Turbulent flows as generalized Kelvin–Voigt materials: Modeling and analysis

Abstract

We perform a new modeling procedure for a 3D turbulent fluid, evolving towards a statistical equilibrium. This will result to add to the equations for the mean field (v,p) the term −α∇⋅(ℓ(x)Dvt), which is of the Kelvin–Voigt form, where the Prandtl mixing length ℓ=ℓ(x) is not constant and vanishes at the solid walls. We get estimates for mean velocity v in Lt∞Hx1∩Wt1,2Hx1∕2, that allow us to prove the existence and uniqueness of regular-weak solutions (v,p) to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to consider Reynolds averaged equations and pass to the limit in the quadratic source term, in the equation for the turbulent kinetic energy k. This yields the existence of a weak solution to the corresponding Navier–Stokes Turbulent Kinetic Energy system satisfied by (v,p,k).