Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications

High-order flux reconstruction thermal lattice Boltzmann flux solver for simulation of incompressible thermal flows

In this paper, a high-order solver combining the flux reconstruction (FR) method and the thermal lattice Boltzmann flux solver (FRTLBFS) is developed for accurately and efficiently simulating incompressible thermal flows. The conservative differential equations recovered from Chapman-Enskog analysis of the thermal lattice Boltzmann equation are solved by the high-order FR method. The thermal lattice Boltzmann method is only applied to reconstruct the local solution used for evaluating fluxes at the solution and flux points. Unlike the traditional Navier-Stokes-Boussinesq (NSB) solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current FRTLBFS are coupled and computed uniformly. In comparison with the recently developed high-order flux reconstruction thermal lattice Boltzmann method, the FRTLBFS holds advantages such as high-order accuracy, good stability, and compactness but is more efficient and low storage, since only macroscopic flow variables including density, velocity, and temperature are stored and evolved. In addition, the physical boundary conditions in FRTLBFS can be directly implemented by using the same method as in conventional NSB solvers. Numerical validations of the proposed method are implemented by simulating (a) the porous plate problem, (b) natural convection in a square cavity, (c) unsteady natural convection in a tall cavity, and (d) thermal lid-driven cavity flow. Numerical results demonstrate that the present solver is an attractive tool to simulate incompressible thermal flows due to its high-order accuracy, stability, and low memory cost.

Particles on demand for flows with strong discontinuities

Particles-on-demand formulation of kinetic theory [B. Dorschner, F. Bösch and I. V. Karlin, Phys. Rev. Lett. 121, 130602 (2018)0031-900710.1103/PhysRevLett.121.130602] is used to simulate a variety of compressible flows with strong discontinuities in density, pressure, and velocity. Two modifications are applied to the original formulation of the particles-on-demand method. First, a regularization by Grad's projection of particles populations is combined with the reference frame transformations in order to enhance stability and accuracy. Second, a finite-volume scheme is implemented which allows tight control of mass, momentum, and energy conservation. The proposed model is validated with an array of challenging one- and two-dimensional benchmarks of compressible flows, including hypersonic and near-vacuum situations, Richtmyer-Meshkov instability, double Mach reflection, and astrophysical jet. Excellent performance of the modified particles-on-demand method is demonstrated beyond the limitations of other lattice Boltzmann-like approaches to compressible flows.

Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has two key steps: collision and streaming. In a conventional LBM, the streaming is exact, where each distribution function is perfectly shifted to the neighbor node on the uniform mesh arrangement. This advantage may curtail the applicability of the method to problems with complex geometries. To overcome this issue, a high-order meshless interpolation-based approach is proposed to handle the streaming step. Owing to its high accuracy, the radial basis function (RBF) is one of the popular methods used for interpolation. In general, RBF-based approaches suffer from some stability issues, where their stability strongly depends on the shape parameter of the RBF. In the current work, a stabilized RBF approach is used to handle the streaming. The stabilized RBF approach has a weak dependency on the shape parameter, which improves the stability of the method and reduces the dependency of the shape parameter. Both the stabilized RBF method and the streaming of the LBM are used for solving three benchmark problems. The results of the stabilized method and the perfect streaming LBM are compared with analytical solutions or published results. Excellent agreements are observed, with a little advantage for the stabilized approach. Additionally, the computational cost is compared, where a marginal difference is observed in the favor of the streaming of the LBM. In conclusion, one could report that the stabilized method is a viable alternative to the streaming of the LBM in handling both simple and complex geometries.

A Strong-Form Off-Lattice Boltzmann Method for Irregular Point Clouds

Radial basis function generated finite differences (RBF-FD) represent the latest discretization approach for solving partial differential equations. Their benefits include high geometric flexibility, simple implementation, and opportunity for large-scale parallel computing. Compared to other meshfree methods, typically based upon moving least squares (MLS), the RBF-FD method is able to recover a high order of algebraic accuracy while remaining better conditioned. These features make RBF-FD a promising candidate for kinetic-based fluid simulations such as lattice Boltzmann methods (LB). Pursuant to this approach, we propose a characteristic-based off-lattice Boltzmann method (OLBM) using the strong form of the discrete Boltzmann equation and radial basis function generated finite differences (RBF-FD) for the approximation of spatial derivatives. Decoupling the discretizations of momentum and space enables the use of irregular point cloud, local refinement, and various symmetric velocity sets with higher order isotropy. The accuracy and computational efficiency of the proposed method are studied using the test cases of Taylor–Green vortex flow, lid-driven cavity, and periodic flow over a square array of cylinders. For scattered grids, we find the polyharmonic spline + poly RBF-FD method provides better accuracy compared to MLS. For Cartesian node layouts, the results are the opposite, with MLS offering better accuracy. Altogether, our results suggest that the RBF-FD paradigm can be applied successfully also for kinetic-based fluid simulation with lattice Boltzmann methods.

Multiscale semi-Lagrangian lattice Boltzmann method

We present a multi-scale lattice Boltzmann scheme, which adaptively refines particles' velocity space. Different velocity sets of lower and higher order are consistently and efficiently coupled, allowing us to use the higher-order model only when and where needed. This includes regions of high Mach or high Knudsen numbers. The coupling procedure of discrete velocity sets consists of either a projection of the higher-order populations onto the lower-order lattice or lifting of the lower-order populations to the higher-order velocity space. Both lifting and projection are local operations, which enable a flexible adaptive velocity set. The proposed scheme is formulated for both a static and an optimal, co-moving reference frame, in the spirit of the recently introduced Particles on Demand method. The multi-scale scheme is validated with an advection of an athermal vortex and in a jet flow setup. The performance of the proposed scheme is further investigated in the shock structure problem and a high-Knudsen-number Couette flow, typical examples of highly non-equilibrium flows in which the order of the velocity set plays a decisive role. The results demonstrate that the proposed multi-scale scheme can operate accurately, with flexibility in terms of the underlying models and with reduced computational requirements.

Analysis of Aeroacoustic Properties of the Local Radial Point Interpolation Cumulant Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has recently been used to simulate wave propagation, one of the challenging aspects of wind turbine modeling and simulation. However, standard LB methods suffer from the instability that occurs at low viscosities and from its characteristic lattice uniformity, which results in issues of accuracy and computational efficiency following mesh refinement. The local radial point interpolation cumulant lattice Boltzmann method (LRPIC-LBM) is proposed in this paper to overcome these shortcomings. The LB equation is divided into collision and streaming steps. The collision step is modeled by the cumulant method, one of the stable LB methods at low viscosities. In addition, the streaming step, which is naturally a pure advection equation, is discretized in time and space using the Lax–Wendroff scheme and the local radial point interpolation method (RPIM), a mesh free method. We describe the propagation of planar acoustic waves, including the temporal decay of a standing plane wave and the spatial decay of a planar acoustic pulse. The analysis of these specific benchmark problems has yielded qualitative and quantitative data on acoustic dispersion and dissipation, and their deviation from analytical results demonstrates the accuracy of the method. We found that the LRPIC-LBM replicates the analytical results for different viscosities, and the errors of the fundamental acoustic properties are negligible, even for quite low resolutions. Thus, this method may constitute a useful platform for effectively predicting complex engineering problems such as wind turbine simulations, without parameter dependencies such as the number of points per wavelength Nppw and resolution σ or the detrimental effect caused by the use of coarse grids found in other accurate and stable LB models.

Semi-Lagrangian implicit Bhatnagar-Gross-Krook collision model for the finite-volume discrete Boltzmann method

In order to increase the accuracy of temporal solutions, reduce the computational cost of time marching, and improve the stability associated with collisions for the finite-volume discrete Boltzmann method, an advanced implicit Bhatnagar-Gross-Krook (BGK) collision model using a semi-Lagrangian approach is proposed in this paper. Unlike existing models, in which the implicit BGK collision is resolved either by a temporal extrapolation or by a variable transformation, the proposed model removes the implicitness by tracing the particle distribution functions (PDFs) back in time along their characteristic paths during the collision process. An interpolation scheme is needed to evaluate the PDFs at the traced-back locations. By using the first-order interpolation, the resulting model allows for the straightforward replacement of f_{α}^{eq,n+1} by f_{α}^{eq,n} no matter where it appears. After comparing the proposed model with the existing models under different numerical conditions (e.g., different flux schemes and time-marching schemes) and using the proposed model to successfully modify the variable transformation technique, three conclusions can be drawn. First, the proposed model can improve the accuracy by almost an order of magnitude. Second, it can slightly reduce the computational cost. Therefore, the proposed scheme improves accuracy without extra cost. Finally, the proposed model can significantly improve the Δt/τ limit compared to the temporal interpolation model while having the same Δt/τ limit as the variable transformation approach. The proposed scheme with a second-order interpolation is also developed and tested; however, that technique displays no advantage over the simple first-order interpolation approach. Both numerical and theoretical analyses are also provided to explain why the developed implicit scheme with simple first-order interpolation can outperform the same scheme with second-order interpolation, as well as the existing temporal extrapolation and variable transformation schemes.

Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows

In this work, an efficient gas kinetic scheme is presented for simulation of two-dimensional incompressible thermal flows. In the scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of the Boltzmann equation. To compute these fluxes, two distribution functions are involved. One is the circular function, which is used to calculate the numerical fluxes of mass and momentum equations. Due to the incompressible limit, the circle at the cell interface can be approximately considered to be symmetric so that the expressions for the conservative variables and numerical fluxes at the cell interface can be given explicitly and concisely. Another one is the D2Q4 model, which is utilized to compute the numerical flux of the energy equation. By following the process for derivation of numerical fluxes of mass and momentum equations, the numerical flux of the energy equation can also be given explicitly. The accuracy, efficiency, and stability of the present scheme are validated by simulating several thermal flow problems. Numerical results showed that the present scheme can provide accurate numerical results for incompressible thermal flows at a wide range of Rayleigh numbers with less computational cost than that needed by the thermal lattice Boltzmann flux solver (TLBFS), which has been proven to be more efficient than the thermal lattice Boltzmann method (TLBM).

Semi-Lagrangian off-lattice Boltzmann method for weakly compressible flows

The lattice Boltzmann method is a simulation technique in computational fluid dynamics. In its standard formulation, it is restricted to regular computation grids, second-order spatial accuracy, and a unity Courant-Friedrichs-Lewy (CFL) number. This paper advances the standard lattice Boltzmann method by introducing a semi-Lagrangian streaming step. The proposed method allows significantly larger time steps, unstructured grids, and higher-order accurate representations of the solution to be used. The appealing properties of the approach are demonstrated in simulations of a two-dimensional Taylor-Green vortex, doubly periodic shear layers, and a three-dimensional Taylor-Green vortex.

Meshless lattice Boltzmann method for the simulation of fluid flows

A meshless lattice Boltzmann numerical method is proposed. The collision and streaming operators of the lattice Boltzmann equation are separated, as in the usual lattice Boltzmann models. While the purely local collision equation remains the same, we rewrite the streaming equation as a pure advection equation and discretize the resulting partial differential equation using the Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin scheme based on augmented radial basis functions in space. The meshless feature of the proposed method makes it a more powerful lattice Boltzmann solver, especially for cases in which using meshes introduces significant numerical errors into the solution, or when improving the mesh quality is a complex and time-consuming process. Three well-known benchmark fluid flow problems, namely the plane Couette flow, the circular Couette flow, and the impulsively started cylinder flow, are simulated for the validation of the proposed method. Excellent agreement with analytical solutions or with previous experimental and numerical results in the literature is observed in all the simulations. Although the computational resources required for the meshless method per node are higher compared to that of the standard lattice Boltzmann method, it is shown that for cases in which the total number of nodes is significantly reduced, the present method actually outperforms the standard lattice Boltzmann method.

Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh

A numerical model of the lattice Boltzmann method (LBM) utilizing least-squares finite-element method in space and the Crank-Nicolson method in time is developed. This method is able to solve fluid flow in domains that contain complex or irregular geometric boundaries by using the flexibility and numerical stability of a finite-element method, while employing accurate least-squares optimization. Fourth-order accuracy in space and second-order accuracy in time are derived for a pure advection equation on a uniform mesh; while high stability is implied from a von Neumann linearized stability analysis. Implemented on unstructured mesh through an innovative element-by-element approach, the proposed method requires fewer grid points and less memory compared to traditional LBM. Accurate numerical results are presented through two-dimensional incompressible Poiseuille flow, Couette flow, and flow past a circular cylinder. Finally, the proposed method is applied to estimate the permeability of a randomly generated porous media, which further demonstrates its inherent geometric flexibility.

Domain-decomposition method for parallel lattice Boltzmann simulation of incompressible flow in porous media

The lattice Boltzmann method has proven to be a promising method to simulate flow in porous media. Its practical application often relies on parallel computation because of the demand for a large domain and fine grid resolution to adequately resolve pore heterogeneity. The existing domain-decomposition methods for parallel computation usually decompose a domain into a number of subdomains first and then recover the interfaces and perform the load balance. Normally, the interface recovery and the load balance have to be performed iteratively until an acceptable load balance is achieved; this costs time. In this paper we propose a cell-based domain-decomposition method for parallel lattice Boltzmann simulation of flow in porous media. Unlike the existing methods, the cell-based method performs the load balance first to divide the total number of fluid cells into a number of groups (or subdomains), in which the difference of fluid cells in each group is either 0 or 1, depending on if the total number of fluid cells is a multiple of the processor numbers; the interfaces between the subdomains are recovered at last. The cell-based method is to recover the interfaces rather than the load balance; it does not need iteration and gives an exact load balance. The performance of the proposed method is analyzed and compared using different computer systems; the results indicate that it reaches the theoretical parallel efficiency. The method is then applied to simulate flow in a three-dimensional porous medium obtained with microfocus x-ray computed tomography to calculate the permeability, and the result shows good agreement with the experimental data.