An Approximate Solution of the Riemann Problem for a Realisable Second-Moment Turbulent Closure Christophe Berthon, Frédéric Coquel, Jean-Marc Hérard, Markus Uhlmann

An approximate solution of the Riemann problem for a realisable second-moment turbulent closure Christophe Berthon, Frédéric Coquel, Jean-Marc Hérard, Markus Uhlmann

To cite this version:

Christophe Berthon, Frédéric Coquel, Jean-Marc Hérard, Markus Uhlmann. An approximate solution of the Riemann problem for a realisable second-moment turbulent closure. Shock Waves, Springer Verlag, 2002, 11 (4), pp.245-269. 10.1007/s001930100109. hal-01484338

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2 C. Berthon et al.: An approximate solution of the Riemann problem

1 Introduction above, many papers referenced in the work by Uhlmann (1997) address this problem. Thus, the basic two ideas We present herein some new results, which are expected underlined in the work are actually the followingones. to be useful for numerical studies of turbulent compress- Assumingthe second-moment closure approach is indeed ible flows usingsecond order closures. These seem to be a suitable one, how should one try to compute unsteady attractive from both theoretical and numerical points of flows including discontinuities (shocks and contact dis- view. Numerical investigation of turbulent compressible continuities) when applyingthis kind of closures to the flows is usually carried out usingone- or two-equation averaged Navier-Stokes equations? And even more: may turbulent compressible models, such as the well-known one exhibit analytical solutions of such a model, at least K-ε closure, or computingthe governingequations for all in a one-dimensional framework (though accounting for Reynolds stress components. Differential transport mod- three spatial dimensions of the physical space)? The lat- els are potentially capable of predictingthe influence of ter problem is indeed a delicate one since the occurrence a shock wave on the anisotropy of the Reynolds stress of non-conservative terms in the first order set of partial tensor, as idealized studies of shock wave/turbulence in- differential equations renders the problem of formulating teraction suggest (Uhlmann 1997). Actually, both first or- jump conditions much more difficult than in the purely der and second-moment models involve non-conservative conservative framework (see the basic works of Dal Maso first order differential systems. Two-equation turbulent et al. 1995; Le Floch 1988; Le Floch and Liu 1992 on that closures have been recently investigated (Forestier et al. specific topic). 1995, 1997; Hérard 1995b; Hérard et al. 1995; Louis 1995); In the first three sections, governing equations and these references suggest two different ways, namely, (i) a some basic properties of the whole viscous system are Godunov-type approach alike Godunov’s (1959) original briefly described. Approximate jump conditions are then scheme (Forestier et al. 1997), and (ii) a flux-difference- proposed, which are based on a linear path with respect splittingas originallyproposed by Roe (extended to non- to some non-conservative variable. This allows construct- conservative systems, for instance, by Hérard 1995b), to ingthe solution of the 1-D Riemann problem, applying compute convective fluxes and non-conservative terms, so for the entropy inequality, and restrictingto weak shocks. that turbulent shock-tube problems might be numerically The solution of the Riemann problem satisfies realisabil- simulated. Both ways have been proven to be suitable ity requirements. These results enable us to propose in the tools for that purpose, and enable to preserve Riemann in- fourth section a simple but efficient way to compute time- variants in linearly degenerate fields. It was also confirmed dependent solutions includingrarefaction waves, shocks numerically (Page and Uhlmann 1996) that standard Eu- and contact discontinuities. It should be emphasised here ler Riemann solvers are not adequate for that goal. Turn- that other ways to provide approximate Riemann solvers ingnow to second-moment closures, preceedingremarks based on strongly coupled upwinding techniques may be still hold; moreover, as underlined recently by Uhlmann suggested, but the main purpose of the present contribu- (1997), Euler-type Riemann solvers can no longer be used tion is not to exhibit the “ultimate” scheme but to take ad- to compute this kind of closures, since they provide un- vantage of the continuous analysis to exhibit an expected stable computational results in some situations, which are meaningful compromise between efficiency and accuracy. mainly due to the occurrence of two new waves, which do In particular, the comparison with commonly used Roe’s not exist in the Euler framework. Though emphasis is put method may be found in the work by Uhlmann (1997), on a simple objective and strongly realisable model arising though the latter scheme does not comply exactly with from the literature, the present analysis extends to more so-called Roe’s condition (which corresponds to the consis- complex closures proposed by Fu et al. (1987), Shih and tency with the integral form of the conservation law, when Lumley (1985), Shih et al. (1994). focusingon conservation laws). The fifth section will thus It must be emphasized here that a priori suitability be devoted to the presentation of sample computational of second-order compressible closures to predict the be- results of turbulent shock tube problems, which confirm haviour of flows includingshocks (or even contact discon- the capabilities of the scheme, even when the turbulent tinuities) is beyond the purpose of the present paper, and Mach number is close to one. might generate a huge discussion. Moreover, it is abso- lutely not claimed here that the approach is the most adequate one. In particular, one should at least distinguish the 2 Set of equations adequacy of second-moment closures in compressible flows includingshock waves in terms of two distinct concepts. Before we examine the solution of the one-dimensional First of all, one may wonder whether continuous solutions Riemann problem associated with the whole governing set satisfy some basic “intrinsic” concepts such as realisability of equations, and then introduce the approximate Rie- (Lumley 1978), invariance under some frame rotation or mann solver which is used to compute turbulent com- translation (Speziale 1979), or some constraint pertaining pressible flows with shocks, we first recall some basic re- to thermodynamics; this will be briefly discussed below, sults connected with a simple strongly-realisable second- and many references to previous works will be recalled. moment closure, which is based on standard Favre’s (1965) A second way to evaluate these models is related to their averaging. All commonly used averaging symbols (over- ability to predict accurately unsteady complex situations tilda for the Favre averaging) have been dropped herein encountered in well-documented experiments; as recalled to avoid confusion between instantaneous and averaged C. Berthon et al.: An approximate solution of the Riemann problem 3 values of functions. Brackets – which are present in where τ represents some turbulent time scale. Basic con- some places – refer to the Reynolds averaging of variables, straints are recalled now. First of all, solutions of the whole instead of standard overbar notations. The latter will be closed set should be such that (8a–c) hold. In particular, used for the arithmetic average when discussing numerical (8a) gives: implementation in finite-volume codes. The nomenclature Rii ≥ 0(i =1, 2, 3) ; is intended to help readers who are not very familiar with R R − R2 ≥ 0(1≤ i ≤ j ≤ 3) ; these notations. Hence, the set of equations is (see the ii jj ij work by Vandromme and Ha Minh (1986) for a general presentation): 2 2 R11R22R33 +2R12R13R23 − R33R12 − R11R23 (ρ),t +(ρUi),i =0; (1) −R R2 ≥ 0 . 22 13 (ρU ) +(ρU U ) +(pδ ) +(R ) = − Σv (2) i ,t i j ,j ij ,j ij ,j ij ,j Any positive quantity θ (chosen amongthe fundamental (E) +(EU ) +(U (pδ + R )) = − U Σv minors of the Reynolds stress tensor R) should behave ,t j ,j i ij ij ,j i ij ,j as follows when vanishing(see Fu et al. 1987; Hérard k 1994, 1995a, 1996; Lumley 1978, 1983; Pope 1985; Shih p Φjj + σ + − (up) ; (3) and Lumley 1985; Shih et al. 1993): E ρ 2 i ,i j ,j ,k θM =0⇒{(Dtθ)M = 0 and (Dt(Dtθ))M ≥ 0} . (Rij),t +(RijUk),k + RikUj,k + RjkUi,k = −ui p,j 2 ε Lumley (1978) proposed a simple model for the pressure- −up + Φ − trace (R)δ +(Φk ) , (4) j ,i ij 3 I ij ij ,k strain term Φij, which is in agreement with the previous constraints. We thus focus on this closure, which accounts with for the return-to-isotropy process and fulfils the objec- R = ρuu ; (5) ij i j tivity requirement (Hérard 1994; Lumley 1978; Speziale 2 1979). Hence: Σv = −µ U + U − U δ . (6) ij i,j j,i 3 1,1 ij ε δ Iδ Φ = − 2+f 3 R − ij , Moreover, it is assumed that the perfect gas state law ij I I3 ij 3 holds. Hence: where δ denotes the product of the three eigenvalues of ρU U 1 3 p =(γ − 1) E − j j − R . (7) the Reynolds stress tensor, i.e. 2 2 jj δ3 = λ1λ2λ3 , Given some (normalised) vector n in IR3, admissible states should comply with the over-realisability constraints and I stands for the trace of the Reynolds stress tensor:

t Rnn(x,t)=n R(x,t)n > 0 (8a) I = Rii . The exact form of the function f is given in the original and be such that paper by Lumley (1978). ρ(x,t) ≥ 0 ; (8b) Finally, it is assumed that the turbulent velocity ﬁeld almost follows a Gaussian distribution, and we thus ne- p(x,t) ≥ 0 . (8c) glect the triple velocity correlation: k Φij =0. We also need to introduce:

t Un(x,t)=U (x,t)n . (9) 3 Some basic results

Above, ρ stands for the mean density, U is the mean ve- 3.1 Entropy inequality locity, E denotes the mean total energy and p is the mean pressure. The Reynolds stress tensor is R, and K is the We introduce the so-called “conservative” state variable: turbulent kinetic energy (K = trace (R)/2); γ (greater than one) is the ratio of speciﬁc heats and σE is a positive Wt =(ρ, ρU, ρV, ρW, E, R ,R ,R ,R ,R ,R ) . function; µ is the molecular viscosity. A standard equation 11 22 33 12 13 23 governs the motion of the mechanical dissipation ε. The Focusingon regularsolutions of set (1-4), the following turbulent mass ﬂux is modelled accordingto the proposi- may be easily derived: tion of Ristorcelli (1993): Proposition 1.

Rij Define: u = τ ρ , −γ i ρ2 ,j η(W)=−ρ log(pρ ) 4 C. Berthon et al.: An approximate solution of the Riemann problem and The parameter θ is assumed to lie in [0,1]. Numerical sim- nv fη (W)=Uη. ulations of the system involved in step I will be depicted later setting θ to zero. Note that all second order terms Regular solutions W of set (1–4) are such that: in step II are in conservative form, unless turbulent mass η +∇·(f nv(W))+∇·(f v(W, ∇W)) = S (W, ∇W) ≤ 0 . fluxes are accounted for. Regular enough solutions of step ,t η η η I agree with The source term may be written as: γ − 1 1 η +(U η) = −θ − Σv (U + U ) ≤ 0 , ,t j ,j T 2 ij i,j j,i γ − 1 Sη(W, ∇W)=− T the non-viscous limit of which being σ 1 × E T T + ε − Σv (U + U ) T ,i ,i 2 ij i,j j,i η,t +(Ujη),j ≤ 0 . τ In a similar way, smooth solutions of step II satisfy −(γ − 1) ρ,iRijρ,j ρ2 v η,t + (fη )j (notingthat T = p/ρ). The viscous flux is such that ,j v γ − 1 σ 1 − θ fη (W, ∇W)=0if∇W = 0. Thus, the entropy inequality E v = − T,iT,i + ε − Σij(Ui,j + Uj,i) reduces to T T 2 nv −σ[η]+ fη ≤ 0 −(γ − 1)uj ρ,j . in the non-viscous limit. It will be helpful to connect states through genuinely non-linear fields in a physical way, when This results in neglecting viscous effects. This entropy inequality is simi- η,t ≤ 0 lar to one arisingwhen investigatingone-and two-equati- in the non-viscous limit, whenever turbulent mass fluxes on models (Forestier et al. 1997; Hérard 1995b). are neglected or chosen to be in agreement with the former 2 proposition: ui = τ(Rij/ρ )ρ,j. This approach may be extended to the framework of some more complex closures 3.2 A conservative entropy-consistent where so-called rapid pressure-strain correlations are ac- splitting-up technique counted for in a suitable way. This is not discussed herein. From now on, we focus on the most difficult part, We now introduce the followingsplitting-uptechnique, namely step I. which requires solvinga first-order differential system in- cludinga very small amount of laminar viscous effects, and a second step which treats both source terms and second 3.3 Hyperbolicity criteria order terms. The main advantage is that this technique for the non-conservative convection system enables includingmass flux contributions (as described above), if needed. Another interestingpoint is that it iso- We focus now on statistically two-dimensional turbulence, lates terms arisingfrom turbulence modellingin step II, and hence assume that whereas step I only involves non-controversial contributions. Hence, we set: R13 = R23 =0,W=0 Step I: and also ϕ,3 =0 (ρ),t +(ρUi),i =0; whatever ϕ stands for. We also introduce a “2D conserv- (ρU ) +(ρU U ) +(pδ ) +(R ) = −θ Σv ; i ,t i j ,j ij ,j ij ,j ij ,j ative” state variable: (E) +(EU ) +(U (pδ + R )) = −θ U Σv ,t j ,j i ij ij ,j i ij ,j t Z =(ρ, ρU, ρV, E, R11,R22,R12,R33) . (Rij),t +(RijUk),k + RikUj,k + RjkUi,k =0. The non-conservative convective subset arisingfrom (1–4) Step II: (or step I) reads:

2 2 (ρ),t =0; nc v Z,t + (Fi(Z)),i + Ai (Z)Z,i =0. (10) (ρUi),t = −(1 − θ) Σij ; ,j i=1 i=1 p (E) = −(1 − θ) U Σv + σ This enables to derive ,t i ij ,j E ρ j ,j Proposition 2. − (up) ; Deﬁne: i ,i 1/2 2 ε γp Rnn (R ) = −up −up + Φ − trace (R)δ . c1 = +3 ; ij ,t i ,j j ,i ij 3 I ij ρ ρ C. Berthon et al.: An approximate solution of the Riemann problem 5

R 1/2 4 An approximate solution c = nn . 2 ρ of the one-dimensional Riemann problem (i) The two-dimensional non-conservative first order par- From now on we restrict ourselves to shocks of small am- tial differential set (10) is a non-strictly hyperbolic system plitude. We afterwards connect states (ρ−1, U, V , p, R , if (8a–8c) holds. Eigenvalues are: 11 R22, R33, R12) with a linear path (see Dal Maso et al. λ = U − c ; 1995; Le Floch 1988; Le Floch and Liu 1992; Sainsaulieu 1 n 1 1995a,b) across discontinuities. If σ stands for the speed λ = U − c ; of a travellingdiscontinuity, the followingapproximate 2 n 2 r jump conditions arise (we set [ϕ] = ϕr − ϕl and ϕ = λ = λ = λ = λ = U ; l lr 3 4 5 6 n (ϕr + ϕl)/2): r r λ = U + c ; −σ[ρ]l +[ρU]l =0; 7 n 2 −σ[ρU]r + ρU 2 + R + p r =0; λ8 = Un + c1 . l 11 l r r (ii) In a one-dimensional framework, the 1-wave and the −σ[ρV ]l +[ρUV + R12]l =0; 8-wave are genuinely non-linear; other fields are linearly −σ[E]r +[U(E + R + p]r +[VR ]r =0; degenerate. l 11 l 12 l r r r The computation of eigenvalues and right eigenvectors −σ[R11]l +[UR11]l +2(R11)lr[U]l =0; is tedious but straightforward. We refer to the books by −σ[R ]r +[UR ]r +2(R ) [V ]r =0; Godlewski and Raviart (1996), Smoller (1983) for basic 22 l 22 l 12 lr l concepts and definitions to investigate hyperbolic systems r r r r −σ[R12]l +[UR12]l + (R11)lr[V ]l + (R12)lr[U]l =0; under conservation form. It seems worth notingthe occur- r r rence of two new contact discontinuities with respect to −σ[R33]l +[UR33]l =0. the Euler framework. It must be noted that the eigenvalue Obviously, these approximate jump conditions turn out problem is “ill-conditioned”, due to the fact that the tur- to be the exact ones when the turbulence vanishes (i.e. 1/2 bulent Mach number, which is proportional to (K/p) is setting R11 = R22 = R33 = R12 = 0). Owingto the usually much smaller than one. A sketch of waves is given entropy inequality in Fig. 1. nv −σ[η]+ fη ≤ 0 , it may be proven that the 1D Riemann problem with suf- ficiently weak shocks has a unique entropy-consistent solution. Proposition 3. Define:

1/2 γpi Xi = ρi (γ−1)/2 3−γ 1/2 ρi a (R ) a da × 1+3 nn i 0 ρi γpi ρi a

Fig. 1. Sketch of waves: GNL – genuinelynon-linear wave; LD for i = L, R. Assume that initial data is in agreement – linearlydegenerate wave with (8a–c). Then, the one-dimensional Riemann problem associated with the non-conservative system (10), above- mentioned approximate jump conditions, and initial data Let us note also that, due to the fact that in most practical applications the turbulent intensity K, which varies Z(x<0 ,t=0)=ZL , as Rnn, is small as compared with either the sound propa- Z(x>0 ,t=0)=ZR gation in the laminar case or even with the mean velocity has a unique solution in agreement with conditions (8a) if U in the turbulent case, intermediate states Z2 and Z6 will be diﬃcult to distinguish from each other; this will be con- the following condition holds: ﬁrmed by numerical experiments. This remark still holds in the multi-dimensional case, whenever one uses struc- (Un)R − (Un)L

Remark 1. The above-mentioned condition of existence scheme (Harten et al. 1983), or its extension such as the and uniqueness is the counterpart of the well-known con- HLLC scheme (Toro 1997) in order to account for the dition of vacuum occurrence when investigating the one- three distinct contact discontinuities, are also natural can- dimensional Riemann problem in gas dynamics (Smoller didates for that purpose. Sample results with the help of 1983) and restrictingto the perfect gasstate laws. It must the Rusanov scheme and an approximate form of the Roe be emphasized that the realisability of the Reynolds stress scheme will be very briefly discussed as well. tensor is preserved, through contact discontinuities, but In order to simplify the presentation, we now consider also through genuinely non-linear fields, whenever these statistically one-dimensional flows in the x-direction, and turn out to be shocks or rarefaction waves (see Appendix thus define a “conservative” vector state Zt =(ρ, ρU, ρV, E, A5). R11,R22,R12,R33), and introduce a matrix B1(Z): Remark 2. It is clear that the solution of the one-dimensi- dF B (Z)= 1 (Z)+Anc(Z) . (12) onal Riemann problem is close to that associated with 1 dZ 1 the Riemann problem arisingin one- or two-equation turbulent models (Forestier et al. 1997). In the present case, We recall that the leadingvariables are, in fact, ρ, U , p, R ; this means   n nn ρU that for given initial data for ρ, Un,p,Rnn on both sides    ρU 2 + p + R  of the initial interface, the solution is completely deter-  11   ρUV + R  mined, independently of the values for the remainingcom-  12    ponents. This is due to the fact that the above quantities  U(E + p + R11)+VR12  remain constant through the two new contact discontinu- F1(Z)=  . (13)  UR11  ities, since we get, using notations introduced in Fig. 1:    UR22  2 2 2 2   [ρ]1 =[Un]1 =[p]1 =[Rnn]1 =0;  UR12  7 7 7 7 [ρ]6 =[Un]6 =[p]6 =[Rnn]6 =0. UR33 We introduce a constant time step ∆t, integrate Eq. (10) 5 An approximate Riemann solver over cell Ωi, which provides: We focus here on the computation of step I using θ =0. Z(x, tn+1)dx − Z(x, tn)dx

The computation of viscous fluxes, which are present in Ωi Ωi the right hand side of equations (2) and (3), requires only tn+1 the implementation of central schemes. Moreover, suitable nc + (F1(Z(x, t))),x + A1 (Z(x, t))Z,x dt dx =0. ways to implement source terms in step II (or, equiva- n t Ωi lently, the so-called “slow terms” in the right hand side of The explicit (first-order) numerical scheme is: equation (4)) were discussed in a previous work (Hérard 1995a). vol(Ω ) Zn+1 − Zn i i i num n n +∆t F1 (Z )dΓ + ∆tSi(Z )=0, (14) 5.1 Introductory remarks Γi We obviously restrict ourselves here to finite-volume tech- where: niques (Eymard et al. 2001) due to the great complexity S (Zn) of the whole non-linear system. On the basis of the above- i  mentioned result, a classical Godunov scheme – in the 0   limit of weak shocks – based upon the linear path, may  0    be developed to compute convective effects (Forestier et  0    al. 1997); this, however, requires a tremendous amount of  0    CPU time. Thus, if we intend to avoid complexity, the = ˆ ˆ  2 R11 Ui+1/2 − Ui−1/2  only schemes of practical interest seem to be approxi-  i   ˆ ˆ  mate Riemann solvers. Since no exact Roe-type Riemann  2 R12 Vi+1/2 − Vi−1/2   i  solver (Roe 1981) based on a state average may be exhib-  ˆ ˆ ˆ ˆ   R11 Vi+1/2 − Vi−1/2 + R12 Ui+1/2 − Ui−1/2  ited in the present case (Uhlmann 1997), we present now i i an extension of the approximate Riemann solver called 0 VFRoe which was recently proposed in order to deal with (15) complex hyperbolic systems (Gallouët and Masella 1996; Masella et al. 1999). Actually, other extensions of stan- The most suitable way to compute the so-called “source” dard schemes to the framework of non-conservative sys- terms, which is based on the recent proposals by Forestier tems might be used for computational purposes, but we et al. (1997) and Masella (1997), is described below. We nonetheless restrict our presentation to VFRoe with non- now present the main features of the approximate Rie- conservative variables. It seems clear that either the HLL mann solver. C. Berthon et al.: An approximate solution of the Riemann problem 7

∗ 5.2 A simple Riemann solver Y = YL if λ1 > 0; ∗ Though the scheme presented below is slightly different Y = Y1 if λ1 < 0 and λ2 > 0; ∗ from the original VFRoe scheme (see Gallouët and Masella Y = Y2 if λ2 < 0 and λ3 > 0; 1996), it has been shown that it provides similar rates ∗ of convergence, when focusing on the isentropic or full Y = Y6 if λ6 < 0 and λ7 > 0; ∗ Euler set of equations. On the basis of a series of ana- Y = Y7 if λ7 < 0 and λ8 > 0; lytical test cases involvingshocks, contact discontinuities ∗ and rarefaction waves, it was checked in this framework Y = YR if λ8 < 0 , that the present VFRoe scheme is as accurate as Roe’s where scheme. This new approach has previously been used to λ1 = U − c1 ; λ2 = U − c2 ; compute some two-phase flows (in this case, no Roe-type λ = λ = λ = λ = U ; approximate Riemann solver can be constructed, see the 3 4 5 6 work of Declercq-Xeuxet 1999 and Faucher 2000), using λ7 = U + c2 ; λ8 = U + c1 ; finite-volume techniques. and also First, using(12), Eq. (10) may be rewritten as follows: 1/2 c1 = τ γp +3R11 ; Z(Y),t + B1(Z(Y)) (Z(Y)),x =0, 1/2 c2 = τR11 . (restrictingto regularsolutions) where The numerical flux in the first integral on the left hand side of (14) is then set as: Yt =(ρ−1,U,V,p,R ,R ,R ,R ) . 11 22 12 33 num n ∗ F1 (Z )=F1 (Z(Y )) . Thus we have The most obvious way to compute integrals in (15) is to −1 apply central schemes: (Y),t + Z (Y)(B)1(Z(Y))ZY(Y)(Y),x =0, Y ˆ n ˆ n n or equivalently: φi = φi and φi+1/2 = φi + φi+1 /2 . This approach is suitable when usingRoe-type schemes (Y),t + C1(Y)(Y),x =0. (16) (modified to account for non-conservative terms, Hérard The 1D Riemann problem associated with the non-linear 1995a,b). However, the most stable scheme when applied hyperbolic system (16) and initial conditions to Godunov-type schemes is obtained via ∗ ∗ φi−1/2 + φi+1/2 y(x<0,t=0)≡ Y = Y(Z ) , φˆ = and φˆ = φ∗ L L i 2 i+1/2 i+1/2 y(x>0,t=0)≡ Y = Y(Z ) , R R (to our knowledge, first proposed by Masella 1997). is linearized as follows: Remark 3. The latter form (Masella 1997) to account for non-conservative terms is the straightforward counterpart (Y),t + C1(Y)(Y),x =0, (17) of the Godunov scheme introduced previously to compute two-equation turbulent compressible closures (Louis where Y =(YR + YL)/2. System (17) contains five dis- 1995). tinct linearly degenerate fields, and its solution is triv- ial, since it only requires computingeight real coefficients Remark 4. It should be emphasized that the present VFRoe scheme turns out to be the original Godunov scheme when noted αi (for i = 1 to 8), setting consideringthe scalar Burgersequation rewritten in non- 8 conservative YR − YL = αiˆri , form, provided that the “star” value is the exact solu- i=1 tion of the Riemann problem on the initial interface ξ = x/t =0. where (ˆri) represents the basis of right eigenvectors of the Remark 5. Obviously, an entropy correction is required matrix C1(Y). Intermediate states are then uniquely de- fined (see Appendix B for details) as: (Buffard et al. 1998, 1999, 2000; Gallouët and Masella 1996; Masella 1997; Masella et al. 1999) to compute shock

Y1 = YL + α1ˆr1 ; tube problems when one sonic point is present in the 1- rarefaction wave (respectively, in the 8-rarefaction wave). Y = Y + α ˆr + α ˆr ; 2 L 1 1 2 2 Remark 6. An interestingproperty of the approximate Rie- Y6 = YR − α8ˆr8 − α7ˆr7 ; mann solver may be exhibited (see Buﬀard et al. 2000 for a straightforward counterpart in the Euler framework): Y = Y − α ˆr ; 7 R 8 8 when consideringa singlediscontinuity, it occurs that the and hence the state Y∗ at the initial location of the data jump conditions associated with (17) are equivalent to discontinuity is given by those detailed in Sect. 4. 8 C. Berthon et al.: An approximate solution of the Riemann problem

6 Turbulent shock tube problems and a rather fine uniform mesh (5000 nodes in the x direction). This is a difficult test case, since both the tur- Shock tube problems are of prime importance to validate bulent Mach numbers and the anisotropy of the Reynolds the numerical treatment of convective terms. Before we stress tensor are by no way negligible. Thus, there exists a examine the capabilities of the scheme to predict the be- strongcouplingnot only between various Reynolds stress haviour of turbulent shock tube flows, we briefly depict its components, but also between mean variables and second speed of convergence when applied to the simulation of the moments. The time step is in agreement with the above- Euler equations with the perfect gas state law (the ratio mentioned CFL condition. of specific heats γ is assumed to be equal to 7/5). When Similar to the pure gas dynamics case, a subsonic rar- dealingwith the Euler equations, the results obtained with efaction wave propagates to the left, and a shock wave the present scheme are in excellent agreement with those travels to the right. The behaviour of the normal (axial) reported by Gallouët and Masella (1996), Masella (1997), component of the velocity U is quite similar to its counter- Masella et al. (1999) and were also compared with those part in gas dynamics; however, the tangential (cross) ve- obtained usingRoe’s scheme or the Godunov scheme. Ac- locity V no longer remains null (Fig. 2b) due to the strong tually (Buffard et al. 2000), the real rate of measured con- shear induced by the non-zero component R . Riemann vergence, using L1 norm, is around 0.82 for both velocity 12 invariants U, p + R are well preserved through the 2- and pressure variables when usingthe first order scheme 11 wave, the 3-4-5-6 wave and the 7-wave (see Fig. 2a, b). It (respectively, 0.98 when usinga MUSCL reconstruction should also be emphasized that the mean density (Fig. 2c) on primitive variables) and a bit lower (approximately and the Reynolds stress component R (Fig. 2d) do not 0.66) for density. This is due to the fact that both U 11 vary across the 2-wave and the 7-wave, accordingto the and p remain constant through the contact discontinu- respective theoretical results detailed in Appendix A. Nu- ity of the Euler set of equations and only vary through merical results obtained usinga coarse mesh are of very genuinely non-linear fields. The scheme has also been ap- poor quality, since the two new waves (the 2-wave and the plied in order to compute the Euler equations with real 7-wave) can hardly be distinguished. gas equations-of-state, using either structured or unstruc- Some other notations are introduced in Fig. 2g: tured meshes (Buffard et al. 2000). The behaviour of the R R scheme is also fairly good when applied to shallow-water V + = V + 12 ; V − = V − 12 . 1/2 1/2 equations, even when “vacuum” occurs in the solution (ρR11) (ρR11) or when dam-break waves are computed (Buffard et al. Let us recall that V + (respectively, V −) is a Riemann 1998). The extension of the scheme to non-conservative invariant of the 2-wave (respectively, the 7-wave), see Ap- hyperbolic systems was first performed by Buffard et al. pendices A2, A3. Figure 2f shows that the mean pressure (1999). p varies considerably through the contact discontinuity All computations discussed below have been obtained associated with the 3-4-5-6 wave, which is a direct con- with a CFL number equal to 1/2: (∆t/h) max (|λ |, |λ |)= 1 8 sequence of the presence of turbulence (this has already 0.5 . This enables to ensure that non-linear waves do not been pointed out by Hérard (1995a,b) and Louis (1995) interact within each cell. In practice, almost all compu- when consideringtwo-equation models). It appears clearly tations might be performed using a higher value (around that the 2-wave and the 3-4-5-6 wave can hardly be dis- 0.9), except probably when very strongshock waves oc- tinguished even on this rather fine mesh; this results in a cur or when strongdouble rarefaction waves develop. The rather strange behaviour of V + due to the fact that the unit normal vector is assumed to be tangent to the x-axis: turbulent component R on the left hand side of the con- n =(1, 0, 0)t. Thus, U and U = U · n are identical here 11 n tact discontinuity associated with the eigenvalues λ − λ (a similar remark holds for V and U = U · τ). 3 6 τ in Proposition 2 is indeed very small (as compared with We present now the numerical results obtained with the left initial value of R ). All Reynolds stress compo- the initial conditions which are similar to those pertain- 11 nents (except R ) are plotted together in Fig. 2h. Figure ingto the Sod shock tube problem, consideringthe basic 33 2e enables to check that the determinant R R − (R )2 set of “laminar” variables and addingnew initial condi- 11 22 12 remains positive in the (x, t) plane. tions on each side of the initial interface for “turbulent” The next computational results presented in this sec- variables, i.e. R ,R , R , R . Figure 2 displays the 11 22 33 12 tion, which were also obtained on a uniform grid with computed solution of a Riemann problem associated with 5000 nodes, are displayed in Fig. 3. The CFL number is (10), usingthe basic first order time-space scheme, the still equal to 0.5. The initial data, which is now followinginitial data        ρ−1 =1 ρ−1 =8 ρ−1 =1 ρ−1 =1          U =0   U =0   U = 100   U = −100           V =0   V =0   V =0   V =0           5   4   5   5   p =10   p =10   p =10   p =10  YL =  3  ; YR =  3  , YL =  5  ; YR =  5  ,  R11 =10   R11 =10   R11 =10   R11 =10   3   3   4   4   R22 =10   R22 =10   R22 =10   R22 =10   2   2   3   3   R12 =5· 10   R12 =5· 10   R12 =5· 10   R12 =5· 10  3 3 4 4 R33 =10 R33 =10 R33 =10 R33 =10 C. Berthon et al.: An approximate solution of the Riemann problem 9

a : P + R11 b : axial and cross (−−) velocity 300.0 140000.0

120000.0 200.0

100000.0

80000.0 100.0

60000.0

40000.0 0.0

20000.0

0.0 −100.0 0.0 2000.0 4000.0 0.0 2000.0 4000.0

c : density d : R11 1.0 10000.0

0.8 8000.0

0.6 6000.0

0.4 4000.0

0.2 2000.0

0.0 0.0 0.0 2000.0 4000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 2a–h. The ﬁrst test case: a Sod-type shock tube problem 10 C. Berthon et al.: An approximate solution of the Riemann problem

e : R11*R22 − R12*R12 f : pressure 15000000.0 100000.0

80000.0

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20000.0

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g : Riemann invariants V+, V− h : R12 , R22 (−−) 100.0 3000.0

50.0

2000.0

0.0

1000.0

−50.0

−100.0 0.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 2. (continued) C. Berthon et al.: An approximate solution of the Riemann problem 11

a : P + R11 b : axial and cross (−−) velocity 280000.0 100.0

260000.0 50.0

240000.0 0.0

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c : density d : R11 1.30 160000.0

1.25

1.20 140000.0

1.15

1.10 120000.0

1.05

1.00 100000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 3a–h. The second test case: a strong double shock wave 12 C. Berthon et al.: An approximate solution of the Riemann problem

e : R11*R22 − R12*R12 f : pressure 1800000000.0 130000.0

1600000000.0

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1000000000.0

800000000.0 100000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0

g : Riemann invariants V+, V− h : R12, R22 (−−) 40.0 12000.0

30.0

20.0 10000.0

10.0

0.0 8000.0

−10.0

−20.0 6000.0

−30.0

−40.0 4000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 3. (continued) C. Berthon et al.: An approximate solution of the Riemann problem 13

generates a strong double shock wave. The high initial ra- the stress component R11 around the characteristic line tio R11/p has been chosen intentionally, in order to check x/t = 0, all three contact discontinuities tend to merge, the capability of the scheme to handle high turbulent as may be seen in Fig. 4g. Mach numbers, which seems to be compulsory when one In Fig. 5 we provide some comparison between the nu- aims at investigating the behaviour of jets impinging on merical results obtained when applyingthe above scheme wall boundaries. When lookingat the predicted values of (VFRoe) and an extension of the Rusanov (1961) scheme the mean density and some Reynolds stress components, to the framework of non-conservative systems (Appendix a small glitch located at the position of the initial discon- C), followingthe initial proposition by Hérard (1995b). tinuity is visible. It is due to the fact that a perturbation Similar results may also be found in the work of Périgaud initially created there is hardly smoothed out since the and Archambeau (2000). The main purpose is to demon- (null) eigenvalue associated with λ3−6 vanishes there dur- strate that rather simple schemes may provide meaningful ingthe whole computational time. This test case is indeed results, if the overall eigenstructure is accounted for. The interestingsince it enables to predict the behaviour of the initial conditions of the test case are the same as those scheme when computingjets impingingonwall bound- of the previous one. It is clear that both computational aries in two- or three-dimensional geometries. This is also results are very similar. The Rusanov scheme performs in one of the best test cases to illustrate the failure of (un- this test case rather well since neither contact discontinu- coupled) approximate Riemann solvers based on the wave ity nor shock wave are present in the solution. The small structure of the Euler set of equations (Uhlmann 1997). density spike seen in the solution is more smeared by the Results obtained with an approximate version of the Roe extended Rusanov scheme (Fig. 5d). It may be noted that scheme are also briefly described in Appendix D (Fig. 6). this robust scheme enables to handle vacuum occurrence The third series of figures (Fig. 4) corresponds to the (see the review on the basis of the Euler equations by computation of a strongdouble rarefaction wave (still us- Seguin 2000 and Gallouët et al. 2000). inga uniform gridwith 5000 nodes) with the help of the VFRoe scheme of Sect. 5.2. The CFL number is again equal to 0.5, and the initial data 7 Conclusion     ρ−1 =1 ρ−1 =1 The main aim of the paper is to present an approximate      U = −1000   U = 1000  solution of the one-dimensional Riemann problem associ-      V =0   V =0  ated with an objective realisable second-moment closure     which is valid for compressible flows. Basic analysis of the  5   5   p =10   p =10  non-conservative first order differential system obviously YL =  5  ; YR =  5   R11 =10   R11 =10  indicates that the structure of internal waves is quite dif-      R =103   R =103  ferent from its counterpart in non-turbulent flows (the Eu-  22   22  3 3 ler equations). This analysis is crucial since the amount of  R12 =10   R12 =10  R =103 R =103 physical diffusive effects in these turbulent models is much 33 33 smaller than that in two-equation models. generates a strong double rarefaction wave. This may rep- An algorithm has been suggested in order to compute resent a schematic view of the flowfield behind a bluff this second-moment closure. Numerical solutions confirm body. Although it is not the case here, we recall that these that the present finite-volume technique provides rather initial conditions may lead to the occurrence of vacuum satisfactory results (in particular, diagonal components as 2 when the modulus of the normal velocity is high enough well as fundamental minors, such as R11R22 − (R12) , re- (see the condition arisingin Proposition 3 in Sect. 4). The main positive). However, it becomes clear that accurate mean axial velocity (Fig. 4b) is not a linear function of computations of the two specific “slow” waves associated x/t, unless γ is equal to 3, since the effective celerity c with λ2 and λ7 require usingeither fine meshes or at least varies as a second-order extension. As might have been expected, approximate Riemann solvers based on the Euler set of 1/2 P ρ γ−1 (R ) ρ 2 equations should not be used since they may provide spuri- c = γ L +3 11 L ρL ρL ρL ρL ous solutions (Uhlmann 1997), for instance when comput- ingflows in the vicinity of wall boundaries, and hence are 1/2 P 1/2 ρ (γ−1)/2 3 (R ) ρ 3−γ not suitable to handle computations of compressible tur- = γ L 1+ 11 L ρL ρL γ PL ρL bulent flows usingthe second-order approach. The extension of the Roe-type scheme introduced by Hérard (1995b) in the 1-rarefaction wave (respectively, in the 8-rarefac- to compute non-conservative hyperbolic systems also pro- tion wave). The cross velocity is non-zero, as may be vides rather good results when computing single-phase checked in Fig. 4b. The small variation of the mean den- turbulent compressible flows with shocks usingsecond- sity (Fig. 4c) around the initial interface is characteristic moment closures. This was recently demonstrated by Uhl- of the Godunov approach. Again, although the minimum mann (1997). A sample result is described in Appendix D. value of the product of eigenvalues is indeed small as com- Moreover, the simple extension of the Rusanov scheme has pared with the left (or right) values, no loss of positivity also been shown to provide rather nice results (Périgaud is observed. Due to the very small order of magnitude of and Archambeau 2000). 14 C. Berthon et al.: An approximate solution of the Riemann problem

a : P + R11 b : axial and cross (−−) velocity 200000.0 1000.0

150000.0 500.0

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0.0 0.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 4a–h. The third test case: a double rarefaction wave C. Berthon et al.: An approximate solution of the Riemann problem 15

e : R11*R22 − R12*R12 f : pressure 100000000.0 100000.0

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g : Riemann invariants V+, V− h : R12, R22 (−−) 10.0 1000.0

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−10.0 0.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 0.0 1000.0 2000.0 3000.0 4000.0 5000.0 Fig. 4. (continued) 16 C. Berthon et al.: An approximate solution of the Riemann problem

Fig. 5a–hThe third test case: a double rarefaction wave. Results for the Rusanov (a–f – dashed lines and g) and VFRoe (a–f – solid lines and h) schemes. Dashed and solid lines in g and h correspond to the Riemann invariants V+ and V−, respectively C. Berthon et al.: An approximate solution of the Riemann problem 17

Fig. 5. (continued) 18 C. Berthon et al.: An approximate solution of the Riemann problem

In the present paper we were not aimingat the “ul- A1. Eigenvalues and right eigenvectors timate” or the most well-suited scheme to deal with this non-conservative set of equations. Other extensions to the non-conservative framework of the well-known schemes It may be easily checked that the eigenvalues of the matrix such as the Osher scheme, the HLL scheme (Harten et al. associated with the above-mentioned first-order differen- 1983), or a suitable counterpart of the HLLC scheme (Toro tial system are et al. 1994; Toro 1997) in order to account for the three contact discontinuities, the AUSM scheme (Liou 1996), or λ1 = U − c1 ; λ2 = U − c2 ; (A1.1) other approximate Godunov schemes, such as a counterpart of PVRS scheme (Toro 1997), are certainly poten- λ = λ = λ = λ = U ; (A1.2) tially good candidates, too. 3 4 5 6 An ongoing effort is currently directed towards the comparison of several suitable schemes for the present pur- λ7 = U + c2 ; λ8 = U + c1 , (A1.3) pose. Obviously, a MUSCL-type second order extension of 1/2 1/2 the present scheme combined with second order time in- with c1 =(γpτ +3τR11) and c2 =(τR11) . tegration (using the RK2 scheme, for instance) is easily Right eigenvectors are as follows: feasible, using( ρ, U, p) variables and Reynolds stress com-     ponents. Implementation of the scheme in a finite volume 1 0 code usingunstructured meshes is in progress.          0   0           0   0  Appendix A          0   0      r3 =   ; r4 =   ; We first introduce τ =1/ρ. We also recall that  0   0              ρUjUj 1  0   0  p =(γ − 1) E − − Rjj .     2 2      0   0  The followingcan be then derived usingthe left hand sides 0 1 of (1)–(4): τ,t + Ujτ,j − τUj,j =0;     0 0     Ui,t + UjUi,j + τp,i + τRij,j =0;      0   0      p + U p + γpU =0;     ,t j ,j j,j  0   0          R + U R + R U + R U + R U =0,  1   0  ij,t k ij,k ij k,k ik j,k jk i,k     r5 =   ; r6 =   ;  −1   0  when consideringregularsolutions. We now restrict our-     selves to a two-dimensional framework, use invariance un-         der rotation, and neglect transverse variations (y-direction);  0   1      hence, the local one-dimensional convective system re-  0   0  duces to:     0 0 τ,t + Uτ,x − τU,x =0;

U + UU + τp + τR =0;     ,t ,x ,x 11,x 0 0     V + UV + τR =0;     ,t ,x 12,x  0   0          p + Up + γpU =0;  c2   c2  ,t ,x ,x          0   0  R11,t + UR11,x +3R11U,x =0;     r2 =   ; r7 =   ;  0   0      R22,t + UR22,x + R22U,x +2R12V,x =0;          −2R12   2R12      R12,t + UR12,x +2R12U,x + R11V,x =0;      −R11   R11  R33,t + UR33,x + R33U,x =0. 0 0 C. Berthon et al.: An approximate solution of the Riemann problem 19

  τ usingthe followingnotations:   τ c2{a, s ,s } da   ϕ = exp 2 1 1 2 ;  c1  2   0 (γp +2R11){a, s1,s2} a  −1   2R12c1(γp +2R11)  τ   c1{a, s1,s2}R12 da   θ = −2 .  −γp  0 (γp +2R11){a, s1,s2} a   r1 =   ;  −3R   11    A3. Shocks and contact discontinuities  −(R +4R2 (γp +2R )−1)   22 12 11     −2R c2(γpτ +2τR )−1  We start from:  12 1 11  −σ[ρ]+[ρU]=0; −R 33 −σ[ρU]+[ρU 2 + R + p]=0;   11 −τ −σ[ρV ]+[ρUV + R12]=0;      c1  −σ[E]+[U(E + R11 + p)]+[VR12]=0;    −1   2R12c1(γp +2R11)  −σ[R11]+[UR11]+2R11[U]=0;      γp  −σ[R22]+[UR22]+2R12[V ]=0;   r8 =   . (A1.4)  3R  −σ[R12]+[UR12]+R11[V ]+R12[U]=0;  11     R +4R2 (γp +2R )−1  −σ[R33]+[UR33]=0.  22 12 11    Contact discontinuities:  2R c2(γpτ +2τR )−1   12 1 11  (i) Through the 2-wave, we have R33 2 2 2 2 [U]1 =[p]1 =[ρ]1 =[R11]1 =0, (A3.1)

A2. Riemann invariants so that 2 2 Riemann invariants may now be computed; ρ(U − σ)[V ]1 +[R12]1 =0; 2 2 1-wave (GNL – genuinely non-linear): (U − σ)[R22]1 +2R12[V ]1 =0; 2 2 τ (U − σ)[R12]1 + R11[V ]1 =0. γ 3 c1(a, s1,s2) φ = s1 = pτ ,s2 = R11τ ,U − da, a Hence 0 σ = U − (R /ρ)1/2 (A3.2) 2 4 11 (R11R22 − (R12) )τ ,R12ϕ, V + θ, R33τ ; (A2.1) and 2-wave (LD – linearly degenerate): 2 [V ]2 = − (R ρ)−1/2R ; (A3.3) 1 11 12 2 1 φ = τ,U,p,R11,R11R22 − (R12) , 2 2 2 2 R11[R22]1 − 2R12[R12]1 =0⇔ [R11R22 − (R12) ]1 =0. −1/2 V + R12(ρR11) ,R33 ; (A2.2) (A3.4)

3-4-5-6-wave (LD): Obviously 2 [R33]1 =0. (A3.5) φ = {U, V, p + R ,R } ; (A2.3) 11 12 (ii) A similar result holds through the 7-wave: 7-wave (LD): [U]7 =[p]7 =[ρ]7 =[R ]7 = 0 ; (A3.6) 6 6 6 11 6 φ = τ,U,p,R ,R R − (R )2, [R ]7 = 0 ; (A3.7) 11 11 22 12 33 6 7 V − R (ρR )−1/2,R ; (A2.4) 7 −1/2 12 11 33 [V ]6 = (R11ρ) R12 ; (A3.8) 6 8-wave (GNL): R R − (R )2 7 = 0 ; (A3.9) 11 22 12 6 1/2 τ c (a, s ,s ) σ = U +(R11/ρ) . (A3.10) φ = s = pτ γ ,s = R τ 3,U + 1 1 2 da, 1 2 11 a 0 (iii) Through the 3-4-5-6-wave, we get: 2 4 (R11R22 − (R12) )τ ,R12ϕ, V − θ, R33τ , (A2.5) 6 6 6 6 [U]2 =[V ]2 =[p + R11]2 =[R12]2 =0. (A3.11) 20 C. Berthon et al.: An approximate solution of the Riemann problem

2 R Shocks: [R11R22 −(R12) ]7 R (iv) Through the 1-wave, we connect states “L” and “1” [ρ]7 usinga real parameter z (z>1) and β =(γ +1)/(γ − 1): =2 (R11)7 +(R11)R (R22)7 +(R22)R ρ7 +ρR ρ = zρ ; (A3.12) − (R ) +(R ) 2 ; (A3.26) 1 L 12 7 12 R βz − 1 p = p ; (A3.13) 1 β − z L ρU − σρ 1 V 2z − 1 7,R 7,R R (R11)1 = (R11)L ; (A3.14) R 2 − z (R11)7,R U 7,R − σ +[U]7 (R12)R U1 = UL − (z − 1) × ρU 7,R − σρ7,R 1 V7 1/2 = .(A3.27) β +1 pL 3 (R11)L R × + . (A3.15) (R11)7,R U 7,R − σ − [U]7 (R12)7 (β − z)z ρL (2 − z)z ρL The speed of the 1-shock is: A4. Solution of the one-dimensional Riemann problem 1 U − U σ = U + U +(ρ + ρ ) L 1 . (A3.16) First, let us notice that U, p, ρ as well as R remain un- 2 1 l 1 L ρ − ρ 11 L 1 changed through the 2-wave and the 7-wave. This may be Besides, we get: veriﬁed using(A2.2), (A2.4), (A3.1), and (A3.6). Thus, in this subsection, we focus on the above subset of variables (R33)1 = z(R33)L ; (A3.17) and omit the 2-wave and the 7-wave; this simply means that at ﬁrst we are only lookingfor the values of interme-

2 1 diate states “1” and “7”. Moreover, a glance at (A3.11) [R11R22 − (R12) ] L shows that both U and p + R11 satisfy 1 [ρ]L =2 (R11)L +(R11)1 (R22)L +(R22)1 ρ +ρ U1 = U7 ; L 1 (A4.1) 2 − (R12)L +(R12)1 . (A3.18) (p + R11)1 =(p + R11)7 . We may introduce (z ,z ) such that The remainingtwo variables V and R12 are the solutions 1 2 of ρ1 = z1ρL ; (A4.2) ρU 1,L − σρ1,L 1 V1 ρR = z2ρ7 . 1 (R11)1,L U 1,L − σ +[U]L (R12)1 Then, owingto (A2.1), (A2.5) as well as (A3.13), (A3.14), ρU 1,L − σρ1,L 1 VL = . (A3.19) (A3.21), (A3.22), we may parametrize both p and R11 as 1 follows: (R11)1,L U 1,L − σ − [U]L (R12)L p1 = pLh1(z1) , (A4.3) (v) Through the 8-wave, we connect states “7” and “R” p = p h (z ); usinga real parameter z (z<1): R 7 2 2 (R ) =(R ) · g (z ) , ρ = zρ ; (A3.20) 11 1 11 L 1 1 R 7 (A4.4) βz − 1 (R11)R =(R11)7 · g2(z2); p = p ; (A3.21) R β − z 7 usingformula (A3.13), (A3.14) if z1 is greater than 1 (re- 2z − 1 spectively, (A2.1) if z1 < 1), and formula (A3.21), (A3.22) (R11)R = (R11)7 ; (A3.22) 2 − z if z2 < 1 (respectively, (A2.5) if z2 > 1). Normal velocities are linked through the following relations: UR = U7 +(z − 1) 1/2 β +1 p7 3 (R11)7 U1 = UL + f1 (z1; ρL; pL;(R11)L); × + . (A3.23) (A4.5) (β − z)z ρ7 (2 − z)z ρ7 UR = U7 + f2 (z2; ρR; pR;(R11)R) , The speed of the 8-shock is: owingto (A2.1), (A2.5), (A3.15), and (A3.23). Then, elim- 1 UR − U7 ination of (A4.3), (A4.4), and (A4.5) and substitution into σ = U7 + UR +(ρ7 + ρR) . (A3.24) (A4.1) provides: 2 ρR − ρ7

We also get: UR − UL = f1(z1; ρL; pL;(R11)L) (R33)R = z(R33)7 ; (A3.25) +f2(z2; ρR; pR;(R11)R) ; (A4.6) C. Berthon et al.: An approximate solution of the Riemann problem 21 −1 τ pLh1(z1)+(R11)L · g1(z1)=pRh (z2) L 2 (R33)1 =(R33)L ≥ 0 . −1 τ1 +(R11)R · g2 (z2) . (A4.7) Otherwise, if the 1-wave is a shock wave, we notice that, This coupled system with two unknowns admits a unique −1 owingto (A3.14), (A3.17) and (A3.18): solution (z1,z2)in[0,m]×[m , +∞] (where m is deﬁned as m = max{2, (γ +1)/(γ − 1)} ), provided that the fol- 2z1 − 1 lowingcondition holds: (R11)1 = (R11)L ≥ 0; 2 − z1 UR − UL

1/2 γpi [R R − (R )2]1 ≥ 0 ⇒ (R R − (R )2) Xi = 11 22 12 L 11 22 12 1 ρi 2 ≥ (R11R22 − (R12) )L ≥ 0 . 3−γ 1/2 ( a )(γ−1)/2 1+3(R11)i a ρi ρi γPi ρi A5.2. In the linearly degenerate 2-wave, we have, using × da, previous results: 0 a (A4.9) (R11)2 =(R11)1 ≥ 0; where i is either “L” (left state), or “R” (right state). Here 2 2 we indicate only the main lines of the proof. First of all, (R11R22 − (R12) )2 =(R11R22 − (R12) )1 ≥ 0; taking(A4.7) into account one may relate z1 to z2: (R33)2 =(R33)1 ≥ 0 . dz1(z2) z = z (z ) , where ≤ 0 . A5.3. Through the 8 rarefaction wave, we have, on the 1 1 2 dz 2 basis of (A2.5): Moreover, it may be checked that f is a decreasingfunc- 1 tion of z , and f is an increasingfunction of z . Thus, τ 3 1 2 2 (R ) =(R ) R ≥ 0; f1(z1(z2); ρL; pL;(R11)L)+f2(z2; ρR; pR;(R11)R)isanin- 11 7 11 R τ7 creasingfunction of z2. Eventually, the notion that 4 XL = lim f1(z1(z2); ρL; pL;(R11)L) , 2 2 τR z2→+∞ (R11R22 − (R12) )7 =(R11R22 − (R12) )R ≥ 0; τ7 XR = lim f2(z2; ρR; pR;(R11)R) z2→+∞ τ (R ) =(R ) R ≥ 0 . enables to conclude the proof. The reader is referred to 33 7 33 R τ Forestier et al. (1995), Herard (1995b), Louis (1995) for a L similar result. In the 8-shock wave, we get, using(A3.22), (A3.25), (A3.26):

A5. The Reynolds stress tensor is realisable −1 2z2 − 1 (R11)7 =(R11)R ≥ 0; We have to check now that Reynolds stress components 2 − z2 satisfy the followinginequalities: −1 (R33)7 =(R33)R z1 ≥ 0; R11 ≥ 0;

2 R11R22 − (R12) ≥ 0; 2 R 2 [R11R22 − (R12) ]7 ≤ 0 ⇒ 0 ≤ (R11R22 − (R12) )R 2 R33 ≥ 0 . ≤ (R11R22 − (R12) )7. We assume that (initial) left and right states fulﬁll the above mentioned conditions. A5.4. In the linearly degenerate 7-wave, we have, using results in A5.3: A5.1. Thus, in the 1-rarefaction wave, we have, on the basis of (A2.1): (R11)6 =(R11)7 ≥ 0; 3 τL 2 2 (R11)1 =(R11)L ≥ 0; (R11R22 − (R12) )6 =(R11R22 − (R12) )7 ≥ 0; τ1

4 (R33)6 =(R33)7 ≥ 0 . 2 2 τL (R11R22 − (R12) )1 =(R11R22 − (R12) )L ≥ 0; This completes the proof. τ1 22 C. Berthon et al.: An approximate solution of the Riemann problem

Appendix B notingthat cˆ2 = τR B1. The four intermediate states are given as follows: 2 11 and Y = Y + α ˆr ;     1 L 1 1 0 0 Y = Y + α ˆr + α ˆr ;     2 L 1 1 2 2      0   0  Y6 = YR − α8ˆr8 − α7ˆr7 ;         Y = Y − α ˆr .  cˆ2   cˆ2  7 R 8 8         Coefficients associated with the “GNL” fields are:  0   0      1 τ rˆ2 =   ;ˆr7 =   . α = [U]R − [p + R ]R ;  0   0  1 2ˆc L 2ˆc2 11 L     1 1     1 τ  −2R   2R  α = [U]R + [p + R ]R  12   12  8 L 2 11 L     2ˆc1 2ˆc1      −R11   R11  with 2 0 0 cˆ1 = τ(γp +3R11) and B2. Considering, first, the 2-wave, and owingto the fact   τ that   Y = Y + α ˆr ,   2 1 2 2  cˆ1    we immediately obtain that the numerical (approximate)  −1   2R cˆ γp +2R  intermediate states satisfy  12 1 11       −γp  [U]2 =[p]2 =[ρ]2 =[R ]2 =0, rˆ =   ; 1 1 1 11 1 1    −3R11    as they do in the continuous case. A similar result holds  2   − R +4R (γp +2R )−1  for the 7-wave:  22 12 11     2 −1  [U]7 =[p]7 =[ρ]7 =[R ]7 =0.  −2R12cˆ1 γp τ +2τR11  6 6 6 11 6

−R33 Turningthen to the 3-4-5-6 wave, we have that the numer-   ical values of the intermediate states “2” and “6” agree −τ with     [U]6 =[p + R ]6 =[V ]6 =[R ]6 =0.  cˆ1  2 11 2 2 12 2    −1   2R cˆ γp +2R  Hence, all these Riemann invariants are numerically pre-  12 1 11    served.    γp  rˆ =   . 8    3R11    Appendix C  2   R +4R (γp +2R )−1   22 12 11    We briefly present here the extension of the Rusanov sche-  2R cˆ2 γp τ +2τR −1   12 1 11  me which has been used to compute the non-conservative system R33 Z +(F (Z)) + Anc(Z)Z =0. (C.1) Coefficients associated with the two new “LD” fields are: ,t 1 ,x 1 ,x The integration over the cell Ω and the constant time 1 2R −cˆ i α = 12 2 [U]R +[p + R ]R step ∆t provides: 2 2R γp +2R τ L 11 L 11 11 cˆ vol (Ω ) Zn+1 − Zn + ∆t FRusanov(Zn)dΓ + 2 [V ]R − [R ]R ; i i i 1 τ L 12 L Γi +∆tπ (Zn)=0, (C.2) i 1 −2R12 cˆ2 R R α7 = [U]L +[p + R11]L where 2R11 γp +2R11 τ Rusanov n n n cˆ F (Z )= F1(Z )+F1(Z ) 2 R R 1 i i+1 + [V ]L +[R12]L , n n τ −ri+1/2(Zi+1 − Zi ) /2 . (C.3) C. Berthon et al.: An approximate solution of the Riemann problem 23

a : P + R11 b : axial and cross (−−−) velocity

280000.0 100.0

260000.0 50.0

240000.0 0.0

220000.0 −50.0

200000.0 −100.0 0.0 10.0 20.0 30.0 0.0 10.0 20.0 30.0

c : Mean density d : R11

160000.0

1.25

1.20 140000.0

1.15

1.10 120000.0

1.05

1.00 100000.0 0.0 10.0 20.0 30.0 0.0 10.0 20.0 30.0 Fig. 6a–d. The second test case: a strong double shock wave. Results for the extension of the Roe scheme to the non-conservative system 24 C. Berthon et al.: An approximate solution of the Riemann problem

We recall that the continuous flux is given by: The state Y is the same as in the main text: Yt =(ρ−1,   U, V , p, R11, R22, R12, R33). The continuous flux is still given by ρU     ρU  2   ρU + p + R11       2     ρU + p + R11   ρUV + R12           ρUV + R12   U(E + p + R )+VR     11 12    F1(Z)=  . (C.4)  U(E + p + R )+VR   UR   11 12   11  F1(Z)=  . (D.4)    UR     11   UR22           UR22   UR12       UR12  UR33 UR33 The spectral radius rspec (B )ofB (Z)=dF (Z)/dZ + 1 1 1 Λ(Z) and Ω(Z) represent the diagonal matrix of eigen- Anc(Z) is computed on each side of the interface and the 1 values and the matrix of right eigenvectors of the matrix resultingcoefficient is written as nc of convective effects B1(Z)=dF1(Z)/dZ + A1 (Z), respectively. The integration of the non-conservative part ri+1/2 = max{rspec ((B1)i) rspec((B1)i+1)} . (C.5) n πi(Z ) is exactly the same as in Appendix C related to Integration of the non-conservative part leads to: the Rusanov scheme. The initial conditions associated with the computa- n πi(Z )= tional results displayed in Fig. 6 are those of the Riemann   problem which corresponds to a double shock wave (see 0   the main text). Hence:    0    Yt = 1, 100, 0, 105, 105, 104, 5 · 103, 104 ;   L    0  Yt = 1, −100, 0, 105, 105, 104, 5 · 103, 104 .   R    0  The uniform mesh contains only 500 nodes. The CFL   =   .   number is equal to 0.5.  2(R11)i(U i+1/2 − U i−1/2)       2(R12)i(V i+1/2 − V i−1/2)    Acknowledgements. The authors would like to thank the re-   (R ) (V − V )+(R ) (U − U ) viewers for their useful remarks. The part of the work con-  11 i i+1/2 i−1/2 12 i i+1/2 i−1/2  cerning the VFRoe scheme has benefited from fruitful dis- 0 cussions with Dr. ThierryBuffard (Université Blaise Pascal), Dr. Isabelle Faille (IFP), Prof. ThierryGallouët (Universitéde (C.6) Provence) and Dr. Jean-Marie Masella (IFP), who are greatly acknowledged herein. Frédéric Archambeau (EDF-DRD) and Appendix D Guillaume Périgaud are also acknowledged for their help. We briefly present here the extension of the Roe scheme which has been used to compute the non-conservative system References nc Z,t +(F1(Z)),x + A1 (Z)Z,x =0. (D.1) Buffard T, Gallouët T, Hérard JM (1998) A na¨ıve scheme to We still denote the constant time step as ∆t. The straight- compute shallow water equations. C. R. Acad. Sci., Paris, forward integration over cell Ω again provides I-326, pp. 385–390 i Buffard T, Gallouët T, Hérard JM (1999) A na¨ıve Riemann solver to compute a non-conservative hyperbolic system. n+1 n Roe n vol (Ωi) Zi − Zi + ∆t F1 (Z )dΓ Int. Series on Numerical Mathematics 129: 129–138 Γi Buffard T, Gallouët T, Hérard JM (2000) A sequel to a rough n +∆tπi(Z )=0, (D.2) Godunov scheme: application to real gases. Computers and Fluids 29(7): 813–847 where Dal Maso G, Le Floch PG, Murat F (1995) Definition and weak stabilityof non-conservative products. J. Math. Pures Roe n n n Appl. 74: 483–548 F (Z )= F1(Z )+F1(Z ) 1 i i+1 Declercq-Xeuxet E (1999) Comparaison de solveurs numé- −Ω(Z(Yi+1/2))|∧|(Z(Yi+1/2)) riques pour le traitement de la turbulence bi fluide. PhD thesis, Université EvryMarne la Vallée, Paris, France, June −1 n n Ω (Z(Yi+1/2))(Zi+1 − Zi ) /2 .(D.3) 23 C. Berthon et al.: An approximate solution of the Riemann problem 25

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