On the Plane Couette Flow of a Viscous Compressible Fluid with Transpiration Cooling

ON THE PLANE COUETTE FLOW OF A VISCOUS COMPRESSIBLE FLUID WITH TRANSPIRATION COOLING

J. L. BAI~AL AND N. C. JAIN [Oepartment of Mathematics, University of Jodhpur, Jodhpur {India)] Received Dr 13, 1973 (Communicated by Prof. J. N. Kapur, v..~.sc.)

Ansr~cr The exact solution for the plane couette flow of a viseous compressible, heat conducting, perfect gas with the same gas injection at the stationary plate and its corresponding removal at the moving plate has been studied. It is found that the gas injection is very helpful in reducing the temperature recovery factor. Effects of injection on the shearing stress at the lower plate, longitudinal velocity pro¡ and the enthalpy arr shown graphically.

1. NOMENCLATURE

E Eckert number, = (~ -- 1 ) M| ~ TI . T1 -- To b distance between the plates; h* defined by equation (30); H enthalpy, = cpT ; H* = H/Ha ; Ha* dimensionless adiabatie wall enthalpy; M,, Maeh number ; P pressure; p* dimensionless pressure, = p/plU ~; Pr Prandfl number ; Ha* -- 1 rŸ recovery factor, = (7- 1)M,, ~ ;

A l~July, 74 2 J. L. BANSAL AND N. C. JAtN

Re Renolds number, = piUb//zx ; T tcmpcrature;

U vclocity componcnt along the x axis;

U* dimensionless axial velocity, = u/U;

U vclocity of the moving platc;

q) vclocity componcnt perpendicular to the x-axis dimensionless trartsverse velocity, = v/U ;

X axial distance from the entrance of the chanrtel;

Y perpendicular distance from the x-axis; Greek Symbols :

7 isentropic exponent, = c~/cv ; dimensionless transverse distance, = y/b ; injection parameter, ---- povob/tz~ ;

tz viscosity;

t. dimensiouless viscosity, = /~/tzl ;

P density ; p* dimensionless density, = p/pl ;

T o dimensionless shearing stress de¡ by equation (16);

Subscripts : 0, 1 values of the variables at the stationary and moving plate respectively.

2. INTRODUCTION IT is well known that mi exact solution of the Navier-Stokes equations exists for a viscous incompressible fluid in the case of plane Couette flow. This solution has been generalized by Nahme, ~ Hausenblas 8 and Groff 4 when the viscosity of the fiuid depends on tempemture. The further exten- sion to a compressible fluid was given by Illingworth5 and Morgan. ~ Aja Plane Couette Flow of a Compressible Fluid

important problem in the development of missiles, satellites and spaceships is the excessive heating of the skin of these vehicles by friction in the high velocity air stream. A method which appears very effective is called transpiration cooling. Although the nature of the problem is purely of boundary layer, effort has also been made by Eckert 7 to describe the reduction of heat transfer by Couette flow in the case of an incompressible fluid by injecting the fluid into the flow field from the stationary plate and corres- pondingly removal from the moving plate. Moreover, the Couette flow model proved to be very helpful, as has been reported by Rannie 8 and Mickley,9 to describe the conditions in the laminar sublayer of a turbulent boundary layer. Since at high speeds the compressibility of the gas should also be taken hato account and the viscosity of the gas depends on temperature the corresponding Couette flow needs ah investigation.

In the present paper an exact solution for tho Gouette flow of a viscous compressible fluid with the same fluid injection at the stationary plate and its removal at the moving plate has been studied. The assumption made in the present study are (i) the specific heat of the gas is constant (ii) for solid walls the Prandtl number Pr is constant but arbitrary (iii) for porous walls the value of Pr has been taken as 3/4, which is ah important special case since it is very near to the value of air and (ir) the viscosity of the gas is directly proportional to the temperature.

3. STATEMENT OF THE PROBLEM

Let us consider the two-dimensional laminar flow of a viscous compressible fluid between two parallel plates, one in uniform motion and the other at rest with uniform injection of the same ¡ at the stafionary plate anda corresponding uniform suction at the moving plate. The axis of x is taken along the stationary plate and y is measured at right angle to it. In such a case all variables of the gas motion can be taken as timctions of y oniy. The equations of continuity, momentum and energy therefore are:

~y(pV) = O, (1 a)

of pv = constant, (1 b)

pv du~ = dy(/Z~)d , (2) 4 J.L. BANSAL AND N. C. JAIN

p~~=-~+~d~ dp d(43~'~ dv) , (3)

1 d(dH) 4 (dv~' (du\ 2 dH dp (4) Pr dy tZ -dy +3/z\~/ +q = PV -df -- v -- and the equation of state is 1 p = p Y- H, (5) where u and v ate the velocity components along x, y directions respectively, H = cpT and the other symbols have their usual meaning.. The boundary conditions are

y =0: u =0, v =yo, H =Ho, p =po, .) (6) y=b: u=U, v--v~~176H =Hz, p =p~.~ Pz

The normal velocities v0 and vopo/px are prescribed in such a manner so that the equation of continuity (1) is satis¡ i.e.

pv = constant = Poro, (7) Let us introduce the following non-dimensional quantities:

u v* = v ~ H p, = u*=o' O' '7= , H*=~, m'

tz* = t~ p* = P ;t -- P~176 (injection parameter), /zz ' P~ ' /-

Re- pzUb (Reynolds number), q where bis the distance between the plates, U the velocity of the moving plate and the subscripts 1 and 2 denote the values of the variables at the stationary plate and at the moving plate respectively.

Equations (1) to (5) in non-dimensional form are:

p* v* = R--e' (8) du* d du*'~ (9) a --g~-~ = it~ ( t~ * ~d~ s ' Plane Couette Flow of a Compressible Fluid 5

~~~d,/ =--Re--+~%" ~ (~~.~~'~a~71' oo>

d [Pr 1 aH*] 4/~, (dv*'~ 2 (au*'~ ~ (r -- 1) M., 2#* -d~-~ .! 4- ~ k--d~-] +/~* \-~/

_ ,~ dH* Re v* dl)_* (11) (~, -- 1) M,** d~/ d~/ ' and p, = p* H* 9,M** z , (12) where

U ~ = O' -- 1) M. ~" (13)

The boundary conditions (6) reduce to =0: u*=0, v*=vo*, H*=Ho*,) (14) =1" u* =1, v* R-~'~ I-I* 1.1 4. ANALYSIS Equation (9), on integration, with proper boundary conditions yields

i~ , du*d~7' = ~u* + ~'o*, (15) where du*~ o* = (~* --~ ]~0' (16) is the dimensionless shearing stress on the stationary plate, to be determined. Now eliminating p* between equations (10) and (11), we get d 1 /~, dH*] tdv *'2 (du*'~ ~ d~- [Pr (7, -- 1) M| ~ --~-~ .] + ~ ~*t,~) + /~* ~~/

-- ~v* [d~*~k- ]

ffi ~ di'I* -- v* d (4 dv*'~ (17) (7'-- 1)M"~ ~ ~ k3#* -~ J" 6 J.L. BANSAL AND N. C. JAIN

Taking the help of (1'5), the equation (17) can be written as

d[ 1 dH* 4 , ,dv*] Pr(9"- 1)M. '~* d~ + ~~ v ~j [ H* v**1 -- a Cd (V -- 1) M** 2 + -2-J

*'~ du ~ =--(au*+-o ) ~. (18)

Integrating equation (18), we get

~ 4Pr(9,--1) M**~ + --A (y_ 1) M**2q- ;~u*Z = A 2 %* u*, (19) where A is a constant of integration.

Case I.~For ;~ = 0 (i.e., in the absence of perosity) and for arbitrary values of the Prandtl number Pr the equations (15) and (19) reduce to

tz , du* -d-~ = ~~ (20)

and ,dH* /z -~-~- = Pr (~, -- 1) M| ~ (A -- r0* u*). (21)

Equafion (21) with the help of (20), becomes

dH* Pr (V 1) M*** (A' -- u*), (22) du ~ ~-

where A' is now the unknown constant.

Integrating the equation (22), aad applying the boundary conditions on u* and H* from (1:,4), we get

Pr H* -- Ho* = (1 -- Ho*) u* q- ~- (9' -- 1) M.. ~ u* (1 -- u*), (23) Plane Couette Flow of a Compressible Fluid 7 or in terms of temperature

T To -- u* + ~ u* (1 -- u*), (24) T1 -- To g., where

E = (r -- 1) M,,2 TI (Eckert number). (25) Ti -- To

One can easily see that for ah incompressible fluid tx* = 1 and therefore from (20), with proper boundary conditions, we get

u* = ~/ and ~'o* = 1. (26)

In this case the expression in (24) reduces to the well known result for the temperature distribution, Schlichting, ~ when the frictional hea-t is taken into account.

Fora compressible fluid let us considera linear relationship between t~ and T, i.e.

tz * ----- H*. (27)

Therefore from (20), using (23), on integration, we find u *~ Pr /'u .2 ~~'o* = He*u* +(1- Ho*) ~- -4 -2-- (y-- 1) M**~k -~ ~~), (28) since at

~7 =0, u* =0.

Now, in order to find out ~'0" we apply the condition ~ = 1, u* = 1 in (28) and find Pr "ro* = 1[1 + H0*+ ~ (,-- 1) M.. ~] , (29)

Further details may be fourtd in Liepmann and Roshko. lo 8 J.L. BANSAL AND N. C. JAIN

Case H.--A q: O. In order to get a compact solution of the equation (19) we make an assumption that Pr = 3/4 which is an important special case, sinee it is very near to the value for air.

Let

H* V*~ = h*. (30) (y--1) M.~ 2 + -~-

Hence equation (19), with the help of (15), becomes

dh* 3 A* 3 (C -- A* u .2 --2 u*) (31) du* 4 A* u* +~1 h* ----" 74 (A* u* + 1) " wh~e A A* ~" **'0 (32) and C is now an unknown constant.

Equation (31) is a linear differential equation and its solution with the boundary conditions (14) is

h* = aa (A* u* + 1)s/4 _ a~ (A* u* V 1)2 _ ax ,... (33) where __ hi* -- ho* (A* + l)S/4 __ a~ {(A* + 1)8/4 _ (A* + 1)*} -1 -- (A* + 1)s/,_ 1

as =ho*+al+al

~-~o~ ~o ~~ ho*--(?_l) M.l+ 2 ' and 1 hx* = (7'-- 1) M.. s + ~ (~).), t Plane Couette Flow of a Compressible Fluid 9

Now in order to determine the distribution of transverse velocity, we inte- grate the equation (10), 4 ,dv* # ~=Av*+Rep*§ (35)

where D is a constant of integration.

Equation (35), with the help of the equations (12), (8), (30) and (33) may be written as 4 (A* u* + 1" dv*

Y i 1 = A'v* + v* 7 las (A* u* + 1)3/, _ a2 (A* u* + 1)2

-- aJ" -- ~-~-'] -)- I)~'o*' (36a)

O1"

3 ( 1-~ )v,~.+ . __{aa(A,u,+l)a,a_a,(h,u,+l)2 al} § do* 2 i* 27_ 1 A, 7--1y -ro* J du* -- 2 (A* u* + 1)

(36 b)

As no exact solution of the equation (36 b) exists it can be integrated only numerically. Applying Picard's method of successive integration, we lind

v *~ + )t* 7 -- ! {aa (A* u* + 1)sn _ as (A* u* + 1)~ -- al! 7 v*s = Vo*S -Ÿ '.S (,X* u* +'I')

'ro@ du* (37) + (~* ~176u* + 19 10 J.L. BANSAL AND N. C. JAtN whose first approximation is v*2=v0*2+~3 [--~~[y q- 1 v0.2 In(A'u* + 1) q_ V Y 1_i4 aa(A*u* + 1)s:4--72a21 (a'u* + 1)'

--alln(,~*u* + 1)} + D---v~ ln()~*u* + 1)--y--1 {4as_ ~}] (38) "ro* 12" y ' Now applying the boundary conditiop.s (14), we finally have v .2 = a7ln (A* u* q- 1) -}- a6 -- a5 (A* u* + 1)* + a, (A*u* + 1)s/,, (39) where

a4=2azV-- 1, 7 3 y--1 a5 =7~a~-- 7 , as : Va *~ -- a4 + as,

-- a6 q- a5 (I* 1)2 _ a4 (A* q- 1)a/4 a7 = In (a* + 1) (40)

Although second and higher order approximations can be obtained in a similar manner we shall confine ourselves only to the first as the analysis becomes increasingly complicated. We have yet to determine the longitudinal velocity distribution and for this we again consider the assumption (27). Equafion (15) in view of (27) and (30), may be written as

(y--1) M,,' [h* -- V;'] du*~ = Qt* u* + 1) ~'0". (41)

Substitufing the values of h* and v .2, from equations (33) and (39) respectively, in equation (41) and on integration with the boundary conditions (14), we find ~~'o* (r -- 1) M** 2

= ~-all {()t* u* + 1)s/, -- 1} _ alo~ {('~* u* + 1)* -- 1}

-- a9~ {In (A* u* + 1)}2 _ ~as In (h* u*. + I), (42) Plane Couette Flow of a Compressible Fluid 11 wherr

ae av as z al q- ~-, a9 =-~,

1 (43) alo=-2 (a2--~), all = ~

Now, in order to determine the dimensionless shearing stress %* at the stationary plate, we apply the boundary condition ~ = 1, u* = 1 in (42) and fmd

"ro* 1)3/4 (~' _ l) M| ~ § ~,x {(A* § _ 1} _ alo {()~, § 1)2 _ 1}

a9 A* {In (A* § 1)}~ -- ~, In (A* q- 1).

Thus fora given values of A*, M**, ~, v0*, A/Re and H0* the value of ro* can be calculated from (44) and then (32) will give the corresponding value of the parameter A from which foUows the value of Re. The longitudinal and transverse velocity profiles can be calculated from equations (42) and (39) respectively. After knowing u* and v* the enthalpy distribution will be given by (30) and (33).

It can be easily seen that for an incompressible fluid, (p*= 1, t~* = 1, i.e., H* = 1)M,, ~ 0, from (30) and (33)

Lim al (~- 1)M** ~ = -- 1, (45) Moo-I, 0

Therefore, from (43) and (44), we get ~qA* To* = In (A* u* + 1), (46) and A* ~o* = In (A* + 1). (47)

Now, in view of equation (32), from equations (46) and (47), we get e~x- I A u* = e-x _ 1 and 1ro* ---- e-x _ 1 , (48) which ate in conformity with the results obtained by Eckert, ~ 12 J.L. BANSAL AND N. C. JArN

5. RECOVERY FACTOR The recovery factor r iis defined as

Ha*- 1 (49) rt = 89 t) M- 2' where Ha* being the dimensionless adiabatic wall enthalpy.

Case L--~ ---- 0 and for arbitrary values of the Prandtl number Ir, we ¡ from equation (23) that (bH*/aV),~.0 = 0 if

Pr Ha* = 1 + ~ (7- 1)M,, 2, (50) where Ho* has been replaced by Ha*.

I-Ir ry = Ir. (51)

Case II.--a~ 0 and Pr = 88 From equation (30), with the help of equations (33) and (37), we fmd that (aH*/b~/),l=o = 0 ir 89[a7 -- 2a5 + 88a,] = [ 88a a-- 2a21. (52)

Substitutmg the values of a's from (34) and (43) in equation (52) we will get the value of Ha* and hence the recovery factor in th~ case, is given by

ry[7~ 1 1 3 1 ] In (a* + 1) + 7t 7 {(a* q- 1,)3/4 _ 1} 3 (~~-1 --1~ 7 ,)

v~ § 4~, I0)~ *~ (~)~~,+1~~ 1~] In (~* + 1)

~.~~11( a V - a,{(a* + 1)'- 1}]

7-- 1 t 3 1 ] (53) [ 7 In (a* + 1) + a, y {(a* + 1)s/, _ 1) Plane Couette Flow of a Compressible Fluid 13

6. NUMERICALDlSCUSSlON The calculations have been carried out for air for which ~, = 1.4. The Prandtl number Pr is already ¡ in the analysis and has the value 0"75. Taking )t/Re = 0.01, Ho* = 1 "0, yo* = 0.001 and M** = 2.0 the method adopted is as foUows:

(i) We have calculated first the values of a's, h.~* and ho* for various values of ~,* and then evaluated %* from equation (44). Once for a given value of )t*, To* is known, the corresponding value of )t is obtained from the relation )t = ~* 70* [cf equation (32)] and are given in Table I.

TABLE I

)~*-+ 0 5"0 10"0 30"0 80'0 A -+ 0 1"9669 2"6279 3"7482 4"7025 ro*-+ 1" 1 0"3934 0"2627 0" 1249 0"0588

(ii) The velocity distribution across the channel, i.e., against ~ is easily determined from equation (42) for a given value of )t* or A.

(iii) For a given value of )t* or )t the value of h* for different values of u* or the corresponding ~/is calculated from equation (33) between 0 and ls. The temperature distribution then follows from equation (30).

In Fig. 1 the dimensionless shearing stress is plotted against the injection parameter )t for the values given in Table I. For the sake of compari- son a curve from Eckert's analysis 7 for incompressible flow is also drawn. It is observed that the compressibility of the fluid increases the shearing stress on the stationary plate, which goes on decreasing as the value of injection parameter A increases and tend to zero asymptotically.

The longitudinal velocity profiles are plotted against ~ in Fig. 2 for various values of the injection parameter )t and are compared with the pro- filr of incompressible flow. It is being noted that except )t = 0 the dimensionless velocity u* in the compressible flow at any point between the plates is more as compared to the incompressible flow and this difference incr,:ases with the increase in )t, wherr for )t = 0, it is less near the upper plate. 14 J.L. BAlqSAL AND N. C. JArN

I'2 t'1"~ I'0

--M.= ,,2"0 t \\ -0-0 0'4

0"2

0 I 0 I'0 2'0 3"0 4"0 5.0

F[o, 1. The variation of the shearing stress plotted against the ª162 paramr for various values of the Mach numbr I'0

(7) r, - - - 1Eckert 0'8 I. Yi /Ÿ ~ ~' =0"0 = I' 9669 0"6 = 2" 9467 -- 3"7482

~f 0"4 q

0'2 1 ,4. J /~f Ÿ ~ 0 0 0"2 0"4 0"6 O'B I'0

Fro. 2. The vclocity distribution pl~tted against the perpendicular disiance for various values of the inj(r parameter h. Plane Couette Flow of a Compressible Fluid 15

In Fig. 3 the ertthalpy is plotted across the secfion of the channel when the walls ate kept at the same temperature (H0* : 1-0), for various value, of the irtjection parameter )t. It is seen that the enthalpy first increases

I-2

I'15 --Ir = O ~~~- =1-9669 TI 2" 9467 H* I'1 -3-7482

I" O~

I-0 0 0.2 0"4 0.6 0"8 I'0 0"6 7/ , -.~

FIG. 3. The enthalpy distribution plotted againstthe perpendicular distance from the stationary plate for various values of the injection parameter ,~.

0"8 0"75'

0-6 1o.4 i

0"2

0 0 I'O 2"0 3' 0 5"0

FIG. 4. The variatiort of the temperature recovery factor plotted against the injeclion parameter X. 16 J.L. BANSAL AND N. C. JAIN attain its maximum and then decreases. Moreover, the maximum value of the enthalpy increases as the injection parameter ~ increases and the maxima point shifts towards the upper plate. For ;~ = 0, the maximum enthalpy occurs in the middle of the charmel.

One of the important characteristic parameter of the compressible flow is the temperature recovery factor rf, which has been plotted against ;~ in Fig. 4. It may be observed that a considerable reduction in r s even for small fluid injection at the stationary plate and its corresponding removal at the moving plate takes place. For ;~ = 0, rf = Pr which is well known.

REI~~NC~

1. Sohlichting, II. .. Boundary Layer Theory, New York, Mr Book Company, Inc., 1968.

2. Nahme, R. .. "Beitdtge zur hydrodynamischen Theorie der I.agerreibung, Ing.-Arch., 1940, 11, 191-209.

3. Hausonblas, H. .. "Die nicht isotherme str6mung einer z~ihen flª durch engo spalto und kapillarrohren," Ibid., 1950, 18, 151.

4. Groff, H. M. DE .. "'On viscous heating," JAS, 1956, 23, 395-96.

5. Illingworth, C. R. .. "Some solutions of the equations of flow of a viscous com- prossible fluid," Proc. Cambr. PhiL Soc., 1950, 46, 469-78.

6. Morgan, A. J. A. .. "Oj the couette flow of a compressible viscous, heat condur i~ng, perfer gas," JAS, 1957, 24, 315-16.

7. Eckort, E. R. G. .. Iteat and Mass Transfer, New York, McGraw-Hill Book Company, 1959.

8. Rannio, W. D. .. "A simplified theory of porous wall cooling jet propulsion Lab.," Progr. Rept., 1957, pp. 4-50.

9. Mickely, H. S. et al. .. "Heat, mass and momentum transfer for flow over a flat plata with blowing of suction," NACA.TN., 1954, p. 3208.

10. Liepmann, H. W. and Elements of Gas Dynamics, New York, Jobn Wiley & Sons, Roshko, A. Inc., 1956.