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

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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 HAL Id: hal-01484338 https://hal.archives-ouvertes.fr/hal-01484338 Submitted on 10 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 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; H´erard 1995b; H´erard 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 H´erard 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 ade- quate 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; H´erard 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 (H´erard 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 stat- es 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 .
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