Asymptotic freedom in the lattice Boltzmann theory.

Asymptotic freedom is a feature of quantum chromodynamics that guarantees its well posedness. We derive an analog of asymptotic freedom enabling unconditional linear stability of lattice Boltzmann simulation of hydrodynamics. We further demonstrate the validity of the derived conditions via the special case of the equilibrium based on entropy maximization, which is shown to be uniquely renormalizable.

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.

Entropic equilibrium for the lattice Boltzmann method: Hydrodynamics and numerical properties.

The entropic lattice Boltzmann framework proposed the construction of the equilibrium by taking into consideration minimization of a discrete entropy functional. The effect of this entropic equilibrium on properties of the resulting solver has been the topic of discussions in the literature. Here we present a rigorous analysis of the hydrodynamics and numerics of the entropic equilibrium. We demonstrate that the entropic equilibrium features unconditional linear stability, in contrast to the conventional polynomial equilibrium. We reveal the mechanisms through which unconditional linear stability is maintained, most notable of which are adaptive propagation velocity of normal modes and the positive-definite nature of the dissipation rates of hydrodynamic eigenmodes. We further present a simple local correction to considerably reduce the deviations in the effective bulk viscosity.

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'.

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.

Thermokinetic lattice Boltzmann model of nonideal fluids.

We present a kinetic model for nonideal fluids, where the local thermodynamic pressure is imposed through appropriate rescaling of the particle's velocities, accounting for both long- and short-range effects and hence full thermodynamic consistency. The model features full Galilean invariance together with mass, momentum, and energy conservation and enables simulations ranging from subcritical to supercritical flows, which is illustrated on various benchmark flows such as anomalous shock waves or shock droplet interaction.

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.

Particles on Demand for Kinetic Theory.

A novel formulation of fluid dynamics as a kinetic theory with tailored, on-demand constructed particles removes restrictions on flow speed and temperature as compared to its predecessors, the lattice Boltzmann methods and their modifications. In the new kinetic theory, discrete particles are determined by a rigorous limit process which avoids ad hoc assumptions about their velocities. Classical benchmarks for incompressible and compressible flows demonstrate that the proposed discrete-particles kinetic theory opens up an unprecedented wide domain of applications for computational fluid dynamics.

Fluid-structure interaction with the entropic lattice Boltzmann method.

We propose a fluid-structure interaction (FSI) scheme using the entropic multi-relaxation time lattice Boltzmann (KBC) model for the fluid domain in combination with a nonlinear finite element solver for the structural part. We show the validity of the proposed scheme for various challenging setups by comparison to literature data. Beyond validation, we extend the KBC model to multiphase flows and couple it with a finite element method (FEM) solver. Robustness and viability of the entropic multi-relaxation time model for complex FSI applications is shown by simulations of droplet impact on elastic superhydrophobic surfaces.

Entropic multirelaxation-time lattice Boltzmann method for moving and deforming geometries in three dimensions.

Entropic lattice Boltzmann methods have been developed to alleviate intrinsic stability issues of lattice Boltzmann models for under-resolved simulations. Its reliability in combination with moving objects was established for various laminar benchmark flows in two dimensions in our previous work [B. Dorschner, S. Chikatamarla, F. Bösch, and I. Karlin, J. Comput. Phys. 295, 340 (2015)JCTPAH0021-999110.1016/j.jcp.2015.04.017] as well as for three-dimensional one-way coupled simulations of engine-type geometries in B. Dorschner, F. Bösch, S. Chikatamarla, K. Boulouchos, and I. Karlin [J. Fluid Mech. 801, 623 (2016)JFLSA70022-112010.1017/jfm.2016.448] for flat moving walls. The present contribution aims to fully exploit the advantages of entropic lattice Boltzmann models in terms of stability and accuracy and extends the methodology to three-dimensional cases, including two-way coupling between fluid and structure and then turbulence and deforming geometries. To cover this wide range of applications, the classical benchmark of a sedimenting sphere is chosen first to validate the general two-way coupling algorithm. Increasing the complexity, we subsequently consider the simulation of a plunging SD7003 airfoil in the transitional regime at a Reynolds number of Re=40000 and, finally, to access the model's performance for deforming geometries, we conduct a two-way coupled simulation of a self-propelled anguilliform swimmer. These simulations confirm the viability of the new fluid-structure interaction lattice Boltzmann algorithm to simulate flows of engineering relevance.

Grid refinement for entropic lattice Boltzmann models.

We propose a multidomain grid refinement technique with extensions to entropic incompressible, thermal, and compressible lattice Boltzmann models. Its validity and accuracy are assessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal, and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the setups of turbulent channel flow, flow past a sphere, Rayleigh-Bénard convection, as well as the supersonic flow around an airfoil. Special attention is paid to analyzing the adaptive features of entropic lattice Boltzmann models for multigrid simulations.

Invariance principle and model reduction for the Fokker-Planck equation.

The principle of dynamic invariance is applied to obtain closed moment equations from the Fokker-Planck kinetic equation. The analysis is carried out to explicit formulae for computation of the lowest eigenvalue and of the corresponding eigenfunction for arbitrary potentials.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.

Conjugate heat transfer with the entropic lattice Boltzmann method.

A conjugate heat-transfer model is presented based on the two-population entropic lattice Boltzmann method. The present approach relies on the extension of Grad's boundary conditions to the two-population model for thermal flows, as well as on the appropriate exact conjugate heat-transfer condition imposed at the fluid-solid interface. The simplicity and efficiency of the lattice Boltzmann method (LBM), and in particular of the entropic multirelaxation LBM, are retained in the present approach, thus enabling simulations of turbulent high Reynolds number flows and complex wall boundaries. The model is validated by means of two-dimensional parametric studies of various setups, including pure solid conduction, conjugate heat transfer with a backward-facing step flow, and conjugate heat transfer with the flow past a circular heated cylinder. Further validations are performed in three dimensions for the case of a turbulent flow around a heated mounted cube.

Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation.

We present in detail the recently introduced entropic lattice Boltzmann model for compressible flows [N. Frapolli et al., Phys. Rev. E 92, 061301(R) (2015)PLEEE81539-375510.1103/PhysRevE.92.061301]. The model is capable of simulating a wide range of laminar and turbulent flows, from thermal and weakly compressible flows to transonic and supersonic flows. The theory behind the construction of the model is laid out and its thermohydrodynamic limit is discussed. Based on this theory and the hydrodynamic limit thereof, we also construct the boundary conditions necessary for the simulation of solid walls. We present the inlet and outlet boundary conditions as well as no-slip and free-slip boundary conditions. Details necessary for the implementation of the compressible lattice Boltzmann model are also reported. Finally, simulations of compressible flows are presented, including two-dimensional supersonic and transonic flows around a diamond and a NACA airfoil, the simulation of the Schardin problem, and the three-dimensional simulation of the supersonic flow around a conical geometry.

Lattice Kinetic Theory in a Comoving Galilean Reference Frame.

We prove that the fully discrete lattice Boltzmann method is invariant with respect to Galilean transformation. Based on this finding, a novel class of shifted lattices is proposed which dramatically increases the operating range of lattice Boltzmann simulations, in particular, for gas dynamics applications. A simulation of vortex-shock interaction is used to demonstrate the accuracy and efficiency of the proposed lattices. With one single algorithm it is now possible to simulate a broad range of applications, from low Mach number flows to transonic and supersonic flow regimes.

Entropic lattice boltzmann method for multiphase flows.

A novel thermodynamically consistent lattice Boltzmann model that enables dynamical effects of two-phase fluids is developed. The key innovation is the application of the entropic lattice Boltzmann stabilization mechanism to control the dynamics at the liquid-vapor interface. This allows us to present a number of simulations of colliding droplets, including complex phenomena such as the formation of a stable lamella film. Excellent agreement of the simulation with recent experiments demonstrates the viability of the present approach to simulation of complex dynamic phenomena of multiphase fluids.

Entropic lattice Boltzmann model for compressible flows.

We present a lattice Boltzmann model (LBM) that covers the entire range of fluid flows, from low Mach weakly compressible to transonic and supersonic flows. One of the most restrictive limitations of the lattice Boltzmann method, the low Mach number limit, is overcome here by three fundamental changes to the LBM scheme: use of an appropriately chosen multispeed lattice, accurate evaluation of the equilibrium, and the entropic relaxation for the collision. The range of applications is demonstrated through the simulation of a bow shock in front of an airfoil and the simulation of decaying compressible turbulence with shocklets.

Multispeed entropic lattice Boltzmann model for thermal flows.

An energy-conserving lattice Boltzmann (LB) model based on the entropic theory of admissible higher-order lattice is presented in detail. The entropy supporting 'zero-one-three" lattice is used to construct a model capable of reproducing the full Fourier-Navier-Stokes equations at low Mach numbers. The proposed direct approach of constructing thermal models overcomes the shortcomings of existing models and retains one of the most important advantages of the LB methods, the exact space discretization of the advection step, thus paving the way for direct numerical simulation of thermal flows. New thermal wall boundary condition capable of handling curved geometries immersed in a multispeed lattice is proposed by extending the Tamm-Mott-Smith boundary condition. Entropic realization of the current model ensures stability of the model also for subgrid simulations. Numerical validation and thermodynamic consistency is demonstrated with classical setups such as thermal Couette flow, Rayleigh-Bénard natural convection, acoustic waves, speed of sound measurements, and shock tube simulations.

Gibbs' principle for the lattice-kinetic theory of fluid dynamics.

Gibbs' seminal prescription for constructing optimal states by maximizing the entropy under pertinent constraints is used to derive a lattice kinetic theory for the computation of high Reynolds number flows. The notion of modifying the viscosity to stabilize subgrid simulations is challenged in this kinetic framework. A lattice Boltzmann model for direct simulation of turbulent flows is presented without any need for tunable parameters and turbulent viscosity. Simulations at very high Reynolds numbers demonstrate a major extension of the operation range for fluid dynamics.