Propagation of a curved weak shock

Abstract

Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an infinite system of transport equations along these rays. For a two-dimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial differential equations in a ray coordinate system, which consists of two independent variables (ζ, t) where the curves t = constant give successive positions of the shock and ζ = constant give rays. The equations show that shock rays not only stretch longitudinally due to finite amplitude on a shock front but also turn due to a non-uniform distribution of the shock strength on it. These changes finally lead to a modification of the amplitude of the shock strength. Since discontinuities in the form of kinks appear on the shock, it is necessary to study the problem by using the correct conservation form of these equations. We use such a system of equations in conservation form to construct a total-variation-bounded finite difference scheme. The numerical solution captures converging shock fronts with a pair of kinks on them – the shock front emerges without the usual folds in the caustic region. The shock strength, even when the shock passes through the caustic region, remains so small that the small-amplitude theory remains valid. The shock strength ultimately decays with a well-defined geometrical shape of the shock front – a pair of kinks which separate a central disc from a pair of wings on the two sides. We also study the ultimate shape and decay of shocks of initially periodic shapes and plane shocks with a dent and a bulge.

Upstream Propagating Curved Shock in a Steady Transonic Flow

“…What is important theoretically is that the scaling, introduced in (10) and (11), gives a flow field in which (i) waves, which are almost perpendicular to the x-axis, are trapped in such a way that the angle θ, which the normal to the wavefront makes with the x-axis, remains small and will have to be appropriately scaled; (ii) the first two terms in (2) and (3)i.e., La 0 and n β Lq β0 become small and of the same order as the third term Lw so that the wavefront turns slowly and remains trapped in the transonic region for a time interval of the order l/(a * τ ); and (iii) we are able to follow the complete history of the nonlinear wavefront as it transverses the entire transonic region. In this case, K and Ω on the right hand side of (5) are also of the same order as that of w. We introduce a scaled angleθ and scaled amplitudew bȳ…”

“…Derivation of the equations of SRT for a shock of arbitrary strength is extremely cumbersome. However, SRT for a weak shock can be derived from the equations of WNLRT by using a theorem (section 9.2, see also Monica and Prasad, 2001). A weak shock front is followed by one parameter family of nonlinear waves belonging to the same characteristic field.…”

“…Following the procedure given in the section 10.1 (see also Monica and Prasad, 2001), we deduce from the equations (15) - (17) the equations of the SRT. We first introduce notations…”

“…(GΘ) | shock and these six equations are to be solved simultaneously (unlike the problem dealt by Monica and Prasad (2001)). Given initial position of the shock (from where Θ may be calculated), the initial shock strength µ and initial value of µ 1 , we can numerically solve these equations.…”

“…Given initial position of the shock (from where Θ may be calculated), the initial shock strength µ and initial value of µ 1 , we can numerically solve these equations. Procedure for setting up initial value is same as that in section 10.3 (see also Monica and Prasad, 2001). …”