Discontinuous Galerkin Methods for Extended Hydrodynamics.

[摘要] This dissertation presents a step towards high-order methods for continuum-transition flows.In order to achieve maximum accuracy and efficiency for numerical methodson a distorted mesh, it is desirable that both governing equations and correspondingnumerical methods are in some sense compact. We argue our preference fora physical model described solely by first-order partial differential equations calledhyperbolic-relaxation equations, and, among various numerical methods, for the discontinuousGalerkin method. Hyperbolic-relaxation equations can be generated asmoments of the Boltzmann equation and can describe continuum-transition flows.Two challenging properties of hyperbolic-relaxation equations are the presenceof a stiff source term, which drives the system towards equilibrium, and the accompanyingchange of eigenstructure. The first issue can be solved by an implicittreatment of the source term. To cope with the second difficulty, we develop aspace-time discontinuous Galerkin method, based on Huynh’s ;;upwind momentscheme.” It is called the DG(1)–Hancock method.The DG(1)–Hancock method for one- and two-dimensional meshes is described,and Fourier analyses for both linear advection and linear hyperbolic-relaxation equationsare conducted. The analyses show that the DG(1)–Hancock method is notonly accurate but efficient in terms of turnaround time in comparison to other semiandfully discrete finite-volume and discontinuous Galerkin methods. Numericaltests confirm the analyses, and also show the properties are preserved for nonlinearequations; the efficiency is superior by an order of magnitude.Subsequently, discontinuous Galerkin and finite-volume spatial discretizationsare applied to more practical equations, in particular, to the set of 10-moment equations,which are gas dynamics equations that include a full pressure/temperaturetensor among the flow variables. Results for flow around a micro-airfoil are comparedto experimental data and to solutions obtained with a Navier–Stokes code,and with particle-based methods. While numerical solutions in the continuumregime for both the 10-moment and Navier–Stokes equations are similar, clear differencesare found in the continuum-transition regime, especially near the stagnationpoint, where the Navier–Stokes code, even when implemented with wall-slip, overestimatesthe density.

[发布机构] University of Michigan

[效力级别] Extended Hydrodynamics

[关键词] Discontinuous Galerkin Method;Extended Hydrodynamics;High Order Method;Hyperbolic-relaxation Equation;Hancock Method;Space-time Discretization Method;Aerospace Engineering;Engineering;Aerospace Engineering and Scientific Computing