Session GM: Supersonic/Hypersonic I

Fluid-structure interaction of converging shocks in water

Numerical simulations of shock focusing in a convergent water-filled geometry with various types of elastic solids (rubber, plastic and metal) as the surrounding material have been performed. The fluid deforms the solid, generating elastic waves, which in turn affect the liquid; thus creating a coupled fluid-structure problem. Here, we use the Overture suit, a code for solving partial differential equations on curvilinear overlapping grids using adaptive mesh refinement. The Euler equations with a stiffened equation of state are used in the fluid domain and linear elasticity is assumed in the solid domain. Preliminary results indicate that the wave speed of the material has a significant influence on the behavior of the converging shock. Comparisons between numerical and experimental results are presented and have the potential to aid in the design of marine structures with convergent sections subjected to dynamic loading events.   

Mixing analysis of PLIF images in a multi-stream injection nozzle

We present quantitative analysis of image sequences of multi-stream injection nozzle flows with several different injection geometries. Image sequences were acquired by A.M. Ragheb and G.S. Elliott (UIUC) using planar laser-induced fluorescence (PLIF) in iodine to visualize flow mixing. The injection nozzle was comprised of a slot, ejector and injector block, with rows of ejector and injector holes along the slot length. The ejector flow exits in an underexpanded state so that upon expanding it forces the slot and injector flows together to enhance mixing. For this study, the diameter and geometry of ejector holes were varied to assess their effect on mixing. Two configurations of ejector holes were used, each with two different diameters for a total of 4 cases with data collected at downstream stations. We carried out a quantitative mixing analysis for these configurations, using two parameters to quantify the mixing. The first parameter, the mixing quality criterion, is assessed from the statistics of the PLIF image intensity histograms, which are bimodal for poorly-mixed flows and have a single peak in well-mixed flows. The second parameter is mixing interface length. Our analysis shows that one injection scheme significantly enhances mixing by stretching the mixing interface.   

Analysis of DNS database of canonical shock/turbulence interaction

A set of databases generated by direct numerical simulation of isotropic turbulence passing through a shock wave is analyzed. Averages conditioned on the local instantaneous shock strength are used to elucidate the structure of the shock/turbulence interaction through the strongest and weakest points on the shock. For sufficiently strong turbulence there exists completely smooth profiles through the shock-region. The unsteady shock-motion is analyzed and linked to the incoming turbulence.   

A Derivation of New Regularized Euler Equations from Basic Principles

Both turbulence and shock formation in inviscid flows are prone to high wave number mode generations. This continuous generation of high wavemodes results in energy cascade to ever smaller scales in turbulence and creation of shocks in compressible flows. This high wavenumber problem is often remedied by the addition of a viscous term in both compressible and incompressible flows. The author's group recently reported a regularization technique for the Burgers equation (Norgard and Mohseni 2008) which is now extended to one-dimensional compressible Euler equations (Norgard and Mohseni 2009). This investigation presents a formal derivation of these equations from basic principles. We will extend our previous results to multidimensional compressible and incompressible Euler equations. We expect this technique to simultaneously regularize shocks and turbulence. Numerical simulation demonstrating the shock regularization properties of these equations will be presented.   

A low-dissipation and dispersion finite volume method for large eddy simulation of compressible flow on arbitrary unstructured grids

One way to develop stable solvers for large eddy simulation with minimal numerical dissipation is to use so-called summation-by-parts (SBP) operators. These discrete operators mimic the integration-by-parts property of the continuous equations, leading to discrete stability by the energy method. Unfortunately, the application of large eddy simulation to compressible flows of engineering interest often involves complex geometries and consequently unstructured grids. In the present work, a method for constructing a fast, explicit compressible flow solver for large eddy simulation is presented that uses standard polynomial reconstruction techniques to build accurate finite volume operators, which are subsequently modified based on the extent to which they are not SBP. Because the SBP property is a property of the operators and not the solution, this can be performed as a pre-processing step in a large eddy simulation. Several examples will be presented demonstrating the robustness and accuracy of this compromise approach, including sound radiated from transonic and supersonic jets.   

High Resolution Direct Numerical Simulations of Compressible Isotropic Turbulence

We present results from a systematic study of Direct Numerical Simulations of forced compressible turbulence. The simulations explore the $M_t$--$\chi$ parameter space where $M_t$, the turbulent Mach number, varies from 0.02--0.6, and $\chi$, the ratio of dilatational to solenoidal energy, varies from 0.01--10, on up to $1024^3$ meshes, with maximum Taylor Reynolds numbers of $R_\lambda>300$. Thus, the study covers the weakly to moderate compressibility effects regime as reflected in the turbulent Mach number values, as well as the low to strong dilatational effects regime that may arise independently from the Mach number effects, e.g. due to exothermic reactions. The forcing method is designed to control the statistically stationary state values of the dissipation (thus the Kolmogorov scale) and the ratio of dilatational to solenoidal dissipation. This ensures that the simulations are both stable and well resolved. The DNS results are used to examine the spectral properties of the solenoidal and dilatational velocity fields and highlight changes in the turbulence properties due to compressibility and dilatational effects.   

Development of a numerical code for the study of a supersonic planar wake

The fully-developed supersonic planar wake represents a canonical high-speed flow occurring in many aeronautical applications. The goal of the current research program is to perform a high-quality direct numerical simulation in order to thoroughly compare the statistics with classical experimental data and gain a better understanding of the structures present in the far-field of a supersonic planar wake. In order to study this flow a code is under development using a very efficient modified MacCormack-type scheme to solve the governing equation set. The main drawback of this numerical method is the large dispersive errors occurring in regions of sharp gradients which can occur in as shocklets in highly compressible flow. To this effect, a study of the numerical properties of this scheme is done using classical one-dimensional test cases such as the Shu-Osher and the Sod problem. The scheme compares very favorably to typical compressible schemes such as the Pade and Roe solvers but shows a very significant advantage in terms of memory usage and speed.   

Effects of Turbulence on Taylor-Sedov Blast Waves in Radially-Symmetric Geometries

Progress in extending studies of the classical Taylor--Sedov blast wave problem by incorporating effects due to turbulence is reported. Investigations consist of the analytical development and initial numerical findings describing the evolution of large and instantaneous energy releases from point explosions (in radially-symmetric systems) while coupling turbulent instabilities. The closure of the Reynolds-Favre averaged mean flow equations is accomplished using a $K$-- $\epsilon$ model in the gradient diffusion approximation. To reduce the complexity of the problem, self-similar analysis is used to reduce the space-time dependent system of partial differential equations to coupled, nonlinear ordinary differential equations in the self-similarity variable. Preliminary approximations considered in the problem are also discussed.