Wavelet-based adaptive large-eddy simulation with explicit filtering

Abstract

Stochastic coherent adaptive large-eddy simulation is a novel approach to the numerical simulation of turbulence, where the coherent energetic eddies are solved for, while modeling the influence of the less energetic coherent/incoherent background flow. The formal separation between resolved and unresolved field is obtained by wavelet threshold filtering that is inherent to the adaptive wavelet collocation numerical method. A new explicit wavelet filtering strategy is introduced and tested, by considering two different filtering levels: the physical level, which controls the turbulence model, and the numerical level that is responsible for the accuracy of numerical simulations. The theoretical basis for wavelet-based adaptive large-eddy simulation with explicit filtering and consistent dynamic modeling is given. Numerical experiments are presented for unsteady homogeneous turbulence, demonstrating the existence of grid-independent solutions.

Introduction

The stochastic coherent adaptive large-eddy simulation (SCALES) method is a novel approach to the numerical simulation of turbulent flows, where the more energetic coherent eddies are solved for, while discarding the dynamics of the less energetic background flow [1]. The decomposition of turbulence into resolved coherent and residual coherent/incoherent motions is obtained through the application of wavelet threshold filtering (WTF). The space–time evolution of the coherent velocity field is governed by the wavelet-filtered Navier–Stokes equations, where, similarly to any other large-eddy simulation (LES) approach, the effect of the unknown residual stresses is modeled.

In order to solve the SCALES governing equations in a computationally efficient manner, the adaptive wavelet collocation method (AWCM) is used, e.g. [2]. The method is a variable high-order finite-difference numerical simulation that exploits the same wavelet filtering procedure to automatically adapt the computational grid to the solution, in both location and scale. The choice of a wavelet thresholding level for the grid adaptation automatically introduces a formal separation between eddies that are resolved by the wavelet collocation grid and unresolvable sub-grid eddies. Up to now, the filtering effect induced by the use of the AWCM has been used to define the filtered-velocity in the SCALES approach, without distinguishing between physical (model) filtering and numerical filtering that is related to truncation of wavelet basis and corresponding computational mesh. Thus, there has been no practical reason, so far, to discern between explicit and implicit wavelet-filtering. Along this line, the residual stresses have been referred to and treated as subgrid-scale (SGS) stresses, e.g. [3], [4].

The implicit approach makes it impossible to separate between filtering, modeling and numerical issues. This is a fundamental problem in LES methods [5] that becomes even more important in the wavelet-based formulation, where the numerical solver allows for the automatic mesh refinement in flow regions with inadequate modeled dissipation. Furthermore, SCALES method with implicit filtering is particularly sensitive to numerical errors since the choice of a relatively high thresholding level unavoidably affects the accuracy of the AWCM-based solution.

In this work, following a strategy that is sometimes adopted in classical non-adaptive LES, e.g. [6], [7], a new wavelet-based adaptive LES method with explicit filtering is introduced and tested. Two different filtering levels are considered: the physical level, which controls the level of turbulence model, and the numerical level that is solely responsible for the accuracy of calculations. The superposition of explicit filtering allows to practically control the computational errors, while keeping apart the filtering and the numerical issues.

The theoretical basis for SCALES with explicit filtering is given and numerical experiments are carried out for freely decaying homogeneous incompressible turbulence. The use of explicit wavelet filtering is studied in order to address the potential of the method to obtain grid-independent numerical solutions, as found for classical LES [8].

In the next section, the wavelet-based adaptive LES approach is briefly reviewed. The new explicit filtering SCALES method is introduced in Section 3, along with the consistent dynamic procedure for modeling the residual stresses. In Section 4, the numerical experiments are presented and discussed. Finally, in Section 5, some concluding remarks are made.