ANALYTICAL EVALUATEON OF THE EFFECTS
OF HEAT TRANSFER AND SKIN FRECTION 0N THE
EFFTCEENCY 01" THE EXPANSION OF A GAS HAVING
A PRANDTL NUMBER 0F ONE THROUGH A
CGNVERGENG ‘ DWERGTNG NOZZLE OF
LARGE MEWS WETH CONSTANT
TEMPERATURE WALLS

Thesis for The Degree of DB. D.
MICHIGAN STATE UNIVERSITY

Richard Lee Ditsworth
1958

THESIS

This is to certify that the

thesis entitled
ANALYTICAL EVALUATION OF THE EFTEOTS OF HEAT TRAIISFPR AND
SKIIHT FRICTION OZI TEE EFFIC EJI‘ECY OF THE EX AIFSIOE‘T OF A GA
HAVIZEG A PRAlIDTL IITILIBER OF OZJE THROUGH A COIWERGI?IG-
DIVERGII‘JG I‘JOZZLE OF LARGE RADIUS WITH CONSTANT TIE-*fPEfiIM‘I‘URF
L‘JALLS.
presented by

RICHARD L. DITS‘IORTH

has been accepted towards fulfillment
of the requirements for

Doctor of Philosonhv degree‘inITeCha-Vfical Erlgineeifing

 

n Q
Date August 24, 195;

   
   

1, [BR A R Y
Talichigan State

""3 b mvcrsxty

'V - r s
’ I-I”.mi‘!'-m

 

ANAHTICAL EVALUATION OF THE. bFFdCTS OF HEAT TRANSFER
AND SKIN FRICTION ON THE EFFICIENCY CF THE EXPANSIQQ
OF A GAS HAVING A PPANDTL NUMBER OF ONE T'rfiiOUGH A
CONVERGING-DIVETGING NOZZLE OF LARGE RADIUS WITH
CONSTANT TWPERATWE WALLS

By

Richard Lee Ditsworth '

AN ABSTRACT

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Lgriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILCBOPHY

Department of Mechanical Engineering

Approved 9%” a 9

Richard Lee Ditsworth

AN ABSTRACT

The effects of skin friction and heat transfer were evaluated for
a gas expanding through a nozzle. This study was lindted to visc0us
and compressible gaseS‘with.Prandtl number of‘unityu. The nozzle Size
was taken to be sufficiently large so that boundary'layer thiCknesses
could be neglected, the nozzle was conically-shaped in both converging
and diverging sections, and the nozzle walls were assumed to have a
constant temperature,

The evaluation of effects on exit mach number were obtained using
a generalized one-dimensional approach. The skin friction and heat
transfer were established from three-dimensional considerations
(axially-symmetric flow). This required consideration of velocity and
temperature gradients occurring in a boundary layer between the wall
and the main potential flow; Laminar boundary layer differential
equations for continuity, energy and momentum Were taken as the baSiS
for boundary layer analysis. A series of transformations, due to
Iangler and to Stewartson, were used to reduce equations.from a three-

dimensional compressible case to a form similar to a two-dimenSIOnal

and essentially incompressible case. An additional transformation,

including an exponential velocity distribution for the potential flow,

Was used to obtain a set of ordinary differential equations of the

same.form previously solved and tabulated in NACA Report 1293 by Cohen

and Reshotko.

(3

Richard Lee Ditsworth. 4.

In order to utilize the temperature and velocity gradient infer-ma-

tion from boundary layer considerations, expressionS'were develOped in

a generalized one-dimensional coondinate system. Solution of these

expressions gave the effects of skin friction and heat transfer on

nozzle efficiency.

Nozzle efficiencies were given in graphical fknmzas a function of

a parameter dependent on initial conditions. Fuwnn these results of the

analysis, the following conclusions were reached:

1.

Heat transfer effects were dominant in the converging
section, and this dominance may extend slightly into
the diverging region for much colder walls.

Skin friction effects become more important in the
diverging section.

Combined effects influence reduction in efficiency of
a mach 3.0 nozzle of the order of 0.2 per cent.

The effect of the boundary layer thickness on.flmw
area, though small for nozzles of large radius, should
be considered along with Skin friction and heat trans-

fer effects to improve reliability of nozzle efficiency

predictions.

ANAHTICAL WALUATION OF THE EFFEIITS OF HEAT TRANSFER
AND SKIN FRICTION ON THE EFFICIENCY OF THE EXPANSION
OF A GAS HAVING A PRANDTL NUMBER OF ONE THROUGH A
CONVERGING-DIVERGING NOZZLE OF IARGE RADIUS WITH
CONSTANT TEMPERATURE WALLS

By
Richard Lee Ditsworth

A THESIS

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DCETOR OF PHILOSOPHY

Department of Mechanical Engineering

1958

U , D 2", 25'”

‘1
HJgeet

ACNNOTUEKHNENT

The author wishes to express his deep
appreciation to Dr. halph M. Rotty for his
suggestion of the topic of research and his
most helpful criticism and encouragement

throughout the course of this investigation.

VITA

Richard Lee Ditsworth
candidate for the degree of

Doctor of Philosophy

Final examination, August 223 1958, 10:15 A.M., Room 113,
Olds Hall

Dissertation: Analytical Evaluation of the Effects of Heat
Transfer and Skin Friction on the Efficiency of
the Expansion of a Gas Having a Prandtl Number
of One Through a ConvergingéDiverging Nozzle of
Large Radius with Constant Temperature Walls

Outline of Studies

major Subject: Mechanical Engineering
Minor Subjects: mathematics, Physics

Biographical Items

Born: April 15, 1925, Algona, Iowa

Undergraduate Studies: Iowa State College, 19h2-h3;
cont. 19h6-h9 .

Graduate Studies? Iowa State College 1951-52; Michigan
State University, 1953-58

Experience: Navigator, Army Air Corp, 19h3-h6; Engineer,
Chance Vought Aircraft, l9h9-Sl; Instructor in
Applied Mechanics, Iowa State College, 1952-53;
Instructor in mechanical Engineering, Michigan
State University, 1953-58

Member of Phi Kappa Phi, Sigma Xi, Pi Tau Sigma, Sigma Pi Sigma

TABLE OF CONTENTS

NOL‘IEJCMTURE

INTRODUCTION . . . . . . . . .

THEORETICAL ANALYSIS . . . . . . .
Objectives. . ,. . . . . . .

Assumptions . . . . . . . .
Mathematical Development . . . .

Generalized One-dimensional Flow .

potential Flow 0 O O O O 0
Boundary Layer Flow . . . .

PROCEDURE IN CAICUIA‘I‘IONS . . . . .
PRESENTATION AND DISCUSSION OF HhBULTS.
CONCLUSIONS. . . . . . . . . .
APPENDIX. . . . . . . . . . .

HSTOFREFhHianS . . . . . . .

Fag e

Cplr'wwH

17
3h
39
53
Sh
76

IJST OF FIGURES

Figure Page
1 Converging-diverging nozzle with space
coordinates and their origins . . . . h
2 Control surfaces. . . . . w . . . . 8

3 Sketch of symmetric nozzle with sink and

source points for radial flow . . . . 15
h Sketch of boundary layer in duct . . . . 20
5 Comparison of ‘3me values for flat
plate and given nozzle laminar boundary flow to
6 Graphical presentation of values of each

term in Equation (111), and variation
with LEA O C O O O C C 0 O O O hl

7 Variation of nozzle efficiency with
parameter’zfis for exit MSA = 1.5 . '. . h5

8 Variation of nozzle efficiency with
parameter‘éfi for exit MSA - 2.0 . . . N6

9 Variation of nozzle efficiency with .
parameter A for exit MSA - 2.5 . . . 147

10 Variation of nozzle efficiency with

parameterzCS :for exit MSA = 3.0 . . . ha
11 Comparison of nozzle efficiency based on
same exit area or pressure . . . . . h9

118T OF TABLES
Table _ Page

1 Values of ParameterzC>|.for Different Given
Conditions..........h3

A

cc

Cl: cl

EffnP

NOMENCLATURE

Symbols used:

cross section area

sonic velocity, ( = 3’ FT)
arbitrary constants
local skin friction coefficient (See Eq. ('7))

specific heat at constant pressure ( = —————

specific heat at constant volume
nozzle efficiency based on same exit area

nozzle efficiency based on same exit pressure

force

function of variable 7' related to stream function

 

enthalpy per unit mass

boundary layer Stagnation enthalpy <; CpT + E.

constant in velocity distribution (See Eq. (78))
thermal conductivity

Sutherlands constant

arbitrary length

length of convergent section (See Fig. l)

length of divergent section (See Fig. 1)

.5 u

mach number 8-;

constant exponent in velocity distribution (See Eq. (69))

Prandtl number (- 32-124..)

static pressure

velocity in cylindrical coordinate system
heat energy transfer

rate of heat transfer at wall

cylindrical coordinate - radial length
nozzle radius

specific gas constant

Reynolds number (- 11935.)

enthalpy function =.—_.s-l
static temperature

longitudinal velocity component

velocity in one dimensional system

normal velocity component

mass flow rate

length coordinate along wall (See Fig. 1)

first transformed length coordinate (See Eq. (53))
second transformed length coordinate (See Eq. (59))
normal length coordinate (See Fig. 1)

first transformed normal length coordinate (See Eq. (Sh))

37 I!

B
U
8
A:
A
7
it
/\
/u
Q
)0
)Du
’1’

'I’

second transformed normal length coordinate

(See Eq. (60))

axial coordinate of nozzle

sonic velocity ( -‘ XIII)

 

2m
t ’ E . a '
cons ant in q (73) ( m + 1
0
ratio of specific heats (-«32)
v

thickness of boundary layer

parameter of initial conditions (See Eq. (ll2))
small interval

variable (See Eq. (79))

nozzle wall angles (See Fig. l)

viscosity proportionality constant (See Eq. (62))
dynamic viscosity

kinematic viscosity (/K/)’)

mass density

mass velocity

shear stress (See m. (88))

stream function

Subscripts used:

e

min

local.flow outside boundary layer
minimum

polar coordinates
radial.direction

isentropic process

isentropic process based on same area

SP

St

Others:

la:

(—1
O
t_1

Il‘

(—‘I

isentropic process based on same pressure
stagnation values

axial direction

wall‘value

conditions where M a l

stagnation value. In boundary layer equations, refers
to free stream stagnation value

converging section
diverging section
single prime refers to first transformed coordinates
double prime refers to second transformed coordinates

origin of coordinate system (x, y)

sub bar means vector quantity
order of magnitude of
approximately equal to

average quantity for given interval

INTRODUCTION

The purpose of a nozzle is to control the expansion of a fluid to
a lower pressure region in order to obtain a high velocity. If the
medium is a compressible fluid, it is possible for acceleration from
subsonic to supersonic speeds to occur if the nozzle has a convergent
section followed by one that is divergent. In such a case the speed at
the minimum cross section or throat is identical to the speed of sound
for the state of the medium at the throat and the Mach number is equal
to one.

From an idealized thermodynamic standpoint, the expansion of a gas
through a nozzle is regarded as occurring without.friction or transfer
of heat, and is described as a reversible and adiabatic proceSs. Hence,
using the assumption of an isentropic process, an ideal exit velocity
may be calculated when the state of the upstream gas and the exhaust
region pressure is given. Since the actual velocities Obtained for the
same operating conditions vary somewhat from the ideal, the engineer
uses a factor'relating the above two velocities. In many applications,
performance evaluated in this manner is sufficiently accurate, espe-
cially when there is an abundance of empirical test data for accepted
nozzle shapes and sizes tested at different pressure ratios and essen-
tially adiabatic conditions.

In the case of an exhaust nozzle for a jet engine, the situation
is somewhat different. The size is necessarily large, and the up-

stream temperature is high. The thrust performance, resulting from the

2.

change of the momentum of the gas, is particularly dependent on an ac-

curate measure of the efficiency of the nozzle and a difference of one

per cent or less may mean the difference between success or failure.

If the inlet temperature is in the order of 3600°R, the external walls

of the nozzle must be cooled, and hence the effects of heat transferred
should be considered. Since throat diameters are large, perhaps one to
two feet, the difficulties of obtaining reliable experimental informa-

tion is magnified and an analytical evaluation of such losses as those

due to.friction and heat transfer at the wall occurring as a viscous

compressible fluid expands to supersonic speeds, has increased value.

THEORETICAL ANALYSIS
Objectives

The purpose of this investigation was twofold: first, to evaluate
quantitatively the relative magnitude of skin friction and heat losses
in a viscous compressible fluid expanding to supersonic speed through a
nozzle with straight walls at a constant temperature; and second, to
evaluate the effects of skin friction and heat losses on the exit vel-
ocity of the fluid. In addition to providing a better understanding
of flow phenomena, the information could be used to establish theoret-
ical maximums in nozzle efficiency coefficients. Not all of the causes
of change from ideal performance by an actual gas have been considered:
variation of specific heats, displacement thickness of boundary layer,
turbulence, and dissociation are some factors which have been omitted

in this study .

Assumptions

A converging-diverging nozzle was considered as follows:

1. The walls were conical with the converging section slightly
rounded at the throat, as shown in Fig. l.

2. The radius was considered as being large, e.g., a throat
diameter of at least six inches.

3. The wall surface was smooth

 

 

 

 

Fig. 1. Converging-diverging nozzle With space coordinates
and their origins .

The system undergoing the expansion process was assumed to be
a compressible pure substance with these properties:
1. The applicable equation of state was p - )9 ET.
2. Specific heat values were constant.
3. Viscosity was dependent on temperature only, and suitably
described by the Sutherland formula.

1.. The value of Handtl number (02% ) was unity.
k

S. The ratio of specific heats ( K ) was equal to 7/5 for
purposes of obtaining numerical results.

6. No phase change occurred.

The flow phenomena resulting in supersonic speeds in the gas in
the absence of body forces, was considered from two points of view.
The first was that of a steady and continuous one-dimensional.flow
occurring with surface forces or wall friction and external heat trans-
fer. The term "one-dimensional" means the fluid properties and vel-
ocity are constant over each cross section. Thermodynamic and fluid
dynamic reasoning may be utilized to Obtain.mathematical expressions
which include terms for skin friction and heat transfer. In order to
solve these differential equations and evaluate the state and velocity
of the fluid at the exit section, additional information was needed
concerning the skin friction and heat transfer terms. This information
is the velocity and temperature gradients in the fluid at the wall.
This required consideration of the flow immediate to the solid boundary,
and hence the second point of view. It is generally accepted that flow
in a channel or around a body may be considered as consisting of a thin

layer of fluid next to the wall, where viscous and inertia terms in the

equation of motion are taken as having the same order of magnitude,
and the main body or "core" of fluid with negligible viscous forces
and heat transfer located external to the boundary layer. Analysis of
an axial-symmetric'boundary layer with large external axial pressure
gradients and heat transfer at the wall was necessary to Obtain the
desired gradients of velocity and temperature at the wall.
Assumptions for the generalized one-dimensional flow were as
follows:
1. The process was steady and continuous.
2. The velocity, pressure, and temperature are uniform at any
given cross section.
3. A frictional shearing stress and heat transfer occur at the
nozzle wall area.
Assumptions made for the "core", or potential flow external to
the boundary layer were:
1. The process was frictionless, adiabatic, steady, and cone
tinuous. -
2. An apparent sink-source type flow was taken to describe the
mass velocity ( r9 He) distribution in the converging and
diverging sections respectively, with uniform properties

at any given r , except in the region close to the throat.

P

3. The potential flow does nottiepend on the boundary layer
flow in regard to first order effects.

The considerations for the boundary layer flow were:

1. The flow was assumed to be laminar. This was reasonable

in view of the large negative pressure gradient existing

in the potential flow, a favorable condition to forestall
a transition to turbulent flow. Further, it has been

found by Lees1

that cooling the fluid noticeably increases
the stability of the laminar layer.

The usual boundary layer assumptions for laminar flow were
taken; that is, the large velocity change from wall to the
potential flow occurred in a very thin layer, 8 , and

inertia and viscous stresses were of the same order of

magnitude.

In all cases the flow phenomena must behave according to the fol-

IOWing basic physical laws:

1. Equation of state.
2. Continuity equation.
3. Energy equation.

h. MOmentum.equation.

 

l

Lees, L. The Stability of the Laminar Bound Layer '
Compressible FluIdT NICK TechC_Note, No. I360, I927.

IS
In:

 

 

Mathematical Development

Generalized One-dimensional Flow

 

A control volume analysis is used to Obtain the one-dimensional

expressions that are needed. The control surface is shown in Fig. 2.

 

 

 

Fig. 2. Control Surfaces

A basic expression of continuity for steady flow is in the form

fJVL-dgso (1)

and when integrated for constant )9 and V at each cross section, one

of a surface integral:

obtains:

W = FIVIAI " 202va2 (2)

or in logarithmic differential form:

 

 

o.dw .dr .__<.1_V..+dA (3)
w )0 V A

In the absence of shear work and with negligible body forces, the
First Law of Thermodynamics for steady flow may be written:

2

W011 + ll... 2

)+ wQ . w(h+‘_;_) (h)

or, considering the changes occurring through a distance dz , the

energy equation is :

de - w(dh + _.__......_dV2 ) (5)
2

The definition of constant pressure specific heat:

db
0 .. .._... (6)
P dT

where h is a function of temperature only, is used to restate Eq. (5)

 

 

as follows:
dQ _ dT . d(V2/22 (7)
OPT T cpif

The basic momentum theorem for the fluid flowing steadily through

the control surface is:

E: =- {m an (8)

The forces on the fluid within the control volume are due to

normal and sidewall pressures and a sidewall friction. These effect

10.

a change of momentum flux expressed as:

dp+IwifilL+ flVdVBO (9)

The equation of state is:

p - )0 FT (10)

and may be written:

p )0 T

 

The definition of sonic velocity is:

a a {($13 (12)

X
Since p = const ( )0 ) for the assumed gas undergoing an isentropic

 

process, the speed of sound is also represented as:
a = D’II'T (13)

Mach number is defined as the ratio of stream velocity to speed

of sound:

M2 ' V2 a __V2___ _ 1’4
c:2 OFT ( )
and is written:
2
c1112 dV dT
7 ' “‘5" - —— (15)
V ‘1‘
Reference states such as isentropic stagnation condition and at

M - l are useful in control surface analysis. Stagnation refers to

conditions of zero velocity, and the isentropic stagnation state of a

11.

fluid denoted by subscript (o) is that obtained by an adiabatic and re-
versible slowing-down.process to negligible motion. Hence, the energy
equation, (Eq. h), written for two control sections so chosen to make

the external work equal to zero:

 

2 2 2
. V V v
Q + cpTl + 21 a cpT2 + £2- = cpT + 2 (16)

can be used to obtain the stagnation temperature for state one and two,

 

as:
V 2
Tel a T]. + 1 (l7)
2 cp
and: .
V 2
2
T02 8 T2 + ‘ (18)
2 0p

If a process is isentropic or adiabatic, the stagnation tempera-
ture remains constant, but decreases in the presence of cooling since
Q is negative. 'When reference is made to a fluid at a condition of
M a 1, a subscript (*) is used.

'When the relation for specific heat for a perfect gas:

5/ ._
Op, b2 1 R (19)

is used with the definition of Mach number, note that:

v2 = I120? =1.12( x - no}; (20)

and hence, EQuations (7) and (9) may be written respectively as:

2
d.Q a dT + 5’ - 1 M2 dV
OFT T 2 v2

 

(21)

12.

 

 

and:
up ._ KM? dV2 _ ”(w dAw (22)

Simultaneous. solution of the Equations of continuity, state, definition
of Mach number, energy and momentum which are (3), (11), (15), (21),

and (22) respectively, yields:

X’ - l
sz .-2(1+-————2———-—-M2) (1A. + 1+XM2 (1Q

.12— l-Mé A l-M2 cT+

2(1 +-——L2-1-LM2) a (SAW
1-M2 A

 

(23)

This equation may be used to evaluate the actual exit Mach number
when expressions for cooling and wall friction effects are available

as function of M and a space coordinate. It is noted that in a fric-

 

2
tionless and adiabatic flow, 41%:— may be evaluated using the d: term,

and the exit Mach number obtained represents an ideal for the given
exit area. The presence of the denominator (1 - M2) in Equation (23) is
. instructive. When M < 1, M increases for area decrease, heating and
friction. For M > 1, M increases for cooling and area increase and
decreases with friction.

The integration of Equation (23) is usually accomplished numer-
ically in stepwise fashion with attendant difficulties in the region
of M = 1 because of the denominator l - MZ tending to zero. This was

eliminated by rearranging Equation (23) to:

 

 

l-M2 an? ._d_A_+#l+YM2 dQ+
2(I+_.§.§:.1..M2) MZ A 2(1+—1-2-'—'—l-M2) cpT
7’ df‘w
W A (2b)

where the left hand term was integrable:

 

 

M ~ \ b’+l
1_M2 and? _l1+"§2":‘L"’2i ZW-l)
uni—zine) m2 "ML 1r 1 at.
I 2 1+ 2 M1):

 

M1
7. )

Upon integration of the right hand term of Equation (2h) the final
actual M may be obtained.

One additional property was needed to establish the final state.
The stagnation temperature was convenient. From Ehuation (18), it

follows:

v K-l
TO=T+ 20 I'T l'0'-"'---'E-'-"-"'-l'12 (26)

 

OI‘:

 

dTo dT +l X-I on? 1+ r_1M2 (27)

'r T\2 M2 2

 

When Equations of energy, definition of Mach number and definition of
stagnation temperature, which are (7), (20), and (26) respectively,

are combined:

dTo . 1 . dQ (28)
To 1+_.I_§:.];.MZ CT

 

 

 

Integration of Equation (28) evaluates To'
Nozzle efficiency is a ratio of actual exit kinetic energy to
ideal exit kinetic energy. It may be based on the same exit area or '

pressure. Nozzle efficiency based on exit area is:

 

 

 

 

Eff "' T ' (29)
and using Equation (26)
X.- 1
a MZ 1’ +__--— 2 MBA T02
Eff “"TM (30)
2
Nozzle efficiency based on same exit pressure is
5’2 - 1
M2 1 + To

EffnP “MS: 2 (31)

 

 

M5?) 1 + ::_LJ;.M2 T01

Which requires obtaining the actual pressure ratio «2- , for the actual
M and evaluating “SP for the same pressure ratio. An expression for

2- maybe derived from Equations (2), (10), (lb), and (26):

3

PI

 

T-l 2

 

 

 

~3— ' AIM)” 1 + 2 Ml—e T02 (7:2)
2

and for isentropic processes

15.

p T O‘- l
___...p . ___...T (33)
l l

and hence 1651: may be obtained using Equation (26) with Equation (33)

in the following form:

 

 

l 3’
2 If - l
34- 1+ Yfi-IMSP (3n)

Potential Flow

The "core“ flow outside the film was assumed to be adiabatic and
reversible. Because of the conical sections, the flow was assumed to

be a radial type from apparent sink and source points located as shown

in Figure 3 .

  
    

/\

1f

11.

 

 

4"’I,,,/’/'::l§23flerr* VFZ Z %l
,fI-r l - f...— f -
' / “V70, arenf

.____- apparent VP. 5,,«7/2'0 Pom'f

Source pomf

Fig. 3. Sketch of symmetric nozzle with sink and source points
for radial flows .

16.

‘With symmetry about the z—axis spherical coordinates were used.

The continuity equation is:

9( puerg)

arp 0 (35)

 

Because of dependence of ue and other properties on rp only, the flow
is irrotational and hence may be called a potential flow.
Integrating Equation (35) gives:

const

We " TI, (36)

In accelerating subsonic flow there is a minimum area at which "choked"

flow occurs. The polar radius for condition of M - l, is denoted as

 

rp* . Equation (36) may then be written for steady flow as:
2
r
P *u# * n
_T . _£__ . _£__<I__ . m (37)
r you }?u )ouc
P”
1 1/2
T* — 1 a* T ‘ ,
with—é:— .(-'-1‘-— T and a - (Ti) , uat10n(37)

 

 

 

 

becomes:
2 U. + l X' + 1
rp _ 1 T* 2( 3’ - 1) . 1 T* To 2( If - l) (38)
r2 M T M ToT
Pa

and using it with Equation (26) since the process is adiabatic:

K + 1

2 W
rp,_1 2 +w—1 “7"”
1.2“ M 3+1 (1 2 M2) (39)

 

17.

The energy equation is a restatement of constant stagnation enthalpy

throughout the adiabatic flow control surface:
2 _
‘n = h + V (to)

 

Therefore, the stagnation temperature is constant in the potential

flow. The momentum or Euler equation is:
)0” du 3 _ (12 (b1)
dr . dr

The expressions given in this section on potential flow were needed in
order to specify the velocity and pressure gradients in the direction

of flow at the outer boundary of the laminar layer.

Boundary Layer Flow

Axially symmetric flow for a viscous compressible fluid is des-
cribed by the following basic equations in cylindrical coordinates
(r, z, {6) and velocities represented by q.

1. Equation of state:

 

 

p = )0 ET (142)
2 . Continuity equation:
l “91”“) + .1. 800%) (L3)
r 3 r r 8 z

3 . Energy equation:

ah+q 3h .q&2+q QB
P[qr8r 92] rar 92+

18.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1:5)
q :qz + qz ng . _ ; a) + l 3
r r 2 ED 2 {/9 z
ngié, 2 29(rqg) 2 E9‘1z
[9436; E? z - 3r é; :7 - —‘ ;>:5/> +

 

‘ aq aq\
r z
--——--- r ‘ + (In)
)Dr C7r[/a :2 arjj
The difficulty of solution of these differential equations is ob-

vious. Applying laminar boundary layer theory, the equations may be

19.

reduced in complexity when considering a low viscosity fluid in a thin
layer along a wall (region of high Reynolds number). It was convenient
at this point to change the coordinate system to (x, y) with veloc-
ities (u, v) as shown in Figure 3, and rewrite Equations (1:3), (ML),
(15), and (’46) in dimensionless form.

The folIOWing boundary layer assumptions were then applied:

1. v/u - [O] S/x

where S is the boundary layer thickness,
2. inertia and viscous forces are of the same order of magnitude:

Dugr a an
>0u 93‘ [Olav/“9y

 

Retaining only high order magnitude terms, the resulting boundary layer
equations were:
1. Equation of state:
p . )0 m (147)

2 . Continuity equation:

1 30011:) + 9603) ,
r ax 3y 0 (148)

3 . Energy equation:

2
3h+8hg 32+Q,_3T+au
)0 9x vay 3x 9y( 2y)/u(9y)
(1:9)

 

h . Momentum equations:

 

 

 

 

u—Q—L'tvau =-_._1__.__a_2_.+ 1 a all)
33: ay )0 ex )0 ay flag)
0

20.

$13-2— [o] 1 (51)
9V

The transformation formulae used are given in Appendix A. Because
of the length of the expressions, the complete derivation was not in-
cluded. The above expressions are identical to those for axially
symmetric boundary layer flow over bodies of revolution of large radius
given in Pail. Pressure is considered to be a function of x only since
Equation (51) states that the change of p in the y direction is neglig-
ible since y S. 3 .

It is possible to reduce the boundary layer Equations (14?), (148),
(1:9), and (50) to a form identical for two-dimensional boundary flow
for a compressible fluid when the duct radius is large relative to
thickness of boundary layer as shown in Figure 1;, (so that r may be

replaced by R) .

 

M.-- . '— Fo+enrm/ [/aou

My
_-.__'

 

- ,_._-___l.-____i--__f+_..i...- a f .

Fig. )4. Sketch of boundary layer in duct.

 

lPai, Shih-I. Viscous Flow Theog, Vol. ;. laminar Flow. D. Van
Nostrand Company, Inc., Hinceton, . J., I936, p. 1131.

21.

This was done using the following Mangler Transformationzl

x/L
x' - L .112. d(X/L) (53)
L2
O
u ,._E_
y L y (Sh)

where L is an arbitrary length.

The resulting basic equations are:

 

 

 

1.13" man (55)
2. amour) . c)(/°'v') .0 (56)
E)x' £93”
2
an:
3. a I ,._____ + ' '._...._._ +
)011 ax: va a)" a x' ”/4 21311)
.57.... 103.11.
8y. u( ay' (57)

 

)4. It____a____u' + 'v'_}?____u' ._ 390+ C) '3“!
VII—— 3;" )0 9y! ax: 3y: (# ay.

(58)

Transformation formulae used are given in Appendix B.

Further coordinate changes may be made with a modified Stewart-

2

son's transformation. This was used by Cohen and Reshotko to obtain

I

lLow, George M. Simplified Methods for Calculation of Compressible

Laminar Bound Layer with Arbitragfi— Free-Stream Pressure —Gradient.
NECK TN 9,51 p. 18.

2Cohen, Clarence and Eli Reshotko. Similar Solutions for the Com-
pressible laminar Bound La er with Heat Transfer and Treasure Grad-
ient. NACK Report [595, £935.

 

 

 

 

 

 

 

 

22.

a solution of the steady two-dimensional compressible laminar boundary

layer equations . The transformation equations are:

x'/L
x" . L A 3232. d(x'/L) (59)
poao

y|
y" =- _C.E§_ L dy' (60)
a0 Jf’o
o

The viscosity lawl used was:

,dx' _ ’x T'
/‘° To

TW To + Ksu
/\ - \I T (62)
° Tw + Ksu

Equations (61) and (62) match values from the Sutherland formula for

 

 

(61)

where :

 

viscosity at the solid boundary. Note that )\ is a constant when Tw

is constant.

An enthalpy function was defined as

 

 

h .
3'1?“ (63)
. (u')2 .
With hst' 2 + h equal to the local stagnation enthalpy.
1

Chapman, Dean and Morris N. Rubesin. Temperature and Velocity
Profiles in the Compressible Laminar Boundary Layer with Arbitrary Dis-
tribution—3f_§urface Temperature. Jour. Aero. Sci., Vol. 16, No. 9,
September I5h9, pp. 557-565.

23.

The transformed two-dimensional boundary layer equations (See

Appendix C for additional transformation formulae) are:

 

 

 

l. p" = WET" (61;)
2.12.". +2.12. .
9x“ 3y" 0 (65)
3. up ‘2)3 +'v" 15>S a 2)0
3x" 93’" pr
3'-
8 a s) _ (1 _ Pr) ”M:
33'" 9y" 1 + %-L lmez

 

 

 

L
h.u"__._au" +vn______311"=u" 9“ e(1+s)+

‘9 29 (—§:;fz . (66)
737'; cDV"
7)

€9u"

 

a1" 3y" 3 9X" 0 3y" 33'"
(67)

A point of interest about the modified Stewartson transformation
is that 'E'quation (65) no longer contains density as in Equation (56).
Hence, the set of differential equations appear similar, in part, to
those for incompressible flow and are identical for an adiabatic core
with Pr - l.

The applicable boundary conditions to Equations (61:), (65), (66),

and (67) are:

u"(x", o) = 0 lim S = O
7"?”
V"(X", 0) g 0
lim u"- ue " (68)

S (x00, 0) a 3' 7'.»

2b.

Cohen and Reshotko:L next made a final transformation of the boundary
layer equations using:

u". = can)!“ (69)

8:3(1) (70)

 

 

(I; f‘Y ) \lz Uoue'hc" (71)

m+l

7 = y" \rngl ‘5?" (72)

 

 

 

and obtained:

93f r 92f (af)2 1 s 3
_.............+ - ___.____ .. .. )
973 fig? 3 9‘7 (7

a? ———g—-—"1Mez
S +prf_‘2_§_=(1-Pr)

 

 

__2.___e
2
Dr :33: + 821* (7b)
37 0f 9%
whereB=—-?-m—l—.
111-.-

Hence, a set of ordinary differential equations were obtained,
dependent only on the variable «7 , if the right hand member of
Equation (71;) were taken as zero or a function of ”)L . The case of
Pr = 1 was taken, and results of their solution tabulated for different
values of 3w (dependent on wall and potential stream stagnation temp-

erature) and B (a measure of pressure gradient).

 

1:Cohen and Reshotko, op. cit.

25.

The boundary conditions for Equations (73) and (71;) when Pr = l

are:
f(o) . 0 11m .QJ. . 1
+00
affo) : 0 7

9‘7 . =0

11m S
3(o) = sw 7" (75)

 

The values of m in ue" = c(x")m that would be needed to describe
the subsonic potential velocity in the given nozzle are as follows:

1. At low Mach numbers

(111 «I
Where ‘er 3-1 0 then m = o
2. At M increasing to one
where u8 = finite value then m >> 1

An expression for m may be obtained by differentiating Equation (69)

with respect to x":

 

 

 

 

 

du " - mu I.
dx: = mc(X")In 1' ‘3 ----——x"6 (76)
and
d I.
m a ue l xlt
dx" u N
or x
m = due - R2 . l . 29.3. 39 aepe R2 dx (77)
dx I2 ue ae Pe a p L2

00

26.

 

 

II
where due _ Ooh 1 p0 due
ax! . H2
dx L2
x
GePe R2
x" g A . __ . dx
aoPo L?
o
a
u " ‘._9 u
e 3e e

Equation (77) may be used to obtain values for m. In the.first

 

example, if ue and x are not zero, and 3:9 g 0, then m g 0. This

 

is analogous to flat plate flow where ue = constant and 3:3 = O.

 

In the second example, :9 _;. a... as Me—v- 1, hence m becomes

very large. Since m must be a constant, the transformation Equations
(69), (70), (71), and (72) were not useful as such for the given
problem.

It was found that if the following potential velocity distri-

bution was assumed x“

KI"
ue" = cle L

(78)

and was used with following transformations

'I’" ” mp [Ln °L (79)

K1
3 = 3(7)
27)OL

y" 3
7 nu."

 

 

 

 

27.

in Equations (6h), (65), (66), and (67), the transformed equations

 

 

 

became
331‘ 921' gr 2
-—-———+f = 2 __ -(l+S) (80)
973 an? (a?)
323 as (ar-1)M2
_._......._..... + PI‘f a (1 .. Pr) 8
r7 2 J + r — 1
7 V 1 *T‘Mz
2
g; a3r + (921‘) (81)
97 973 9:12
'with the following boundary conditions
f(o) = 0 Que) __, 0
8(0) 33” 9?
limS=O 11‘“ af=1 (82)
7%» 7—4»

The transformations and detailed substitutions to obtain Equations (80),
(81) are given in Appendix D.

Since the transformed Equations (80), (81) are the same set of
simultaneous non-linear ordinary differential equations of fifth order
with the same boundary conditions as in Equations (73), (7b) with B = 2
and Pr = 1, the solutions for f and S are the same.

The one remaining point to be satisfied was that the velocity
distribution as defined in Equation (78) described the given potential
flow to be investigated. The derivation required that Kl be a constant.
Further discussion of this point may be found in the Procedure Section.

Temperature and velocity gradients were desired at the wall. They

may be expressed as follows:

28.

Since .52.}... = .1137... , the velocity gradient is:

3‘7 ue

(I 2 b:
.5211. =__‘."_2.,__..S3.j..-__2i.fi (53)
a y do &yfl2 f0 L

and further, at the wall,
2
(i3...) .( “8) (”L 11 " (fl?) 21‘ .) Kine" R (at)
93' w “0 >00 e 07‘? w 27) L L

An expression for temperature is needed. Using the following:

2

 

 

 

2 - l 2
3.8.1396 +__(.2,...u' Tq’To (1+3) (1+ K_l 2)
Te 662 Tog Me
T +213.
T3,____3_32
Te Te

2
3" .. _ u 2
hence .i'; a (1 + S)(l + ___..__1.M82) X l e u

 

e 2 2 2:5 ue
U - 1,12 2
01" L a (1 + S) .. — (2.5.) (85)
To 1 + #Mez 9"?

Therefore, the temperature gradient is
a T _ T 3 3
83* x ° 33* x

(9T) 3T (93)” “6% 2 x1...» (86)
ayw o 3’1 “on L 2720L

 

 

 

 

 

29.

since - 0 at the wall.

 

31"
(9

One method used to check some of the data obtained was to evaluate

the dimensionless parameter cfw)’ Raw and compare with available

data for flat plate flat. The following expression was derived in the
x,y coordinate system for wall conditions.

The skin friction coefficient is defined as

7:.
cf '(1/2)/ow “e2 (87)

 

and shear stress in the laminar flow is defined as

7/

     

(88)

Substituting Equations (83), (88) into (87) for wall conditions:
2
M 3363) “eh—«2 92 (KM ’0 5-
° T av; 21>]. p0 L w
l/2 )Dw (ue" )2

Cf '

(89)

The definition of Reynolds number is

m...- 33:- (90)
w

and combining Equations (89) , (90)

2
a6 9 r); Klue" . ue x
A/(o 130(3- all) us "(9:2 V2 90 L 9W
of " Raw = 2 (91)
w 1/2 fl, (1.16")

 

 

 

30.

Since K1 in the velocity distribution is

 

 

 

K In nan/e1 1n ue"/c1
l I a ’ 92
x"/L x/L p R2 ( )
a2 8 ._- d L
o A aOpO 112 (X/ )

Equation 91 may be written

0:. Wr(‘“‘3‘v}f“)

W

 

 

x/ L . 32: . .3- F34)
R 01 Ta
- - I ' ° (93)
L V1; 3 b’ — 1
'2( ""-"‘1 2
< 32‘ T ) 5-5 d(x/L)
To L

 

dQ and I; W
cpT A

were needed for

 

Finally, expressions for

use in Equation (21:) . These terms depend upon velocity and temperature
gradients in the fluid adjacent to the wall. The gradients were ob-
tainable upon solution of the laminar boundary layer equations .

Fourier's Law of heat conduction was used

 

 

 

 

91'
q n—k (9h)
w (9y)w
to obtain
«(2»
dQ , _ Y dA
cT wcT W (95)

31.

The following substitutions may be used:

 

 

 

dAw 311127} R dx
I L: 6
A feueu - cos #)R2 (9 )
k ./¢ (97)
°p
p = D6 (98)

along with Equations (10), (1h), (61), (86), (92) and isentropic p - T

relations to obtain the heat transfer term:

 

 

 

 

 

 

 

 

 

dQ ,_ _ sin}2 7}
CDT 1 - cos 79
“I
7)o A In 113"
2m. 'c‘i'
x/L 3 r — 1
2( 2f - 1)
Me < 3} \ E; d(x/L)
+ — 1 L
1 2 112/
O
. 93 d L (99)
( 9"?)w (X/ )

32.

The skin friction term may be obtained using Equations (11:), (61) ,
(8h), (88) and (96):

73am I'KN? . “fig—(Te).

 

 

 

 

 

 

 

 

 

A l - cos?" To
720 A In :1 1
21% 1
x/L 3 K" - 1
2( U - l) 2
Me < 1 \ 7 d(X/L)
1 + Y E l 112/
0
(49—33%) d(x/L) (100)
a’L w

Hence Equation (2h) may be written:

1 - M2 dM2 dA sin2 7"

 

 

 

 

 

 

 

 

 

:(l + U2“ 1M2) M2 - T 1 - COS??- .
90 A In ue"
2130 3i"
I x/L 3 ‘6' -_l
2( ‘0' - l)
l R
—-d(x/L)
O

 

 

 

Kn.)
LO
0

When a constant wall temperature, inlet stagnation conditions,

and nozzle angles are given, the following parameters are fixed: 7)o,

A9391» K 9 (fi), (cjzf) ,and fl. Fquation(101)
° 97w 91%

is then usable to find M for a given change in 3:.

Knowing variation of Mwith 1:, Equations (28) and (99) may be

 

 

 

 

 

 

 

 

 

combined:
dTo . _ 81112 & 1 a
To 1—cos735 1+ E-IMZ
2
00A 1 113:
n
2L“o c1 # A (a S) d(x/L)
3 I - I
x/L V 2( 1r .. 1) ”I W
1 R2
M d L
e 1 + 75 21.112 :2 (X/ ) (102)
2

to allow solution for To.

HimmURE IN CALCULATIONS

For convenience, data was evaluated by specifying the ideal Mach
number and then obtaining values for displacement, area, etc.. This
‘was done because of the difficulty in solving for “SA explicitly in
most of the gas dynamic equations.

The origin_for x, displacement along the wall, was chosen at

M = 0.001, where fluid properties had essentially stagnation values.

Using this initial condition, the constant c in Equation (78) must be

 

equal to ue"x . The ratio of 11e" in Equation (92) may be written:
'L a 0 c1
ue" ue" ue' a0 '
T‘u“ 'Z—E—T—=_—— (103)
l e x e e .E,. O
1'0 L

With reference to Figure l, useful relations between Space co-

ordinateS'were:

R1 = (rpl* + L1 - x) sin 91
R2 = (rpz-x» + x - L1) sin 92 (10h)

'where subscripts l and 2 refer to converging and diverging sections

respectively.

For A1* = A2* , then

 

r22* 1 - 008 91
rplfi l - COS 92 (105)

and displacement in terms of area ratios become:

 

x A A
" '- - -— for M < 1 (105)
rpl* (A* >_____x '0 Ax.

r131}

1 - cos 0
X ' V l A; - l + L1 for M ) l (107)

For calculation purposes, such as in Equation (101), the arbitrary

 

 

 

 

length Lwas considered as rpl* .

Area ratios were obtained using Equation (39). Since the term

 

1 appeared regularly, it was convenient, from Equation
1 4- _.9:__:_3_-M2
2
TSA
(26), to tabulate its equivalent as T- .
o

Logarithmic and anti-logarithmic data were obtained by use of
tables1 and the logarithmic series for 1n (1 + x).

For purposes of evaluating effects of losses, the nozzle entrance
was taken where MSA = 0.1. However, the integral under the radical
sign in Equations (101) and (102) had to be evaluated from the origin
x = O. In Appendix E, Table I, small intervals were taken and data
evaluated to obtain this integral and the variation of K1. The inte-
gration was approxinmted by summing the product of the average value
of the integrand, and the displacement, for a small interval. The
change in the integral was negligible after Me > 0.5. These values ob-
tained by the previously described method of integration were found to

be quite satisfactory when checked in the interval of 0.001 ( M < 0.1

 

1Table of Natural Logarithms, Vol. I, National Bureau of Standards,
19h1.

36.

 

 

 

11.0
(where the large change occurs and(.¥£) has little change) using
0
3'= l.h‘With:

X/rpi* x/rpr.
.0 .0 2

Te R2 x T 1‘ . A x (108)

d - e 81112 0 K35 d
0 p1 pl* 0

O

and integrating the right hand expression:

(is x -(A w w x
F; 91 A* rp1* F rph, rpli,

(109)

 

 

 

 

The K1 variation may be seen in column ( 7) of Table I, Appendix E.
Although not of constant value, it was considered that the variation
was not excessive in the range of Mach number to be considered, es-
pecially with primary interest in if ) 1. K1 was considered as a con-
stant and average values were used for given small intervals. There-
fore, the integration results1 of Equations (73) and (71:) for B - 2 and
Pr = l were taken as applicable to Equations (80) and (81) for Pr 1

The data used were:

 

 

a 2 '
37%;)” - 1.6870 and 3;)” " 0 when :w - 1.0

.. 1.3329 - 0.23014 ° - 0.6
=- 0.91.80 . 0.11331 =- 0.2
(110)

lCohen and Reshotko, op. cit.

37.

As ShOWn in Equation (30) it was convenient to express efficiency
in terms of actual exit Mach number M. To obtain this, Equation (101)
had to be integrated. Note that if substitutions arexnade in Equation
(101), using Equations (103) and (110), the only variables are M and x.
Numerical methods could be used to integrate this equation. However,
calculations were based on the assumption that very small error would
be introduced by using ideal mach.numbers in the coefficient of d(x/L)
in place of actual M, since effects of friction and heat transfer were
expected to be quite small compared with the change in deue to area
change.

From Equation (101), for an adiabatic and frictionless process,
the integrated expression.for actual area change is identical to that
given in Equation (25) in terms of ideal Mach number (MSA)’ based on

area change only. Hence, a solution of Equation (101) was taken to be:

 

 

 

 

,. . 4.1.1...
1 + 'r - 1M? 2('K'- 1)
-111 2 ”SA - 81112 9 &01\ .
3’- 1 2 M (1 - cos 9) 2La
l +.___....MSA o
L 2 J

 

 

23') A (x/L) - (111)

'where (fi) - average value of B for the (x/L) interval

”3A

 

 

,_ 1n0.001 11 .
x/L 3r-1
T 21—3 ) 2
“SA (if) r If} d(x/L)

o

 

 

_ 1 *n‘SAZ _ (as) + X ”3:12 3%
_.___ 1..“ all _____2
2(1+._§.§:_1MSA2) 9‘7 w 1+__.....X2" 11431.2 071 w
and L = rpl*

Intervals of 0.05 for MBA were taken between 0.1 i M S 3.0.

Calculated data using Equation (106) for one set of initial conditions,
92 = 15°, 10°, and 5° and Tnx/ To = 1.0, 0.6, 0.2 are given in
Appendix E, Table IV.. Stagnation temperature Equation (102)

was evaluated in a similar manner, with calculated data given in

Appendix E, Table IV. Results of T02 , M actual and Effna are
To
1

given in Table V, with the nozzle efficiency values obtained by using
Equation (30).

Some values of nozzle efficiency based on same exhaust region
pressure were obtained by:

1. obtaining p/pl using M actual in Equation (32).

2. calculating MSp.for same pressure ratios using Equation (3b).

3. evaluating Effnp.for values of “SP: M.Actua1, and T02/ T01

using Equation (31).
Values of the parameter cfvafie : were evaluated using previously

obtained data in Equation (93) given in Appendix E, Table I.

PRESENTATION AND DISCUSSION OF RESULTS

Using calculation procedures previously described, results were
obtained for:
1. c " Re
fw W
2. Nozzle efficiency based on same exit area for several
values and constant wall temperature, exit angle and

parameter A .

3 . Nozzle efficiency based on same exit pressure.

These values were dependent on the effects of skin friction and
heat transfer at the smooth walls of large radius occurring in a com-
pressible fluid of K i 1.11 and Pr = 1.0 expanding in a continuous
manner to the exhaust region pressure.

The comparison of cfw Rew for the given flow to that of flat
plate flow where external velocity is constant and pressure gradient
zero, shown in Figure S , was considered as adequate. It is reasonable
that the initial flow in the entrance of the converging section ap-
proaches flat plate flow behavior since the axial pressure gradient is
essentially zero at those cross sections. The effect of cold walls in
decreasing the cfw W was consistent since of W becomes very small
in larger M as the wall Reynolds number must increase rapidly due to
increases in x and ue.

In Figure 6, a graphical presentation of values of the three terms

in Equation (111) at various downstream locations

Call;

0./

0

 

 

 

 

(Shapiro, P' los ¢)

1 l ' I

Where: ‘V =
Calcu/afea’ dafa — \
Ha r plate f/ow - — ‘

 

 

0

/.0
M

2.0

Fig. 5. Comparison of cfw VRew values for flat plate
and given nozzle laminar boundary flow.

to.

Heaf Transfer, 5km flvchon Terms ”7- 54.0”).

 

+003/0 l T T F
I
f00’80 __ Example 9'08"; I / _
. #325. I //
,Pg/g" /
A~ -. a
:3X/O '7. /
k' 0‘9 I /
\
M0460 — "9 1° P‘p/ / ~
(0
/ /
Mal-V0 *- Skm frtclion \ / / ._
./ /
1:04.20 *- _
0.000 d- j—i’" / #-

-o.o,,20 >—

-o.o,,¢o .—

—0,0,60

' 0° 0’80 r—

 

-o.o.m

 

 

  
 

 

fleaf frans {e r

  
  

Egan/Ion ////J (an/fer? as:

;a. '
' 'H-r—‘Mz 2w.)
‘Ifl 7 fl”. :XC‘OQHQ)A%— + EOCKQ A 5.
/+ gins; M ‘
z -

= 60’ ’7‘. fem 4- {nchon {arm

 

 

 

Fig.

6. Graphical presentation of values of each term in
Equation (111), and variation With ”SA-

141.

E2.

was of interest. In both cases of cold walls, the heat transfer term
had the dominant role of influence on actual Mach number in the sub-
sonic flow, and with lower temperature conditions, extended its in-
fluence into the supersonic region. At a point of zero net effects,
it should be noted that the state is not the same as the ideal because
of the frictional effects on pressure even though Mach numbers are the
same.

A parameter A was defined because of the many possible combina—
tions of other parameters in Equation (Ill). The defined parameter

used consists of
A .90" sin92 (1-cos 91)
- ZQDGOH-cosgz)

 

 

(112)

Resultant Avalues for various possible operating and given
conditions are presented in Table 1. The choice of 10 psia low
pressure would be consistent with a condition of operation of jet ex-
haust nozzles at high altitudes where the exhaust region pressure is
quite low relative to sea level atmoSpheric conditions. The following
summarizes the changes in A parameter resulting from changes in in-
dividual parameters: ‘

Parameter increased A
throat radius (Rmin) decreased

 

 

Table 1

Values of Parameter A: .for Different Given Conditions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O

a 1.0

 

 

Rmin ' 3 in.
T 92 I 15° 02 I 5°
0 .
po 3 10 p0 I 600 po 3 10 p0 . 600
1200 0.0389931 0.031L610 0.0215616 0.0319571
2500 0.0213171 0.0317395 0.0223398 0.0330207
3600 0.0213u7h 0.0321135 0.0228u27 0.0336699
h200 0.0217750 0.0322915 0.0230822 0.0339791
R‘min ' 6 in.
—-—~T 92 a 15° 92 . 5°
0
p0 = 10 p0 - 600 Po = 10 p0 - 600
1200 0.0363591 0.0h82095 0.02110h2 0.0313838
2500 0.039527 0.0312300 0.02165h5 0.0321300
3600 . 0.0211577 0.031h9h5 0.0220101 0.0325950
1200 0.0212551 0.0316203 0.022179h 0.0328136
Rmin ' 15 in.
To 02 a 15° 02 - 5°
po-lo po-600 po-lO po=600
1200 0.03h0218 °°°h51921 0.0369837 0.0h8752h
2500 0.0360258 0.0137793 0.02101161: 0.0313509
3600 0.0373213 0.0h9h519 0.0212713 0.0316h12
h200 0.0379380 0.03102h8 0.021378h 0.0317795
sin 0 (l - cos 0 )
where A. I ?)o/\ 2 _ 1
2 Rmin “o (l " cos 92)
92 325° K 'llt 8353.14 To inoR
TM, ksu = 198.6 pO in psia

’43.

Exhaust section angle (02) decreased
Stagnation pressure (p0) decreased
Stagnation temperature (To) increased
Viscosity constant (In) increased

(Variation not shown in Table 1)

Effects of skin friction and heat transfer on exit velocity were
expressed in terms of effects on nozzle efficiency. Values of effic-
iency, based on design area, for ideal exit mach numbers of 1.5, 2.0,
2.5, and 3.0 are given in Figures 7, 8, 9, and 10 respectively.

Other calculations substantiate that the linear curves may be extended
in increasing A direction to at least 0.023. The resulting curves
show the effects of friction and heat transfer at walls of different
temperatures. In all cases, efficiencies decreased with increases in
105 parameter. At smaller exit angles of 02 the effects are more dis-
tinct as a result of the increased wall area due to greater length.

For a given .Z§.parameter value, the effects of various Tvv /To con-
ditions for Mach numbers greater than twenty were negligible, which
leads one to the conclusion that heat transfer effects were negligible
or that skin friction was extremely dominant. Under such conditions
the effect of heat transfer as an energy loss was negligible. It would
not follow that adiabatic and non-adiabatic walls for 02 2— 15° have the
same flow behavior, for cold walls can.definite1y influence the velocity
gradients and hence skin friction in the boundary layer.

Figure 11 expresses the relation between the nozzle efficiencies

established on different bases. The value based on same pressure at

 

 

 

 

 

 

 

 

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0.9985 _ -
A .. .9o1\ szga-Cosv?)
“ ZRmm 49(1'5055)
0.9980 but 1 ' J '
0.0.. 0.035 0. 02/

A

Fig. 7. Variation of nozzle efficiency with parameter
for exit M311 ' 1.5.

145.

 

 

“.
\ ~\\
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0.9985 -

 

0.9980 fi

Fig. 8.

* T«I/T.=I.0

 

:616 -----
:02‘-——--

 

 

A , dehsflO-acw
- 2 ern a.(l'c°51%)

0,035

A

Variation of nozzl-
for exit MBA ‘ 2.0.

efficiency

1:6.

 

- 4.1

0. 02/

with parameter

 

 

 

 

 

 

L7.

 

 

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’ 0.9980 ‘ J L
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Fig. 9. Variation of nozzle efficiency with parameter

for exit Mg); = 2.5.

 

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A

Fig. 10. Variation of nozzle efficiency with parameter
for 6X11} MSA : 3000

1:8.

 

 

   
 
   
  

 

 

 

 

 

 

Nozzle Eff/CIency

Fig. 11. Comparison of nozzle efficiency based on same exit

area or pressure.

_ ~ __ .. ' \' new
(16 Same. on? area ‘
Same emf pressure
. f _
.x. \ .
Rm“1 g g _ M54: 2.5
0.4 - \ .4
Based in: \
.. 7; :O.z
- 2-5 /To‘* - _x -_ Mg=20
0.2 — A .a.802600 / -
- - 9
0.0 i 1 1 J 1
49990 Q9994 04998 L0

50.

exit section for an actual and an ideal nozzle is convenient in thermo-
dynamic studies since thermodynamic systems, the processes undergone,
and resultant states are important. From the design standpoint, where
physical.equipment and effects of changes thereof are of primary inter-
est, efficiencies based on area can be very useful. In the given ex-
ample, comparison shows the'pressureebased efficiency to be higher,and
by about 30% of the remaining change from area-based efficiency to
100%. The pressureébased efficiency was expected to be higher because,
using a given area, when frictional pressure drop occurs the exhaust
pressure would not be reached in the fluid at exit section, and further
expansion to this slightly lower pressure would allow M‘to be slightly
larger in the corresponding pressure-based value.

The effects of skin friction and heat transfer as demonstrated in
the change of nozzle efficiency appear quite small. However, it is
“well to restate that other phenomena known to occur under certain
circumstances and difficult to evaluate analytically and experiment-
ally, were not included.

The calculated data would say that for (1) a M = 3 nozzle with
30—inch diameter, stagnation pressure of 600 psia, stagnation tempera-
ture of 3600°R and exhaust angle (92) of 5°, the nozzle efficiency
based on area is 0.9998 or(2) a M = 3 nozzle with 6-inch diameter,
stagnation pressure of 10 psia, stagnation temperature of h200°R, and
exhaust angle of 5°, the efficiency is in the order of 0.997.

These values would be the estimate_of efficiency of a nozzle with

gas of Fr - l expanding with

51.

l. smooth walls such that surface irregularities do not
cause a disturbance in the flow to promote transition
from laminar to turbulent boundary layer

2. exit pressure identical to exhaust regions

3. no discontinuities in the flow due to shocks

h. wall cooling to retain laminar layer

5. straight walls of large radius such that boundary layer
thickness of fluid was negligible.

Shapirol

states "well-designed nozzles with straight axes . . . .
operated at design pressure ratios and at high Reynolds numbers, . . .
have efficiencies for 9h-99 per cent, and even higher for sizable wind
tunnel nozzles."

The author believes that the values obtained of nozzle efficiency
including skin fricition and heat transfer effects are reasonable, and
if the effect of boundary thickness on area available for flow is in-
cluded, the combined analysis should constitute a reasonable approach
to predicting nozzle efficiencies of the given type. One author2
measured a linear growth of a laminar layer in supersonic wind tunnel

at an average of 0.7°. Such growth could cause appreciable area change

and affect a change in efficiencies as much as one per cent.

 

1Shapiro, Ascher. The gynamics and Thermodynamics of Compressible
_ Fluid Flow, Vols. I and II. First Ed., New York. The Ronald Press
Company ,7953, p. 99.

2McLellan, Charles, Thomas William, Ivan Beckwith. Investigation
of the Flow Through a Single Stage TWOADimensional Nozzle in the
Langley Eleven Inch Hypersonic Tunnel. NACA TV 2223, 1951.

 

 

 

52.

Extension of these results may not be made directly to a gas
with Pr f l, e.g., air which has a Pr}?! .75. However, a number of
investigators have found that variation of Prandtl number has neg-
ligible effect on skin friction coefficients, and recently, methods
have been used to modify Nusselt number (dimensionless heat transfer
expression) for Prandtl number variation. Such modification in the

given method would necessarily start with Equation (99).

CONCLUSIONS

This investigation dealt With the flow of compressible gas

(Pr = 1) through a converging-diverging nozzle of large radius and

straight walls of constant temperature.

The conclusions drawn on the basis of results obtained were:

1.

The heat transfer effects relative to skin friction
effects were dominant in magnitude in the subsonic
region, and in the case of considerable cooling (TVV /TO -
0.2) this effect extended into the supersonic region.
Skin friction effects were very dominant in supersonic
region, to the point that heat energy losses may be con-
sidered negligible. If cooling at walls occur, however,
the effects on boundary layer development must be con-
sidered in order to establish velocity gradients and
hence skin friction effects.

Effects of skin friction and heat transfer on nozzle
efficiency were small, resulting in a difference from
100 per cent of between 0.2 of one per cent to zero.

The effect of boundary layer thickness on nozzle effic-
iency should be included before expecting the results to
accurately describe the best performance a'wellqdesigned

nozzle might have.

APPEND Di

 

APPENDIX A

Transformation formulae used to change basic differential Equations

(113), (111:), (1:5), and (1:6), from cylindrical to . space coordinates
(x: Y: 79' ).

 

 

 

/ t
f L
Z
_/._._-._ WI.” - - y
20 "’4
Velocity relations: Space Coordinate relations:
qz -ucosz9' +vsin75" xa-xo+(z+zo)cosfl— +rsin2"
qr aIusin'zfi- ~vcosz“ y=(z+zo) sinfl -rcosz"

Transformation Formulae:

3 Si 9x+a &1
9r ax 9r air or

, hence

a - a sinwa - cosifl'

 

 

 

:9
ar ax DY

 

 

.9...

6’2

2......

56.

y , hence

QJQ)

NR
+
Q)
Q)

91:

c9 _ Q a
92 9x c037} + &y sin?“

 

 

‘APPENDIX B

Transformation formulae to change basic differential Equations (1:8),

(149), and (50) into two-dimensional form using Mangler Transformation.

Mangler Transformations based on R g r where R - R(x):

 

x/L Defining:
.. 32 89" &W
x Lf?d(x/L) muffle clan--70”
o
aw'. 1| a¢‘._ Iv!
y' '«E'y a) y: 70 u <9 x: ’0
p'-p h'=h

L - fixed reference length

Resultant Transformation Formulae

 

 

 

 

 

 

2
a ate—w“; a:
9x :5 ax' R dx 6}"
dv'__I3
& .33 9 E;- L
CDIY I. a; y”
911.1111}.
dx R dx
and some examples of formulae used:
$111 _ R2 au'+y_'__g_§ an.
3:: L?- axI R dx ay'
&u .i $u'
«93' L ay'

3')

 

 

 

Relation of u' to u and v' to v:

u'-u

v'u-I: xii-1933,11
R Rdx

 

 

58.

APPENDIX C

Transformation formulae using modified Stewartson expressions to
change two-dimensional Equations (56), (57), and (58) to essentially

incompressible form .

Stream functions used in the two coordinate systems:

In x', y' system:

42.42.. W , ._Q.‘£’.'.__;e_'ir_-'_
93' fo' .9 1' 21'

In x", y" system:

34!» “1.. , aw"_ v
93'” 9;"

Transformation formulae:

(ac-:1. -(—-—-——)..(————) Mtg-5%):

(.59...) . (a...) :21:
9}" .I’X' 9y" X" “0 f0

The transformations were made assuming:
1. A - constant, since 'wall temperature was constant.

2. External potential flow, hence “e and Pa were dependent on

1 dPe _, u Clue
- e

f’e dx

 

 

 

3: only, and Euler's equation ( ) was

applicable to describe conditions at outer limit ( 8 ) of

boundary layer .

60.

3. CI) and Pr as constant.
1;. p =1 pe for any given value of x.
S. p" 3pl am T" gT'.

Relation of u' to u":

fiaw, 290(av’u) AM) .332 «W 3333
a?" 3.3!}:

 

 

(I

APPENDIX D

Transformations and substitutions used to change Equations (65), (66),
and (67) .to a set of ordinary differential equations as a function of

7onlyWith I! = 2 for Pr 3 1:
Equations to be transformed

l . Continuity

 

 

 

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a If" 9y" 9x" 3 x" a y"2

 

 

 

 

 

It a ue a 3 Q’ I:
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2 . S
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62.

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aw") (219.1: a:
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a y" x" o

( 25’1") (:2)... (1.3—) ‘3»:
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APPENDIX E
TABLE V
Final Results from All Data Calculated

Based on 01 - 25°, Y- 1.1;, L - rp1*, 31‘: . 0 at M - 0.001

 

 

 

 

 

 

 

and A . /»0’\ sin 92(1 cos 91)
2% 8min <1 - 92>
T
02 "SA A 02 Mactual Effnet
o T01
15 1.5 0.0h78115 . l.h99981 0.9999827
2.0 1.999965 0.9999806
2.5 2.h9993h 0.9999765
3.0 2.999887 0.9999731
10 1.5 1.899975 0.9999772
2.0 7.999951 0.9999728
2.5 2.199907 0.9999669
3.0 2.999831 0.9999598
5 1.5 1.h99961 0.99996h1
2.0 1.999908 0.9999h89
2.5 2.h9981b 0.9999339
3.0 6 2.999668 0.9999209
15 1.5 0.0h88026 .99999026 1.h99990 0.9999811
2.0 .99998963 1.999978 0.999977h
2.5 .99998895 2.8999505 0.9999713
3.0 .99998828 2.9999137 0.9999676
10 1.5 .99999009 1.h99989 0.9999900
2.0 .9999891h 1.999967 0.9999708
2.5 .99998811 2.899930 0.9999632
3.0 .99998709 2.999871 0.9999563
5 1.5 .99998957 1.899977 0.999968h
2.0 .99998767 1.999933 0.999950h
2.5 .99998559 2.h99858 0.9999351
3.0 .99998355 2.9997h1 0.9999219
15 1.5 0.0310718 .99997770 1.500007h 0.99998h5
2.0 .99997627 1.999993 0.999972h
2.5 .99997h71 2.h99973 0.9999651
3.0 .99997317 2.9999h7 0.9999606

 

TABLE V (cont .)

 

 

 

 

MSA A 02 Mactual Effna
O T01
10 l .5 .99997731 1 500005 0 .9999819
2.0 .99997515 1.999986 0.9999678
2.5 .99997278 2.899957 0.9999575
3.0 .99997085 2.999935 0.9999550
5 1.5 .99997613 1.899999 0.9999750
2.0 .99997179 1.999963 0.9999512
2.5 .99996703 2.h99911 0.9999358
3.0 .99996235 2.999828 0.9999208
15 1.5 0.032670h 1.00000000 1.8999330 0.999938h
2.0 1.999882 0.99993h
2.5 2.899780 0.999922
3.0 2.9996213 0.9999098
10 1.5 1.899916 0.9999228
2.0 1.999838 0.9999078
2.5 2.899683 0.9998873
3.0 2.999h38 0.9998593
5 1.5 1.h99869 0.9998795
2.0 1.999690 0.9998278
2.5 2.h99378 0.9997788
3 .0 2 .9988985 0 .9997377
15 1.5 0.0329311 0.99996753 1.89997h 0.9999h36
2.0 0.999965812 1.999926 0.99992h3
2.5 0.99996316 2.h99880 0.9999063
3.0 0.999960922 2.9997015 0.9998899
10 1.5 0.99996695 1.199961 0.9999310
2.0 0.999963881 1.999888 0.9999017
2.5 0.99996035 2.899766 0.9998771
3.0 0.999956967 2.999570 0.9998586
5 1.5 0.9999652h 1.h9992h 0.999895h
2.0 0.999958909 1.999775 0.9998380
2.5 0.99995198 2.h99527 0.9997838
3.0 0.999915171 2.9991375 0.9997398
15 1.5 0.0335725 0.99992568 1.500025 0.9999887
2.0 0.999920908 1.999982 0.9999109
2.5 0.99991569 2.h99912 0.999888h
3.0 0.99991052 2.9998236 0.9998686
10 1.5 0 399921137 1 .500016 0.9999391
2.0 0.99917152 1.999952 0.9998905
2.5 0.99990927 2.h99866 0.9998616
3.0 0.999901512 2.999787 0.9998508
5 1.5 0.99992085 1.h99993 0.9999180
2.0 0.999905966 1.99987h 0.9998350
2.5 0.99989010 2.199695 0.999781?
3.0 0.99987h523 2.999hlh 0.9997350

 

TABLE V (cont .)

T02

 

 

 

83A 2: T h“actual Effna
o 01
15 1.5 0.0378811 1.899796 0.9998128
2.0 1.999688 0.9998088
2.5 2.899375 0.9997777
3.0 2.998868 0.9997298
10 1.5 1.899785 0.9997655
2.0 1.999502 0.9997233
2.5 2.899037 0.9996575
3.0 2.998316 0.9995989
1.5 1.899708 0.9997315
2.0 1.999076 0.9998866
2.5 2.898139 0.9993380
3.0 2.996688 0.9992098
15 1.5 0.0388026 .99990259 1.899893 0.9998082
2.0 .99989632 1.999783 0.9997758
2.5 .99988989 2.899567 0.9997355
3.0 .99988277 2.999185 0.9996792
10 1.5 .99990087 1.899883 0.9997933
2.0 .99989181 1.999663 0.9997082
2.5 .99988107 2.899290 0.9996286
3.0 .99987091 2.998710 0.9995637
5 1.5 .99989573 1.89977 0.9996883
2.0 .99987678 1.999328 0.9995038
2.5 .99985596 2.898578 0.9993888
3.0 .99983555 2.997815 0.9992198
15 1.5 0.0210717 .99977705 1.500078 0.9998851
2.0 .99976278 1.999938 0.9997261
2.5 .99978709 2.899785 0.9996568
3.0 .99973171 2.999871 0.9996058
10 1.5 .99977311 1.500089 0.9998182
2.0 .99975188 1.999856 0.9996715
2.5 .99972781 2.899580 0.9995785
3.0 .99970857 2.999361 0.99955285
5 1 .5 .999 76137 1 .8999 76 0 .9997393
2.0 .99971790 1.999625 0.9995096
2 .5 .9996 7035 2 .899079 0 .9993829
3.0 .99962362 2.998282 0.999205
15 1.5 0.0216615 1.00000000 1.899570 0.9996086
2.0 1.999221 0.9995838
2.5 2.898612 0.9995063
3.0 2.997605 0.9998298
10 1.5 1.899870 0.9995080
2.0 1.998885 0.9993808
2 .5 2 .897965 0 .9992761
3.0 2.996850 0.9991580

 

TABLE V (cont .)

 

 

 

 

02 702
MSA A T Mactual Ififfna
0 01
15 1.5 0.0216615 1.00000000 1.899570 0.9996086
2.0 1.999251 0.9995838
2.5 2.898612 0.9995063
3.0 2.997605 0.9998298
10 1.5” 1.899870 0.9995080
2.0 1.998885 0.9993808
2.5 2.897965 0.9992761
3.0 2.996850 0.9991580
5 1.5 1.89932 0.9993787
2.0 1.998053 0.9989179
2 .5 2 .896068 0 .9986006
3.0 2.993009 0.9983328
15 1.5 0.0218557 0.00079868 1.899832 0.9996802
2.0 0.99978186 1.999529 0.9995198
2.5 0.00076708 2.899005 0.9998133
3.0 0.99975287 2.998199 0.9993280
10 1.5 0.99979101 1.89975 0.9995612
2.0 0.99977108 1.999290 0.9993767
2.5 0.99978929 2.898502 0.9992166
3.0 0.99972787 2.997281 0.9990802
5 1.5 0.99978019 1.899528 0.9993826
2.0 0.99978015 1.998587 0.9989551
2.5 0.99969635 2.896998 0.9985858
3.0 0.99965330 2.998586 0.9983533
15 1.5 0.0222598 0.99953008 1.500059 0.9995883
2.0 0.99989987 1.999859 0.9998216
2.5 0.99986690 2.899868 0.9992778
3.0 0.9998388 2.998886 0.9991692
10 1.5 0.99952173 1.5000181 0.9995388
2.0 0.99987618 1.999696 0.9993073
2.5 0.99982627 -2.899112 0.9991107
3.0 0.99937728 2.998653 0.9990567
5 1.5 0.99989698 1.899988 0.9998892
2.0 0.99980537 1.999208 0.9989656
2.5 0.99930516 2.898060 0.9986155
3 .0 0 .99920668 2 .996309 0 .99 83277

 

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The Ronald Press Company: 19-5737.

 

 

 

E 088%.

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1293 03071 4749

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