On the implicit large eddy simulations of homogeneous decaying turbulence

Abstract

Simulations of homogeneous decaying turbulence (HDT) in a periodic cube have been used to examine in a detailed and quantitative manner the behaviour of high-resolution and high-order methods in implicit large eddy simulation. Computations have been conducted at grid resolutions from 32³ to 256³ for seven different high-resolution methods ranging from second-order to ninth-order spatial accuracy. The growth of the large scales, and dissipation of kinetic energy is captured well at resolutions greater than 32³, or when using numerical methods of higher than third-order accuracy. Velocity increment probability distribution functions (PDFs) match experimental results very well for MUSCL methods, whereas WENO methods have lower intermittency. All pressure PDFs are essentially Gaussian, indicating a partial decoupling of pressure and vorticity fields. The kinetic energy spectra and effective numerical filter show that all schemes are too dissipative at high wave numbers. Evaluating the numerical viscosity as a spectral eddy viscosity shows good qualitative agreement with theory, however if the effective cut-off wave number is chosen above k_max/2 then dissipation is higher than the theoretical solution. The fifth and higher-order methods give results approximately equivalent to the lower order methods at double the grid resolution, making them computationally more efficient.

Introduction

As current computational power does not allow direct numerical simulation (DNS) of complex flows, LES has emerged as a viable alternative where the time dependent behaviour of the flow must be resolved. Conventional LES, where an explicit subgrid model is added to the averaged Navier–Stokes equations, has been employed successfully in many prototype flows, however it is known to provide excessive dissipation in flows where the growth of an initially small perturbation to fully turbulent flow must be resolved [1], [2]. It has been recognised that some numerical schemes gain good results in complex flows without the explicit addition of a subgrid model [1]. This occurs when the subgrid model is implicitly designed into the limiting method of the numerical scheme, based on the observation that an upwind numerical scheme can be rewritten as a central scheme plus a dissipative term (see [3], [4], [5] and references therein). Such implicit subgrid models fall into the class of structural models, as there is no assumed form of the nature of the subgrid flow thus the subgrid model is entirely determined by the structure of the resolved flow [6].

Using implicit LES (or ILES), excellent results have been gained in simulation of flows as varied as Rayleigh–Taylor and Richtmyer–Meshkov instability [7], [8], Free jets [9], [10], channel flow [10], open cavity flow [11], [3], geophysical flows [12], [13], delta wings [14] and decaying turbulence [15], [16], [17], [18], [19], [20]. Attempts to formalise the development of ILES numerical schemes is hindered by the inherent complexity of theoretical analysis of non-linear schemes, however, recent developments show some good agreements between truncation errors due to the numerical scheme and the required form of the subgrid terms [21], [5].

Several of these flows are of mixed compressible and incompressible nature, where a compressible method is required to capture certain flow features (e.g. shock waves), yet the turbulent vortices are near-incompressible. In experimental studies [22] the turbulent Mach number rarely exceeds 0.2. Thus it is of importance to assess the performance of Godunov-type schemes applied to low Mach number turbulence.

This paper assesses the performance of high-order Godunov-type methods for these applications, via simulations of low Mach number homogeneous decaying turbulence. The study does not intend to prove that ILES is a better approach than standard LES, based on explicit subgrid scale models, for the flow in question. It is intended as a starting point for future development by identifying quantitatively the strengths and weaknesses of high-resolution methods used in ILES by comparing the ILES results with experimental studies, DNS and previous conventional LES. It is a complementary extension of the work of Garnier et al. [23], where the ability of shock-capturing schemes was tested for resolutions up to 128³ and for six extrapolation methods from second to fifth-order. The authors concluded that the dissipation rate of the ILES methods is too high, and that the behaviour of the schemes is more akin to a low Reynolds DNS than an LES. In the present paper, the extrapolation methods employed are less diffusive and range from MUSCL second-order through to WENO ninth-order accurate. These are finite volume methods which differ in behaviour from the finite difference and flux limiting methods employed in [23]. Each of these extrapolation methods have been run on grids from 32³ to 256³ to examine the behaviour and convergence (if any) of turbulent statistics and spectra.

The layout of the paper is as follows. Section 2 details the numerical scheme employed, the form of the implicit subgrid model, and the method used to initialise a homogeneous, isotropic turbulent field. The effect of non-zero compressibility in the flow field is discussed. Section 3 compares the quantitative behaviour of the seven ILES variants in terms of fundamental properties of a turbulent flow field; growth of the integral length scale; decay rate of turbulent kinetic energy; time variation of enstrophy; skewness and flatness of the velocity derivative; velocity increment and pressure fluctuation probability distribution functions; kinetic energy spectra; effective numerical filter and spectral numerical viscosity. Section 4 concludes this paper and discusses the areas for future development.