Optimizing free parameters in the D3Q19 Multiple-Relaxation lattice Boltzmann methods to simulate under-resolved turbulent flows
Introduction
Over the last decades, the Lattice Boltzmann method (LBM) has become an alternative to traditional discretization techniques of the Navier–Stokes (NS) equations to simulate fluid flows. Thanks to the efficient parallelization [2], the LBM has a high computing performance being competitive against NS approaches for many applications [3], [4], in particular for the simulation of turbulent flows [5], [6]. The LBM is capable of simulating situations where sound and flow interact, such as aeroacoustic generation [7], [8]. Additionally, it is used to simulate a wide range of multiphase [9], [10] and multicomponent [11] flows.
The collision operator is responsible of modeling the physics correctly, and has a strong effect in the numerical stability of the scheme [12], [13]. Several collision operators have been proposed to extend the range of applicability of the LBM. Recently, Coreixas et al. [14] proposed a formalism that encompasses all these approaches within a common mathematical framework. In this work, we focus into the single BGK and multiple-relaxation MRT time collision operators.
The most popular collision operator, the single-relaxation time, is based on the BGK [15] approximation. In this model a unique relaxation time is considered for all the probability distribution functions. As a result of its simplicity, it has severe stability limitations [16], precluding its use at low viscosities, large Mach numbers or under-resolved simulations.
The multiple-relaxation time with raw moments (MRT-RM) collision operator [17] was introduced in an attempt to improve the BGK stability limitations. This operator enables different relaxation times for each probability distribution function. The increased complexity with respect to BGK resulted in an improved numerical stability [18].
Despite the enhancement in the stability of the MRT-RM with respect to the BGK collision operator, the MRT-RM can still show instabilities for small fluid viscosities [19]. Consequently, the MRT with central moments (MRT-CM) collision operator was introduced [20]. Premnath and Banerjee [21] showed that the cascaded LBM with central moments is consistent with the Navier–Stokes equations via a Chapman-Enskog expansion, and that in this approach, Galilean invariance is naturally enforced based on the relaxation central moments, see also [22]. Using central moments for the collision operator, it is possible to enhance the numerical stability by increasing the numerical dissipation at high wavenumbers [1].
Von Neumann stability analyses [23] enable the quantification of numerical errors in numerical schemes with periodic boundary conditions. Sterling and Chen [24] were the first to apply this analysis to the LB BGK approach. Then, Lallemand and Luo [18] used it to compare the enhanced stability of the MRT-RM over the BGK approach. Subsequently, Siebert et al. [25] included high order terms into the equilibrium distribution of the D2Q9 model to improve the linear stability of the scheme (as shown depicted Fig. 2 in [25]). Malaspinas [26] proposed a new version of the BGK with improved stability and based on recursive relations and regularization for the LB posed as Hermite series, which has been subsequently validated by Mattila et al. [27] and Coreixas et al. [28]. Later, in our previous work [1], we showed how the MRT-CM is more dissipative at higher wavenumbers compared with BGK and MRT-RM, providing better numerical stability. Also, we explored higher order terms in the fluid velocity, following Malaspinas’ work [26] to observe that our proposed approach provides similar results, with comparable stability, and for similar range of Mach numbers.
In addition to quantifying the numerical stability, this technique can also be used to provide insights into dispersive and dissipative errors. Marié et al. [29] compared BGK and MRT-RM collision operators. Dubois et al. [30] studied the numerical stability of the relative velocity (MRT-RM and MRT-CM) D2Q9 schemes with two different set of moments, proposed by Lallemand and Luo [18] and Geier et al. [19]. They concluded that MRT-CM with Geier's moment basis [19] had better stability properties. Recently, Gauthier et al. analyzed the interaction between modes [31] to understand the reason of numerical instabilities.
As mentioned, both the MRT-RM and the MRT-CM relax each hydrodynamic moment with a different relaxation time but all combinations lead to the same macroscopic state through the Chapman-Enskog expansion [32]. The relaxation times that are not fixed by the physics of interest become free parameters that can be optimized to enhance particular numerical aspects. Lallemand and Luo [18] optimized these parameters (for the D2Q9 lattice scheme) maximizing the Galilean invariance of the scheme, while reducing numerical errors (i.e. dispersion and dissipation). Similarly, Xu and Sagaut [7], [33] proposed an optimization to minimize dispersion/dissipation errors for the MRT-RM in D2Q9 scheme. A different approach is to adjust the high order relaxation parameters to enhance the stability for under resolved simulations. This approach was used by Ning et al. [34] to improve the stability of a 2D central moment LB method for a lid driven cavity flow problem. However, the authors of that work stated that their results were problem-dependent, and that additional work was required in 3D to optimize the relaxation parameters of the cascaded MRT LBM by means of a Fourier linear analysis. Adam et al. [22] proposed a cascaded LBM for the D3Q19 in the context of non-Newtonian flows. In their approach the free parameter were set to 1 (i.e. equilibrium), although it was already noticed that these could be adjusted independently to improve numerical stability by means of a Fourier linear stability analysis. These ideas were recently exploited by the authors [1], where the D2Q9 MRT-CM scheme was optimized to increase dissipation only for high under-resolved wavenumbers (above the k-1% dispersion-error), leaving low wavenumbers (i.e. well resolved scales) unchanged. This was tested successfully in the double periodic shear layer test.
In this work we extend our previous work to improve the MRT-CM for three-dimensional flows. The optimization strategy used previously for the D2Q9 lattice scheme [1] is now extended to the D3Q19 scheme. Again, instead of minimizing dissipation errors for all wavenumbers, we propose to maintain numerical dissipation for well resolved wavenumbers whilst increasing dissipation for under-resolved wavenumbers. The optimization is inspired in the rule of k-1 dispersion-error presented by Moura et al. [35] in the context of high order numerical methods. They suggested that waves are only accurately resolved if the dispersion error (difference between theoretical and numerical) is below 1. The wavenumber at which the error becomes 1 was named “k-1 dispersion-error” and lead to the “1 rule”. Following this rule, all waves above the k-1 should be dissipated since these are poorly resolved and may pollute the solution. We follow the idea of damping under-resolved waves and apply it to the LBM for the first time in a 3D lattice scheme.
To assess the D3Q19 optimized MRT-CM, we simulate the Taylor-Green vortex (TGV) case [36] that includes starting transitional flow followed by decaying homogeneous turbulence. This case enables the quantification of vortex stretching/pairing processes leading to energy cascading from large to small eddies, allowing the study of the dynamics of transition from laminar to turbulent flow and subsequent turbulent energy decay. This test-case has been widely used to study dissipation errors of numerical schemes, of high order type, e.g. [37], [38], [39] and for LB schemes, e.g. [40], [41], [42].
The remaining of this text is organized as follows. First, in Section 2, we describe the numerical methodology which is divided in two parts: first, the Lattice Boltzmann method with the different collision operators and second the optimization strategy. Then, in Section 3, the results of the optimized approach are tested for the turbulent Taylor Green Vortex case. Finally, in Section 4 conclusions are presented.
Section snippets
Methodology
In this section the numerical methodology used in this work is presented. First, the Lattice Boltzmann method (LBM) is introduced. Special attention is paid to the definition of the collision operator. Secondly, an optimization method based on linear stability analysis is shown. The optimization aims at maximizing the robustness of the scheme for under-resolver simulations without penalizing its accuracy. The final objective is to improve the performance of the scheme for turbulent flows,
Numerical validation
In this section the new optimized scheme is compared with previous approaches found in the literature. The aim of this section is to show that the proposed scheme provides increased stability without penalizing the accuracy.
In order to study the effect of the present optimized MRT-CM on a three-dimensional turbulent configuration, the decaying Taylor-Green vortex (TGV) [36] has been simulated. It is a fundamental test case used as prototype for vortex stretching and production of small-scale
Conclusions
In this paper, the free parameter (relaxation times) of the D3Q19 MRT-CM are optimized to enhance robustness for under-resolved simulations. The optimization strategy aims at maximizing the dissipation of the numerical scheme for under-resolved flow features. The limit between well- and under-resolved waves is based on von Neumann linear stability analyses and the so called k-1% dispersion-error rule. The D3Q19 optimized MRT-CM scheme is compared with standard BGK and MRT-CM schemes by means of
Conflicts of interest
None declared.
Acknowledgment
This project has received funding from the European Unionś Horizon 2020 research and innovation programme under grant agreement No 785549 (FireExtintion: H2020-CS2-CFP06-2017-01).
The authors acknowledge the computer resources and technical assistance provided by the Centro de Supercomputación y Visualización de Madrid (CeSViMa).
I’m an aeronautical engineer with a PhD in aerospace engineering. Currently, I am working as an teaching assistant at the E.T.S.I. Aeronautics and Space Engineering of the Universidad Politécnica de Madrid. My main research topic is focused on the analysis and application of the Lattice Boltzmann (LBM) method to fluid mechanics. Besides, the use of tools based in finite element and high order methods applied to aerospace industry applications.
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