On Alternative Setups of the Double Mach Reflection Problem

Abstract

The double Mach reflection (DMR) problem is an important test case for the assessment of the resolution of Euler codes. The initial and boundary conditions specified in the conventional setup of the problem result in formation of undesirable numerical artefacts which interfere with the solution. In this study, two alternative setups are proposed to prevent the manifestations of such artefacts. The first setup involves modifying the computational domain to simplify the boundary conditions while the second setup involves modifying the initial conditions to make the problem more conducive to run on uniform Cartesian grids and foregoing the problematic boundary conditions altogether. Both setups employ a modified two-step initialization procedure to obtain clean, artefact-free results. These methods are robust and easy to implement. The DMR problem was simulated using a 7 th order finitevolume WENO-based solver developed using OpenFOAM to demonstrate that the proposed methods can be used for the assessment of higher order methods. The proposed methods have yielded good results and complete removal of the artefacts is observed in some cases. The simulations were performed with two different numerical flux schemes to determine the dependency of the flux schemes on the performance of the setups. Lastly, the second setup was found to be superior than the first one in terms of effectively removing the numerical artefacts.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

“…Shock-cylinder interaction was studied in Ou and Zhai (2019). Alternative set-ups of the double Mach reflection problem were discussed in Vevek, Zang and New (2019). Partial characteristic decomposition for multi-species Euler equations was used in Wang, Pan, Hu and Adams (2019a).…”

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

“…The solution êh amounts to a smeared shock profile on the (x, y) plane which, at time t, is centered on the straight line x = x s (y, t) where x s (y, t) = x r + Us sin(θ) t + y tan(θ) . This shock is followed by some back-trailing oscillations, as has been observed in [17,43]. In order to eliminate these artifacts from the initial condition, a diagonal strip of N d points centered around the shock location is selected.…”

“…More precisely, we enforce zero values on the derivative of e with respect to the direction parallel to the shock, along both the complete upper computational boundary and the region 0 ≤ x ≤ x r (that is, left of the ramp) on the lower computational boundary. This method can be considered as a further development of the approach proposed in [43], wherein an extended domain in the oblique direction was utilized in conjunction with Neumann conditions along the normal direction to the oblique boundary.…”

“…Additionally, following [43], we utilize a numerical viscous incident shock as an initial condition in order to avoid well-known post-shock oscillations, as noted in [17], that result from the use of a sharp initial profile. Such a smeared shock is obtained in our context by applying the FC-SDNN solver to the propagation of an oblique flat shock on all of space (without the ramp) up to time T = 0.2, including imposition of oblique Neumann boundary conditions throughout the top and bottom boundary.…”

A carbuncle cure for the Harten-Lax-van Leer contact (HLLC) scheme using a novel velocity-based sensor

“…The double Mach reflection (DMR) problem proposed by Woodward and Colella [26] involves an M = 10 shock impinging on an inclined ramp to form a system of complex shock reflections. The DMR problem was solved on a modified computational domain proposed by Vevek et al [28] to prevent the formation of numerical artefacts. A coarse mesh of the modified domain is shown in Fig.…”