Ellen M. Taylor

A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence

Two new formulations of a symmetric WENO method for the direct numerical simulation of compressible turbulence are presented. The schemes are designed to maximize order of accuracy and bandwidth, while minimizing dissipation. The formulations and the corresponding coefficients are introduced. Numerical solutions to canonical flow problems are used to determine the dissipation and bandwidth properties of the numerical schemes. In addition, the suitability and accuracy of the bandwidth-optimized schemes for direct numerical simulations of turbulent flows is assessed in decaying isotropic turbulence and supersonic turbulent boundary layers.

Stencil Adaptation Properties of a WENO Scheme in Direct Numerical Simulations of Compressible Turbulence

Weighted essentially non-oscillatory (WENO) methods can simultaneously provide the high order of accuracy, high bandwidth-resolving efficiency, and shock-capturing capability required for the detailed simulation of compressible turbulence. However, rigorous analysis of the actual versus theoretical error properties of these non-linear numerical methods is difficult. We use a bandwidth-optimized WENO scheme to conduct direct numerical simulations of two- and three-dimensional decaying isotropic turbulence, and we evaluate the performance of quantitative indicators of local WENO adaptation behavior within the resulting flow fields. One aspect of this assessment is the demarcation of shock-containing and smooth regions where the WENO method should, respectively, engage its adaptation mechanism and revert to its linear optimal stencil. Our results show that these indicators, when synthesized properly, can provide valuable quantitative information suitable for statistical characterization.

Assessment of STBLI DNS Data and Comparison against Experiments

The direct numerical simulation data of a Mach 2.9 turbulent boundary layer flowing over a 24-degree compression ramp are assessed. A summary of the flow features and the comparison of the simulation data against experiments are given. Of main interest are discrepancies found in the wall-pressure distribution. The flow characteristics of the separation shock foot are studied to better understand the wall-pressure prediction, and the effect of the numerical shock capturing technique on the data is considered.

Optimization of Nonlinear Error Sources for Weighted Essentially Non-Oscillatory Methods in Direct Numerical Simulations of Compressible Turbulence

Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions. We therefore construct and evaluate WENO schemes that also reduce dissipation due to two independent nonlinear error sources: (i) the smoothness measurement that governs the application of stencil adaption away from the linear optimal stencil, and (ii) the numerical accuracy (e.g. order-of-accuracy and bandwidth) properties of the less favorable stencils that take over when adaption engages. Direct numerical simulations (DNS) include one-dimensional test cases and three-dimensional compressible isotropic turbulence. Although efforts to address the second source listed above fail to meaningfully alter WENO performance, the smoothness measurement modification inspired by the first source both significantly enhances numerical accuracy and generates negligible additional computational expense. Moreover, this technique appears to be broadly effective regardless of flow configuration.

Numerical Investigation of Shock-wave/Isotropic Turbulence Interaction

†We conduct direct numerical simulations (DNS) of shock/isotropic-turbulence interactions (SITI), in which the turbulence is highly compressible. We find, consistent with previous studies using less-compressible turbulence, that turbulent kinetic energy and transverse vorticity fluctuations become persistently amplified upon passage through a shock wave and that the Taylor microscales and Kolmogorov lengthcale all diminish. Jumps in thermodynamic quantities fall short of their laminar magnitudes. In general, all of these effects tend to intensify with decreasing strength of upstream turbulence and with increasing strength of the normal shock. Comparison of individual terms for the vorticity variance budget shows that the amplification of vorticity is dominated by the compression term. Reynolds stress budgets show that downstream of the interaction, the pressure term acts to decrease the streamwise Reynolds stress and increase the transverse. Two-dimensional energy spectra show that the interaction leaves the spectrum with more energy in scales smaller than the original energetic scales. For the conditions chosen, the interaction corresponds to sharply-defined shocks across the entire wrinkled shock surface rather than distorted or broken shock fronts with regions of smooth compression. The shock structure follows the similarity scaling based on turbulent and convective Mach numbers as proposed by Donzis. (Donzis, Diego A., “Similarity Scaling in Shock-Turbulence Interactions,” Presented at the 63rd Annual Meeting of the American Physical Society Division of Fluid Dynamics, Long Beach, CA, November 21-23, 2010.)

On Synchronization of Weighted Essentially Non-Oscillatory Methods

Weighted essentially non-oscillatory (WENO) methods have been developed to simultaneously provide robust shock-capturing in compressible fluid flow and avoid excessive damping of fine-scale flow features such as turbulence. Under certain conditions in compressible turbulence, however, numerical dissipation remains unacceptably high even after optimization of the linear component that dominates in smooth regions. We demonstrate that a significant nonlinear source of dissipation error is due to a WENO implementation defect that we call the “synchronization deficiency,” and we develop and evaluate a preliminary technique to mitigate its eects. Direct numerical simulations (DNS) include one-dimensional test cases and three-dimensional compressible isotropic turbulence. Although we find the current formulation of our mitigation technique to be quite successful for the one-dimensional problems, its ecacy is more modest in the turbulence simulations, and its cost-to-benefit ratio is uncomfortably high. We believe, however, that with further work it will be possible to alleviate these drawbacks.

Preliminary Study of the Turbulence Structure in Supersonic Boundary Layers Using DNS Data

Direct numerical simulation data are used to visualize coherent structures in turbulent boundary layers at Mach numbers from 0.3 to 7. Different criteria to identify the threedimensional turbulence structure are selected. We find that using the discriminant of the velocity gradient tensor, the swirling strength and the λ2 criteria give nearly identical results, with λ2 identifying more structures very close to the wall.

Local Adaption and Dissipation Properties of a Weighted Essentially Non-Oscillatory Scheme

We perform several direct numerical simulations (DNS) of decaying isotropic turbulence with various initial turbulent mach numbers Mt using a modified 1 weighted essentially non-oscillatory (WENO) method. We then present a procedure for the identification of shock-containing and smooth regions where the WENO scheme should respectively engage its adaption mechanism and revert to its linear optimal stencil. Weirs 1 has previously proposed the centrality index CI and nonlinearity index NI as two quantitative measures of stencil adaption, and we analyze these values within the regions of interest in the DNS flow fields. Our preliminary results indicate that these indices are suitable for assessments of the local adaption and dissipation properties of the WENO method; however, further studies must be conducted over a wider range of compressible flow conditions.

Evaluation of Traditional and Shock-Confining LES Filters using DNS Data of Compressible Turbulence

uniformly highorder-accurate numerical methods for approximating the convective terms of the NavierStokes equations: the introduction of new, non-physical extrema near discontinuities. Extensive investigations have been conducted into myriad types of shock-capturing methods, which adaptively decrease their order of accuracy within shock-containing regions in order to maintain accuracy and stability over time. Inspired by such work, Grube et al. (AIAA 2007-4198) have proposed the conceptually similar idea of “shock-confining” filtering (SCF) for compressible large-eddy simulations (LES), in which a class of linear filters is subject to adaptive adjustments based on smoothness information provided by a weighted essentially non-oscillatory (WENO) method. In the current work, we utilize the turbulent flow fields produced by direct numerical simulations (DNS) of shock/isotropic-turbulence interaction to perform preliminary a priori testing of these shock-confining filters against the corresponding linear filters. Although additional study, both a priori and a posteriori, is required to verify the necessity and suitability of SCF for LES, we conclude that linear filtering can indeed create serious qualitative discrepancies in global flow characteristics, discrepancies that are avoided by the use of SCF.