The reflexion of a plane shockwave from a heat-conducting wall J. F. By Clarke Department of Aerodynamics, The College of Aeronautics, Cranfield, Bedford {Communicated by M. J. Lighthill,Sec.R.S.—Received 12 October 1966) The situation following reflexion of a discontinuous plane shockwave from a coplanar, heat- conducting wall is analysed by the method of matched asymptotic expansions. Temperature jump effects are included. Two terms of the outer expansion are calculated ((i) the ‘ideal’ inviscid, nonheat conducting solution; (ii) the displacement effect) as well as two terms of the inner series ((i) constant pressure thermal boundary layer; (ii) first correction term resulting from pressure and temperature changes due to the displacement effect). The results agree reasonably well with experimental observations and it appears that accommodation effects may account for some observations of reflected-shock trajectories.
222 J. F. Clarke Goldsworthy result, neither are they seriously at variance with it. The uncertainties are to some extent due to the inherent difficulties of pressure measurement on sub-microsecond time scales.
Reflexion of a plane shockwave froma heat-conducting wall 223 Together with equations (5) this enables us to rewrite (1) through (4) as follows: 0 , - -pP, = ^(M 0A+iPr(r (7) U1+ = fPr ^ (p/mx)x’ (8) rx p2ux +pt = 0 or u = - \ p~2ptdy, (9) P=p (10) where R = a^tjK^, = AJp^Cp, (11) Pr = yttooGp/Aoo. (12) Equations (7) through (9) are valid if u(t, 0) is zero, a condition which is maintained throughout. A,*, and p^ are the reference values of conductivity and viscosity, while A = A'/Aoo, = a,. (13) R in (11) is a Reynolds number and Pr in (12) is a Prandtl number. We propose to simplify the analysis somewhat by assuming that both A' and p' are proportional to T'. This permits us to write, with the aid of (5), (10) and (13) pX — pp = pO = p. (14) It is easy to eliminate p from (7) and (9) with the aid of (10). The result is three equations, for the variables p, 6 and u, which are in a form suitable for further developments, namely, 1 h-1id 1 4Pr (15) "i 1 p^l~~ R ^ ^ x )x 3j£ ^ > ^ 1 4 Pr ut+yPx = ~jR^ux)r (16) (17) M=Jo E(0Ip)Ax- The solid material in x' <0 is assumed to have constant thermal properties. In particular its thermal conductivity A' and its thermal diffusivity ks are constant. The temperature T'sof the solid satisfies the simple diffusion equation, dT's,d2T's (18) It is advantageous to define a dimensionless length variable £, where l = x'lfKtoo), (19) and also to put 0s = T'JTn. (20) Then, if we use t (defined in (5)) equation (18) becomes simply &st ~ 0s%- (21).
224 J. V. Clarke Before the initial instant the solid is at a uniform temperature . It follows from (20)that 6s(t0, £ < 0) = c. (22) Subsequent to the initial instant 6 > 0, ^ —> — oo) -> (23) The conditions to be satisfied at the interface £ = 0 = are as follows. Because the interface is assumed to be impermeable it is necessary to put ,t(u0) = 0 (all (24) and this is in fact implied in (15), (16), and (17). Despite this it is necessary to ensure that (24) is not violated in the subsequent analysis. Since energy must not accumu late at the interface we must put (in dimensional form) dT' 3T' Translated into dimensionless form with the aid of (5), (6), (11), (19) and (20) this condition reads ^ 0^ , 0) = p{t, 0) dx{t, 0), (25) where Qs ia ratio of the thermal properties of solid and gas, namely ^-s j^<x> Q (26) Arrt \i K c When accommodation effects are significant the temperature jump at the interface is related to the temperature gradient in the gas as follows r ^ ((',0) = T 'ro)-T »',0).
Reflexion of a plane shockwave from a heat-conductin2g2 w5all specific information about the temperature dependence of r we further propose to treat A as a constant. Although the latter may not be an especially sophisticated assumption it has the merit that it preserves the essential physics of the problem whilst leading to tractable analytical results. In cases where 6 does not vary greatly it is also probably quite accurate.
226 J. F. Clarke a shock travelling into the gas behind the incident wave at a (constant) velocity 0rt' Whe“ («„ - (»a -Ut) = ~ <■ (30) t/r0 has been called the ‘ideal’ shock velocity in the Introduction. The velocity of the reflected wave relative to the moving gas ahead of it will be written as Us0.
Reflexion of a plane shockwave from a heat-conducting wall 227 Any attempt to define an outer series like (33) for 6S, coupled with an attempt to satisfy the conditions given in (21), (22), (23), (25) and (28) would fail because of the singular character of the solution in (34). For example, (28) would yield (for ds0o(t,) = 00(t,0) = 1 while (23) and (25) would require — °o)^c, 0) = 0.
228 J. F. Clarke But (34) shows that pQ(t, 0) = 1 = 60{t, 0). It therefore follows that Po(t,f) = h (42) UH{t, ft) = ©o (43) ©o(*,°°) = 1- (44) The solution given in (42) permits considerable simplification of (39), reducing it in fact to the homogeneous diffusion equation ©o< = ©o#-- (45) Equation (44) provides the outer boundary condition for (45). Since the remaining conditions involve the temperature of the solid we must now consider that quantity. Before doing so, however, we note that the solution for U0 given in (43) can be simplified with the aid of (45) to read U0( t , f t ) = ©o^f, - ©o^ (M>). (46) This solution for U0 satisfies the boundary condition that there shall be no gas velocity at the interface.
Reflexion of a plane shockwave from a heat-conducting wall 229 5. Thermal boundary layer: first approximation The first inner approximation requires that we should solve (45) and (48) (with n = 0) under the conditions imposed by (44), (49) (with n = 0), (50) and (51). This set of conditions is complete with one notable exception; there is no initial condition for ©0 and it is not possible to find a value for ©0(0, ijf) from the matching techniques which led to equation (44), for example. The reason for this deficiency must be sought in the inadequate description of the true state of affairs near to the initial instant. The shock wave is not of vanishing thickness on the scale of events near to this point in time, and indeed the methods that we are using are not valid for time intervals shorter than those characterized by an Rt of order unity. Recognizing these facts it seems entirely reasonable, on physical grounds, to adopt as an initial condi tion for 0O the requirement thatf 0o(O,f)=l. (53) We note that 0o(O, ifr-> oo) must in any case approach unity in order to be consistent with (44).
230 J. F. Clarke First of all we apply Van Dyke’s matching principle to a two term outer series and a one term inner series for the gas velocity u. Using (33), (37), (38), and (46) it transpires that we must find #i(i2) ux(t, 0) = co). (59) The matching is therefore effected by setting dx(R) — iR~ (60 a) and u1(t,0) = U0(t,co ) . ( 6 0 6) The quantity ux(t, 0) is readily found (from (58)) to be given by (1 -c)Q %(£, 0) = - (61) (iT W x exp It may be noted that ux(t, 0) is negative when < 1 (all other variables being essentially positive). Physically this corresponds to a flux of gas from the outer flow into the interface as a result of the hot gas in the thermal layer being cooled and contracted by contact with the cold solid. It is also interesting to note that 0,0) is finite for all nonzero r. In the absence of accommodation effects r is zero (see (56) and (28) with A = 0) and ux(t, 0) behaves like t~i. This is the case considered by Goldsworthy and compared with experimental observations by Sturtevant & Slachmuylders and by Baganoff. We comment on these matters at greater length below in § 8.