skip to main content

This content will become publicly available on April 1, 2023

Title: Multidimensional transonic shock waves and free boundary problems
We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.
Authors:
;
Award ID(s):
2054689 1764278
Publication Date:
NSF-PAR ID:
10324137
Journal Name:
Bulletin of Mathematical Sciences
Volume:
12
Issue:
01
ISSN:
1664-3607
Sponsoring Org:
National Science Foundation
More Like this
  1. The behaviour of low-viscosity, pressure-driven compressible pore fluid flows in viscously deformable porous media is studied here with specific application to gas flow in lava domes. The combined flow of gas and lava is shown to be governed by a two-equation set of nonlinear mixed hyperbolic–parabolic type partial differential equations describing the evolution of gas pore pressure and lava porosity. Steady state solution of this system is achieved when the gas pore pressure is magmastatic and the porosity profile accommodates the magmastatic pressure condition by increased compaction of the medium with depth. A one-dimensional (vertical) numerical linear stability analysis (LSA) is presented here. As a consequence of the pore-fluid compressibility and the presence of gravitation compaction, the gradients present in the steady-state solution cause variable coefficients in the linearized equations which generate instability in the LSA despite the diffusion-like and dissipative terms in the original system. The onset of this instability is shown to be strongly controlled by the thickness of the flow and the maximum porosity, itself a function of the mass flow rate of gas. Numerical solutions of the fully nonlinear system are also presented and exhibit nonlinear wave propagation features such as shock formation. As applied tomore »gas flow within lava domes, the details of this dynamics help explain observations of cyclic lava dome extrusion and explosion episodes. Because the instability is stronger in thicker flows, the continued extrusion and thickening of a lava dome constitutes an increasing likelihood of instability onset, pressure wave growth and ultimately explosion.« less
  2. When a plane shock hits a two-dimensional wedge head on, it experiences a reflection-diffraction process, and then a self-similar reflected shock moves outward as the original shock moves forward in time. The experimental, computational, and asymptotic analysis has indicated that various patterns occur, including regular reflection and Mach reflection. The von Neumann's conjectures on the transition from regular to Mach reflection involve the existence, uniqueness, and stability of regular shock reflection-diffraction configurations, generated by concave cornered wedges for compressible flow. In this paper, we discuss some recent developments in the study of the von Neumann's conjectures. More specifically, we discuss the uniqueness and stability of regular shock reflection-diffraction configurations governed by the potential flow equation in an appropriate class of solutions. We first show that the transonic shocks in the global solutions obtained in Chen-Feldman [19] are convex. Then we establish the uniqueness of global shock reflection-diffraction configurations with convex transonic shocks for any wedge angle larger than the detachment angle or the critical angle. Moreover, the solution under consideration is stable with respect to the wedge angle. Our approach also provides an alternative way of proving the existence of the admissible solutions established first in [19].
  3. Abstract

    Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first-order reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two- and three-dimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structure-preserving WENO-like schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this classmore »of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

    « less
  4. We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the nonexistence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties.
  5. Abstract

    This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. Themore »methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PNPM-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times.

    « less