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A critical comparison of explicit vs traditional algebraic stress models of turbulence is made in an effort to clear up the confusion that appears to have been generated by the recently published literature on the subject, in which disparate approaches are adopted. The only way that general second-order closures can formally lead to fully explicit algebraic stress models, in a global sense, is in the limit of equilibrium homogeneous turbulence. When these fully explicit models are then applled to turbulent flows that are far from equilibrium, a singularity can arise, which can be removed by a systematic regularization. When solved explicitly either analytically or numerically, the traditional, implicit algebraic stress models have either multiple solutions or singularities, which tends to explain why they have had problems in applications to complex flows. Thus, it is argued that traditional algebraic stress models are intrinsically ill-behaved and should be abandoned in future applications in favor of regularized, explicit algebraic stress models. It is furthermore argued that these should be based on the homogeneous equilibrium hypothesis, which allows for more general second-order closures to be used to obtain single-valued models.

We present an explicit algebraic stress model (EASM) based on the more current pressure-strain model of Speziale et al. (the SSG model). The SSG model represents a recent upgrade of the Launder et al. model that does not require wall reflection terms. The ability of the proposed model to predict nonequilibrium flows over an airfoil and a wing will be established. The ISAAC Navier-Stokes code is used to simulate the flows considered.

During the past few years, explicit algebraic stress models have been developped that are formally consistent with full second-order closures in the limit of homogeneous turbulence in equilibrium. These models allow for the solution of complex turbulent flows with a substantially reduced level of computation compared with full second-order closures, since they constitute two-equation models.The purpose of the present note is to show how the effects of anisotropic dissipation can be systematically incorporated into these explicit algebraic stress models by a simple readjustmentof the coefficients.

The explicit algebraic stress model of Gatski and Speziale is considered. This constitutes a two-equation model with an anisotropic eddy viscosity that is systematically derived from the SSG second-order model via the algebraic stress approximation for equilibrium turbulent flows.

Dimensional similarity arguments proposed by Millikan are used with the Morkovin hypothesis to deduce logarithmic laws for compressible turbulent boundary layers as an alternative to the traditional van Driest analysis. It is shown that an overlap exists between the wall layer and the defect layer, and this leads to logarithmic behavior in the overlap region. The von Karman constant is found to depend parametrically on the Mach number based on the friction velocity, the dimensionless total heat flux, and the specific heat ratio. Even though it remains constant at approximately 0.41 for a freestream Mach number range of 0 to 4.544 with adiabatic wall boundary conditions, it rises sharply as the Mach number increases significantly beyond 4.544. The intercept of the logarithmic law of the wall is found to depend on the Mach number based on the friction velocity, the dimensionless total heat flux, the Prandtl number evaluated at the wall, and the specific heat ratio. On the other hand, the intercept of the logarithmic defect law is parametric in the pressure gradient parameter and all of the aforementioned dimensionless variables except the Prandtl number. A skin friction law is also deduced for compressible boundary layers. The skin friction coefficientmore » is shown to depend on the momentum thickness Reynolds number, the wall temperature ratio, and all of the other parameters already mentioned. 26 refs.« less

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.

The ability of two-equation models to accurately predict separated flows is analyzed from a combined theoretical and computational standpoint. Turbulent flow past a backward facing step is chosen as a test case in an effort to resolve the variety of conflicting results that were published during the past decade concerning the performance of two-equation models. It is found that the errors in the reported predictions of the k-epsilon model have two major origins: (1) numerical problems arising from inadequate resolution, and (2) inaccurate predictions for normal Reynolds stress differences arising from the use of an isotropic eddy viscosity. Inadequacies in near wall modelling play a substantially smaller role. Detailed calculations are presented which strongly indicate the standard k-epsilon model - when modified with an independently calibrated anisotropic eddy viscosity - can yield surprisingly good predictions for the backstep problem.

A variety of two-equation turbulence models,including several versions of the K-epsilon model as well as the K-omega model, are analyzed critically for near wall turbulent flows from a theoretical and computational standpoint. It is shown that the K-epsilon model has two major problems associated with it: the lack of natural boundary conditions for the dissipation rate and the appearance of higher-order correlations in the balance of terms for the dissipation rate at the wall. In so far as the former problem is concerned, either physically inconsistent boundary conditions have been used or the boundary conditions for the dissipation rate have been tied to higher-order derivatives of the turbulent kinetic energy which leads to numerical stiffness. The K-omega model can alleviate these problems since the asymptotic behavior of omega is known in more detail and since its near wall balance involves only exact viscous terms. However, the modeled form of the omega equation that is used in the literature is incomplete-an exact viscous term is missing which causes the model to behave in an asymptotically inconsistent manner. By including this viscous term and by introducing new wall damping functions with improved asymptotic behavior, a new K-tau model (where tau is identical with 1/omega is turbulent time scale) is developed. It is demonstrated that this new model is computationally robust and yields improved predictions for turbulent boundary layers.

Near-wall modeling of the dissipation rate equation is investigated and its asymptotic behavior is studied in detail using a k-epsilon model. It is found that all existing modeled dissipation rate equations predict an incorrect behavior for the dissipation rate near a wall. An improvement is proposed and the resulting near-wall dissipation rate distribution is found to be similar to that given by numerical simulation data. To further validate the improved k-epsilon model, it is used to calculate flat-plate turbulent boundary-layer flows at high- as well as low-turbulence Reynolds numbers, and the results are compared with measurements, numerical simulation data, and the calculations of three different two-equation models. These comparisons show that all the models tested give essentially the same flow properties away from the wall; significant differences only occur in a region very close to the wall. In this region, the calculations of the improved k-epsilon model are in better agreement with measurements and numerical simulation data. In particular, the modeled distribution of the dissipation rate is significantly improved and a maximum is predicted at the wall instead of away from the wall. Furthermore, the improved k-epsilon model is found to be the most asymptotically consistent among the four different two-equation models examined.

The applicability of linear and quadratic models for the fluctuating pressure-strain correlations is extended to flows with a system rotation by using the intrinsic spin tensor in place of the coordinate-dependent vorticity tensor. The extension is tested here for flow in a channel rotated about its spanwise axis but is equally applicable to all other models of rotation