CORE Metadata, citation and similar papers at core.ac.uk Provided by Archive Ouverte en Sciences de l'Information et de la Communication An exact Riemann solver for multicomponent turbulent flow Emmanuelle Declercq, Alain Forestier, Jean-Marc Hérard, Xavier Louis, Gérard Poissant To cite this version: Emmanuelle Declercq, Alain Forestier, Jean-Marc Hérard, Xavier Louis, Gérard Poissant. An exact Riemann solver for multicomponent turbulent flow. International Journal of Computational Fluid Dynamics, Taylor & Francis, 2001, 14, pp.117-131. hal-01580049 HAL Id: hal-01580049 https://hal.archives-ouvertes.fr/hal-01580049 Submitted on 23 Dec 2019 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. 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Part I An exact Riemann solver for a multicomponent turbulent ow y zx k Emmanuelle Declercq Alain Forestier JeanMarc Herard Xavier Louis yy Gerard Poissant key words Multicomp onent turbulence mo del Entropy characterization Riemann solver abstract This contributions topic is the resolution of the hyperb olic system which describ es a multicomponent turbulent ow The mo del is written for an isentropic gas We compute the exact solution of the Riemann Problem RP asso ciated to the hyperb olic system It is comp osed of constant states separated by rarefaction waves or sho ck waves and a contact discontinuity The selection of the admissible part of the sho ck curve is obtained by an entropic criterion Compressive sho ck means entropic sho ck for only one of the two mathematical entropies found This entropy is the total energy of the system With these existence and uniqueness prop erties we compute the exact solution of RP by a Smollers kind of parameterization Introduction The recent need for computation of complex systems of non linear PDEs such as those arising when investigating turbulent phenomena has motivated the development of adequate solvers Actually hyperb olic systems arising in the framework of single phase turbulent compressible CEMIF rue du Pelvoux Courcouronnes Evry Cedex France declercqworldonlin efr y CEA Saclay DRNDMTSEMT GifSurYvette Cedex France xaviernsemtsmtsceafr z CEA Saclay DRNDMTSEMT GifSurYvette Cedex France alainforestierceafr x CEMIF rue du Pelvoux Courcouronnes Evry Cedex France EDF LNHDER quai Watier Chatou France JeanMarcHerardderedffr k CMI UMR CNRS Universitede Provence rue Joliot Curie Marseille CEA Saclay DRNDMTSEMT GifSurYvette Cedex France forsemtsmtsceafr yy CEMIF IUT GMP cours Mgr Romero Evry France GPoissantiutunivevryfr mo dels contain dierent scales of pressure elds The standard mean pressure accounts for mi croscopic eects whereas the mean turbulent kinetic energy fo cusing on Kepsilon type mo dels stands for some counterpart of the mean pressure at a macroscopic level This was recently demonstrated by several workers see for instance Co quel and Berthon or who hence prop osed various upwinding schemes for practical purp oses This is true for one or twoequation mo dels but it is even more convincing when turning to socalled secondmoment closures In this case the very small amount of viscous eects urges investigating basic solutions of homo geneous convective systems Though the decoupled approaches are still often used in industrial co des recent examples of computation of impinging jets on wall b oundaries have shown that the coupled approach should b e preferred for stability reasons We will fo cus in this work on the tight coupling b etween the mean pressure eld and turbulent kinetic energy when computing multicomponent compressible rstorder turbulent closures One of the main ob jectives here is to derive exact or approximate Riemann solvers for our sp ecic problem and b eyond to compare b oth eciency accuracy and stability of resp ective schemes The pap er is thus organized as follows In the rst part the turbulent mo del used to describ e the ow is briey presented Since b oth viscous and source terms may b e easily computed applying standard Finite Volume schemes on structured meshes at least emphasis is given on the analysis of the convective ho mogeneous problem which is hyperb olic but is not under conservative form Studying Riemann invariants entropy inequality and assuming some approximate jump conditions hold enables to derive an existence and uniqueness result for the solution of the onedimensional Riemann problem asso ciated with the convective problem provided that the initial data agrees with some condition This result is made p ossible by using the admissible part of the sho ck curves owing to the entropy inequality It also requires that the strength of sho cks is suciently weak The second part of the pap er is devoted to the construction of a Go dunov type solver which accounts for nonconservative terms and to comparison with some rough Go dunov scheme and also with the adaptation to the frame of non conservative systems of the rough but robust Rusanov scheme A turbulence mo del to describ e multicomponent ows Governing equations We b egin with Euler equations for an average compressive multicomponent ow see The gas k v l are assumed to b e isentropic like in the Psystem We dene by the mean density of the mixture the volume fraction of the v ow in the mixture P the pressure and U the velocity of the mixture We use Favres average to deal with compressive ows U U U k v l k k k k k k k U U U v v l v l v l We introduce the mass fraction Y and the relative velocity V k X v V U U Y V YV Y V Y Y k k k k v l v k 2 2 X V V v v 2 2 2 2 Y Y V U U U Y k v l k v l Y Y k rU t Y rY U rY V t v S 2 V v s 2 K U rU P I rY t Y v v R tensor In the two dimensional frame The kinetic turbulent energy K is the trace of the we write X v 2 2 2 2 K Tr R R u v u v ii v v v v v v i s v l 2 2 U U Remark that we have noted K K K v l v l 2 U But that is not the turbulence of the melting ow K K mo del for isentropic multicomponents ows To close the mo del we derive a supplementary equation for the kinetic turbulent energy in the v 2 2 ow To compute this equation we subtract the equation of U from the equation of U v v v v We introduce the deviator D such that Tr R R I D Prop osition The evolution equation of a discontinuous by phases turbulent ow is v v v t 3 2 U K rK D U r U U U rP U v a n K rU r v t v v v v i i v v v v v Pro of 2 3 U I r U U rP t v v v v v 2 3 2 U U U rP I U r U v a n v v v v v i i v t v v v 2 3 t d U r U r R U U rP I U M t v v v v v v v v v 2 v 2 v 2 R U U U K tr v v v v v v v 3 t v 3 U U R U U rP I U rP I r K r v v v v v v t v 2 U U v a n U U U v a n P a n P r v v i i v v v v v i i v v i v v v v 3 3 2 3 3 U U U U U R U U Tr v v v v v v v v v v v v U rP U rP U P r U P a n rP U rP U rP U v v i v v v v v 2 2 2 U U v a n U U U v a n U U v a n U U v a n v v i i v v v v v i i v v v i i v v v i i v v v v z 0 v v t v 3 2 K rTr R U r R U r U U U rP U v a n t v v v v v v i i v v v v v R Tr t t v v v U r R U r R U rTrR U r D U rTr v v v v v t v v R rU rTrR U RrU r D U Tr Tr v v v v v K v t rK U rU r D U v v v v K v v t 3 2 D U r U U rP U U v a n K rK U rU r v v v v i i v t v v v v v Then we make some simplications to close the system At rst we neglect area source terms and o dd correlations 2 U U v a n v v i i v v 3 r U v v After we assume an isotropic turbulence so the Reynolds tensor is diagonal and isotropic It is v describ ed through K v v K R ij ij In a two dimensional framework we obtain k k k K rK U K rU U rP t v v s To close the S system we add the K evolution equation This one is obtained by summation k of K over phases We supp ose that the ows have the same velocity From now on we s neglect the average symbol and set K for K We give the system here obtained adding the viscous terms and are p ositive quantities dep ending on the choice of the turbulence mo del t We recall that the melting gas is isentropic with a pressure law P Y known ef f lam t is the turbulent dissipation which is mo deled see for example the one equation turbulence mo del of or rU t Y rY U t S t 2 U rU I r rU rU I K P rU t ef f K t K rKU I rU r r K rU rU rU rU t t Setting W C K we are interested in the rst order convective system S which is conser c vative in C Y U variable but not in K variable C rF C K t S c K rU K rKU t Exact Riemann solver From a D problem to the D Riemann Problem It is well known that Finite Volume upwinding schemes are ecient metho ds to solve no linear hyperb olic systems The most natural nite volume metho d is the Go dunovs metho d which requires getting the exact solution of the Riemann Problem at the interface