Topological structure of shock induced vortex breakdown

Abstract

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier–Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ (x, t) = 1/ρ(∂ρ/∂t+∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If λ > 0, the sectional streamlines spiral inwards in the near region of the centre. If λ < 0, the sectional streamlines spiral outwards in the same region. If λ = 0, the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown.

Heuristical and numerical considerations for the carbuncle phenomenon

“…In the region upstream of the shock, he picks the middle slice of the computational grid and artificially sets the velocity to zero as sketched out in Figure 1. This closely corresponds to the experimental and theoretical investigations by Kalkhoran and Smart [24] and by Zhang et al [66]. It turns out that the carbuncle-like structure which comes with the Godunov-scheme is similar to the experimental results.…”

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

“…• ENO and WENO schemes were used to simulate the interactions of shocks with vortices, which involve both strong discontinuities and complicated smooth structures, especially suitable for using high-order non-oscillatory schemes. See Erlebacher, Hussaini and Shu (1997), Shu (2005, 2006b), Zhang, Jiang, Zhang and Shu (2009a), Zhang, Zhang and Shu (2009b) and Zhang et al (2013). Effects of shock waves on Rayleigh-Taylor instability were studied in Zhang, Shu and Zhou (2006c).…”

On high-order accurate weighted essentially non-oscillatory and discontinuous Galerkin schemes for compressible turbulence simulations

“…WENO schemes are particularly suited for the simulation of shock-vortex interactions because of their simultaneous capability of high-order solution and non-oscillatory shock transition. Examples of such applications can be found in earlier studies [41][42][43][44][45][46][47].…”