On the conservativity of the Particles-on-Demand method for solution of the Discrete Boltzmann Equation (original) (raw)
Related papers
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.
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.
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.
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.
Comparison between isothermal collision-streaming and finite-difference lattice Boltzmann models
International Journal of Modern Physics C
We present here a comparison between collision-streaming and finite-difference lattice Boltzmann (LB) models. This study provides a derivation of useful formulae which help one to properly compare the simulation results obtained with both LB models. We consider three physical problems: the shock wave propagation, the damping of shear waves, and the decay of Taylor–Green vortices, often used as benchmark tests. Despite the different mathematical and computational complexity of the two methods, we show how the physical results can be related to obtain relevant quantities.
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.
Numerical solutions of the Boltzmann equation: comparison of different algorithms
European Journal of Mechanics - B/Fluids, 2008
In the paper we compare different algorithms for numerical solutions of the Boltzmann equation. For this comparison we have taken the standard problem of the shock wave structure in a mono-atomic rarefied gas. Different parameters characterizing the shock structure have been calculated by a Monte Carlo simulation (DSMC), a second order time-relaxed Monte Carlo method (TRMC2), a fully deterministic discrete velocity method (DV), a discrete velocity method with Monte Carlo calculations of collision integral (DVMC) and a molecular dynamics method (MD). Results of these calculations have been compared with the shock wave structure obtained in experiments in a shock tube. The results of the comparison are not conclusive. We have observed general agreement between numerical and experimental results but there is no single numerical method which fits best to the experimental measurements.
Computers & Fluids
A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al. [12], which has been found, via different test cases, to be unstable for supersonic regimes. The improvements consist of: (1) a new discretization of the lattice closure correction term making possible to properly simulate supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized formalism. The result is a hybrid method that resolves the mass and momentum equations with an LBM algorithm, and resolves the entropy-based energy equation with a finite volume method. This approach fully recovers the physics of the Navier-Stokes-Fourier equations with the ideal gas equation of state, and is valid from subsonic to supersonic regimes. It is then successfully assessed with both smooth flows and flows involving shocks. The proposed model is shown to be an efficient, accurate, and robust alternative to classic Navier-Stokes methods for the simulation of compressible flows.
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.
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...