A direct method for computation of nonequilibrium flows with detached shock waves

Supersonic Flow over Convex and Concave Shapes with Radiation and Ablation Effects

“…The -4 unsteady form of conservation equations is everywhere hyperbolic; thus the problem has been converted from a mixed hyperbolic elliptic boundary Value problem to a hyperbolic initial-boundary value problem which is mathematically more tractable. This approach has been previously applied successfully to aerodynam ical problems by several investigators [5] [6] [7] [8] [9].…”

“…However, as demonstrated in Refs. 5,6, and 10,. they? remain use ful for the computation of shocked flows, probably because mag nitudes of discontinuities are decreased in the conservation form.…”

“…(l)- (4) and (5)- (8) are singular along the axis of'symmetry; equations valid along this line are obtained from (l)- (4) and (5)- (8) after division by sine or y respectively and passage to the limit e -ff or y 0 [5]. In the evaluation of these limits the symmetry of the field and L'Hospital's rule are used.…”

“…In cylindrical coordinates the axisymmetric flow is described by the following system of equations [5]: The remarks made about Eqs. (l)- (4) apply here also.…”

“…In spherical polar coordinates with the azimuthal (4) variation suppressed, the governing differential equations for the conservation of mass, momentum, and energy are [5][6]: a (pr 2 sine) + -(pur 2 sine) + --(pvr sine)…”