An Improved Lattice Boltzmann Method for Steady Fluid Flows

2004 ◽  
Author(s):  
Aditya C. Velivelli ◽  
Kenneth M. Bryden
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equation ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Flow Problems ◽  
Computational Performance ◽  
Boltzmann Method

The use of the lattice Boltzmann method in computational fluid dynamics has been steadily increasing. The highly local nature of lattice Boltzmann computations have allowed for easy cache optimization and parallelization. This bestows the lattice Boltzmann method with considerable superiority in computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size. The time step size is limited by the Courant-Friedrichs-Lewy (CFL) condition and local gradients in the solution, the latter limitation being more extreme. This paper describes a novel explicit discretization for the lattice Boltzmann method that can perform simulations with larger time step sizes. The new algorithm is applid to the steady Burger’s equation, uux = μ(uxx + uyy), which is a nonlinear partial differential equation containing both convection and diffusion terms. A comparison between the original lattice Boltzmann method and the new algorithm is performed with regard to time for computation and accuracy.

scholarly journals Large-scaled simulation on the coherent vortex evolution of a jet in a cross-flow based on lattice Boltzmann method

2015 ◽  
Vol 19 (3) ◽  
pp. 977-988 ◽  
Author(s):  
Yanqin Shangguan ◽  
Xian Wang ◽  
Yueming Li
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Flow Characteristics ◽  
Cross Flow ◽  
Hairpin Vortex ◽  
Graphic Processing Units ◽  
Computational Performance ◽  
Good Ability ◽  
Secondary Vortices ◽  
Boltzmann Method

Large eddy simulation (LES) is performed on a jet issued normally into a cross-flow using lattice Boltzmann method (LBM) and multiple graphic processing units (multi-GPUs) to study the flow characteristics of jets in cross-flow (JICF). The simulation with 8 1.50?10 grids is fulfilled with 6 K20M GPUs. With large-scaled simulation, the secondary and tertiary vortices are captured. The features of the secondary vortices and the tertiary vortices reveal that they have a great impact on the mixing between jet flow and cross-flow. The qualitative and quantitative results also indicate that the evolution mechanism of vortices is not constant, but varies with different situations. The hairpin vortex under attached jet regime originates from the boundary layer vortex of cross-flow. While, the origin of hairpin vortex in detached jet is the jet shear-layer vortex. The mean velocities imply the good ability of LBM to simulate JICF and the large loss of jet momentum in detached jet caused by the strong penetration. Besides, in our computation, a high computational performance of 1083.5 MLUPS is achieved.

A Meshfree-Based Lattice Boltzmann Approach for Simulation of Fluid Flows Within Complex Geometries

2018 ◽  
pp. 188-222 ◽  
Author(s):  
Sonam Tanwar
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Moving Boundary ◽  
Fluid Flows ◽  
Computational Domain ◽  
Time Step ◽  
Boundary Problems ◽  
Complex Process ◽  
Element Free ◽  
Boltzmann Method

This chapter develops a meshless formulation of lattice Boltzmann method for simulation of fluid flows within complex and irregular geometries. The meshless feature of proposed technique will improve the accuracy of standard lattice Boltzmann method within complicated fluid domains. Discretization of such domains itself may introduce significant numerical errors into the solution. Specifically, in phase transition or moving boundary problems, discretization of the domain is a time-consuming and complex process. In these problems, at each time step, the computational domain may change its shape and need to be re-meshed accordingly for the purpose of accuracy and stability of the solution. The author proposes to combine lattice Boltzmann method with a Galerkin meshfree technique popularly known as element-free Galerkin method in this chapter to remove the difficulties associated with traditional grid-based methods.

Seismic finite‐difference modeling with spatially varying time steps

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1290-1293 ◽  
Author(s):  
Ekkehart Tessmer
Keyword(s):  
Finite Difference ◽  
Seismic Velocity ◽  
Finite Difference Methods ◽  
Low Velocity ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Large Velocity ◽  
Numerical Grid ◽  
Spatially Varying

Numerical seismic modeling by finite‐difference methods usually works with a global time‐step size. Because of stability considerations, the time‐step size is determined essentially by the highest seismic velocity, i.e., the higher the highest velocity, the smaller the time step needs to be. Therefore, if large velocity contrasts exist within the numerical grid, domains of low velocity are oversampled temporally. Using different time‐step sizes in different parts of the numerical grid can reduce computational costs considerably.

scholarly journals A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2284
Author(s):  
Endre Kovács ◽  
Ádám Nagy ◽  
Mahmoud Saleh
Keyword(s):  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Second Order ◽  
Stiff Systems ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Two Stage ◽  
Time Step Size ◽  
New Methods

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

scholarly journals An efficient algorithm for simulating ensembles of parameterized flow problems

2018 ◽  
Vol 39 (3) ◽  
pp. 1180-1205 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Initial Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Flow Problems ◽  
Viscosity Coefficients ◽  
Deviation Ratio ◽  
Theoretical Results ◽  
Parameter Deviation

Abstract Many applications of computational fluid dynamics require multiple simulations of a flow under different input conditions. In this paper, a numerical algorithm is developed to efficiently determine a set of such simulations in which the individually independent members of the set are subject to different viscosity coefficients, initial conditions and/or body forces. The proposed scheme, when applied to the flow ensemble, needs to solve a single linear system with multiple right-hand sides, and thus is computationally more efficient than solving for all the simulations separately. We show that the scheme is nonlinearly and long-term stable under certain conditions on the time-step size and a parameter deviation ratio. A rigorous numerical error estimate shows the scheme is of first-order accuracy in time and optimally accurate in space. Several numerical experiments are presented to illustrate the theoretical results.

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method

2002 ◽  
Vol 452 ◽  
pp. 61-96 ◽  
Author(s):  
K. SANKARANARAYANAN ◽  
X. SHAN ◽  
I. G. KEVREKIDIS ◽  
S. SUNDARESAN
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Volume Fraction ◽  
Computational Cost ◽  
Computational Experiments ◽  
Mass Force ◽  
Virtual Mass ◽  
Time Step ◽  
Mass Coefficient ◽  
Boltzmann Method

We present closures for the drag and virtual mass force terms appearing in a two-fluid model for flow of a mixture consisting of uniformly sized gas bubbles dispersed in a liquid. These closures were deduced through computational experiments performed using an implicit formulation of the lattice Boltzmann method with a BGK collision model. Unlike the explicit schemes described in the literature, this implicit implementation requires iterative calculations, which, however, are local in nature. While the computational cost per time step is modestly increased, the implicit scheme dramatically expands the parameter space in multiphase flow calculations which can be simulated economically. The closure relations obtained in our study are limited to a regular array of uniformly sized bubbles and were obtained by simulating the rise behaviour of a single bubble in a periodic box. The effect of volume fraction on the rise characteristics was probed by changing the size of the box relative to that of the bubble. While spherical bubbles exhibited the expected hindered rise behaviour, highly distorted bubbles tended to rise cooperatively. The closure for the drag force, obtained in our study through computational experiments, captured both hindered and cooperative rise. A simple model for the virtual mass coefficient, applicable to both spherical and distorted bubbles, was also obtained by fitting simulation results. The virtual mass coefficient for isolated bubbles could be correlated with the aspect ratio of the bubbles.

What is the Small Parameter ε in the Chapman-Enskog Expansion of the Lattice Boltzmann Method?

2012 ◽  
Vol 134 (1) ◽  
Author(s):  
Minoru Watari
Keyword(s):  
Small Parameter ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Viscosity Coefficient ◽  
Second Order ◽  
Time Step ◽  
Order Accuracy ◽  
The Difference ◽  
Boltzmann Method

The lattice Boltzmann method (LBM) is shown to be equivalent to the Navier-Stokes equations by applying the Chapman-Enskog (C-E) expansion, which has been established by pioneer researchers. However, it is still difficult for elementary researchers. There is no clear explanation of the small parameter ε used in the C-E expansion. There are several expressions for the viscosity coefficient; some are unclear on the relationship with ε. There are two expressions on the LBM evolution equation. Elementary researchers are perplexed as to which is correct. The LBM achieves second order accuracy by including the numerical viscosity within the physical viscosity. This is not only difficult for elementary researchers to understand but also sometimes leads senior researchers into making errors. The C-E expansion of the LBM was thoroughly reviewed and is presented as a self-contained form in this paper. It is natural to use the time step Δt as ε. The viscosity coefficient is expressed as μ∝Δxc(τ − 1/2). The viscosity relationship and the second order accuracy were confirmed by numerical simulations. The difference in the two expressions on the LBM evolution is simply one of perspective. They are identical. The difference between the relaxation parameter τD for the discrete Boltzmann equation and τ for the LBM was discussed. While τD is a quantity of time, τ is genuinely nondimensional, which is sometimes overlooked.

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