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TECHN l CAL HEHORANDmfi.S

lTATIOUAL ADVISORY COMMITTEE FOR AERonAUTICS

Ho . 1032

HEAT TRANSFER 1£T THE TURBULBFT BOUNDARY LAYER

OF A COMPRESSIBLE GAS AT HIGH SP~EDS

By F . Frn.n k l

[!,nd

FRICTION IN THE TURBULENT BOUnDARY LAYER

OF A OOMPRESSIBLE GAS AT HIGH SPEEDS

By F. Frankl and V . Voishe1

Central Aero-Hydrodynamica1 Institute

Washington October 1942

58032 https://ntrs.nasa.gov/search.jsp?R=20050009880 2018-10-18T21:54:58+00:00Z

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rATIONAL ADVISORY CO!UUTTZE FO:3. A:E"BONATJTICS

TECHNICAL MEMORANDUl~ NO. 1032

HEAT T:a.A.NSFEl1 III THE TUlU3ULENT BOUNDA.R~ LAnu:t

OF A COUP:a.ESSIBLE GAS AT HIGH SP"R]IDSl

By F. Frankl

SUHH.A1l.Y

The Aeynolds law of heat transfer from a "raIl to a turbulent stream is extended to the case of flow of a compressible gas at high speeds. The analysis is based on the modern theory of the turbulent boundary layer with laminar sublayer. The investigation is carried out for the case of a plate situated in a parallel stream. The results are obtained independently of the velocity dis­tributio~ in the turbulent boundary layer. The heat transfer q in unit time is a function of t1"~e frictional

coefficient 2T ~ = =-,

pu2 the free stream temperature

the temperature at the plate T.. the free-stream velocity u and the density p. the velocity n.t the boundary of the laminar sublayer and the basic turbulent layer ul' and the Prandtl number for the given gas

The problem of heat transfer at high velocities has at the present time become a very pressing one not~ in connection with heat technolo~~ and high-speed aviation. 3.ecently this problem has been encountered i~ the computa­tion of the surface of a wing radiator. The eAisting papers on heat transfer from the wall to the gas in part refer to small speeds (::ae~molds, Prandtl, and others) "Ihile 'lid tl: regard to high-speed flo'l;l some (Bus ere an t Frankl) consider only the laminar regiEe, others (Crocco) neglect the lar:.inar sublayer in the turbulent layer, while still others (Guc~man. Shirokov) start out fron

laeport No. 240, of the Central Aer~Hydrodynamical Insti­tute, Hoscow, 1935.

I

L

2 NACA T~chnic~l Uenorandum No. 1032

relatively rough hydraulic assumptions. In the present paper the problem is treatedfron the point of view of the Bodern turbulence theory.

We shall consider the case of plane parallel flow and in particular the boundary layer of a plate in a stream :;.)arallel to it. '''e shall first revie-,'l SOHe basic kn 01.-In res u 1 t s •

The instantaneous velocity components u , v and the values of the state variables P, p, T are broken up into their Dean values u, v, ... and fluctuations about the Lleall. values u l • vI, Ive assu;'le tl:'.'lt t~!.e turJ1J.1e":t fhlctul1.tions of the density pI are s;;u).ll by co;::parisol1 with the nean density p. Ye then have the followinc equations for the turbulent stresses T xx ' Txy ' Tyy and

the heat transfer per unit tine through unit area

i Txx pU l2 I = I

._-------- ( Txy = pulv t i (

I

------- I

pv' "" I

Tyy -- j

------ -] q-- -- c (lu'T'

, ~ ..... 1)1.

>-' '" ( ------, 'i q,- = c p'~' mt p' • ... I .! J

where t:le bars denote the time-averaced val'_:es, ancl the equatio~s of motion

d'.l op d T ---. c) T -."

I AA -~ P -dt CJ :..:: CJX 0:/

"-- '::>T ~~ f dT oj;) l) X;)" P =

dt oy d -r o~7"

d( au) "' ( \ I +

o pD., = 0 ---d}:-- - dy--

cl ( _ 'L:;:' +;2" ul;; + ;12 \ P - ~ Jc T + ------ + - --- J'

dt \ P 2 2

() r-

0<1:;: "- .,

0 yT -r-)

I oq", ! ( 1:T-c---

- \ ( ~lT xy J I = + ~T :c:" ) + + ---"-

dX -,--'\. oy ; ,;t 0-,- oy i

L J

]. )

.-, 1 c:.., )

•

.. •

,i'

..

..

• '.

ITACA Technical r:emorar..d-QlJ1 :;0. 1032 3

The viscosity forces are here r..eglectedby comparisor.. with the turbulent friction.

We now proceed to the Case of the plate. In this case p has a constant value. We choose the edge of .the ;:late for the x-axis. We may then neglect tlce deriva­tives l7ith respect to y, and v by comparison "lith u. 'lie aSs-"'1.me, :finally, that the average energy of the fl1.1.c­tuatir..g turbulent velocities has a constant value. Equa­tions (3) and (5) the~ become

cln P dt = ( 3 f )

(51 )

For the determination of T xy and Cly '"ie start

from the following generally accepted aSsuIDytions. ~he deviating of a body o-f fluid arriving at the }Joint (xy) from the point with ordinate y + ~y' is given by

Similarly we obtain

U f = 1. y ctu dy

aT T' = Zy Oy

(6)

( ? )

Substituting these eApressions in (1) and (2) we obtain

T = --f-Z 0-\1 - A du xy pv y oy - Oy ( 6 )

( 9 )

uhere A is the coefficient of t'.lrbulent apparer.t vis­COSity, A the coefficient of turbulent heat conductivity. From the equations

A = pvf ty

A = cppv l ty

L

4 lUCA Technical I:lemora.ncl'J.m lio. 1032

there follows directly the well-known equation of Prandtl

(10)

Substituting equations (8) and (9) in equat~ons (3 1 ) and (5") anc1 drOI)};,ing the averaging signs, since in \<!hat follows we shall deal only with mean values, we obtain

dt",

d i u2" p (JcT+-

2)'=

(It \ P dY

au \ y)

( , ,

o u 2 \ I

I'A ,·,.cm+-~I ... d:l \V :p - 2 ) _i

Thus the velocity and total energy u 2

Jc T + -P 2

( 11 )

satisfy the same linear differential equation. All bound­ary conditions may therefore be satisfied by setting

u 2 Jc T + - = au + b

p 2

where a and b are independent of was first derived by Crocco.

(13)

This relatiorl

We now proceed with our problem, requiring from the above relations only equations (10) and (13). We shall usc tho following notation:

u, T, ~, ..• the values of u, T, e, ... in the free flow u 1 ' T1 , 8

1, the values at the boundary of the lamine,r sub-

layer and t~rbulent layer, T*, p*, u* the values of T, p. ~ at the wall.

Equation (13) may then be rewritten in ~he forD

In the laminar sublayer the tangential stress Txy

and the heat transfer qy are deternined by the formulas

•

..

. 1

;

l

•

..

-.

RAGA.. Technical Memorandum No. 1032

1'xy == Ou

~oy

oT qy = ),,-oy

where ~ is the coefficient of viscosity and A the coefficient of heat conductivity. From the physical significance of Txy and qy it follows that they

5

should be continuous over the entire region of flow. (The schematic representation of completely laminar and co~pletely turbulent sublayers leads therefore to a dis-

continuity of eu dY and in passing from the laminar

to the turbulent layer since ~ is conSiderably less than A.. and Ales s than A).

As a first approximation we assume a linear distri­bUtion of velocity and temperature in the l~inar sub­layer. We then have

T h u

1 == !J..

(15 )

Tl !jl == 'A

q h - -* ...

(16 )

where T and. q are the values of Txy and qy directly at the wall and h is the thickness of the laminar sublayer. From the assumption of the continuity of Txy and qy it then follows that

(17)

(18 )

From equations (10), (I?). and (18) we obtain the relation

I

L

6 JAOA Technical Uemorandum No. 1032

oT ::= ~_ OU oy 1 Cp T oy

We now differentiate equation (14) with respect to y for y::= h and obtain

oe e-e c1..u ::=

____ l..

oy u -u 1 dy

1 1

( 20)

Substituting the value from (18) we obtain

Ja. u

1 .+ -- =

T

~_=_ .. ~l.. ~ - u 1

Finally substituting the value of and (16)

r:1

"'1'

( 21).

fr om (15)

there is obtained the following relation between q, T,

U, T, T * and u 1 :

q l-J. * '\ u :2 ul.;;:; u \ +--

A.* T 1) 2 2

Solving this equation for q we obtain

-- (U-U_\2 C (T - T ) + ----~-

P * 2J q = T

or (li - u )2 : T * + _____ .1. __ _

2J cp p u -----.------------* '. . u

. -1 +. ( P r - l)? u

..

. "

.. ,

.~

••

.. . . - .

eo

NACA Technical Memorandum No. 1032

CplJo Pr = -X- (= 0.8 ~or diatomie gases).l

The heat trans~er through unit area per unit time can therefore be found when the temperature in the free flow and at the wall, the velocity of the flow, the coefficient of friction, and the density at the vall are known. The velocity u 1 on the right side of the equa-

tion can be determined in the following manner. In an incompressible fluid according to the hypothesis of Ka.rma.n we have

where s is an absolute constant. According to the tests of Nikuradse it has the value

( 25)

s = 11.6 (26)

7

In a compressible gas we have the analogous relation

(25 ' )

where s may in general depend also on the nondimensional

paramet ers T

P and

Since, however, s enters only into terms that are comparatively $mall at not too high (in particular, sub­sonic) velocities we may as a first approximation use the value s = 11.6. From (25') we have

(27 )

so that equation (29) may be written also in the form

lIn all our formulas cp denotes the heat capacity of a unit mass at constant pressure. If unit weight instead of unit mass is used then cp is replaced by gcp •

8 NACA Technical ~emorandum No. 1032

I + (Pr - 1)8' j£ \f -2

CORRECT IO:;

( 25 )

These papers had already gone to ~ress when it wac pointed out to the author by 1:. Shirokov that according to the unpublished results obtained by him it is not correct to neglect the dissip~tion of energy in the laminar subla~rer. As a result the curve of temlJerature distribution in the laminar sublayer cannot be considered a straight line. Instead is obtained the ~arabola

T ==

3quation (23) above is therefore replaced by the equation

"T) -u 2 + (Pr - 1) 11 2 _______ 1

2J q==T ___ _

and similarly equation (12) in the second articlo is replaced by the equation

1 = 1 + 1r - 1 r r'l + s (p r - I)J - ------= 'Y [ ~;z - (p r - 1) s 2] •

- T 2k cr Pr

Correspondingly there is a change in the re~aining conclusions of our articles: in particular, thoro is a quantitative change in the dependence of the friction coefficient on Ba. We may also note that A. Guchman ancl his c01.'lorkers (Leningrad) found experisentally a considerable lowerinG in the friction coefficient in pipes with increasing Ea.

,

:

.'

' ..

.t

· ..

· . · · .

NACA Technical Memorandum No. 1032

FRIOTION IN THE TURscrLENT»OUNDARY LAYER

OF A OOKPRESSI»LE GAS AT HIGH SfEEDS

By,,! Frankl and V. Voishel

9

The Prandtl-Karm~n theory of turbulent friction, the s~called "Mischungswegft theory, is extended to cover the case of high-speed motion of a compressible gas with heat transfer from the wall. A correction of the K~rman fric­tion law is derived that depends on the Bairstow nunber (the compressibility effect) and on the ratio of the a.bso­lute temperatures in the free stream and at the wall. The analysis is based on the heat transfer law developed ina preceding article by F. Fr~kl.

The airfoil drag is made up of two part s, nanely, the drag of a plate of the SaDe chord and the -forn drag.' that is, the excess drag above that of the plate. The effect of the compressibility on these two drag components depends on a numper of fact ors. The form drag coefficient varies mainly with change in the velocity distribution about the airfoil while the plate drag coefficient depends only on the change in the fr i ct fon coefficient. In the present paper we shall investigate the effect of the compressibil­ity. on the frictional coefficient and also the effect of heating the wall, having in mind the case of the wing radiator. ~e restrict ourselves to the case of the turbu­lent boundary layer of a plate situated in a flow of con­stant velocity at zero angle of attack. For this purpose we generalize the new theory of the turbulent boundary layer, the so-called mix'in:g path (Uischungsweg) theory.

Before writi~g down the fundamental differential equations we pass to nondimensional magnitudes-by intro­ducing measures for the lengths, velocities, and gas state variables. The nondimenalonal velocities and state vari­ables as functions of nondil1{ensional coordinates will then depend on parameters which are nondimensional combi­nations of the measures, the<boundary condi ti ons at the plate, and the physical COllstal'l'ts of the gas. As boundary data ,ve may take the frictional stress T the heat trane­fer per unit time q and the Btii..~ e variabl es of the gas P. P"" ap!i T. at the plate. Whe pressure p, as

,;' "

.. .

10 "NAOA"TechnicaJ. Hemorandum No. 10~52

usual in ·th'e boun'a:ary layer theory. is ass-arued constant. i'le now introduce p* aI).d T* as measures for p and T.

As a meaSure of the velocity we shall take the "friction ve10cit;'rrl1

v* -V,,·/-:r-- -p*

and as measure of the length

"The-re l-L* is the value of the viscosit~r coefficient at the plate. By introducing the above measures the param­eters of the solution may now be only nondimensional COD­binations of the boundary data and the ~hysical constants.

~ . Together with Karman we aSsusa that only the end con-

ditions at points very near the wall produce an effect.

In the absence of compressibility and heat transfer such cOllbinations do not exist, as a result of which fact thero is obtained as is known a universal velocity dis­tribution not dependent on any parameters. We shall therefore in what follows denote these measures as "uni-versal. " The only such independent parameters are

T 'Y = :p

~ =

( ~ \ u J

and the magnitudeR depending only on the physical constants

I: = 52 ( 5) Cv

.. c-ofJ. PJ;" -- _.>.- ( 6 ) A.

",t

· ..

• · .. .. · ~.

ll.tOA Technical :iemo_randum No. 1032 11

For diatomic gases we have k:= 1.4 and the Prandtl number Pr = 0.8. (In the above cp ' Cv are the spec!£iC heats at constant pressure and volume. respectively, ~ the heat conduction coe£ficient o£ the gas, A* the value of ~ at the wall.)

. We shall now investigate the turbulent boundary layer of the plate. We shall take the edge of the plate for the x-axis; the variable velocity in the boundary layer will be denoted by u; the other notation is the same as for the preceding article "Reat Transfer".

We now assume that for the compressible ~as the fundamental differential equation of the Xarman turbulence theory remains valid:

where K is an absolute constant equal to 0.4 and T is the frictional stress at a distance y from the plate. When the coordinate x (tbat is, the distance from the forward edge of the plate) is sufficiently large, however, we may neglect the forces of inertia. ~ecause of this and from the constancy of the pressure p it follows that T does not depend on y.

We now proceed to the "universal ft measures and in­troduce the nondimensional magnitudes:

'11 = y.;p;T

(8 ) J.L*

u cp :=

v* (9)

p T* C' = :=

p* T (10)

Equation (7) then assumes the form

12 RACA Technical Me~orandum ~o. 1032

( 11 )

In order to obtain the differential equation for ~

it remains only to express a in terms of w. Ihis is possible with the aid of equations (14) and (21) of the prececling article. From these there is obtained an equa­tion analogous to equation (23) of the ~rece~ing article:

q = T ( \

U + u Pr - 11 1

which after substituting the universal measures is re­duced to the form

6 ' 1 = 1 + -' Cf' + s (Pr - 1) I C5 Pr '-

lr - 1 2 i ,.. .- ( 12)

where s = u1/v* according to the aboVe considerations

can depend only on Y, ~,k, and Pr. 1

Substituting expression (12) in (11) ~e obtain the differential equation for 9 which is easily solve~ in terms of elementary functions.

Io deteruine the boundary conditions we the hypothesis of linear distribution of u laEinar sublayer. This gives

U =

or

T

!-L*

cP = T)

and hen c e cp 1 = cp (s) = s.

'IT v

star t fr Or-1

in the

( 13)

------------------------------------------------------

lSee Correction on p. 8.

, ..

• ..

, ..

.j..

, . . ,

!iACA !j;echni cal lIe:oor andum No. 1032 13

As regards the value of O~/O~ in the turbulent layer for y = s (that is at the boundary of the laminar sublayer) we have

= f ( 14)

where f, like s, can depend only on Y.~. k and Pl'. In an inco:opressible fluid in the absence of heat trans­fer f has the constant value

f = 0.289 (15)

Our problem now is to find the coefficient of fric­tion $* (see preceding article) as a function of the Bairstow number, the ratio of the temperature in the free stre~ to that at the wall and the Reynolds nu:ober at a given point of the plate. that is.

where

( 17)

.J (= a. velocity of sound at the wall)

Instead of T*/T it is convenient in what follows to introd.uce the parameter w determined by the equation

= (18 ) T 2

In what follo'lrn:: 'IrTe shall assume that w is of the order of magnitude of unity. The physical significance

14 lJACA Technical Henorandum Ho. 1032

of this assumption lies in the fact that T* T is of the .same order of magnitude as the temperature differ­en~e that occurs with friction and without heat tr~nsfor. For wing radiators at large velocities this assumption may be considered as satisfied.

". It is now necessary to d6termine ~* in the forn

For this purpose it is necesse,ry first of all to ex­press tile paraLleters 'Y and v in the velocit;l d.istri-·­but ion la\v in t er:P.1.s of \jJ *; 3a* and. w.

For 'Y ;,'0 obtain i!Jilediat ely

k 'Y -- '-\1* 3a,..2 "2 ~

and for • fro~ equation (28) of the preceline article

J~ Pr-\j ~

\ 2 ~ _ --___________ ~--=_~. Ea 2

~ 2 * 1 + (Pr - 1)SV~

'->

r-

Iw / '-,- ,,\8l

I 1 ;'V:±:~ ': i (20) \ 2 ) ..;

We now substitute (19) and (20) in the solution of equations (11) and (12) for the boundary conditions (13) an(l (14). If this solution is expandeo. in a Taylor series in powers of 3a*2 and only the zero and first power terDs retained there is obtained

T1 = s + -------------Kf

-1),.~ (m _ cp)2-1-

L 2 l

n \- ( 21)

whero m, nand r depend only on In the case Ba* = 0 we obtain the velocity distribution.

w, Pr, $*, ~ and s. well-kno""i: locarit:li"i!ic

I~

:)

, , , . , -

'","

I I.

•

..

• . ~ ...

!rAOA Technical Eemorandum lro.· '1032

For the solution of tnis problem it is still re­quired that the velocity distribution in the well-known integral condition of Karman be substituted:

~

~ f pu (u - u) dy = - T dx

o

or in absolute measure

15

d~ == 2 (22 )

where

Oarrying out the integration of equntion (22) on the assumption that· f and s maintain t~e constant values s == 11.6 and f = 0.289 and setting

we obtain

~ == P1(z) + eZP2(Z)

where

e + + O~ ln z + §

~ + 04 E i ( Z) + $10 (w) E i (z - c».

)

p (z) == 9542 - 3134z + 246z 2 1

p~ (z) == 1.31 (Z2 - 4z + 6) ""

( 23)

16 NAOA Technical Memorandum 30. 10~2

•

s 0 ( w) -- 8 6 21 7 - 26 76 (w - 1) + 16 8 (w - 1 F'

1) - 38.0(w _1)2

s 2( w) == - [25353 - 4567 (w - 1) + 173(CJ - 1 F- J

S3(UJ) = 373151 + 4380(w-l)- 162(UJ - 1)2

S4 (w) = [175174 + 6211 ( ,,, ,.I - 1) - 222 (u - 1 r3 ]

8 5 (w) - - 1. 011 + 3.669 (u - 1) - 0.C55 (w - IF'

S6 (w) = 1 . OO~) 0.965 (c --1 ) + 0.105 (c - 1)2

S7(U) - Cl.131 + 0.131 ( I"~ • .1 - 1 ) - 0.0(;) ( (J -- 1)2

SeCW) = 86.187 - 1.723 (w - 1) - 0.048 (0 - 1)~

[144.71 + 3.26 (w - 1) + 0.51 (0 -1)2J

° -- 186231 8

0 3 = -58.416

--, (,,) ....: i ~ ..

z ( . eZclz

= I J Z

- co

We t~nu:; have an e(}_1;.:),ti0!~ in t:"lo fOl'1!l ~ == ~ (t *' 3a,tI<, Cj)

anc. Oi.'-r funclamel1tal l')roblem is solveci.. It reE"laiES to j::c:'te ()-rl_j t:''='t the ae-r ,e11ae-'1ee nf the "ite-r "olr1n Tu.Y1") 0" -:::0=

~·~~6j-v ... __ ;'L' ~::.- t;,"B~~~~,"-a:"d.J w • c~l/liJ::;:,~i~G- ~e~rC~-~J:.~l~r lou:::c: .•

TraLslati~~ by S. ~8iss, National A~viscry COffiillittoo for Aeronautics.

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