@phdthesis{Rieper2008, author = {Rieper, Felix}, title = {On the behaviour of numerical schemes in the low Mach number regime}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus-5747}, school = {BTU Cottbus - Senftenberg}, year = {2008}, abstract = {This work is concerned with the study of two-dimensional inviscid flow. The main concern is the so-called "accuracy problem" of first-order upwind schemes in the low Mach number regime. The thesis is divided into two parts. In a preliminary chapter the governing thermodynamic and mechanical equations are introduced. The Navier-Stokes equations for viscous flow are presented because the concepts of fluid viscosity and Reynolds number are later needed in the context of artificial viscosity of numerical schemes. The Euler equations are introduced along with a single-scale asymptotic analysis for small Mach numbers. In the first part the behaviour of various first-order upwind schemes in the low Mach number regime is analysed with respect to a one-dimensional model case, where the two-dimensionality only remains in the shear waves of the local Riemann problem. The concepts of numerical viscosity and numerical Reynolds number are first introduced for the upwind method for the scaler linear advection equation. These concepts are then applied to Roe's scheme and the scheme by Harten-Lax-van Leer (HLL). The aim is to show that the accuracy problem can only be avoided, if all characteristic waves are resolved by the upwind scheme. Otherwise, the artificial viscosity on these waves is of the wrong order of magnitude dx/M and grows with decreasing Mach numbers, instead of being order dx. In the following chapter, various flux vector splitting methods are analysed in this respect. The analysis presented is verified with a number of numerical results. Comparisons with analytical solutions give a direct measure of the accuracy of the scheme; to this end we use the flow around a cylinder. As standard test case the flow around a NACA0012 aerofoil is used. The second part of the thesis is dedicated to two-dimensional flow. The numerical scheme of choice is based on Roe's approximate Riemann solver, which does not show the accuracy problem in the one-dimensional setting. Part II begins with broad numerical studies, which path the way to the analysis by giving answers to the following questions. How does the numerical behaviour of the scheme change, when the cells in a structured, body-fitted grid around a cylinder are continuously transformed from (almost) squares, via trapezoids to triangles? How important is the grid structure to the accuracy problem? Later we restrict the numerical investigations to structured grids with two different cell geometries: squares and rectangular triangles, which originate from the squares by adding a diagonal edge. On these types of finite volume cells the steady flow around a rectangle and the dissipative behaviour of an oblique contact layer is investigated. The moving Gresho vortex is simulated as an example of an unsteady flow simulation. All numerical studies suggest two things: firstly, the accuracy problem is linked to the momentum transport in shear flow and, secondly, this problem is absent if the finite volume cells are of triangular shape -- irrespective of the grid being structured or unstructured. The analysis begins with heuristic considerations. We investigate a special case of a divergence free flow with constant pressure and density on a Cartesian and a related triangular grid. Already here, the different form of the artificial viscosity term on square and triangular finite volume cells becomes evident. On Cartesian grids there is a decoupling of the artificial viscosity into horizontal and vertical direction. For non-trivial velocity fields the viscosity terms of order dx/M cannot vanish and cause the numerical error to grow for decreasing Mach numbers. On triangular grids there is no such decoupling: the artificial viscosity of order dx/M contains all jumps of the normal component of the velocity at the three cell interfaces. This suggests that the accuracy problem is absent because these jumps can completely vanish without constraining the velocity field to a trivial flow. The thesis closes with a proof of this proposal for the Roe scheme for steady low Mach number flow. Using asymptotic analysis of the semi-discrete equations, it is explicitly derived that the accuracy problem is avoided, if the Roe scheme is used on triangular finite volume cells. The accuracy of the scheme is accompanied by a constraint: the normal velocity does not jump at cell interfaces. This constraint leaves enough degrees of freedom for the velocity field as is shown with graph theoretic arguments.}, subject = {Numerische Str{\"o}mungsmechanik; Zweidimensionale Str{\"o}mung; Upwind-Verfahren; Schwach kompressible Str{\"o}mung; CFD; Numerische Str{\"o}mungsmechanik; Upwind Schemata; Numerische Viskosit{\"a}t; Weakly compressible flow; CFD; Upwind schemes; Numerical viscosity; Low Mach number flow}, language = {en} } @phdthesis{Puetz2023, author = {P{\"u}tz, Michele}, title = {Numerical investigation and extension of quadrature-based moment methods for population balances}, doi = {10.26127/BTUOpen-6575}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-65751}, school = {BTU Cottbus - Senftenberg}, year = {2023}, abstract = {Particulate systems can be described by a number density function (NDF) with respect to a vector of internal coordinates. The evolution of the NDF is governed by the typically high-dimensional population balance equation (PBE). A common approach to reduce the dimensionality of the problem is to solve only for a set of moments instead of the NDF. The derived system of moment equations, however, includes unclosed integral terms that still contain the unknown NDF. One way to close the system of moment equations is to approximate the unclosed integral terms using a Gaussian quadrature computed from the moments. The procedure of taking a set of moments to compute a Gaussian quadrature, which is, in turn, used to close the moment equations, is known as the quadrature method of moments (QMOM). It gave rise to an entire family of methods, the quadrature-based moment methods (QBMMs), which are the primary focus of this work. The presented research can be divided into three major parts. The first part involves the formulation of a common Lagrangian droplet breakup model for QBMMs and the numerical investigation with the QMOM as well as the more sophisticated extended QMOM (EQMOM). The results indicate that the approximations are reasonably accurate when at least six moment equations are solved, with the EQMOM providing no advantages for the investigated configurations. In the second part, a quadrature-based moment model for the effects of fluid turbulence on particle velocities is formulated. The resulting moment equations contain non-smooth integrands that are the source of large errors when using common QBMMs. As an alternative, the Gauss/anti-Gauss QMOM (GaG-QMOM) is proposed that uses the average of a Gaussian and an anti-Gaussian quadrature. Numerical studies show that the GaG-QMOM is able to significantly reduce the previously observed large errors. Another novelty proposed in this context is the modification of the second-order strong-stability preserving Runge-Kutta method to guarantee the preservation of moment realizability in the presence of phase-space diffusion. The third part is concerned with the numerical exploration of the core algorithm of most QBMMs in terms of performance and accuracy. The algorithm consists of, first, computing the recurrence coefficients of the orthogonal polynomials associated with a set of moments, second, solving a symmetric tridiagonal eigenvalue problem to obtain the quadrature nodes and weights, and third, evaluating the integral terms in the moment equations. The results indicate that the contribution of the first step to compute the recurrence coefficients from moments to the overall computational costs is negligible. Instead, the primary focus should be on the fast solution of the eigenvalue problem and, possibly, on the efficient implementation of the moment source term evaluation, which becomes important when second-order processes are considered.}, subject = {Numerical algorithms; Computational fluid mechanics; Orthogonal polynomials; Turbulence; Population balance equations; Numerische Algorithmen; Numerische Str{\"o}mungsmechanik; Orthogonale Polynome; Populationsbilanzgleichungen; Turbulenz; Numerische Str{\"o}mungssimulation; Str{\"o}mungsmechanik; Numerisches Verfahren; Orthogonale Polynome; Turbulente Str{\"o}mung}, language = {en} }