Near-wall Reynolds Stress Closure Research Articles

Reynolds-Stress Model Dual-Time-Stepping Computation of Unsteady Three-Dimensional Flows

An efficient and robust implicit methodology for the integration of the unsteady three-dimensional compressible Favre-Reynolds-averaged Navier-Stokes equations with near-wall Reynolds-stress closure is developed. The five mean-flow and seven turbulence transport equations are discretized on a structured deforming grid, using an O(Δx 3 ) finite volume upwind-biased MUSCL scheme. Time integration uses an implicit O(Δt 2 ) dual-time-stepping procedure with alternating-direction-implict subiterations (with approximate Jacobians designed to minimize the computing time requirements of the implicit phase for the Reynolds stresses, so that the computational overhead of the Reynolds-stress seven-equation closure, compared to a two-equation closure, is less than 30% per iteration), based on a dynamic criterion of subiterative convergence. Grid-deformation velocities associated with solid-wall displacement are computed using a Laplacian operator. The method is validated by comparison with experimental data for 1) two-dimensional pitching oscillations of a NACA-0012 airfoil and 2) three-dimensional shock-wave oscillation in a transonic channel. The influence of the various parameters of the method is analyzed in detail

Wall-Normal-Free Reynolds-Stress Closure for Three-Dimensional Compressible Separated Flows

A near-wall Reynolds-stress closure that is independent of the distance from the wall and of the normal to the walldirection isdevelopedandvalidated.Particularattention wasgiven intheapplicabilityofthemodeltocomplex three-dimensional e ows with shock waves and boundary-layer separation. In the separated e ow region, measurements andmodelcomputations indicatethatthe e atnessparameter Aoftheanisotropy tensor aij approaches unity. Therefore, controlofseparation isachieved in themodel through theparticularfunctional dependenceoftherapid pressure‐strain isotropization of production model coefe cienton A. Echo termsaretreated by replacing geometric normalsanddistancesbyfunctionsofthegradientsofturbulencelengthscaleandanisotropytensorinvariants.The modelisinitiallycomparedwithmeasurementsforcompressiblee at-plateboundary-layere ows.Itisthenvalidated by comparison with experimental data in a two-dimensional compression corner oblique shock-wave/boundarylayer interaction at Mach 3. Finally the model is applied to the computation of the three-dimensional interaction of a Mach 1.5‐1.8 strong shock wave with the boundary layers of a rectangular channel e tted with a swept bump on the lower wall, and results are compared with measurements. One important advantage of the proposed model is its robustness in complex three-dimensional e ows. A detailed discussion of therange of validity of the model and possible improvements is presented.

A geometry independent near-wall Reynolds-stress closure

Models proposed for high-Reynolds-number Reynolds–stress closures are, in general, geometry independent. Therefore, they are most suitable for flows with complex geometries. With the advent of supercomputers, the near-wall flow can be calculated with greater efficiency to give more accurate results on surface properties. However, with modifications made to the various models to account for near-wall behavior, the closures are no longer geometry independent. The cause is the presence of wall distance and wall unit normals in the proposed corrections for either the velocity–pressure-gradient correlation tensor, the dissipation rate tensor, the dissipation rate equation, or all of them. In this paper, a proposal is put forward where the modifications made to these tensors and equation are free of wall distance and wall unit normals. The closure is validated against a wide variety of simple flows and good agreement with data is obtained. Further, the closure is applied to calculate backstep flows and developing square duct flows. The results are compared with direct numerical simulation and experimental data. In addition, they are also compared with the calculations of another near-wall Reynolds–stress closure with wall unit normal dependence and is known to give good correlations with backstep flow and square duct flow data. The present closure is found to yield results that are in good agreement with the data and the calculations of the model with wall unit normals. As a result, a viable geometry independent near-wall Reynolds–stress closure is available.

Morkovin Hypothesis and the Modeling of Wall-Bounded Compressible Turbulent Flows

Modelingofcompressiblewall-boundedturbulente owsreliesonthehypothesisofMorkovin,who suggestedthat compressibility effects on turbulence could be accounted for by the mean density variations alone. This hypothesis has been shown to yield good results for the mean velocity and mean temperature e elds when the incompressible turbulence models are extended directly to calculate compressible turbulent boundary layers. However, its applicability for the turbulence e eld has been less closely scrutinized. The reason is the lack of sufe ciently detailed compressible turbulence data for comparison. Such data are now becoming available. Therefore, the purpose here is to assess the applicability of the Morkovin hypothesis to the turbulence e eld using direct numerical simulation data of a supersonic, e at plate boundary layer. A near-wall Reynolds-stress closure based on a quasi-linear pressure-strain model is used to calculatethis supersonic, boundary-layer e ow. Comparisons between calculations and direct numerical simulation data show that the Morkovin hypothesis is just as applicable for the turbulence e eld and there is a dynamic similarity between the near-wall turbulence e eld of an incompressible and a compressible wall-bounded turbulent e ow. In addition, the validation of this model is reported for compressible e ow calculations covering a wide range of Mach numbers with adiabatic and constant-temperature wall boundary conditions. These results show that the model yields good predictions of e at-plate turbulent boundary layers up to a Mach number of 10.31.

Near-Wall Reynolds-Stress Three-Dimensional Transonic Flow Computation

A computational method for the Favre-Reynolds-averaged three-dimensional compressible Navier-Stokes equations using near-wall Reynolds-stress closure is described. The near-wall Reynolds-stress closure uses the Launder-Shima formulation for the Reynolds stresses and the Jones-Launder-Sharma modified dissipation (e*) equation. The mean-flow and turbulence-transport equations are discretized using a finite volume method based on MUSCL Van Leer flux-vector-splitting with Van Albada limiters. The mean-flow and turbulence equations are integrated in time using a fully coupled approximately factored implicit backward Euler method. The resulting scheme is robust and achieves optimal convergence with local-time-step Courant-Friedrichs-Lewy = 50. The turbulence closure is validated by comparison with classic flat-plate boundary-layer data. Results are presented for the three-dimensional Delery transonic channel test case and compared with k-e computations. An analysis of the limitations of the closure is attempted, and possible improvements are suggested.

Modeling Reynolds-Number Effects in Wall-Bounded Turbulent Flows

Recent experimental and direct numerical simulation data of two-dimensional, isothermal wall-bounded incompressible turbulent flows indicate that Reynolds-number effects are not only present in the outer layer but are also quite noticeable in the inner layer. The effects are most apparent when the turbulence statistics are plotted in terms of inner variables. With recent advances made in Reynolds-stress and near-wall modeling, a near-wall Reynolds-stress closure based on a recently proposed quasi-linear model for the pressure strain tensor is used to analyse wall-bounded flows over a wide range of Reynolds numbers. The Reynolds number varies from a low of 180, based on the friction velocity and pipe radius/channel half-width, to 15406, based on momentum thickness and free stream velocity. In all the flow cases examined, the model replicates the turbulence statistics, including the Reynolds-number effects observed in the inner and outer layers, quite well. Furthermore, the model reproduces the correlation proposed for the location of the peak shear stress and an appropriately defined Reynolds number, and the variations of the near-wall asymptotes with Reynolds numbers. It is conjectured that the ability of the model to replicate the asymptotic behavior of the near-wall flow is most responsible for the correct prediction of the Reynolds-number effects.

Assessment of a Reynolds Stress Closure Model for Appendage-Hull Junction Flows

A multiblock numerical method, for the solution of the Reynolds-Averaged Navier-Stokes equations, has been used in conjunction with a near-wall Reynolds stress closure and a two-layer isotropic eddy viscosity model for the study of turbulent flow around a simple appendage-hull junction. Comparisons of calculations with experimental data clearly demonstrate the superior performance of the present second-order Reynolds stress (second-moment) closure over simpler isotropic eddy viscosity models. The second-moment solutions are shown to capture the most important features of appendage-hull juncture flows, including the formation and evolution of the primary and secondary horseshoe vortices, the complex three-dimensional separations, and interaction among the hull boundary layer, the appendage wake and the root vortex system.

Calculations of a Curved-Pipe Flow Using Reynolds Stress Closure

It has been observed that as a fully developed turbulent flow enters a curved bend the anisotropy of the normal stresses near the outer bend (furthest from the centre of the bend curvature) increases. According to the arguments of vorticity generation, a sudden increase in the anisotropy of the normal stresses may lead to the formation of a secondary flow of the second kind. If this secondary motion is to be calculated, then a near-wall Reynolds stress closure that can mimic the anisotropic turbulence behaviour near a wall has to be used. This study presents the results of just such an attempt. In addition, two high Reynolds number closures assuming wall functions in the near-wall region are tested for their ability to replicate the behaviour of the normal stresses in a curved-pipe flow. These two closures differ in their modelling of the pressure-strain terms. Consequently, the effects of near-wall and pressure-strain modelling on curved-pipe flow calculations can be examined. Comparisons are also made with recent curved-pipe flow measurements. The results show that pressure-strain modelling alone is not sufficient to predict the rapid rise of the anisotropy of the normal stresses near the outer bend, and hence the formation of the secondary flow of the second kind. Overall, the near-wall Reynolds stress closure gives a more accurate prediction of the measured mean flow and turbulence statistics, and a realistic calculation of the secondary flow of the second kind near the outer bend.