A study of the structure of the contact region has been made taking into account the effects of viscosity, heat conduction and radiative heat transfer. Analytical solutions for the temperature, velocity and pressure distributions in a uniformly moving contact region have been obtained under the optically thick-gas approximation when the thermal conductivity and absorption coefficients are given by power laws. Applying the analysis of the contact region to the situation when a plane shock is reflected from a plane heat-conducting wall it has been shown that the reflected shock is attenuated due to the combined effects of molecular heat conduction and radiative heat conduction

Pandey, B. C.

English

Defence Science Journal

STRUCTURE OF THE CONTACT REGION AND ITS APPLICATION TO THE REFLEXION OF A PLANE SHOCK WAVE FROM A HEAT-CONDUCTING WALL B. C. PANDEY* Indian Institute of Science, Basgalore (~echived 20 August 1969 ; Revised 23 January 1970) A study of the structure of the contact region has been made taking into account the effects of viscosity, heat conduction and radiative heat transfer. Analytical solutions for the temperature, velocity and pressure distributions in a uniformly moving contact region have been obtained under the optically thick-gas approximation when the thermal conductivity and absorption coeffisients aregiven by power laws. Applying the analysis of the contact region to the situatiop when a plane shock ia reflected from a plane heat-conducting wallit has been shown that the reflected shock is attenuated due to the combined effects of molecular heat conduction and radiative heat conduction. A contact region can occur in compressible fluid flow in different physical situations. In ideal gas theory this is regarded as a discontinuous front across which density, tempe- rature and entropy undergo jumps but the pressure is the same on both sides of the front so that it moves with the fluid. But in fact it is, hke a shock layer, a region of small thick- ness where dissipative mechanisms are effective. Goldsworthyl studied the structure of the contact region on the basis of the concepts associated wiih the Prandtl's boundary layer theory. He showed that the pressure across it remains constant approximately. Hall2 ih his study of a uniformly moving contact region, experimentally verified that very little pressure change occurs across it. Goldsworthyl applied his analysis of the contact region to the situation when a plane shock is reflected from a plane heat-conducting wall. He showed that the shock is attenuated due to the presence of the contact region adjacent to the wall, in which the effects of viscosity and heat conductioa were considered. b Sturtevant & Slachmuylders~ have experimentally measured the position and the velocity of the-reflectdd shook and found that Goldsworthy's approach to the problem of shock reflexion from a heat-conducting wall is in a quite reasonably agreement with their observations. Further they concluded that temperature jump effects a t  the interface are not significant when the reflected shock is more than ten shock thicknesses distant from the wall. Baganoff's4measurements of the pressure rise a t  the wall are, as he states, not conclusively in favour of Goldsworth$s results, neither are they seriously at variance with it. The cause of the uncertainties is attribbted to the inherent difficulties of pressure measurement on sub-microsecond scale. Clarke5 has included the temperature jump effects in his study of the problem using the technique of matched asymptotic expansions and his results agree reasonably well with experimental observations. He concludes that the temperature jump effects may account for some observations of reflected shock tra- jectories which, according to Sturtevant & Slachmuylders3 is attributed to experimental inaccuracies. Goldsworthyl considered the effects of viscosity and heat conduction in his analysis of the structure of the contact region. I, following him, has studied this problem including radiative heat transfer under the optically thick-gas approximation. To *Present Address :-Department of Mathematics, Bihar Institute of Technology, Sindri, Dhanbad, Bihar. L 143 144 DEF. SUI. J., VOL. 20, JULY 1970 illustrate the modified theory, which incorporates the general problem of an accelerating " contact region, I have obtained analytical solutions for the temperature, velocity and pressure distributio~s in a uniformly moving contact region when the thermal conductivity and jabsorption coefficients are given by pdwer laws. Under the caption "Structure of a Uniformly Moving Contact Region", I have obtained analytical solution to the problem for any ratio Ta/Tl of temperatures. A-series solution has been developed for the case when the ratio (I;-T2)/Tl is much less than unity. It is found that the effect of radiative heat transfer is, under the thick-gas approximation, similar to that of molecular heat conduction as expected. Under the eaption "Normal Refle$o~ of a Shock from a Heat- Conducting Wall", I have applied the theory of the contact region to determine the flow set up when a plane shock is reflected from a heat-conducting wall. I have found that the combined effects of radiative heat conduction and molecular heat conduction decrease the strength of the reflected shock. E Q U A T I O N  G O V E R N I N G  T H E  F L O W  I have considered e uniform gas of infinite extent which is initially at rest and in the region y < 0 it is heated a t  a rate depending only on the distance y from the fixed plane y = 0 and the t i e  t. This situation involves a shock wave which is propagated into the non-heated gas. The shock wave is followed by a contact surface which separates the heated gas from the non-heated gas. If viscous and heat conduction effects are neglected, the motion and cond,itions at  the two sides of the cohtact disconti- nuity can be determined given the rate of generation of energy in the ffuid. It is assumed that this 'ideal-gas' solution is known. The one-dimensional gas-dynamic equations including radiant heat transfer but disregarding the radiation energy and pressure are p = RPT, (4) where u is the fluid velocity, p the pressure, p the density, T the temperature, Fr the radiative flux in the positive y direction, y the distance measuted from the fixed plane y = 0, yo ( t )  the position at  time t  of the particle initially at  the origin, H (yo--) the Heaviside unit function which is zero for y > yo (t), Q (y, t )  the rate at  which energy is generated per unit mass of the fluid, p the coefficient of viscosity, kg the coefficient of thermal conductivity, c, the specific heat at  constant pressure and R the gas consb&nt. \ Since the contact surface moves with the fluid equations '(1) to (3) are rewritten in the Lagrangian frame of reference for the sake of convenience P ~ E Y  : Structure of the Contmt Region 145 where x is the initial position'of a particle which is at  position y at  time t ,  6 the initial constant density of the gas and a,h and i are the new independent variables. Writing equation (5)  in the form " and differentiating (8) with respect to t keeping i,b constant, the velocity u of a fluid par- ticle is found where u = (%)+ and uo ( t )  is the velocity at lime t of the particle initially at the origin. Writing the actual pressure and gas velocity i i  the contact region in the form and substituting (10) in (9) a ~ d  (6), we obtaih * ' 5  where P, and U denote the 'ideal-gas' solutions for the pressure, density and velocity. We use suffices 1 and 2 to label quantities in the heated and non-heated parts, of the gas adjacent to the contact region respectively. We assume that the thickness 6 of the contact region is small and neglect dpl ( which is 0 f ( I - 5) 8 3 ) and the uis- c* cous term in equation (7)  following Goldsworthyl.' Then the pressure in equation (7)  can be replaced by the ideal gas,pressure P, (t)  evaIuated at  j = 0. He1 also replace Q ($, t )  by Qo ( t )  on the assumption that the rate of generation of heat in the fluid is not too strongly dependent upon the temperature. Now the equation (7) becomes 140 - ' Dm. SOI. S.i-Rori 20, JWLY 19W * Under the optieally thich-ga? approeat ion,  . , -  -.. & , > *  -." , > , .  M 1 6 ~ ~ ~ -  aT F T = r .  - ' L *3kr . -  - w  * (14) wS&, on using (5),. takes the ,form . . L 16oPp3 8 ~  F , . = - - -  - ,  3% a$ (15) where kr is the absorption coefficient ~ e r '  unit volume and a the Stefan-Boltzmann constant. - .   - - We solve (13) subject to the boundiq conditions T  = Tl at / = - a, T  = T2 at + = a, where Tl and TB are, the temperqture~ of the gas on either side of the contact disdontinuity and they, ' using the relation (16)) satisfy the equations 1 (Y-1) EFP, -- -' Qo (t) db yP0 .  * dl T 1 = - - -  OP " (16) \ . - dTa - (Y-1) ' @ o  ' dt YP" ' dt T 2 = 0 .  ,. . .  . ' .  (17) - putting T  = TIB (/, t) or T  = T10 + T2, the equation (13) using (16) and (17) reduces to r 3 , - (18) Making use of (16) in (18) we have 8 * .  - Equation (19), in general, can be solved numerically once the laws governing the variatiws of kj and k;, are prescribed. In what follows we have solved (19) analytically when kg and kT are given by power laws.' Once the temperature is known as a function of $.and t, the positign y oy a particle is determined from equation,(8) whicli can be written J By using equations (11)) (12)) (16), f-l8).and. (20) the expressions for the velocity and pres- sure are obtained. , ,  PAXDEY : Structure of the Conteot Region . % * .  147 < - where P, is the Prandtl number, f ( t )  a ~ d  g (t) are arbitrary functions of time, Y is the position of a particle at  ti& t giverby 'ideal-gas' flow theory and. y the corresponding position of the particle when the effects of viscosity, heat conduction and yadiative-heat conduction are considered. The functions f ( t )  and g ( t )  are determined by considering the effect of the contact region on the external ideal-gas flow. For a uniformly moving contact surface which we have considered, aT dPO 0, - + 0 at the edges of the contaat -a- = at kegion. As such, in this case, it follows from (21) and (22) that u +- U and p += P at both-edges of the contact region iff ( t )  = 0, g ( t )  3 0. . S T R U C T U R E  OF A UNIJ?OR&XLP M O V I N G  C O N T A C T  R E G I O N  Si~ce  the contact region is assurned,to move witb constant velocity, . the ratio T2!T1 of temperatures across it remains constant. It is assumed that k, and kr vary according 1 to the power laws. < *, kg = clT, kr = c , p P  (533) where cl and c, are constants. By making use of (23) and T = Tl 6 eqnatio~ (19) reduces to which in terms of the similarity variable defined by . a , '  ' . ' v = *l(tP)* p) where I . '  * J  ! 148 DEZ. 801. J., VOL. 20, JULY 1970 transforms t o t  - .  = 0. (27) i '  Equation (27) under the boundary conditions - .  . . > I e ( - m ) = l ,  e (00) -, - T l  admits the solution "-) e. (+). (28) dP0 Equations (21) and (22) o n  wing (28), - = 0, at f ( t )  = 0 and g (t) = 0 . *- * ," ' give t& velocity and pressure distributions: in a uniformly rno-vingieontact- respectively. Series @oluticrrzs It is assumed that Ic, and kT vary according to pwer  laws given by kg = kl T, kr = k2W2, (29) 8 4  in order tct. obtain analytical solution, the ratjo of temperatures across the contact region has to be restricted. It is assumed that 1 Tl -:5 << 1. . T l  ' By making .use of (29J and T = @'+ T$ tbe equation (19) reduces to , a e - =  at - - The boundary conditions are where e =  €80+"62&1&. .'. . (32) On substituting (32) in (30) and on equating coefficients of equal powers of r we obtain PAND~Y : Struotuce of the Cbdttao~Be~ion ' 149 The boundary conditioas (31) with the help of (32) take the. for& (i) 80 (00) = 8, (00) = 0, (ii) 80 (- w) = 1, 8, (- a) = 0 1 (35) < Zeroth Order Appodmatim - " The zeroth order equation (33) in terms of the simiBrity.v&iable defined-by where @ ,  Pok1 1 6uT8 a=- + ------ , cpB 3k2cp (37) reduces to aaeo 1) ct8 a+ + a + = O , '  - -  - (38) which under the boundary conditions 0, (00) = 0, eo (- 00) = 1, admits the following solution - - " - 8, = ( 1 ~  erfc (5) . (39) Erst  Ordw Approximation, i The first order equation (34) by using (35) reduces to I.e, p a81 = - 16~2'~- h2 2 37i.@,Pl (40) which by using (39) and the boundary conditions admits the following solution - ?(Ti= $ [l - ( erf (+)i2 + J q erfc (+) erf (+) dq - m where . . .Isur, &=--. - - , 3k.2131~~ N O R & A &  R E F L E X I O N  .OF S H O C K  .&OM A H E A T  CONDUCTI IN^ 'WAL-L - 1 1 ,  It is assumed that at  time t = O, rt plane shoe& gf given strength is reflected fro111 the face, 3: = 0, of a wall which otxmpies the regton y > 0.. If viscosity, heat  ond duct ion and radiation heat conduction are neglected, the veloci%y 'U8 of the reflected shock han be - 160 .DBF. S a t  Z., VOL. SO,, J ~ L Y  1 9 7 0  determined. A<contact region near the face of the wall exists which influences the external idd-gas flow. fn w b t  follows we determine the distribution of thmpertiture, presswe and velocity in the'contact region and thereby show how the shock is affected by the conducting wd. We use the subscripts 2 and 3 to denote .&e flow variables ahead of and behind the reflected shook respectively. . - . ,  . . .  In the gas (in the region y < 0) adjacent to the wall, the" temperature satisfies equation (19) which we rewrite in the form which for kg = qT and hT = 4 P  Tq reducgs to In the region y > 0, the temperature.of the solid satisfies the diffusion equation where kw is the thermal conductivity, Pw the density and cw the specific heat of the wall. The initial temperature of the wall is denoted by TI; Equation (43) nnder.the boundary condition 2'+T,as$-+-co, admits the qolution where and B is the constant of integration. Equation (44) under the boundary condition ." T + T l a s y + f c o ,  admits the sohtion where and A is the constant of integration. Further, by using the condition the constants A and B are'&te&ed where I .  The velocity distribution in the gas is determined from (21) by using the condition that the particle velocity must be zero at  the wall, hence ' where T is given by (45).  aT *Outside the contact region - + 0 and theref&, the velocity there is ad =-  (Ts - TI)., (P4-14 '(m -I? 1) V)! ' Expression (22) for pressure contains the dn@own function 9 ( t )  which is determined on the assumption that the efiect of the contact region on the external ideal-gas flow is small. Substituting in the ideal-gas flow equations and on neglecting the squares and higher powers of the perturbed quantities the wave-equation for u' which admits the solution where A, is the isentropic sound speed behind the reflected shock. The perturbed pressure p' is given by P' = - wsA3 [ F (4 + 9) - G.(A3t - y):]. (a) &king use of the boundLGcondition (52) a t  the edge of the oontact region (at the = 01, we obtain whese At the unpeptmbed shook poaitioa given By y k -- U,b /Us shock speed); the vebci ty and pressure perturbations are related by the equation where . . and 2M8 - M' ' 21 [See Appen*, eqn. (lo)] - ' )= [(3y - 1) M z  f (3 - y)] . . - Substituting expressions (54) and (56) in equation (57) we obtain . . A - whicB' ban be put in the form- ' 3(&)~=--iW6!(hS), \ whqe I is a variable, ' N =  I 1++i&f.dT 1-4 llw - .  , - .  I * &  \ n d  . . . A=-=. A -  - We put E = A3t in (56) and on substitutingfor P (6) from (59), we obtain .the equa- tion for G (6) #- 1 - + d (ti = .- z ( BS,{ j~ , (60) which has the soiuti& 1; '(= {N/(h)4 ] - l <+%/@ * (61) - .  . . Making use of (59) we bbtain . .  _ F ( f )  = -4 - 'N .. - (A&)* At l ( ~ l ~ ~  - 2 - (62) -. 'Expressions (54) and (55) on using .(61) and (62) are - - It is noted that the function 9 ( t )  in (22) is equal to the perturbed pressure at the edge of the contact region, so that from (64) and (22) we- have From the perturbed shock squa%ions we obtain the perturbed shock speed Us' . l . J ; = # ( M 8 ) % ' ,  (66) where ,(y - 1) hi: f 2 Y + l  4 (Ms)  = -j-- p l [See Appendix, eqn. ( l l ) ]  (3Y - 1) -M: + (3 - Y )  Making use of (63) at y = - U8t (66) takes the form u,' = - ( N - l )  1 (1  +Ma)* (( Njh* ) - 1 )  ' -F (67) which shows that the perturbed shock speed varies inversely as the square root of time so that as t += co the refleeted shock apeed "pprmhes the valie based on ideal-gas theory. ,Hence the reflected shock is attenuated due to the combined effects of molecular heat con- duction and radiatiw beat conduction. A P P E % D I X  r Taking into account the small perturbations in the flow variables behind the reflected shock due to the presence of the contact region, we write the Ranking-Hugoniot conditions wz (- u, - u8' + ~ 2 )  = (03 + p i )  (- Us - Us' + u;), (1)- W~(-U,-U,'+~)"+~=C~%+PP~)~-U~-U:+%')~+~~$.~;, (2) r " + - ( , ,  - 1) p3 + ~ 3 '  I ( Y + ~ )  w* - =: p2 (3)  - Y Neglecting second and higher powea of perturbed quantities, we deduce from equation (3) Similarly, from equation (1 )  we obtain / ) and from equation (2 )  we have - at = 2q,U8ocflt - U&'. (6) , Using the relation - -- ( y - 1 ) M , 2 + 2  *g . ( Y - I - ~ ) ~ ?  1 (7 )  in equation (5) we get From equations (4) and ( 7 )  we obtain (Y + 1) M: 2YM,2 - ( y  - lj- >(9) . horn equations (6),  (7) and (9) we 4educe, 6 , pa' = w4Aa+ (Me) u' &ere r - 2 ~ , ( ( y  - 1) M.P + 2 )  4 0%) = * (10) I (3Y - 1) d- (3 '- Y )  Eliminating p i ,  and between equaifions (61, (71, (8) and (lo), we obtain, u; = (11) ci . . ilc estrern* grsWuCto Wifessor P. L. Bhatnagw for '$is kind help and the prepawtiop of this paper. R;E%EBEMCES 1. GOLDBWOBTIIY, B. A., J: FW aech., 5 (1959), 164. oronto, Institute of Aerophysiy, No. 26 (1954). , E., Phya. Fluids, 7 (1@4), 1201. 4. @AGANOZE, D,, J.  8 k i d  Me~h.,# (1965)fl096 4. Crz&as~, J. F., Pm. Boy. Sot. (London), A 299 (1$67), 221. 

Structure of the Contact Region and its Application to the Reflexion of a Plane Shock Wave from a Heat Conducting Wall

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Structure of the Contact Region and its Application to the Reflexion of a Plane Shock Wave from a Heat Conducting Wall

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