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.

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

“…Regarding the implementation of the derivative calculations, the Fourier coefficients in (17) are first computed using a fast Fourier transform (FFT) and point-values of the derivatives in (20) are obtained using an inverse FFT. Both steps, FFT and inverse FFT, can be computed with a complexity of O(N log(N )).…”

“…To accurately capture all regimes in such complicated flow structures it is necessary to account adequately for both sharp shock discontinuities (without introduction of oscillatory behavior near shocks) as well as complex smooth flow structures. A well-known highly effective approach for the solution of the compressible Navier-Stokes equations governing such flows is based on a highorder weighted essentially non-oscillatory (WENO) finite difference method; in particular, high-order WENO algorithms have been used to produce successful simulations of the Rayleigh-Taylor instability [19] and Richtmyer-Meshkov instability [20,21] in two and three space dimensions.…”

A study of planar Richtmyer-Meshkov instability in fluids with Mie-Grüneisen equations of state

“…Various definitions exist for mixing layer amplitude. 11,50 We define amplitude by a position weighted integral of the initial volume fraction for solids modeled by isotropic Mie-Grüneisen equations of state and mass fraction for perfect gases. Before roll up occurs, we define the interfaces centerline as y cd ðx; tÞ ¼ Ð 1 À1 ywðx; y; tÞð1 À wðx; y; tÞÞdy Ð 1 À1 wðx; y; tÞð1 À wðx; y; tÞÞdy ; (49) where w is the initial volume fraction.…”

Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

“…A well-known alternative for the solution of such flows is based on a high-order weighted essentially nonoscillatory (WENO) finite difference method; in particular, high-order WENO algorithms have been used to produce successful simulations of the Rayleigh-Taylor instability [24] and Richtmyer-Meshkov instability [25,26] in two and three space dimensions. Finite volume version of the high-order WENO scheme have been used in shock wave bubble interactions in three dimensions [36].…”