Erlend Magnus Viggen

Viscously damped acoustic waves with the lattice Boltzmann method.

Acoustic wave propagation in lattice Boltzmann Bhatnagar–Gross–Krook simulations may be analysed using a linearization method. This method has been used in the past to study the propagation of waves that are viscously damped in time, and is here extended to also study waves that are viscously damped in space. Its validity is verified against simulations, and the results are compared with theoretical expressions. It is found in the infinite resolution limit k→0 that the absorption coefficients and phase differences between density and velocity waves match theoretical expressions for small values of ωτν, the characteristic number for viscous acoustic damping. However, the phase velocities and amplitude ratios between the waves increase incorrectly with (ωτν)2, and agree with theory only in the inviscid limit k→0, ωτν→0. The actual behaviour of simulated plane waves in the infinite resolution limit is quantified.

Implementation of LB Simulations

After reading this chapter, you will understand the fundamentals of high-performance computing and how to write efficient code for lattice Boltzmann method simulations. You will know how to optimise sequential codes and develop parallel codes for multi-core CPUs, computing clusters, and graphics processing units. The code listings in this chapter allow you to quickly get started with an efficient code and show you how to optimise your existing code.

Lattice Boltzmann for Advection-Diffusion Problems

After reading this chapter, you will understand how the lattice Boltzmann equation can be adapted from flow problems to advection-diffusion problems with only small changes. These problems include thermal flows, and you will know how to simulate these as two interlinked lattice Boltzmann simulations, one for the flow and one for the thermal advection-diffusion. You will understand how advection-diffusion problems require different boundary conditions from flow problems, and how these boundary conditions may be implemented.

The Lattice Boltzmann Equation

After reading this chapter, you will know the basics of the lattice Boltzmann method, how it can be used to simulate fluids, and how to implement it in code. You will have insight into the derivation of the lattice Boltzmann equation, having seen how the continuous Boltzmann equation is discretised in velocity space through Hermite series expansion, before being discretised in physical space and time through the method of characteristics. In particular, you will be familiar with the various simple sets of velocity vectors that are available, and how the discrete BGK collision model is applied.

Numerical Methods for Fluids

After reading this chapter, you will have insight into a number of other fluid simulation methods and their advantages and disadvantages. These methods are divided into two categories. First, conventional numerical methods based on discretising the equations of fluid mechanics, such as finite difference, finite volume, and finite element methods. Second, methods that are based on microscopic, mesoscopic, or macroscopic particles, such as molecular dynamics, lattice gas models, and multi-particle collision dynamics. You will know where the particle-based lattice Boltzmann method fits in the landscape of fluid simulation methods, and you will have an understanding of the advantages and disadvantages of the lattice Boltzmann method compared to other methods.

Boundary Conditions for Fluid-Structure Interaction

After reading this chapter, you will have insight into a large number of more complex lattice Boltzmann boundary conditions, including advanced bounce-back methods, ghost methods, and immersed boundary methods. These boundary conditions will allow you to simulate things like curved boundaries, flows in media with sub-grid porosity, rigid but moveable objects immersed in the fluid, and even flows with deformable objects such as red blood cells.

Basics of Hydrodynamics and Kinetic Theory

After reading this chapter, you will have a working understanding of the equations of fluid mechanics, which describe a fluid’s behaviour through its conservation of mass and momentum. You will understand the basics of the kinetic theory on which the lattice Boltzmann method is founded. Additionally, you will have learned about how different descriptions of a fluid, such as the continuum fluid description and the mesoscopic kinetic description, are related.

MRT and TRT Collision Operators

After reading this chapter, you will have a solid understanding of the general principles of multiple-relaxation-time (MRT) and two-relaxation-time (TRT) collision operators. You will know how to implement these and how to choose the various relaxation times in order to increase the stability, the accuracy, and the possibilities of lattice Boltzmann simulations.

Boundary and Initial Conditions

After reading this chapter, you will be familiar with the basics of lattice Boltzmann boundary conditions. After also having read Chap. 3, you will be able to implement fluid flow problems with various types of grid-aligned boundaries, representing both no-slip and open surfaces. From the boundary condition theory explained in this chapter together with the theory given in Chap. 4, you will be familiar with the basic theoretical tools used to analyse numerical lattice Boltzmann solutions. Additionally, you will understand how the details of the initial state of a simulation can be important and you will know how to compute a good initial simulation state.

Analysis of the Lattice Boltzmann Equation

After reading this chapter, you will be familiar with many in-depth aspects of the lattice Boltzmann method. You will have a detailed understanding of how the Chapman-Enskog analysis can be used to determine how the lattice Boltzmann equation and its variations behave on the macroscopic Navier-Stokes level. You will know a number of such variations that result in different macroscopic behaviour from the standard lattice Boltzmann equation. Necessary and sufficient conditions that serve as stability guidelines for lattice Boltzmann simulations will be known to you, along with how to improve the stability of a given simulation. You will also have insight into the accuracy of both general simulations and lattice Boltzmann simulations. For the latter, you will understand what the sources of inaccuracy are, and how they may be reduced or nullified.