Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

A discontinuous Galerkin method for the solution of compressible flows

This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation. The 1D Euler equations are used to assess the performance and stability of the discretisation. The explicit time restriction is derived and it is established that the optimal polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since the method is characterised by minimal diffusion, it is particularly well suited for the simulation of the pressure wave generated by train entering a tunnel. A novel treatment of the area-averaged Euler equations is proposed to eliminate oscillations generated by the projection of a moving area on a fixed mesh and the computational results are validated against experimental data. Attention is then focussed on the development of a 2D DG method implemented using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance. An artificial diffusion term is implemented to stabilise the solution of the Euler equations in transonic flow with discontinuities. To speed up the convergence of the explicit method, a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as an artificial diffusion sensor, to assign appropriate polynomial degrees to each element of the domain. The zonal solver uses a modification of a method for matching viscous subdomains to set the interface conditions between viscous and inviscid subdomains that ensures stability of the flow computation. Both the p-adaption and the zonal solver maintain the high-order accuracy of the DG method while reducing the computational cost of the simulation

Shock interaction in sphere symmetry

We investigate the interaction of two oncoming shock waves in spherical symmetry for an ideal barotropic fluid. Our research problem is how to establish a local in time solution after the interaction point and determine the state behind the shock waves. This problem is a double free boundary problem, as the position of the shock waves in space time is unknown. Our work is based on a number of previous studies, including Lisibach's, who studied the case of plane symmetry. To solve this problem, we will use an iterative regime to establish a local in time solution.Comment: arXiv admin note: text overlap with arXiv:1907.03784, arXiv:2202.08111 by other author

A bounded upwinding scheme for computing convection-dominated transport problems

A practical high resolution upwind differencing scheme for the numerical solution of convection-dominated transport problems is presented. The scheme is based on TVD and CBC stability criteria and is implemented in the context of the finite difference methodology. The performance of the scheme is investigated by solving the 1D/2D scalar advection equations, 1D inviscid Burgers’ equation, 1D scalar convection–diffusion equation, 1D/2D compressible Euler’s equations, and 2D incompressible Navier–Stokes equations. The numerical results displayed good agreement with other existing numerical and experimental data

Computation Of 2d Inviscid Compressible Flow Using An Entropy Consistent Euler Flux

The Roe flux function is one of the most established flux functions for inviscid flow and widely used in many CFD codes. However, the Roe flux suffers from shock instability which is usually observed for very strong shocks as in hypersonic flow. This maybe because the Roe flux does not strictly adhere to the second law of thermodynamics which is critical to capture shocks. The entropy consistent Euler flux (EC) is a new shock capturing method developed by Ismail & Roe in an attempt to overcome the above mentioned deficiencies of the Roe flux. The EC flux is designed to discretely satisfy the basic conservation laws, besides satisfying the second law of thermodynamics (entropy control). Most of the previous studies on the entropy schemes are based mainly on academic test problems using one-dimension. The main objective of this study is to determine the performance of the EC flux function relative to the Roe flux under subsonic, transonic, supersonic and hypersonic flow. This new flux function is tested for 1D and 2D test problems. The 1D test problem is based on Sod’s shock tube problem, whereas the 2D problems include the Mach 3 flow over a staircase, a steady flow over a cylinder and a steady flow over a NACA 0012 airfoil. In order to solve these 1D and 2D problems, an in-house CFD research code based on inviscid compressible flow was developed for a general 2D curvilinear geometry using structured grids. The grid is generated using GAMBIT software and then fed into a self-written converter software to ensure that the grid configurations are compatible with the newly developed CFD solver

Hyperbolic Conservation Laws

[no abstract available

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos