A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

Abstract

Two-dimensional, oblique detonations induced by a wedge are simulated using the reactive Euler equations with a detailed chemical reaction model. The focus of this study is on the oblique shock-to-detonation transition in a stoichiometric hydrogenair mixture. A combustible, gas mixture at low pressure and high temperature, corresponding to the realistic, inflow conditions applied in oblique detonation wave engines, is presented in this study. At practical flight conditions, the present numerical results illustrate that oblique detonation initiation is achieved through a smooth transition from a curved shock, which differs from the abrupt transition depicted in the previous studies. The formation mechanism of this smooth transition is discussed and a quantitative analysis is carried out by defining a characteristic length for the initiation process. The dependence of the initiation length on different parameters including the wedge angle, flight Mach number, and inflow Mach number is discussed. Despite the hypothetical nature of the simulation configuration, the present numerical study uses parameters we deem relevant to practical conditions and provides important observations for which future investigations can benefit from in reaching toward a rigorous theory of the formation and self-sustenance of oblique detonation waves. C 2015 AIP Publishing LLC. [http://dx

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

“…Different from the conventional semi-Lagrangian schemes, the present scheme has rigorous numerical conservation. With both the trajectory and numerical flux being evaluated with the third-order Runge-Kutta method, we expect that the present scheme has a numerical accuracy comparable with the Eulerian variant in [44]. To demonstrate this, we compare the total errors, dissipation errors, and dispersion errors between Eulerian MCV-WENO4 scheme [44] and the present CIP-CSL-WENO4 scheme through some benchmark tests for linear advection Equation (1).…”

“…A more general procedure that converts different moments to a set of mean values and constructs the WENO projection under the conventional FVM framework is reported in [40]. A WENO reconstruction, the so-called simple WENO limiter [41][42][43], was derived for the DG method, which uses the local DOFs over a compact stencil of only three cells.We have recently proposed a new scheme, called multi-moment constrained finite volume with WENO limiter of fourth-order (MCV-WENO4) method [44], by incorporating the WENO limiter to the multi-moment constrained FVM of third-order (MCV3) scheme. The new WENO limiter in MCV-WENO4 scheme distinguishes itself from the existing ones by taking fully use of the local nodal values (PV) in the WENO interpolation, which leads to a compact computational stencil involving merely the target cell and its adjacent cells.…”

A note on non‐negativity correction for a multimoment finite‐volume transport model with WENO limiter

“…With an adjustable slope parameter, it is possible to remove the non-physical oscillations in the results of the high-order models through using the properly selected formulations to calculate the slope parameters based on known PVs and VIAs. A formulation based on the weighted essentially non-oscillatory (WENO) concept is proposed in Sun et al (2015) to calculate the slope parameter, which works well in the problems with the smooth structures or the discontinuities.…”

A non‐oscillatory multimoment finite‐volume global transport model on a cubed‐sphere grid using the WENO slope limiter

“…Recently, a new formula for the slope limiter in CSL3 interpolation was designed using the WENO concept (Sun et al , ). With this new limiter, the multimoment scheme can achieve fourth‐order accuracy and is effective in dealing with discontinuities, as verified by numerical tests of both advection and the Euler system in Sun et al s*().…”

“…Recently, a new formula for the slope limiter in CSL3 interpolation was designed using the WENO concept (Sun et al , ). With this new limiter, the multimoment scheme can achieve fourth‐order accuracy and is effective in dealing with discontinuities, as verified by numerical tests of both advection and the Euler system in Sun et al s*(). The WENO limiter is implemented using a very compact stencil involving three adjacent cells in the one‐dimensional case and it is convenient to implement on a cubed sphere, though it involves a limiter, which is determined over more than one computational element, making the proposed scheme less scalable than “local” high‐order multimoment schemes built following the Multi‐Moment Constrained Flux Reconstruction (MMC‐FR) framework (Xiao et al , ).…”