A dynamic subgrid-scale tensorial Eddy viscosity model

Abstract

In the Navier-Stokes equations the removal of the turbulent fluctuating velocities with a frequency above a certain fixed threshold, employed in the Large Eddy Simulation (LES), causes the appearance of a turbulent stress tensor that requires a number of closure assumptions. In this paper insufficiencies are demonstrated for those closure models which are based on a scalar eddy viscosity coefficient. A new model, based on a tensorial eddy viscosity, is therefore proposed; it employs the Germano identity [1] and allows dynamical evaluation of the single required input coefficient. The tensorial expression for the eddy viscosity is deduced by removing the widely used scalar assumption of the high-frequency viscous dissipation and replacing it by its tensorial counterpart arising in the balance of the Reynolds stress tensor. The numerical simulations performed for a lid driven cavity flow show that the proposed model allows to overcome the drawbacks encountered by the scalar eddy viscosity models. In numerical simulations of turbulent hydrodynamic field quantities different levels of the representation of the frequency spectrum of hydrodynamic quantities can be pursued. A full representation of this spectrum, achieved by Direct Numerical Simulation (DNS), needs a numerical discretization of the continuum hydrodynamic field that is able to resolve all turbulent high-frequency microscales; in this scheme the viscous stresses dissipate the turbulent kinetic energy into heat. The representation of the high frequency turbulent eddies, thus, imposes massive computational efforts and restricts DNS to simple flows and low Reynolds numbers. At the opposite extreme is the statistical approach, based on an averaging process of the Navier-Stokes equations (Reynolds Averaged Navier-Stokes, RANS); it defines the filtered hydrodynamic fields and removes the turbulence length-scales except those of the macroscales whose dimension is comparable with that of the domain in which the equations are solved. As a consequence of the averaging process, the non-linear convective terms of the Navier-Stokes equations cause the appearance of a turbulent stress tensor related to the removed turbulent fluctuating velocities. Since this tensor is an additional unknown, suitable closure assumptions must be introduced. The low-frequency fluctuating velocities are predominantly dependent on the averaged hydrodynamic fields and simple closure conditions may be introduced. However, as the intended resolution of the frequency spectrum is increased to higher frequencies, more complex closure models are required to accomodate for the increased resolution of the subgrid fluctuations. They, in general, also require the determination of a large number of flow dependent parameters. It is important that these equations satisfy the necessary form invariance under Euclidean transformations [2]. Large Eddy Simulation (LES) [3] is a compromise approach between DNS and RANS since it defines a turbulence mesoscale that allows to distinguish the large energy-carrying scales that are numerically resolved