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Study of the $$k$$-epsilon model of turbulence for compressible flows. (Étude du modèle $$k$$-epsilon de la turbulence pour les écoulements compressibles.) (French) Zbl 0922.76217
Rocquencourt: Institut National de Recherche en Informatique et en Automatique (INRIA). Paris: Univ. de Paris VI, ii, 123 p. (1991).
This Ph.D. dissertation deals essentially with the development and numerical solution of stable and numerically efficient $$k$$-$$\epsilon$$ turbulence models for compressible flows. It includes a main contribution dealing with the derivation and theoretical analysis of two-equations model derived from the standard $$k$$-$$\epsilon$$ model ($$\phi$$-$$\theta$$ model), in order to obtain positiveness and boundedness of $$k$$ and $$\epsilon$$. It also develops specific time discretizations of the dynamic part of these models that ensure in most cases the positiveness of computed $$k$$ and $$\epsilon$$.
The overall discretization of the models considered is performed by splitting the variables into ”turbulent” and ”physical” ones. To solve for the physical variables, an upwind Galerkin finite volume solver is used. The solution for the turbulent variables is performed by splitting the transport and diffusion operators. The transport step is solved by a Galerkin-characteristics formulation in a $${P}_1$$ finite element context. This step includes the integration of the source terms. The discretization of the diffusion operator is also performed by the Galerkin method, using mass lumping to ensure the positiveness of turbulent variables.
All these contributions are used to analyse a two-layer approach, using a low-Reynolds $$k$$-$$\epsilon$$ model for flow far from solid walls, and a $$k$$-$$L$$ model for wall flow. This model is tested in some specific cases of interest, and compared to the pure $$k$$-$$\epsilon$$ model , including wall laws. It turns out to be considerably stable and grid-independent, and is able to simulate attached as well as detached flow with slight adaptations.

##### MSC:
 76F05 Isotropic turbulence; homogeneous turbulence 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics