Existence of Solution to Nonlinear Elliptic Systems Arising in Turbulence Modelling

Abstract

We study some nonlinear elliptic systems governing the steady-state of a two-equation turbulence model that has been derived from the so-called k-ε model. Two kinds of problems are considered: in the first one, we drop out transport terms and we deduce the existence of a solution for N ≥ 2; in the second one we take into account all transport terms; in this case, the existence result holds for N = 2 or 3. Positivity and L ∞ -regularity of the scalar quantities are also shown here.

Analysis of a turbulence model related to that of k-epsilon for stationary and compressible flows

“…ii) We allow the viscosity parameters C θ , C ϕ to depend on x, θ, ϕ whereas in the previous works these parameters were taken constant (except in [20,21] where however an artificial regularization was made, and only the incompressible situation was studied).…”

“…Another important property attempted for a turbulence model is its capability to predict the possible steady states. In the previous works only the evolutive version of (P) was studied (except in [21] where however only the incompressible situation, with pertubated viscosities was considered), and the results obtained cannot predict the existence or non-existence of steady states.…”

Analysis of a Singular Convection Diffusion System Arising in Turbulence Modelling

“…Another important property attempted for a turbulence model is its capability to predict the possible steady states. In the previous works (except in [10]) only the evolutive version of ( ) was studied under very restrictive assumptions. In [10], however, the stationary problem is studied, but it is simplified by considering a perturbation of the viscosities that artificially cancel the singularity of the system.…”

“…In the previous works (except in [10]) only the evolutive version of ( ) was studied under very restrictive assumptions. In [10], however, the stationary problem is studied, but it is simplified by considering a perturbation of the viscosities that artificially cancel the singularity of the system. Hence, in this paper we shall study the stationary version of ( ) on a bounded domain Ω ⊂ R , = 2 or 3, on which we impose the boundary conditions = , = on Ω.…”