Curved shock theory (CST) is introduced, developed and applied to relate pressure gradients, streamline curvatures, vorticity and shock curvatures in flows with planar or axial symmetry. Explicit expressions are given, in an influence coefficient format, that relate post-shock pressure gradient, streamline curvature and vorticity to pre-shock gradients and shock curvature in steady flow. The effect of pre-shock flow divergence/convergence, on vorticity generation, is related to the transverse shock curvature. A novel derivation for the post-shock vorticity is presented that includes the effects of pre-shock flow non-uniformities. CST applicability to unsteady flows is discussed.

Curved shock theory

Curved shock theory is introduced and applied to calculate the flow behind concave shock waves. For sonic conditions, three characterizing types of flow are identified, based on the orientation of the sonic line, and it is shown that, depending on the ratio of shock curvatures, a continuously curving shock can exist with Type III flow, where the sonic line intercepts the reflected characteristics from the shock, thus preventing the formation of a reflected shock. The necessary shock curvature ratio for a Type III sonic point does not exist for a hyperbolic shock so that it will revert to Mach reflection for all Mach numbers. A demonstration is provided, by CFD calculations, at Mach 1.2 and 3.

Flow behind concave shock waves

Novel relations are developed for the tangential and normal derivatives of the pressure, density, and velocity components just downstream of a regular point on a curved shock wave. The perfect gas flow is three dimensional with a nonuniform freestream and the flow may be unsteady. The analysis starts with data in a fixed, laboratory-frame Cartesian coordinate system. By means of an orthogonal transformation, a coordinate system is introduced via a solid-body translation and two solid-body rotations. This Cartesian system is shock based and limits the analysis to a selected shock point at a given instant of time when the flow is unsteady. The analysis is a local one in that Taylor series expansions are utilized for the configuration of the shock and the upstream gradients of the velocity, pressure, and density. The coefficients in the last three expansions must satisfy Euler equation constraints. These expansions are performed in both Cartesian systems, with occasional preference given to the laboratory-frame expansions where there are known data. The derivatives are in terms of a third coordinate system that is shock based and utilizes the flow plane at the selected shock point. Exact, explicit results are provided for the tangential and normal derivatives in as simple a form as possible. In the Taylor series expansion version for the tangential derivatives, the relative importance of various factors can be assessed. These are the upstream velocity, pressure, and density gradients, curvatures, and shock-shape terms not associated with curvatures. A different type of discussion is provided for the more involved normal derivatives.

Derivatives on the downstream side of a moving, curved shock

Microfluidics is a multidisciplinary area founding applications in several fields such as the aerospace industry Microelectromechanical systems (MEMS) are mainly adopted for flow control, micropower generation and for life support and environmental control for space applications Microflows are modeled relying on both a continuum and molecular approach In this paper, the compressible Navier–Stokes (CNS) equations have been adopted to solve a two-dimensional unsteady flow for a viscous micro shock-channel problem In microflows context, as for the most gas dynamics applications, the CNS equations are usually discretized in space using finite volume method (FVM) In the present paper, the PDEs are discretized with the nodal discontinuous Galerkin finite element method (DG–FEM) in order to understand how the method performs at microscale level for compressible flows Validation is performed through a benchmark test problem for microscale applications The error norms, order of accuracy and computational cost are investigated in a grid refinement study, showing a good agreement and increasing accuracy with reference data as the mesh is refined The effects of different explicit Runge–Kutta schemes and of different time step sizes have also been studied We found that the choice of the temporal scheme does not really affect the accuracy of the numerical results

https://www.mdpi.com/2226-4310/5/1/16/pdf

Discontinuous Galerkin Finite Element Investigation on the Fully-Compressible Navier–Stokes Equations for Microscale Shock-Channels

We consider the evolution of a shock wave generated by an impulsively accelerated, two-dimensional, almost planar piston with a sinusoidally corrugated surface of amplitude. We develop a complex-variable formulation for a nonlinear theory of generalized geometrical shock dynamics (GGSD) (Best, Shock Waves, vol. 1, issue 4, 1991, pp. 251-273; Best, Proc. R. Soc. Lond. A, vol. 442, 1993, pp. 585-598) as a hierarchical expansion of the Euler equations that can be closed at any order. The zeroth-order truncation of GGSD is related to the equations of Whitham's geometrical shock dynamics (GSD), while higher-order corrections incorporate non-uniformity of the flow immediately behind the piston-driven shock. Numerical solutions to GGSD systems up to second order are coupled to an edge-detection algorithm in order to investigate the hypothesized development of a shock-shape curvature singularity as the rippled shock evolves. This singular behaviour, together with the simultaneous development of a Mach-number discontinuity, is found at all orders of the GGSD hierarchy for both weak and strong shocks. The critical time at which a curvature singularity occurs converges as the order of the GGSD system increases at fixed, and follows a scaling inversely proportional to at sufficiently small values. This result agrees with the weakly nonlinear GSD analysis of Mostert et al. (J. Fluid Mech., vol. 846, 2018, pp. 536-562) for a general Mach-number perturbation on a planar shock, and suggests that this represents the universal behaviour of a slightly perturbed, planar shock.

/pdf/evolution-of-a-shock-generated-by-an-impulsively-accelerated-39i42rn87c.pdf

J.C. Pant

Papers

Some aspects of unsteady curved shock waves