The purpose of this study is to investigate the accuracy and applicability of the Flowfield Dependent Variation (FDV) method for large-eddy simulations (LES) of decaying isotropic turbulence.
In an earlier paper, the FDV method was successfully demonstrated for simulations of laminar flows with speeds varying from low subsonic to high supersonic Mach numbers. In the current study, the FDV method, implemented in a finite element framework, is used to perform LESs of decaying isotropic turbulence. The FDV method is fundamentally derived from the Lax–Wendroff Scheme (LWS) by replacing the explicit time derivatives in LWS with a weighted combination of explicit and implicit time derivatives. The increased implicitness and the inherent numerical dissipation of FDV contribute to the scheme’s numerical stability and monotonicity. Understanding the role of numerical dissipation that is inherent to the FDV method is essential for the maturation of FDV into a robust scheme for LES of turbulent flows. Accordingly, three types of LES of decaying isotropic turbulence were performed. The first two types of LES utilized explicit subgrid scale (SGS) models, namely, the constant-coefficient Smagorinsky and dynamic Smagorinsky models. In the third, no explicit SGS model was employed; instead, the numerical dissipation inherent to FDV was used to emulate the role played by explicit SGS models. Such an approach is commonly known as Implicit LES (ILES). A new formulation was also developed for quantifying the FDV numerical viscosity that principally arises from the convective terms of the filtered Navier–Stokes equations.
The temporal variation of the turbulent kinetic energy and enstrophy and the energy spectra are presented and analyzed. At all grid resolutions, the temporal profiles of kinetic energy showed good agreement with t(−1.43) theoretical scaling in the fully developed turbulent flow regime, where t represents time. The energy spectra also showed reasonable agreement with the Kolmogorov’s k(−5/3) power law in the inertial subrange, with the spectra moving closer to the Kolmogorov scaling at higher-grid resolutions. The intrinsic numerical viscosity and the dissipation rate of the FDV scheme are quantified, both in physical and spectral spaces, and compared with those of the two SGS LES runs. Furthermore, at a finite number of flow realizations, the numerical viscosities of FDV and of the Streamline Upwind/Petrov–Galerkin (SUPG) finite element method are compared. In the initial stages of turbulence development, all three LES cases have similar viscosities. But, once the turbulence is fully developed, implicit LES is less dissipative compared to the two SGS LES runs. It was also observed that the SUPG method is significantly more dissipative than the three LES approaches.
Just as any computational method, the limitations are based on the available computational resources.
Solving problems involving turbulent flows is by far the biggest challenge facing engineers and scientists in the twenty-first century, this is the road that the authors have embarked upon in this paper and the road ahead of is very long.
Understanding turbulence is a very lofty goal and a challenging one as well; however, if the authors succeed, the rewards are limitless.
The derivation of an explicit expression for the numerical viscosity tensor of FDV is an important contribution of this study, and is a crucial step forward in elucidating the fundamental properties of the FDV method. The comparison of viscosities for the three LES cases and the SUPG method has important implications for the application of ILES approach for turbulent flow simulations.
This work was made possible, in part, by a grant of high performance computing resources and technical support from the Alabama Supercomputer Authority. Support for the first author was provided in part by the National Space Research & Development Agency (NASRDA), Nigeria, and the Mechanical and Aerospace Engineering Department at the University of Alabama in Huntsville through a graduate teaching assistantship.
Olatoyinbo, S., Rani, S. and Frendi, A. (2017), "Large-eddy simulation of decaying isotropic turbulence using the flowfield dependent variation method", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 27 No. 1, pp. 235-262. https://doi.org/10.1108/HFF-07-2015-0290Download as .RIS
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