Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

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 non-existence 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.Comment: 20 pages, 4 figures, To appear in: SIAM Journal of Mathematical Analysis, 202

Computations of Viscous Compressible Flows in h, p, k Finite Element Framework with Variationally Consistent Integral Forms

This thesis presents mathematical models for time dependent and stationary viscous compressible flows based on conservation laws, constitutive equations and equations of state using Eulerian description. In the presence of physical viscosity, conductivity and other transport properties, the mathematical models are well recognized Navier-Stokes equations. Variable transport properties as well as ideal and real gas models are considered for equations of state. The mathematical models are a highly non-linear coupled partial differential equations in space and time. The mathematical and computational infrastructure using finite element method is presented for obtaining numerical solutions of the Boundary Value Problems and Initial Value Problems associated with the mathematical models. This infrastructure is based on h, p, k (h-characteristic length, p-degree of local approximation, k-order of approximation space) as independent computational parameters with an additional requirement that the integral form be variationally consistent in case of Boundary Value Problems and space-time variationally consistent in case of Initial Value Problems. All methods of approximation except Least Squares and Space-Time Least Squares Processes are Variationally Inconsistent. Variational Consistency and Space-Time Variational Consistency of integral forms ensure unconditionally stable computational processes. A variety of numerical studies are presented for Initial Value Problems as well as Boundary Value Problems. 1-D transient viscous form of Burgers equation, 1-D Riemann shock tube with ideal and real gas models and Boundary Value Problems in 2-D compressible flow : Carter's plate with Mach 1, 2, 3 and 5 flows and Mach 1 flow past a circular cylinder are used as model problems. Shock evolution, propagation, interactions and reflection are quantified based on the rate of entropy production using Air as a medium for 1-D Riemann shock tube. It is clearly established that rarefaction shocks are not possible for FC70 for any choice of initial conditions. In all studies evolution of a shock is presented (unlike the published work). Its existence and sustained propagation is established based on Sr, the rate of entropy production per unit volume. In case of transient Burgers equation it is demonstrated that time accurate evolutions can be computed for any finite Reynolds number. Contrary to the common belief, the work presented here shows that solutions of Boundary Value Problems in compressible flows present no special problems. In Summary : (i) the mathematical models for the compressible flow are based on Navier-Stokes equations. (ii) computational infrastructure is based on $hpk$ and unconditionally stable integral forms with higher order global differentiability in space and time. (iii) All numerical studies utilize actual transport properties of the medium. (iv) Up-winding methods such as SUPG, SUPG/DC, SUPG/DC/LS are neither needed nor used. (v) existence of shocks is established through evolution and not using Rankine-Hugoniot relations. (vi) Governing Differential Equations in the mathematical models are neither linearized nor altered in any form during the entire process of formulation and computations. The work presented here clearly demonstrates that the numerical simulations of Boundary Value Problems and Initial Value Problems based on Navier-Stokes equations describing viscous compressible flows can be done in a straight forward manner in $h, p, k$ framework with Variational Consistent and Space-Time Variationally Consistent integral forms. The computational processes always remain unconditionally stable. The mathematical models based on Euler's equations lack physics, computational methods for Euler's equations use problem dependent up-winding methods which lack mathematical basis and rigor and thus in our view are of little merit if at all for numerical simulations of Boundary Value Problems and Initial Value Problems in compressible flows

Numerical simulation of liquid sloshing in a partially filled container with inclusion of compressibility effects

A numerical scheme of study is developed to model compressible two-fluid flows simulating liquid sloshing in a partially filled tank. For a two-fluid system separated by an interface as in the case of sloshing, not only a Mach-uniform scheme is required, but also an effective way to eliminate unphysical numerical oscillations near the interface. By introducing a preconditioner, the governing equations expressed in terms of primitive variables are solved for both fluids (i.e. water, air, gas etc.) in a unified manner. In order to keep the interface sharp and to eliminate unphysical numerical oscillations in unsteady fluid flows, the non-conservative implicit Split Coefficient Matrix Method (SCMM) is modified to construct a flux difference splitting scheme in the dual time formulation. The proposed numerical model is evaluated by comparisons between numerical results and measured data for sloshing in an 80% filled rectangular tank excited at resonance frequency. Through similar comparisons, the investigation is further extended by examining sloshing flows excited by forced sway motions in two different rectangular tanks with 20% and 83% filling ratios. These examples demonstrate that the proposed method is suitable to capture induced free surface waves and to evaluate sloshing pressure loads acting on the tank walls and ceiling

An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations

In this article we propose a new symmetric version of the interior penalty discontinuous Galerkin finite element method for the numerical approximation of the compressible Navier-Stokes equations. Here, particular emphasis is devoted to the construction of an optimal numerical method for the evaluation of certain target functionals of practical interest, such as the lift and drag coefficients of a body immersed in a viscous fluid. With this in mind, the key ingredients in the construction of the method include: (i) An adjoint consistent imposition of the boundary conditions; (ii) An adjoint consistent reformulation of the underlying target functional of practical interest; (iii) Design of appropriate interior--penalty stabilization terms. Numerical experiments presented within this article clearly indicate the optimality of the proposed method when the error is measured in terms of both the L2-norm, as well as for certain target functionals. Computational comparisons with other discontinuous Galerkin schemes proposed in the literature, including the second scheme of Bassi and Rebay, the standard SIPG method outlined in [Hartmann,Houston-2006], and an NIPG variant of the new scheme will be undertaken