.53 8.30 22.03 63.60 235.20 1087.20
1.60 1.40 8.35 7.91 19.38 60.60 192.00 909.60
1.65 1.28 8.14 7.10 16.97 56.40 175.20 806.40
1.70 1.37 7.40 7.06 16.76 55.20 163.20 774.00
1.75 1.40 7.48 7.30 16.90 54.60 150.60 582.60
1.80 1.44 7.60 7.50 17.48 79.20 144.00 381.60
1.85 1.50 8.92 8.20 22.82 86.40 157.80 418.80
1.90 1.60 10.10 8.60 25.05 100.20 177.00 490.80
Table 28. Relaxation factor (FS) vs. computing time*, h-1/20
Relaxation Reynolds number
factor 20 50 100 200 500 1000 2000
0.90 9.96 45.00 74.60 186.60 + + +
1.00 7.97 39.85 62.16 164.40 + + +
1.10 7.47 38.50 54.18 159.60 + + +
1.20 7.09 37.74 50.73 150.00 + + +
1.30 6.64 35.05 45.63 135.00 625.80 + +
1.40 6.20 32.30 39.35 120.00 529.20 904.80 +
1.50 5.91 30.29 34.61 112.80 439.80 792.60 3314.40
1.60 5.40 29.50 31.96 93.60 372.00 688.80 2529.00
1.65 5.22 29.00 29.03 85.20 336.00 620.40 2012.00
1.70 5.11 28.24 28.70 79.80 303.00 577.80 1870.80
1.75 5.05 27.05 28.05 76.20 283.20 541.80 1716.00
1.80 5.30 31.50 27.40 72.00 252.60 504.00 1530.00
1.85 5.50 32.80 29.70 97.20 269.40 513.00 1752.60
1.90 5.90 36.20 32.20 112.80 326.40 528.00 2206.80

 

‘* Computing time is measured in minutes.
-+ The run wasn't attempted, because the trend was obvious.

13.118:
‘BCCO

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93

 

 

 

 

 

Table 29. Relaxation factor (FV) vs. computing time*, h-1/15
Relaxation Reynolds number
factor 20 50 100 200 500 1000 2000
0.90 1.70 16.90 9.30 37.00 76.80 228.00 442.80
1.00 1.30 13.00 7.13 16.60 61.80 144.00 381.60
1.05 1.20 8.22 6.06 17.12 + + +
1.10 1.11 5.80 4.12 17.76 + + +
1.15 1.04 5.03 6.72 + + + +
1.20 1.00 4.66 7.50 + + + +
1.25 0.84 5.12 10.11 + + + +
1.30 0.78 5.66 13.76 + + + +
1.35 0.76 6.16 + + + + +
1.40 0.74 7.46 + + + + +
1.45 0.81 9.20 + + + + +
1.50 0.92 10.65 + + + + +
1.60 1.23 16.10 + + + + +
1.70 1.80 30.10 + + + + +
1.80 2.60 67.05 + + + + +
1.90 3.55 73 78 + + + + +
Table 30. Relaxation factor (FV) vs. computing time*, h-1/20
Relaxation Reynolds number
factor 20 50 100 200 500 1000 2000
0.90 6.20 34.20 34.80 83.60 354.00 642.00 1878.00
1.00 5.01 26.60 26.83 67.30 282.20 504.00 1530.00
1.05 4.94 24.30 24.04 80.30 + + +
1.10 4.88 22.05 20.33 99.20 + + +
1.15 4.39 19.80 19.02 + + + +
1.20 4.15 17.66 20.90 + + + +
1.25 3.77 15.55 26.54 + + + +
1.30 3.40 13.60 35.10 + + + +
1.35 2.92 17.38 + + + + +
1.40 2.66 23.05 + + + + +
1.45 2.48 26.90 + + + + +
1.50 2.35 30.00 + + + + +
1.60 3.01 34.50 + + + + +
1.70 4.50 40.50 + + + + +
1.80 8.01 50.00 + + + + +
1.90 8.60 52.90 + + + + +

 

* Computing time is measured in minutes.

‘6 The numerical solution did not converge for these entries.

 

able 3

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Table 31. Weighting factor (KS) vs. computing time*, h-1/15

weighting Reynolds number
factor 20 50 100 200 500 1000 2000
0.00 0.80 4.30 4.60 19.75 72.60 184.20 408.00
0.05 0.81 4.50 4.83 17.89 70.80 172.80 390.00
0.10 0.82 4.60 5.13 18.05 69.00 152.40 381.00
0.15 0.84 4.82 5.28 18.30 72.60 129.60 376.80
0.25 0.86 5.05 5.46 18.90 75.60 135.65 367.80
0.25 0.87 5.29 5.59 20.02 76.82 147.58 301.80
0.30 0.88 5.42 5.70 21.00 79.80 180.00 325.80
0.40 0.95 5.78 6.62 22.00 91.85 252.60 367.80
0.50 0.99 6.00 7.02 24.35 99.60 304.80 444.00
0.60 1.05 6.46 7.76 26.52 105.55 + +
0.70 1.30 7.86 8.70 34.55 + + +
0.80 1.75 10.03 11.76 43.65 + + +
0.90 1.91 16.25 17.10 60.66 + + +
1.00 2.35 24.89 26.90 89.20 + + +

Table 32. Weighting factor (KS) vs. computing time*, h-1/20

Weighting Reynolds numbers
factor 20 50 100 200 500 1000 2000
0.00 2.25 14.90 32.70 60.00 291.00 1251.00 1974.00
0.05 1.79 14.30 25.41 49.20 288.60 1024.80 1850.40
0.10 1.20 14.00 20.91 46.30 283.80 922.20 1746.00
0.15 1.32 14.80 21 80 49.80 276.60 643.20 1679.40
0.20 1.50 16.00 22.38 57.00 319.80 564.00 1637.40
0.25 1.55 16.30 22 72 58.20 330.60 829.80 1590.00
0.30 1.65 16.80 23 26 59.40 349.20 1143.60 1573.20
0.40 2.05 20.95 23 56 63.00 381.70 1645.80 2430.60
0.50 2.50 21.00 26.70 64.80 469.20 2412.00 3133.80
0.60 3.10 21.66 31.23 67.20 580.20 + +
0.70 3.47 25.98 36.03 69.60 + + +
0.80 4.15 34.90 42.31 93.00 + + +
0.90 4.96 52.33 59.95 115.20 + + +
1.00 6.05 73.55 80.70 138.90 + + +

 

'* Computing time is measured in minutes.
-+ The run wasn't attempted, because the

trend was obvious.

MC :0'

Table 3

..L
3 04 § 9 J n V J H U 3 \J '1'
0 .14. 0 .14 O .14 a .11. .0. ..1J 01‘ .3 film. .3 a... u 5 «MU 1U s . PL .11... :J «1. .s J u . flu C u
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I . u .h .1.- . 1 a . u. e .41- I. o Ilka I .v u .4

Ihq1111val...nnl1v.qu. .nv..1..

95

 

 

 

 

 

Table 33. Weighting factor (KV) vs. computing time*, h-l/15

Weighting Reynolds number
factor 20 50 100 200 500 1000 2000
0.10 + + + + + + +
0.15 + + + + + + +
0.20 + + + + + + +
0.25 + + + + + + +
0.30 1.48 6.20 + + + + +
0.35 1.15 5.90 + + + + +
0.40 0.60 5.50 8.42 20.36 + + +
0.45 0.62 3.31 7.38 18.27 + + +
0.50 0.65 3.57 6.26 17.51 72.60 219.60 429.00
0.55 0.77 3.72 7.71 16.90 66.60 174.00 385.20
0.60 0.86 3.93 8.98 17.71 58.20 135.00 303 00
0.65 0.87 4.14 9.20 17.92 61.20 139.20 438.00
0.70 0.88 4.38 9.60 18.03 64.80 147.00 492.00
0.75 0.91 5.21 10.10 20.56 70.80 160.80 543.00
0.80 0.93 5.96 10.50 23.66 75.00 169.80 618.00
0.85 1.40 6.87 10.80 27.72 106.80 225.60 756.00
0.90 1.80 7.55 11.05 33.30 150.00 289.80 948.00
1.00 2.52 8.78 14.05 39.24 215.80 390.40 1205.00

Table 34. Weighting factor (KV) vs. computing time*, h-1/20

Weighting Reynolds number

factor 20 50 100 200 500 1000 2000
0.10 + + + + + + +
0.15 + + + + + +' +
0.20 + + + + + + +
0.25 + + + + + + +
0.30 + + + + + + +
0.35 + + + + + + +
0.40 2.30 14.05 + + + + +
0.45 2.01 10.20 + + + + +
0.50 2.19 6.05 17.35 49.20 + + +
0.55 2.25 7.90 13.41 40.25 + + +
0.60 2.30 10.01 14.80 34.80 252.00 1173.00 1536.00
0.65 2.33 11.40 15.20 37.20 238.20 822.00 1422.00
0.70 3.37 12.05 17.03 39.00 223.80 564.00 1344.00
0.75 4.70 14.81 19.15 43.80 276.00 912.00 1506.00
0.80 5.41 18 52 20.83 46.80 348.00 1251.60 1782.00
0.85 7.90 25.70 34.50 66.00 363.00 1362.00 1968.00
(3.90 9.25 32.80 46.40 73.80 385.80 1494.00 2232.00
1.00 14.70 40.15 60.05 86.92 412.70 1710.00 2668.00

 

* Computing time is measured in minutes.
‘9 Time numerical solution did not converge for these entries.

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96

 

 

Table 35. Factors of reduction in the computing time

Mesh Reynolds number

size 20 50 100 200 500 1000 2000
h

1/15 3.0 4.0 3.5 2.5 1.5 2.5 3.6
1/20 6.6 6.6 2.0 4.6 2.8 2.0 2.5

 

 

 

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[f '- E

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Forward step Backward step

 

 

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Figure 3. Finite step with notations.

(x,y-l-h) '3

 

(it-ML (m) (NM)
4 h i 2

 

(x,y-h) 5

Figure 4. Finite difference representation of basic equations.

 

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Figure 10. The streamwise pressure gradient. (Re) "-20.

 

 

 

 

 

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Figure 14. Streamlines in the vicinity of the forward step located
in the profile—development region, (Re "=20.

 

1.0 1.0 0.0 ~05

Figure 15. Streamlines in the vicinity of the forward ste located
in the profile-development region. (Rah-=20 .

been

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Figure 16. Streamlines in the vicinity of the forward ste located
in the profile-development region. (Re)"-20 0

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Fonda step with height of 0 4H
H-Upetrean channel height
-0.10
-0.00
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o 1-
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0.50 0.25 000 0.15 0.10 0.115 0.00
V"
Figure 17. Effect of eSep gositian on separation
region. Re ..1- 00.
Fatwa-d step with height of 0 4H
H-Upstrean enamel height
1-0.20
I ..
A
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‘ A inviscid ears
. I’ l U V—I I
0.0 0.0 0.7 0.0 0.5 0.4 0.3 0.2 0.1 0.0
X!"

F' 18. Effect 0 'tio se ration
igure region.°(R:s:E-£000. n on pa

‘

106

 

Forward step with height (a) in the inviscid core
H-Upetreom channel Wt

 

-O.15
l .
1E—OJO >
. F
O
A
° A -0.05
. 2
e e-OAGH A _
o e-OAGi . O
0 m ‘

 

 

 

Fi ure 19. Effect of hei ht on so oration
g region. (Raw-208. p

 

 

 

 

 

Forward step with hei¢1t(o) in the inviscid core.
H-Upstreom channel height
-0.25
e
-0.20
e i’
. 1
l
' ‘ -0.15 i
O o A b
C o ‘
e o ‘ A -0.10
A
. ' o A .
e 9 A A
e «0.4014 e 0 A A "0'05
a 90.401 . e 0 0 A A _
A M . . O 2 A
A e-OJG-i O
V T I I I
.10 0.110 0.70 0.50 0.30 0.10
V"

F1 ure 20. Effect 0 hei ht on se oration
9 region. (Rage-10&. p

107

 

. Forward step in the inviscid oorw rodeo

 

 

 

 

 

 

 

 

 

 

0.05-
‘ A—A Reott. (Yr)
0,00 . I . I . I H0.5-2'131'14
0.50 0.35 0.40 0.45 0.50
Step height (a)
Figure 21. Effect of step height .on se oration
and reattachment paints. e)" -200
0.”
Forward step in the inviscid core region
0.00-
0.704
0.60-
g 4
° 0.50-
i
0.40-1
q
0.30-
0.20‘ 4.4
; +* -
f H Mar)
'.i—‘i'i'l‘i'i'i‘i'
0.30 0.33 0.35 0.30 0.40 0.43 0.45 0.40 0.50

 

Stop Misfit (a)

Figure 22. Effect of step hei ht on a on
and reattachment9 ”huffing-11000.

Xs/o . Yr/o

Separation point (Xe)

108

 

002 -‘

d

Forward step with height (a) in the inviscid core

A—A Rsott.(Yr/o)
H Sepo.(Xs a

 

 

0.0

 

0

‘l

I I I I I I I I I ' I I I I I I I I I
200 400 500 000 i 000 i 200 1400 i 000 1000 2000 2200
Reynolds number (Re) I.

Figure 23. Separation and reattachment points for
various Reynolds numbers.

 

0.7-1

0.0-1

0.5-1

0.4-

0.3 -

0.2-

001‘

0.0

H Present work
H Nunerlcol [30]
H W0 [27]

 

 

 

T F f
400 000 1 200 l 050 2000
Reynolds number (Re)u

Figure 24. Comparison of separation point for a
forward step with numerical results.

Reottochment point (Yr)

Y/H

 

 

 

 

 

 

 

e—e Present work
H Wed [30]
0.3-*
.1
0.2-1
-1
0.1 1
-1
0'0 I . I I I I I I
U 400 800 1200 1800 2000
Reynolds mber (Re)"
Figure 25. Comparison of Ireattochment point for a
forward stop with numerical results.
0.20
more step with height of 0.2H
‘ l-l-Downstrearn channel height
0.15-4
i 0
0.10.1
8
.1
0.054 8
‘ A My—dsv
e Profile—dew
000 a “sold oorw
' 0.00 t 0.1): ' 0.10 0.1: 030 0.5:
X/H

Figure 26. Effect of op position on separation
region. (Re .- 0.

 

Y/H

Y/H

110

 

 

 

 

 

 

 

 

 

0.30 -
Bockwad step with height of 0.3H
H-Downstreom channel height
0.25
O
0.20 0
‘ 8
0.15-
I
‘ O
0.10-
8
‘ O
0.05-1
A Fully-dsv
‘ e Profile-dew
o inviscid core
00°C I’ I W i 1‘ i I ‘ r i I ‘ I
0. 30 0.05 0.1 0 0.1 5 0.20 0.25 0.30 0.35 0.40
X/H
Figure 27. Effect of op position on separation
region. (Re 11" 0.
0.4
Backward step with height of 0.4H
H-Downstreom chains! height
0
0.3-. 9 3
I
0.2-1 o .
O
-1 I
O
0.1-1 g
.. A Fwy-dew
e Proer—dsv
0.0 o inviscid core
I I I l T r s —r—H'.—Ti u r r
- 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.7
V"

Figure 28. Effect of

s sition on s ration
region. (Roger-£3. 09°

Y/H

”If/H

0.5
‘ Backward step with height of 0.4H
HuDownstreom channel height
0.4
13'988
1 8888303
00083
fighs
0.3-1 OO
00 28
I .1 o 88
> 00 g
o x,
0 2'1 O O
’ °o “X.
00 A0.
.1
A
0 A1.
0
0.1 "1 A.
0 e
4 0 A'e Almmqu
e Profile-dew
Inviscid cars
go , I . I . T .7 e—dhe I° 47 I
0.0 0.4 0.8 1.2 1.8 2.0 2.4
X/H
Figure 29. Effect of step position on separation
region. Re)..= 00.
0.30
‘ Backward step with hold“ of 0.2H
H-Downstreorn channel height
0.25..
0.20 .8880
0338
o C
O
: 0AAX
\.0J51 0 g
>- 00 XX
1 o X. .
0 AA'.
0.10-4 A .
° A e
1 90 AA.
0 A
0°05'1 O
.1 A Filly—dew
. e Proer-dsv
o inviscid core
0.00 r i r r r l I I eff—TH r I r
0.0 0.5 i .0 1.5 2.0 2.5 3.0 3.5 4.
km

111

 

 

 

 

 

 

 

 

 

 

Fi ure 30. Effect of ste osition on separation
9 region. (Re) “£2000.

1 1 2
( 4.111s) (1.1.1.0)

v-ees
were
.1— see
lit-0.20
1:11:

 

0.0 l.0 2.0

Figure 31. Streamlines in the vicinity of the backward step located
in the profile—development region. (Re)"=20.

$- 0.70
$- 0.50

0.20
0.10
E...

 

0.0 1.0 2.0

Figure 32. Streamlines in the vicinity of the backward step located
1n the profile-development region. (Re)"=200.

(Mi-0)

3:83:

p—oes
fi-oas

3:01:

    

0.0 1.0 2.0

Figure 33. Streamlines in the vicinity of the backward ste located
In the profile-development region. (Re)"=-200

Y/H

113

 

 

 

 

 

 

0.5
Backward step in the inviscid region
-1 H-Downstrsorn channel height. o-step height
41
0.4-1 A
A
1 A
11 5
0.3-1 A ‘
A
A ‘
A
0.2 s A
s A A
0
0H ° A A
A o-0.51-i
'1 o . A 0 0.0.“.
s a-O.3H
A o o-0.2H
000 r r o-r—-f——’ t 1 Ii- 1' ‘ I I I
0.0 0.1 0.2 0.3 0.4 0 5 0 6 0.7
X/H
Figure 34. Effect of step height on separation
region. (Re). -20.
0.6

 

 

“
‘0
0
‘ A pas-1
A A GIG“
. «0.31
o o-OJ-l
0.0- —r—T-O-r—r-O-r—r6—r—rkr r r i_ r I_,__..
0.0 0.5 1 .0 i .5 2.0 2.5 3.0 3.5 4.0

 

Backward step in the inviscid region
H-Downstreom dunnei height. a-Step height

 

 

Fi ure 35. Effect of hei ht on separation
9 region. (Rx-208.

I\)

o.
fi0\I.PV «...-01 COxoOLUQ‘h

0w
0

0.1

0.5 ‘

 

114

Backward step in the inviscid region
H-Downstrsan channel heidst. o-Step height

 

 

 

 

 

 

 

 

 

:r
> R
“11.3%A
A
. . AA A o-OJI-l
0 e o-OJH
o a-0.2H
0'0 l I 1—0._ I r I
0.0 0.5 i .0 1.5 2.0 2.5 3.0 3.5 4.0
X/H
Fi ure 36. Effect of op hei ht on separation
9 region. (12:11. «508.
1.2
Backward step in the inviscid core region
1.0— . A
A___i is
"o‘ + — .
} M:
b d
i 0.8-1
S
a
3 .
0.0-
‘ A—A “Me's”
s-e (Itch-200
e-e (Re -20
0.4 I r l s I I g '
0.1 0.2 0.3 0.4 0.5 0.0
Step height (a)

Figure 37. Separation

oint vs. step height
for differen

Reynolds numbers.

 

0.6

0.!

RV.

0.
any stain I00“

0

0.5

0.4

I\>

 

115

 

 

 

 

 

 

 

 

 

0.0
‘ Backward step in the inviscid region
0.5-
J
0.4-
3 4
E
'3 0.3d
3
g +
0.2d
4
0.1-
e -500
O -200
0.0 ,.,,,.,-°,-
0.0 0.5 l .0 1.5 2.0 2.5 3.0
Reattachment point (Xr)
Figure 38. Reattachment point vs. stageheight
for different Reynolds num rs.
0.5
Backward step in the fully-developed region
‘ a-0.3H. H-Davmstrearn channel height
0.4-
0.3 . . .
O
A
3%342‘t.‘ . ’ e
e A '
O . AA “ . Q
0.2J 0 A ‘ '
A ‘A ’
d A ‘
e A ‘
e A A e
C . A (ROM-2W
* A A A (Rah-100
O (Rah-30
0.0—W r I t W
0.0 0‘ 0.8 1 .2 l .0 2.0 2.4
V"

Figure 39. Separation and reattachment points
for vanaus Reynolds numbers.

116

 

0.3

Backward step in the inviscid core region
a-0.2H. H-Channel heiwt

 

 

 

 

 

a (Rah-2000
e (Nah-1000
e (no)
A e s (Mg-zoo
O (Rah-i“
°~°"—"°—H—r—-r—fi—v—f . r . o
0.0 0.4 0.0 1.2 1.0 2.0 2.4 2.0
V“
Figure 40. Separation and reattachment paints
for various Reynolds numbers.
1.2
Backward step in the inviscid redon with height
4 a-0.2i-l. H-Downstream channel Wt.
1.0‘ _——-Q— a

 

 

Separation paint (Ys/a)
8
4

 

 

on . f . . r . . . . .
.5 E 450 550 :50 1600 12'oo 1100 1030 1300 zo'oo
Reynoidenumber(Re)..

Figure 41. Variation of separation paint with
Reynolds number.

 

 

Reattochment paint (Xr/o)

 

Backward step in the inviscid region
o-0.2H. H-Downetream enamel hel¢1t

HPreeentwork
e-eRecchefM]

”World-8|
' I ' I ' I ' I

25 50 75 1 00 i 25 1 {IO
Reynolds number (Re)a

 

Figure 42. Com arison of reattachment point with theoreotscal results

for eynolds numbers in the range of (4—

 

Reottochment point (Xr/a)

Backward step In the inviscid region
a-0.2H. H-Downstreom chmnel height

 

 

 

r v ' ' I ' I ' i ' I
1&3 260 360 400 500 m 700 800

Reynolds number (Re)¢

Figure 43. Corrngarieon of reattachment point with ex timeoritol data

eynolds numbers in the range of (10

 

Y/H

Y/H

118

 

 

 

 

 

 

 

 

 

0.40
0.35 Fhitestepintheinviecidccrsregianwith
‘ height of 0.3H. H-Channel height
4
0.30-I
O 0
0.25 g
« 8
0.20-‘ o 8
.i
0.15« 6 A
I o
0.10-4 .
4 O
0.05-J ‘ ‘ I u «a
4 C W -2H
0.00 . , . , we. , ° “3"“ ""
0.00 0.50 l .00 i .50 2.00
X/H
Figure 44. Effect of step lengzh n downstream
separation region. Re “-200.
0%
0 35 Finite step in the inviscid core region with
' ‘ height of 0.3H. H-Chmnel height
.1
' e
‘ e
0.25- I .
. 0
and .0
. . . .
0.15.. '
I 0 .
0.10-4 8
0.05d . s Fay-dev
a O Praile—dsv
0.00 I T I I r I #r-_T_—.1—F-E‘_h
0.00 0.50 i 00 1.50 2.00 2.50 3.00
V"

Figure 45. Effect of step rosi 'on on downstream
separation reg on. Re)..-500.

 

pl!

In:

m
m

Fm
Hm

N15
Mm
y.

km

No:

(All!

119

(“Al-0) (i SJ 0)

 

 

p-ua
fi-ue

i—w/
{gm/I:

 

 

 

 

 

 

 

 

0.3
I . .

——7 f V v V f r——r v v V v V V V

 

 

-I'.o 0.9 1.0

V v

Figure 46. Streamlines in the vicinity of the finite step located
in the profile—deveIOpment regIon. (Re "=20.

p-csc
t-eao

p-asa
p—m

0.10
0.00

    

-i .0 0.0 I .0 2.0 3.0

Figure 47. Streamlines. in the vicinity of the finite step located
In the profile—development regIon. (Re),,-200.

(4.34.0) (3.5.1.9)

3:3:

f-aec

#- om
fi-cao

 

-i.0 0.0 1.0 2.0 M

Figure 48. Streamlines in the vicinity of the finite step located
In the profile-development region. (Re),,-1300

I\\..p

5.2.

ill.

.1

_ WM. —4

 

 

 

 

 

 

 

 

 

0.3
Finite step in the inviscid core region
I H-Chmnel Wt. o-Step height
A ‘
A
0.4- ‘
A
‘ A
ii
A
A
0.3- A
A A
I A
> A ‘
0
0.2 e ‘
A
0
O A ‘
0.1- o ‘
o A A III-0.5"
. A A c-0.4l-f
J O a-OJH
e a-0.2i~i
0.0 —|'—r'°'I'—[—-r.-1—s—'—A.— l . fir—fi T I ‘
0.0 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9
V"
Figure 49. Effect of step heigiet n downstream
separation region. Re3.-20.
0.6 ‘
Finite step in the inviscid core region
H-Chmnei helmt. a-Step height
0.5
A
A ‘ “
A
A
A
0.4 A ‘ ‘
A
AAAA A
A A
A A
i 0.3 A ‘
>' A
... A A
e. A A
A A
0.2 ‘ A A
O . A
.1 O O A A
0 0
0.1- O A ‘
0 A A a-0.5H
. o A 5 A a-0.4H
O a-OJH
whim-AT . ° "0'2"
0.0 0.5 1.0 I.5 2.0 2.5 3 a 335 4.0

X/H

Figure 50. Effect of step height
separation region. (Re

 

 

n downstream
“-200.

 

Separation point (Ye/o)

Reettachment point (Xr/a)

1.1

 

 

 

 

 

 

 

 

Finite step in the inviscid core region

1.0—

J
0.9-4

1
0.0-

d
0.7-1

e-e (R‘L‘m

0.5 . fl . I , r . 7" (R‘L'EL.

0.1 0.2 0.3 0.4 0.5 0.0

Step heidtt (a)
Figure 51. Downstream separation point vs. step height
for various Reynolds numbers.

6.5

‘ Finite step in the inviscid core region.
5.5-
4.5-

0

3.5-

4 e
2.5-
1.5d

//0. (“0"200
0.5 . , . , . , . t" (hr—29...
' 0.1 0.2 0.3 0.4 0.5 0.0

Step height (a)

Figure 52. Downstream reattachment point vs. step height
for various Reynolds numbers.

122

 

 

 

0.:
Finite step in the inviscid cars radon
H-Chmnel Wt
AA ”In
4 A A I.
o A a!
00 .. A A‘ BBB
§ 03" ° 0 A a B:-
A
‘ O . A A a .
O A D -
O O A A Do.-
°-"‘ 0 I e (Re -l300
A ‘ I
A (Re -200
e (its -100
a -50
0.0—WW'

V"

Figure 53. Effect of Reynolds number on downstream separation
and reattachment points. a-0.3H.

 

4 Finite step in the inviscid core region
c-O.3H. l-l-Chonnel height.

Reattochment point (Xr/a)
O
L

 

 

 

6's'o'ibo'iio'zéo'zio'scio'aio'wo

Reynolds numebr (Re)¢

Figure 54. Downstream reattachment point vs. Reynolds number.

.5

Y/H

Y/X

123

 

 

 

 

 

Step with one mesh size length and height of 0.3H
0.4- H-Chonnei height
" O O O O
o o
0.: °
0
I 8 O
X o
.2. x ,
A O
I
A
. O
0.1-« A
’ ° e rely-«Ia
e Praer-dev
a Inviscid core
o'a'i'l‘iflri'ifir'r'i'.
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.7 0.0 0.9 i .0
X/H

Figure 55. Effect of step position on downstream
separatIon regIon. (Re),.-20

 

 

 

 

Step with one mesh size length and height of 0.3H
0,4... H-Chcnnel height
I I I
aa-ls ' ' I
I
I
d .
I
0.2- '
I
. O
0.1- 0
A
‘ . A Elly-dot!
e Pious—dew
0.0 a hviscld cone
1 ‘ I' 1 t r
0.0 0.5 fro its 2T)
V"

Figure 56. Effect of step posit'on on downstream
separotIon regIon. (Rah-200.

124

.o Nuzflomv 606...: 0.50 EomSE or: E

23002 noun a. Em of co 3._Eo_> or: E moE_Eoobw .mm Snot

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9n 0.0
SW
3 .
W.
i\
A. 13.3 8. 3.9..
MN": Aomv 6203 200 EomSE or: E
93002 ago a. Em or: A0 >3EE> on» E moE_Eoobm Km 939...
ad . 0H0
. . . .
do
H “
8.5.8 . 8.36:.

. 3.9. s

2.? s
8.9....

. 89....

2.9. s
8.9.. s

125

 

0.3 -'

0.4

0.3

Y/l-i

0.2

0.1

 

 

 

Sin¢e step in the inviscid core region
"......" i-l-Channei height
I...-
I...
I.-
I.-
I
.I
I
.I
.I
A A A .
...... s
A ‘ ‘ g
A I
AAA ‘A‘ '.
AA . .
AA ‘A CE
.. AA “ Cb
e “A A on o e-osu
e A A a A a-0.4i-l
.. A A A a-OJH
e o-0.2H
a o-0.fH
‘ I if r I‘h—fk—f—Er ‘ u
0 5 1.0 1.3 2.0 2.3 3 0 3 3
X/H

Figure 59. Effect of step height n downstream
separotIon region. Raga-200.

 

 

Single step in the inviscid core region
H-Channel height

 

 

 

 

‘A
A A
O . A A“
.. A
O . ‘
I A ‘A
_ o A A
0.1 3
A
o . A ‘ ‘ Win-5W
. I (R. -m
e (Its -100
I I ‘BIi-so i
Dec 1 I f‘HF—fi—f ' I l
0.0 D 5 i 0 i 5 2.0 2.3 3.0 3.5
xI"

figure 60. Effect of Reynolds number on downstream
separation region. a-0.3H.

126

 

P
u-
1

L

P
‘
1

A

Step height (a)
o
i..-
l

0
Li
1

0.1-

 

Single step in the inviscid core region

 

 

0.0

U r

i T I I
0.5 i .0 i .5 2.0
Reattachment paint (Xr)

Figure 61. .
reattachmen pom . (Re)..-200.

i
2.5

Effect of ste hei ht on downstream

3.0

 

‘
O
I

L

l

Reattochment point (Xr/a)
1‘ 7‘

N
1

Single step in the inviscid core region. a-O.3H

 

 

' v I r ' 1

I
25 so 75 160 155
Reynolds number (Re)a

130

 

i 75

Figure 62. Downstream reattachment point vs. Reynolds number.

Rsattoohment point (Xr/a)

Separation point (Ye/a)

127

 

 

 

 

 

 

 

 

 

 

 

2B4 Steps in the inviscid core region. a-O.3H
2.- /
20-1
10-4
.4
12d
a.l
4.4
H 901!- M
4 H M [45:
o r V r r T r V r I r T fi. mu ‘VLb
O 50 1% 150 200 250 SE) 0 400
Reynolds number (Re),
Figure 63. Comparison of downstream reattachment points.
1.2
Steps in the inviscid core region
1.0- i + .t :0
0.8-
0.6-
‘ o—e Skye
H Finite
0 A H W
0" I I I I I I l T I U I I U l
l) 50 100 150 200 250 300 350 400

Reynolds number (Re)a

Figure 64. Comparison .of downstream separation
poin s for different steps.

Reattachment point (Xr/a)

128

 

 

 

 

. Steps in the inviscid core region
iZ-i
ad
1
4...
HSnge
Hassle-ad
0 e-eflnite
r'rrffrfi 'Tfr'i'i
a so 100 150 150 250 300 350 400-

Reynolds number (Re)o

Figure 65. Comparison of downstream reattachment
points for different steps.

F3. W. KS. KY

F8. FV. KS. KV

129

 

 

 

 

2.0 A
in O o
a O 0
1.0
1.4
4
1.2-1 0
4 e
1.0- 0 0 0 0
0.8-q
0.0-1 A A A A
inn
0.4
A KV
0.2 ‘ A KS
‘ 5 e N
A e rs
Ono r f U r r fr 0 l' t r U l I fir“ l—r r f r V
200 0 600 800 1000 1200 14001000 1800 2000 2200 2400 2600

(R.)u

Figure 66. Optimum accelerating parameters vs. Reynolds number. h-1/15.

 

 

 

 

2.0
1.0 O o O o
1.6
1.4-t
«O
1.21 .
1.0- O O O O
0.8-
-« A A A
0.6 A
MA
0.4
A A IN
0.2 A A K8
‘ ‘ e N
00 e rs
e. r I’ ‘ I I I I T r I l V i V l’ T I r I r I I I I I T
0 200 400 000 800 10001200140010001000 2000 2200 2400 2000

(R°)u

Figure 67. Optimum accelerating parameters vs. Reynolds number. h-1/20.

KS.KV

130

 

 

 

 

 

 

 

 

 

2.0
1.0 .‘ ‘ ‘ I I
A A A
1.0
1.4-in
‘I
1240
I
4 O
1.0-l s e e e . FV.h-1/20
4 A FS.h-1/20
e FV.h-1/15
08 A FS.h-1/13
O V r r T T r T I t r l' r 1 r f r T l’ r l' r r f rf
0 200 400 600 800 1000 1200 1400 1000 1800 2000 2200 2400 2000
(R°)u
Figure 68. Optimum over-relaxation factors vs. Reynolds number.
0.8
Ii
0.7-* A A A
0.0- A A A A
a A A
0.3- AA
...12"
0.3- 0
i O
0.2-1 I
. I O
A KV.h-1/20
0.1 O O O A Kv.h-1/15
o e KS.h-1/20
. o KS.h-1[15
0'0 T‘i‘i'i'irl'i'i‘i'i
0 200 400 600 000 1000 1200 i400 1000 1000 2000

“‘00:

Figure 69. Optimum weighting factors vs. Reynolds number.

BI BLI OGRAPHY

10.

11.

12.

13.

14.

15.

16.

17.

18.

BIBLIOGRAPHY

Schiller, L. and Angew, 2., Math. Mech., Vol. 2, P. 96 (1922).

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Campbell, W.D. and Slattery, J.C., J. Basic Engg., Vol. 85, P. 41
(1963).

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Schlichting, H., ZAMM, Vol. 14, P. 368 (1934).

Schlichting, H., Boundary Layer Theory, 6th ed. McGraw-Hill Book
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Sparrow, E.M., Lin, 8.8. and Lundgren, T.S., Phys. Fluids, Vol.
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J. and Liron, N., Appl. Sci. Res., Vol. 19,

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Smith, F.T., J. Fluid Mech., Vol. 78, P. 709 (1976).

Smith, F.T., J. Fluid Mech., Vol. 79, P. 631 (1977).

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(1979).

Dimopoulos, H.G.
303 (1980).

and Hanratty, K., J. Fluid Mech., Vol. 33, P.

Smith, F.T., Proc. Roy. Soc. Land. A 356, P. 433 (1977).

Smith, F.T. and Stewartson, K., Proc. Roy. Soc. Lond. A-332, P. l

(1973).

Smith, F.T., J. Fluid Mech., Vol. 90, Part 4, P. 725 (1979).

Smith, F.T. and Duck, P.W., J. Fluid Mech., Vol. 90, Part 4, P.

727 (1980).

Blowers, J., Independent Report, Mech. Dept., MSU (1981).

Dennis, S.C. and
393 (1980).

Smith, F.T., Proc. Roy. Soc. Lond., A-372, P.

Hung, T.K. and Macagno, E.O., Houille Blanche, Vol. 21, No. 4, P.

391 (1966).

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(1972).

D. thesis, State U. of New York at Buffalo

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(1970).

Nallasamy, M., Ph.D. thesis IIsc Bangalore (1975).

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81., Academic Press, N.Y. (1969).

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Friedman, M., J.

of Eng. Math., Vol. 3, P. 21 (1969).

of Engg. math., Vol. 6, No. 3, P. 285 (1971).

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in Physics, Vol. 8, Springer-Verlag, New York (1971).

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(1970).

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(1986).

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(1970).

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E, P. 171 (1962).

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433 (1978).

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(1980).

134
62. Schrader, M.L., Ph.D. thesis, Stanford University (1981).

63. Bodoia, J.R., Ph.D. thesis, Carnegie Institute of Technology,
July (1959).

APPENDICES

APPENDIX A

OOOOGOOGOOO GOO

nnnnnnnnnnnnnnn

Appendix A

Computer program for entrance region of the channel

15

534

REAL*8 F(601,51),Fs(601,51),Q(601,51),Qs(601.51)

2.U(601,51).Y(51).X(601).PF(601,51).PC(601.51)

REAL*8 ERF,ERQ,EFF,EQQ
OPEN(UNIT=1,FILE='OUTPRE',STATUS='NEW',FORM=

2'FORMATTED')

OPEN(UNIT=2,NAME=INLET,TYPE='OLD')
F.Fs ARE THE CALCULATED STREAM FUNCTION AND
VORTICITY
FS.QS ARE THE STORED STREAM FUNCTION AND
VORTICITY
U IS A STREAMWISE VELOCITY
Y IS A NORMAL COORDINATE
x IS A STREAMWISE COORDINATE
PF IS A STREAMWISE PRESSURE GRADIENT AT THE WALL
PC IS A STREAMWISE PRESSURE GRADIENT AT CENTER
ERF,ERQ ARE THE MAXIMUM RELATIVE ERRORS
IN STREAM FUNCTION AND VORTICITY
FOR INNER ITERATIONS
EFF,EQQ ARE THE MAXIMUM RELATIVE ERRORS
IN STREAM FUNCTION AND VORTICITY
FOR OUTER ITERATIONS
READ(2,15)ITMAX,M,N,H,RE,AKS,AKV
FORMAT(3110.F5.2,F10.2,2F4.2/)
ITMAX IS A PRESELECTED NO. OF ITERATIONS
M IS A NO. OF GRID POINTS IN STREAMWISE DIRECTION
N IS A NO. OF GRID POINTS IN NORMAL DIRECTION
H IS A MESH SIZE EQUAL-IN X-AND-Y DIRECTIONS
RE IS REYNOLDS NUMBER BASED ON THE CHANNEL HEIGHT
FS IS OVER-RELAXATION FACTOR FOR POISSON EQ.
Fv IS OVER-RELAXATION FACTOR FOR N.S. EQS.
AKS IS A WEIGHTING FACTOR FOR STREAM FUNCTION
ARV IS A WEIGHTING FACTOR FOR VORTICITY
ITERF IS A NO. OF INEER ITER. FOR STREAM FUNCTION
ITERQ IS A NO. OF INEER ITER. FOR VORTICITY
ITER IS A NO. OF OUTER ITERATIONS FOR-STREAM
FUNCTION AND VORTICITY
COMPUTE OVER-RELAXATION FACTOR
******************************
PI=4.*ATAN(1.)
MM1=M-1
NM1=N-1
ALPHA=COS(PI/M)+COS(PI/N)
FS=(8.-4.*SQRT(4.-ALPHA**2))/ALPHA**2
WRITE(1.534)N,M,FS
FORMAT(10X.'TOTAL GRID Y-DIR.='.IS.10X,

135

0000

00

GD

0000

888
814

136

Z'TOTAL GRID X-DIR.=',IS,'FS=',F10.7/)

COMPUTE COORDINATES FOR GRID POINTS
***t*****t'k************************
x(1)=0.

DO 1 I=2,M

X(I)=x(I-1)+H

Y(1)=0.

DO 2 J=2,N

Y(J)=Y(J-l)+H

A. STREAM FUNCTION BOUNDARY CONDITIONS

F(I,1)=o.

F(I,N)=1.

2. DOWNSTREAM CONDITION

DO 4 J=2,NM1
F(M;J)=3.*Y(J)**2-2.*Y(J)**3

3. UPSTREAM CONDITION (ACTUAL PROFILE)
3120.055

Y1(S)=0.17

F(1,2)=0.03

F(1,3)=0.0725

F(1,4)=0.1175

F(1,5)=Y1(5)

DO 888 J=6,17

Y1(J)=Y1(J-l)+Hl

DO 814 J=6,17

F(1,J)=Y1(J)

F(1,18)=0.8825

F(1,19)=0.9275

F(1,20)=o.97

3. UPSTREAM CONDITION (UNIFORM PROFILE)
DO 5 J=2,NM1

F(1,J)=Y(J-1)+H

4. INTERIOR REGION (LINEAR INTERPOLATION)

--------------—-----------“-------------

DO 6 J=2,NM1
F(I.J)=Y(J)
VORTICITY BOUNDARY CONDITIONS

Q(1,2)=-0.225
Q(1,3)=-0.4
Q(1,4)=-0.5

331

no

34

34

400

no

15

GOO

18

0000

80

on

20

ran

137

Q(1,5)=—1.5

DO 331 J=6,16

Q(1,J)=o.

Q(1,17)=-Q(1,5)

Q(1,18)=-Q(1,4)

Q(1,19)=-Q(1,3)

Q(1,20)=-Q(1,2)

1. UPSTREAM CONDITION (UNIFORM PROFILE)
DO 34 J=2,NM1

Q(1,J)=0.

2. DOWNSTREAM CONDITION

DO 35 J=2,NM1

Q(M,J)=12.*Y(J)-6.

3. INTERIOR REGION CONDITION

DO 400 I=2,MM1

DO 400 J=2,NM1

Q(I,J)=o.

STORING STREAM FUNCTION AND VORTICITY

DO 15 I=2,MM1

DO 15 J=2,NM1

Fs(I,J)=F(I,J)

QS(I,J)=Q(I,J)

BEGIN OUTER ITERATION FOR STREAM FUNCTION
AND VROTICITY ---------------------------
ITER=0

ITER=ITER+1

SOLVING POISSON EQUATION FOR STREAM FUNCTION

ITERF=0

ITERF=ITERF+1

ERF=0.

COMPUTE STREAM FUNCTION FOR INNER REGION
****************************************

DO 20 I=2,MMI

DO 20 J=2,NM1

FOLDF=F(I,J)

F(I,J)=F(I,J)+ FS/(2./H**2+2./H**2)*((F(I+1,J)

2+P(I-1,J))/H**2+(F(I,J+1)+F(I,J-1))/H**2+Q(I,J)
3-(2./H**2+2./H**2)*F(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABS((F(I,J)-FOLDF)/EEEF))
TEST STREAM FUNCTION FOR CONVERGENCE

IF(ERF.LE.0.000001) GO TO 75

0000 0000

an

0M1

75
48

175

32

500

600

700

800

900
21

138

IF(ITERF.GT.5000 ) GO TO 999
GO TO 80
END OF INNER ITER. FOR STREAM FUNCTION

RECALCULATE F(I,J) USING WEIGHTING FACTOR
*tttt************************************

DO 48 I=2,MM1

DO 48 J=2,NM1
P(I,J)=AKS*FS(I,J)+(1-AKS)*F(I,J)

SOLVING NAVIER-STOKES EQUATIONS FOR VORTICITY

ITERQ=0

ITERQ=ITERQ+I

ERQ=0.

4. LOWER AND UPPER WALLS CONDITIONS

DO 32 I=2,MM1
Q(I,1)=(F(I,1)-F(I,2))*3./H**2-(0.5*Q(I,2))
Q(I,N)=(F(I,N)-F(I,NM1))*3./H**2-(0.5*Q(I,NM1))
DO 21 I=2,MM1

DO 21 J=2,NM1

FOLDQ=Q(I,J)

A=F(I+1,J)-F(I—1,J)

B=F(I.J+1)-F(I.J-1)

REA=0.5*A*RE

REB=0.5*B*RE

IF((A.GE.0.).AND.(B.GE.0.)) GO TO 500
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 600
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 700
IF((A.LT.0.).AND.(B.LT.0.)) GO TO 800
Q(I,J)=(1-FV)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*

20(I-I,J)+(I.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA+REB))

GO TO 900
Q(I,J)=(1-FV)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+

20(1-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))

GO TO 900
Q(I.J)=(l—FV)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*

2Q(I-1,J)+O(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))

GO TO 900
Q(I,J)=(1-FV)*Q(I,J)+Fv*(((1.-REB)*Q(I+1,J)+

20(1-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-l))/(4.-REA-REB))

EEEQ=Q(I,J)+0.00001
ERo=DMAx1(ERQ,DABS((Q(I,J)-FOLDQ)/EEEQ))
CONTINUE

TEST VORTICITY FOR CONVERGENCE

IF (ERQ.LE.0.000001) GO TO 85
IF(ITERQ.GT.5000 ) GO TO 998

GO TO 175

CECE

GOOD

85
111

93

94

nnnn

105
777

139

END OF INNER ITERATION FOR VORTICITY

RECALCULATE Q(I,J) USING WEIGHTING FACTOR
*****************************************
DO 111 I=2,MM1

DO 111 J=2,NM1
Q(I,J)=AXV*QS(I,J)+(1-AKV)*Q(I,J)

EFF=0.

EQQ=0.

Do 93 I=2,MM1

DO 93 J=2,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
EQQ= DMAx1(EQQ, DABs((Q(L J)-QS(I ,J))/EEEQQ))
TEST RECALCULATED VALUES FOR CONVERGENCE
ETA=0.000001
IF((EFF.LE.ETA).AND.(EQQ.LE.ETA)) GO TO 105
IF(ITER.GT.ITMAx) GO TO 205

D0 94 I=2,MM1

DO 94 J=2,NM1

FS(IIJ)=F(IPJ)

QS(I.J)=Q(I.J)

GO TO 18 .

END OF OUTER ITERATION

DO 777 I=2,MM1

Do 777 J=2,NMI
U(I,J)=(F(I,J+l)-F(I,J-1))/(2.*H)
WRITE(1, 2002)

2002 FORMAT(3x, 'X' 5x, 'X/RE', 6x, 'dP/dx(W)', 3x,

C
C

0(1

2' dP/dX. RE/12',,3X, 'dP/dX(C)', 'dP/dX. RE/12' /)

COMPUTE PRESSURE GRADIENT AT THE WALL
*************************************

DO 2101 I=2,MM1

X(I)=X(I-l)+H

XF=X(I)

XD=XF/RE

RESl=1./(RE*H)

RESZ=1./(RE*B**2)

RES3= 1. /(RE*H**3)

PU(I, 1)=RESZ*( —U(I, 4)+4*U(I, 3)- -5*U(I, 2)+2*U(I, 1))
PDU2=PU(I,1)*RE/12

PF(L 1)=. 5*RES3*(- 3*F(I, 5)+14*F(I, 4)- -24*F(I, 3)+

218*F(L 2)-5*F(I,1))

PDF2=PF(I,1)*RE/12
COMPUTE PRESSURE GRADIENT AT THE CENTERLINE

*******************************************

140

PC(I,11)=RE82*(-U(I+3,ll)+4*U(I+2,11)-5*
2U(I+1,11)+2*U(I,11))+RES2*(-U(I,14)+
34*U(I,13)-5.*U(I,12)+2.*U(I,11))-0.S*U(I,ll)/H
4*(-U(I+2,11)+4*U(I+l,11)-3.*U(I,11))

PDC2=Pc(I,11)*RE/12

WRITE(1,9888)XF,XD,PF2,PDF2,PC2,PDC2

9888 FORMAT(2x,F5.3,2x,F5.3,4F13.4/)

2010 CONTINUE

WRITE(1,666)ITER,EFF,EQQ,RE

666 FORMAT('NO. OF ITER.=',15,'EFF=',E14.7,10x,'
2EQQ=',E14.7,10x,'RE=',F10.2//)

WRITE(1,808)ITERF,ITERQ

808 FORMAT(10x,'ITERF=',I5,10x,'ITERQ=',I5//)

WRITE(1,180)

180 FORMAT(on,' STREAM FUNCTION VALUES'/)

DO 620 J=1,N

620 WRITE(1,621)(F(I,J),I=1,MM1)

621 FORMAT(1x,'F(I,J)=',12F10.8//)

WRITE(1,190)

190 FORMAT(10x,'VORTICITY VALUES '/)

DO 535 J=1,N

535 WRITE(1,536)(Q(I,J),I=1,MM1)

536 FORMAT(1X,'Q(I,J)=',12F10.6//)

WRITE(1,170)

170 FORMAT(15x,'VELOCITY DISTRIBUTION ')

DO 445 J=1,NM1

445 WRITE(1,446)(U(I,J),I=1,MM1)

446 FORMAT(1x,'U(I,J)=',12F10.6//)

WRITE(1,729)

729 FORMAT(10X,'CHCEK VELOCITY FROM 99 PER.
20F DEVELOPED VELOCITY'//)

DO 730 I=2,MM1

IF(U(I,11)-1.485) 731,732,732

732 WRITE(1,733) X(I),U(I,11)

733 FORMAT(10x,'PARABOLIC VELOCITY PROFILE
2AT DISTANCE =',F10.5//,10x,'VALUE OF
3CENTERLINE VELOCITY IS =',F10.5//)

731 XXI=0.

730 CONTINUE

GO TO 333

999 WRITE(1,555)

555 FORMAT(10x,'POISSON EQUATION PROBLEM')

GO TO 333

998 WRITE(1,656)

656 FORMAT('NAVIER-STOKES EQUATIONS PROBLEM')

205 WRITE(1,767)

767 FORMAT(10x,'OUTER ITERATIONS PROBLEM')

CLOSE(UNIT=2)

333 CLOSE(UNIT=1)

STOP

END

 

SE
Q

141

FLOW CHART FOR ITERATIVE PROCEDURE

 

 

GUESS VALUES FOR STREAM
FUNCTION F AND VORTICITY
Q AND STORE AS F5 AND OS

A

 

—

 

~

 

SET
FS=F

 

 

SET
QS=Q

7

SOLVE POISSON EQUATION
FOR STREAM FUNCTION F

 

 

 

 
   
 

IF I(F-Fs)/FI< 10*
NO

 

USING WEIGHTING FACTOR AND
DENOTE BY F1
1.1
L '1
CALCULATE VORTICITY AT
THE WALL USING F1

1

SOLVE NAVIER-STORES EQS.
FOR VORTICITY USING F1
AND DENOTE BY Q

[RECALCULATE STREAM FUNCTION,

 

 

 

 

 

 
   
 

IF |(Q-QS)/Ql< 10-
NO

 

RECALCULATE VORT I CITY USING
WE I GET I NG FACTOR AND DENOTE.

BY_Q1

 
 

  
    

    

 

IF I(F1-FS)/FII< 10-
IF I(QI-Qs)/QII< 10*

 

CALCULATE VELOCITY AND PRINT
‘S'I’REAM FUNCTION AND VORTICIT

 

APPENDIX B

8
8

CCCCCCCCCCCCCC «(k-II.CCCCqILFLPk~LCP¥FEhLCk

 

00000000000000

nnnnnnnnnnnnnnn

88

Appendix B
Computer program for the forward step

REAL*8 F(2601,21),FS(2601,21),Q(2601,21),

2QS(2601,21),U(2601,21),Y(21),X(2601),Z(21)

REAL*8 ERF,ERQ,EFF,EQQ
OPEN(UNIT=1,FILE='OUTCON',STATUS='NEW',FORM=

2'FORMATTED')

OPEN(UNIT=2,NAME=CONDATA,TYPE='OLD')
F,Q ARE THE CALCULATED STREAM FUNCTION
AND VORTICITY
FS,QS ARE THE STORED STREAM FUNCTION
AND VORTICITY
ERF,ERQ ARE THE MAXIMUM RELATIVE ERRORS IN
STREAM FUNCTION AND VORTICITY
FOR INNER ITERATION
EFF ARE THE MAXIMUM RELATIVE ERRORS IN
STREAM FUNCTION AND VORTICITY
FOR OUTER ITERATION
U IS A STREAMWISE VELOCITY
X IS A STREAMWISE DIRECTION
Y IS A NORMAL DIRECTION
2 IS A DISTANCE FROM THE TOP OF THE STEP
READ (2,88) ITMAX,M,N,L,MA,H,RE,AXS,AXV
FORMAT (5I5,4F10.5)
ITMAX IS A PRESELECTED NO. OF ITERATIONS
M IS A NO. OF GRID POINTS IN STREAMWISE DIRECTION
N IS A NO. OF GRID POINTS IN NORMAL DIRECTION
L IS A STEP POSITION
MA IS A STEP HEIGHT
H IS A MESH SIZE EQUAL IN X-AND Y-DIRECTION
RE IS REYNOLDS NUMBER BASED ON CHANNEL HEIGHT
AKS IS A WEIGHTING FACTOR FOR STREAM FUNCTION
AXV IS A WEIGHTING FACTOR FOR VORTICITY
FS IS AN OVER-RELAXATION FACTOR FOR POISSON EQ.
Fv Is AN OVER-RELAXATION FACTOR FOR N.S.EQS.
ITERF IS A NO. OF INNER ITER. FOR STREAM FUNCTION
ITERQ Is A NO. OF INNER ITER. FOR VORTICITY
INER IS A NO. OF OUTER ITER. FOR STREAM
FUNCTION AND VORTICITY
NM1=N-1
MM1=M—1
LL=L-1
LR=L+1
MA1=MA+1
MA2=MA~1

142

534

333i

CCQC

1%.

C

 

C
C

nnnn

C

00

()0

143

COMPUTE OVER-RELAXATION FACTOR
******************************
PI=4.*ATAN(1.)
ALPHA=Cos(PI/M)+COS(PI/N)
FS=(8.-4.*SQRT(4.-ALPHA**2))/ALPHA**2
PRINT 534,N,M,FS

534 FORMAT(10X,'TOTAL GRID Y-DIR.=',IS,10X,

Z'TOTAL GRID X-DIR.=',IS,'FS=',F10.7/)

PRINT 3333,L,MA,RE

3333 FORMAT('L=',I3,'MA=',I2,10X,'RE=',F8.1//)

l

2

3

4
5

40

6

7

8

9

10

COMPUTE COORDINATE FOR GRID POINTS
**********************************
x(1)=0.

DO 1 I=2,M

X(I)=X(I-1)+H

Y(1)=0.

DO 2 J=2,N

Y(J)=Y(J-1)+H

z(MA)=0.

DO 3 J=MA1,N

z(J)=z(J-1)+1./(N-MA)

A. STREAM FUNCTION BOUNDARY CONDITIONS

DO 4 I=1,L

F(I,I)=0.

DO 5 J=2,MA

F(L,J)=0.

DO 40 I=LR,M

F(I,MA)=0.

2. UPPER WALL CONDITION
DO 6 I=1,M

F(I,N)=1.

3. UPSTREAM CONDITION
D0 7 J=2,NM1
F(1,J)=Y(J-1)+H

4. INTERIOR REGION CONDITION

-------_---------—p---------.

DO 8 J=2,NM1

F(I,J)=Y(J)

DO 9 I=L,MM1

DO 9 J=MA1,NM1

F(I,J)=z(J)

5. DOWNSTREAM CONDITION

DO 10 J=MA1,NM1
F(M,J)=3.*Z(J)**2-2.*Z(J)**3

cccc

 

0000

11

12

13

an

14

15

16

000

300

no

80

0000

17
C
C

144

B. VORTICITY BOUNDARY CONDITIONS

1. INTERIOR REGION CONDITION

DO 11 I=2,LL

DO 11 J=2,NM1

Q(I,J)=o.

DO 12 I=L,MM1

DO 12 J=MA1,NM1

Q(I,J)=0.

2. UPSTERAM CONDITION

DO 13 J=2,NM1

Q(1,J)=0.

3. DOWNSTREAM CONDITION

DO 14 J=MA1,NMI

Q(M,J)=12.*z(J)—6.

STORING STREAM FUNCTION AND VORTICITY

DO 15 I=2,LL

DO 15 J=2,NM1

Fs(I,J)=F(I,J)

os(I.J)=Q(I,J)

DO 16 I=L,MM1

DO 16 J=MA1,NM1

Fs(I.J)=F(I.J)

QS(I.J)=Q(I.J)

BEGIN OUTER ITERATIONS FOR STREAM FUNCTION
AND VORTICITY -----------------------------
ITER=0

ITER=ITER+1

BEGIN INNER ITERATION FOR STREAM FUNCTION
ITERF=0

ITERF=ITERF+1

ERF=0.

SOLVING POISSON EQUATION FOR STREAM FUNCTION

COMPUTE STREAM FUNCTION ON THE LEFT OF STEP
*******************************************
DO 17 I=2,LL

DO 17 J82,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)+

2F(I,J-l)+F(I,J+l)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001)
ERP=DMAX1(ERF,DABS((F(I,J)-FOLDF)/EEEF))

COMPUTE STREAM FUNCTIONS ON THE TOP OF STEP
*******************************************

23

21

24.

18

23
24

244

I 145

DO 18 I=L,MM1

DO 18 J=MALNM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I—1,J)+F(I+1,J)+

2F(I,J—1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABs((F(I,J)-FOLDF)/EEEF))
CHECK STREAM FUNCTION FOR CONVERGENCE
IF(ERF.LE.0.00001) GO TO 75
IF(ITERF.GT.2000 ) GO To 999

GO TO 80

END OF INNER ITERATION FOR STREAM FUNCTION

DO 19 J=2,NM1
F(I,J)=AXS*Fs(I,J)+(1-AXS)*F(I,J)

DO 20 I=L,MM1

DO 20 J=MA1,NM1
F(I,J)=AXS*FS(I,J)+(1-AXS)*F(I,J)

BEGIN INNER ITERATION FOR VROTICITY

ITERQ=0

ITERQ=ITERQ+1

ERQ=0.

4. UPPER WALL CONDITION

DO 21 I=2,MM1
Q(I,N)=(F(I,N)-F(I,NM1))*3./H**2-(0.5*Q(I,NM1))
5. LOWER WALLS CONDITIONS

DO 23 J=2,MA2
Q(L,J)=(F(L,J)-F(LL,J))*3./H**2-(0.5*Q(LL,J))
DO 24 I=LR,MM1.
Q(I,MA)=(F(I,MA)-F(I,MA1))*3./H**2-(0.5*Q(I,MA1))
Q(L,1)=0.

DO 244 I=2,LL
Q(I,1)=(F(I,l)-F(I,2))*3./H**2-(0.5*Q(I,2))
Q(L,MA)=-(1/H**2)*(F(L,MA1)+F(LL,MA))
COMPUTE VORTICITY ON THE LEFT OF THE STEP
*************_**********.******************
DO 26 I=2,LL

DO 26 J=2,NM1

FOLDQ=Q(I,J)

A=F(I+1,J)-F(I-1,J)

B=F(I,J+1)-F(I.J-1)

REA=0.5*A*RE

REB=0.5*B*RE

IF((A.GE.0.).AND.(B.GE.0.)) GO TO 500
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 600
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 700

T.

 

11]4.\.‘

‘..
I:~

4 I I hm \U. C. NJ

4 ...C
\d

C
C

(In

146

IF((A.LT.0.).AND.(B.LT.0.)) GO TO 800
500 Q(I,J)=(1-FV)*Q(I,J)+Fv*((Q(I+1,J)+(1.+REB)*
20(1-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA+REB))
GO TO 900
600 Q(I,J)=(1-Fv)*Q(I,J)+Fv*((Q(I+1,J)*(1.-REB)+
ZQ(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
GO TO 900
700 Q(I,J)=(1-Fv)*Q(I,J)+Fv*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))
GO TO 900
800 Q(I,J)=(1-Fv)*Q(I,J)+Fv*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+O(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA-REB))
900 EEEQ=Q(I,J)+0.00001
ERQ=DMAX1(ERQ,DABs((Q(I,J)-FOLDQ)/EEEQ))
26 CONTINUE

COMPUTE VORTICITY ON THE TOP OF THE STEP
*****************************************

DO 27 I=L,MM1
DO 27 J=MA1,NM1
FOLDQ=Q(I,J)
A=F(I+1,J)-F(I-1,J)
B=F(I,J+1)-F(I,J-1)
REA=0.5*A*RE
REB=0.S*B*RE
IF((A.GE.0.).AND.(B.GE.0.)) GO TO 5000
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 6000
IF((A. LT. o. ). AND. (B.GE.0.)) GO TO 7000
IF((A. LT. 0. ). AND. (B. LT. 0. )) GO TO 8000
5000 Q(I, J)= (1- -FV)*Q(I, J)+FV*((Q(I+1,J)+(1. +REB)*
2Q(I——1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-—1))/(4.+REA+REB))
GO TO 1900
6000 Q(I,J)=(l-FV)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+
2Q(I-1,J)+(1.+REA)*O(I,J+1)+Q(I,J-1))/(4.+REA—REB))
GO TO 1900
7000 Q(I,J)=(1-Fv)*Q(I,J)+Fv*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))
GO TO 1900
8000 Q(I.J)=(1-FV)*Q(I,J)+FV*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA—REB))
1900 EEEQ=Q(I,J)+0.00001
ERQ=DMAX1(ERQ,DABS((Q(I,J)-FOLDQ)/EEEQ))
27 CONTINUE
CHECK VORTICITY FOR CONVERGENCE
IF (ERQ.LE.0.000001) GO TO 85
IF(ITERQ.GT.5000 ) GO TO 998
GO TO 175
END OF INNER ITERATION FOR VORTICITY

85 DO 28 I=2,LL

DO 28 J=2,NM1

"‘4

 

 

28

29

30

31

32

33

OD

105
38

39
666

170

445
446

‘447
4148

147

o(I,J)=AKV*QS(I.J)+(1-AKV)*F(I,J)

DO 29 I=L,MM1

DO 29 J=MA1,NM1
Q(I,J)=AKV*QS(I.J)+(1-AKV)*F(I,J)

EFF=0.

EQQ=O.

Do 30 I=2,LL

DO 30 J=2,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
EQQ=DMAX1(EQQ,DABS((Q(I,J)-QS(I,J))/EEEQQ))
DO 31 I=L,MM1

DO 31 J=MA1,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-FS(I,J))/EEEFF))
EQQ=DMAX1(EQQ,DABs((Q(I,J)-Qs(I,J))/EEEQQ))
CHECK FOR OUTER CONVERGANCE

ETA=0.000001
IF((EFF.LE.ETA).AND.(EQQ.LE.ETA)) GO TO 105
IF(ITER.GT.ITMAX) GO TO 205

D0 32 I=2,LL

Do 32 J=2,NM1

FS(I,J)=F(I,J)

Qs(I.J)=Q(I.J)

DO 33 I=L,MM1

DO 33 J=MA1,NM1

QS(I.J)=Q(I,J)

GO TO 300

END OF OUTER ITERATION

Do 38 I=2,LL

DO 38 J=2,NM1
U(I,J)=(F(I,J+1)-F(I,J-1))/(2.*H)

Do 39 I=L,MM1

DO 39 J=MA1,NM1
U(I.J)=(F(I,J+1)-F(I,J-l))/(2.*H)
WRITE(1,666)ITER,EFF,EQQ

FORMAT(10X,'NO. OF ITER.=',15,10X,'EFF=',E14.7,

2'EQQ=',El4.7//)

WRITE(1,170)

FORMAT(15X,'VELOCITY DISTRIBUTION ')
DO 445 I=2,LL
WRITE(1,446)(U(I,J),J=2,NM1)
FORMAT(1X,'U(I,J)=',10F11.8//)

DO 447 I=L,MM1
WRITE(1,448)(U(I,J),J=MA1,NM1)
FORMAT(1X,'U(I,J)='.15F7.4//)

 

180

620
621

622
623

190

533
538

535
536

999
555

998
656
205
767

333

148

WRITE(1,180)

FORMAT(10x,' STREAM FUNCTION VALUES'l)
DO 620 I=2,LL
WRITE(1,621)(F(I,J),J=2,NM1)
FORMAT(1X,'F(I,J)=',10F11.8//)

DO 622 I=L,MM1
WRITE(1,623)(F(I,J),J=MA1,NM1)
FORMAT(1X,'F(I,J)=',10F11.8//)
WRITE(1,190)

FORMAT(10X,'VORTICITY VALUES '/)

DO 533 I=2,LL
WRITE(1,538)(Q(I,J),J=1,N)
FORMAT(1X,'Q(I,J)=',10F11.5//)

DO 535 I=L,MM1
WRITE(1,536)(Q(I,J),J=MA,N)
FORMAT(1X,'Q(I,J)=',10F11.5//)

GO TO 333 -

WRITE(1,555)

FORMAT(10X,'POISSON EQUATION PROBLEM')
Go To 333

WRITE(1,656)

FORMAT(10X,'NAVIER-STOKES EQUATIONS PROBLEM')
WRITE(1,767)

FORMAT(10X,'OUTER ITERATIONS PROBLEM')
CLOSE(UNIT=2)

CLOSE(UNIT=1)

STOP

END

 

APPENDIX C

C
C

88

on

Appendix C
Computer program for the backward Step

RBAL*8 F(3201,21),FS(3201,21),Q(3201,21).

20$(3201,21),U(3201,21),Y(21),X(3201),Z(21)

REAL*8 ERF,ERQ,EFF,EQQ
OPEN(UNIT=1,FILE='OUTMOD',STATUS='NEW',FORM=

2'FORMATTED')

OPEN(UNIT=2,NAME=MODDATA,TYPE='OLD')
ALL PARAMETERS HAVE THE SAME DEFINATIONS AS
IN FORWARD STEP PROGRAM

READ (2,88)ITMAX,M,N,,MA,L,RE,H,AKS,AKV
FORMAT(5I10,4F10.4/)

NM1=N-1

MM1=M-1

LL=L~1

LR=L+1

MA1=MA+1

MA2=MA-1

COMPUTE OVER-RELAXATION FACTOR
******************************
PI=4.*ATAN(1.)

ALPHA=COS(PI/M)+Cos(PI/N)
FS=(8.-4.*SQRT(4.-ALPHA**2))/ALPHA**2
PRINT 534,N,M,FS

534 FORMAT(10X,'TOTAL GRID Y-DIR.=',15,10X,

C
C
1

2

<10

2'TOTAL GRID X-DIR.=',I5,'FS=',F10.7/)

COMPUTE COORDINATE FOR GRID POINTS
**********************************
x(1)-0.

DO 1 I=2,M

x(I)=X(I-1)+H

Y(1)=0.

DO 2 J=2,N

Y(J)=Y(J-1)+H

z(MA)=0.

DO 3 J=MA1,N

Z(J)=Z(J-1)+1./(N—MA)

A. STREAM FUNCTION BOUNDARY CONDITIONS

DO 40 I=LR,M
F(I,1)=0.

149

 

00

10

0000

11

12

13

14

15

.16

nrur;

150

2. UPPER WALL CONDITION

DO 7 J=MA1,NMI
F(1,J)=z(J)
4. INTERIOR REGION CONDITION

DO 8 J=MA1,NM1

F(I,J)=z(J)

DO 9 I=LR,MM1

DO 91J=2,NM1

F(I,J)=Y(J)

5. DOWNSTREAM CONDITION

DO 10 J=2,NM1
F(M“J)=3.*Y(J)**2-2.*Y(J)**3
B.VORTICITY BOUNDARY CONDITIONS

1. INTERIOR REGION CONDITION
DO 11 I=2,L

DO 11 J=MA1,NM1

Q(I,J)=0.

DO 12 I=LR,MM1

DO 12 J=2,NM1

Q(I,J)=0.

2. UPSTERAM CONDITION

DO 13 J=MA1,NM1

Q(1,J)=0.

3. DOWNSTREAM CONDITION

DO 14 J=2,NM1
Q(M,J)=12.*Y(J)-6.

STORING THE VALUES

DO 15 I=2,L

DO 15 J=MA1,NM1
FS(I,J)=F(I,J)
QS(I,J)=Q(I,J)

DO 16 I=LR,MM1

DO 16 J=2,NM1

Fs(I,J)=F(I,J)
QS(I.J)=Q(I,J)

BEGIN OUTER ITERATION FOR STREAM FUNCTION
AND VORTICITY ----------------------------

 

r)

300

0000

80

00

17

18

75
19

20

0(1f10

.175

0(3

21

151

ITER=0
ITER=ITER+1
SOLVING POISSON EQUATION FOR STREAM FUNCTION

ITERF=0

ITERF=ITERF+1

ERF=0.

COMPUTE STREAM FUNCTION ON THE TOP OF STEP
******************************************
DO 17 I=2,L

DO 17 J=MA1,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)

2+F(I,J-1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABs((F(I,J)-FOLDF)/EEEF))
COMPUTE STREAM FUNCTIONS ON THE RIGHT OF STEP
*********************************************
DO 18 I=LR,MM1

D0 18 J=2,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)+

2F(I,J-1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABs((F(I,J)-FOLDF)/EEEF))
CHECK STREAM FUNCTION FOR CONVERGENCE
IF(ERF.LE.0.0001) GO TO 75
IF(ITERF.GT.5000) GO TO 999

GO TO 80

END OF INNER ITERATION FOR STREAM FUNCTION
DO 19 I=2,L

DO 19 J=MA1,NM1
F(I,J)=AKS*Fs(I,J)+(1-AKs)*F(I,J)

DO 20 I=LR,MM1

DO 20 J=2,NM1
F(I,J)=AKS*Fs(I,J)+(1-AKS)*F(I,J)

SOLVING NAVIER STOKES EQUATIONS FOR VORTICITY

ITERQ=0

ITERQ=ITERQ+1

ERQ=0.

4. UPPER WALL CONDITION

DO 21 I=2,MM1
Q(I,N)=(F(I,N)-F(I,NM1))*3./H**2-(0.5*Q(I,NM1))

 

 

"‘1

152

C 5. LOWER WALLS CONDITIONS

DO 23 J=2,MA2

23 Q(L,J)=(F(L,J)-F(LR,J))*3./H**2-(0.5*Q(LR,J))
DO 24 I=LR,MM1

24 Q(I,1)=(F(I,1)—F(I,2))*3./H**2—(0.5*Q(I,2))
Q(L,1)=0.
DO 244 I=2,LL

244 Q(I,MA)=(F(I,MA)—F(I,MA1))*3./H**2-(0.5*Q(I,MA1))

Q(L,MA)=-(1/H**2)*(F(L,MA1)+F(LR,MA))

C COMPUTE VORTICITY ON THE TOP OF THE STEP
C **************t*************************
DO 26 I=2,L

DO 26 J=MA1,NM1
FOLDQ=Q(I,J)
A=F(I+1,J)-F(I-1,J)
B=F(I,J+1)-F(I,J-1)
REA=0.S*A*RE
REB=0.5*B*RE

IF((A.GE.0.).AND.(B.GE.0.)) GO TO 500
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 600
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 700

IF((A.LT.0.).AND.(B.LT.0.)) GO TO 800
500 Q(I ,J)=(l-FV) *Q( I ,J)+FV*( (Q( I+1,J)+(1.+REB)*
20(1-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-l))/(4.+REA+REB))
GO TO 900
600 Q(I,J)=(1-Fv)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+
2Q(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
GO TO 900
700 Q(I,J)=(l-FV)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))
GO TO 900
800 Q(I,J)=(1-Fv)*O(I,J)+Fv*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.—REA-REB))
900 EEEQ=Q(I,J)+0.00001
ERQ=DMAX1(ERQ,DABs((Q(I,J)-FOLDQ)/EEEQ))
26 CONTINUE
C COMPUTE VORTICITY ON THE RIGHT OF THE STEP
C *************that****************************
DO 27 I=LR,MM1
DO 27 J=2,NM1-
FOLDQ=Q(I,J)
A=F(I+1,J)-F(I-1,J)
B=F( I 1J+1)-F( I pJ'l)
REA=0.S*A*RE
REB=0.5*B*RE
IF((A.GE.0.).AND.(B.GE.0
IF((A.GE.0.).AND.(B.LT.0
IF((A.LT.0.).AND.(B.GE.0
IF((A.LT.0.).AND.(B.LT.0
£5000 Q(I,J)=(1-FV)*Q(I,J)+FV*

) GO TO 5000
) GO TO 6000
) GO TO 7000
) GO TO 8000
Q(I+1,J)+(1.+REB)*

A. O O O

153

 

ZQ(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA+REB))
GO TO 1900
6000 Q(I,J)=(l-FV)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+
2Q(I-I,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
GO TO 1900
7000 Q(I,J)=(1-FV)*Q(I,J)+Fv*((Q(I+1,J)+(1.+REB)*
20(1-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-l))/(4.—REA+REB))
GO To 1900
8000 Q(I,J)=(1-Fv)*Q(I,J)+Fv*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA-REB))
1900 EEEQ=F(I,J)+0.00001
ERQ=DMAX1(ERQ,DABS((Q(I,J)-FOLDQ)/EEEQ))
27 CONTINUE
C CHECK VORTICITY FOR CONVERGENCE
C ...............................
IF (ERQ.LE.0.00001) GO TO 85
IF(ITERQ.GT.5000 ) GO TO 998
GO TO 175
C END OF INNER ITERATION FOR VORTICITY

85 DO 28 I=2,L
DO 28 J=MA1,NM1
28 Q(I,J)=AKV*QS(I,J)+(l-AKV)*F(I,J)
DO 29 I=LR,MM1
DO 29 J=2,NM1
29 Q(I,J)=AKV*FS(I,J)+(1-AKv)*Q(I,J)
EFF=0.
EQQ=0 .
DO 30 I=2,L
DO 30 J=MA1,NM1
EEEFF=F(I,J)+0.00001
EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
30 EQQ=DMAX1(EQQ,DABs((Q(I,J)-QS(I,J))/EEEQQ))
DO 31 I=LR,MM1
DO 31 J=2,NM1
EEEFF=F(I,J)+0.00001
EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
31 EQQ=DMAX1(EQQ,DABS((Q(I,J)-Qs(I,J))/EEEQQ))
CHECK OUTER ITERATION FOR CONVERGANCE
ETA=0.000001
IF((EFF.LE.ETA).AND.(EQQ.LE.ETA)) GO TO 105
IF(ITER.GT.ITMAX) GO TO 205
D0 32 I=2,L
DO 32 J=MA1,NM1
FS(I,J)=F(I,J)
32 QS(I,J)=Q(I,J)
DO 33 I=LR,MM1
DO 33 J=2,NM1

00

33

00

105
38

39
666

170

445
446

447
448

180

620
621

622
'623

190

533
538

535
536

999
555

998
656

767
333

154

Fs(I,J)=F(I,J)

Qs(I.J)=Q(I,J)

GO TO 300

END OF OUTER ITERATION

Do 38 I=2,L

DO 38 J=MA1,NM1
U(I,J)=(F(I,J+1)-F(I,J-1))/(2.*H)

DO 39 I=LR,MM1

DO 39 J=2,NM1
u(I,J)=(F(I,J+1)-F(I,J-1))/(2.*H)
WRITE(1,666)ITER,EFF,EQQ,RE,AKS,AKV,L
FORMAT(‘N0.0F ITER.=',Is,'EFF=',EI4.7,'EQQ='.El
24.7,‘RE=',F10.2,'AKS='F4.2,'AKV=',F4.2,15//)
WRITE(1,170)

FORMAT(15X,'VELOCITY DISTRIBUTION ')
DO 445 I=2,L -
WRITE(1,446)(U(I,J),J=MA1,NM1)
FORMAT(lx,'U(I,J)=',12F8.4//)

DO 447 I=LR,M
WRITE(1,448)(U(I,J),J=2.NM1)
FORMAT(1X,'U(I,J)=',10F10.4//)
WRITE(1,180)

FORMAT(on,' STREAM FUNCTION VALUES'/)
D0 620 I=1,L
WRITE(1,621)(F(I,J),J=MA,NM1)
FORMAT(IX,'F(I,J)=',13F8.4//)

DO 622 I=LR,M
WRITE(1,623)(F(I,J),J=2,NM1)
FORMAT(IX,'F(I,J)=',10F10.4//)
WRITE(1,190)

FORMAT(10X,'VORTICITY VALUES '/)

DO 533 I=1,L
WRITE(1,538)(Q(I,J),J=MA,N)
FORMAT(1X,'Q(I,J)=',13F9.4//)

DO 535 I=LR,M
WRITE(1,536)(Q(I,J),J=1,N)
FORMAT(IX,'Q(I,J)=',10F10.4//)

GO TO 333

WRITE(1,555)

FORMAT(‘POISSON EQUATION PROBLEM')

Go To 333

WRITE(1,656)

FORMAT(‘NAVIER—STOKES EQUATIONS PROBLEM')
WRITE(1,767)

FORMAT(lox,'OUTER ITERATION PROBLEM')
CLOSE(UNIT=2)

CLOSE(UNIT=1)

STOP

END

 

 

 

 

000

no

(2
C:

00

Appendix D
Computer program for the finite step

REAL*8 F(1580, 21), Fs(1580, 21) ,Q(1580, 21),
2QS(1580, 21), U(1580, 21) ,X(1580) ,Y(21), z(21)
REAL*8 ERF ,ERQ, EFF ,EQQ
OPEN(UNIT=1,FILE='OUTSTEP' ,STATUS='NEW' ,
2FORM= ' FORMATTED' )
OPEN ( UNIT=2 , NAME=STEPDATA , TYPE= ' OLD ' )
READ(2,88)ITMAX,M,N,L,K,MA,RE,H,AKS,AKV
88 FORMAT(6I6,4F10.5/)
K IS THE END OF THE STEP
ALL OTHER PARAMETERS HAVE THE SAME
DEFINATINS AS IN FORWARD STEP PROGRAM
MMI=M-1
NM1=N-1
LL=L-1
LR=L+1
KL=K-1
KR=K+1
MA1=MA+1
MA2= MA- I
COMPUTE OVER- RELAXATION FACTOR
******************************
PI=4.*ATAN( 1.)
ALPHA=COS(PI/M)+COS(PI/N)
FS=(8 . -4 . *SQRT( 4 . -ALPHA**2) )/ALPHA**2
PRINT 4444,RE,L,K,MA
4444 FORMAT(‘RE=',F5.1,'L=',I3,'K=',IS,'MA=',I2/)
PRINT 4445,AKS,AKV
4445 FORMAT(10X, 'AKS=' ,F10.4,'AKV=',F10.4/)
WRITE(1, 534)N, M, FS
534 FORMAT(10X, 'TOTAL GRID Y-DIR.=',,15 10X,
2' TOTAL GRID X-DIR. = ,15, 'FS=', F10. 7/)
COMPUTE COORDINATES FOR GRID POINTS
**********************************
x(1)=0.
DO I I=2,M
1 x(1)=x(I-—l)+H
Y(1)=0.
Do 2 J=2,N
2 Y(J)=Y(J-l)+H
z(MA)=0.
DO 3 J=MA1,N
3 z(J)=z(J-1)+1./(N-MA)
A. STREAM FUNCTION BOUNDARY CONDITIONS

155

 

40
41
42

91

00

10

0000

11

12

121

156

1. LOWER WALLS CONDITIONS

DO 5 J=2,MA

F(L,J)=0.

DO 40 I=LR,KL
P(I,MA)=0.

DO 41 J=2,MA

F(K,J)=0.

DO 42 I=KR,M

F(I,1)=0.

2. UPPER WALL CONDITION

DO 7 J=2,NM1
F(1,J)=Y(J-1)+H
4. INTERIOR REGION CONDITION

DO 8 I-2,LL

DO 8 J=2,NM1
F(I,J)=Y(J)

DO 9 I=L,K

DO 9 J=MA1,NM1
F(I,J)=z(J)

DO 91 I=KR,MM1
DO 91 J=2,NM1
F(I,J)=Y(J-1)+H

DO 10 J=2,NM1
F(M,J)=3.*Y(J)**2-2.*Y(J)**3
B. VORTICITY BOUNDARY CONDITIONS

DO 11 I=2,LL

DO 11 J=2,NM1
Q(I,J)=0.

DO 12 I=L,K

DO 12 J=MA1,NM1
Q(I,J)=0.

DO 121 I=KR,MM1

DO 121 J=2,NM1
Q(I,J)=0.

2. UPSTERAM CONDITION

 

13

14

00

15

16

161

000

300

0000

80

00

17

18

157

Do 13 J=2,NM1

Q(1,J)=0.0

3. DOWNSTREAM CONDITION

DO 14 J=2,NM1

Q(M,J)=12.*Y(J)-6

STORING STREAM FUNCTION AND VORTICITY

DO 15 I=2,LL

DO 15 J=2,NM1

FS(IIJ)=F(IIJ)

Qs(I.J)=Q(I,J)

DO 16 I=L,K

DO 16 J=MA1,NM1

Fs(I,J)=F(I,J)

Qs(I.J)=Q(I.J)

DO 161 I=KR,MM1

DO 161 J=2,NM1

Fs(I,J)=F(I,J)

QS(I.J)=Q(I,J)

BEGIN OUTER ITERATION FOR STREAM FUNCTION
AND VORTICITY ----------------------------
ITER=0

ITER=ITER+1

SOLVING POSSISON EQUATION FOR STREAM FUNCTION

ITERF=0

ITERF=ITERF+1

ERF=0.

COMPUTE STREAM FUNCTION ON THE LEFT OF STEP
************tt*****t***********************
DO 17 I=2,LL

DO 17 J=2,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)+

2F(I,J-1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF:F(I,J)+0.00001
ERF=DMAX1(ERF,DABS((F(I,J)-FOLDF)/EEEF))
COMPUTE STREAM FUNCTION ON THE TOP OF THE STEP
**********************************************
DO 18 I=L,K

DO 18 J=MA1,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)+

2F(I,J-1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABS((F(I,J)-FOLDF)/EEEF))

 

 

C
C

0000

00

181

75
19

20

120

175

23

24

244
242
245

158

COMPUTE STREAM FUNCTION ON THE RIGHT OF THE STEP
************************************************
DO 181 I=KR,MM1

DO 181 J=2,NM1

FOLDF=F(I,J)
F(I,J)=F(I,J)+0.25*FS*(F(I-1,J)+F(I+1,J)+

2F(I,J-1)+F(I,J+1)-4.*F(I,J)+H*H*Q(I,J))

EEEF=F(I,J)+0.00001
ERF=DMAX1(ERF,DABs((F(I,J)-FOLDF)/EEEF))
CHECK STREAM FUNCTION FOR CONVERGENCE
IF(ERF.LE.0.000001) GO TO 75
IF(ITERF.GT.5000 ) GO TO 999

GO TO 80

END OF INNER ITERATION FOR STREAM FUNCTION

RECALCULATE F(I,J) USING WEIGHTING FACTOR
*****************************************

DO 19 I=2,LL

DO 19 J=2,NMI
F(I,J)=AKS*FS(I,J)+(1-AKS)*F(I,J)

DO 20 I=L,K

DO 20 J=MA1,NM1
F(I,J)=AKS*Fs(I,J)+(1-AKS)*F(I,J)

DO 120 I=KR,MM1

DO 120 J=2,NM1
F(I,J)=AKS*Fs(I,J)+(1-AKS)*F(I,J)

BEGIN INNER ITERATION FOR VORTICITY

ITERQ=0

ITERQ=ITERQ+1

ERQ=0.

4. UPPER WALL CONDITION

DO 21 I=2,MM1
Q(I,N)=(F(I,N)-F(I,NM1))*3./H**2-(0.5*Q(I,NM1))
5. LOWER WALLS CONDITIONS

DO 23 J=2,MA2
Q(L,J)=(F(L,J)-F(LL,J))*3./H**2-(0.5*Q(LL,J))
Q(K,1)=0.

Q(L,1)=0.

DO 24 I=LR,KL
Q(I,MA)=(F(I,MA)-F(I,MA1))*3./H**2-(0.5*Q(I,MA1))
DO 244 I=2,LL
Q(I,1)=(F(I,l)-F(I,2))*3.0/H**2-(0.5*Q(I,2))
DO 242 J=2,MA2
Q(K,J)=(F(K,J)-F(KR,J))*3./H**2-(0.5*Q(KR,J))
DO 245 I=KR,MM1
Q(I,1)=((F(I,1)-F(I,2))*3./H**2-(0.5*Q(I,2))

159

Q(L,MA)=-(1./H**2)*(F(L,MA1)+F(LL,MA))
Q(K,MA)=-(1./H**2)*(F(K,MA1)+F(KR,MA))
C COMPUTE VORTICITY ON THE LEFT OF THE STEP

C *****************************************
DO 26 I=2,LL
Do 26 J:2,NM1
FOLDQ=Q(I,J)
A=F(I+1,J)—F(I-1,J)
B=F(I,J+1)-F(I,J-1)
REA=0.5*A*RE
REB=0.5*B*RE
IF((A.GE.0.).AND.(B.GE.0.)) GO TO 500
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 600
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 700
IF((A.LT.0.).AND.(B.LT.0.)) GO TO 800
500 Q(I,J)=(1-FV)*Q(I,J)+FV*((Q(I+1,J)+(l.+REB)*
2Q(I—1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA+REB))
GO TO 900
600 Q(I,J)=(1-Fv)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+
2Q(I—1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
GO TO 900
700 Q(I,J)=(1-Fv)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*
20(I-I,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB)
GO TO 900
800 Q(I,J)=(1-FV)*Q(I,J)+FV*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-l))/(4.-REA-REB))
EEEQ=Q(I,J)+0.00001

900 ERQ=DMAX1(ERQ,DABS((Q(I,J)-FOLDQ)/EEEQ))
26 . CONTINUE

C COMPUTE VORTICITY ON THE TOP OF THE STEP
*****************************************
DO 27 I=L,K

DO 27 J=MA1,NM1
FOLDQ=Q(I,J)
A=F(I+1,J)-F(I-1,J)
B=F(I'J+l)-F(IIJ‘1)
REA=0.5*A*RE
REB=0.5*B*RE
IF((A.GE.0.).AND.(B.GE.0.)) GO TO 5000
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 6000
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 7000
IF((A.LT.0.).AND.(B.LT.0.)) GO TO 8000
5000 Q(I,J)=(1-FV)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA+REB))
GO TO 1900
6000 Q(I,J)=(1-Fv)*Q(I,J)+FV*((Q(I+1,J)*(1.-REB)+
2Q(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
Go To 1900
7000 Q(I,J)=(1-Fv)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+Q(I,J+l)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))

 

 

 

 

160

GO TO 1900
8000 Q(I,J)=(1-Fv)*Q(I,J)+FV*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA-REB))
EEEQ=Q(I,J)+0.00001
1900 ERQ=DMAX1(ERQ,DABs((Q(I,J)-FOLDQ)/EEEQ))
27 CONTINUE
C COMPUTE VORTICITY ON THE RIGHT OF THE STEP
C ******************************************
DO 272 I=MER,MM1
DO 272 J=2,NM1
FOLDQ=Q(I,J)
A=F(I+1,J)-F(I-1,J)
B=F(I,J+1)-F(I,J-1)
REA=0.5*A*RE
REB=O.5*B*RE

IF((A.GE.0.).AND.(B.GE.0.)) GO TO 5001
IF((A.GE.0.).AND.(B.LT.0.)) GO TO 6001
IF((A.LT.0.).AND.(B.GE.0.)) GO TO 7001

IF((A.LT.0.).AND.(B.LT.0.)) GO TO 8001
5001 Q(I,J)=(1-FV)*Q(I,J)+FV*((Q(I+1,J)+(1.+REB)*
20(1-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J41))/(4.+REA+REB))
GO TO 2900
6001 Q(I,J)=(1-FV)*Q(I,J)+Fv*((Q(I+1,J)*(1.-REB)+
2Q(I-1,J)+(1.+REA)*Q(I,J+1)+Q(I,J-1))/(4.+REA-REB))
GO TO 2900
7001 Q(I,J)=(1-Fv)*Q(I,J)+Fv*((Q(I+1,J)+(1.+REB)*
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA+REB))
, GO TO 2900
8001 Q(I,J)=(1-Fv)*Q(I,J)+FV*(((1.-REB)*Q(I+1,J)+
2Q(I-1,J)+Q(I,J+1)+(1.-REA)*Q(I,J-1))/(4.-REA—REB))
EEEQ=Q(I,J)+0.00001

2900 ERQ=DMAX1(ERQ,DAas((Q(I,J)4FOLDQ)/EEEQ))
272 CONTINUE

‘ C CHECK VORTICITY FOR CONVERGENCE
1 C -------------------------------
‘ IF (ERQ.LE.0.00001) GO TO 85
‘ IF(ITERQ.GT.5000 ) GO TO 998
, GO TO 175
C END OF INNER ITERATION FOR VORTICITY
C . ------------------------------------
C RECALCULATE Q(I,J) USING WEIGHTING FACTOR
C ***t*************************************

85 DO 28 I=2,LL
DO 28 J=2,NM1

28 Q(I,J)=AKV*QS(I,J)+(1-AKv)*Q(I,J)
DO 29 I=L,K
DO 29 J=MA1,NM1

29 Q(I,J)=AKV*QS(I.J)+(1-AKv)*Q(I,J)

DO 129 I=KR,MM1
DO 129 J=2,NM1

nnnh

129

30

31

131

32

33

133

105
38

39

161

Q(I,J)=AKV*QS(I,J)+(1-AKV)*Q(I,J)

EFF=0.

EQQ=0.

DO 30 I=2,LL

DO 30 J=2,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
EQQ=DMAX1(EQQ,DABS((Q(I,J)-QS(I,J))/EEEQQ))
DO 31 I=L,K

DO 31 J=MA1,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-Fs(I,J))/EEEFF))
EQQ=DMAX1(EQQ,DABS((Q(I,J)-QS(I,J))/EEEQQ))
DO 131 I=KR,MM1

‘DO 131 J=2,NM1

EEEFF=F(I,J)+0.00001

EEEQQ=Q(I,J)+0.00001
EFF=DMAX1(EFF,DABS((F(I,J)-FS(I,J))/EEEFF))
EQQ=DMAX1(EQQ,DABs((Q(I.J)-QS(I.J))/EEEQQ))
CHECK OUTER ITERATION FOR CONVERGENCE
ETA=0.000001
IF((EFF.LE.ETA).AND.(EQQ.LE.ETA)) GO TO 105
IF(ITER.GT.ITMAx) GO TO 205

D0 32 I=2,LL

DO 32 J=2,NM1

FS(I,J)=F(I,J)

QS(I.J)=Q(I.J)

DO 33 I=L,K

DO 33 J=MA1,NM1

FS(I,J)=F(I,J)

QS(I.J)=Q(I.J)

DO 133 I=KR,MM1

DO 133 J=2,NM1

Fs(I,J)=F(I,J)

Qs(I.J)=Q(I.J)

GO TO 300

END OF OUTER ITERATION

.---b-----------------

COMPUTE STREAMWISE VELOCITY
***************************

DO 38 I=2,LL

DO 38 J=2,NM1
LHI,J)=(F(I,J+1)-F(I,J-1))/(2.*H)
DO 39 I=L,K

DO 39 J=MA1,NM1
U(I,J)=(F(I,J+1)-F(I,J-1))/(2.*H)
DO 139 I=KR,MM1

139
666

170

445
446

447
448

449
450

180

620
621

622
623

664
665

190

533
681

535
536

537
538

999
555

998
656
205
767

333

162

DO 139 J=2,NM1
U(I,J)=(F(I,J+1)-F(I,J-l))/(2.*H)
WRITE(1,666)ITER,EFF,EQQ,RE

FORMAT(10X,'NO. OF ITER.=',I5,10X,'EFF=',E14.7

2,10X,'EQQ=',E14.7,10x,'RE=',F10.2//)

WRITE(1,170)

FORMAT(15X,'VELOCITY DISTRIBUTION ')
DO 445 I=2,LL
WRITE(1,446)(U(I,J),J=2,NM1)
FORMAT(IX,'U(I,J)=',10F10.3//)

DO 447 I=L,K
WRITE(1,448)(U(I,J),J=MA1,NM1)
FORMAT(IX,'U(I,J)=',10F10.4//)

DO 449 I=KR,MM1
WRITE(1,450)(U(I,J),J=2,NM1)
FORMAT(1X,'U(I,J)=',10F10.3//)
WRITE(1,180)

FORMAT(10X,‘ STREAM FUNCTION VALUES'/)
DO 620 I=2,LL
WRITE(1,621)(F(I,J),J=2,N)
FORMAT(IX,'F(I,J)=',10F10.6//)

DO 622 I=L,K
WRITE(1,623)(F(I,J),J=MA,N)
FORMAT(1X,'F (I,J)=',11F9.6//)

DO 664 I=KR,M
WRITE(1,665)(F(I,J),J=2,N)
FORMAT(IX,'F(I,J)=',10F10.6//)
WRITE(1,190)

FORMAT(10X,'VORTICITY VALUES'/)

DO 533 I=1,LL
WRITE(1,681)(Q(I,J),J=1,N)
FORMAT(IX,'Q(I,J)=',10F10.3//)

DO 535 I=L,K
WRITE(1,536)(Q(I.J),J=MA,N)
FORMAT(1X,'Q(I,J)=',11F9.3//)

DO 537 I=KR,M
WRITE(1,538)(Q(I,J),J=1,N)
FORMAT(1X,'Q(I,J)=',10F10.3)

GO TO 333

WRITE(1,555)

FORMAT(‘POISSON EQUATION PROBLEM')
GO TO 333

WRITE(1,656)

FORMAT(‘NAVIER-STOKES EQUATIONS PROBLEM')
WRITE(1,767)

FORMAT(10X,'OUTER ITERATION PROBLEM')
CLOSE(UNIT=2)

CLOSE(UNIT=1)

STOP

END