The Gauss‐Seidel line relaxation method is modie ed for the simulation of viscous e ows on massively parallel computers. The resulting data-parallel line relaxation method is shown to have good convergence properties for a seriesoftestcases.Thenewmethodrequiressignie cantlymorememorythanthepreviouslydevelopeddata-parallel relaxation methods, but it reaches a steady-state solution in much less time for all cases tested to date. In addition, the data-parallel line relaxation method shows good convergence properties even on the high-cell-aspect-ratio grids required to simulate high-Reynolds-number e ows. The new method is implemented using message passing on the Cray T3E, and the parallel performance of the method on this machine is discussed. The data-parallel line relaxation method combines the fast convergence of the Gauss ‐Seidel line relaxation method with a high parallel efe ciency and thus shows promise for large-scale simulation of viscous e ows.

Data-Parallel Line Relaxation Method for the Navier -Stokes Equations

Rayleigh–Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II

Progress in shock wave/boundary layer interactions

Review and assessment of turbulence models for hypersonic flows

This review highlights the profound and unexpected ways in which viscosity varying in space and time can affect flow. The most striking manifestations are through alterations of flow stability, as established in model shear flows and industrial applications. Future studies are needed to address the important effect of viscosity stratification in such diverse environments as Earth's core, the Sun, blood vessels, and the re-entry of spacecraft.

Instabilities in Viscosity-Stratified Flow

The RANS (Reynolds averaged Navier–Stokes) equations can yield significant error when applied to practical flows involving shock waves. We use the interaction of homogeneous isotropic turbulence with a normal shock to suggest improvements in the k–e model applied to shock/turbulence interaction. Mahesh et al. [J. Fluid Mech. 334, 353 (1997)] and Lee et al. [J. Fluid Mech. 340, 22 (1997)] present direct numerical simulation (DNS) and linear analysis of the flow of isotropic turbulence through a normal shock, where it is found that mean compression, shock unsteadiness, pressure-velocity correlation, and up-stream entropy fluctuations play an important role in the interaction. Current RANS models based on the eddy viscosity assumption yield very high amplification of the turbulent kinetic energy, k, across the shock. Suppressing the eddy viscosity in a shock improves the model predictions, but is inadequate to match theoretical results at high Mach numbers. We modify the k equation to include a term due to s...

Modeling shock unsteadiness in shock/turbulence interaction

Reynolds-averaged Navier-Stokes (RANS) methods often cannot predict shock/turbulence interaction correctly. This may be because RANS models do not account for the unsteady motion of the shock wave that is inherent in these interactions. Previous work proposed a shock-unsteadiness correction that significantly improves prediction of turbulent kinetic energy amplification across a normal shock in homogeneous isotropic turbulence. We generalize the modification to shock-wave/turbulent boundary-layer interactions and implement it in the κ-e, κ-ω, and Spalart-Allmaras models. In compression-comer flows, the correction decreases the turbulent kinetic energy amplification across the shock compared to the standard κ-e and κ-w models. This results in improved prediction of the separation shock location, delayed reattachment, and slower recovery of the boundary layer on the ramp. For the Spalart-Allmaras model, the modification amplifies eddy viscosity across the shock, moving the separation location closer to the experiment.

Modeling the effect of shock unsteadiness in shock/ turbulent boundary-layer interactions

A series of experiments was conducted in the Princeton University Mach 8 Wind Tunnel to study shock interactions on axisymmetric double-cone geometries. Schlieren images and surface-pressure data were taken. Two models were tested, which were expected to produce steady Type VI and Type V shock interactions. The experiments are compared to computational fluid dynamics calculations, and the features of these complicated flowfields are discussed. The comparison is excellent for the laminar Type VI shock interaction. The computations accurately reproduce the size of the separation zone and the surface pressure. However, for the Type V interaction the laminar computation overpredicts the size of the separation region. In addition, the experimental results for the Type V interaction show that the size of the separation region decreases with increasing Reynolds number, whereas the laminar computations predict the opposite trend. Turbulent computations show much better agreement with experimental data and reproduce the experimentally observed relationship between the size of the separation region and the Reynolds number, indicating that the reattachment shocks cause transition to turbulence in these flows

Numerical and experimental investigation of double-cone shock interactions

Reynolds-averaged Navier-Stokes prediction of shock wave/turbulent boundary layer interactions can yield significant error in terms of the size of the separation bubble. In many applications, this can alter the shock structure and the resulting surface properties. Shock-unsteadiness modification of Sinha et al. (Physics of Fluids, Vol.15, No.8, 2003) has shown potential in improving separation bubble prediction in compression corner flows. In this article, the modification is applied to oblique shock wave interacting with a turbulent boundary layer. The challenges involved in the implementation of the shock-unsteadiness correction in the presence of multiple shock waves and expansion fans are addressed in detail. The results show that a robust implementation of the model yields appreciable improvement over standard k-ω turbulence model predictions.

Shock-unsteadiness model applied to oblique shock wave/turbulent boundary-layer interaction

High-speed turbulent flows with shock waves are characterized by high localized surface heat transfer rates. Computational predictions are often inaccurate due to the limitations in modelling of the unclosed turbulent energy flux in the highly non-equilibrium regions of shock interaction. In this paper, we investigate the turbulent energy flux generated when homogeneous isotropic turbulence passes through a nominally normal shock wave. We use linear interaction analysis where the incoming turbulence is idealized as being composed of a collection of two-dimensional planar vorticity waves, and the shock wave is taken to be a discontinuity. The nature of the postshock turbulent energy flux is predicted to be strongly dependent on the angle of incidence of the incoming waves. The energy flux correlation is also decomposed into its vortical, entropy and acoustic contributions to understand its rapid non-monotonic variation behind the shock. Three-dimensional statistics, calculated by integrating two-dimensional results over a prescribed upstream energy spectrum, are compared with available data from direct numerical simulations. A detailed budget of the governing equation is also considered in order to gain insight into the underlying physics.

Krishnendu Sinha

Papers

Modeling shock unsteadiness in shock/turbulence interaction

Modeling the effect of shock unsteadiness in shock/ turbulent boundary-layer interactions

Numerical and experimental investigation of double-cone shock interactions

Shock-unsteadiness model applied to oblique shock wave/turbulent boundary-layer interaction

Turbulent energy flux generated by shock/homogeneous-turbulence interaction