On the implicit large eddy simulations of homogeneous decaying turbulence
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.
Section snippets
Governing equations
For all simulations in this paper it is considered that the Kolmogorov scale is significantly smaller that the mesh size, equivalent to stating that the viscous effects are negligible. Therefore, the Reynolds number Re = ∞ and the Navier–Stokes equations reduce to the Euler equations. The three-dimensional compressible Euler equations can be written in conservative variables and Cartesian co-ordinates aswhere
Turbulent isotropy
It is important to quantify turbulent isotropy, as turbulent theory relies on this assumption to derive analytical expressions for kinetic energy decay rates and growth of the length scales. The integral length was calculated from the longitudinal and transverse energy spectra using [49], [50]whereand f and g are the second-order longitudinal and lateral correlation functions relative to the
Conclusions
The ability of high-order finite volume Godunov-type ILES schemes to simulate isotropic, homogeneous decaying turbulence at low Mach number has been investigated quantitatively using a number of different parameters. The homogeneous isotropic flow field is initialised using the divergence of a vector potential to minimise the compressible component of the kinetic energy spectrum.
It has been demonstrated that the behaviour of the large scales is captured well at resolutions greater than 323, or
Acknowledgements
The authors thank David Youngs, Robin Williams, and Anthony Weatherhead (AWE, Aldermaston) and Evgeniy Shapiro (Fluid Mechanics and Computational Science Group, Cranfield University) for their advice and suggestions whilst developing the computer code. Additionally, they thank the reviewers for their insightful comments and suggestions. Finally, they acknowledge the financial support from EPSRC, MoD and AWE through the EPSRC(EP/C515153)-JGS (No. 971) project and the EPSRC-AWE PhD Case award.
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