SUMMARYThe ÿrst of a two-paper series, this paper introduces a new decomposition not of the hyperbolic ux vector but of the ux vector Jacobian. The paper then details for the Euler and Navier-Stokes equations an intrinsically inÿnite directional upstream-bias formulation that rests on the mathematics and physics of multi-dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the di erential equation level, before the spatial discretization, within a characteristics-bias governing system. Through a decomposition of the Euler ux divergence into multi-dimensional acoustics and convection-acoustics components, this characteristics-bias system induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable-strength upstreaming that correlates with the spatial distribution of characteristic velocities.

SUMMARYThe second of a two-paper series, this paper details a solver for the characteristics-bias system from the acoustics-convection upstream resolution algorithm for the Euler and Navier-Stokes equations. An integral formulation leads to several surface integrals that allow e ective enforcement of boundary conditions. Also presented is a new multi-dimensional procedure to enforce a pressure boundary condition at a subsonic outlet, a procedure that remains accurate and stable. A classical ÿnite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier-Stokes equations, an approximation that remains multi-dimensional independently of the orientation of the reference axes and computational cells. The time-dependent discrete equations are then integrated in time via an implicit Runge-Kutta procedure that in this paper is proven to remain absolutely non-linearly stable for the spatially-discrete Euler and Navier-Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics-convection upstream resolution algorithm generates essentially non-oscillatory solutions for subsonic, transonic and supersonic ows, encompassing oblique-and interacting-shock ÿelds that converge within 40 time steps and re ect reference exact solutions.

No abstract

No abstract

SUMMARYAn exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.