 TU Darmstadt ULB TUbiblio

A high-order Discontinuous Galerkin solver for incompressible and low-Mach number flows

Abstract

In this work, we present a high-order Discontinuous Galerkin Method (DGM) for simulating incompressible and variable density flows at low-Mach numbers. For steady cases, we apply the SIMPLE algorithm to solve the non-linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae and the SIMPLE algorithm is applied to solve the non-linear system in each time step. The proposed method is implemented in the in-house software library BoSSS. The solver is extensively tested with respect to temporal and spatial convergence rates, performance and stability by simulating various test cases.

In the first part of this work, we describe the discretization and algorithm for incompressible flows. Using a mixed-order formulation for the spatial discretization, we obtain convergence rates of k+1 for velocity and k for pressure for various test cases, where k and k-1 are the orders of the approximation polynomials for velocity and pressure, respectively. Applying pressure stabilization for the equal-order formulation, the convergence rates are approximately the same, while the absolute error is smaller. By simulating the Orr-Sommerfeld problem we investigate the stability of the proposed method. The solver is validated by studying the two- and three-dimensional flow past a square cylinder. Main parts of this work concerning the solver and numerical results for incompressible flows have been published before by the author of this thesis in [KLEIN, B., KUMMER, F., OBERLACK, M. (2013): A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. Journal of Computational Physics 237, 235–250] and [KLEIN, B., KUMMER, F., KEIL, M., OBERLACK, M. (2015): An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows. International Journal for Numerical Methods in Fluids 77, 10, 571–589].

In the second part of this work, the solver is extended to variable density flows at low-Mach numbers. An intermediate step in the development of the solver for low-Mach number flows is a method for simulating multiphase flows with a smooth interface approach and without surface tension. The solver for low-Mach number flows is based on the low-Mach number equations, which are an approximation of the compressible Navier-Stokes equations in the limit of zero Mach number. To the best of the author's knowledge, it is the first time that the DGM is applied to the low-Mach number equations. For spatial discretization the mixed-order formulation is applied. Various test cases confirm the high accuracy of the method also for multiphase flows and low-Mach number flows.

Item Type: Ph.D. Thesis
Erschienen: 2015
Creators: Klein, Benedikt
Title: A high-order Discontinuous Galerkin solver for incompressible and low-Mach number flows
Language: English
Abstract:

In this work, we present a high-order Discontinuous Galerkin Method (DGM) for simulating incompressible and variable density flows at low-Mach numbers. For steady cases, we apply the SIMPLE algorithm to solve the non-linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae and the SIMPLE algorithm is applied to solve the non-linear system in each time step. The proposed method is implemented in the in-house software library BoSSS. The solver is extensively tested with respect to temporal and spatial convergence rates, performance and stability by simulating various test cases.

In the first part of this work, we describe the discretization and algorithm for incompressible flows. Using a mixed-order formulation for the spatial discretization, we obtain convergence rates of k+1 for velocity and k for pressure for various test cases, where k and k-1 are the orders of the approximation polynomials for velocity and pressure, respectively. Applying pressure stabilization for the equal-order formulation, the convergence rates are approximately the same, while the absolute error is smaller. By simulating the Orr-Sommerfeld problem we investigate the stability of the proposed method. The solver is validated by studying the two- and three-dimensional flow past a square cylinder. Main parts of this work concerning the solver and numerical results for incompressible flows have been published before by the author of this thesis in [KLEIN, B., KUMMER, F., OBERLACK, M. (2013): A SIMPLE based discontinuous Galerkin solver for steady incompressible flows. Journal of Computational Physics 237, 235–250] and [KLEIN, B., KUMMER, F., KEIL, M., OBERLACK, M. (2015): An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows. International Journal for Numerical Methods in Fluids 77, 10, 571–589].

In the second part of this work, the solver is extended to variable density flows at low-Mach numbers. An intermediate step in the development of the solver for low-Mach number flows is a method for simulating multiphase flows with a smooth interface approach and without surface tension. The solver for low-Mach number flows is based on the low-Mach number equations, which are an approximation of the compressible Navier-Stokes equations in the limit of zero Mach number. To the best of the author's knowledge, it is the first time that the DGM is applied to the low-Mach number equations. For spatial discretization the mixed-order formulation is applied. Various test cases confirm the high accuracy of the method also for multiphase flows and low-Mach number flows.

Place of Publication: Darmstadt
Uncontrolled Keywords: BoSSS
Divisions: 16 Department of Mechanical Engineering
16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
Exzellenzinitiative
Zentrale Einrichtungen
Zentrale Einrichtungen > University IT-Service and Computing Centre (HRZ) > Hochleistungsrechner
Zentrale Einrichtungen > University IT-Service and Computing Centre (HRZ)
Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE)
Exzellenzinitiative > Graduate Schools
Date Deposited: 29 Nov 2015 20:55 View Item