Computational fluid dynamics

An overview of computational fluid dynamics (CFD) activities at the Langley Research Center is given. The role of supercomputers in CFD research, algorithm development, multigrid approaches to computational fluid flows, aerodynamics computer programs, computational grid generation, turbulence research, and studies of rarefied gas flows are among the topics that are briefly surveyed

On the plane wave Riemann Problem in Fluid Dynamics

The paper contains a stability analysis of the plane-wave Riemann problem for the two-dimensional hyperbolic conservation laws for an ideal compressible gas. It is proved that the contact discontinuity in the plane-wave Riemann problem is unstable under perturbations. The implications for Godunovs method are discussed and it is shown that numerical post shock noise can set of a contact instability. A relation to carbuncle instabilities is established.Comment: 27 pages, 1 figur

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

Applied aerodynamics: Challenges and expectations

Aerospace is the leading positive contributor to this country's balance of trade, derived largely from the sale of U.S. commercial aircraft around the world. This powerfully favorable economic situation is being threatened in two ways: (1) the U.S. portion of the commercial transport market is decreasing, even though the worldwide market is projected to increase substantially; and (2) expenditures are decreasing for military aircraft, which often serve as proving grounds for advanced aircraft technology. To retain a major share of the world market for commercial aircraft and continue to provide military aircraft with unsurpassed performance, the U.S. aerospace industry faces many technological challenges. The field of applied aerodynamics is necessarily a major contributor to efforts aimed at meeting these technological challenges. A number of emerging research results that will provide new opportunities for applied aerodynamicists are discussed. Some of these have great potential for maintaining the high value of contributions from applied aerodynamics in the relatively near future. Over time, however, the value of these contributions will diminish greatly unless substantial investments continue to be made in basic and applied research efforts. The focus: to increase understanding of fluid dynamic phenomena, identify new aerodynamic concepts, and provide validated advanced technology for future aircraft

Regularity of weak solutions of the compressible isentropic Navier-Stokes equation

Regularity and uniqueness of weak solution of the compressible isentropic Navier-Stokes equations is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $\gamma>1$ when $N=2,3$ and $P(\rho)=a\rho^{\gamma}$. In a second part we prove a condition of blow-up in slightly subcritical initial data when $\rho\in L^{\infty}$. We finish by proving that weak solutions in \T^{N} turn out to be smooth as long as the density remains bounded in L^{\infty}(L^{(N+1+\e)\gamma}) with \e>0 arbitrary small

Long-Time Behavior of a Point Mass in a One-Dimensional Viscous Compressible Fluid and Pointwise Estimates of Solutions

We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid--point mass system is governed by the barotropic compressible Navier--Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass, and it gives pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values. As a corollary, it is shown that the velocity $V(t)$ of the point mass satisfies a decay estimate $|V(t)|=O(t^{-3/2})$ --- a faster decay compared to $t^{-1/2}$ known for the motion of a point mass in the viscous Burgers fluid~[J.~L.~V{\'{a}}zquez and E.~Zuazua, Comm. Partial Differential Equations \textbf{28} (2003), 1705--1738]. The rate $-3/2$ is essentially related to the compressibility and the nonlinearity. As a consequence, it follows that the point mass is convected only a finite distance as opposed to the viscous Burgers case. The main tool used in the proof is the pointwise estimates of Green's function. It turns out that the understanding of the time-decay properties of the transmitted and reflected waves at the point mass is essential for the proof.Comment: Ver. 4: Although the logic of the proof is unchanged, some notations have been changed and several typos are corrected. The abstract and the introduction have been modified and some remarks are adde