ARPA ORDER NO :88 1
^
Oi
?0
IMORANDUM
M-5089-ARPA
fiJtJLY 1967
EXACT SIMILAR SOLUTIONS OF
THE LAMINAR BOUNDARY-LAYER EQUATIONS
C. Forbes Dewey, Jr., and Joseph F. Gross
D D C
MAR8
1968 _
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PREPAP ED FOR:
ADVANCED RESEARCH PROJECTS AGENCY
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ARPA ORDER NO. 189-1
MEMORANDUM
RM-5089-ARPA
JULY 1967
EXACT SIMILAR SOLUTIONS OF
THE LAMINAR BOUNDARY-LAYER EQUATIONS
C. Forbes Dewey, Jr., and Joseph F. Gross
This resoarci) is supported by ihc AHvanreH Rrscarch Projects Agency under Conlrart
No. DAHClTi 67 C 0141. Any views or conclusions contained in tliis Meinorandiun
should not be interpreted as representing the official opinion or policy of ARPA.
DISTRIBUTION STATEMENT
Distriluition oi this document is unlimited.
mo,
•7%€ rS"H I I Ly^it^MÄ»*
(700 HAtN it
• J»NtA MOMK* • CAlltOtNl*-
•lit-
PREFACE
The purpose of this Memorandum is to present a comprehensive and
accurate table of values for the similar solutions of the laminar
boundary-layer equations for a homogeneous gas.
These values may be
used to determine skin friction, heat transfer, and appropriate bcundary-layer thickness parameters for a wide variety of physical situations.
They are applicable to such problems as stagnation-point flov.s,
flow over a wedge or a flat plate, and hypersonic viscous interaction,
and may also be used as good approximation for the determination of
nonsimilar flows.
Well over 1800 solutions are tabulated, of which
about five-sixths have not been published before.
of RAND's work for ARPA on reentry aerodynamics.
The study is part
C. Forbes Dewey, Jr.
is an Assistant Professor in fhe Department of Aerospace Engineering
Sciences at the University of Colorado and a consultant to The RAND
Corpjration.
SUMMARY
The purpose of this Memorandum Is to present a comprehensive and
accurate table of values for the similar solutions of the laminar
boundary-layer equations for a homogeneous gas.
These values may be
used to determine skin-friction, heat-transfer, and appropriate boundary-layer-thickness parameters for a wide variety of physical situations.
They are applicable to such problems as stagnation-point flows,
flow over a wedge or a flat plate, and hypersonic viscous interaction,
and they may also be used as good approximations for the determination
of nonsimilar flows.
The general boundary-layer equations for a thermally and calorically perfect binary gas mixture are derived together with a discussion of the numerical integration procedure used to obtain solutions
to the equations, the concept of local similarity, and solutions for
large values of the pressure-gradient parameter.
The tabulated solutions include effects of pressure gradient
2
(0 ^ ß ^ 5 and ß = »), Mach number (0 s u /I'H
< 1) temperature-
viscosity law (0.5 ^ a- < 1 or 0 < s ^ 0.2), leading-edge sweep
vü.l
5 t
9
^1) wall temperature (0 ^ t
W
^2), Prandtl number
2 2
(0.5 ^ Pr ^ 1), local streamwise velocity (0 ^ u e /u öD ^ 1), and
mas;-! transfer at the surface (-1.6 ^ f
w
^0).
-vli-
ACKNOWLEDGMENT
The authors wish to express their gratitude to Susan Fredlund
!
i
j
and Jeannine McGann-Lamar for their expert assistance in performing
the numerical calculations of this Memorandurru
The original numeri-
cal procedures were developed by Kathrine Purdom, now with Lockheed
Missiles and Space Company, Sunnyvale, Calilornia and James I,
Carlstadt of The RAND Corporation.
Special thanks go to Mrs. Alice
Jefts for her precise work in the compilation of the tables.
We have been privileged to receive a review of this manuscript
by Nelson H. Kemp of the AVCO/Everett Research Laboratory, Everett,
Massachusetts.
■IK-
COLTENTS
PREFACE
iii
SUMMARY
v
ACKNOWLEDGMENT
vii
LIST OF SYMBOLS
xi
Section
I. INTRODUCTION
II.
III.
IV.
V.
VI.
VII.
SIMILARITY EQUATIONS FOR THE LAMINAR BOUNDARY LAYER
General Equations
Conditions for Similarity
Equations for Solution
Special Classes of Reduced Equations
Range of Solutions and Parameters
3
3
9
12
17
22
NUMERICAL RESULTS
24
NUMERICAL INTEGRATION PROCEDURE
39
APPLICATIONS OF THE CONCEPT OF LOCAL SIMILARITY
General Discussion
Asymptotic Expansion of the Boundary-Layer Equations ...
Determination of f^ß,^) by Successive Approximations ..
43
43
45
48
SOLUTIONS FOR LARGE VALUES OF THE PRESSURE-GRADIENT
PARAMETER ß
The Outer Limit Equations
The Inner Limit Equations
56
57
61
DISCUSSION
64
LIST OF TABLES
REFERENCES
I
1
,
78
151
BLANK PAGE
i
-xi-
SYMBOLS
A. ... A,
■
constants defined in Eqs. (105), (115), and (120)
a
=
function defined in Eq. (109); also function defined in
Eq. (154)
B
■
function defined in Eq. (86)
C
"
Chapman-Rubesin constant
C(a)
-
limit of g(a,t) as t -• »
Cf
«
2
local skin-friction coefficient; C- » 2T/P U
f
» »
C.h
■
local heat transfer coefficient; q/p
U (H
- Hw )
%? oo or.N aw
c
a
mass concentration of i
c
=
specific heat of the i
D17
■
binary diffusion coefficient
E
=
function as defined in Eq. (102)
6
•
numerical differencf, function defined in Eq. (82)
•e
■
function as defined in Eq. (86)
F.
»
similarity function defined in Eq. (23)
f'
-
transformed fluid velocity as defined in Eq. (12)
f
■
wall constant defined by Eq. (22)
3
=
function defined in Eq. (83)
G
=
dimensionless enthalpy function, (H/H )
g
■
transverse-velocity function, w/w
Ü
■
numerical difference function defined in Eq. (8^)
H
=
total enthalpy of mixture - h + (l/2)(u2 + w2)
component
component
e
-xii-
■
numerical difference function defined in Eq. (85)
h
■
chemical enthalpy of i
>
«
integrals defined by Eqs. (74) to (78)
J (t)
■
function defined as the n
tion
j
«■
geometrical index in boundary layer equations; also
Reynolds analogy ratio 2C /C.
n f
k
=
thermal conductivity of the mixture
Le
«■
Lewis number p D,« c /k
12 p
M
=
Mach number
m
■
constant exponent defined in Eq. (34)
Pr
■
Prandtl number, c
p
■
fluid pressure
q
=
heat flow
R
■
gas constant
r
•
radial coordinate define
S
=
Sutherland constant
Sc
■
Schmidt number, ^/p D^
s
=
S/T
■
transformed velocity function defined in Eq. (150)
W
component
h
l
2
IjO)
1^2)
1^3)
s
integral of the error func-
|i/k
in Fig. 1
o
-xiii-
temperature of the fluid
free-stream stagnation temperature
transformed similarity variable defined in Eq. (150); transformed variable defined by Eq« (108)
aw
dimensionless adiabatic wall temperature, c - t
for Ö^O) - 0
Eckert reference temperature defined in Eq. (53c)
t
dimensionless sweep parameter defined by 'sq« (32)
- 1 - (U200/2fl (£» ) sin2 A
dimensionless wall enthalpy ratio defined by Eq. (31)
T /T
w o
free-stream velocity
u
flow velocity in the x-direction
V
flow velocity in the y-direction
W
constant defined in Eq. (141)
w
flow velocity in the z-direction (transverse velocity)
X
coordinate in direction of flow
y
coordinate normal to the surface
h
dimensionless concentration function defined in Eq. (15)
z
dimensionless function, (1 - Z.); also, coordinate transverse
to the flow direction
9
pressure gradient parameter defined in Lq. (34b)
Y
adiabatic constant > c /c
P v
boundary layer displacement thickness defined by Eq. (80)
e
asymptotic parameter defined in Eq. (98)
similarity variable defined In Eq. (11)
e
boundary-layer momentum thickness defined In Eq. (81)
-xlv-
9
■
dimenaionless enthalpy function, (H - H )/(H
H
■
trial function for solution of first-order equation deflued by Eq. (122)
A
■
sweep angle
- H )
X. - dimensionless density-viscosity product, (1/C)(PM>/P P- )
V. ■ viscosity
§
•
transformed x-coordinate defined in Eq* (10)
p
■
fluid density
hypersonic parameter (U /2H )
modified hypersonic parameter, (U 2 /2H ) • (u /u ) 2
modified hypersonic parameter,
- (I^f2& )C{u /u )2 cos2 A + 8ln2A]
-e
e «
total skin friction
skin friction in spanvise direction
skin friction in x-dlrection
function defined in Eq. (105)
variable defined by Eq. (141)
X(a)
X
o
Kx,y)
♦i ••' ♦a
function Uefined in Eq. (112)
constant defined in Eq. (141)
stream function defined by Eq. (9)
functions defined in Eqs. (117), (118), and (121)
1
exponent in the viscosity-temperature law, p. ~ Tv
constant defined In Eq. (53b)
-XV-
Subscripts
(
)
(
)
(
)
w
■
function at the edge of ehe boundary layer
"
function at the wall
a
function evaluated in the free stream
-1-
I.
INTRODUCTION
Daring the last decade, many similar solutions to the laminar
boundary-layer equations have been obtained.
Most of these results
have been for specific exact conditions, such as stagnation-point
flow or flat-plate flow, and are based on the assumptions of a linear
temperature-viscosity law and a Prandtl number, Pr, of unity.
It is
well recognized, however, (cf. the citations of Refs. 1 to 5) that an
accurate estimate of skin friction, heat transfer, and boundary-layer
thickness in compressible flow usually requires the use of a realistic temperature-viscosity law and retention of the dissipation term
which appears in the energy equation.
The purpose of this Memorandum is to present comprehensive, systematic, and accurate tables of solutions to the laminar boundarylayer similarity equations for a perfect homogeneous gas.
These solu-
tions are applicable to the classical cases wherein the requirements
for similarity are satisfied exactly.
The solutions may be used also
in applying local similarity methods to cases that do not meet the exact requirements for similar flows.
In particular, the sensitivity of
the numerical values of the wall derivatives (which govern heat transfer and skin friction) and the values of the integral thickness parameters to changes in the similarity variables may be accurately determined.
For many physical situations, relaxing one or more of the conditions
required for exact similarity will not lead to large errors when the
local similar solutions are used to estimate boundary-layer properties.
Solutions are included to illustrate the effects of ieading-edge
sweep, mass transfer at the surface, pressure gradient, wall temperature.
free-stream Mach number, local external flow velocity, the Prandtl number, and the viscosity-temperature law of the fluid.
The tables in-
clude most of the numerical solutions obtained by previous authors
(recomputed to the accuracy of the present program) and also new solutions,
"'hese tables do not include solutions with ß < 0 (decelerat-
ing flows) or f
> 0 (suction).
Section II presents the similarity equations in their general
form, including the equation for species concentration.
Transforma-
tions applicable to compressible flows are given, as are equations
for the computation of skin friction, heat transfer, and integral
boundary-layer thicknesses.
Classes of reduced equations are also
discussed; these classes represent cases where one or more of the parameters Pr, u), 0, t , t , Ü/2K and u /u are either zero or unity.
s
w
»' e
e oo
'
Section III gives a concise guide to the solutions of this Memorandum.
Section IV presents the numerical method of computation used
in this program.
The concept of local similarity is discussed in Sec-
tion. V, where criteria are presented for judging the applicability of
similar solutions in situations where the similarity parameters vary.
The analysis is extended to large positive values of the pressure-gradient parameter, ß, in Section VI.
Our experience indicates that values of the skin-friction and heattransfer derivatives and the boundary-layer thickness integrals may be
estimated to high accuracy for intermediate values of the similarity
parameters by careful cross-plotting of the present exact solutions.
The accuracy varies witn the particular quantity and the specific values of the .oimilarity parameters, but it is generally between 0.5 percent and 2.0 percent.
In particular, use of the solutions for ß -• oo
allows estimation of all quantities for all positive ß.
II.
SIMILARITY EQUATIONS FtR THE LAMINAR BOUNDARY LAYER
The purposes of this section are three.
First, we shall display
the laminar-boundary-layer equations for a binary mixture of perfect
gases in their most general form, lis^ng the general conditions that
lead to similar solutions.
Second, the specific equations treated here
are deduced from the general equations, and relations are given for com■/uting heat transfer, skin friction, and integral-thickness parameters.
Finally, special reduced forms of the general equations (Pr • 1, etc.)
are discussed.
GENERAL EQUATIONS
The equations that describe the conservation of mass, momentum,
and energy in a laminir boundary layer are well known.
They can be ex-
pressed as follows:
9
^ (pvrJJ)
— (purJ1) +—
dx
dy
0
(1)
Du
äu
pu —+pv— =
^x
By
op
1 3 / . üu\
-—+ —— LrJ—I
ax
rJ dy \
3y/
(2)
öw
öw
pu — + pv— 3x
3y
19/
äw\
— — Lrj —)
rJ 9y \
9y/
(3)
0
(4)
ap
=
~ =
ay
pu
an
äH
i ö / . n
— + pv — m —J — [rJ
9x
öy
r
dy \
i a
+ ——
OH
\ is
. /
)+ —— rVll
Pr By /
.
1
^ äy
/
1
\
:^- 1--
rJ öy
Le I
Pr \
ac.
ac.
i
a
ax
ay
rJ ay \
i \ —
a /u
+w
I
I-
\
r-
Pr ' dy \
Sc/
Y h. -^
^
1
/ .
(5)
Sy
ac. \
pu
(6)
ay
/
j
=
o
two-dimensional flow
(7a)
j
=
1
axisymmetric flow
(7b)
The coordinates x, y, and r are defined in Fig. 1.
Other symbols are
defined as follows:
c.
=
mass fraction of i
D,«
=
binary-diffusion coefficient
H
-
2
2
total enthalpy of mixture = h + (l/2)(u + w )
h.
=
chemical enthalpy of i
k
=
thermal conductivity of fluid
Le
=
Lewis number, P D,« cp/k
Pr
■
Prandtl number, c
p
=
local fluid pressure
Sc
=
Schmidt number, ^/p D^
u,v
=
flow velocities in the x, y directions, respectively
w
=
flow velocity in direction normal to x, y plane
\i
=
viscosity coefficient of fluid
p
=
local fluid density
i
2
P
constituent
constituent
|J./k
-5-
E
>>
w
3J
iJ
C
•f-l
T3
l-i
O
O
o
u
o
<T3
M
n)
•T3
C
3
O
OQ
I
I
-6-
These equations neglect thermal diffusion and diffusion-themo effects.
If the fluid is a mixture of thermally perfect gases, Eq. (4) integrates to
P(x, y)
-
Pe(x)
=
PeRTe
(8)
A stream function Y(x,y) is introduced that satisfies the continuity
equation:
ÖY/äx
-
- pvrJ
(9a)
at/dy
-
+ purj
(9b)
and a transformation for the independent variables is introduced:
5
• rocw.3rkj
1
ptJ äy
dx
" fat
(10)
(ID
where subscript e denotes any property at the edge of the boundary layer and r, denotes a characteristic radius.
The quantity C, by defi-
nition, is a combination of physical properties that are functions of
x only.
In reducing the partial differential equations in T\ and § to ordinary differential equations in the single variable T|, all derivatives
with respect to ? within the boundary layer are neglected.
The fluid
velocity, the enthalpy, and the concentration are then defined in terns
of the transformed similarity variable T|:
-7-
u/u
=
df(^;/dTl
=
f'Cl)
(12)
w w
e
■
sC7!)
(13)
H/He
-
G(T1)
(14)
=
Z
(15)
/
C /C
i
iw
i^>
(The dependent variable f appearing in Eq. (12) is simply equal to
/5T times the stream function i.)
Then the boundary-layer equations,
(2), (3), (5), and (6) for a two-component boundary layer may be
written as a set of coupled ordinary nonlinear differential equations.
2§ du
ZS
[X~f"|
+ff"
u
/ _
p \
£ (f/2 ._£
d§ \
p /
(16)
r2-1
(17)
.2j
"77 — 1 + fG ' .2j Pr
2
Sfc-'ll^h'f) 'lw
1
s
2
iit- )^-^.'
r2J 1
^-zl1+fz;
=
0
A + gg ' sin
(18)
(19)
2
A
-8-
where
u
A
™
leading-edge sweep angle
X
-
(l/C)((Wp n )
e e
-
free-stream flow velocity in the x-direction
The above equations are applicable to the two-component boundary
layer of an axisymmetric body or a swept two-dimensional surface.
The
boundary conditions for these equations are given at the surface of the
body by the no-slip condition and the requirement of similarity for
the wall concentration and enthalpy:
f '(0)
=
0
(20a)
g(0)
=
0
(20b)
G(0)
=
G
w
(20c)
Z^O)
-
1
(20d)
At the outer edge of the boundary layer, the velocity and enthalpy must
match the values in the free stream.
The concentration of any material
injected at the surface is assumed to be zero in the external flow.
f'O»)
-
1
(21a)
g(»)
-
1
(21b)
G(»)
»
i
(21c)
Z^»)
=
0
(21d)
The final boundary coidition is obtained by considering the 2-parcicles
at the wall.
This yields the well known Eckert-Schneider rendition
represented by the equation:
r v2
w
x
Iw
k
From the continuity equation, the injection parameter is related to
the similarity coordinates by the equation
w
(pv) w /5T
rj
CPe^eUe
rkJ
all ß
(22b)
w
r-
1
du
u
dx
?)-e
ß
T
e
e
■1/2
e
CP k u r,2J
e e e k
^
o
CONDITIONS FOR SIMIIARITY
In order that Eqs. (16) through (19) be similar, severe restrictions must be placed on some of the terms.
2§ du
ue d|
^
These are:
p
N
£(f/29 ._^
.
V
F(T1)
(23)
p ^
x = m)
1
2, 2
/r
k
either Pr
(24)
(25)
F2(T1)
=
1
(26)
2
or u /u
2
■
const, and Pr
=
Pr(1)
•10Le
=
Le(T1)
(27)
2 2
G , c, , and r /r. are constants
w
Iw
w k
f
=
f(0)
=
(28)
N
const.
(?9)
Equations (23) and (25) represent the most severe restrictions on the
system because the external velocity distribution and the body surface
must satisfy the requirements shown.
It may be verified that Eq. (23)
has tb-. following form:
2§ du
F]01)
T
ts
(
,
1
r
(1 tw)6 (1 t8)8 on)
+
" 7^'t r 7 ^ " ' "
M
(30)
where
9
T
-
(H - H V(H - H )
we
w
=
free-stream stagnation '
erature
and
t
w
=
H /H
w e
-
T /T
w o
(31)
(adiabatic and calorically perfect 1 iviscid flow) and
Y - 1 2
9
1 + J~-i M* cosz A
2
<*>
2
00
(32)
$■""
It should be noted that either j » 0 and O^t
t
s
=1.
If t
requires Chat
s
^1, orj=l and
and t are considered to be free parameters, Eq. (30)
w
-11-
21 du
T
-St
8
u d§ T
e
e
£
-
i AT\)
(33)
1
Tf we assume Che relationship between u
and £ to be given h/
m
u e ~ /Til
§'
e o
(34a)
then
2§ due /To\
to
s
»
2m s
ß
(34b)
where ß is the "edified Falkner-Skan parameter.
Using Eq. (34) to define the parameter ß, the equations assume
the following similarity form:
r2'
X^Tf"!
+ff"
=
f^-ir (i
ß
- tw)e - (i - t3)g^ + tw
(35)
t
r
k
r2J
(36)
V
cos
r?,r
K
Iw
H
J
r2J
' I1
L-
2
A + gg' sin
GO
\
(37)
9
A
-12-
r2J 1
-jj — Z^l + IZ^
" r2^ Sc
-
0
(38)
7
In Eq. (37,', a =» (U /2H ) and ii given the designation of hypersonic
parameter.
Two modified hypersonic parameters can be defined, as
follows
•■ • (M
(39a)
and
a2
The parameter a
■ ft)' "■'"•'■-]
=
|
1|| — ]
cos'A+sin
A|
(39b)
includes the shock-angle effect on the energy equation;
it is a times the ratio of the velocity at the edge of the boundary
layer to the free-stream velocity.
The parameter a. is a generaliza-
tion of a. including the effect of sweep angle.
When a = 0, the ratio
of kinetic energy in the flow to the stagnation enthalpy of the free
stream is zero; as a result the dissipation term in the energy equation
is neglected.
As a increases, the dissipation term plays an ever-in-
creasing role in the energy equation.
EQUATIONS FOR SOLUTION
Subject to simplification regarding the flows to be considered,
Eqs. (35) to (38) are those equations for which solutions are presented
in this Chapter.
These simplifications are:
■13-
I.
The Chapman-Rubesin constant is defined as
C
»
p M- /p p.
w w e e
so that the quantities s and X become
5 - r rkj pwv e
dx
w w
2.
Curvature effects are neglected:
(r/rk)2
3.
-
1
The product (Le - 1) • (h„ "hi) is assumed to be zero, and
thermal diffusion is neglected.
A heuristic justification of this
simplification for equilibrium air is »iven by Beckwith and Cohen,
Appendix A of Ref. 6.
The Lewis number has been taken to be unity; i.e., Pr = Sc.
As
a final step, the concentration variable Z.. is transformed to the dependent variable
c
Z
»
1 - Z,
1
=
- c.
1
Iw
c
iIw
Substitution of 9 for G and z for Z1 yields the equations that
were solved exactly to prepare the tables of solutions:
(Xf") ' + ff"
=
ßjf/2 - — [(1 - tw)e - (1 s
2
ts)g
+ tji
(40)
-14-
(XgV • fg' - 0
(41)
(x-\ +fe'-——(-- i\ f'f/M
\ Pr/
((1 - tw)\Pr
(-0
This is a 9
\*J
/[
cos
2
A + gg' sin
2
A
+ iz'
(42)
(43)
-order nonlinear set of ordinary differential equations.
Nine boundary conditions are required, five at the wall and four at
the edge of the boundary layer.
The boundary conditions to be satis-
fied are:
At the wall when T) ■ 0:
f'(0)
=
g(0)
=
9(0)
-
2(0)
0
(44)
and the Eckert-Schneider condition which relates the conservation of
2-particles at the wall:
f(0)
=
'lw
S
1
«
-
z'(0)
(45)
C
lw>
In obtaining Eq. (45) from Eq. (22a), we have made use of the fact
that (X)
■ 1.
At the outer edge of the boundary layer, the condi-
tions are:
f'Ol)
- gOU
- eon
- zOU
- i
e
e
e
e
The value of Tj
(46)
is chosen to be large enough to insure that the bound-
ary conditions given by iiq. (46) are satisfied asymptotically to a high
degree of accuracy (see Section IV).
-15-
The similarity conditions are satisfied by:
or (u /u )
Sc
2
j3
■
constant
X
=
P^/P„M.„
w w
-
X(T1)
either Pr
=
1
=
-
(47)
constant and Pr
=
(48)
Pr(Tl) for Le
and c.
are constants
t
f(0)
=
constant
=
I
Pr(Tl)
Sc(TO [ =
=
(49)
1]
(50)
(51)
f
(52)
The viscosity-temperature relationship may be characterized by
the power-law expression:
|J. = AT .
This yields \ ~ T
.A value
u) ■ 0.7 corresponds to conventional wind-tunnel conditions, '.'hile
u) = 0.5 represents conditions encountered in hypersonic flight.
In terms of the similarity variables, X may be written in the
form
(D-l
X
=
— {(1 - tw)e - (1 - t8)g2 - a1 (cos2 A)f/2 + tj
t
w
If u) - 1, then X = 1 and pp, - constant.
In the literature, this lat-
ter assumption has often been made in order to simplify the boundarylayer equations.
The authors have demonstrated in a previous paper
(4)
that bound-
ary-layer characteristics calculated by means of a Sutherland viscosity
-16-
law can be approximated almost exactly using the power-law relation
p. ~ T
, provided the empirical exponent uo
is suitably chosen.
Suther-
land's law may be written
s + 1
-
t3/2
=
^o
where t = t/T
o
(53a)
s + t
and s = S/T .
o
The Sutherland constant, S, is a charac-
teristic temperature for the gas, and ^
the stagnation temperature T •
a-
r
is the viscosity evaluated at
The empirical equations for calculating
are
t
u)
r
t
The quantity t
=
r
0.5 (t
2
e
-«■
+ s
E
in t r
+ t ) + 0.22 (1 - t )
w
w
(53b)
(53c)
is sometimes referred to as the Eckert reference tem-
perature.
Solution of Eqs. (40) to (43) subject to the hour.'ary conditions
of Eqs. (44) to (46) requires the specification of eight independent parameters.
£
w
These are
f(0)
=
blowing parameter
Pr
=
Prandtl number, taken to be constant = Sc
t
=
sweep parameter
t
=
normalized wall temperature
=
local streoTwise velocity
ratio
•*
(u
/u 00')
* g
2
■17ß
2
a - U /2H
no
e
j' or s
=
pressure-gradient parameter
=
hjrpersonic parameter
=
temperature-viscosity law
All eight must be independent of the streatnwise coordinate § for sirai2
larity to hold exactly. If Pr = 1, the parameters a and (u /u ) are
e
^
not required.
It should be emphasized that thermal diffusion has been neglected
and the product (Le - l)(h
- h ) has been assumed zero.
In this ap-
proximation, solution of the three equations for f, 6, and g are independent of the diffusion equation and, consequently, are independent
of the value of the Schmidt number.
SPECIAL GLASSES OF REDUCED EQUATIONS
Two-Dimensional Flow
If r
2i
2J
/r. J = 1, the ordinary two-dimensional boundary-layer equa-
tions result:
(Xf") ' + ff"
=
ß
k-M
(i - t )e - (i - t )g' + t
w
s
>
WJI
ag')' + fg' » o
/
6' '
(
v
2aX
N
/I
v f
W
'Iw
(54)
(55)
/u2 \
Icos
A + gg' sin
A
"
1
TSZ- ).
^■h^
(56)
■18-
(-3
+ fZj'
-
0
(57)
Equations (54) to (57) are valid for axisynrmecric flow if A is set
equal to zero and j is unity.
When the free-stream-flow direction coincides with the x-direction,
then A ■ 0 and t
= 1.
For this case, the boundary-layer equations re-
duce to:
/
(XfV + ff"
-
V
(/I
e
ß[f'2 - (1 - tw)e + tj
v
/u2 \
2CJ\
(58)
)'
'
+ llü _/_ . A^ i . hJZ.'
z
I
H
L
ISc \Le
e
/
(59)
J
Z' ■'
fä)^
(60)
No Mass Transfer
In the special case where no mass transfer takes place in the
boundary layer, Z. = 0 and c.
=0, and the boundary conditions (in-
eluding the Eckert-Schneider conditicn^ are ignored:
Although the solution for the concentration ratio Z, is Z. • 0,
the solution of Eq. (43) for the normalized concentration z has a nonvanishing solution and a nonzero gradient z' at the wall.
•19-
(xf'V'+ ff" - ß[f'2 - (i - tw )e - tw ]
^r
/l
\
\pr
/ (1 - t )
(61)
/u2\
2aX
+ fe'
(62)
V/
Equations (61) and (62) are the ordinary two-dimensional Brandt 1 boundary-layer equation.
tx ~ T
If the viscosity is directly proportional to the temperature,
>
£
1, and Eqs. (61) and (62) reduce to:
f" + ff = ß{f/2 - (i - tw)e - tj
/e7 /
_
^Pr /
Pr
+fe'
=
L
ll
v 2af/f/ /u^ v
--i
-|
\pr
/ (1
- t ') \u or, '
^
(63)
(64)
w
1
=
Equations (63) and (64) reduce to a form in which the dissipation
term of the energy equation is zero:
f* + ff"
2
= ßlf'
- (1 - tw )e - tw 1J
1
e" + fe' « o
(65)
(66)
g ■ 0
Finally, if the pressure-gradient parameter is zero, the momentum
equation is uncoupled from the energy equation; Eq. (65) becomes
-20-
t"+
ff
=
0
(67)
and by inspection, the solution of the energy equatio- is
6
=
f'
(68)
Computation of Boundary-Layer Properties
Heat-transfer rat>js and skin friction are related by the similarity transformations to the derivatives 9'(0), f'^O), and £'(0).
The
heat transfer from the stream to the wall is given by
% = +kwOT/9y)w
(69a)
After the proper substitutions have been vide, the expression for the
heat transfer in similarity coordinates is
%
k p u H (1 - t )r1J
w k „e(0)
t,
+ w w e e
P
w
s
c /gr
(69b)
and the local heat transfer coefficient is
Ch = %/^aPJ\v - Hw)l
(70)
The skin friction for the x-dtrection can be described in similarity
coordinates as follows:
i
2
r,kJ Pw u e
T
f"(0)
=
u
"
r^ p u w
^w^-^g'CO)
w
yir
(71)
The spanwise coefficient is
T
z
(72)
-21-
The total component of skin friction is the vector sum of the two components
and
T
T
=
:
T
a
k
w
2
[{f.(0)]2u2 + {g/(o)} w2-]
we
(73)
M
and the local skin-friction coefficient is defined by
C.
JL
-
2T/P
00
U'00
Several integral relations have also been tabulated.
I1
-
l
"
2
(74)
They are
(1 - t^Cl^l) - 1^2)] - (1 - tJl^S) +1^2)
f'Cl - f') dTl
(75)
(76)
where
I^D
1^2)
=
(1 - g') dTl
(77)
(1 - f/2) dT]
(78)
1^3)« r ^ -e) ^
(79)
These integrals can be related to the boundary-layer-thickness parameters expressed in similarity coordinates.
Displacement thickness:
-22-
6
(80)
o \
p u /
e e'
p u rMl /
e e k
\T / "^
Msmentum thickness:
e e
\
u /
pur.
e e k
RANGE OS SOLUTIONS AMD PARAMETERS
The parameters for the systems of equations under consideration
here are
pressure-gradient parameter
0 ^ ß ^ 5 and ß » «
Prandtl number
Pr = 0.5, 0.7, 1.0
Schmidt number
Sc » Pr (I.e., Le - 1)
temperature-viscosity-law parameter
U)
sweep-angle parameter
0 ^ t
mass-transfer parameter
-0.6 ^ f ^ 0
w
wall temperature
0 ^ t
shock-angle parameter
(u £ /u00 )
hypei4.onic parameter
*-a - ü'"/2H
'■ 0, 0.5, 1.0
CO
g
- 0.5, 0.7, 1.0 or
.01 ^ s ^ .3
s
w
S ?
* 1.2
2
■ 0, 0,5, 1.0
The quantities tabulated are
f"
=
normalized chcirdwise velocity gradient at wall, [f'/(0)']
6'
=
normiiized total enthalpy gradient at wall, [6/(0)]
g'
«
normalized transverse velocity gradient at wall, [g'(0)]
■23-
plus the integrals 1^ 12, 1^1). 1^2), 1^3) as defined by Eqs. (75)
to (79).
-24-
III,
NUMERICAL RESULTS
This section provides a guide to the present solutions of the
laminar boundary-layer equations.
schematically in Keys 1 to 6.
The available solutions are shown
Each key shows all of the values of the
eight independent parameters covered by the correspondingly numbered
table found at the end of this Memorandum.
If the key indicates (by
an x in the parameter matrix) that a solution is available, then the
numerical values are given in the correspondingly numbered table.
The
key also provide; the reader with a graphic display of the solutions
available in the neighborhood :f his range of interest.
Inner solu-
tions for the limit £5 -• oo are given in Table 7 which is too short to
warrant a separate key.
Approximately a sixth of the solutions listed have been reported
by other authors.
All values listed have been computed, or recomputed,
according to the numerical procedures described in Section IV.
Some
differences exist between the present values ai ! those of earlier investigators, but the present values are believed to be accurate within the limits prescribed in the succeeding three sections (generally,
correct to four significant figu.-es).
The most extensive of the earlier tabulations of solutions to the
similarity equations foi a laminar boundary layer of a perfect gas may
tt found in the works of Beckwith,
Nagamatsu,
Dewey.
and others.
Cohen and "eshotko,
Beckwith and Cohen,
Li and
Reshotko and Beckwith,
and
Solutions of high accuracy have been obtained by Smith^
'
Early contributions to the understanding of the role of
fluid properties may be found in the works of Busemann,
(12)
' Kantian and
-25-
Tsien,
Crocco,
Enimons and Lexgh
Young and Janssen,
(18)
report a large number of solutions with the sur-
face mass-transfer parameter f
pendiums
and van Driest.
w
other than zero.
Several recent com-
on boundary-layer theory may be consulted for a more
comi-iete description of previous contributions.
Since this Memorandum is restricted to positive values of the
pressure-gradient parameter ß and values f
^0 corresponding to mass
injection into the boundary layer, a list of sources is offered in
which solutions for ß < 0 and f
> 0 may be found.
Emmons and Leigh
have obtained solutions for 3=0 and values of f /T equal to 10, 6,
,
^, 3, 2.1, 2, 1.5, 1.4, 1.3, ... .5, .45, .40, ... 0.1, and 0.05, assuming that Pr = 1 and the viscosity-temperature law is linear. Suc(^3)
{94)
tion results are reported also by Spalding and Evans,
Pretsch,
Watson,
I
f25^
and Bussman,
Eckert, Donoughe, and Moore,
/OR)
Mangier,
Koh and Hartnett/
(2.9)
Schaefer,
^26^
(20)
Thwaites,
(ID
Schlichting
Emmons and Leigh,
(18)
and
'
Negative values of the pressure-gradient parameter ß have been
considered by Hartree,
Reshotko,
(32)
Beckwith,
Stewartson,
(21 33)
(11 12^
'
Smith,
'
Cohen ?nd
and Hufen and Wuest.
Solutions fcr nega-
tive values of 0 differ from their counterparts for positive ß in two
ways.
First, two sets of "proper" solutions exist for each value of
ß; second, every value of the wall derivative f" and its corresponding
value of 6', etc., will satisfy the boundary conditions f
and f' -» I as 71 -• oo.
>» 0 at Tj » 0
The "proper" solution must, therefore, be defined
as that solution for whlt,.i f
(see Refs. 3, 8, and 21).
approaches unity most rapidly from above
•26-
Key 1
Solutions for
Similai
U) -
Pr - 1
t
f
8
o.o^
0.25
0.2857
0.3
0.4
0.5
w
0.0
-0.1414
-0.2828
-0.4243
-0.5657
-0.7071
-0.7782
-0.8485
-0.8755
0.0
0.0
0.0
0.05
0.1
0.2
s
all
a
0
.4
.6
.8
1.0
X
X
ii
X
X
ii
X
X
ti
X
X
n
X
X
X
X
•JL
X
II
II
II
2.0
X
0.0
0.0
-0.2
-0.4
-0.6
0.0
0.0
-0.2
-0.4
-0.6
0.N>
.2
X
X
0.0
.15
w
.5
X
1.0
1.0
0.1539
0.3333
0.6250
1.0
0.1000
1.0
1.0
1.0
1.0
-0.5
0.75
t
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1.0
1.0
1.0
1.0
1.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0.1000
0.1539
0.3333
0.6250
1.0
0.1539
0.3333
0.6250
1.0
0.1000
0.3333
1.0
X
y
X
X
X
•
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
(a) All solutions for Pr
X
X
X
X
X
X
1 and 9 ■= 0 are linear in t
X
X
X
-27-
Key 1, cont.
t
0
fw
1.0
0.0
t
s
0.1000
0.1539
0.2500
0.3333
0.5000
0.6250
1.0
-0.5
0.1539
0.3333
0.6250
0.8333
1.0
-1.0
0.1539
0.3333
0.6250
1.4
0.0
0.1000
1.5
0.0
0.1539
0.3333
0.6250
1.8
2.0
0.0
0.0
0.1000
0.1000
Ö.1539
0.3333
0.6250
2.4
0.0
0.1539
0.3333
0.6250
2.8
0.0
0.1000
0.1539
0.3333
0.6250
1.0
1.0
1.0
1.0
3.4
4.0
5.0
0.0
0.0
0.0
1.0
1.0
1.0
1.0
0
X
X
X
X
X
X
X
X
X
X
.15
2
.4
X
X
.6
.8
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
y
X
X
X
X
X
w
.5
X
X
X
X
X
X
X
X
X
X
X
X
X
1.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
A
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
2.0
X
X
X
X
X
-28-
Key 2
Similar Solutions for a Power-Lav; Viscosity Relation
Pr - 0.7, f - 0
w
B
0.0
OJ
t
0.5
all
0.7
s
all
0
hV'
y
0.0
0.5
1.0
•«•
•••
0.0
•••
•••
o.s
...
1.0
1.0
0.1
0.2
0.0
0.5
1.0
.6
1.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
.05
•••
■
■
•
•••
•*•
1.0
1.0
1.0
.1
X
X
0.5
1.0
1.0
0.7
0.3333
0.5000
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
*«•
X
0.5
0.3333
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
X
X
X
X
0.3333
0.5000
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
•••
X
X
X
X
X
0.3333
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
••«
X
X
X
0.3333
0.6250
1.0
1.0
1.0
0.0
0.5
1.0
1.0
1.0
•••
•••
•••
X
X
X
X
X
X
0.3333
0.5000
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
•••
X
X
X
X
X
X
X
1.0
1,0
•••
X
X
X
0.7
1.0
0.2857
all
t«
.15 .4
0.5
0.7
1.0
* « ■
■
•
•
X
X
1.1
t
aw
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
v
X
X
X
X
X
X
X
X
X
29-
Key 2, cont.
t
e
ÜÜ
0.4
0.5
t
s
a
0.3333
1.0
0.6250
0.7
0.3333
0.6250
1.0
1.0
0.3333
0.6250
1.0
(?)
.05
.1
0.0
0.5
1.0
.15
w
.4
.6
1.0
1.1
X
X
X
X
X
0.0
0.5
1.0
X
X
0.0
0.5
1.0
•••
•••
•••
X
1.0
0.0
0.5
1.0
X
X
X
X
X
X
0.0
0.5
1.0
X
X
X
X
X
X
1.0
1.0
0.0
0.5
K0
1.0
1.0
0 n
X
X
X
X
X
X
X
X
X
X
X
•••
X
■ « ■
X
X
X
X
X
X
X
1.0
1.0
X
0.7
1.0
0.75
0.3333
1.0
X
X
X
X
X
1.0
1.0
1.0
0.0
0.25
0.5
0.6
0.8
1.0
0.5
••*
X
X
•••
X
X
X
X
0.0
0.5
1.0
•• »
0.0
1.0
0.5
1.0
0.5
0.7
1.0
0.0
0.5
...
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
:;
'A
X
X
X
X
X
• • ■
...
...
X
X
X
X
X
X
X
•••
••*
•«•
X
X
X
1.0
0.5
aw
X
X
0.5
t
X
X
X
X
X
X
-30-
Key 2 , cont,
2
/u \
k)
B
U)
t
s
a
1.0
0.5
0.3333
0.6250
1.0
1.0
1.0
0.0
0.25
0.5
0.6
0.8
1.0
0.0
0.0
1.0
1.0
0.0
0.5
0.9
1.0
0.0
0.0
0.1539
0.3333
0.6250
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.7
1.0
0.3333
0.6250
1.0
1.4
0.7
1.0
0.6
1.5
0.5
1.0
0.7
1.0
0.0
0.25
0.5
0.0
1.8
0.5
1 0
0.6
2.0
0.5
1.0
0.7
1.0
0.0
0.25
0.0
0.5
0.7
1.0
1.0
0.0
0.0
3.0
•
•
•
•
•
•
•
•
•
•
•
•
t
.05
X
X
X
X
X
X
X
X
.1
X
X
.15
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
w
.4
.6
1.0
1.1
t
aw
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
•31-
Key 3
Similar Solutions for Pr ■ 0.7. t
s
« 1. f
w
^ 0
t
B
X
0.0
0.5
0.7
f
w
0.4
0.5
0.7
.6
1.0
t
aw
X
X
X
X
0.0
0.0
0.5
1.0
X
0.0
0.5
1.0
X
0.0
0.5
1.0
X
0.0
0.5
1.0
X
0.0
-0.2
-0.4
-0.6
1.0
1.0
1.0
1.0
X
X
X
X
0.0
-0.2
-0.4
-0.6
1.0
1.0
1.0
1.0
0.0
-0.2
-0.4
-0.6
1.0
1.0
1.0
1.0
0.0
0.0
0.5
1.0
-0.6
0.7
w
.4
1.0
1.0
1.0
1.0
-0.4
0.5
I
.15
0.0
-0.2
-0.4
-0.6
-0.2
0.2
ff
.5
-0.2
-0.4
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0.0
0.5
1.0
X
X
X
X
0.0
0.5
1.0
X
X
X
X
X
X
X
-32-
Key 3, cont.
t
B
(jü
0.4
0.7
0.5
0.75
1.0
0.7
0.7
0.5
0.7
1.0
f
w
-0.6
a
l
.5
0.0
0.5
1.0
.15
w
.6
X
X
X
X
X
X
X
X
t
.4
1.0
aw
X
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-0.2
-0.4
-0.6
0.5
0.5
0.5
0.5
X
X
X
X
0.0
-0.2
-0.4
-0.6
0.0
0.0
0.0
0.0
X
X
X
X
0.0
-0.2
-0.4
-0.6
-0.8
-1.2
-1.4
-1.6
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
X
X
X
X
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.5
-1.6
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
•33-
Key 3, cont,
UJ
2.0
5.0
aw
0.5
0.0
-0.4
0.0
0.0
0.7
-0.2
-0.4
-0.6
0.0
0.0
1.0
0.0
-0.2
-0.6
1.0
1.0
0.5
■■-j-
-34-
Key 4
Similar Solutions for t
B
0.0
U)
0.5
^1
0.0
0.5
1.0
0.7
0.0
0.5
1.0
Pr
0 .05
0.7
0.5
0.7
0.5
0.7
1.0
X
0.0
0.5
1.0
.6
.8
1.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
r -5
X
X
X
X
X
X
X
X
0.5
0.7
1.0
1.0
1.0
0.0
1.0
0.7
0.7
1.0
0.7
0.15
0.7
0.0
0.7
0.2
0.5
0.7
1.0
1.0
1.0
0.0
1.0
0.7
0.7
1.0
0.7
0.0
0.5
1.0
1.0
0.0
1.0
0.7
0.7
0.7
0.7
1.0
0.7
Ü.0
1.0
aw
X
0.7
l.n
0.7
0.10
t
X
X
0.7
1.0
0.5
0.7
0.5
0.7
1.0
2.0
X
X
X
1.1
X
X
1.0
1.0
tw
.5
X
0.0
0.3
.4
X
1.0
0.7
1.0
.2
= 0
X
0.05
0.2857 0.5
.15
w
X
0.7
1.0
1.0
.1
= 1, f
s
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
<
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
4V
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
>.
X
X
X
X
r
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
•35-
Kty 4, cone
t
8
0.4
uu
0.5
0.7
1.0
a
Pr
0.0
0.5
1.0
0.7
0.7
0.7
0.0
0.5
1.0
0.7
0.7
0-7
0.0
0.7
1.0
0.7
0.7
l
0.5
1.0
0.5
0.5
0.0
0.25
0.5
0.6
0.8
1.0
0.7
0.0
0.5
1.0
1.0
0.0
1.0
0.75
1.0
1.0
0.7
0.7
0.7
1.0
0.5
0.0
0.5
0.7
1.0
0.25
0.5
0.6
0.8
1.0
0.7
0.7
0.7
0.7
0.7
.15
.2
.4
.5
.6
.8
1.0
1.1
t
2.0
X
X
X
X
X
X
X
X
X
X
\
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
*v
X
aw
X
X
0.5
0.7
1.0
0.7
0.5
0.7
0.5
0.7
1.0
0.5
0.7
.1
X
0.5
0.7
1.0
0.7
0.7
0.7
0.7
0.7
0.5
J.O
0.5
0.0
0.5
0.7
0 .05
w
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
XXX
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
A
X
X
X
X
X
X
X
X
X
X
X
X
X
1
-36-
Key 4, cont,
t
9
cu
1.0
0.7
1.0
1.4
0
i
0.0
Pr
L
0.5
0.7
1.0
0.5
0.9
1.0
0./
0.7
0.7
0.0
0.5
0.7
1.0
■5
.1
.15
.2
X
X
X
w
.4 .5
>8
1.0
1.1
X
X
X
X
X
X
X
X
X
X
.6
X
X
X
X
X
X
X
X
X
X
A
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
0.0
0.25
0.5
0.7
0.7
0.7
0.7
1.0
0.0
0.0
0.7
1.0
0.5
0.6
0.8
0.7
0.7
0.0
0.5
0.7
0.7
X
X
X
X
X
X
X
X
X
X
X
X
0.5
0.5
0.7
0.5
X
X
X
J.O
0.0
1.0
X
2.4
1.0
0.0
1.0
X
X
A
2.8
1.0
0.0
1.0
X
X
X
3.0
0.5
0.7
0.0
0.0
0.7
0.7
3.4
1.0
0.0
1.0
X
X
X
4.0
1.0
0.0
1.0
X
X
X
5.0
1.0
0.0
1.0
X
X
X
1.5
1.8
2.0
0.5
0.5
0.25
0.7
0.0
aw
X
X
0.7
0.7
0.7
0.7
t
X
0.6
0.8
0.6
0.5
2.0
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
i7-
Key 5(a)
Similar Solutions for a Sutherland Viscosity—Temperature Relation,
Pr . 0.7, t
1.0
aw
o.o
0.01
0.03
0.05
0.0
0.0
0.5
1.0
-0.2
-0.6
0.0
0.0
1.0
1.0
0.1
0.0
0.0
0.0
0.3
0.0
0.0
0.5
0.0
0.5
1.0
-0.2
-0.6
0.0
0.0
0.0
1.0
0.0
CO
0.0
0.0
0.0
0.0
0.0
0.0
0.4
0.05
0.5
0.01
-0.2
-0.6
0.02
0.3
0.0
0.0
-0.2
-0.6
1.0
0.01
0.0
-0.2
0.05
0.1
0.2
0.0
0.0
0.0
-0.5
2.0
0.9
1.0
0.0
0.0
0-0
0.0
0.3
0.0
0.0
0.9
0.01
0.0
0.0
Two sets of answers
represents the Sutherland
the second set represents
temperature relation with
are given for each case. The first set
solution for the indicated value of s, while
the solution using a power-law viscosityx as defined by Eq. (53b)
u)
Solution independent of s,
■38-
Key 6
Outer Solutions for (3 - »
hS0
Pr
uu
.15
.5
1.0
ft)
0.5
0.0
1.0
0.0
0.5
0.7
0.7
0.7
0.3333
0.9
1.0
1.0
0.0
0.5
0.7
0.7
0.7
1.0
0.9
1.0
1.0
0.0
0.5
0.9
1.0
1.0
X
X
X
X
X
X
X
X
X
X
X
X
0.7
0.7
0.7
X
X
X
X
X
X
X
1.0
0.0
0.5
0.7
0.7
0.7
X
X
X
X
X
X
X
X
X
0.9
1.0
1.0
0.0
0.5
0-7
0.7
0.7
X
X
X
X
X
X
X
X
X
0.1000
0.9
1.0
1.0
0.0
0.5
0.7
0.7
0.7
X
X
X
X
X
X
X
X
0.3333
0.9
1.0
1.0
0.0
0.5
0.
0.
0.
X
X
X
X
X
X
X
X
X
1.0
0.9
1.0
1.0
0.0
0.5
0.7
0.7
0.7
X
X
X
X
X
X
X
X
X
-0.2
1.0
0.9
1.0
1.0
0.0
0.5
0.7
0.7
0.7
X
X
X
X
X
X
X
X
X
-0.4
1.0
0.9
1.0
1.0
0.0
0.5
0.7
0.7
0.7
X
X
X
X
X
X
X
X
X
0.1000
0.1538
0.3333
0.6250
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
■0.4
1.0
0.9
1.0
X
X
X
•0.2
0.7
0.1000
0,0
0.0
1.0
X
X
X
X
X
X
X
X
X
X
X
X
-39-
IV.
NUMERICAL INTEGRATION PROCEDURE
The program for the numerical integration of the system nf equations (Eqs. (40) to (43)) was written at RAND in FORTRAN for the IBM
7044.
The system was treated as a two-point boundary-value problem,
and the Runge-Kutta method was employed for the numerical integration.
The sequence of operations is shown schematically in Fig. 2.
Inputs
to the program are values for the eight parameters and initial guesses
for the four wall derivatives f"(0), $'(0), g^O), and z'(0).
lutions are then separated into two categories:
layer solutions and adiabatic wall solutions.
The so-
ordinary boundaryThe latter requires a
subprogram that searches for and obtains solutions in which the adiabatic wall condition is met.
In either category, the Runge-Kutta meth-
od is used to integrate the equation to a given value of Tl
.
On the
basis of experience u. Is value was selected to yield an acceptable
asymptotic solution within the limits of the variable values at Tl
The integration stepsize was varied to insure the proper accuracy, and
the values of the variables at Tl
were held to within 10
ma""
,
When the integration procedure is completed, the final values at
v
'
are compared with the required values and the Newton-Raphson
scheme is used to recalculate the initial condition.
as follows:
Let
The procedure is
-40INPUT DATA:
PARAMETERS
& INITIAL
CONDITION
GUESSES
COMPUTATION
OF TW BASED
ON INPUTS
YES
TAW CASE
-KD
NO
COMPUTE
CONSTANT
EXPRESSIONS
-KD
SET UP
RUNGE-KUTTA
-<3)
INTEGRATION
INTEGRATE
ERRORCORRECTION
ROUTINE
RECOMPUTE
YES
PUNCH CASE
DATA & PRINT
PROFILE
IF THE INITIAL GUESSES
ARE SUCH THAT THE
INTEGRATION CANNOT
BE COMPLETED THE
PROGRAM ALTERS F"(0)
& BEGINS AGAIN
I
Fig. 2 -- Protram flow chart.
>
•41-
9'(o)]
=
f'(T\nax) "
^[f'CO), g'(ü), z'CO), e'(o)]
=
8(Tlmax) "
1
(83)
= a[f"(0). g'(O), z'CO), e'(O)]
-
6(71^) - 1
(84)
M = W[f"(0), g'(O), z'CO). B'CO)]
=
zC]^) - 1
(85)
e
= e.if'\o), g'(0),
3 J>
z'(o),
1
(82)
and
ß
•6
\ax>
B
•
£"(0)
f/(
g'CO)
■
and S
(86)
e
e'(o)
(Vx>
z'CO)
It is required that -e ■ «(e,!,^, H) - 0.
so that -c = -c(S) = B(S) - 1 = AB.
Now d-e = dB = M • dS, where
5f
3f
öf
of"(0)
ög'O)
dz ''(0)
b
^b
M"{0)
b
b
a
^b
Sg'CO)
*b
39'(0)
dz '(0)
5e
3e
de^o)
dz '(0)
M
(87)
öe
b
af"(o)
"b
98 '(0)
b
b
ä2
d
32
3z
af"(o)
ag^o)
39'(o)
^'(o)
b
h
b
b
The integration is performed using the initial guesses for S.
This gives B and < (S).
follows:
The initial guesses are then perturbed as
•42-
£[£■'(0) +Af"(0), g'^O), e'(0). z'COn
(88)
3[f"(0), g'CO) +Ag'(0). 9'(0),
Z'(0)]
(89)
*Cf"(0), g^O), e'(0) +A9/(0), z^O)]
(90)
M[f"(0), g'tO). 9'(0). z^O) +AZ/:0)1
(91)
These integrations give the columns of M.
The new guesses for the ini-
tial condition are corrected by calculating:
AS
=
-M-1 -e
o
(92)
It was established that convergence was very difficult for cases involving high & (ß ^ 2).
In these cases, the 10
and this is indicated in each table as needed.
limit accuracy was relaxed,
■43-
V.
APPLICATIONS OF THE COKCEPT OF LOCAL SIMILARITY
GENERAL DISCUSSION
The numerical results tabulated in this Memorandum are rigorously
applicable only when the numerous similarity requirements listed in
Section II are satisfied.
In considering the diverse applications of
compressible laminar boundary-layer theory, it is a rare occurrence
Indeed when all of these conditions are met.
tion then arises:
A most important ques-
What approach should be used in predicting the be-
havior of the nonsimilar laminar boundary layer?
The copious literature relating to this question offers four basic types of approach.
The first is to abandon the similar solutions
entirely and adopt approximate techniques such as integral and series
solutions containing free parameters.
The most successful of the in-
tegral approaches appears to be that developed by Tani.
(35)
cendental approximation proposed by Hanson and Richardson
The transand the
f 37)
"improved approxitu ^ion" technique of Yang^
also appear very promising.
Related to the integral methods is the powerful "strip meth
/og\
od" proposed by Pallone,
whichi follows
fol
closely the inviscid flow
integral method of Belotserkovskl.
(39)
A second type of nonsimilar calculation employs a strictly numerical approach.
The complete nonsimilar boundary-layer equations are
used and a new set of calculations is performed for each particular
problem.
Examples of this approach may be found in the works of Smith
and Clutter^
and Fl:igge-Lot2 and Baxter.
'
^
Although numeri-
cally satisfying, these calculations are extremely expensive and intractable to generalization.
-44(43) has been raost
The third type of approach, suggested by Lees,
successful in capturing the spirit of the use of similar solutions in
situations where exact similarity does not exist.
He observed that un-
der certain circumstances, notably when there is a highly-cooled body
in hypersonic flow, the local pressure-gradient parameter, ß, had a
negligible effect on the heat transfer to the surface.
Many elabora-
tions of this approach have been proposed to improve Lees' simple result to provide more accurate numerical estimates of heat transfer,
skin friction, and boundary-layer thickness.
summary of one group of these results.
Moore
(22)
gives a lucid
Additional ideas for modifying
local similarity not discussed by Moore may be found in the papers of
Smith,
(44)
Kemp, Rose, and Detra,
(45)
and Beckwith and Cohen.
(6)
This third approach is saddled with one significant difficulty:
it
is necessary to make one or more implicit ad hoc approximations regarding the contribution of the nonsimilar terms in the complete boundarylayer equations.
In each of the papers cited in the previous paragraph,
the question is not "Are similar solutions applicable?" but rather "Which
similar solution should be used?"
A number of methods have been proposed
for choosing the similar solutions most appropriate to the local inviscid
flow conditions, local wall temppraturc, and boundary-layer history.
In
the context of the present discussion, this question necessitates the judicious choice of values for the eight similarity parameters.
These val~
ues are usually determined (e.g., Ref. 6) by satisfying one or more integral conservation equations exactly using assumed profiles obtained from
similar solutions.
Such procedures are closely related to the integral
method of Thwaites
ind R(
and to the more recent use by Lees and
Reeves
of a family of similarity profiles generated by Stewartson.
(33)
-45-
The fourth, and in many ways most satisfying, type of approach to
nonsimilar boundary-layer calculations may be traced to the work of
Meksyn.
(19)
His underlying premise is very powerful:
if the boundary
layer is at all times very nearly described by a similar solution, then
the direct effects of the nonsimilar terms may be calculated by asymptotically expanding the full boundary-layer equations in terms of small
parameters which measure the departure of the solutions from similarity.
In this way, the accuracy of local similarity methods is explicitly determined by using the full nonsimilar equations.
We shall demonstrate
shortly that the linearized equations governing the departure from similarity depend only on the local similarity parameters and, consequently,
need be computed only once.
ASYMPTOTIC EXPANSION OF THE BOUNDARY-LAYER EQUATIONS
For purposes of illustration, we shall limit our consideration to
the "incompressible" momentum equation
f
wi
+ £
%
+
1
K^
=
- V
2§C
V^ "
1
VTIH
(93)
where §, T|, and ß(§) are as defined in Section II and the subscripts T|
and § denote derivatives.
Equation (93) may be obtained from the gen-
eral equations by assuming Pr = uu = t
f
= t
=1.
We also assume that
=0, making the three boundary conditions for Eq. (93)
f(?,0)
f^CS.O)
ys,*)
=
0
(94)
-66-
Merk
(49)
was Che first to expand the complete nonsimilar momentum
equation in terms of a small parameter.
Subsequently Bush
pointed
out that Merk's derivation neglected important terms in the correction
equation.
The remainder of this section will be concerned with Bush's
equations and their approximate solution.
In the spirit of Meksyn's approximation, we look for solutions to
Eq. (93) when the right-hand side is small.
The key to an appropriate
expansion is the inversion introduced by Merk.
We change variables
from [$,T|; P(§)] to [ß,T|; §((3)1 so that the streamwj.se momentum equation becomes
f
wi+ "TüI
+ 0C1
- f*] "
e(P)C
¥ffrVTiTi]
f(ß,0)
-
f^ß.O)
-
0
;
f^ß,»)
e(ß)
-
2§P'(§)
-
2§(e)/§'(iJ)
-
(95)
1
(96)
where
(97)
If e is zero, the momentum equation reduces to the Falkner-Skan similarity equation with ß as a single parameter.
For small e we may per-
form an asymptotic expansion of f((ä,T)) of the form
f(M)
-
f0(M) + e^f^M) + ....
(98)
Then the derivatives f- and fQ are
1
P
S ' ^o^
+ e
(P><fi)Tl
+
"•;
(99)
-47-
+ e/(e)f1 + ...
(100)
As Bush pointed out, the term e'(ß) is, in general, of order unity and
may be expressed as
1
d
e'Cß)
ß'd) dl
C2§ß/(S)] = 2[1 + E(ß)]
(101)
where
E(ß) = ^"(D/ß'CS) = -Uß)§"(ß)/[§,(ß)]2
(102)
Substitution of the asymptotic sequence [Eq. (98)] into the momentum equation and boundary conditions produces a hierarchy of equations,
the first two of which are (primes on f o and x,1 denote differentiation
with respect to T|) :
order Unity
2
f0 + f o f"o + ßCl - (f')
x
0 ]J = 0
fo(ß.0)
=
f^ß.Q)
= 0
;
fo'(ß,»)
\I
!
i )
=
(103)
Order e
f + f f
T
o ; - h^i+ A2f;fi =
*<M)
|
(104)
f^ß.O)
»
f^ß.O)
-
,
f1 (ß,»)
=
0
)
-48-
In Eqs. (104) the terms (A^ A , i) are
Lj
-
2 + 2ß + 2E
3 + 2E
(105)
HM - f^ - (fo)ßf;
Note that two independent parameters (|3,E) appear in the first-order
equation.
Equations (103) represent t\2 similar solutions of Falkner and
Skan.
Equations (104) represent the first correction f. to the ve-
locity profile which arises from nonsimilar terms.
For example, the
skin-friction derivative f'^ß.O) is expressed as
f'O.O)
-
f"(ß,0) + ei."(ß,0) + ...
(106)
where f"(0,O) is the local similarity solution corresponding to the local value of ß, and f''(ß,0) is the correction obtained as a solution of
Eqs. (104).
It is apparent that the correction will be of ord?.r e as
long as f''(ß,0) ic of the same order ap f''(ß,0) and e is smal1.
DFTERMINATION OF f^ß.TI) BY SUCCESSIVE APIROXIMAtLONS
We proceed to a consideration of the first-order equations for
f1(ß,Tl), Eqs. (104).
The differential equation is linear with homoge-
neous boundary conditions and may be solved numerically.
The primary
difficulty arises in computing the inhotncgeneous term $(ß,T|) which contains derivatives of f
with respect to both T] and ß.
This is a diffi-
cult term to calculate numerically because a number of similarity
-49-
solutions in the neighborhood of & must be known with high precision.
The exact mimerlcal calculation of f. appears possible but has not boen
attempted.
The technique adopted here is to substitute a transcendental approximation for f (ß,fl) which allows the coefficients of the different
tlal
of known functions.
sent f
to be
".lation and the forcing function Hß.Tl)
expressed in terms
Following an earlier paper by Bushj (52) we repre-
by the relation
f^CiM)
=
(107)
erf(t)
where
t
- all
(108)
1/2
a(ß)
1 + ß
(109)
(■•:)
This relation is found to be in excellent agreement with exact solutions
for f and appears quite adequate for our present purposes.
It is con-
venient to transform coordinates from (ß,T1) to (a,t) and define a new
dependent variable g(a,t) accorc'ing to the relation^
a
= a 'CB)(-- + -)
9t
00
fj^O.TI)
\ a at
-
(UO)
aa '
x(a)g(a,t)
(HI)
0,23084
(112)
x(a;
VnTT \Me'
dß
•50-
Substitution of these transfonnations into Eq. (104) results in a
linear differential equation for g(a,t) which contains derivatives of
g with respect to t up to third order.
The boundary conditions for
this equation are (primes denote differentlition with respect to t):
g(a,0)
=
g'U.O)
=
g'U,»)
=
0
(113)
The differential equation for g(a,t) is now integrated formally
three times with respect to t, using the boundary conditions given by
Eq. (US).
g(a,t)
-
The resulting integral equation is
— g'(a,0) - j J1(t)S(a,t) dt + ^ J J J0(t)g(a,t) dt
2
o
^o
-A4 J J J J_1(t)g(a,t) dt + ^(t)
(114)
The coefficients A (a) and A.(a) are given by
3
*>
A3(a)
-
4 + 2E + 20
and the terms J (t) are the n
n
;
:L0
1.2...
(X)
-I
Jl&lM
6 + 4E + 2ß
(115)
2
e X
" vw '
1 + n
The term ^.(t) is defined by
»
integrals of the error function:
2
J
A (a)
J (x)
o
-
erf(x)
xj (x) + - J .(x) n
n-1
Ä
n.'V^"
(116)
-51-
1
1
V^
^(t)
1
- --J,(^t)
- - J,(t) - -1
J0(t)J (t) +- ^.(t)
l
(117)
S(t:)
*-
» — Jo(l5t) - - J (t) 1 + —J (c)
V2
2
I
2 "
J
(118)
ifl
2
S
4^
where
A method of successive approximations is now applied to Eq. (114),
substituting trial functions g(a,t) for g(a,t) in the three integrals
and continuing until a trial function g is found that agrees satisfactorily with the function g computed from the integral equation.
This
scheme differ» from Picard's method in tha.: the sequence of trial
functions g used in the integrals are suitably-chosen integrable functions of t rather than the functions g(a>t) obtained in the previous
iteration step.
Picard's method converges absolutely, but in practice
it ujually cannot be continued analytically beyond one or two iterations.
In the present scheme, convergence depends on the choice oil
trial functions g but the integrals may be evaluated in closed form.
Evaluation of the Velocicy Profile
In this analysis based on the simplified momentum equation [Eq.
(93)] we are ir.oet interested in the correction f"(ß,0) to the velocity
gradient at the wall.
It is therefore more convenient to work with
the first derivative of Eq. (114), which is (after some rearrangement)
g^a.t)
-
tg"(a,0) - A5J](t)g(a,t) + A6 J
+ A4 J J Jo(t)g'(a,t) dt + t3(t)
J1(t)g'(a,t) dt
(119)
-■72-
vMere
\5
»
3 + 2E
1
Kit)
3
2 + 2E
1
,^ 20
- " J/^t)
- - J (t) - —
j (t)
2
(120)
l
2
1
8
(121)
0
In following the technique of successive approximations, the terms
g and g' on the right-hand side of Eq. (119) must be replaced by the
trial function g and its derivative g'.
The behavior of g and g' wy
be inferred from the boundary conditions g(a,0) - g/(a,0) ■ g^a,^) - 0
and is sketched in Fig. 3.
9 (a,oo) = corifant
Fig. 3 -- Behavior of g and g',
■53-
The behavior of t»(t) and the two integrals appearing in Eq. (119) may
be inferred from the behavior of g' and the quantities J (t).
Suppose we decompose the trial function g(a,t) into two parts, so
that
g(a,t)
-
C(a)H(t)
(122)
We then define K(t) to be unity as t -• «, making g(a,<») - C(a) j 0.
The boundary conditions g(a,0) - 0 and g'(a,0) - 0 are automatically
satisfied for all suitable trial functions g(a,t); t1-? boundary condition g^a,«) = 0 serves to evaluate both C(a) and g"(a,0).
The terms
appearing on the right-hand side of Eq. (119) have the following asymptotic forms:
lim
t-HOD
lim '
[j^tOKa.t)] = C(a) (t - — )
(123)
nt
[J JxC^g'U.t) dt] = C(a)Y1
^[flV^«^'0 dt] "
C(a)
(124)
^ - Y31
(125)
Ji!
ri',(t)]
- -(v^- Dt -t—
3
2
(126)
8
The terms (Y,, V , Y ) are numerical constants which are determined by
the choice of the trial function K(t).
Substituting Eqs. (123) to (126) into Eq. (119) and applying the
boundary condition g/(a,ao) ■ 0, we obtain the following formulas for
C(a) and g"(0):
-54-
C(a)
+ A,
8
g"(a,0)
.Vn
(*•'.)
V3
= C(a)C(l + A5) - A^] - - (V? - 1)
(127)
(128)
We have chosen several functions H(t) »nd used Eqs. (127) and
(128) to determine C(a) and g"(a,0).
The Integrals appearing in Eq.
(119) have been evaluated in closed form and the profiles g/(a,t) have
been determined for each trial function.
Products and sums of the functions J (t) are useful in generating
successively mere accurate trial functions H(t) because the integrals
are easily evaluated.
Typical examples of H(t) are:
H(t)
--:J-i(t)l
(V2 - 1)
(129)
Mt)
2
and there are othars.
Weighted sums of the functions listed above were also used to effect a close fit to g'(a,t).
The computed profiles g^a.t) differed
in magnitude with the different functions H(t), but both the qualitative behavior and the quantity g"(a,0) were relatively insensitive to
the form of H(t).
-55For j? = E = 0, g'(a,t) has a vnaximum near C ■ 0.84 and decreases
to about 12 percent of its niaximum value at t a 2.0.
g"(a,0) is approximately -0.055.
The derivative
For 3=0, the term a(ß) computed
from 2q. (109) is 1/2, and
B ■» E - 0
f'^O.O)
-
a2x(a)g'/(a,0)
-.025
The Blasius solution to Eqs. (103) for ß - 0 is f"(0,0) ■ 0.46960, and
the final estimate of the correction to the s.cin fr1' ction coelficient
for ß » E - 0 is
f"(0,0)
a:
fQ(0,J)[l - 0.053 e + ... ]
(130)
The physical interpretation of this result agrees qualitatively
with the sign of the correction-
If e > 0, ß is increasing and the
local similarity value, f'^ß^), would neglect "relaxation" effects
and overestimate the shear at the wall.
The correction reduces the
skin-friction value by an amount proportional to e.
The small tragni-
tude («053) of the correction term is somewhat surprising, but this
value is probably accurate to within about ± 20 percent.
Bush esti-
mated the correction ten- to be 0.204 but his result was obtained by
asymptotically expanding an approximate solution rather than obtaining
an apfeoxlmate solution to the exact first-order equation.
-55VI.
SOLUTIONS FOR LARGE VALUES OF THE PRESSURE-GRADIENT PARAMETER B
In using the coucept of local similarity in highly accelerated
Clows, it is useful to have solutious of the laminar boundary-layer
equations for lar^e values of j3.
The mathematical difficultieo en-
countered in the 11 it ß » 1 may be illustrated by rewriting the
streamwise (Domentum equatirr. [Eq. (40)] in the form
- [xf")' + ff] - If2 - — [d - tw)e - (i - t8)g2 + tw]| - o
(i3i)
S
Here X la redefined as the density-viscosity racio at the wall (pp-Zpw u.w ).
in the limit ß -• », Eq. (131) reduces frow third to first order:
lim f/
(
1
I
w
)e - (i - t
+
^ 'vlj
1/2
(132)
The transverse-momentum equation and the energy equation remain of second order.
As originally pointed out by Coles
ated upon by Beckwith and Cohen.,
(53)
and as later elabor-
' 'zhis leads to a singular perturba-
tion problem in which the thickness of the velocity layer is of crder
ß
-1 2
with respect to the total enthalpy layer and transverse ve,.orlty
layers of order unity.
The method of solution is similar to that em-
ployed in deriving a uniformly valid approximation to the Navier-Stokes
^55)
equations in the limit of large Reynolds number [see Kaplun,'
Lagerstrom and Cole,
(56)
and the recent book by Van Dyke
(57>
].
Discussicn of this problem also appears in an ibbrevtated version
(54)
in Lagerstrom's article.
-57THE OUTER LIMIT EQUATIONS
Since the raathematical justification of this singular perturbation
solution (more popularly called ar. imier- and outer-expansion procedure)
has been discussed in detail by C ies^"* ' cur purposes will be served
by a cursory development of the governing equations.
analysis extends the work of Beckwith and Cohen
The present
to include a power-
law temperature-viscosity relation, a constant but nonunit Prandtl number, and the mass transfer at the surface.
Assume that appropriate outer representations of the dependent
variables f, 9, g are of the following forms:
f
=
f
o
1
+—
f, +
1
Vß
1
e =
ö 0 +—
e
8
8
8
'
, + •••
Vß
o
+
^
1
;
\
(133)
+
Substituting these representations into the momentum equations and the
energy equations and dropping all terms of order ß
-1/2
and smaller,
*
the following "o'-.ter" equations are obtained:
*
The outer limit equations are properly obtained by applying ehe
limit ß -» oo to the full equations expressed in outer variables, with
T| held fixed.
This gives identical results to those cited here.
-58-
f
o -
1
<1^
172
IT-^-^O^ -^^«!!
,
(_e'i +fe
+|
^— |i-—|
D0
\Pr V
Ul - t ) \
Pr/
'2-
2
8?
, V)
cos2 A + ""S sin2 A | >
2
j )
(Xg0') ' + fX -
=.
0
(135)
0
(136)
The appropriate boundary conditions are found (a) by requiring
that the outer equations satisfy the exact ou-er boundary conditions,
and {t> by exact matching r I the inner representacions with the outer.
The results may be written
f (0)
o
" fw
;
e (0)
o
" go(0) "
0
;
9 (0B)
o
- go(06)
= 1
(137)
Only one boundary condition on f may be satisfied by the outer
equations, because the outer limit equation for f
order.
reduces to first
The exact boundary condition f/-»la8Tl-»ool8 automatics lly
satisfied by Eq. (134).
The no-slip condition, f' -• 0 as T) ~» 0, must
be satisfied by an inner solution which is valid in a region of extent
ß
l
with respect to the scale of the outer solution.
In this approximation, the density-viscosity ratio X = (p^/p M- )
in the outer layer becomes
-59-
(1 tw)eo (1 ts)8 + tw
(138)
" 17 (T^) l ' " ' ° If'
where
a«
= | —- 111—^1 cos
A + sin
A
As 1) - 0, X -* [(1 - O/t fl because f'(0) - (t /t )1/2 and f'(0) is
£
nonzero in general,.
s
o
WS
The energy and transverse momei cum equations are
coupled through the implicit appearance of 00th 6
a d (136).
o
IfPr«l,6
and g
in Eqs. (135)
"g» and the number of coupled ordinary dif-
ferential equations is reduced from three to two.
Setting both the vis-
cosity-temperature exponent uu and the Prandtl number Pr equal to unity
reproduces the equations of Eeckwith and Cohen.
the IOCAI Mach number is zero), t
s
If a« = 0 (i.e.,
■ 1, and the transverse momentum and
energy equations are again uncoupled.
One interesting case was pointed out by Coles and we extend his result to generalized compressible flow.
a, - (U2/2H )(u /u )2 s
a
Let t
Then X » (1 -
W
1
=t
1
CT,) ""
0
=l,A=0so that
and f' = 1 for all T\.
2
-1
In taking this limit with Pr 4 1, the product of (f' ) and (1 - t )
approaches G7 so that the energy equation becomes
1
9"o + e'01
+ fw ) Pr (1 - a,)
"^! - (1,(1
- Pr)]"1
o
1
i
=
0
(139)
it
It should be noted that the inner and outer expansion procedure
breaks down in the limit t -«0, because the outer solution for f satw
o
Isfies the exact boundary conditions for the complete equations and the
inner solution for f' is simply zero.
-60with the boundary conditions
Tl - 0
,
eo(0)
-
0
;
Ti - oo
,
e (»)
-
i
,
(140)
If we define the new variable x and the constants x
x
■v7-x»
and W by
;
w
(141)
\/17
W
-
U)-l.-l
1
[1 - a^l - Pr)][Pr(l - a^""
]
then Eqs. (139) and (140) are satisfied by the soluti on
6
=
[erf(x/V2) + et-f(x0/V2)]/[l + erf(x /V2)]
;
(142)
For xc < 0, we note the identity
erf(-x)
=
-erf(x)
(143)
Equations (134) to (136) with the boundary conditions of Eq. (137)
represent a two-point boundary-value problem for the complete solution
of the outer limit equations.
The derivatives 6'(0) and g'(0) along
with the integrals I., 1,(1), 1,(2), and 1,(3) evaluated using f , 9 ,
/
and g
x
X.
1
o
in place of f, 9, and g are given in Table 6.
is identically equal to zero.
The integral I
Beckwith and Cohen^ ' calculated several
of these quantities for the case of Pr ■ U) - 1, £
values of t .
s
o
»0, and nonunit
-61-
THE IHNER LIMIT EQUATIONS
Inasmuch as the order of Che energy and transverse momentum equations Is not reduced in taking ehe limit ß -• a*, the outer equations
[Eqs. (134) to (136)] represent complete solutions for the total enthalpy and transverse velocity profiles for large ß.
The terns f
and
(f7 )', which appear in the outer equations, differ from the exact so2
-1/2
lutions f and (f' ) ' only in a region which is ß
smaller in extent
than the region of applicability of the outer equations.
Therefore,
the outer equations asymptotically represent the complete solutions
for 6 <- id g as ß -• », and in this limit, the inner solutions for 9 rnd
g are identically zero.
The no-slip condition £'(0) - 0 is satisfied by the inner limit
equation for i; the inner equations for 6 and g, as noted previously,
reduce to 9 = g » 0.
To examine the inner streamwise-momentum equa-
tion, it is necessary to introduce the new independent variable Tf and
an inner representation for f:
/T^l
;
f
-
f
+-:
f0(Ti) + ...
(144)
Vß
so that
fgCTl)
-
f^(Tl)
-
u
—
u
e
(145)
After rearrangement, the inner equation for f' becomes
o
1
,
t
(Xf") ' + - f f" - (f')
+
-^
>
o'
o o
o'
.
ft
-
0
(146)
■62and dropping terms of order ß
-1/2
and smaller, the result is
2
(Xf!) ' - (f')
+—
o
t
(147)
where
[a. COB
2
A
m
«lu)-l
(148)
The boundary conditions on f' are found from the exact bour.dary conditions at the wall and the matching condition
that the inner and outer
representations of f agree in the limit Vß -• " with if large but fixed.The following boundary conditions for the inner equation result:
0
^(0)
fo'(.) - /TTF
(149)
Since Eqs. (147) and (148) involve only f' and not f , chey represent a well posed, second-order, two-point boundary-value problem.
A
more symmetrical form may be obtained by one final transformation:
t)
•o<
■V? ^
(t /t ) 1/4
ws
(150)
Then the problem may be written
(xip '-•; + !
-
0
(151)
The matching condition applied here Is elaborated by Van Dyke:
Inner representation of "outer representation" - Outer representation
of "Inner representation".
-63with the boundary conditions
8o(0)
-
0
;
8o(»)
-
1
(152)
and the definitions
X -
[1 - as^r1
^3)
a
^ cos' A/t8
(154)
-
Values of sj«)) satisfying Eqs. (151) to (153) are given in Table
For the special case of u = 1,
7 for several values of u,, a, and A.
Coles
pointed out that an analytic solution may be found in the form
so
=
1-3 sech2 (t/l6"+ tanh"1 ^2/3)
(155)
Using either the analytic solution for a; = 1 or the numerical solutions
for U) ^ 1, the surface skin friction derivative f" is then found by reversing the previous transformations
f
3/4
t
^ ; - *{f)
<(o)
(156)
s
In general 8^(0) depends on the three parameters a,,
üU,
and t ; f or
s
u) ■ 1, the value of s^O) is 4/3.
Our thanks go to J. Aroesty of The RAND Corporation for suggesting a simple numerical quadrature technique for solving Eq. (151).
-64-
VII.
DISCUSSION
The large number of the solutions in Tables 1 to 7 (p. 78 ff.)
suggests many possibilities for numerical correlations and comparisons
with approximate results.
Although an exhaustive discussion of these
possibilities goes beyond our present purposes, we present selected
examples in Figs. 4 to 14 (p. 72 ff.) of the use of the present solutions in understanding the influence of the similarity parameters on
heat, mass, and momentum transfer.
The local skin-friction and heat-transfer coefficients c. and C,,
h
t
are defined by Eqs. (70) and (73).
Following Ref. 59, they may be
placed in a more symmetric form by using the Reynolds number
Re e.x
p u x/>
e e
e
(157)
and the dimensionless streamwise coordinate
^wv^"1
=
/ ("^
2i , \ /
2j -1
p u. u r. J dx
p M- u r. Jx
\J Vw e k
/ vw w e k /
(158)
With these definitions, the quantities Ch and Cf become
cos A
V^
e,x
2
/P..K,\1/
/P~u
Vw\1/2 /Ve\
,p a /
e e
\p.u_/
e'(l
- t )
wv
y'
Pr(t
N
aw
_2_cos A /Pw^\l/2/Peu,
te
e,x
XP p
p
e^-
oo on
(159)
- t )
w
cos2 A + (gj)2 sin2 A
1/2
oo
(160)
-65-
In Eqs. (159) and (160), only the terms appearing in the square
brackets depend upon the similarity parameters; the terms preceding
the square brackets represent the external flow conditions and waJ1
temperature.
The Reynolds analogy function
2Ch/Cf
(161)
is simply the ratio of the square brackets appearing in C. and Cf and
consequently depends only upon the similarity parameters.
In Fig. 4, the expected increase in the skin-friction coefficient
with increasing pressure-gradient parameter Ö is shown.
The relative
increase in Cr/(Cf)„ _ is much greater fcr higher wall temperatures.
This occurs because the boundary layer is thicker and the response to
increasing ß is proportionately higher.
Figure 5 shows similar behavior for the heat-tra isfer coefficient.
We have used the wall-gradient
parameter 9/r(l - t )/(t
- t )! in
0
w
w
aw
w
(4)
forming the heat-transfer coefficient. As shown previously,
9
varies greatly with ß for t > 0.6 because t
varies with ß If a j* 0.
^
w
aw
*
The percentage increase here is less than for the skin friction but the
trends with ß ind t
are the same.
Lees
(43)
argued chat the high den-
sity in the boundary layer near the wall for low wall temperatures "insulates" tine wall from pressure-gradient effects and the influence of
It Is particularly important to determine the, proper adiabatic
wall temperature ir cases with mass injection because t
decreased by injection (see Table 3).
t
3W
is greatly
The assumption Pr =• 1 predicts
«1 under all conditions, which is seriously in error for large
aw
'
injection and high local Mach numbers.
•66-
ß on C. is greatly reduced.
This behavior is shown with decreasing
values of t .
w
The next two figures deal with the Reynolds analogy function
j - 2Ch/Cf.
Li and Gr
(60)
showed earlier that large deviations
from unity could occur In a hypersonic boundary layer even for
i
t
w
« 1 and ß - 0.
Figure 6 demonstrates that j decreases with increasing ß, the de-
crease being faster with increasing wall temperature.
This result can
be easily explained by examining the limiting equations for P -» «,
The
heat-transfer coefficient C. approaches an asymptote as ß -» », whereas
h
.1/2
for large ß the skin-friction coefficient increases as ß
.
The ap-
proach to this limitij^ behavior is more rapid with larger wall temperatures.
Similar behavior is shown in Fig. 7 for values of the hyper-
sonic parameter a ■ 0, 0.5, and 1.0Figure 8 shows thp well known boundary-layer property that heat
transfer is decreased by mass injection at the surface.
transfer reduction caused by a given value of f
creasing ß.
The heat-
decreases with in-
The reason will be explained below in the discussion on
limiting solutions for large ß.
Finally, the sweep angle parameter, t , is varied in Fig. 9.
s
When
the sweep angle A is zero, there is skin friction only in the x-directton
(see Eq. (160)).
As the sweep angle increases, the spanwise term,
g,(0)sin A, contributes Increasingly to the skin-friction coefficient.
If the free-stream direction is sufficiently oblique, the :;-component
of C-, f"(0) (u /u ) cos A, no longer exerts an appreciable Influence.
L
e
The quantity
-67-
{l + [g"(0) sin A/f"(0) cos
A]
, -1/2
J
plotted in Fig, 9 Is a relative measure of the contribution of the two
skin-frictlon components, and approaches a limiting value as t -•0.
Whalen
reported a similar result for the displacement thickness for Pr = ou = 1.
It is very difficult to obtain exact numerical solutions of the
laminar-boundary-layer equations for values of 3 grea.er than 2.
The
reason is simply that the singular behavior of f'(T]) near the wall as
ß -• » becomes dominant even for moderate values of ß.
Conversely, the
results obtained for ß = » should be good representations of the behavior of the boundary layer for large but finite values of ß.
One
of the Important results we wish to demonstrate is chat, by combining
the exact numerical results for ß ^ 5 given in Tables 1 to 5 and the
limiting solutions for ß -• » given in Tables 6 and 7, it is possible
to estimate accurately the skln-rrlctlo.i and heat-transfer derivatives
f , g',
and 9' for all positive values of ß.
w 0w
w
The skln-frlctlon results for ß ■ ■» are displayed in Fig. 10.
The Influences of the two parameters m and a on the inner solutions
for s'(0) are seen to be ralatlvely small.
o
For 0.5 £
ID
^ i.o and
0 ä a ^ 1.0, the value of »'(0) may be found by interpolation co
better than 0.25 percent.
*% //
1 /«
The skln-frlctlon parameter £*ß'
v
limit of a'{0) as ß -• <»..
o
(t /t )" '
ws
approaches the
The difficulties that were observed previ-
ously in calculating exact solutions for 3^2 imply that this limit
is approached very rapidly with increasing ß.
This supposition is
borne out by Figs. 11 to 13, wheie the skin friction parameter is
-60-
shown as a function of ß
-1/2
,
The limit parameter (J
-1/2
is suggested
by the ordering procedure used to obtain the inner and outer equations.
Solid lines indicate e*9ct numerical solutions and dashed lines are
extrapolations.
Figure 11 illustrates the approach of the skin-friction parameter
to its asymptotic limit "»'(0) for different wall temperatures.
The
limiting value is approached most rapidly for high vdll temperatures.
This result is to be expected from the behavior of the outac equations.
Large values of t
increase the magnitude of the velocity difference
across the inner layer, wherea« for t
-• 0, the dlstinctlcn between
the inner and outer layer« breaks down aud no proper limit is obtained.
It has also been found empirically that exact solutions are more difficult to obtain for large
t .
0
w
The approach of the skin-friction parameter to its limiting value
with increasing ß is illustrated in Fig. 12 for several values of the
sweep parameter t .
From a numerical point of view, the accuracy of
the present extrapolation procedure increases with increasing sweep«
It is of interest to examine the behavior of the inner and outer
equations with mass inject'on at the wall.
Applications of these re-
sults include mass-transfer cooling of rocket nozzles (Back and
(57)
Witte)v ' and blunt hypervelocity vehicles.
The outer equations de-
termine the heat-transfer derivative 6', and they contain the boundary
condition f (T| -» 0) - f .
o
w
Thus, surface mass transfer (£ < 0) acts
'
* w
to reduce 6' and, consequently, surface-heat transfer even in the limit
f5 -• ».
On the contrary, the skin friction derivative f" is found from
the inner equation for ?'(0), Eq. (147), a
tion is independent of the value of f .
the solution of this »• ua-
In highly accelerated flows,
•69-
therefore, the effects of bloving on skin friction become negligible
in the limit 0 -> m with f
w
fixed.
Figure 13 illustrates the behavior of the skin-friction parameter
as a function of the injection parameter f .
cmoothly with decreasing values of j3
-1/2
The limit is approached
for all f , and accurate esti-
mates of f" may be obtained for all ß by comparing the exact solutions
for 0 < 5 and the inner limit solutions of Fig. 10.
The wall heat transfer is related by the modified Stewartson and
riowarth-Dorodnitsyn transformations (Eqs. (10) a- i (11)) to 9'.
w
mach as 6
anf g
Inas-
represent the complete solutions tor 9 and g as
0 -• <», the values of 9'(0) ard g/(0) represent the asymptotic limits
of 9' and e
as B -• <».
.t should be emphasized that 9' and g' be-
come independent of ß as S -♦ », demonstrating that the heat transfer
predicted by a local similarity analysis ir highly accelerated flows
approache- a limiting value.
Figure 14 shows the typical behavior of the heat-transfer derivtive 9' with increasing values of ß.
w
The approach of 9' to its limitw
ing value for $ — <*>, is seen to be smooth brH (at least for the case
of a, = 0) monotonic.
In an earlier paper,
we demonstrated that
the proper parameter to use in comparing different heat-transfer calculations is 9/[(l - t )/(t
- t )1.
w
w N aw
w
adiabatic wall temperature t
For c
= 0 and t
s
= 1, the
is unity fur all Fr so that the heat
transfer parameter reduces to 9 '.
Although we have nc
prove it analytically, it appears that t
Pr, t , a), and a in the limit ß = oo.
and should be examined further.
1
been able to
= 1.0 for all values of
This is a very S'-*-prising result
•70*
The boundary-layer displacement thickness 6
is defined by the
relation [Eq. (79)1
pu
/2l 11
v
< ■- . o V '-—K
p u /'
rr 'i- \X
. J VT^ /
p u„ r,
e e k
e
e e
where the integrals I1 and I
ß = oo, the integral I
T
o
I
/ 2
are given in the list of symbols.
is identically zero and the integral I
For
[Eq.
(75)]
^
is recorded in Table 6.
f'a - f') dti
o
In the absence of sweep, the velocity profile
is monotonic and 0 < f' < 1, so that the integral I„ is positive and
the displacement thickness is negative.
With sweep (t
< 1), there
is often an overshoot in the velocity profile, so that f' > 1 for some
range of T], and 1. becomes negative.
With sweep, therefore, the dis-
placement thickness may be either positive or negative, depending upon
the particular parameters being considered.
Numerical comparison be-
tween the calculated results for moderate ß and the present results
for 3 ^ o» suggests that 6
monotonically decreases with increasing ß.
In the special case when ß = » and t
the order of ß
-1/2
- t
■ 1, f
■ 1 and 6
is of
times the scale of the boundary displacement layer
thickness for ß => 0,
The numerical results giv-m ».n Table 5 display the variation of
boundary-layer properties for various Prandtl numbers near unity.
These results may be used to estimate recovery factors for Pr > 1 by
-71-
combining the present results with the asymptotic solution of Narasimha
and Vasantha
for Pr » 1.
Although these authors explicitly solved
the problem of a flat plate (ß " 0) with (u - 1 and low Mach numbers,
there is no reason why a similar analysis could not bo. conducted for
B > 0 and general compressible flow.
As Narasimha and Vasantha demon-
strate, interpolations accurate to better than 3 percent in the recovery factor can be made between first-order asymptotic solutions for
Pr » 1 and exact numerical results for Pr of order unity.
In concluding this discussion, it is advisable to point out that
the numerical values listed for the wall derivatives f", 6', and g/
W
W
\rf
allow reconstruction of the complete velocity and total enthalpy profiles by standard numerical integration techniques.
Whereas the bound-
ary-layer equations themselves, including the boundary conditions at
T) ■ 0 and T| -» », represent a two-point boundary-value problem, one may
solve the equations as an ititial-value problem if the wall derivatives t"t Ö » and g/ are known.
W
W
v/
Furthermore, by using the wall derl-
vatives given here, multi-term expansions of the boundary-layer equations may be constructed for small f\.
This property is useful in
cases where additional "nonclassical" behavior occurs near the surface,
for example, in MKD boundary layers where an electrostatic potential
sheath exists for small T|.
U
l^ 2.0
0.5
1.0
1.5
2.0
2.5
3.0
ß
Fig.4—Variation of the skin friction coefficient with pressure gradient
= 0.6
Fig.5-Variation of the heat transfer coefficient with pressure gradient
■73Pr=0.7
fw = 0
ts -1.0
<T, =0
^
tw = 0.15
fw = 0.6
0
u.z
u.4
O-A
nft
1,0
Fig.6—Variation of the Reynolds analogy function with pressure gradient
o}=0
»1=0.5
a
i = 1.0
^-a.--^/ tw =0.15
^•^zi-sr tw =0.60
0.2
0.4
ß
0.6
0.8
1.0
Fig.7—Variation of the Reynolds analogy function with pressure gradient
-74-
0.8 0.6 u u
0.4
/3=1.0
0.2
-0.2
-0.4
-0.6
-0.8
fw
Fig.8—Variation of the heat transfer coefficient with mass transfer
CN
in
O
U
«5
c
+
Fig.9—Variation of skin friction coefficient with sweep angle
-751.25
Limit solutions
/3——«
CO
I
> CN
II
O
J
1.15
I
I
I
I
I
I
L
0.5
1.0
2
o -
0\ cos A
Fig. 10—Inner solut'ons for ß—**,
o
2.0
I
1
1
1
t
1
1
i
i
1
i
r
>"
Pr = <ü = ts = 1
fw = 0
I
tw = 0.4//
^
y^^l^^ ^^
1.5 :
Hi
E
o
^
""^
Sü^^^
8.
^
c
o
Z
^
1.0
. -i
0
1
I - i
Limit
1
I
—~*> : 1.1547
1
1
1
1
0.5
Limit parameter, ß
t
1
I
1.0
-1/2
Fig. 11-Variation of skin friction with wall temperature for large ß
1.5
-762.0
T—i—i
—\
r
1
1—r
1
1
r
I
CO
I
0)
>
I
c
o
I
1.0
0.5
V5
Limit parameter, ß
-1/2
Fig. 12—Variation of skin friction with ts for large ß
1.61—i—i—i—i—r
n
i .
* -
1.4
I
E
I
1.0
Pr =
tw=
t« =
o2=
c
o
T
c
0.8 1-
0.6
Limit/3-—CO: 1.1547
■?--—'
1.2 b
i.
i
-j
<u = 0.7
0.6
1
0
i
i
I
0.5
i
i
i—i—J
1.0
-1/2
Limit parameter, ß
1
i
i
u.
Fig. 13- Variation of skin friction with injection for large ß
1.5
•77-
i—i—i—i—i—i—i—r
J
0.5
Limit parameter, ß
1.0
i
i
L
1.5
-1/2
Fig. 14—Variation of heat trcnsfer with wall temperature for large ß
-78-
11 ST OF TABLES
Table
s
Pr ■ 1
1.
Similar Solutions for tu
68
2.
Similar Solutions for a Power-law Viscosity Relation,
Pr - 0.7, f w -0
80
3.
Similar Solutions for Pr = 0.7, f
97
4.
Similar Solutions for
5.
6.
Similar Solutions for
Relation, Pr - 0.7.
'
Solution of the Outer
7.
Solution of the Inner Limit Equations for ß-»oc
^ 0, t
« 1
' w
s
t - 1, £ - 0
106
3
' w
a Sutherland Viscosity—Temperature
t - 1
126
s
Limit Equations for ß-«»
134
139
i
-790)
ß
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0
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in in
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in in
in in
r^ r^
CNICM
n ri
cim
'nin
ooco
00 00
CO 00
C^ <N
>£> vO
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o o
o o
o^ o
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u
(fl
p
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(^ as
CM JM
l-i
in
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in in
c^
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VI
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60
nj
a
^^
cs
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sO vO
vD vD
ro CI
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in m
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CMCM
McM
f^cn
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vgvO
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-151REFERENCES
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"""
DOCUMENT CONTROL DATA
1 ORIGINATING ACTIVITY
2a. REPORT SECURITY CLASSIFICATION
UNCLASSIFIED
2b. GROUP
THE RAND CORPORATION
3. REPORT TITLE
EXACT SIMILAR SOLUTIONS OF THE LAMINAR BOUNDARY-LAYER EQUATIONS
4. AUTHOR(S) (Laatnam«, first nomt.initiol)
Dewey, C. Forbes, Jr. and Joseph F. Gross
5. REPORT DATE
July 1967
7. CONTRACT OR GRANT No.
DAHC15 67 C 0141
SoTOTALNo. OFPAGES
170
8. ORIGINATOR'S REPORT No.
RM-5089-ARPA
9o. AVAILABILITY/ LIMITATION NOTICES
DDC-1
10. ABSTRACT
Tabl^-.-'Tf llOO values that may be used to
deternino skin friction, heat transfer.
ar. i arprcnriate boundary-layer thickness
parameters for reentry vehicles.
Assuminr -} perfect bjmoReneous pas atmosphere.
the values were computed to an accuracy
of r')ur significant figures in terms of
eight parameters:
positive pressure gradient. Mach number, temperature-viscosity
law, leading-edge sweep, wall temperature.
Prandtl number, local streamwise velocity.
and mass transfer at the surface (positive
only).
The accuracy requirements were relaxed for very h i ^h accelerations.
More
than five-sixths of the values are presented for the first time; the rest were
recor.putfd to the accuracy of the present
program.
A key to the tables displays
graphically the solutions available.
numerical procedures are given in detail.
For application to cases in which not all
of the eight parameters are similar, the
direct effects of the nonsimilar terms may
be calculated by asymptotically expanding
the full boundary-laye
equations in terms
of the departures from similarity.
6b. No. OF REFS.
£ ,
61
9b. SPONSORING AGENCY
Advanced Research Projects Agency
II. KEY WORDS
Aerodynami cs
Hypersonic flow
^luid dynamics
Reentry vehicles
Heat transfer
Mathematical physics
Numerical methods and
processes