Oleg Schilling

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

Weighted essentially non-oscillatory (WENO) simulations of the reshocked two-dimensional single-mode Richtmyer-Meshkov instability using third-, fifth- and ninth-order 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 Convergence Study of Nearly Incompressible, Inviscid Taylor-Green Vortex Flow

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor–Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it is shown that for the Taylor–Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.

High-order WENO simulations of three-dimensional reshocked Richtmyer-Meshkov instability to late times: Dynamics, dependence on initial conditions, and comparisons to experimental data

The dynamics of the reshocked multi-mode Richtmyer-Meshkov instability is investigated using 513 x 257{sup 2} three-dimensional ninth-order weighted essentially nonoscillatory shock-capturing simulations. A two-mode initial perturbation with superposed random noise is used to model the Mach 1.5 air/SF{sub 6} Vetter-Sturtevant shock tube experiment. The mass fraction and enstrophy isosurfaces, and density cross-sections are utilized to show the detailed flow structure before, during, and after reshock. It is shown that the mixing layer growth agrees well with the experimentally measured growth rate before and after reshock. The post-reshock growth rate is also in good agreement with the prediction of the Mikaelian model. A parametric study of the sensitivity of the layer growth to the choice of amplitudes of the short and long wavelength initial interfacial perturbation is also presented. Finally, the amplification effects of reshock are quantified using the evolution of the turbulent kinetic energy and turbulent enstrophy spectra, as well as the evolution of the baroclinic enstrophy production, buoyancy production, and shear production terms in the enstrophy and turbulent kinetic transport equations.

High-resolution simulations and modeling of reshocked single-mode Richtmyer-Meshkov instability: Comparison to experimental data and to amplitude growth model predictions

The reshocked single-mode Richtmyer-Meshkov instability is simulated in two spatial dimensions using the fifth- and ninth-order weighted essentially nonoscillatory shock-capturing method with uniform spatial resolution of 256 points per initial perturbation wavelength. The initial conditions and computational domain are modeled after the single-mode, Mach 1.21 air(acetone)/SF6 shock tube experiment of Collins and Jacobs [J. Fluid Mech. 464, 113 (2002)]. The simulation densities are shown to be in very good agreement with the corrected experimental planar laser-induced fluorescence images at selected times before reshock of the evolving interface. Analytical, semianalytical, and phenomenological linear and nonlinear, impulsive, perturbation, and potential flow models for single-mode Richtmyer-Meshkov unstable perturbation growth are summarized. The simulation amplitudes are shown to be in very good agreement with the experimental data and with the predictions of linear amplitude growth models for small times, and with those of nonlinear amplitude growth models at later times up to the time at which the driver-based expansion in the experiment (but not present in the simulations or models) expands the layer before reshock. The qualitative and quantitative differences between the fifth- and ninth-order simulation results are discussed. Using a local and global quantitative metric, the prediction of the Zhang and Sohn [Phys. Fluids 9, 1106 (1997)] nonlinear Pade model is shown to be in best overall agreement with the simulation amplitudes before reshock. The sensitivity of the amplitude growth model predictions to the initial growth rate from linear instability theory, the post-shock Atwood number and amplitude, and the velocity jump due to the passage of the shock through the interface is also investigated numerically.

Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. I. Comparison to experimental data

A 1152×760×1280 direct numerical simulation (DNS) using initial conditions, geometry, and physical parameters chosen to approximate those of a transitional, small Atwood number Rayleigh–Taylor mixing experiment [Mueschke et al., J. Fluid Mech. 567, 27 (2006)] is presented. In particular, the Atwood number is 7.5×10−4, and temperature diffusion is modeled by mass diffusion with an equivalent Schmidt number of 7. The density and velocity fluctuations measured just off of the splitter plate in this buoyantly unstable water channel experiment were parametrized to provide physically realistic, anisotropic initial conditions for the DNS. The methodology for parametrizing the measured data and numerically implementing the resulting perturbation spectra in the simulation is discussed in detail. The DNS is then validated by comparing quantities from the simulation to experimental measurements. In particular, large-scale quantities (such as the bubble front penetration hb and the mixing layer growth parameter αb), ...

Measurements of molecular mixing in a high-Schmidt-number Rayleigh-Taylor mixing layer

Molecular mixing measurements are reported for a high-Schmidt-number ( Sc ~ 10 3 ), small-Atwood-number ( A ≈ 7.5 × 10 −4 ) buoyancy-driven turbulent Rayleigh–Taylor (RT) mixing layer in a water channel facility. Salt was added to the top water stream to create the desired density difference. The degree of molecular mixing was measured as a function of time by monitoring a diffusion-limited chemical reaction between the two fluid streams. The pH of each stream was modified by the addition of acid or alkali such that a local neutralization reaction occurred as the two fluids molecularly mixed. The progress of this neutralization reaction was tracked by the addition of phenolphthalein – a pH-sensitive chemical indicator – to the acidic stream. Accurately calibrated backlit optical techniques were used to measure the average concentration of the coloured chemical indicator. Comparisons of chemical product formation for pre-transitional buoyancy- and shear-driven mixing layers are given. It is also shown that experiments performed at different equivalence ratios (acid/alkali concentrations) can be combined to obtain a mathematical relationship between the coloured product formed and the density variance. This relationship was used to obtain high-fidelity quantitative measures of the degree of molecular mixing which are independent of probe resolution constraints. The dependence of molecular mixing on the Schmidt and Reynolds numbers is examined by comparing the current Sc ~ 10 3 measurements with previous Sc = 0.7 gas-phase and Pr = 7 (where Pr is the Prandtl number) liquid-phase measurements. This comparison indicates that the Schmidt number has a large effect on the quantity of mixed fluid at small Reynolds numbers Re h 3 . At larger Reynolds numbers, corresponding to later times in this experiment, all mixing parameters indicated a greater degree of molecular mixing and a decreased Schmidt number dependence. Implications for the development and quantitative assessment of turbulent transport and mixing models appropriate for RT instability-induced mixing are discussed.

Analysis of spectral eddy viscosity and backscatter in incompressible, isotropic turbulence using statistical closure theory

The spectral eddy viscosity and backscatter viscosity in three-dimensional, incompressible, unforced, nonhelical, isotropic turbulence are decomposed into a sum of contributions corresponding to the Reynolds and cross-stresses, and studied numerically as a function of different assumed kinetic energy spectra. The eddy viscosities and backscatter viscosities are computed using the kinetic energy transfer obtained from the eddy-damped quasinormal Markovian (EDQNM) closure model as a function of k/kc (where kc is the cutoff wave number) using the sharp Fourier cutoff filter. The behavior of the Reynolds and cross-contributions is studied using a Kolmogorov kinetic energy spectrum, a family of spectra with small wave number scaling proportional to k, and a spectrum from an EDQNM calculation that includes both a k4 energy production subrange and a dissipation subrange. The principal results of this theoretical investigation and sensitivity study are (1) the main contributions from the Reynolds and cross-compon...

Investigation of Rayleigh–Taylor turbulence and mixing using direct numerical simulation with experimentally measured initial conditions. II. Dynamics of transitional flow and mixing statistics

A 1152×760×1280 direct numerical simulation (DNS) using initial conditions, geometry, and physical parameters chosen to approximate those of a transitional, small Atwood number, nonreacting Rayleigh–Taylor mixing experiment was presented in Paper I [Mueschke and Schilling, Phys. Fluids 21, 014106 (2009)]. In addition, the DNS model of the experiment was validated by comparing quantities from the simulation to experimental measurements, including large-scale quantities, higher-order statistics, and vertical velocity and density variance spectra. In Paper II of this study, other quantities not measured in the experiment are obtained from the DNS and discussed, such as the integral- and Taylor-scale Reynolds numbers, Reynolds stress and dissipation anisotropy, two-dimensional density and velocity variance spectra, hypothetical chemical product formation measures (similar to those used in reacting shear flow experiments), other local and global mixing parameters, and the statistical composition of mixed fluid...

Influence of subgrid scales on resolvable turbulence and mixing in Rayleigh-Taylor flow

The energy transfer process and the interaction of different scales in a flow induced by the variable-density Rayleigh–Taylor instability in miscible fluids is investigated using a three-dimensional direct numerical simulation database with a spatial resolution of Nx×Ny×Nz=512×512×2040. The method used to study the transfer of energy between the supergrid and subgrid scales in the homogeneous planes, determined by partitioning the modes into resolved and unresolved scales defined by a two-dimensional cutoff wave number kc in Fourier space, is applied to the kinetic energy evolution equation. The treatment of the flow inhomogeneity in the direction z parallel to the acceleration is analogous to that used in the analysis of incompressible wall-bounded flows, including channel flow and Rayleigh–Benard convection [J. A. Domaradzki et al., Phys. Fluids 6, 1583 (1994); J. A. Domaradzki and W. Liu, ibid. 7, 2025 (1995)]. Using a sharp Fourier cutoff filter, the kinetic energy transfer is decomposed into (1) the ...

Large-eddy simulation of very large kinetic and magnetic Reynolds number isotropic magnetohydrodynamic turbulence using a spectral subgrid model

A spectral eddy viscosity and magnetic resistivity subgrid-scale model based on the eddy-damped quasi-normal Markovian (EDQNM) kinetic and magnetic energy transfers is used in large-eddy simulation (LES) of asymptotically large kinetic and magnetic Reynolds number magnetohydrodynamic (MHD) turbulence. The model is assessed a posteriori on three-dimensional, incompressible, isotropic, nonhelical, freely decaying MHD turbulence. Using LES initialized with spectra such that the Alfven ratio of kinetic to magnetic energy equals unity, it is shown that the kinetic energy and magnetic energy spectra exhibit k−5∕3 Kolmogorov inertial subrange scalings consistent with the EDQNM model.