A second-order accurate flux splitting scheme in two-dimensional gasdynamics

Comparison of coordinate-invariant and coordinate-aligned upwinding for the Euler equations

“…Let this section be closed with the remark that the formal framework developed in Eqs. (17)(18)(19)(20)(21)(22)(23)(24) also applies to shocks with the high-pressure side lying at A rather than B; just interchange the subscripts A and B.…”

“…(21) exceeds some threshold E 0 > 1 and satisfies some other constraints, its value is used in a fixed-point iteration to determine the shock Mach number M as a solution to Eq. (20). As to additional constraints, the solution at node A, still assuming it designates the low-pressure side of a shock, is also required to satisfy the inequality (22) This makes sure that the flow on the low-pressure side is indeed supersonic unless it is an unsteady shock that runs into fluid at rest, as it happens in shock-tube problems.…”

“…Montagne 20 proposed an unsteady flow problem that tests the capability of an Euler solver to simulate a rapid progression of Downloaded by UNIVERSITY OF MICHIGAN on February 3, 2015 | http://arc.aiaa.org | DOI: 10.2514/3.12175 markedly dissimilar flow patterns involving regular and irregular shock reflections as well as interactions between shocks, expansion fans, and shears. In this problem, a flow in a two-dimensional channel, whose two sections of constant cross-sectional area are joined through a ramp of 20% slope (contraction ratio: 0.8), is started from uniform supersonic conditions (M = 1.6).…”