Supersonic flow calculation using a Reynolds-stress and an eddy thermal diffusivity turbulence model

A second-order model for the velocity field and a two-equation model for the temperature field are used to calculate supersonic boundary layers assuming negligible real gas effects. The modeled equations are formulated on the basis of an incompressible assumption and then extended to supersonic flows by invoking Morkovin's hypothesis, which proposes that compressibility effects are completely accounted for by mean density variations alone. In order to calculate the near-wall flow accurately, correction functions are proposed to render the modeled equations asymptotically consistent with the behavior of the exact equations near a wall and, at the same time, display the proper dependence on the molecular Prandtl number. Thus formulated, the near-wall second order turbulence model for heat transfer is applicable to supersonic flows with different Prandtl numbers. The model is validated against flows with different Prandtl numbers and supersonic flows with free-stream Mach numbers as high as 10 and wall temperature ratios as low as 0.3. Among the flow cases considered, the momentum thickness Reynolds number varies from approximately 4,000 to approximately 21,000. Good correlation with measurements of mean velocity, temperature, and its variance is obtained. Discernible improvements in the law-of-the-wall are observed, especially in the range where the big-law applies

A compressible near-wall turbulence model for boundary layer calculations

A compressible near-wall two-equation model is derived by relaxing the assumption of dynamical field similarity between compressible and incompressible flows. This requires justifications for extending the incompressible models to compressible flows and the formulation of the turbulent kinetic energy equation in a form similar to its incompressible counterpart. As a result, the compressible dissipation function has to be split into a solenoidal part, which is not sensitive to changes of compressibility indicators, and a dilational part, which is directly affected by these changes. This approach isolates terms with explicit dependence on compressibility so that they can be modeled accordingly. An equation that governs the transport of the solenoidal dissipation rate with additional terms that are explicitly dependent on the compressibility effects is derived similarly. A model with an explicit dependence on the turbulent Mach number is proposed for the dilational dissipation rate. Thus formulated, all near-wall incompressible flow models could be expressed in terms of the solenoidal dissipation rate and straight-forwardly extended to compressible flows. Therefore, the incompressible equations are recovered correctly in the limit of constant density. The two-equation model and the assumption of constant turbulent Prandtl number are used to calculate compressible boundary layers on a flat plate with different wall thermal boundary conditions and free-stream Mach numbers. The calculated results, including the near-wall distributions of turbulence statistics and their limiting behavior, are in good agreement with measurements. In particular, the near-wall asymptotic properties are found to be consistent with incompressible behavior; thus suggesting that turbulent flows in the viscous sublayer are not much affected by compressibility effects

A near-wall four-equation turbulence model for compressible boundary layers

A near-wall four-equation turbulence model is developed for the calculation of high-speed compressible turbulent boundary layers. The four equations used are the k-epsilon equations and the theta(exp 2)-epsilon(sub theta) equations. These equations are used to define the turbulent diffusivities for momentum and heat fluxes, thus allowing the assumption of dynamic similarity between momentum and heat transport to be relaxed. The Favre-averaged equations of motion are solved in conjunction with the four transport equations. Calculations are compared with measurements and with another model's predictions where the assumption of the constant turbulent Prandtl number is invoked. Compressible flat plate turbulent boundary layers with both adiabatic and constant temperature wall boundary conditions are considered. Results for the range of low Mach numbers and temperature ratios investigated are essentially the same as those obtained using an identical near-wall k-epsilon model. In general, the numerical predictions are in very good agreement with measurements and there are significant improvements in the predictions of mean flow properties at high Mach numbers

A near-wall two-equation model for compressible turbulent flows

A near-wall two-equation turbulence model of the K - epsilon type is developed for the description of high-speed compressible flows. The Favre-averaged equations of motion are solved in conjunction with modeled transport equations for the turbulent kinetic energy and solenoidal dissipation wherein a variable density extension of the asymptotically consistent near-wall model of So and co-workers is supplemented with new dilatational models. The resulting compressible two-equation model is tested in the supersonic flat plate boundary layer - with an adiabatic wall and with wall cooling - for Mach numbers as large as 10. Direct comparisons of the predictions of the new model with raw experimental data and with results from the K - omega model indicate that it performs well for a wide range of Mach numbers. The surprising finding is that the Morkovin hypothesis, where turbulent dilatational terms are neglected, works well at high Mach numbers, provided that the near wall model is asymptotically consistent. Instances where the model predictions deviate from the experiments appear to be attributable to the assumption of constant turbulent Prandtl number - a deficiency that will be addressed in a future paper

A review of near-wall Reynolds-stress

The advances made in second-order near-wall turbulence closures are summarized. All closures examined are based on some form of high Reynolds number models for the Reynolds stress and the turbulent kinetic energy dissipation rate equations. Consequently, most near-wall closures proposed to data attempt to modify the high Reynolds number models for the dissipation rate equation so that the resultant models are applicable all the way to the wall. The near-wall closures are examined for their asymptotic behavior so that they can be compared with the proper near-wall behavior of the exact equations. A comparison of the closure's performance in the calculation of a low Reynolds number plane channel flow is carried out. In addition, the closures are evaluated for their ability to predict the turbulence statistics and the limiting behavior of the structure parameters compared to direct simulation data

Fully-Coupled Simulation of Cosmic Reionization. I: Numerical Methods and Tests

We describe an extension of the Enzo code to enable fully-coupled radiation hydrodynamical simulation of inhomogeneous reionization in large $\sim (100 Mpc)^3$ cosmological volumes with thousands to millions of point sources. We solve all dynamical, radiative transfer, thermal, and ionization processes self-consistently on the same mesh, as opposed to a postprocessing approach which coarse-grains the radiative transfer. We do, however, employ a simple subgrid model for star formation which we calibrate to observations. Radiation transport is done in the grey flux-limited diffusion (FLD) approximation, which is solved by implicit time integration split off from the gas energy and ionization equations, which are solved separately. This results in a faster and more robust scheme for cosmological applications compared to the earlier method. The FLD equation is solved using the hypre optimally scalable geometric multigrid solver from LLNL. By treating the ionizing radiation as a grid field as opposed to rays, our method is scalable with respect to the number of ionizing sources, limited only by the parallel scaling properties of the radiation solver. We test the speed and accuracy of our approach on a number of standard verification and validation tests. We show by direct comparison with Enzo's adaptive ray tracing method Moray that the well-known inability of FLD to cast a shadow behind opaque clouds has a minor effect on the evolution of ionized volume and mass fractions in a reionization simulation validation test. We illustrate an application of our method to the problem of inhomogeneous reionization in a 80 Mpc comoving box resolved with $3200^3$ Eulerian grid cells and dark matter particles.Comment: 32 pages, 23 figures. ApJ Supp accepted. New title and substantial revisions re. v

Modeling of Fluctuating Mass Flux in Variable Density Flows

The approach solves for both Reynolds and Favre averaged quantities and calculates the scalar pdf. Turbulent models used to close the governing equations are formulated to account for complex mixing and variable density effects. In addition, turbulent mass diffusivities are not assumed to be in constant proportion to turbulent momentum diffusivities. The governing equations are solved by a combination of finite-difference technique and Monte-Carlo simulation. Some preliminary results on simple variable density shear flows are presented. The differences between these results and those obtained using conventional models are discussed

Reasoning with comparative moral judgements: an argument for Moral Bayesianism

The paper discusses the notion of reasoning with comparative moral judgements (i.e judgements of the form “act a is morally superior to act b”) from the point of view of several meta-ethical positions. Using a simple formal result, it is argued that only a version of moral cognitivism that is committed to the claim that moral beliefs come in degrees can give a normatively plausible account of such reasoning. Some implications of accepting such a version of moral cognitivism are discussed

Extended Optical Model Analyses of Elastic Scattering, Direct Reaction, and Fusion Cross Sections for the 9Be + 208Pb System at Near-Coulomb-Barrier Energies

Based on the extended optical model approach in which the polarization potential is decomposed into direct reaction (DR) and fusion parts, simultaneous $\chi^{2}$ analyses are performed for elastic scattering, DR, and fusion cross section data for the $^{9}$Be+$^{208}$Pb system at near-Coulomb-barrier energies. Similar $\chi^{2}$ analyses are also performed by only taking into account the elastic scattering and fusion data as was previously done by the present authors, and the results are compared with those of the full analysis including the DR cross section data as well. We find that the analyses using only elastic scattering and fusion data can produce very consistent and reliable predictions of cross sections particularly when the DR cross section data are not complete. Discussions are also given on the results obtained from similar analyses made earlier for the $^{9}$Be+$^{209}$Bi system.Comment: 5 figure

Extended Optical Model Analyses of Elastic Scattering and Fusion Cross Sections for 6Li + 208Pb System at Near-Coulomb-Barrier Energies by using Folding Potential

Based on the extended optical model approach in which the polarization potential is decomposed into direct reaction (DR) and fusion parts, simultaneous $\chi^{2}$ analyses are performed for elastic scattering and fusion cross section data for the $^{6}$Li+$^{208}$Pb system at near-Coulomb-barrier energies. A folding potential is used as the bare potential. It is found that the real part of the resultant DR part of the polarization potential is repulsive, which is consistent with the results from the Continuum Discretized Coupled Channel (CDCC) calculations and the normalization factors needed for the folding potentials. Further, it is found that both DR and fusion parts of the polarization potential satisfy separately the dispersion relation.Comment: 6 figure