A Moment Solution of Comprehensive Kinetic Model Equation for Shock Wave Structure

Transcription

1 A Moment Solution of Comprehensive Kinetic Model Equation for Shock Wave Structure K.Maeno *, R.Nagai *, H.Honma *, T.Arai ** and A.Sakurai ** * Chiba University, ** Tokyo Denki University, Japan Abstract. We consider the classic problem of a onedimensional steady shockwave solution of the Boltzmann kinetic equation utilizing a new type of moment approimation proposed by Oguchi (998). The model, unlike previous ones, epresses the collision term in an eplicit function of the molecular velocity. We can thus obtain moment integrals directly because of its eplicit epression. The principal value is utilized for the moment integral to cope with the singularity, and we can have five relations for five unknown functions to be determined as functions of the coordinate. These can be reduced to a firstorder differential equation that can be solved to provide the familiar smooth monotonic transition from the upstream supersonic state to the subsonic downstream state. Computed values of shock thickness for various shock Mach numbers compare well with some eisting results obtained by different models and methods to certain Mach numbers beyond which no solution eists. Some problems concerning the eistence of solution are considered in Appendi. INTRODUCTION We consider the classic problem of a onedimensional steady shock wave structure. Study of this problem has a long history, beginning at almost the same time when research started on shock wave phenomenon. We can recall such early works by Riemann [] in Euler s equation, Taylor [] for weak shock solution of the NavierStokes equation, and Becker [] for an eact solution of the NavierStokes equation. We can recall also the first measurement of shock profile by Talbot & Shermann [4] and many other early contributions; we will refer to some of them below as they are needed. However, the current status now is far from satisfactory, and it is still necessary to study its essential features. Besides its study based on continuum mechanics represented by the NavierStokes equation, here we are interested in the approach by the Boltzmann kinetic model. Notice that the need to apply a kinetic model to this problem was mentioned as early as Becker s paper []. There are various different ways of solving the Boltzmann kinetic equation. A recent standard one is Direct Simulation Monte Carlo (DSMC) (Bird [5]). However, the classical way of seeking a distribution function close to a Mawell distribution by perturbation (Chapmann and Cowling [6]) is still useful for the problem we have now, to which we can apply a moment equation derived from the basic Boltzmann equation. Here, we utilize a new type of moment approimation (Omodel) proposed by Oguchi [7]. The model, unlike the previous ones originated by Grad [8], epresses the collision term in an eplicit function of the molecular velocity, where its dependency on spatial coordinates is given through macroscopic quantities such as temperature, pressure, density, fluid velocity, stress tensor and thermal flu as functions of the spatial coordinates to be determined. This fact enables us to eamine directly the nature of the singularity in the distribution function with respect to this particular problem caused by the vanishing molecular velocity component (Beylich [9] ; Cercignani et al., [0]). First, we postulate a distribution function in which the space derivative is bounded. It turns out that this leads to only a trivial result of a uniform flow without a shock layer. Net, we allow the collision term to be finite at the crucial point. The Omodel collision term can be utilized to compute moments directly from integrals using its eplicit epressions of the molecular velocity. The principal value of the integral is introduced to cope with the singularity since it is only a simple pole. As a result, we can derive five equations for five unknown functions of space coordinate. These can be reduced to a firstorder differential equation that can be solved to provide the familiar smooth monotonic transition from its upstream supersonic value to its subsonic downstream value. Computed values of shockfront thickness for various shock Mach numbers compare well with results obtained by different models and methods to a certain Mach number

2 beyond which no solution eists. We consider the molecular velocity distribution function F( v,, t) of time t, the molecular velocity v, and the spatial coordinates whose components are given as v ( v, vy, vz ) and ( yz.,, ) We use the Boltzmann kinetic equation for F in a new type of moment approimation proposed by Oguchi [7], which is given in the simplest case of the Mawell molecular model as + Ê F F nn V È p + Ê ij t T T Î pt VV qv i i V v ep i j, () / ( p ) 4 pt 5 T with V v u, V V, n n(, t) Fdv, u u v v (,t) Fd, T T t n ( ) n FV, d v, p p(, t) nt, pij pij (, t) FVV i jdv d ij p, q q t FV Vd i i( ), i v, where ij,,, stand for yz,, ; d ij is for the Kronecker d ; and the double inde is for summation. All quantities are in dimensionless form based on the representative collision frequency n 0, the temperature T 0, the pressure p 0, the mean free path l 0 and the density n 0 as n 0 t Æ t, / l 0 Æ l0 / ( RT0 / n 0 ), F/{ n0( RT0) } Æ F, v/ RT0 Æ v, V/ RT0 Æ V, n/ n0 Æ n, T / T 0 Æ T, p/ p 0 Æ p, p ij / p 0 Æ p ij, qi / { RT0 p0} Æ qi, n / n0 Æ n. SHOCKWAVE SOLUTION A onedimensional steadyflow case of the shock structure problem reduces the above equation to dimensionless equation F n V p v T T pt V V qv V Ê È Ê + Ê n pt T ep P, () / ( p ) Î 4 5 where V V y + V z and the relation pyy pzz 05. p from the symmetry are utilized, and n n( ) Fdv, u u Fv d ( ) v, T T n ( ) n FV d v, p nt, p p ( ) FV d p v, q q ( ) FV Vdv. Here, we have the shock structure problem for which we consider a steady plane shock wave whose normal is in the dimension, and the distribution function F becomes the Mawell distribution function F ± in the uniform subsonic and supersonic regions as Ʊ or n V F( v, ) Æ F ep Ê / ( T ) ± ± T, p ± ± V v u ±, V v u v y v ( ± ) + + z as Ʊ, () where n ±, T ± and u ± are the values at Ʊ, They are related with each other through the RankineHugoniot condition as Ï Ô nu nu + + Ô nu + p nu Ì p+, (4) Ô Ônu 5 + pu nu pu + + Ó where we assume the gas is ideal and consists of particles of a single kind so that the ratio of its specific heat is g 5/. We use the Mach number M of the flow field given as

3 M M( ) 6 u u and M 6 5 T 5 T, (5) so that u+ n M + u n+ 4M, p + 5M. p 4 (6) We may use the quantities at as reference values to produce n p T, M 6/ 5u. We look for a solution F of eq.() subject to the boundary condition of eq.(). eq.() is still of integrodifferential type as is the original Boltzmann kinetic equation, but its collision term is epressed in an eplicit function of V with its dependency on being given through five moments, n, u, p (or T ), p and q in eq.() as functions of only. Accordingly, we may derive relations between these moments by integrating eq.() over v with certain weights, from which we may determine these. We can now obtain three relations by integrating eq.() over v, using the weights, v and v and utilizing the conservation laws of mass, momentum and energy as Ï Ô nu C Ô Ì nu + ( p + p ) C, (7) Ô Ô 5 nu + pu + up + q C Ó where C nu n+ u+, C n u p n u p +, C n u 5 p u n u p+ u+ from eq.(4). These relations as given in eq.(7) can conveniently be rewritten as C u, (8) n p C C u p, (9) u q up 4C Z, Z ( u u)( u u+ ) ( n n)( n+ n). (0) n n+ We also have p+ p u u+ ( n+ n), p q 0 at ±. () We need two more equations to determine the five unknowns. For this, we have to see the nature of the singularity of the solution F at v 0 in eq.(), which is the kind appearing particularly in steadyflow problems of the Boltzmann kinetic equation. Notice in this connection that the collision term in this model of eq.() is epressed eplicitly in variable V, so that we can directly eamine the nature of the singularity through this eplicit epression. We allow P of eq.() to be finite at v 0 and F/ P / v to have a pole there. The principal value (PV) is utilized for the moment integral to cope with this singularity. Besides three moment equations for five quantities of ( n, u, p, p, q ) given in eqs.(8), (9) and (0), we consider two additional moment integrals given as F d PV P v ( ) v d v, () F V d PV V v ( ) P dv. () v Using definitions for n and p in eq.() We have from eqs.() and () n Fdv, p V Fdv.

4 Ï dn n È T I p Ê I I q I d T p + Ê ( ) n T ( ) ( 0) () Ô ( / ) Ô p Î 4 5 Ì, ( 5) Ô d Ê n T I p p I I T I q d n È Ê ( 4) T p + ( ) ( 0) ( / ) p 4 + Ê ( ) T ( ) I I 5 T Î ÓÔ where I n V V n ( ) ( PV) ep Ê T V + u dv. (4) We have ( I n ) T n / n ( d ui n ) Ê V ep( z ) z z z T and V ( 0) dv I PV sinh( ) ( ) ep Ê h ep ep d I( ) T ( ) (h ) h p, V + u 0 h u T, h v T. Thus, eq.(4) is simplified to dn nn È Ï I I up + Ê Ì q Ap Bq d pt Ó + Ï Ê Ì Ó +, (5) Î 4 5 dp È u T dn nn Ï I ( + ) + Ì up + q ( I ) Cp + Dq, (6) d Î d p Ó8 5 from which we have the equation dp Cp + Dq. (7) dn Ap + Bq Notice the essential similarity of the relation between p and q and dn/d given in eq.(5) together with eqs.(8)(0) with the linear stressstrain relation of Newton s law, which is one of the bases of the NavierStokes equation. Since u, T, I, p and q are all given by the previous three relations (8), (9) and (0) as functions of p and n, eq.(7) is an ordinary differential equation for pn ( ), n n n+, and we look for its solution pn ( ) subject to the boundary condition: pn ( ± ) p±. (8) Since both p and q should vanish at the two ends n n ±, which correspond to ± as given in eq.(), eq.(7) becomes singular at n n ± as 0/0. This fact makes it difficult to start the solution from either n n or n +. We may start instead from a point n n, which is a little larger than n with an adjustable p value towards n + to get p + close enough to be p + at a point n n ª n. The process is repeated for various M values to a critical M value of about.7, beyond which the process breaks down. Results are shown in Fig. as pn curves for some M values. It is remarkable that these curves are almost straight lines. Inspired by this fact, we tried to construct an approimate solution of p(n) as p p p + u u+ ( n n ). (9) which satisfies the boundary condition of eq.(8) obviously at n n and also at n n +, as seen in the Rankine Hugoniot condition given in eq.(). We note here that this epression of eq.(9) corresponds to the MottSmith solution (MottSmith [] ; Sakurai []) in the sense of that it is an interpolating formula connecting boundary conditions at two ends. We also have from eqs.(8), (9) and (0) that for p p C p u Z, q C Z, (0) and Ê dp Cp Dq S + C ud uu +, dn Ap + Bq A ub p p

5 FIGURE. Numerical solution of p(n) for M.05,. and.5. We find that S is small in the entire range of n n n+ and accordingly demonstrates the validity of the approimations of p p as given in eq.(9). Notice also that the factor Z is canceled out there, as are the singularities at n n ± in eq.(7). In fact, for the limiting weakshock case of M Æ, we have C ud S Æ S u u M Ê 5 ( + ) ª ª , A ub M and the starting value of dp/dn at n n for small M is found to be nu, which coincides eactly with the value of u u + in the limit of M Æ to u ( 5/ 6) 5/. Using the p(n) function given above in eq.(9), the function n(), < < can be derived from eq.(5) combined with boundary conditions of n( ± ) n ± as given in eq.(). eq.(5) is now reduced by eq.(0) to dn nn Ï I Ap + Bq + Ê ( ) Ì CZ d pt Ó, u 5 4 T, which can be rewritten as dn En ( n)( n+ n), d C k I E Ï + Ê ( ) Ì n n pt Ó, () where we have used eq.(0) and the Mawell molecular model for n to yield n kn, k 5 p. () Then the mean free path of hardsphere molecule l0 6 m 0 / ( 5n0m prt0 ) of upstream flow is selected as standard of length. Since Tp/n, u C / n, p p + u u+ ( n n ) from eqs.(), (8) and (9), and u/ T, E can be regarded as a given function of n. We can thus epress eq.() in an integrated form as n dn, n 0 n( 0) ( n + n+ ), () n En ( n)( n+ n) 0 where we have used boundary conditions n( ±) n ±. As can be observed in eq.(), as long as E is bounded positive, this provides the density function n(), < < in the familiar shock layer profile of a monotonic transition of n from n at to n + at +. In

6 fact, we can see this feature more eplicitly in the limiting case of a weak shock wave for which we have approimately E E for small M and n dn n n E n (4) + n0 ( n n)( n+ n) E ( n+ n)log n+ n with k 5 Ï4 5 6 E ( E) Ì I( 5/ 6) 6 6 Ó from n p T, u 5/ 6 of eq.(6), () or n n + ( n+ n) [ + ep {E( n+ n) } ]. (5) Shock thickness t given by the maimum slope of dn/d defined as t Ï ma dn ÌÓ ( n+ n ) (6) d for M Æ, from eq.() 0 77 t { }( ) E E ma ( n n )( n + n ) n+ n ( n + 4 n ) Æ. ( M ) (I( 5/ 6).089). (7) In general, we can find a profile by integrating eq.() over n as long as E is bounded positive. This is violated at about M 8., to which E in eq.() becomes zero, and the solution does not eist beyond this M values of.8. This value can be compared with the corresponding Grad s [8] value of M.65, beyond which his moment solution does not eist, and for which the theoretical limit of M.85 for the convergence of the Hermite polynomial epansion solution was given by Holway []. FIGURE. Numerical solution of the reduced distribution function G ( M.5). The solution F considered above can have a singularity of a pole at v 0. A purely numerical solution is given by a stepbystep integration of an unsteady flow equation, for which the term F/ t is retained and starts from a discontinuous step function type condition. An eample of the solution reaching a steady state after many steps is shown in Fig., where we can indeed notice a polelike singular nature at v 0. The details of this numerical study will be published soon in another paper. We added some standard theoretical data such as by Taylor (/t 0.( M )) and the NavierStokes solution by Becker (/t 0.7( M )). These can be compared directly with

7 our result given in eq.(7) as /t 0.77( M ). These comparisons show the consistency of our approach with the conventional approaches. CONCLUSION We attempted to practically apply the new moment approimation model to the shock structure problem and obtained some satisfactory results. In particular, the obtained shock profile has familiar features, and its thickness is reasonably consistent with other data acquired by various methods. The eplicit epression of the collision term of this model based on the variable of molecular velocity is found to be useful, especially in studying the nature of a singularity caused by vanishing molecular velocity in the smoothness question concerning the molecular velocity distribution function. ACKNOWLEDGMENTS The authors are indebted to Prof. Oguchi (Prof. Emeritus of the University of Tokyo) for his useful advice in theoretical model equation and to the Japanese Ministry of Education for the support of a Grantinaid (No ). APPENDIX. REMARK ON THE EXISTENCE PROBLEM We have eqs.(7), (8) for p(n) as dp Cp + Dq. (A) dn Ap + Bq As noticed above, eq.(a) becomes singular as 0/0 at two ends of n n ±, since both p and q should vanish there, which correspond to ±. We eamine the nature of the solution there. First, we utilize eq.(7) combined with the RankineHugoniot relation of eq.(7) to have p C Cu p ( p p+ ) C( u u+ ) ( p p ) C( u u ), q up 4C Z, Z ( u u)( u u+), Ê C + ud p C DZ dp u dp 4, dn C dn Ê A + ub p 4C BZ and transform the above equation (A) into the following two different forms for p(u) with the independent variable u, u + u u as Ê C + ud p p C u u C DZ dp ( p ) C du ( u ) {( ± ) + ( ± )} + 4 ±, pu ( ± u Ê ± ) p±. (A) A + ub ( p p ) + C ( u u ) C DZ { ± ± } + 4 Each of these equations in (A) represents a singular initial value problem of the type dy c + dy, ( y, ) ( 00, ), (say), d a + by After some algebra following the standard procedure of dealing with this type of singular initial value problem of a differential equation [4], we have found that the nature of the solution near u u + is of node type while it is of saddle point type near u u. Accordingly (i) we have two regular solution curves starting from (p, u ) of which one stretches towards to an area near (p +, u + ) while (ii) any solution curve near u u + approaches to the point (p +, u + ).

8 (iii) Further, solution of (A) passing a point ( pu, ) of u + < ũ < u, p < p < p + is regular and unique in u + < u < u as far as the magnitude of the parameter M is close to or M is small enough, since from eq.(6) u+ M, p ( M u 4 M ), p 4 from which we have (a, b, c, d) above are almost constant to the present case of eq.(a). Combine all three facts of (i), (ii), (iii) above, we can see that there eist a unique smooth solution of eq.(a) starting from (p, n ) and ending at (p +, n + ). Once these five moments upnp,,,, q are given as functions of, higher moments can be obtained from these five moments above since we have simply from eq.(), () i F wh v w d H () i V P d V 0 for i > i0 ( 5, say ), where H () i i 0,,... are the Hermite Polynominals H () i ( V ) [8] of with the weight function w. So that we have formally () F a i () i  H, (A) i 0 which satisfies the boundary condition of eq.(). Alternatively, the solution F can be obtained simply by integrating both sides of eq.() over from to to yield v ( F F ) P d. It is not certain here that we have Pd v ( F+ F ) to satisfy the boundary condition of F at. However, at least Pd can be finite because of the requirement for p and q Æ 0 as Ʊ. REFERENCES. Riemann, B., Goett. Ges. d. Wiss. 8 (860).. Taylor, G. I., Proc. Roy. Soc. London, A84, 777 (90).. Becker, R., Z. f. Physik 8, 6 (9). 4. Talbot, L., Shermann, F. S., Rep. No. HE507, Institute of Engineering Research, University of California, (956). 5. Bird, G. A., Molecular gas dynamics and the direct simulation of gas flows, Clarendon Press, Oford, Chapmann, S., Cowling, T. G., The mathematical theory of nonuniform gases, Cambridge at University Press, Cambridge, Oguchi, H., Soga, T., Computation of rarefied gas flows at nearequilibrium state by discretevelocityordinate method in Rarefied Gas Dynamics, edited by C. Shen, et al, Peking University Press, Beijing, 997, pp Grad, H., Commun. Pure Appl. Math. 5, 5700 (95) ; Grad, H., Commun. Pure Appl. Math., 50 (949). 9. Beylich, A. E., Phys. Fluids,, (999). 0. Cercignani, C., Frezzotti, A., Phys. Fluids, (999).. MottSmith, H., Phys. Rev. 8, (95).. Sakurai, A., J. Fluid Mech., 5560 (957) ; A note on MottSmith solution of the Boltzmann equation for a shock wave II, Res. Rep. 6, pp.495, Tokyo Denki Univ., Japan (958).. Holway, L. H., Phys. Fluids 7, 99 (964). 4. Coddington, E. A., Levinoon, N., Theory of Ordinary Differential Equations, McGrawHill, New York,955.