Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of non-equilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second (and higher) order non-hydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from $10^4$ to $10^6$, and where a thorough analysis of the case at $Re=3\times 10^4$ is conducted. In the latter, results obtained using both regularization steps are compared against the BGK-LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.Comment: Accepted for publication as a Regular Article in Physical Review

Regularized characteristic boundary condition for the Lattice Boltzmann methods at high Reynolds number flows

This paper reports the investigations done to adapt the Characteristic Boundary Conditions (CBC) to the Lattice-Boltzmann formalism for high Reynolds number applications. Three CBC formalisms are implemented and tested in an open source LBM code: the baseline one-dimension inviscid (BL-LODI) approach, its extension including the effects of the transverse terms (CBC-2D) and a local streamline approach in which the problem is reformulated in the incident wave framework (LS-LODI). Then all implementations of the CBC methods are tested for a variety of test cases, ranging from canonical problems (such as 2D plane and spherical waves and 2D vortices) to a 2D NACA profile at high Reynolds number (Re = 100,000), representative of aeronautic applications. The LS-LODI approach provides the best results for pure acoustics waves (plane and spherical waves). However, it is not well suited to the outflow of a convected vortex for which the CBC-2D associated with a relaxation on density and transverse waves provides the best results. As regards numerical stability, a regularized adaptation is necessary to increase the Reynolds number. The so-called regularized FD adaptation, a modified regularized approach where the off-equilibrium part of the stress tensor is computed thanks to a finite difference scheme, is the only tested adaptation that can handle the high Reynolds computation

A linear stability analysis of compressible hybrid lattice Boltzmann methods

An original spectral study of the compressible hybrid lattice Boltzmann method (HLBM) on standard lattice is proposed. In this framework, the mass and momentum equations are addressed using the lattice Boltzmann method (LBM), while finite difference (FD) schemes solve an energy equation. Both systems are coupled with each other thanks to an ideal gas equation of state. This work aims at answering some questions regarding the numerical stability of such models, which strongly depends on the choice of numerical parameters. To this extent, several one- and two-dimensional HLBM classes based on different energy variables, formulation (primitive or conservative), collision terms and numerical schemes are scrutinized. Once appropriate corrective terms introduced, it is shown that all continuous HLBM classes recover the Navier-Stokes Fourier behavior in the linear approximation. However, striking differences arise between HLBM classes when their discrete counterparts are analysed. Multiple instability mechanisms arising at relatively high Mach number are pointed out and two exhaustive stabilization strategies are introduced: (1) decreasing the time step by changing the reference temperature $T_{ref}$ and (2) introducing a controllable numerical dissipation $\sigma$ via the collision operator. A complete parametric study reveals that only HLBM classes based on the primitive and conservative entropy equations are found usable for compressible applications. Finally, an innovative study of the macroscopic modal composition of the entropy classes is conducted. Through this study, two original phenomena, referred to as shear-to-entropy and entropy-to-shear transfers, are highlighted and confirmed on standard two-dimensional test cases.Comment: 49 pages, 23 figure

Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: a direct grid coupling approach

The present study proposes a highly accurate lattice Boltzmann direct coupling cell-vertex algorithm, well suited for industrial purposes, making it highly valuable for aeroacoustic applications. It is indeed known that the convection of vortical structures across a grid refinement interface, where cell size is abruptly doubled, is likely to generate spurious noise that may corrupt the solution over the whole computational domain. This issue becomes critical in the case of aeroacoustic simulations, where accurate pressure estimations are of paramount importance. Consequently, any interfering noise that may pollute the acoustic predictions must be reduced. The proposed grid refinement algorithm differs from conventionally used ones, in which an overlapping mesh layer is considered. Instead, it provides a direct connection allowing a tighter link between fine and coarse grids, especially with the use of a coherent equilibrium function shared by both grids. Moreover, the direct coupling makes the algorithm more local and prevents the duplication of points, which might be detrimental for massive parallelization. This work follows our first study (Astoul~\textit{et al. 2020}) on the deleterious effect of non-hydrodynamic modes crossing mesh transitions, which can be addressed using an appropriate collision model. The Hybrid Recursive Regularized model is then used for this study. The grid coupling algorithm is assessed and compared to a widely-used cell-vertex algorithm on an acoustic pulse test case, a convected vortex and a turbulent circular cylinder wake flow at high Reynolds number.Comment: also submitted to Journal of Computational Physic

Etude des méthodes lattice Boltzmann pour les simulations de systèmes d'air secondaires de turbomachines

Ce manuscrit présente une étude du potentiel des méthodes lattice Boltzmann pour traiter des écoulements circulant dans les systèmes de refroidissement des turbomachines. La combinaison de phénomènes physiques complexes donne naissance à des structures instationnaires, non-axisymmétriques et de période a priori inconnue. Leur bonne modélisation représente un défi pour la simulation numérique en mécanique des fluides. Ce travail peut être divisé en trois sous-parties. Une étude physique des instabilités à l'origine des structures tourbillonnaires est d'abord effectuée par analyse de stabilité linéaire des écoulements. Ensuite, les méthodes lattice Boltzmann sont introduites et leurs problèmes de stabilités numériques sont étudiés via des analyses basées sur l'approche de von Neumann. Enfin, la méthode est évaluée sur des simulations académiques de complexité croissante représentatives des systèmes d'air secondaire, nécessitant des simulations à flux de chaleur conjuguésThis thesis provides an investigation on the use of lattice Boltzmann methods to treat turbomachinery secondary cooling systel flows. The combination of complex physical phenomena (rotating environment with high temperature fluctuations) gives rise to unsteady, non-axisymmetric structures with a priori unknown periodicity. Their modelling, required for a correct heat transfer prediction, represents a challenge for numerical simulations in fluid mechanics. This work can be divided into three sub-sections. A physical study of the instabilities at the origin of unsteady structures is first carried out by analyzing the linear stability of the flows. Lattice Boltzmann methods are then introduced and their numerical stability issues are studied through analyses based on the von Neumann approach. Finally, the method is assessed on academic simulations of increasing complexity representative of secondary air systems, requiring conjugate heat transfer simulation

Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

International audienceWith the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical prefactor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime

An extended spectral analysis of the lattice Boltzmann method: modal interactions and stability issues

International audienceAn extension of the von Neumann linear analysis is proposed for the study of the discrete-velocity Boltzmann equation (DVBE) and the lattice Boltzmann (LB) scheme. While the standard technique is restricted to the investigation of the spectral radius and the dissipation and dispersion properties, a new focus is put here on the information carried by the modes. The technique consists in the computation of the moments of the eigenvectors and their projection onto the physical waves expected by the continuous linearized Navier-Stokes (NS) equations. The method is illustrated thanks to some simulations with the BGK (Bhatnagar-Gross-Krook) collision operator on the D2Q9 and D2V17 lattices. The present analysis reveals the existence of two kinds of modes: non-observable modes that do not carry any macroscopic information and observable modes. The latter may carry either a physical wave expected by the NS equations, or an unphysical information. Further investigation of modal interactions highlights a phenomenon called curve veering occurring between two observable modes: a swap of eigenvectors and dissipation rate is observed between the eigencurves. Increasing the Mach number of the mean flow yields an eigenvalue collision at the origin of numerical instabilities of the BGK model, arising from the error in the time and space discretization of the DVBE. (C) 2019 Elsevier Inc. All rights reserved

Consistent vortex initialization for the athermal lattice Boltzmann method

International audienceA barotropic counterpart of the well-known convected vortex test case is rigorously derived from the Euler equations along with an athermal equation of state. Starting from a given velocity distribution corresponding to an intended flow recirculation, the athermal counterpart of the Euler equations are solved to obtain a consistent density field. The present initialization is assessed on a standard lattice Boltzmann solver based on the D2Q9 lattice. Compared to the usual isentropic initialization, a much lower spurious relaxation toward the targeted solution is observed, which is due to the spatial resolution rather than approximated macroscopic quantities. The amplitude of the spurious waves can be further reduced by including an off-equilibrium part in the initial distribution functions

A hybrid lattice Boltzmann method for gaseous detonations

International audienceThis article is dedicated to the construction of a robust and accurate numerical scheme based on the lattice Boltzmann method (LBM) for simulations of gaseous detonations. This objective is achieved through careful construction of a fully conservative hybrid lattice Boltzmann scheme tailored for multi-species reactive flows. The core concept is to retain LBM low dissipation properties for acoustic and vortical modes by using the collide and stream algorithm for the particle distribution function, while transporting entropic and species modes via a specifically designed finite-volume scheme. The proposed method is first evaluated on common academic cases, demonstrating its ability to accurately simulate multi-species compressible and reactive flows with discontinuities: the convection of inert species, a Sod shock tube with two ideal gases and a steady one-dimensional inviscid detonation wave. Subsequently, the potential of this novel approach is demonstrated in one- and two-dimensional inviscid unsteady gaseous detonations, highlighting its ability to accurately recover detonation structures and associated instabilities for high activation energies. To the authors' knowledge, this study is the first successful simulation of detonation cellular structures capitalizing on the LBM collide and stream algorithm