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During my PhD thesis I worked on compressible turbulent flow, its modeling by means of one-point closures (Reynolds stress models) and the adequate numerical solution of the model equations. This also meant dealing with hyperbolic equations (at least regarding the convective subsystem) containing source terms which in turn later led to my interest in other such cases as the shallow water equations. A lot of the following owes to a very fruitful collaboration with J.M. Hérard from EDF and U. Marseille, France.

- "Solveur pour le système des équations hyperboliques non-conservatives issu d'un modele de transport des tensions de Reynolds", Internal Report 95-2, LMFA, Ecole Centrale de Lyon, France. This document describes for the first time how an approximate Riemann solver can be designed to take into account the production-related source terms in the framework of a Reynolds stress model.
- "Traitement de la partie hyperbolique du système
des équations Navier-Stokes moyennées et des
équations de transport issues d'une fermeture au
premier ordre pour un fluide compressible" by A. Page
and M. Uhlmann (Internal Report 96-1, LMFA, Ecole Centrale
de Lyon, France). Here we
compare various methods of treating the numerical source-term
problem in the framework of two-equation closures (i.e.
k-
*eps*). - "Etude de modèles de fermeture au second ordre et contribution a la résolution numérique des écoulements turbulents compressibles", PhD Thesis, Ecole Centrale de Lyon, France, 1997. Deals with the subjects of engineering-type modeling of compressible, turbulent flow (consistency, realizability, realism) and the adequate numerical solution of the model equations. The application of the model builds up from the simplest homogeneous flows to complex shock-induced separation of boundary layers.
- "An Approximate Roe-type Riemann Solver for a Class of
Realizable Second Order Closures" by G. Brun, J.-M. Hérard,
D. Jeandel, M. Uhlmann appeared in
*Int. J. Comp. Fluid Dyn.*13(3):223-250, 2000. It describes our approximate Riemann solver in detail. The abstract in html is here. - A short
note also
appeared in
*J. Comp. Physics*151:990-996, 1999 under the name "An Approximate Riemann Solver for Second-Moment Closures" by the same authors. - An article describing the analytical solution of the
one-dimensional Riemann problem for this system of equations,
including a proof of existence and uniqueness, has appeared in
*Shock Waves*11(4):245-269, 2002 under the name "An approximate solution of the Riemann problem for a realisable second-moment closure" by C. Berthon F. Coquel, J.-M. Hérard and M. Uhlmann (available here). - An online tutorial on the shallow water equations and numerical methods for treating them is available here The same document in postscript can be downloaded. There is also some source code in FORTRAN for the reference solution and numerical methods of the Godunov and Roe type. Incidentally, there is a variant treating the (scalar) Burgers equation.

markus.uhlmann AT kit.edu

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