Britton Olson

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor-Green vortex, Shu-Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.

Rayleigh–Taylor shock waves

Beginning from a state of hydrostatic equilibrium, in which a heavy gas rests atop a light gas in a constant gravitational field, Rayleigh–Taylor instability at the interface will launch a shock wave into the upper fluid. We have performed a series of large-eddy simulations which suggest that the rising bubbles of light fluid act like pistons, compressing the heavy fluid ahead of the fronts and generating shocklets. These shocklets coalesce in multidimensional fashion into a strong normal shock, which increases in strength as it propagates upwards. The simulations demonstrate that the shock Mach number increases faster in three dimensions than it does in two dimensions. The generation of shocks via Rayleigh–Taylor instability could play an important role in type Ia supernovae.

A mechanism for unsteady separation in over-expanded nozzle flow

Shock wave induced separation in an over-expanded planar nozzle is studied through numerical simulation. These Large-Eddy Simulations (LES) model previous experiments which have shown unsteady motion of the shock wave in flows with similar geometries but offered little insight into the underlying mechanism. Unsteady separation in nozzle flow leads to “side loads” in the rocket engine which can adversely affect the stability of the rocket. A mechanism for the low-frequency shock motion is identified and explained using the LES data. This mechanism is analyzed for a series of over-expanded planar nozzles of various area ratios and nozzle pressure ratios. The effect of grid resolution and Reynolds number on the instability is discussed. A simple reduced order model for the unsteady shock behavior is used to further validate the proposed mechanism. This model is derived from first principles and uses data from the LES calculations to capture the effects of the turbulent boundary layer and shear layer.

Large eddy simulation requirements for the Richtmyer-Meshkov instability

The shock induced mixing of two gases separated by a perturbed interface is investigated through Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). In a simulation, physical dissipation of the velocity field and species mass fraction often compete with numerical dissipation arising from the errors of the numerical method. In a DNS, the computational mesh resolves all physical gradients of the flow and the relative effect of numerical dissipation is small. In LES, unresolved scales are present and numerical dissipation can have a large impact on the flow, depending on the computational mesh. A suite of simulations explores the space between these two extremes by studying the effects of grid resolution, Reynolds number, and numerical method on the mixing process. Results from a DNS are shown using two different codes that use a high- and low-order numerical method and show convergence in the temporal and spectral dependent quantities associated with mixing. Data from an unresolved, high Reyn...

Nonlinear effects in the combined Rayleigh-Taylor/Kelvin-Helmholtz instability

The combined Rayleigh-Taylor/Kelvin-Helmholtz (RT/KH) instability is studied in the early nonlinear regime. Specifically, the effect of adding shear to a gravitationally unstable configuration is investigated. While linear stability theory predicts that any amount of shear would increase the growth rate beyond the Rayleigh-Taylor value, numerical (large eddy) simulations show a more complex and non-monotonic behavior where small amounts of shear in fact decrease the growth rate. A velocity scale for the combined instability is proposed from linear stability arguments and is shown to effectively collapse the growth rates for different configurations. The specific amount of shear that minimizes the peak growth rate is identified and the physical origins of this non-monotonic behavior are investigated.

Directional artificial fluid properties for compressible large-eddy simulation

Abstract An improved methodology for large-eddy simulation (LES) for flows involving shock waves and turbulence is described. This approach provides better shock capturing and enhanced resolution of turbulence while preserving numerical stability on high aspect ratio (AR) grids. The proposed improvements are based on the LES approach which uses artificial fluid diffusivities (shear viscosity, bulk viscosity and thermal diffusivity) to damp the unresolved gradients of turbulence, shock waves and contact discontinuities, respectively. The scalar artificial viscosities are active only in under-resolved regions of the flow and added directly to the physical quantities. On high aspect ratio grids, the length scale disparity of the mesh leads to over dissipation in one or more direction, causing mis-prediction of physical quantities and added numerical stiffness which reduces the stable time step by a factor of 1/AR. Our proposed method allows fluid diffusivities to be independently applied along each grid direction by forming directional quantities, which ensure the method is minimally dissipative. This alternative approach reduces the errors and numerical stiffness associated with over dissipation. Several test cases are presented which demonstrate the improved performance of this approach on high aspect ratio grids and the enhanced numerical stability. Brief results from LES of an over-expanded planar nozzle are given which demonstrate the method’s robustness on practical applications.

Large-Eddy Simulation of an over-expanded planar nozzle

This fluid dynamics video shows visualizations of a Large-Eddy Simulation (LES) of an over-expanded planar nozzle. This configuration represents the experimental setup of Papamoschou et. al. which found the position of the internal shock to be unstable. Our LES calculations confirm this instability and offer a vibrant and dynamic view of the underlying flow physics. The interaction between shock and turbulent boundary layer is shown as is the subsequent separation region downstream. Numerical Schlieren provide a glimpse of the low frequency shock motion and suggest potential mechanisms for the instability. Key features include the asymmetry of the shock structure (with large and small lambda shocks), compression and expansion waves downstream of the shock and large scale flow reversal. Full details of the experiment and the calculation can be found in the references.

Directional artificial bulk viscosity for shock capturing on high aspect ratio grids

An improved shock-capturing method for high-order finite-difference schemes, which alleviates numerical stiffness on anisotropic grids, is proposed. This method is an extension of the artificial bulk viscosity (ABV) scheme, developed to capture shock waves with a minimal dissipation of the physical oscillations present in the flow, such as those associated with turbulence. This modified scheme generalizes the ABV treatment to situations where there are strong spatial inhomogeneities that often require anisotropic grids. To accomplish this, ABV is independently computed and applied along each computational grid direction, therefore taking a multi-valued, directional form rather than a scalar form. Scalar dissipation on high aspect ratio (AR) grids with explicit time-stepping schemes can cause the stable time step to be reduced by a factor of 1/AR. The proposed method removes this constraint and can allow for substantial speedups on high AR grids with shock waves. A two-dimensional test case illustrates the added numerical stability of the method and its ability to capture shock waves more sharply. Brief results from a large-eddy simulation of an over-expanded planar nozzle are given which demonstrate the methods robustness in practical applications.

Low-frequency unsteadiness in nozzle flow separation

A series of large-eddy simulations (LES) have confirmed the existence of a low-frequency shock wave instability in an over-expanded planar nozzle. This numerical simulation models the experiments performed by Papamoschou et.al. 9,21,22 and was conducted to shed light on the underlying physics of the shock-boundary layer interaction and the free-shock separation 20 (FSS) which is present in over-expanded supersonic nozzles and gives rise to large asymmetric side loads. The LES methodology used in this study is presented with emphasis on a modified sub-grid scale model for compressible wall bounded flows. The mechanism which causes the low-frequency shock motion is explained and illustrated using LES data. This mechanism is analyzed for a series of over-expanded planar nozzles of various area ratios and nozzle pressure ratios. We propose and evaluate a reduced order model for the unsteady shock behavior. The model is derived from first principles and uses data from the LES calculations to capture the effects of the turbulent boundary layer and shear layer. The model agrees with the LES data surprisingly well, given that only one-dimensional effects are captured in the model’s dynamics. A new scaling law for the peak frequency of the unsteadiness is proposed which is derived from the reduced order model and collapses data for all cases.