Improved compressible hybrid lattice Boltzmann method on standard lattice for subsonic and supersonic flows (original) (raw)
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LATTICE BOLTZMANN APPROACH TO HIGH-SPEED COMPRESSIBLE FLOWS
International Journal of Modern Physics C, 2007
We present an improved lattice Boltzmann model for high-speed compressible flows. The model is composed of a discrete-velocity model by Kataoka and Tsutahara [Phys. Rev. E 69, 056702 (2004)] and an appropriate finite-difference scheme combined with an additional dissipation term. With the dissipation term parameters in the model can be flexibly chosen so that the von Neumann stability condition is satisfied. The influence of the various model parameters on the numerical stability is analyzed and some reference values of parameter are suggested. The new scheme works for both subsonic and supersonic flows with a Mach number up to 30 (or higher), which is validated by well-known benchmark tests. Simulations on Riemann problems with very high ratios (1000 : 1) of pressure and density also show good accuracy and stability. Successful recovering of regular and double Mach shock reflections shows the potential application of the lattice Boltzmann model to fluid systems where non-equilibrium processes are intrinsic. The new scheme for stability can be easily extended to other lattice Boltzmann models.
Computers & Fluids
A multi-dimensional double distribution function thermal lattice Boltzmann model has been developed to simulate fully compressible flows at moderate Mach number. The lattice Boltzmann equation is temporally and spatially discretizated by an asymptotic preserving finite volume scheme. The micro-velocities discretization is adopted on regular low-symmetry lattices (D1Q3, D2Q9, D3Q15, D3Q19, D3Q27). The third-order Hermite polynomial density distribution function on low-symmetry lattices is used to solve the flow field, while a second-order energy distribution is employed to compute the temperature field. The fully compressible Navier-Stokes equations are recovered by standard order Gauss-Hermite polynomial expansions of Maxwell distribution with cubic correction terms, which are added by an external force expressed in orthogonal polynomials form. The proposed model is validated considering several benchmark cases, namely the Sod shock tube, thermal Couette flow and two-dimensional Riemann problem. The numerical results are in very good agreement with both analytical solution and reference results.
A linear stability analysis of compressible hybrid lattice Boltzmann methods
Journal of Computational Physics
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 oneand 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 re f and (2) introducing a controllable numerical dissipation σ 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.
The lattice Boltzmann method for compressible flows at high Mach number
The lattice Boltzmann method (LBM) is a mesoscopiclevel “particle-based” method representing a density distribution that can be used to solve certain types of partial differential equations (PDEs). More specifically, LBM provides a very simple numerical procedure for simulating the Boltzmann equation at the microscopic level, whose appropriate coarse graining leads to the standard hydrodynamic equations (in the long wavelength limit) that express the conservation of mass (continuity equation) and momentum (Navier Stokes equation) at the continuum level. Owing to the fact that LBM offers numerous advantages such as simplicity, efficiency and accuracy over more traditional methodologies, it has been increasingly used by researchers in recent years for the simulation of turbulence and multi-phase and multi-component flows in porous media. In the initial development of LBM, researchers focused on the simulation of very low-Mach-number incompressible and isothermal flows [2]. These methodologies are not suitable (e.g., unstable and unreliable) for the simulation of fluid flows at larger Mach numbers. Currently, there is no consensus on the “correct” method for application of LBM to the simulation of compressible high-Mach-number flows (which necessarily requires the development of a model for LBM that leads to the correct Newtonian thermohydrodynamics in the long wavelength limit). Nevertheless, there has been some important recent developments on models for LBM that can simulate high-Mach-number compressible flows, most notably the Kataoka-Tsutahara (KT) model [4] and Qu’s model [7]. In this paper, we present and compare results for a number of benchmark test cases of compressible flows obtained with the KT model and Qu’s model. To this purpose, these thermal LBM models have been implemented using a number of different finite volume advection schemes such as the simple upstream differencing scheme (UDS) and various total variation diminishing (TVD) schemes, such as the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the weighted essentially non-oscillatory (WENO) schemes [7, 10]. Typical benchmark tests such as the shock tube problem and the double Mach reflection problem have been implemented in order to validate the two compressible LBM models and the proposed advection schemes used with these models. These simulations demonstrate the capability of LBM in the simulation of compressible flows at high Mach numbers and will provide the guidance for future researchers who are interested in finding the most appropriate LBM models and advection schemes for specific compressible flow applications.
Three-dimensional Lattice Boltzmann model for high-speed compressible flows
2010
A highly efficient three-dimensional (3D) Lattice Boltzmann (LB) model for high speed compressible flows is proposed. This model is developed from the original one by Kataoka and Tsutahara[Phys. Rev. E 69, 056702 (2004)]. The convection term is discretized by the Non-oscillatory, containing No free parameters and Dissipative (NND) scheme, which effectively damps oscillations at discontinuities. To be more consistent with the kinetic theory of viscosity and to further improve the numerical stability, an additional dissipation term is introduced. Model parameters are chosen in such a way that the von Neumann stability criterion is satisfied. The new model is validated by well-known benchmarks, (i) Riemann problems, including the problem with Lax shock tube and a newly designed shock tube problem with high Mach number; (ii) reaction of shock wave on droplet or bubble. Good agreements are obtained between LB results and exact ones or previously reported solutions. The model is capable of simulating flows from subsonic to supersonic and capturing jumps resulted from shock waves.
Two-dimensional lattice Boltzmann model for compressible flows with high Mach number
Physica A: Statistical Mechanics and its Applications, 2008
In this paper we present an improved lattice Boltzmann model for compressible Navier-Stokes system with high Mach number. The model is composed of three components: (i) the discrete-velocity-model by Watari and Tsutahara [Phys Rev E 67,036306(2003)], (ii) a modified Lax-Wendroff finite difference scheme where reasonable dissipation and dispersion are naturally included, (iii) artificial viscosity. The improved model is convenient to compromise the high accuracy and stability. The included dispersion term can effectively reduce the numerical oscillation at discontinuity. The added artificial viscosity helps the scheme to satisfy the von Neumann stability condition. Shock tubes and shock reflections are used to validate the new scheme. In our numerical tests the Mach numbers are successfully increased up to 20 or higher. The flexibility of the new model makes it suitable for tracking shock waves with high accuracy and for investigating nonlinear nonequilibrium complex systems.
Compressible Lattice Boltzmann Method for Turbulent Jet Flow Simulations
2018
Abstract—In Computational Fluid Dynamics (CFD), there are a variety of numerical methods, of which some depend on macroscopic model representatives. These models can be solved by finite-volume, finite-element or finite-difference methods on a microscopic description. However, the lattice Boltzmann method (LBM) is considered to be a mesoscopic particle method, with its scale lying between the macroscopic and microscopic scales. The LBM works well for solving incompressible flow problems, but certain limitations arise from solving compressible flows, particularly at high Mach numbers. An improved lattice Boltzmann model for compressible flow problems is presented in this research study. A higher-order Taylor series expansion of the Maxwell equilibrium distribution function is used to overcome limitations in LBM when solving highMach-number flows. Large eddy simulation (LES) is implemented in LBM to simulate turbulent jet flows. The results have been validated with available experime...
Lattice Boltzmann method for compressible flows with high Mach numbers
Physical Review E, 2000
In this paper we present a lattice Boltzmann model to simulate compressible flows by introducing an attractive force. This scheme has two main advantages: one is to soften sound speed effectively, which greatly raises the Mach number ͑up to 5͒; another is its relative simple procedure. Simulations of the March cone and the comparison between theoretical expectations and simulations demonstrate that the scheme is effective in the simulation of compressible flows with high Mach numbers, which would create many new applications.
Large-eddy simulation of subsonic turbulent jets using the compressible lattice Boltzmann method
International Journal of Numerical Methods in Fluids, 2020
The lattice Boltzmann method (LBM) is a powerful technique for the computational modeling of a wide variety of singles and multiphase flows involving complex geometries. Although the LBM has been demonstrated to be effective for the solution of incompressible flow problems, there are limitations when this methodology is applied to the solution of compressible flows, especially for flows at high Mach numbers. In this article, we investigate strategies to overcome some of the limitations associated with the application of LBM to compressible flows. To this purpose, one of the key contributions of this study is the synthesis and integration of previous efforts concerning the formulation of LBM for the large-eddy simulation (LES) of compressible turbulent flows in the subsonic flow regime. It is shown how certain limitations of applying the LBM to compressible flows can be addressed by using either a higher order Taylor series expansion of the Maxwell–Boltzmann equilibrium distribution function or using the Kataoka and Tsutahara (KT) LBM model formulation for compressible flows. The proposed LBM/LES methodology for compressible flows has been combined with the Kirchhoff integral formulation for computational aeroacoustics and used to simulate the flow and acoustic fields of compressible jet flows at high subsonic speeds with practical relevance for providing a better understanding of problems associated with jet noise. In this context, simulations of the physics associated with the jet flow and concomitant noise in the nearand far-field regimes were conducted using the proposed framework of a compressible LBM/LES and Kirchhoff integral method. The results of the subsonic isothermal and nonisothermal jet flow simulations for the flow and acoustic fields have been compared with available numerical and experimental results with generally good to excellent agreement.
Lattice Boltzmann method and gas-kinetic BGK scheme in the low-Mach number viscous flow simulations
Journal of Computational Physics, 2003
Both lattice Boltzmann method (LBM) and the gas-kinetic BGK scheme are based on the numerical discretization of the Boltzmann equation with collisional models, such as, the Bhatnagar-Gross-Krook (BGK) model. LBM tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process. On the other hand, the gas-kinetic BGK scheme is a finite volume scheme, where the time-dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces. Currently, LBM is mainly used for low Mach number, nearly incompressible flow simulation. For the gas-kinetic scheme, the application is focusing on the high speed compressible flows. In this paper, we are going to compare both schemes in the isothermal low-Mach number flow simulations. The methodology for developing both schemes will be clarified through the introduction of operator splitting Boltzmann model and operator averaging Boltzmann model. From the operator splitting Boltzmann model, the error rooted in many kinetic schemes, which are based on the decoupling of particle transport and collision, can be easily understood. As to the test case, we choose to use the 2D cavity flow since it is one of the most extensively studied cases. Detailed simulation results with different Reynolds numbers, as well as the benchmark solutions, are presented.