SOLUTION OF THE VISCOUS TRANSONIC EQUATION FOR FLOW PAST A WAVY WALL by Martin Sichel The University of Michigan Ann Arbor, Michigan May 1970 This work was sponsored by the U. S. Army Research Office in Durham, North Carolina, under Contract DAHC04-68-C0008 This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of the U. S. Army Research Office-Durham.

SUMMARY The viscous transonic small disturbance equation has been recast into a form which displays the role of the usual inviscid transonic similarity parameter and of a newly defined viscous transonic similarity parameter in the two dimensional flow past slender bodies. Using the approximate method of Hosokawa the solution for viscous transonic flow past a wavy wall has been obtained in analytical form, and displays the role of the above similarity parameters. The influence of the Reynolds number and free stream Mach number on the supersonic pockets which arise at the wall surface has been investigated. ii

TABLE OF CONTENTS Page SUMMARY ii LIST OF FIGURES iv I. INTRODUCTION 1 II. THE VISCOUS TRANSONIC EQUATION AND SIMILARITY 3 III. WAVY WALL PROBLEM-FORMULATION 10 IV. RELATION BETWEEN X, X AND THE PARAMETERS OF THE MATHIEU EQUATICdN 14 V. SOLUTION OF THE MATHIEU EQUATION 18 VI. RESULTS AND DISCUSSION 24 FIGURES 28 REFERENCES 40 APPENDIX A 41 APPENDIX B 42 iii

LIST OF FIGURES Figure 1. Definition of the Coordinate System. Figure 2a. The Variation of Xv, X and L /L with S and M. V 00 V 00 Figure 2b. The Variation of Xv and L /L with the Reynolds Number. Figure 3. Graphical Solution for I. Figure 4. Variation of v with r7 and X 0 Figure 5. Variation of q with X and xv. Figure 6. Variation of Accelerating Sonic Point x* with Xv and x Figure 7. Stable and Unstable Regions in the a - q Plane. Figure 8. Variation of = u1 - u( with x, c =1.00, X = 0.61, q = 3.0. c v Figure 9. Variation of ( =ui - u ( with x, w=1. 00, X = 0 33, q =10. 0. c v Figure 10. The Variation of ~ with x; Hosokawa's Inviscid Solution. Figure 11. Variation of u =u() - u() with x for a Sonic Free Stream. c iv

I. INTRODUCTION Regions of supersonic flow arise near the surfaces of airfoils and nozzles as the free stream Mach number approaches unity. Such supersonic pockets are usually, though not always (Nieuwland and Spee, 1968) terminated by a weak shock wave. Analysis of the flow within and near such pockets, particularly in the neighborhood of the terminating shock is a crucial problem of transonic aerodynamics. A particular difficulty is that the pressure rise across the terminating shock wave is not the same as that obtained from the RankineHugoniot equations due to the interaction between the shock and the boundary layer at the surface and also due to the fact that the weak terminating shock may be sufficiently thick that the shock structure is no longer one dimensional (Sichel, 1968). To gain some insight into this problem Hosokawa (1960a, b), using an approximate method, obtained solutions of the inviscid transonic equation for flow past a wavy wall. With increasing free stream Mach number supersonic pockets do appear at the surface of the wavy wall, and Hosokawa's analysis always requires that such supersonic regions be terminated by a shock like discontinuity. Hosokawa's method was applied by Sichel and Yin (1969) to obtain solutions of the viscous transonic (or V-T) equation for wavy wall flow. The V-T equation, which is discussed in a survey by Sichel (1968), corrects the inviscid transonic equation by an additional term representing the influence 1

of compressive or longitudinal viscosity, but shear viscosity is still neglected. Thus the V-T equation makes it possible to analyze weak shock waves with a two dimensional internal structure. Sichel and Yin's results could only be obtained by numerical integration and hence were limited to a single Reynolds number and only several values of free stream Mach number. In the present paper the V-T equation is first recast into a form which displays the role of the usual inviscid transonic similarity parameter, and a newly defined viscous transonic similarity parameter in transonic flows about bodies in general. Then, using the method of Hosokawa, the solution for V-T flow past a wavy wall is obtained in analytical form so that the influence of Reynolds number and free stream Mach number on the supersonic pockets which arise can be explored in detail. 2

II. THE VISCOUS TRANSONIC EQUATION AND SIMILARITY The V-T small disturbance equation has been derived (Sichel, 1968) from the Navier-Stokes equations by perturbing with respect to a uniform sonic flow. But, a more general equation is obtained if perturbations are with respect to some uniform free stream velocity U near the sonic value but not necessarily equal to it. Consider two dimensional flow past a body, whose surface is given by yw(X) =Sf(x/L) (1) as shown in Fig. 1. Barred quantities are dimensional. S is the maximum thickness, and the body shape function f(x/L) - 0(1). Then using U, p, p U as reference velocity, density and pressure; and the body length, L, as reference length, the expansions U=-= E U + e.P 2 + E +... U M 2' 00 (2) p~ (1) ~v (1) =P =1 + E p( ) +.. u =-= Xv Poo U and the coordinate stretching x=-; y=X y (3) L L when introduced in the Navier Stokes equations, yield the following equations for u() and v 3

(i - 2 it E(1 + rl)u() + (1 ULc Pr? U xx e x 1 ) M 2 ((1) )(1) 2(4) - (y 1) M2 u u +a 0 (4) co x e y u(1) = v(l) (5)1) y x The stretch factor X takes into account the difference in the characteristic x and y dimensions which arises in transonic flow while u(1), p( ), p(1), and v(1) are presumed to be 0(1), and the expansion parameter e <K 1. v" is the compressive or longitudinal viscosity and Pr is the longitudinal Prandtl number. Since the flow is irrotational to the present order of approximation introduction of a potential 4 reduces Eqs. (4) and (5) to 2( -— (+ j$ + - -- + -(y+i) M ~ ~ +-4 =0 Pr/ xxx e xx o x xx E yy (6) To the present order of approximation the boundary conditions will be that (1) dyw df e v (XO x, 0) =Xe 4 (x 0) S (7) y dx where S =S/L, and that u() and v( must remain finite as y - oo. At the same time the pressure coefficient C is given by 4

P - Po C 2=. ~=-2e P 1- U2 2 oo Following Ferrari and Tricomi's (1968) discussion of inviscid transonic similarity let AX =S (8) since both v(1)x, 0) and f'(x) should be 0(1). Using Eq. (8) the V-T equations can then be written in the form -, 21) + ( -M 2) e2 EL E 00 U prt' xxx S2 xxx U L S S (y + 1)M E 0 —- D 4 + 4 =0 (9) S2 x xx yy Now, in V-T flow the four terms of Eq. (9) should be of the same order, and if the expansion and stretching of Eqs. (2) and (3) are appropriate, the (4, 4), yy - 0(1), and it becomes convenient to choose E so that xx x yy (y+ 1) M E3 2/3 =1 or E = -23-/ (10) S2 (Y + 1) M The coefficient of 4~ then becomes xx (1 -M 2)2 (1-M 2) s2/3 M 4/3 ( + 1)2/3=X (11) 5

where X is now the usual inviscid transonic similarity parameter (Ferrari and Tricomi, 1968). The coefficient of the viscous term of Eq. (9) becomes if2 2mi VV E) r -1 __ o(1 +pr^}"' — 2/+ —47^ ---— am1 pT^^v (12) U 2 Prft) ULS2/3 4/3 (.+ 1)2/3 (+prt= X (12) E 0 (+ Y ) where Xv is now a viscous-transonic similarity parameter. The V-T equation can thus be written in the form Xv +X -A + - =0 (13) v xxx oo xx x xx yy with the boundary conditions 4 (x,O) f'(x); y (x boundedas y I -, Ixl- c (14) y y' x The pressure coefficient will be 2S2/3 p (y+1) M(1 Equations (13), (14), and (15), provide the basis for viscous-transonic similarity rules. Since U L/v" is a Reynolds number, Re, based on compressive viscosity, the newly defined V-T similarity parameter, Xv can also be expressed in the form Re S2 M 4 )2 (16) 00 6

Thus Xv varies inversely with the Reynolds number, and the coefficient of the viscous term in Eq. (13) will become very small for large Re. Letting y" 11 l+-l') (17) U s2/3M 4/3 (y+1)3 + Pr' 00 /(Y the V-T similarity parameter Xv becomes Xv =7'/L (18) and 71 is a length of the same order as the thickness of a weak shock wave. Finally, in terms of a viscous length Lv =?"/ao,'V/;\ Pr" xv =o2/3 7/3 + 1)2/3 (19) Since L is of the order of the mean free path, the ratio LV/L is clearly a Knudsen number. It is readily shown that at the sonic point (1 -M 2) (1) (1) (20) c 2 M 4/3 2/3 - (20) t is te c v o t e c u) that is the critical value of the expansion coefficient u() equals the inviscid transonic similarity parameter. 7

The variation of X with S and Mo is shown in Fig. 2(a), while Fig. 2(b) o0 00 shows the variation of Xv with the Reynolds number U L/i" for different values of thickness ratio S, and also shows the variation of L /L7 with Re. Clearly, the use of the continuum theory is questionable when L /L > 0. 1 or when Re < 10 in the present case with Mo 1.0. With Re> 100, v << 1, except for extremely small values of the thickness ratio S, and then solution of the V-T equation (13) becomes a singular perturbation problem. Equation (15) implies the similarity rule that the pressure coefficient C, 2/3 2 varies as S/3 for fixed XO, Xv, at least to first order in (1 - M 2) or E. For geometrically similar bodies it is necessary to know the influence of the physical variables L, M, S, and Lv upon the similarity parameters X and Xv to assess the significance of the V-T similarity rule implied by Eqs. (13), (14), and (15); hence the effects of these variations have been summarized in Table I below. Case 1 in Table I represents flow with constant free stream density, temperature, and velocity, but with variable characteristic length L, which might, for instance, be the airfoil chord. Then even though the inviscid transonic similarity parameter remains constant, the viscous parameter Xv varies inversely with L. Flow past a fixed object with constant p, To but variable Mo is represented by Case 2, and then Xv is fixed but the inviscid parameter X is variable. Cases 3 and 4 show the effects of varying body thickness S, and of varying po and T and hence the viscous length L upon the similarity parameters. Case 5 shows the variation of C when parameters are adjusted to keep Xo and Xv fixed. 8

Table I. Relation Between Physical Variables and Similarity Parameters in V-T Flow. Physical Variables Similarity Parameters Case.. No. Fixed Variable Fixed Variable 1. MS, L L XO X oc L1 2. SL, Lv X X oc L-M V 2 0 I. - ~2 2~ —2/3 1 - M oc2S 9o fi ~9

III. WAVY WALL PROBLEM -FORMULATION The problem of flow past a wavy wall provides a means of examining the effects of the parameters x and Xv on V-T flow since, using the method of Hosokawa (1960), it is possible to obtain approximate solutions of the V-T equation in analytical form. With the ordinate of the wall given by y S sin 27r (21) w a where I is the wavelength, the boundary condition (Eq. (14)) becomes v'i)= y (x, 0) = cos ox (22) where w= 27r L/l. The factor ov determines the choice of the characteristic length L. Thus letting c = 1 implies that L = -/27r, and, as will become evident later, is the most convenient choice. In the present formulation the influence of S and X enters only through the similarity parameters X and Xv in Eq. (13), with the choice of c a matter of computational convenience,. in contrast to the previous formulation (Sichel and Yin, 1969) where the parameters of the problem occur in both the equation and the boundary conditions. Following Hosokawa (1960b) the potential 4 is split into two parts 4 = ~ + g (23) such that f satisfies a linearized V-T equation with the acceleration 6 in the 10

nonlinear term replaced by a constant K so that Xv xxx + xxx - K'x + yy = (24) f is made to satisfy the wavy wall boundary condition (Eq. (22)) so that iLox (x, 0) =wo cos cox = Re (co e ) (25) The function g is a nonlinear correction which in view of Eqs. (13), (24), and (25) must satisfy xg +Xg +(K-^ )^-(gk) -gg +g =0 (26) Xv gxxx + Xx gxx + (K - 4xx) ox - (gx Ox) - gx gxx + gyy 0 (26) gy(X, 0) = 0 A key assumption in the approximate analysis, which is discussed in detail by Hosokawa (1960b) and by Sichel and Yin (1969) is that yy (x, 0) = 0 (27) so that Eq. (26) can be treated as an ordinary differential equation for g at the wall, y =0. Following Hosokawa (1960b) the constant K is chosen as 0 (x, 0) at the accelerating sonic point, e (x, 0) = X, in the linear solution where K > 0. This procedure yields the equations ~x (x* O; X o Xv) c X o (28) xx (x*, O; X, Xv) = K 11

for K and the sonic point x =x*. The justification for the choice of K is discussed by both Hosokawa (1960b) and Sichel and Yin (1969). Equations (23)(28) present the basic elements of Hosokawa's approximate method. Assuming that f is of the form iCVx k=Re [F(y)e ] the solution of Eq. (24) with boundary condition (Eq. (25)) is readily shown to be =Re[- (w/m1/2) exp- ml/2 y cos )exp icx - m 2 y sin - (29) where 1/2 2 4 3 m= X2 4 3 X + XvS+KI) (30) w) X + wK 3=arctan -- 2; 0 3 2 X Letting K X = 0 reduces the linearized equation (24) to that for inviscid linearized subsonic or supersonic flow, and the solution (29) reduces to the well known linearized subsonic or supersonic wavy wall solutioni. Equation (26) for g(x, 0) can be integrated once to yield Xv gxx + Xoogx -2(x +gx) + K =A1 (31) 12

where A1 is a constant of integration. Introduction of the variable (1) W(x) =gx(X o) + x(x, o) - x0 = u (x, 0) - u( (32) for the deviation of u (x, 0), the velocity at the wall, from the critical value and use of the linearized solution for f then yields the following ordinary differential equation for ~: 1 2 Xc A1 m 12 o s 2 (3) "2X' cos + -- -- Cos x+ (33) 2Xv 2X, X X 2 2 2V 2v v v Equation (33) is a Riccati equation, which with the transformation =- X+ +2 4 4 (34) 1 dT (= Xv TTd5 changes to the following second order linear equation for T: T"I + (a - 2q cos 20) T = 0 (35) with 2A - x2 1/2 A oo m 2 2; 22 Xv X Equation (35) is the Mathieu equation whose properties are well known (McLachlan, 1947). The V-T wavy wall flow has thus been reduced to the problem of solving the Mathieu equation and is considered below. 13

IV. RELATION BETWEEN Xv, X, AND THE PARAMETERS OF THE MATHIEU EQUATION The wavy wall solution depends upon the similarity parameters Xv and Xc through their influence on a and q in the Mathieu equation (35). Therefore it is necessary to solve Eq. (28), which relates m and the angle 3 to Xv and Xe Introducing the solution for X, Eq. (28) becomes 2 m X o/2 sin Cox* - I ) = X wo~~~~~~~3 (w ~)=K (36) 1-2 cos x*-2)=K m In solving Eq. (36) it is more convenient to deal with the variable coX + (K/o) U = tan = --- (37) Xco so that 1/2 1/2 21/4 ml/2 = I 1/2 (1 +2) (38) Upon squaring both sides Eq. (36) can be reduced to the following algebraic equation for 71 14

2 2/: 3 1+( - ) W 212 (39) (1 + q) Although Eq. (39) is of sixth order in 71so that an analytical solution is not available, a graphical interpretation of Eq. (39) does provide an insight into the behavior of 7). Let X 2 2/Ix2 3 XX, 2- 2 1 (1 + r2) then solutions of Eq. (39) are the intersections of the curves y1(1) and y2( 7) in the y-r7 plane as shown in Fig. 3. As long as w2/ X 13 > 1. 0 Eq. (39) will have two solutions T and 72 such that r1 < Wo X/Xo< 2' and from Eq. (37) it then follows that K < 0 when 77 = 71 and K > 0 when r = T2 Since K is to be evaluated at the accelerating sonic point, 72 must be the appropriate solution of Eq. (39). Although Eq. (39) cannot be solved for 71 as a function of Xv and X, the equation can be solved for Xv and a function of T] and XO so that Xv T i -j (40) Xv Wn+ v 2)1/2 and the minus sign must be chosen in Eq. (40) to ensure that K> 0. From Eq. (40) it follows that ) < [(a4 / IX6) - 1] if Xv is to be real while 15

4/ 2 1/2 7 > I [(4//lx ) - 1] | if X is to be positive. For purely subsonic flows such that x(x, 0) < Xo for all x the present formulation loses its meaning. The variation of q with Xv and X as determined from Eq. (40) is shown in Fig. 4, for subsonic free stream flow with X0 > 0, and with c = 1.0 for convenience in calculation. Figure 4 reflects the influence of compressive viscosity on the linear solution for p. From the definitions of qi (Eq. (37)) and q (Eq. (35)) it follows that x 11/2 2 1/4 q = ~~2 (1 + 77) (41) WXV and the variation of q with Xv and X is shown in Fig. 5. Remarkably, q varies only slightly with x but is mainly a function of the viscous parameter Xv* For a sonic free stream with Xc =0, Eq. (36) reduces to a cubic equation for K with solution - ---- 1/3 -3 1/3 K-= ^/fTT^/ + 3) 13 while the Eq. (41) for q becomes (X + K)12 1 + X v +Xv q 2 2 when Xv 1 (42) X Xv 16

From Eq. (36) and (37) it can be shown that rI 1 T - 1,-1 X 7 x* = 1 tan 1 7 + tan1 -..x (43) and the variation of the accelerating sonic point x* with Xv and X0 is shown in Fig. 6 for a subsonic free stream with X > 0. 17

V. SOLUTION OF THE MATHIEU EQUATION From Eq. (41) for q and the definition of Xv and X it can be shown that q - 0 as S - 0, that is as the amplitude of the wavy wall decreases to zero. Solutions of the Mathieu Eq. (35) in this limiting case q = 0 will be considered first for guidance to the appropriate choice of the constants of integration. Then T=A2 el + A3 e (44) 2 3 where A, and A are constants of integration and 2 3, -A V/Xc - 2A1 d- a= v From Eq. (34) it then follows that o X- Ax[(A2/A3) e/1- e-1] ( =u1 (x, 0) - X, = A2 3 (45) (A/A3) e - e - and A2/A3 is now the only independent constant of integration. Letting A2/A - 223 (= —X v=- X - 2A and choosing A1 =0 yields = - X (46) 18

corresponding to undisturbed free stream flow. This is the expected result in the limiting case of vanishing amplitude. However, with A1 = 0 three other possible solutions are (A2/A 3)=.O; =- tanh X (47) (A2/A3) =0; =+ X (48) (A/A3)= 1.; -X coth X (49) The solution (47) corresponds to Taylor's (1910) solution for the structure of a weak normal shock wave while the solution (49) diverges at r =0. None of the solutions (47)-(49) are appropriate limiting solutions for vanishing amplitude. When q / 0, it follows from the Floquet theory for second order linear equations with periodic coefficients that solutions of the Mathieu equation can be written in the form (Abramowitz and Stegun, 1964) T =A e- A P() + A3 e/l P(-_) (50) P(~) is a function with period 7r or 2X7 and M = IL(a, q) is the characteristic exponent which may be real, zero, imaginary, or complex, and is a function of the parameters a and q. With /L =0 one of the solutions T will be periodic of 19

period 7T or 2ir and for a given q the values of reseulting in i = 0 are called characteristic values of the Mathieu equation. With Ai real T diverges at either -+ o0 or 4 - - c and the solution is called unstable, while if Ai is imaginary T will be periodic but of period other than r or 27I and the solution is termed a stable solution. The (a, q) plane can thus be subdivided into regions of stable and unstable solutions separated by g(,, q) = 0. The portion of the (a, q) plane pertinent here is shown qualitatively in Fig. 7 which also shows curves of constant i. With q = 0 it has already been shown that = = Xo0/wX and that @ < 0 so that the solution for T lies in the unstable region along the negative a axis (Fig. 7). When q > 0 it is to be expected that the solution will, at least for some range of q, still lie in this unstable region, however i V- i but will depend on both q and a. Then the solution (50) can be expressed in the form (McLachlan, 1947) 00 00 T=A2 e C2r e + A3 e C2r e i (51) r=-co r=-0o with the recurrence relation [ -(2r -il)] C2r-q(C2r 2 +C2r-2) =0 (52) 20

From Eq. (52) it follows that C2r and C 2r are complex conjugates. Then letting i2r -i2r C2r =2r 2r e, C 2r =P2r e, C =2p Equations (34) and (51) yield the following solution for (, the deviation of the velocity u) at the wall from the sonic value: _ A [0 1 2 e + =2rP sin (2r- + 2r 2 scos (2rt +) 2r ) r=l (53) ^ 00 f 11 [ e + ZP + Pr2 2r sin (2rt - 02r) + P12r cos (2rt - 2r) Lr=1 A200 1 [ 0 20 +1 0 + 2r cos (2rt + 2rj + e Po + - 2r cos (2r- r X 0~, 2r 2r) w Xv+ r=l (54) Letting A2/A oo as in the limiting q = 0 case considered above the solution (53) for C becomes 00 PO 2 P2r cos (2rt + 2r) r=1 21

From the recurrence relation (52) it follows that as q - 0, C2r and hence P2r - 0 so that x - - wx v which is identical to the limiting solution (46). The solution (54) has period fr in t or is of period 27r in wx, that is the same period as the wavy wall. If A2/A3 =0 the solution reduces to Eq. (48) in the limit q - 0. When A2/A3 = 1.0 the solution forJ behaves like a weak normal shock wave at the origin with a superimposed oscillation, while the solution will diverge at some value of 5 when A2/A3 < 0. Thus, the choice A2/A3 - co appears to be the only proper one for the wavy wall problem. The constant of integration Al determines the parameter i (Eq. (35)) and hence influences the solution through both the recurrence relation (52) and through the dependence of the characteristic exponent y upon a. With q 4 0 the choice of A1 is thus not as straightforward as in the limiting q = 0 case. A property of Hosokawats (1960) inviscid analysis but not of the present viscous solution is that the correction g vanishes at the accelerating sonic point defined by Eq. (28). In the viscous case gx(x*) will depend upon the integration constant Al; therefore, in order to compare the viscous and inviscid results A1 has been chosen to make gx(x*) =0. Therefore, at the accelerating sonic point, x =x* or ~ = *, where {((*) = 0, the condition 22

00. 2rp2r sin (2* + 02r) j(, q) = r=1 (55) PO + P2r cos (2r* + +2r) r=1 must be satisfied. Equation (55) is an implicit equation for A and hence A1, which can only be solved by trial and error. The series in the solution (54) converge very rapidly so that only three or four terms of the series need to be retained in the numerical evaluation of. For low values of q the values of A for different values of i and q have been obtained from graphs in Abramowitz and Stegun (1964). For large values of q asymptotic expressions given by Erdelyi et al (1955) have been used to determine jL as described in Appendix A. In practice computations are carried out by first choosing a value of q, which, from Fig. 5, is equivalent to fixing the viscous similarity parameter Xv. C is then computed for arbitrary values of ft, and the position of the sonic point, * read from these solutions can be used to determine X from Fig. 6. The computation of C can then be repeated using values of a corresponding to equal increments in X 23

VI. RESULTS AND DISCUSSION The variation of C with x has been determined for Xv = 0. 61 and 0. 33 as shown in Figs. 8 and 9. Hosokawa's (1960a) inviscid solution for several different values of X, described in Appendix B, is shown in Fig. 10. In the case of a sonic free stream with X = 0 the variation of C is plotted in Fig. 11 for Xv =0. 61, 0.48, 0. 33, 0. 27, and for Xv =0 corresponding to Hosokawa's (1960a) inviscid solution. From Figs. 8 and 9 it can be seen that in the viscous theory shock discontinuities terminating regions of supersonic flow are replaced by smooth compressions across which the Rankine-Hugoniot conditions are not necessarily satisfied. Comparison of Figs. 8 and 9 shows the influence of decreasing X or increasing Re upon the solutions. It can be seen that the transition to subsonic flow becomes steeper with decreasing Xv With a sonic free stream x = 0, Xv does not seem to affect the location of the compression wave; in fact, the inflection points of the sonic compressions occur at x = 1. 25 ir which is also the location of the corresponding shock discontinuity in the inviscid solution (Fig. 10). With a subsonic free stream, x > 0, on the other hand the effect of viscosity is to shift the compressions upstream and to shorten the supersonic region as compared to the inviscid solution of Hosokawa (Fig. 10). Comparison of the viscous and inviscid solutions shows that viscosity has a negligible effect on the accelerating portion of the flow preceding the shock transition. 24

Figure 11 shows that the viscous solution approaches the inviscid solution as X - 0 when the free stream flow is sonic (X =0); however, as indicated above this no longer appears to be true when the free stream is subsonic (X > 0). This interesting result, which suggests that it may be essential to include viscous effects in the analysis of certain transonic flows, bears further investigation. Since the theory developed here rests on a number of approximations and assumptions, the result that the subsonic solutions fail to approach the inviscid solutions as X - 0 can only be considered tentative. The sonic solutions have certain special features. For these solutions the characteristic exponent A = 0 so that the sonic solutions correspond to the characteristic solutions of the Mathieu equation. Further, with X =0, the phase angle 3, which arises in the linear solution for f has a value of 7T/2 just as in the inviscid case while x* = r/4 and is independent of Xr* The choice of the condition gx(x*) =0, used here, will have a particularly important influence on the behavior of the subsonic solutions. Setting gx(x) =0 is based in part on the inference of Oswatitsch (1955) that the linearized solution should be valid near the accelerating sonic point, and of course this is also the point at which the constant K coincides with the actual acceleration x of the linearized flow; however, there is no rigorous justification for this choice. gx(x*) could also, for example, be chosen to make the inflection points of the compressions in the viscous solutions coincide with the position of the inviscid 25

shock discontinuities. With this choice all the viscous solutions would probably approach the inviscid solutions as v -0. The flows considered here correspond to low Reynolds numbers. For example, for a body with thickness ratio S = 0. 01 the range 0. 27 < v < 0. 61 corresponds to64> Re > 28, and 0. 016 < (Lv/L) < 0. 035 (Fig. 2(b)). For this value of S, M will be very close to unity and the range 0 < X0 < 0. 8 corresponds to 1 > M > 0. 970. Boundary layers have, of course, been neglected in the present analysis, and the Re is based upon the wavelength or characteristic body length. In an actual flow the behavior of the free stream shock wave will also be influenced by disturbances induced in the boundary layer by the rapid free stream compression. Then an appropriate Reynolds number to describe the behavior of the flow near the shock wave might be more appropriately based upon a length characteristic of the boundary layer thickness rather than the length of the body. It is thus possible even when the Reynolds number based on L is extremely large, that the local Reynolds number governing the shock behavior may be extremely small, that is of the same order as the Reynolds numbers considered here. As compared to the previous paper on wavy wall flow (Sichel and Yin 1969) it has now been possible to obtain solutions in analytical form. The ratio of wall amplitude to wave length and to a viscous length appears here only through the parameters Xv and X whereas these ratios were introduced through the 26

boundary conditions in the previous paper. It is difficult to compare the present results to those obtained by numerical integration in the previous paper although there seems to be some difference. In particular, with a sonic free stream the location of the inviscid shock and the viscous compression do not coincide as in the present paper. As pointed out in Section 5 above there are several solutions for ( which are periodic but only one has the proper limiting behavior as the wall amplitude S - 0. While it is easy to choose the appropriate solution here, the difficulty of discriminating between the different solution branches wfhen numerical integration is used may account for the differences between the previous and present paper. Solutions have been obtained only for sonic and subsonic flow, i. e. for X > 0, in the present paper. However the present analysis can be readily used to evaluate solutions for supersonic free stream flows with X < 0, by letting the constant of integration A2/A3 = 0 in Eq. (53), and by extending the range of the calculations for the relation between Xv, X, and 3. In the limit X - 0 the supersonic solutions will approach the same characteristic Mathieu solution as the subsonic solutions. The present results provide an indication of the role of Xv and X in viscous transonic flow. The boundary layer has been neglected and since experimental results for transonic wavy wall flow are not available there is no way of comparing theory and experiment. 27

V t y=yw(x)=sf( —) Ua' / Figure 1. Definition of the Coordinate System. 28

6.0 12.0 6.0 Variation of Xo 1Variation of Xv \ w/ih Mao and S w/th Lv/L and S 4.0 10.0/ MO =/.O 2.0 Qoq 8.0 0.0 6- 6.0 -2.0- 4.0-c -4.0- 2.0- 0c5 -6.0o -------- a ^ —-0.0 0.8 0.9 1.0 1.10 1.20 0.0 0.2 0.4 0.6 0.8 1.0 MCo Lv/L Figure 2a. The Variation of?, Xoo, and L /L with S and M. V o

1.0 0.1-,,I3.j~-I 0.01___________ 0.01- X Pr" Re S 213 MC4,3(y+)213 0.001 100 1.0 10.0 100 1000 Re Figure 2b. The Variation of x, and L /L with the Reynolds Number. 30

cp~~~~~~~~~~~~-o C) k O \ o \ HI 0,

Variation of Xv with V and Xco 2.00 (O c40) Xv N V 0 0 ra O~~~O ^ c0/ / / / / / ^/^^ CAD 1.00 c5 0* O \ 0) 2 4 6 8 10.0 12 14 16 18 Figure 4. Figure 4.

100.0 Variation of q with Figure 5. Xco and Xv q 10.0 X O/-:xo=0.0, 0. Xao=0.9-/ Xco=0.5 1.0 I I 1 0.10Xv 1.0 33

Variation of Accelerating Sonic Point X* with Xv and Xco 2.0 -o. --- 0x 1.0 —-- 0.20.0 1.20 0 0.2 0.4i0.8 1.0g e 1.2 1.4 1.6 Xv Figure 6.

The a-q Plane A a Unstable Stable 2 3 4 5 q First Unstable l -2-, regioRegion (Wavy /._S. Wall Solutions) -4 L 0. Figure 7. Stable and Unstable Regions in the a - q Plane. 35

3.0 Variation of =u()with X w 1.00 2.0 X,= 0.61 q =3.0 1.0F, Fr 8'O Figure 8.

Variation of = u()-u(I) with X ~3.~0 () - 1.00 Xv = 0.33 { q =10.0 2.0 y,^ 1 \ 2 XX=0.0 10 ^Xoo = 0. 1 -3.0Figure 9. 0.3 -1.0'3.0 Figure 9.

2.0 i/~XCO =0.0 1.0 /I 0.10 40,0y10: 0,20 o _ ^_ -.%rj~-O 30, X 1'.0:0.40'Ob~~ x~o.so ~~" 2.OFigure 10. The Variation of w with x; Hosokawa's Inviscid Solution. 38

(I) Voriation of C u -uc With X for Sonic Free Stream 3.0 2.0 - -Xv = 0.0 0. 27 -0.33 0.48 1.0 - C ^j-0.61 C.0~~~~~~~~~~. -rr A I Ir 2I X -iT iT 2i r T 2.0 F 1 Figure 11.

REFERENCES Abramowitz, M. and Stegun, I. A., 1964, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. National Bureau of Standards, Applied Math. Ser. 55. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., 1955, Higher Transcendental Functions, Vol. III, Sec. 16. 3, McGraw Hill, New York. Ferrari, C. and Tricomi, F.G., 1968, Transonic Aerodynamics, Academic Press, New York. Hosokawa, I., 1960a, J. Phys. Soc. Japan, 15, 2080. Hosokawa, I., 1960b, J. Phys. Soc. Japan, 15, 149. McLachlan, N.W., 1947, Theory and Application of Mathieu Functions, Clarendon Press, Oxford, England. Nieuwland, G. Y. and Spee, B. M., 1968, "Transonic Shock Free Flow, Fact or Fiction?" Transonic Aerodynamics, AGARD Conference Proceedings, No. 35, of Conference held in Paris, France, 18-20 September, 1968. Sichel, M., 1963, Phys. Fluids, 6, 653. Sichel, M., 1968, "Theory of Viscous Transonic Flow-A Survey, " Transonic Aerodynamics, AGARD Conference Proceedings No. 35, of Conference held in Paris, France, 18-20 September, 1968. Sichel, M. and Yin, Y. K., 1969, "Viscous Transonic Flow Past a Wavy Wall, " Fluid Dynamics Transactions, 4, 403-414, Polish Acad. of Sci., Warsaw. Taylor, G.I., 1910, Proc. Roy. Soc. A, 84, 371. 40

APPENDIX A For q ~ 1, a >> 1 the asymptotic expressions for the characteristic value Ai given by Erdelyi et al (1955) can be expressed in terms of complete elliptic integrals. Since these expressions are not, to the author's knowledge, available elsewhere they are presented below. For a < - 2 Iq I the expression for iu can be reduced to =,(2/7) i 2q E (2- ) (A-1) where E( ) is a complete elliptic integral of the second kind (Abramowitz and Stegun, 1964). On the other hand for - 2q < i < 2q cosh ir = cos I! cosh I2 (A-2) where =/2 iii = 4q/2q+ a) m = 4q/(2q+ a) 1 =2 = 2 2 -a [m 2 E(l/m) - (m - 1) m 1/2 K(l/m)] m = 4q/(2q - a) In these expressions K( ) is a complete elliptic integral of the first kind. 41

APPENDIX B Hosokawa's (1960b) inviscid solution when expressed in the variables used here becomes =+ 2 = X + 2K [K - (x*)] -2XOOx (B-l) with the positive sign when x > Xo and the negative sign when x < XO. In the inviscid case - 1/3 1 - = - ~w cos (X - - 3) K -X [ (W4/3/X 2) 1- 1/2 4(x*) =- (X/) [ (4/3/X2) - 1/2 B -arc tan [ (w43/X 2) - 1/2 42

Unclassified. Srecti^tr CtI. ^^^_ 1kafl________ DOCUlENHT CONTROL DATA *R & D Setieelfrt r.1alitl..n'i t:).,...' b.f lh'.fbrrt *"ld l#tke*tn annMolofonn eml. he eiterwd when fhe overall report Io clA#fiNed) 1. QF9IO NA tINO ACTI(VIT (.Coipertf1 t lhor) 2a. REPORT SECURITY CLASSIFICATION Department of Aerospace Engineering Unclassified The University of Michigan 2h. GROUP Ann Arbor, Michigan 481048 1., REPONT TITLE Solution of the Viscous Transonic Eguation for Flow Plast a Wavy Wall 4. OElscItP TIlVr' O.tt-E (type of tepao and Inchtonve dale.) Interim Technical Report S. AU t1 -,S') ( rat name, middle Inltial, taat name) Martin Sichel 6. REPORT ODAT' 7a. TOTAL NO. OF PAGES Ib. NO. OF REFS May 1970 42 11 54. C.N TRACT OR "RANT NO. -e. ORIGINATOR'S REPORT NUMBER(S) DAHC04 -68- -0 008 PbR pwOJEcT No. 01561-1-T. 9b. OTHER REPORT NO(S) (Any other numbers that may be asslgned this report) d. 10. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of the U. S. Army Research Office —Durham. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Research Office-Durham Box CM, Duke Station ______________ |Durham, North Carolina 27706 i)3. ABSTRACT The viscous transonic small disturbance equation has been recast into a form which displays the role of the usual inviscid transonic similarity parameter and of a newly defined viscous transonic similarity parameter in the two dimensional flow past slender bodies. Using the approximate method of Hosokawa the solution for viscous transonic flow past a wavy wall has been obtained in analytical form, and displays the role of the above similarity parameters. The influence of the Reynolds number and free stream Mach number on the supersonic pockets which arise at the wall surface has been investigated. DD,.473 Unclassified Securlty Classification

Unclassified Security Claali ricatioon 14. LANK A LINK W LIN^- C KEYV WORDS | l ROLE WT RO.LE WT ProC: WY: --- --. l I l: lLI I —,-. —— I —-, —-I:_- - - - - -... Unclassified