Effects of WENO flux reconstruction order and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability

Abstract

Weighted essentially non-oscillatory (WENO) simulations of the reshocked twodimensional single-mode Richtmyer-Meshkov instability using third-, fifth-and ninthorder spatial flux reconstruction and uniform grid resolutions corresponding to 128, 256 and 512 points per initial perturbation wavelength are presented. The dependence of the density, vorticity, simulated density Schlieren and baroclinic production fields, mixing layer width, circulation deposition, mixing profiles, production and mixing fractions, energy spectra, statistics, probability distribution functions, numerical turbulent kinetic energy and enstrophy production/dissipation rates, numerical Reynolds numbers, and numerical viscosity on the order and resolution is investigated to long evolution times. The results are interpreted using the implicit numerical dissipation in the characteristic projection-based, finite-difference WENO method. It is shown that higher order higher resolution simulations have lower numerical dissipation. The sensitivity of the quantities considered to the order and resolution is further amplified following reshock, when the energy deposition by the second shock-interface interaction induces the formation of small-scale structures. Lower-order lower-resolution simulations preserve large-scale structures and flow symmetry to late times, while higher-order higher-resolution simulations exhibit fragmentation of the structures, symmetry breaking and increased mixing. Similar flow features are qualitatively and quantitatively captured by either approximately doubling the order or the resolution. Additionally, the computational scaling shows that increasing the order is more advantageous than increasing the resolution for the flow considered here. The present investigation suggests that the ninth-order WENO method is well-suited for the simulation and analysis of complex multi-scale flows and mixing generated by shock-induced hydrodynamic instabilities.

Numerical simulations of a shock interacting with successive interfaces using the Discontinuous Galerkin method: the multilayered Richtmyer–Meshkov and Rayleigh–Taylor instabilities

“…The Discontinuous Galerkin method [6][7][8][9][10] is a numerical method for solving partial differential equations which combines the advantages of the finite element and finite volume methods. In contrast with previous RMI studies using finite difference and finite volume methods [20,22,26,28,34], the numerical solution is represented in each computational cell of the domain with high-order polynomial basis functions. The method is therefore high-order accurate and is superconvergent in the cell averages at a rate of 2N + 1 [2,3], where N + 1 is the number of basis function in each cell.…”

“…Two gases, air and SF 6 , lie in a shocktube at atmospheric pressure, and the interface between the two is sinusoidally perturbed. The properties of air are ρ air = 1.351 kg/m 3 , γ air = 1.276, and those of SF 6 are ρ SF 6 = 5.494 kg/m 3 , γ SF 6 = 1.093 [26]. The initial amplitude and wavelength of the interfacial perturbations are a 0 = 0.183 cm and λ = 5.933 cm, respectively.…”

“…We use a linear polynomial basis (N = 1) for the Discontinuous Galerkin method, resulting in third-order accuracy in smooth regions. An exponential diffusion function is used to initialize a thermodynamically consistent diffuse interface between the gases [26], with a thickness set to 0.5 cm. To avoid a large spatial domain, we add a constant upward velocity calculated from an exact Riemann solver so that the post-shock upward mean velocity is zero and the interfaces remain in the domain.…”

“…For the nominal case, the growth rate is 7.4 m/s [26]. We solve three Riemann problems using an exact Riemann solver to calculate the numerical values of A + and u to determine the growth due to the reflected shock from the second interface: (i) the initial shock interacting with the first interface; (ii) the transmitted shock interacting with the second interface; and (iii) the reflected shock from the second interface interacting with the first interface.…”

“…The canonical RMI, consisting of a single planar shock wave interacting with a single planar interface separating two fluids, has been studied extensively in the past, both experimentally [11,21,29,37] and numerically [20,22,26,31,34]. While some of these studies have considered late-time mixing, most have focused on the early time dynamics.…”