A Viscosity Adaptive Lattice Boltzmann Method (2015)

The present thesis describes the development and validation of a adaption method for the numerical simulation of non-Newtonian fluids on the basis of the Lattice Boltzmann Method (LBM), as well as the development and verification of the related software bundle SAM-Lattice.

By now, Lattice Boltzmann Methods are established as an alternative approach to classical computational fluid dynamics 
methods. The LBM has been shown to be an accurate and efficient tool for the numerical simulation of weakly compressible or incompressible fluids. Fields of application reach from turbulent simulations through thermal problems to acoustic calculations among others. The transient nature of the method and the need for a regular grid based, non body conformal discretization makes the LBM ideally suitable for simulations involving complex solids. Such geometries are common, for instance, in the food processing industry, where fluids are mixed by static mixers or agitators. Those fluid flows are often laminar and non-Newtonian. 

This work is motivated by the immense practical use of the Lattice Boltzmann Method, which is limited due to stability issues. The stability of the method is mainly influenced by the discretization and the of the fluid. Thus, simulations of non-Newtonian fluids, whose kinematic depend on the shear rate, are problematic. Several authors have shown that the LBM is capable of simulating those fluids. However, the vast majority of the simulations in the literature are carried out for simple geometries and/or moderate shear rates, where the LBM is still stable. Special care has to be taken for practical non-Newtonian Lattice Boltzmann simulations in order to keep them stable. A straightforward way is to truncate the modeled range by numerical stability criteria. This is an effective approach, but from the physical point of view the bounds are chosen arbitrarily. Moreover, these bounds depend on and vary with the grid and time step size and, therefore, with the simulation Mach number, which is freely chosen at the start of the simulation. Consequently, the modeled range may not fit to the actual range of the physical problem, because the correct simulation Mach number is unknown a priori. A way around is, to perform precursor simulations on a fixed grid to determine a possible time step size and simulation Mach number, respectively. These precursor simulations can be time consuming and expensive, especially for complex cases and a number of operating points. This makes the LBM unattractive for use in practical simulations of non-Newtonian fluids. 
The essential novelty of the method, developed in the course of this thesis, is that the numerically modeled range is consistently adapted to the actual physically exhibited range through change of the simulation time step and the simulation Mach number, respectively, while the simulation is running. The algorithm is robust, independent of the Mach number the simulation was started with, and applicable for stationary flows as well as transient flows. The method for the adaption will be referred to as the viscosity adaption method (VAM) and the combination with LBM leads to the viscosity adaptive LBM (VALBM).

Besides the introduction of the VALBM, a goal of this thesis is to offer assistance in the spirit of a theory guide to students and assistant researchers concerning the theory of the Lattice Boltzmann Method and its implementation in SAM-Lattice. In Chapter 2, the mathematical foundation of the LBM is given and the route from the BGK approximation of the Boltzmann equation to the Lattice Boltzmann (BGK) equation is delineated in detail. 
The derivation is restricted to isothermal flows only. Restrictions of the method, such as low Mach number flows are highlighted and the accuracy of the method is discussed.

SAM-Lattice is a C++ software bundle developed by the author and his colleague Dipl.-Ing. Andreas Schneider. It is a highly automated package for the simulation of isothermal flows of incompressible or weakly compressible fluids in 3D on the basis of the Lattice Boltzmann Method. By the time of writing of this thesis, SAM-Lattice comprises 5 components. The main components are the highly automated lattice generator SamGenerator and the Lattice Boltzmann solver SamSolver. Postprocessing is done with ParaSam, which is our extension of the
open source visualization software ParaView. Additionally, domain decomposition for MPI
parallelism is done by SamDecomposer, which makes use of the graph partitioning library MeTiS. Finally, all mentioned components can be controlled through a user friendly GUI (SamLattice) implemented by the author using QT, including features to visually track output data.
In Chapter 3, some fundamental aspects on the implementation of the main components, including the corresponding flow charts will be discussed. Actual details on the implementation are given in the comprehensive programmers guides to SamGenerator and SamSolver.

In order to ensure the functionality of the implementation of SamSolver, the solver is verified in Chapter 4 for Stokes's First Problem, the suddenly accelerated plate, and for Stokes's Second Problem, the oscillating plate, both for Newtonian fluids. Non-Newtonian fluids are modeled in SamSolver with the power-law model according to Ostwald de Waele. The implementation for non-Newtonian fluids is verified for the Hagen-Poiseuille channel flow in conjunction with a convergence analysis of the method. At the same time, the local grid refinement as it is implemented in SamSolver, is verified. Finally, the verification of higher order boundary conditions is done for the 3D Hagen-Poiseuille pipe flow for both Newtonian and non-Newtonian fluids.

In Chapter 5, the theory of the adaption method is introduced. For the adaption process, a target collision frequency or target simulation Mach number must be chosen and the distributions must be rescaled according to the modified time step size. A convenient choice is one of the stability bounds. The time step size for the adaption step is deduced from the target collision frequency \(\Omega_t\) and the currently minimal or maximal shear rate in the system, while obeying auxiliary conditions for the simulation Mach number. The adaption is done in the collision step of the Lattice Boltzmann algorithm. We use the transformation matrices of the MRT model to map from distribution space to moment space and vice versa. The actual scaling of the distributions is conducted on the back mapping, because we use the transformation matrix on the basis of the new adaption time step size. It follows an additional rescaling of the non-equilibrium part of the distributions, because of the form of the definition for the discrete stress tensor in the LBM context. For that reason it is clear, that the VAM is applicable for the SRT model as well as the MRT model, where there is virtually no extra cost in the latter case. Also, in Chapter 5, the multi level treatment will be discussed.

Depending on the target collision frequency and the target Mach number, the VAM can be used to optimally use the range that can be modeled within the stability bounds or it can be used to drastically accelerate the simulation. This is shown in Chapter 6. The adaptive LBM is verified in the stationary case for the Hagen-Poiseuille channel flow and in the transient case for the Wormersley flow, i.e., the pulsatile 3D Hagen-Poiseuille pipe flow. Although, the VAM is used here for fluids that can be modeled with the power-law approach, the implementation of the VALBM is straightforward for other non-Newtonian models, e.g., the Carreau-Yasuda or Cross model. In the same chapter, the VALBM is validated for the case of a propeller viscosimeter developed at the chair SAM. To this end, the experimental data of the torque on the impeller of three shear thinning non-Newtonian liquids serve for the validation. The VALBM shows excellent agreement with experimental data for all of the investigated fluids and in every operating point. For reasons of comparison, a series of standard LBM simulations is carried out with different simulation Mach numbers, which partly show errors of several hundred percent. Moreover, in Chapter 7, a sensitivity analysis on the parameters used within the VAM is conducted for the simulation of the propeller viscosimeter.

Finally, the accuracy of non-Newtonian Lattice Boltzmann simulations with the SRT and the MRT model is analyzed in detail. Previous work for Newtonian fluids indicate that depending on the numerical value of the collision frequency \(\Omega\), additional artificial is introduced due to the finite difference scheme, which negatively influences the accuracy. For the non-Newtonian case, an error estimate in the form of a functional is derived on the basis of a series expansion of the Lattice Boltzmann equation. This functional can be solved analytically for the case of the Hagen-Poiseuille channel flow of non-Newtonian fluids. The estimation of the error minimum is excellent in regions where the \(\Omega\) error is the dominant source of error as opposed to the compressibility error.

Result of this dissertation is a verified and validated software bundle on the basis of the adaptive Lattice Boltzmann Method. The work restricts itself on the simulation of isothermal, laminar flows with small Mach numbers. As further research goals, the testing of the VALBM with minimal error estimate and the investigation of the VALBM in the case of turbulent flows is suggested.