Compressible, turbulent flow and other
hyperbolic systems
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
