Lattice Boltzmann model for compressible flows on standard lattices: Variable Prandtl number and adiabatic exponent

Particles on demand method: Theoretical analysis, simplification techniques, and model extensions

The particles on demand method [Phys. Rev. Lett. 121, 130602 (2018)0031-900710.1103/PhysRevLett.121.130602] was recently formulated with a conservative finite-volume discretization and validated against challenging benchmarks. In this work, we focus on the properties of the reference frame transformation and its implications on the accuracy of the model. Based on these considerations, we propose strategies that simplify the scheme and generalize it to include a tunable Prandtl number via quasi-equilibrium relaxation. Finally, we adapt concepts from the multiscale semi-Lagrangian lattice Boltzmann formulation to the proposed framework, further improving the potential and the operating range of the kinetic model. Numerical simulations of high Mach compressible flows demonstrate excellent accuracy and stability of the model over a wide range of conditions.

Discrete unified gas kinetic scheme for continuum compressible flows

In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equations in the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equation for a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equations exactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equations are discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.

Thermal Lattice Boltzmann Flux Solver for Natural Convection of Nanofluid in a Square Enclosure

In the present study, mathematical modeling was performed to simulate natural convection of a nanofluid in a square enclosure using the thermal lattice Boltzmann flux solver (TLBFS). Firstly, natural convection in a square enclosure, filled with pure fluid (air and water), was investigated to validate the accuracy and performance of the method. Then, influences of the Rayleigh number, of nanoparticle volume fraction on streamlines, isotherms and average Nusselt number were studied. The numerical results illustrated that heat transfer was enhanced with the augmentation of Rayleigh number and nanoparticle volume fraction. There was a linear relationship between the average Nusselt number and solid volume fraction. and there was an exponential relationship between the average Nusselt number and Ra. In view of the Cartesian grid used by the immersed boundary method and lattice model, the immersed boundary method was chosen to treat the no-slip boundary condition of the flow field, and the Dirichlet boundary condition of the temperature field, to facilitate natural convection around a bluff body in a square enclosure. The presented numerical algorithm and code implementation were validated by means of numerical examples of natural convection between a concentric circular cylinder and a square enclosure at different aspect ratios. Numerical simulations were conducted for natural convection around a cylinder and square in an enclosure. The results illustrated that nanoparticles enhance heat transfer in higher Rayleigh number, and the heat transfer of the inner cylinder is stronger than that of the square at the same perimeter.

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.

A lattice Boltzmann model for reactive mixtures

A new lattice Boltzmann model for reactive ideal gas mixtures is presented. The model is an extension to reactive flows of the recently proposed multi-component lattice Boltzmann model for compressible ideal gas mixtures with Stefan-Maxwell diffusion for species interaction. First, the kinetic model for the Stefan-Maxwell diffusion is enhanced to accommodate a source term accounting for the change in the mixture composition due to chemical reaction. Second, by including the heat of formation in the energy equation, the thermodynamic consistency of the underlying compressible lattice Boltzmann model for momentum and energy allows a realization of the energy and temperature change due to chemical reactions. This obviates the need for ad-hoc modelling with source terms for temperature or heat. Both parts remain consistently coupled through mixture composition, momentum, pressure, energy and enthalpy. The proposed model uses the standard three-dimensional lattices and is validated with a set of benchmarks including laminar burning speed in the hydrogen-air mixture and circular expanding premixed flame. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

High-order semi-Lagrangian kinetic scheme for compressible turbulence

Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time-step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of discrete velocities in three dimensions, the SLLBM uses only 45 discrete velocities. Based on compressible Taylor-Green vortex simulations we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques other than the filtering introduced by the interpolation, even when the time-step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time-step sizes is dictated by physics rather than spatial discretization.

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.

Mesoscopic kinetic approach for studying nonequilibrium hydrodynamic and thermodynamic effects of shock wave, contact discontinuity, and rarefaction wave in the unsteady shock tube

This paper presents a detailed description of a molecular velocity distribution-based mesoscopic kinetic approach that enables a better understanding of various nonequilibrium hydrodynamic and thermodynamic effects in shock waves, contact discontinuities, and rarefaction waves. This builds on the mesoscopic kinetic approach in a previous investigation into regular reflection shocks by further addressing the mesoscopic physical meaning of kinetic moments from the view of kinetics and the implications of the magnitude and sign of nonequilibrium kinetic moments. To deepen understanding of nonequilibrium effects, this work focuses on the one-dimensional unsteady shock tube problem, which contains the typical and essential features of the discontinuous flows, and has no interference of two-dimensional flow direction. The approach uses a lattice Boltzmann method to solve the flow field, and describes nonequilibrium effects through the nonequilibrium kinetic moments of molecular velocity distribution functions. The mechanism of nonequilibrium effect in discontinuous flows is further probed. This work develops the mesoscopic kinetic approach and clarifies the mesoscopic physics of shock waves, contact discontinuities, and rarefaction waves.

Extended Lattice Boltzmann Model

Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of the temperature. To that end, we propose a unified formulation that restores Galilean invariance and the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at temperatures that are higher than the lattice reference temperature, which enhances computational efficiency by decreasing the number of required time steps. Furthermore, the extended model also remains valid for stretched lattices, which are useful when flow gradients are predominant in one direction. The model is validated by simulations of two- and three-dimensional benchmark problems, including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer over a flat plate and the turbulent channel flow.

Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids

In this paper, a finite-volume discrete Boltzmann method based on a cell-centered scheme for inviscid compressible flows on unstructured grids is presented. In the new method, the equilibrium distribution functions are obtained from the circle function in two-dimensions (2D) and the spherical function in three-dimensions (3D). Moreover, the advective fluxes are evaluated by Roe's flux-difference splitting scheme, the gradients of the density and total energy distribution functions are computed with a least-squares method, and the Venkatakrishnan limiter is employed to prevent oscillations. To parallelize the method we use a graph-based partitioning approach that also guarantees the load balancing. The method is validated by seven benchmark problems: (a) a 2D flow pasting a bump, (b) a 2D Riemann problem, (c) a 2D flow passing the RAE2822 airfoil, (d) flows passing the NACA0012 airfoil, (e) 2D supersonic flows around a cylinder, (f) an explosion in a 3D box, and (g) a 3D flow around the ONERA M6 wing. The benchmark tests show that the results obtained by the proposed method match well with the published results, and the parallel numerical experiments show that the proposed parallel implementation has close to linear strong scalability, and parallel efficiencies of 95.31% and 94.56% are achieved for 2D and 3D problems on a supercomputer with up to 4800 processor cores, respectively.

Lattice Boltzmann simulations of thermal flows beyond the Boussinesq and ideal-gas approximations

In this work, the recent lattice Boltzmann model with self-tuning equation of state (EOS) [R. Huang et al., J. Comput. Phys. 392, 227 (2019)]JCTPAH0021-999110.1016/j.jcp.2019.04.044 is improved in three aspects to simulate the thermal flows beyond the Boussinesq and ideal-gas approximations. First, an improved scheme is proposed to eliminate the additional cubic terms of velocity, which can significantly improve the numerical accuracy. Second, a local scheme is proposed to calculate the density gradient instead of the conventional finite-difference scheme. Third, a scaling factor is introduced into the lattice sound speed, which can be adjusted to effectively enhance numerical stability. The thermal Couette flow of a nonattracting rigid-sphere fluid, which is described by the Carnahan-Starling EOS, is first simulated, and the better performance of the present improvements on the numerical accuracy and stability is demonstrated. As a further application, the turbulent Rayleigh-Bénard convection in a supercritical fluid slightly above its critical point, which is described by the van der Waals EOS, is successfully simulated by the present lattice Boltzmann model. The piston effect of the supercritical fluid is successfully captured, which induces a fast and homogeneous increase of the temperature in the bulk region, and the time evolution from the initiation of heating to the final turbulent state is analyzed in detail and divided into five stages.

Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics

Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell-Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature

With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier-Stokes-Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier-Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman-Enskog analysis and further corroborated through spectral dispersion-dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Semi-Lagrangian lattice Boltzmann method for compressible flows

This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method [Dorschner, Bösch, and Karlin, Phys. Rev. Lett. 121, 130602 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.130602], the method operates in a static, nonmoving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem, and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to nonuniform grids.

Consistent vortex initialization for the athermal lattice Boltzmann method

A 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.

Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

Compressible lattice Boltzmann model on standard lattices [M. H. Saadat, F. Bösch, and I. V. Karlin, Phys. Rev. E 99, 013306 (2019).2470-004510.1103/PhysRevE.99.013306] is extended to deal with complex flows on unstructured grid. Semi-Lagrangian propagation [A. Krämer et al., Phys. Rev. E 95, 023305 (2017).2470-004510.1103/PhysRevE.95.023305] is performed on an unstructured second-order accurate finite-element mesh and a consistent wall boundary condition is implemented which makes it possible to simulate compressible flows over complex geometries. The model is validated through simulation of Sod shock tube, subsonic and supersonic flow over NACA0012 airfoil and shock-vortex interaction in Schardin's problem. Numerical results demonstrate that the present model on standard lattices is able to simulate compressible flows involving shock waves on unstructured meshes with good accuracy and without using any artificial dissipation or limiter.

Arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method for computing two-dimensional compressible inviscid flows over moving bodies

The objective of this study is to develop and apply an arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method (ALE-FVLBM) for solving two-dimensional compressible inviscid flows around moving bodies. The two-dimensional compressible form of the LB equation is considered and the resulting LB equation is formulated in the ALE framework on an unstructured body-fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion. The spatial discretization of the resulting system of equations is performed by a second-order cell-centered finite-volume method on arbitrary quadrilateral meshes and an implicit dual-time stepping method is utilized for the time integration. To stabilize the numerical solution, appropriate numerical dissipation terms are added to the formulation. At first, the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed FVLBM for this unsteady test case which includes shock, expansion wave, and contact discontinuity in the flow domain. Then, the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ALE-FVLBM. The compressible inviscid flow in the transonic regime is then computed around the stationary NACA0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters. Now, the transonic inviscid flow is simulated over the pitching or plunging NACA0012 airfoil to investigate the accuracy and capability of the proposed solution method (ALE-FVLBM) for the computation of the compressible flows over moving bodies. Finally, the pitching or plunging NACA0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil. It is indicated that the solution methodology proposed based on the finite-volume LBM formulated in the arbitrary Lagrangian-Eulerian framework (ALE-FVLBM) is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect.

Extensive analysis of the lattice Boltzmann method on shifted stencils

Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)].PRLTAO0031-900710.1103/PhysRevLett.117.010604 From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity. Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.

Pseudoentropic derivation of the regularized lattice Boltzmann method

The lattice Boltzmann method (LBM) facilitates efficient simulations of fluid turbulence based on advection and collision of local particle distribution functions. To ensure stable simulations on underresolved grids, the collision operator must prevent drastic deviations from local equilibrium. This can be achieved by various methods, such as the multirelaxation time, entropic, quasiequilibrium, regularized, and cumulant schemes. Complementing a part of a unified theoretical framework of these schemes, the present work presents a derivation of the regularized lattice Boltzmann method (RLBM), which follows a recently introduced entropic multirelaxation time LBM by Karlin, Bösch, and Chikatamarla (KBC). It is shown that both methods can be derived by locally maximizing a quadratic Taylor expansion of the entropy function. While KBC expands around the local equilibrium distribution, the RLBM is recovered by expanding entropy around a global equilibrium. Numerical tests were performed to elucidate the role of pseudoentropy maximization in these models. Simulations of a two-dimensional shear layer show that the RLBM successfully reproduces the largest eddies even on a 16×16 grid, while the conventional LBM becomes unstable for grid resolutions of 128×128 and lower. The RLBM suppresses spurious vortices more effectively than KBC. In contrast, simulations of the three-dimensional Taylor-Green and Kida vortices show that KBC performs better in resolving small scale vortices, outperforming the RLBM by a factor of 1.8 in terms of the effective Reynolds number.