Heat transmission in the boundary layer

NACrrlYiA 21














NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL MEMORANDUM No. 1229


HEAT TRANSMISSION IN THE BOUNDARY LAYER*

By L. E. Kalikhman


Up to the present time for the heat transfer along a curved wall in
a gas flow only such problems have been solved for which the heat trans-
fer between the wall and the incompressible fluid was considered with
physical constants that were independent of the temperature (the hydro-
dynamic theory of heat transfer). On this assumption, valid for gases
only, for the case of small Mach numbers (the ratio of the velocity of
the gas to that of sound) and small temperature drops between the flow
and the wall, the velocity field does not depend on the temperature
field.

In 1941 A. A. Dorodnitsyn (reference 1) solved the problem on the
effect of the compressibility of the gas on the boundary layer in the
absence of heat transfer. In this case the relation between the tempera-
ture field and the velocity field is given by the conditions of the
problem (constancy of the total energy).

In the present paper which deals with the heat transfer between the
gas and the wall for large temperature drops and large velocities use is
made of the above-mentioned method of Dorodnitsyn of the introduction of
a new independent variable, with this difference, however, that the
relation between the temperature field (that is, density) and the velocity
field in the general case considered is not assumed given but is deter-
mined from the solution of the problem. The effect of the compressibility
arising from the heat transfer is thus taken into account (at the same
time as the effect of the compressibility at the large velocities). A
method is given for determining the coefficients of heat transfer and the
friction coefficients required in many technical problems for a curved
wall in a gas flow at large Mach numbers and temperature drops. The
method proposed is applicable both for Prandtl number P = 1 and
for P 1 1.

"Gazodinamicheckaya Teoriya Teploperedachi." Prikladnaya Matematika
i Mekhanika, TomX, 1946, pp. 449-474.






NACA TM No. 1229


I. FUNDAMENTAL RELATIONS FOR THE LAMINAR AND TURBULENT BOUNDARY

LAYER IN A GAS IN THE PRESENCE OF HEAT TRANSFER

1. Statement of the Problem


We consider the flow over an arbitrary contour of the type of a
wing profile in a steady two-dimensional gas flow (fig. i). For
supersonic velocities we take into account the existence of an oblique
density discontinuity (compression shock) starting at the sharp leading
edge or a curvilinear head wave occurring ahead of the profile. For
subsonic velocities we assume there are no shock waves (value of the Mach
number of the approaching flow is less than the critical).

We denote by u, v the components of the velocity along the axes
x, y, where x is the distance along the arc of the profile from the
leading edge, y is the distance along the normal, T is the absolute
temperature, p the pressure, p the density, p the coefficient of
viscosity, X the coefficient of heat transfer, e the coefficient of
turbulence exchange, Mt the coefficient of turbulent heat conductivity,
Cp the specific heat, and J the mechanical equivalent of heat.

U2
T* = T + --
2JCp

is the stagnation temperature



r p

is the velocity of sound


K = P
cv

is the adiabatic coefficient
gcp

is the Prandtl number. The remaining notation is explained in the text.
The values of the magnitudes in the undisturbed flow are denoted by the
subscript m, the values of the magnitudes at the wall by the subscript w.

The problem consists in the solution of the system of equations
(reference 2)






NACA fT No. 1229


(au
Pu-


( V+ ) a]


S(pu) + (p~) = o




-V -


( CT*
are^


a+v
+ v /


p = pKT, p = oTn (1.5)

where R is the gas constant, and C and n are constants.

In the solution of the system equations (1.1) to (1.5) we start
out from considerations on the dynamic and thermal boundary layer of a
finite (but variable) thickness. The flow outside the dynamic boundary
layer approaches the ideal (nonviscous) flow, nonvortical in front of
the shock wave and, in general, vortical behind the wave. Equation (1.4)
shows that the pressure is transmitted across the boundary layer without
change, that is, the pressure po(I) and the velocity on the boundary
of the layer U(x) may be considered as given functions of x. The
flow outside the thermal boundary layer we assume to occur without heat
transfer, that is, outside the thermal layer and on its boundary the
total energy 10 is constant:


u22
u2 U= U2
JcpT + = JcpT., + -2- = JcpT +


From this it follows that the stagnation temperature T*
thermal boundary layer has a constant value

T= TOO = -


outside the


Thus the flow outside the thermal boundary layer for small velocities
Is assumed nearly isothermal while for large subsonic velocities isentropic.


(1.6)


+ v al= -C + -
CV) a1


\ a ST)
TY V FY)


(1.7)






NACA TM No. 1229


For supersonic velocities the entropy is constant up to the shock wave
while after the wave the entropy is constant along each flow line of the
external flow about the profile but variable from one flow line to the
next.

The boundary conditions of the problem are:

u = v = O, T* = Tw for y = 0 (1.8)

where Tw = Tw(x) is a given function. In the absence of external heat
transfer (across the wall) we have instead of the second condition of

equation (1.8) the condition I---J = 0. Further


u = U(x) for y = By(x)
(1.9)
T = TOO for y = Y(x)


where 8y and Ay are the values of y referring, respectively, to the
boundary of the dynamic and the boundary of the thermal layer.


2. Fundamental Expressions for the Temperatures

For P = 1 equation (1.3) gives the "trivial integral" T* = Constant.
Taking into account the second condition of equation (1.9), we obtain


T = To00( 12) = (2.1)


The temperature of the wall is equal to the temperature of the adiabatic
stagnation:



Tk = Too = Tl + -(K 1)2 (M= (2.2)

The integral (2.1) corresponding to the case of the absence of external
and internal heat transfer in the boundary layer, ( = = 0 for P = ,

was obtained on the basis of the solution of A. A. Dorodnitsyn (reference 1).






NACA TM No. 1229


For P = 1, U = Constant, and T = Constant, equation (1.3) is
likewise integrated independent of the solution of the remaining
equations of the system and gives the qo-called Stodola-Orocco integral
T* = au + b (2.3)

Imposing the boundary conditions we obtain

T = TOO [l + ( ) (l- : (2.4)

From equation (2.3) we also obtain

t*
-- (t* = T, to = TO Tw) (2.5)
to

The integral (2.4) corresponds to the case where there is similarity of
the velocity field with the field of the stagnation temperature drop
(equation (2.5)) and was used in the solution of the problem of the
flow about a flat plate (reference 3).

In any more general case U V Constant [for ( Y) 1 0 or

T j Constant or P 9 1 the integral of (1.3) is not known in advance.
The existence of the trivial integral is not, however, the required
condition for the solution of the problem and this fact is fundamental
for what follows.

Let the function T*(x, y) or u(x, y) be integral of the system
(1.1) to (1.5) satisfied by the boundary conditions (equations (1.8)
and (1.9)). The temperature at an arbitrary point can then be represented
in the form
T1
T = T OO u0

for (2.6)

T = O l-2 + (T, -1) 1-






NACA TM No. 1229


As is easily seen, the integrals (2.1) and (2.4) are particular cases of
t*
t U
the second form of the relation (2.6) for q = 1 and =-- U
to
respectively. This relation permits expressing together with the
temperature also the density and viscosity as a function of the velocity
and stagnation temperature drop.


3. Expressions for the Pressure, Density, and Viscosity

The pressure p at any point within the boundary layer is determined
by the equation of Bernoulli

K

P = O = 02(1 Uj2) (3.1)



where pO is the pressure on the boundary of the layer (thermal or
dynamic depending on which of them is thicker), p02 is the pressure
on adiabatically reducing the velocity to zero in the tube of flow
passing through the shock wave. The density p at an arbitrary point
within the boundary layer is determined by the equation of state (first
equation of (1.5)), equations (2.6) and (3.1)


K
K-i
P p1 (1 2) (3.2)
S 02 2 + (w 1)


where p02 is the density on adiabatic reduction of the velocity to zero.

The viscosity is determined by the equation


S= O 2 + -l) n (n 0.75) (3.3)


where 400 corresponds to the temperature TOO, that is, is obtained on
the adiabatic reduction of the velocity to zero.







NACA TM No. 1229


For any point ahead of the shock we have


p = PO(1




p = p0ol(


K
-K-1
- 02)


1

- u2 -1


(3.4)


where p01 and p01 are, respectively, the pressure and density on
adiabatically reducing the velocity to zero up to the intersection of the
streamline with the shock.

As a scale of the velocities it is possible instead of the assumed
magnitude j/2i to take the critical velocity a* and the local sound


a. As is known a* = (Ki 1)/(K + 1) /2

and = M we obtain
a


-- -
SU=+ 1--- ,
v + 1


Substituting


S(K -1)M/2
1 + ( K 1)M2/2


M= l 1

( + 1)/2 ( 1 1)12/2


4. Integral Relations of the Momentum and Energy in New Variables

Fram equations (3.1) and (3.2) we obtain


dp dU OUU
S=-poU L- = -POTU


(4.1)


velocity
U 1
*


(2.5)


In order to distinguish various uses of the symbol X herein,
sabscripts 1 and 2 have been added by the NACA reviewer in the
translated version.






NACA TI No. 1229


where p0 Is the density on the boundary of the layer (thermal or
dynamic depending on which is the thicker). We represent equations (1.1)
and (1.2) in the form

(pu2) + (PU) = p0UU' + + 6) Lu




6 (puU) + (pvU) puU' = o


Subtracting the previous equation from the above we obtain


+ UU'p ) + UUt (


-


- + y [(U 4]



= -[( + ') '


(4.2)


From equation (3.2) we find
1
O= 02(1-~-1
P0 = P02(i -19)


P0
1 =
p


2 -2 + ( 1)(1 t*/to*)
i -V2


(4.3)


Integrating equation (4.2) term by term from y = 0 to y = Ay, if Ay > By
and to y = 6y if 8y > Ay (for definiteness we assume that Ay > By;
the same result is obtained if we assume By > Ay), making use of the
relation (4.3), and taking into account the fact that starting from the
boundary of the dynamic layer the velocity u is constant along x and
the friction T is equal to zero, we obtain


u2 L 1- + 2UU'I P 1 -
Ctc1 a (l-i@






NACA TM No. 1229


fUy
3x J


+ UU' I


-U dy + UU'


0y
0


+1


(4.4)


where


+ ) Jy=0


is the frictional stress at the wall.

We now represent equations (1.2) and (1.3) in the form


- (puto*)
Ox


+ (pvto*)


- u dt0*= 0
dx


(put*) + (pvt*) -
y


dto
pu -
dx


= ( + e)-


Subtracting the second equation from the first, integrating the result
term by term from y = 0 to y = AY (assuming as above for definiteness
that Ay > By), and remembering that for y > By the heat transfer


oT
q = X r and the friction T =
of the thermal layer we obtain


d Uto* p
ai 0


d are equal to zero on the boundary


- )o/ dy =


(4.6)


1 -


10y


P 1


UU'(T 1)
+ 1 2


j0A


- t) dy = Tw
o /


(4.5)


UP-


p(1


H dy
U)


V /w


- LOT^
6 (-,


-1)


Ty = [(






NACA TM No. 1229


where








1 Y= C = X (4-7)
=-c y=0 w

is the intensity of the heat transfer at the wall.

In solving the problem of the boundary layer without heat transfer
between the gas and the wall for large velocities and P = 1,
A. A. Dorodnitsyn introduced the change in variables


2K
Td = 1 a ( (4.8)
-0 1 u-

Noting that the function under the integral sign in equation (4.8) agrees
with the expression p/p02 for vw = 1, we introduce a new independent
variable of the analogous equation containing in the function under the
integral sign the expression P/pO2 for the general case according to
equation (3.2)


S= 1 (1 ) dy (4.9)
0 1 U2 + (T 1)(1 t*/to0)


For Tw = 1 the relation between the coordinates n and y depends on
t*
the unknown velocity profile. For -- = equation (4.9) gives the
to U
change in variables applied to the problem of the heat interchange of
the plate with the gas flow. In this case the relation between n and y
likewise depends only on the velocity profile.

As is seen from equation (4.9) in the general case the relation between
the coordinates n and y depends not only on the velocity profile u(x, y:
but also on the temperature-drop profile t (x, y) which likewise is
not initially known but is determined from the solution of the problem.






NACA TM No. 1229


Replacing the deniaty p in equations (4.4) and
expression (3.2) and passing to the variables x, Tj


-i) dT +UU(2


(4.6) by its
we obtain


+ 1 -22)fR (1 -) d
+ -,I U~


+ UUI'k


i+ 1-


d
0x


-t0
-t-
0)


ut+'+,U 1)
dTi + (I


- *
0)


dT = T--
P02


=PC
P02op


(4.10)






(4.11)


where 6 and A are the values of the variable r, referring,
respectively, to the boundary of the dynamic and the thermal layers.

We denote the thickness of the lose in momentum and the thickness
of the displacement in the plane xz, respectively, by


* = H = I d


(4.12)


we introduce the concept of the thickness of the energy loss (in the
plane xTr)



e = to0 dj (4.13)


This magnitude has a clear physical meaning; namely, the magnitude e
characterizes the difference between that total energy which the mass of
the fluid that flows in unit time through a given section of the thermal


U d1
U2 1L (l


A = i0 1 u d






NACA TM No. 1229


boundary layer would have if its stagnation temperature were equal to the
stagnation temperature of the external flow and the true total energy of
this mass. The magnitude e thus represents in length units referred
to the temperature to* the "loss" in total energy due to the heat
transfer. For small velocities the concept of the thickness of energy
loss agrees with the previously introduced concept of the thickness of
heat-content loss (reference 4). The magnitude


A te m* \
A* =ear / i T A
Jo \ to


(4.14)


may be called the thickness of the thermal mixing.

With the aid of the magnitudes defined by equations (4.12), (4.13),
and (4.14) we represent the obtained integral relations of the moment
and energy (equations (4.10) and (4..1)) in the final form


as U'U
d + H + 2 (H + 1)
d U 1-


de U' to*
+ e+ --
ai U to*


U' (T 1 ) 7T
- + --- -
U(1-U2) p U2
02


dt
dx


Po2Ucpt


We note that the integration with respect to y (or n) may be taken
from 0 to a so that the relations (4.15) and (4.16) are general for
the theory of the boundary layer of finite thickness and the theory of
the asymptotic layer.

For small Mach numbers (the effect of the compressibility due to
the temperature drop) the relations (4.15) and (4.16) assume the form


d- U' U' T+
S+ (H + 2) + T 1
dx U U T


HTe = T
p0U2


(4.17)


de U'
-+ -
dx U


to0
to


0 Up0
Po'Vo


(4.15)


(4.16)


d-Tz
dx/






NACA TM Ho. 1229


where to = TO T, To and pO are, respectively, the temperature and
density of the isothermal flow outside the thermal boundary layer (it
follows from equations (2.6) and (3.2) if we set approximately U = 0,
that is, according to equation (3.5) M = 0). The pressure distribution
over the profile is determined by the equation of Bernoulli for an
incompressible fluid.1

As is seen from equations (4.15) and (4.16) and also from what follows,
in the variables x, q the equations of the system (1.1) to (1.3) are
simplified and approach in principle the corresponding equations for the
incompressible fluids. For this reason the fundamental methods of the
theory of the boundary layer in an incompressible fluid may be generalized
to the case of a body in a gas flow with heat interchange.

We give below the generalization of the method of Pohlhausen for
the case of the laminar layer and the logarithmic method of Prandtl-
Kfrm n for the case of the turbulent layer. The proposed method of the
solution of the problems connected with heat interchange permits, of
course, generalization of certain other problems in the theory of the
boundary layer in an incompressible fluid.


II. LAMINAR BOUND LAYER WITH HEAT INTERCHANGE

BETWEEN TEE GAS AND TEE WALL

5. Transformation of the Differential Equations


Assuming in equations (1.1) to (1.3) 6 = 0 and t* = T* TV,
substituting the values p, p, and p according to equations (3.1)
to (3.3), we transform these equations to the new independent vari-
ables x = x and ni, determined according to equation (4.9). The
equations of transformation of the derivatives will be

K


K x + (lx -' )y 1 -- 2 + ( K 1)(1 t*/to*) _


IFram equations (3.1) and (3.5) we have


2-i
-0 = + K-1)M --1 pO 1 + t + 2 +

whence setting M = 0, we obtain

pO + (po 92) = Constant
2an what follows we restrict ourselves to the case P = 1.






NACA TM No. 1229


We obtain (introducing the


+ v =


u
x


00
notation 02 =
P02


1 -2 + (t 1)(1 t*/to*)


+ v02 ( -


a
52)1 a
d)


civ
01
+ I- = 0
OT


-2 + (+ 1) -


(v = v


p
02


dt
-u-
dx


K2(-
v02 (1- 2)


1 t* n-1

to
1


+ (, 1)


If it is assumed approximately that
still further simplified.


n = 1, the obtained system can be


6. Generalization of the Method of Pohlhausen

We represent the velocity profile and the stagnation temperature-
drop profile by the polynomials


S= A ) +


+ A ) 3
v2


+ A


(6.1)


u (5.1)






(5.2)


6a
OX


3t*
u-
ax


+t*
+ V -
on


t

J-9


(5.3)


- a2







NACA TM No. 1229


-= t BI + B-B + B (6.2)


To determine the coefficients of the polynomials we set up the conditions


2 a t
62 u t6 t* + u
(/8)2 u ai(nl/A)to* t (n/s) U


for = 0
8


(6.3)


where


S2-n
B2U'T
X2 =
v02(1 ) .--1


Condition (6.3) follows from the
u = = 0 for = = 0. Further,


a = (1 n)


equation of motion (5.3) since


= 1i u 0
U a(,T/) U '


62
( 2 1 = 0
-(n/B)2 0


for = 1
B


From equation (5-3) for n = 0, v = v = 0 we obtain


=0 for = 0


(6.5)


Similarly to conditions (6.4) for the
profile t*/t0* the conditions


a t
W=A) 0.
b(n/A) to*


a2
J


profile u/U we take for the



t*
= 0 for 1=1 (6.6)


to*
O


(//Ar)' Wo /a


(6.4)


62
6(n/A)2


I


"


Tw TOO
Tw


f a t*
a---- -o
lo6iA to*






16 NACA TM No. 1229

By differentiating equations (5.1) and (5.3) with respect to T
with the subsequent equating of 9 to zero, it is easy to obtain the
conditions for the third derivatives of u/U and t*/t0* at the wall.
These conditions may be useful for various aspects of the method of
Pohlhausen. Using conditions (6.5) and (6.6) we obtain


3 /9 12a
B1 = -


B2 = 6 3B1,


B3 = -8 + 3B1,


B4 = 3 B1


From conditions (6.3) and (6.4) we now find


12 + X2

6 (3 /9 12o)8/A


A2 = 6 3A1,


A3 = -8 + 3A1,


A4 = 3 Al


(6.8)


In relations (4.15) and (4.16) there enter, besides 3(x) and e(x),
the four unknown functions Tw, H, q HT. In constructing the profiles
there were also introduced the auxiliary functions 5 and A. The
required six additional equations are obtained by substituting equa-
tions (6.1) and (6.2) in equations (4.5), (4.7), and (4.12) to (4.14).
We obtain


S= 5 =6 ,
20


T7 = T0 n-l U
w IooTw 5


K- )
(1 Vf~-


-5AI2 + 12A1 + 144
4 = 6 -------
1260


A1, qw = coCp0 'n-1 t- (1
A


8 B1
A* = e = 20


e = A(M1 + NiA1)


BI


- 8b()3 + 9 ,


N1 = b1- 3b22 3b 4)+ b )


24 5B1
4 = 2520


Al =


(6.7)


(6.9)


K
Ki-1


B1 (6.10)


where for


M1= 6b2( )


(6.11)


6 B1
b =60


16 3Bi
b2 = 420


5 B1
3 = 280'






NACA TM No. 1229


for > 1
8

8 B1
M1 20 +


B [-0.4 + 0.2290('7
2K


+ B,(1 0.1 0.1143



N1 = 0.0500 0.0429(


- 0.143c3 + 0.0290(
\A I \A,


A+ 0.0536/ 2



+ 0.0286( 3 -3
6X01


- 0.00961' I



o.oo006/)
V,,'J


+ B1( 2- 0.0167 + 0.0214 )-0.0107 ()


+ 0.o02()


From equations (6.10) and (6.8) it is seen that the point of separation
of the laminar layer for the given boundary conditions is determined
by the condition Xk = -12.

7. Determination of the Initial Conditions
For subsonic flow and also supersonic in those cases where there is
a head wave in front of the profile a critical point U(0) = 0 is
formed at the leading edge (x = 0). The latter is a singular point of
equations (4.15) and (4.16) in which the derivatives do/dx, de/dx,
and so forth, increase to infinity if the initial b, 8 are not
subjected to certain special conditions.
Substituting the expressions for Tv, q, A* from equations (6.10)
and (6.11) into equations (4.15) and (4.16), multiplying the latter
by 6 and A, respectively, and equating to zero the coefficients of 1/U
these conditions are obtained in the form


(7.1)


8--B 0,(A)2 = 0
X,(H + 2) + X2(Tw, l) 20 8-1 = O, r = o






18 NACA TM No. 1229


In the absence of heat transfer (Tw = 1) the first relation (7.1)
is a cubical equation in X2. Of its three roots (7.052, 17.75, and -70)
only the root (X2)x=0 = 7.052 satisfies the physical conditions. This
value of X2 is the one generally assumed initial condition in the theory
of the laminar layer for the equation of Karmtn-Pohlhausen. The equations
(7.1) may be conveniently regarded as a system for determining the
initial values of (X2)x=0 and (A/B)x=0. We present the results of the
computation of these values for 0 < (Tw)xz= < 5 (for n = 0.75).


Tw = 0

X2 = 0


0.05 0.10 0.50 1.o0 1.50 2.00 3.00 4.oo 5.00


0.3 0.7 4.05 7.05 7.5


6.0 3.9 2.6 1.9


= 1.12 1.16 1.18 1.23 1.31 1.50 1.87 2.74 3.70 4.86
8

The graphs of these results are shown in figure 2.


8. Method of Successive Approximations

Simple computation formulas can be obtained by generalizing the
method of H. Lyon (reference 5). At the same time we modify the method
of Lyon with the object of improving the convergence.

We multiply (4.15) by 2T and have


Ut 2(H + 2 -U2) -2 U T -1 -
+ + 2 -U Y2
U 1 U2 U 1-g2


K
2 n-1K-
OP-T(C-U) A
ROv


(8.1)


where


= A =L
L L


Sx dU
x U'=
Ar'


R02


"oLo2
-00


and L is a characteristic dimension. Setting k = 2(H + 2) we try
in the function k to separate a certain principal part constant for a
given value of Tw; that is, we set


k = c1 + (r cl), where c1 = c(i()


2 d
di2







NACA TM No. 1229


After simple transformations we obtain

dY 2T1 UUF a1 U9 2
2 + -- + 2(, L-J2 + 1 + 2(T -1) --


K
n-2 1 K-1 1 2
= Twn-2 (1 )1 2A1 2 (k- c) (8.3)
UR2 5 )


Assuming over a certain part of the boundary layer X1 < x < TXl that
A*
the ratio = h is constant with respect to i and equal to the
mean value for the given segment, we set3

1260(8 BI)
c = cl + 2(T 1)h = cl + 2(Tw 1) 2 (8.4)
L 20(-5A12 + 12A1 + 144)


Multiplying equation (8.3) by UJ we obtain


d (UiUc) + (c 2) 42



K
n-2 c-1 -1 -
= T (1 -U [2A


Considering this equation as linear in b2Jc we write its solution In
the form



S02 c + Tn-2 c-1 (1 f2) -- d (85)
i f-) (8.5)

For TJ Constant there is taken in the exponent c a constant
mean value of Tw for each section.






NACA TM No. 1229


where


0 = 2A1 g- Tw (k cl),


1- -
C -- 02To (1 -_ 2)
x=x0


where C = 0 for XO = O. Substituting under the integral (in the
function C and exponent c) certain initial values of (X2)0 and (A/6)0
we obtain a first approximate function A(x). The arbitrariness in the
choice of the magnitude cl may be utilized for improving the convergence.
For this purpose cl must be chosen such that for each value of Tw the
error due to the assumed (inexact) values of X2 and A/6 is a minimum.


Since ) itself depends little on A/6 it is
the condition of little variation of $ with X2.
small dependence of the functions 6/6 and k on
in the argument Al on which they depend, X2 = 0,
approximate expression


S12 + X=(A0)
6 X2=0o52=Ac


sufficient to set up
Neglecting the relatively
X, that is, setting
we obtain the


2
t-Q X2=0[k) -c1]
TV -W/ XoL =0 11X=


In order that C = Constant the coefficient of X2 must be equal
to zero, whence we obtain


(A1)X2=0Tw
01c = ( k) _
2=0 6(9/6)
S 6X2=0


This dependence of cl on Tw in a wide interval of change of
the argument (and practically independent of A/5) is close to a linear
one. For Tw = 0.1 we obtain cl = 9.35 for = 1 (c, = 9.12 for
=2, c = 9.7 for = 0.5). For T= = 1 we obtain cl = 6.26.
Rounding off the last value to cl = 6 we apply the linear relation


cl = 9.5 3.5Tw


(8.7)


Finally multiplying the equation of energy (equation (4.17)) in nondimensional
form term by term by 202 ( = and integrating as linear we obtain


(8.6)






NACA TM No. 1229


=R2 = ( -' 1)2


K

+ 2B1(TV I2 ) (l -2)1! 1 e
XO


C' = [2RO2b2( -_ 1)2]


and C' = 0 for i = 0.

The computation of the dynamic and thermal
and (8.9) can be carried out in this sequence.
with the parameter X2, the analogous composite


layers by equations (8.5)
We consider, together
parameters


2 20 (1 2- f2


S- ) 2-n

(l (1J2) W


-'2R (-m-2-n
2 e= 2U'T210


By equations (69) and (6.) represented in the form

By equations (6.9) and (6.11) represented in the form


X-5A12 + 12A, + 14 2
= 2 1260


2?e X2A(M1 + NiA1)2


V X2


there are constructed for the given value of T. auxiliary graphs of
the dependence of X2 on X2, for various values of X2A and for the
dependence of X2G on X2e for various values of X2. The magnitudes
taken as initial in the computation by formula (8.5) are determined from


where


(8.9)





NACA TM No. 1229


the initial data. If U/ 0 for 30 = 0, there are taken the values (2)o = 0
and (A/B)0 = 1. If f = 0 for XO = 0, there can be taken as the
initial values the values of (X2)0 and (X2A)0 = (A/)02(2)0 according
to figure 2. By the values of the functions i(i), that is, X20(1) of
the first approximation using the initial value (A/8)0 (for the succeeding
approximations it is convenient to use the initial values of X2.), X2(x)
of the first approximation is found with the aid of the graph. Further,
by equation (8.9) there is computed e(x), that is, X2e(i) of the first
approximation, making use of X2(i) of the first approximation and (A/B)0
(in the succeeding approximations there is used the function X2(i) following
and X2,(x) preceding). From the values of 2g(1) and x2(i) with the aid
of the graph there is obtained X2(i). In those cases where there is a
considerable change in the ratio A*/8 (or the function Tw(E)) the
computation must be conducted over segments. The required data for each
succeeding segment are taken equal to the corresponding values obtained
at the end of the preceding segment. The local Nusselt number (that is,
the coefficient of heat transfer qw/to* reduced to nondimensional form)
and the coefficient of friction are found from the equations



N-= -- = (K M- U -1 (8.10)

S= + (' 1)4 (8.11)
Pum R. UL J

"In particular for 1T = 1 formula (8.5) with (2)0 = 0 gives
(M. < Mcr

9 ~1---+ PC-
poo00 = ---o .- 7 -(u 2 p- l d (c = 6)
2-- 0
1c(l U2) 2
which agrees with the equation of L. G. Loitsiansky and A. A. Dorodnitayn
for the computation of the laminar layer without heat transfer (reference 6).
In the absence of heat transfer and for small Mach numbers we again
obtain from this the quadrature

R = o.47 f dx (:R

earlier derived by us on the basis of bhe method of Karman-Pohlhausen for
the laminar layer in an incompressible fluid. (See Tekhnika Vozdushnogo
Flota, No. 5-6, 1942.)






NACA TM No. 1229


For small Mach numbers equations (8.5) and (8.9) assume the form


n-2 ( c-1
'0~


(8.12)


( 'o oLp
* di (= -U0I-j
\ 0-


/n-1
Tar;


2B1 -2


(T/To 1)2(U/U.)2 J


U 0


(8.13)


9. Dependence of the Reynolds Number R02 on

the Parameters of the Flow


Taking account of the fact that according to the equation of state
p02
= -- we represent the parameter R02 in the form
P01


p02
R02 = R01 -,
P01


where R1 =
-00


(9.1)


The parameter R01 is expressed directly in terms of the Reynolds
U.Lo U,
number Rm =-- and the Mach number M = 2 of the approaching
flow. From equation (33) to (3) we fin a
flow. From equations (3.3) to (3.5) we find


S +1
01 = (C1 I- .2) -1 + [l ( 1)M_. 2(Kl
01 U


( 1()M92

(9.2)


In the case of subsonic subcriticall) velocities we have
R02 = R01 = R00. For supersonic velocities the ratio p02/P01 is
found from the condition of a line of a flow passing through an oblique
shock wave (or a head wave) at the leading edge of the body. Considering
each surface of the profile separately we denote by %0 the angle which
the tangent. to the surface of the airfoil at any point makes with the
direction of the velocity of the undisturbed flow and, by cp the angle
which the normal to the surface of discontinuity makes with the same
direction. From the equations of the oblique shock wave we obtain


2R = U1
(U/U^ )c


( TA


P02
P01


1
2






NACA TM No. 1229


K 1
P02 / 1/2(K + 1)M. 0os2 :-l 2 K- : (
= --- -- M.co (9.3)
Pol \1 + 1/2( 1)M cos2 K + 1 +



1/2(K + 1)M2sin 29
tan (c + Po) = (9.4)
2 1 + 1/2( 1)M2cos2(


In the case of a head wave in front of the body the direction of the
velocity after the discontinuity (for the flow line at the profile) may
be considered to coincide with the direction of the velocity before the
discontinuity; in equation (9.3) there is in this case to be substituted
p = 0.

10. Boundary Layer in the Flow of a Gas with Axial Symmetry

For any axial flow about a body of revolution the integral relations
of the impulse and energy have the following form:


a 6 d dp
S ypu2r dy U pur dy = d 6yr Twr (10.1)
dx -'0 0 dx-




d- cpu(t* t0*)r dy = -qr (10.2)
dx 0


In these equations the usual simplifications were made; x is the distance
along the arc of the meridional section, d the distance along the
normal to the surface, and r(x) the radius of the cross section of the
body of rotation (the change of the radius vector within the boundary
layer is neglected). The boundary conditions of the problem and also the
assumptions with regard to the external flow are taken to be the same
as in section 1. Setting up expressions for T, p, p, and 4 as in
sections 2,and 3 and introducing the new independent variable I by
formula (4.9) we obtain the integral relations of the moment and energy
in thb variables x, T:







NACA TM No. 1229


H + 2 + (H + 1) .----2T +
1 t- 2


U'(TM 1)

U(l -.U2)


r' w_
eHT + -- = -
r POOU2


toI*
e+ -- +
r to*


Qe
oo8=
PooUc to*


(= L, = -,and so forth) (0.4)
SL L


(L is a characteristic dimension, for example, the length of the body
of rotation.) Restricting ourselves to the case P = 1 and taking the
expressions (6.1) and (6.2) we obtain a closed system of equations
(10.3), (10.4), (6.9), and (6.11) the solution of which we write (for the
case where there is no shock wave) in the form


2R00 = 11-2



6201
Tc( ~T2) 22



e2ROO = C
UQ(T -_ 1)2F2 _


JI


r
+
x0


) c
1--+-I
Tn--l (i1 2) 2 K-I 2


Twn-2Bl (1


(10.5)
K- 2)
-fl2) (t,- _1)2j;2 Bdi]


(10.6)


where


1-C
I =c
C = I[2ROCci(1 2 t) 22J
X=1O


C' = [2ROo2(w )2
x=x0


The method of computation does not differ from the case of the
two-dimensional flow. For the coefficients of the heat transfer and
friction the equations (8.10) ani (8.11) remain valid. In the case of
the internal problem (flow in nozzles) the Nusselt number and the
friction coefficient are determined by the equations


K
q -L n-l K-1
Noo = Tw( U 2)--
0oot0*


K
2rw 2 n-1 -o -1 A1
Too = Tw (1 ) -
SPOO o00 U5


di U'


d9 U'
dix U


(10.3)






NACA TM No. 1229


where X00 is the coefficient of heat conductivity corresponding to
the temperature TOO.


III. TURBULENT BOUNDARY LAYER IN TEE PRESENCE OF HEAT

TRANSFER BETWEEN THE GAS AND WALL

11. Fundamental Assumptions


The functions H, HT, Tw, and q. entering equations (4.15)
E.an (4.16) are determined by equations (4.12), (4.14), (4.5), and (4.7)
which express them as functions of ) and e through the medium of the
velocity profile and the stagnation temperature-drop profile. The
present state of the problem of turbulence does not permit representing
th- velocity profile (and also the temperature profile) by a single
equation which holds true from the wall to the boundary-layer limit.
The fundamental dynamic and thermal characteristics of the turbulent
layer can nevertheless be computed with an accuracy which is sufficient
for practical purposes. A fortunate property of equations (4.15)
and (4.16) which can be predicted on the basis of the results with
respect to noncompressible fluids is that the functions H and HT
change very little over the length of the turbulent layer and the
functions Tw and qw connected with and 8 by the equations are
little sensitive to the actual condition which prevail at a given section
of the boundary layer. Hence H and HT (and also magnitudes analogous
'to them) can be taken as constant over x and the relations between Tw
and a (the resistance law) and between qw and e (the heat-transfer
law) can be set up starting from the assumption that the conditions at
the given section of the boundary layer do not differ from the conditions
on the flat plate. On the basis of the derivation of these supplementary
equations we assume the simple scheme of Karmnn according to which the
section of the boundary layer is divided into a purely turbulent
"nucleus of the flow" and a laminarr sublayer" in immediate contact
with the wall. In the latter the turbulent friction and the temperature
drop are small by comparison with the molecular. We assume that in the
turbulent "nucleus" the frictional stress is expressed by the formula
of Prandtl:

2
T = p (11.1)


where I is the length of the mixing path. In other words, in
equations (1.1) and (1.3) we set = p22 dl. It follows directly from
dy






NACA TM No. 1229


this in view of the fact that the turbulence assumption of Prandtl gives
Xt = Cpe that the expression for the heat transfer is


dT 2 du dT
q = n = cppl
dy d pP y dy


(11.2)


The thickness 86y of the laminar sublayer of the dynamic boundary
layer, equal for P = 1 to the thickness Ay6 of the thermal sublayer,
is determined by the critical Reynolds number (the Karman criterion)


(a S 11.5)


(11.3)


where u7 is the velocity on the boundary of the lminar sublayer, pw
and pw are the density and viscosity at the wall.


12. Derivation of the Resistance Law

Assuming that as in the case of the noncompressible fluid a linear
variation of the velocity in the laminar sublayer is permissible on
account of the small thickness, we have


(12.1)


From equations (11.3) and (12.1) we obtain


F B = y


(12.2)


In equation (12.2) we pass to the variable TI. Near the wall on account
of the smallness of the terms u2 and t*/t0* we have


T0
"0


(1 12)1- 1 l2 + (T 1) -


K
dyi %b w(1 -2)1-"


(12.3)


hence


K
B~aT~l-U2)K'1


u15:I4 = <,2


Tw- = ,W Y-


t*
to W






NACA TM No. 1229


where 65 = At is the thickness of the laminar sublayer for the variable T.
Further,


K
P1 (1 U)-1
Pw = 002 (1w


S=
W, = POOT4


Substituting these expressions in equation (12.2) we obtain


= ,
e RB


For the fundamental parameters of the dynamic layer there is here
introduced the notation


RE = U-R02o,



Equation (12.1) is
into the form



T = p027L 2


U 1 2(K-1)
11 U TP) (12
V TV/Po2 TV

with the aid of equations (3.2) and (4.9) transformed


3K

-12 2 (1 U2) (12
[1 2 + (T, 1)(1 t*/t0o]


.5)


.6)


Since for small i the terms U2 and t*/t0* are small and the mixing
path I = ky (k = 0.391) where the coordinate y is expressed according
to equation (12.3), the "generalized" mixing path 1 near the wall is
a linear function of t:


S 1 )2(-1)
' = kii -172 (1 )
TV1/2


(12.7)


In deriving the resistance law in an incompressible fluid a linear
mixing-path distribution and a constant frictional stress are assumed
for the entire section of the boundary layer, from the wall to the
outer boundary. Actually the mixing path increases at a considerably
slower rate than according to the linear law and the friction drops to
S


(12.4)


(61 =






NACA TM No. 1229


zero as the outer boundary of the layer is approached. Ths assumptions
made act in opposite directions and lead to a satisfactory relation
between the parameters RE and t.

Carrying over this fundamental idea of the logarithmic method into
our present theory we set T = T in equation (12.7) and assume
the linear law (equation (12.7)) for the entire section of the boundary
layer. We thus assume that as in the case of the noncompressible fluid
there will be a mutual compensation of the errors committed in the distri-
bution of T and 2. Integrating equation (12.6) between I and B
we obtain the approximate velocity profile:

S1 + n 1 (12.8)
U kl 8

From equations (12.1) and (12.4) we obtain the velocity at the boundary
of the laminar sublayer


U r
ul

The condition of the equality of the velocities of the turbulent and
laminar flows on the boundary of the sublayer gives



Pr = Cl w nekk ( = RO, C=a e-ka = 0.326) (12.10)


Making use of the velocity profile (12.8) in equations (4.12) we obtain


B1 ,-1 1= -- (12.11)
6 Ft k 6 5 kc

Eliminating from equation (12.10) and the first of relations (12.11) the
auxiliary variable 8, we obtain the resistance law:


BR = ClTwnek (l- ) (12.12)


We obtain incidentally also the approximate expression for the
parameter H:
6* 1
H= (12.13)
0 1 2/kt





NACA TM No. 1229


13. Derivation of the Heat-Transfer Law

In this section we shall give a generalization to the case of a
gas moving with large velocities of the heat-transfer law earlier derived
by us (reference 4) for an incompressible fluid.

We construct the function

q* = = q + u (13.1)
dy J

For the turbulent nucleus of the flow we have


dt* dt* 2 du dt*
q* = At -- = c ;--= cpp2 (13.2)
Sdy dy dy dy(13.2)

Transforming this equation to the variable j we obtain

3K
~2 du dt* f2 2-
q* = c P02op 2 = d1 )K-
d= dT [1 U2 + (T- 1)(1 t*/to*]3

(13.3)

Near the wall the function q* behaves like q, ,that is, differs little
from the constant value qw, and the mixing path I depends linearly
on I according to equation (12.7). The common mechanism of the transfer
of heat and the transfer of the momentum in the flows along solid walls
provides a basis in the derivation of the law of heat transfer for
assuming as before a constant value q* = q, and the linear law (equa-
tion (12.7)) for the entire thickness of the thermal layer. Substituting the
expression for du/dij obtained from equation (12.8) and integrating
equation (13.3) from r to A we obtain the approximate profile for the
stagnation temperatures

t* q U
= 1 + In (13.)
t Cpto0*T k







NACA TM No. 1229


From equation (13.1), assuming a linear distribution of the stagna-
tion temperatures in the laminar sublayer, we find


t I*
qw = Cp', -7
Ayz


(13.5)


where t2* is the stagnation temperature at the boundary of the laminar
sublayer.

As the fundamental thermal characteristics of the boundary layer
we introduce the following parameters:


Re = URO2,


RO2 U 1


K
-u2) -1


qoL

xooto


(13.6)


From equations (13.4) and (13.5) we obtain the stagnation temperature
on the boundary of the laminar sublayer


to CT


(13.7)


Equating the stagnation temperatures on the boundary of the laminar
sublayer and making use of condition (12.4) we obtain


R = ClT nexp(kCT)k


(RA = T0RO2)


(13.8)


Substituting the expressions for u/U and t*/to* according to
equations (12.8) and (13.4) in equations (4.13) and (4.14) we obtain


S1+ In
e (. i, kCT


2
k2 CT,


1* 1
A kCT


(13.9)


From equations (12.10) and (13.8) it follows that = exp(k kt)
so that we have


S= 1 ( k2T


(13.10)






NACA TM No. 1229


Eliminating from equations (13.8) and (13.10) the auxiliary parameter A
we obtain the heat-transfer law


Re = C01 n ) exp ktT


We obtain incidentally also the approximate expression for the
parameter HT:


(13.11)


(13.12)


l/kt
1/kC(l 2/kT)


14. Solution of the Equation of the Turbulent

Dynamic Boundary Layer

We represent equation (4.15) in the form


t' (H + 1)
+ -
U(1 t2)


SU'(Ty-1)A* 1 1i
+ =2 =
I(l V2) ( TV


K
(i U-- U 02


(14.1)


We make the change in variables (reference 7):

z = ekC(l 2/ki)k2t2


Differentiating this relation with respect to
we obtain


dz
S= Kz
dx


1 R0
dz


n
- A1


(14.2)


x and using equation (12.12)


S1 2/k k22
(K 1 2/kt + 2/k2t


da


(14.3)






NACA TM No. 1229


From equations (14.1) to (14.3) we obtain


U'K(H + 1)
+
U(1 2)


+ 'K(Ty 1)
U(1 U2)


6* Tw'
z + nK
' TV


02
K00o~k2
Z --
z n+l
SCin+l
ClTw


K
(1 j2)K-1 (14.4)


The magnitudes H and K
layer are assumed constant with
as a known.function of 1 then
respect to z.


which change little along the boundary
respect to i. If 6*/a is considered
equation (14.4) is a linear equation with


Assuming a constant mean value of the ratio --= h
interval Xo < x < 11 (in the first approximation we may
A* 6*
turbulent layer assume h = = = H, which holds for
obtain the solution of equation (14.4) in the form5


(1 )c
M- n-Kc


KRO2k2 px


TwnK-n-lc+l K
-----(1 )K-1

(/ -TO


over a certain
for the entire

the plate),we


dxJ


(14.5)


c +


where


C = [zT.n


(1 u)- ,
/d-uerj
X=X Q~


The constants must be taken equal to



K = 1.20, H = 1.4, h = 1.4,


c = K(H + 1) + K(T, 1)h


(Cl = 0.326; k = 0.391)


See footnote (3) on page 19.






NACA TM No. 1229


After computation of the dynamic and thermal layer new mean values
HTRg
of the variables K, H, h = -H can be found and, if the deviations
"b
from the assumed values are considerable,the computation is repeated.
If, for greater accuracy, the computation is conducted over segments the
values of the constants for each succeeding segment are determined by
the values of RE, C, Re, and tT obtained at the end of the preceding
segment. In integrating from the point of the transition of the laminar
into the turbulent state, the magnitude (z)-=- is determined from the
condition of equality of the initial value of Re to the value of
at the end of the laminar segment. In integrating from the leading
edge, C = 0. By equations (12.2) and (14.2) the auxiliary graphs of the
functions log (RE C1 ) and log z as functions of kt can be
constructed once for all.
The local friction coefficient is found from the equation


1 + ( 1.)
2+


l -l 02 \2 1
i o \Uoo -Cl


For small Mach numbers equation (14.5) assumes the form


KRk2 T 'nK-n-1 c+l
+ --- TO


R UoLp 0
1.0


Re = U R,
U,


15. Solution of the Equation of the Turbulent Thermal Layer

We represent equation (4.16) in the form


dRe 1
d-- + ;1


dt *
Rg = --- fUR(1 2)K-l
di T 02Tw


(15.1)


2Tw
Cf = ----
PU2


2_
Tw


K
U2) K-1


(14.6)


where


(14.7)


C= = (U0 0


Z (T,/(= U1 c
(TW/To),-j(U/Ujj)0


To0 oT







NACA TM No. 1229


Carrying out the change in variables (reference 4)


zT = xep (kT) k kT

and making use of the heat-transfer law (13.1) we find


1 dTv
+ T di
T,* dx


1 dtO*
ZT + KT -* d-
0 ci


k2
ZT = KT kC- +l
ktCiT4


(KT


1- l/kSt
1 2/kT + 2/k2tT


The solution has the former


Kr-p 2k2. r(
C+ i "
'-0


-1)K U
kt


C'= [TTT(t 1)KT
X=Xo


6The equation of
between Re and tT


energy (15.1) in connection with the
in the form (13.11) in the case T =


an equation with separable variables so that together with
in the form (15.4) we can use its accurate solution


exp(ktT)(kT 3) + 2Ei(kT) = C + ROl
Where
where


F, 1(kCT) =


IkT
-e


relation
Constant is
the solution


K
(l tp)14-1 di


- du, C = [exp(kT)(kT 3) + 291(kST)
u X=X0


(for X0 = 0, C = 0)


dzT
di


(15.2)


UR02(1 UL2)"1


(15.3)


ZT = ( 1)KT
w


(1
(1_b2 d.1


(15.4)






NACA TM No. 1229


The constant K' must be taken equal to 1.15 (with subsequent check
by the results of the computation of the thermal layer). The magnitude
((T)i= is determined by equation (13.11) from the condition of the
equality of the value of Re at the start of the turbulent region to its
value at the end of the laminar section. In integrating from the leading
edge, C' = 0. The auxiliary graph of the function log zT against kCT
can be constructed once for all.


The local heat transfer is found from the equation


qWL
N -
__0


n
S1i + 1 (K 1) M21
= T T 2


K
(1 Uj2) 1-1


For small values of the Mach number equation (15.4) assumes the form


1
T = T(T/T9T(T;/T--- KTEC
T (Ty/TO^nT(T /TO 1KT


16. Determination of the Profile Drag for Subsonic Velocities

The drag of a wing of infinite span (over unit length of span) is
obtained from the momentum theorem in the form


COQ = ) =


Pouc1 00U20


wnere g denotes the sum of the momentum-loss thicknesses referred
to the upper and lower surfaces and computed at a great distance from
rh- wing where 8 --> and U---U.. For the drag coefficient we have

1
ep = 2Q 2 m 1 + 4 1)M2 K (16.2)
p L
DoiLTo L -


nKT On-1

W0 (0


+1


-vK


U 1
U00 kt


(15.6)


(16.1)


(15.5)







NACA TM No. 1229


The problem consists in expressing
thermal characteristics of the boundary
Since in the wake behind the body T7 =
and (4.17) for the wake assume the form


H + tf
1- u


di U
d TT


-, in terms of the dynamic and
layer at the trailing edge.
0 and q = 0, equations (4.16)


Tw -
fj2u


(16.3)





(16.4)


1 dt0
+ = 0
S0 dx
0


We introduce the notation


G(i) =
1 U2


Tw 1
+ 1 T
1 --2 a


and represent equation (16.3) in the form


1 di
l di


= -(2 + G) In
dx


(16.6)


Integrating this equation with respect to I from
(denoted by the subscript 1) to i = a, we obtain


in = (2 + 01) In
1 *1


For U, 0,


+ O
91


the trailing edge


In dG
UM


T, = 1 the function 0G() goes over into H(i).


an incompressible fluid however the hypothesis of Squire and Young
(reference 8) on the linear character of the dependence In(U/Ua) on H
holds:


In(U/U.)
Hw- H


In(Ul/U.)
E H1


(16.5)


(16.7)


For


*E2
+ -T 2


; = 0






NACA TM No. 1229


Making the analogous assumption


ln(U/U- )
G--G


ln(Ul/uI,)
G.- -G


(16.8)


we obtain7


In
.1


i2 l
2 Uf.


(16.9)


We write down the expression for a1:


2
HI + U2
01 2
1 Ul


(16.10)


1 -_ U 1
+ -- 1 9T
1-U2 "1


It is easily seen that Ho = 1 and BH~ = 1, hence


2
G =
S1 2
1- Tm


m -1 e,
+ 12 ,
i U2 .0


(16.11)


'The same result can be arrived at from the following elementary
considerations. Equation (16.7) may be represented in the form



ln = (2 + 01) In m + (GC G1) In U
1l Um U.


where Um is a certain mean value of the velocity U that lies between
U1 and U,. For the usual profile shapes however the ratio Ul/ is,
in general, near unity and since the magnitude 2 + 01 exceeds the
magnitude Go _1 by several times,therefore for any choice of the
mean value of Um the relative error in the determination of S, is
not large. Taking the geometric mean m = UVU we again arrive at
equation (16.9).






NACA TM No. 1229 39

Replacing in equation (16.9) G1 and G, by their values, we obtain
finally

= 1 -(16.12)
u./


H1 +5 H1 + 1 U12
2 2 1 12


-2
U- 2
a v


- -

i 2
1 )1


1
2


(TW i)oa/aCO
1-v


From equation (10.4) it follows that 0U(Tw 1) = Constant, hence


(TV" 1) 9- = (if --1)91 U (16.13)
(9 U1 (16.13)

For small Mach numbers equations (16.2), (16.12), and (16.13) assume
the form


cxp = 2 .,


H + 5
2


(Tw


s = 9u1
00 1 \U it


17. Boundary Layer in a Gas Flow with Axial Symmetry
For the turbulent flow about a body of rotation the equations (10.1)
and (10.2) in the variables x, y and the transformed equations (10.3)
and (10.4) remain valid. Restricting ourselves to the case of the
Prandtl number P = 1 and introducing the parameters Rb, R, RE, and
we represent the integral relations in the form


elHT1
'i


(TITW


/ 0


-)


5. = (Tow





NACA TM No. 1229


dE UI(H + 1) 1) 1 1_1
- -+ + +- % + + fi T noo-- 1 W
d U(1 ) U( U) r C2 T
(17.1)


R r
- + -- R
dx r


d to*
+ R-= 0 y(1 -y
t0* T


Making use of the drag law equation (12.12),the heat-transfer law
equation (13.11), effecting the change in variables


z = k22'


ZT = exp (kT) (l- k


assuming the little changing magnitudes H, K, XT constant with respect
to x, and also a constant mean value for the ratio = h, we obtain
a system of linear equations the solution of which has the form


S/(I- :52)
S= 0


XRook2 zT nK--i +,
+ C VcI c (l~ di
Cl Ji0 1- )


(17.3)


S/(1 -_ )0 _


ZT = 1 )T
TVnWTiT(TL T


l 41ook2 fr2)I
c+ 0 m --f


C' = z (rw, 1)Kr


(17.2)






NACA TM No. 1229


For the frictional stress and the Nusselt number, equations (14.6)
and (15.5) remain in force. In the case of the internal problem



,L K- 2 1
N00 U-RooU(1 2) Of = = -(ol -
X oo oT* tT T 00 o


(17.5)


The theory presented, in particular the integral relations of the
moment and energy established in part I, permits determining the
thermal and dynamic characteristics of the boundary layer at a curved
wall in the most general cases, that is, in the presence of external
and internal heat interchange. The computation of the boundary layer
by the equations derived in parts II and III on the assumption of the
Prandtl number P = 1 permits finding directly for arbitrary Mach numbers
(excluding the interval from M = Mcr to M = i):

(1) The coefficients of the heat transfer from the wall to the gas
for a given maintained temperature of the wall through heat supplied
outside the body and the coefficients of heat transfer from the gas to
the wall, that is, required for maintaining the heat conduction within
the body at the given temperature of the wall.

(2) The distribution of the frictional stress along the wall and
the profile drag of the wing (in the case M, < M r) for arbitrary
ratio of the stagnation temperatures and those of the wall.

For small velocities the obtained equations express the dependence
of the heat transfer and the drag on the ratio of the absolute tempera-
tures of the flow and the wall (the effect of the compressibility and
the change of the physical constants due to the heat interchange).

In conclusion we give the results of computation of a single example.
In figure 3 is given the distribution of the velocities of the external
flow for the supersonic flow about a body with two sharp edges. The
contour of the body and the position of the discontinuity are also shown.
The flow was computed by the method of Donov (reference 9). In figure 4
are given the curves for the Nusselt number N which assure the uniform
cooling of the surface up to the temperature Tw = 0.25Too for
RE = 15 x 106, MN = 2, and Mw = 6 for the laminar (lower curves) and
turbulent (upper curves) regimes.

Translated by S. Reiss
National Advisory Committee
for Aeronautics






NACA TM No. 1229


REFERENCES


1. Dorodnitsyn, A. A.: Boundary Layer in a Compressible Gas. Prikladnaya
Matematika i Mekhanika, vol. VI., 1942.

2. Frankl, F. I., Christianovich, S. A., and Alexseyev, R. P.: Fundamentals
of Cas Dynamics. CAHI Report No. 364, 1938.

3. Kalikhman, L. E.: Drag and Heat Transfer of a Flat Plate in a Gas
Flow at Large Speeds. Prikladnaya Matematika 1 Mekhanika, vol. IX,
1945.

4. Kalikhman, L. E.: Computation of the Heat Transfer in a Turbulent
Flow of an Incompressible Fluid. NII-I Report No. 4, 1945.

5. Lyon, H.: The Drag of Streamline Bodies. Aircraft Engineering,
No. 67, 1934.

6. Dorodnitsyn, A. A., and Loitsiansky, L. G.: Boundary layer of a Wing
Profile at Large Velocities. CAHI Report No. 551, 1944.

7. Kalikhman, L. E.: New Method of Computation of the Turbulent Boundary
Layer and Determination of the Point of Separation. Doklady
Akademii Nauk SSSR, vol. 38, No. 5-6, 1943.

8. Squire, H. B., and Young, A. D.: The Calculation of the Profile
Drag of Aerofoils. ARC, R & M No. 1838, 1938.

9. Donov, A.: Plane Wing with Sharp Edges in Supersonic Flight.
Izvestia Akademii Nauk SSSR, Ser. Matematichoskaya, 1939,
pp. 603-626.







WACA TM No. 1229


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