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A resent extension of the nonlinear K–e model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–e model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–e type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–e model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed.

We show that the class of constitutive relations for turbulence models put forward by Wang (1997) in this journal conflicts with dimensional analysis, unless the turbulent Reynolds stresses were to be tied to the molecular viscous stresses everywhere in the flow. We then reiterate, using counter-examples, that the controversial postulate of material frame-indifference is unfounded for turbulence, and is counter-productive in the quest for accuracy. We add a comment on the role of acceleration.

An important consistency condition for non-linear algebraic Reynolds stress models involving their dependence on rotational strains is discussed from a basic theoretical standpoint. A variety of recently proposed models are demonstrated to violate this condition which leads to physically inconsistent results and unrealizable solutions in rotating flows — an undesirable state of affairs that was avoided in the majority of earlier models. It is shown that when non-linear algebraic models are systematically derived from the Reynolds stress transport equation, such inconsistencies are automatically avoided. The implications of these results for turbulence modeling are demonstrated quantitatively by the illustrative example of isotropic turbulence in a rotating frame.

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.

The consistency of second‐order closure models with results from hydrodynamic stability theory is analyzed for the simplified case of homogeneous turbulence. In a recent study, Speziale, Gatski, and Mac Giolla Mhuiris [Phys. Fluids A 2, 1678 (1990)] showed that second‐order closures are capable of yielding results that are consistent with linear stability theory for the case of homogeneous shear flow in a rotating frame. It is demonstrated in this paper that this success is due to the fact that the stability boundaries for rotating homogeneous shear flow are not dependent on the details of the spatial structure of the disturbances. For those instances where they are—such as in the case of elliptical flows where the instability mechanism is more subtle—the results are not so favorable. The origins and extent of this modeling problem are examined in detail along with a possible resolution based on Rapid Distortion Theory (RDT) and its implications for turbulence modeling.

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.

The realizability of Reynolds stress models in homogeneous turbulence is critically assessed from a theoretical standpoint. It is proven that a well known second-order closure model formulated using the strong realizability constraints of Schumann (1977) and Lumley (1978) is, in fact, not a realizable model. The problem arises from the failure to properly satisfy the necessary positive second time derivative constraint when a principal Reynolds stress vanishes-a flaw that becomes apparent when the nonanalytic terms in the model are made single-valued as required on physical grounds. More importantly, arguments are advanced which suggest that it is impossible to identically satisfy the strong from of realizability in any version of the present generation of second-order closures. On the other hand, models properly formulated to satisfy the weak form of realizability—wherein states of one or two component turbulence are made inaccessible in finite time via the imposition of a positive first derivative condition—are found to be realizable. However, unlike the simpler and more commonly used second-order closures, these models can be ill-behaved near the extreme limits of realizable turbulence due to the way that higher-degree nonlinearities are often unnecessarily introduced to satisfy realizability. Illustrative computations of homogeneous shear flow are presented to demonstrate these points which can have important implications for turbulence modeling.

Explicit algebraic stress models that are valid for three-dimensional turbulent flows in non-inertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope (1975) who based his analysis on the Launder, Reece & Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models – as well as anisotropic eddy viscosity models – is theoretically established. A need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

Turbulent channel flow and homogeneous shear flow have served as basic building block flows for the testing and calibration of Reynolds stress models. A direct theoretical connection is made between homogeneous shear flow in equilibrium and the log-layer of fully-developed turbulent channel flow. It is shown that if a second-order closure model is calibrated to yield good equilibrium values for homogeneous shear flow it will also yield good results for the log-layer of channel flow provided that the Rotta coefficient is not too far removed from one. Most of the commonly used second-order closure models introduce an ad hoc wall reflection term in order to mask deficient predictions for the log-layer of channel flow that arise either from an inaccurate calibration of homogeneous shear flow or from the use of a Rotta coefficient that is too large. Illustrative model calculations are presented to demonstrate this point which has important implications for turbulence modeling.

Direct simulations of homogeneous turbulence have, in recent years, come into widespread use for the evaluation of models for the pressure–strain correlation of turbulence. While work in this area has been beneficial, the increasingly common practice of testing the slow and rapid parts of these models separately in uniformly strained turbulent flows is shown in this paper to be unsound. For such flows, the decomposition of models for the total pressure–strain correlation into slow and rapid parts is ambiguous. Consequently, when tested in this manner, misleading conclusions can be drawn about the performance of pressure–strain models. This point is amplified by illustrative calculations of homogeneous shear flow where other pitfalls in the evaluation of models are also uncovered. More meaningful measures for testing the performance of pressure–strain models in uniformly strained turbulent flows are proposed and the implications for turbulence modeling are discussed.

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.

Analytical methods for the development of Reynolds stress models in turbulence are reviewed in detail. Zero, one and two equation models are discussed along with second-order closures. A strong case is made for the superior predictive capabilities of second-order closure models in comparison to the simpler models. The central points are illustrated by examples from both homogeneous and inhomogeneous turbulence. A discussion of the author's views concerning the progress made in Reynolds stress modeling is also provided along with a brief history of the subject.

Consistency conditions for the prediction of turbulent flows in a rotating frame are examined. It is shown that the dissipation rate should vanish along with the eddy viscosity in the limit of rapid rotations. The latter result is also true when the eddy viscosity is anisotropic and formally follows from the explicit algebraic stress approximation as well as from a phenomenological treatment. The former result has been built into the modeled dissipation rate equation of recent turbulence models where the second result has been violated. In fact, some of these models have the eddy viscosity going to infinity while the dissipation rate vanishes, leading to an inconsistency. For consistency, both of these conditions must be satisfied. The implications of these results for turbulence modeling are thoroughly discussed.

Reynolds stress models and traditional large-eddy simulations are reexamined with a view toward developing a combined methodology for the computation of complex turbulent flows. More specifically, an entirely new approach to time-dependent Reynolds-averaged Navier-Stokes (RANS) computations and very large-eddy simulations (VLES) is presented in which subgrid scale models are proposed that allow a direct numerical simulation (DNS) to go continuously to a RANS computation in the coarse mesh/infinite Reynolds number limit. In between these two limits, we have a large eddy simulation (LES) or VLES, depending on the level of resolution. The Reynolds stress model that is ultimately recovered in the coarse mesh/infinite Reynolds number limit has built in nonequilibrium features that make it suitable for time-dependent RANS. The fundamental technical issues associated with this new approach, which has the capability of bridging the gap between DNS, LES and RANS, are discussed in detail. Illustrative calculations are presented along with a discussion of the future implications of these results for the simulation of the turbulent flows of technological importance.

Traditional models for the turbulent dissipation rate assume an equilibrium in which the production by vortex stretching is exactly balanced by the leading order part of the viscous destruction term. In the present study, the effect of allowing for unbalanced vortex stretching is explored in an effort to describe departures from equilibrium. It is found that the presence of a small unbalanced vortex stretching term has a number of profound consequences for the calculation of isotropic decay, homogeneous shear flow, and more complex turbulent shear flows with separation. In the case of isotropic decay it accounts for enstrophy blow-up in the limit of zero viscosity, while for homogeneous shear flow it predicts a production-equals- dissipation equilibrium at large times instead of an unbounded exponential growth of turbulent kinetic energy. Preliminary calculations for turbulent flow over a backward facing step indicate that even a minute imbalance in vortex stretching can have a major influence on the reattachment length.

Turbulence modeling remains a crucial component for the calculation of complex aerodynamic flows. Limitations in computer capacity, for now and the foreseeable future, point to the need for the development of improved turbulence models if the complex turbulent flows of technological interest are to be properly predicted. The turbulence modeling group considered a wide range of topics in turbulence modeling that include Reynolds stress models of the two-equation and second-order closure type, subgrid scale models for large-eddy simulations, and transitional models. The members of the group consisted of: P. S. Bernard, University of Maryland A. 0. Demuren, Old Dominion University L. D. Kral, McDonnell Douglas C. G. Speziale, ICASE S. Thangam, Stevens Institute of Technology Z. Yang, NASA Lewis Close interactions were also maintained with members of the simulation and turbulence theory groups.

Various papers on turbulence are presented. Individual topics addressed include: modeling the dissipation rate in rotating turbulent flows, mapping closures for turbulent mixing and reaction, understanding turbulence in vortex dynamics, models for the structure and dynamics of near-wall turbulence, complexity of turbulence near a wall, proper orthogonal decomposition, propagating structures in wall-bounded turbulence flows. Also discussed are: constitutive relation in compressible turbulence, compressible turbulence and shock waves, direct simulation of compressible turbulence in a shear flow, structural genesis in wall-bounded turbulence flows, vortex lattice structure of turbulent shear slows, etiology of shear layer vortices, trilinear coordinates in fluid mechanics.

A recently developed K-tau model for near wall turbulent flows is applied to a variety of test cases. The turbulent flows considered include the incompressible flat plate boundary layer with adverse pressure gradients, incompressible flow past a backward facing step, and the supersonic flat plate boundary layer at zero pressure gradient. Calculations are performed for this two-equation model using an anisotropic as well as isotropic eddy-viscosity. The model predictions are shown to compare quite favorably with experimental data.

An eddy viscosity model based on the Renormalization Group (RNG) theory of Yakhot and Orszag [J. Sci. Computing, 1(1):3–51, 1986] is applied to the large-eddy simulation of transition in a flat-plate boundary layer. The simulation predicts with satisfactory accuracy the mean velocity and Reynolds stress profiles, as well as the development of the important scales of motion. The evolution of the structures characteristic of the nonlinear stages of transition is also predicted reasonably well.

The effect of rigid body rotations on the structure of turbulence correlations for incompressible two-dimensional turbulent flow is examined. It is proven that the usual turbulence correlations that are constructed from the fluctuating velocity are invariant under rigid body rotations of the fluid while those that are constructed from the fluctuating pressure are not. An explicit transformation rule for rigid body rotations is developed for an important class of turbulence correlations that are built up from the fluctuating pressure. It is shown how these results can serve as a powerful tool in the development of turbulent closure models that are suitable for the study of atmospheric turbulence in the stratosphere.

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri=-2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.

The origin of turbulent secondary flow in pipes of noncircular cross section is examined from a theoretical standpoint. It is proven mathematically that secondary flows result from a nonzero difference in the normal Reynolds stresses on planes perpendicular to the axial flow direction. Furthermore, it is shown that the K-ϵ model of turbulence has no natural mechanism for the development of secondary flow while the currently popular second-order closure models do. The implications that this has on turbulence modeling are discussed briefly.

The effect of an arbitrary change of frame on the structure of turbulence models is examined from a theoretical standpoint. It is proven, as a rigorous consequence of the Navier-Stokes equations, that turbulence models must be form invariant under arbitrary translational accelerations of the reference frame and should only be affected by rotations through the intrinsic mean vorticity. A direct application of this invariance property along with the Taylor-Proudman theorem, material frame-indifference in the limit of two-dimensional turbulence, and Rapid Distortion Theory is shown to yield powerful constraints on the allowable form of turbulence models. Most of the commonly used turbulence models are demonstrated to be in violation of these constraints and consequently are inconsistent with the Navier-Stokes equations in noninertial frames. Alternative models with improved noninertial properties are developed and some simple applications to rotating turbulent flows are considered.

The consistency of the problem of Reynolds stress closure in turbulence with the fundamental principles of modern continuum mechanics is examined. It is shown that Reynolds stress closure is completely consistent with the principles of determinism and material frame-indifference of modern continuum mechanics. The consequences that this has on turbulence modeling are briefly discussed.

The modeling of the pressure gradient–velocity correlation of turbulence is considered. Two distinctly different approaches have been proposed in the turbulence literature: one in which the pressure gradient–velocity correlation is decomposed into a pressure‐strain correlation and a pressure‐diffusion correlation, and another in which the pressure gradient–velocity correlation is split into its deviatoric and isotropic parts. By examining the limit of two‐dimensional turbulence, it is demonstrated that the models obtained from the former approach are inconsistent with the Navier–Stokes equations in a fundamental way, whereas the models obtained from the latter approach are not. Consequently, it appears that the direct modeling of the pressure gradient–velocity correlation in its deviatoric and isotropic parts should be favored. The implications that this result has on turbulence modeling are discussed briefly.

In the present work a solver for the Reynolds averaged compressible Navier-Stokes equations, to compute high speed turbulent flows characterized by interacting shock waves and viscous layers, is presented. A k-epsilon turbulence model that accounts for compressibility effects is developed. The numerical algorithm is based on a finite volume multistage Runge Kutta technique that is explicit for the solution of the mean flow variables, and implicit for the solution of the k-epsilon equations. The model is validated by extensive comparison with experimental results of flows over compression ramps characterized by interacting shock waves/boundary layers.

The invariance of second moment turbulent closure under a change of frame is examined. It is shown that the Reynolds stresses and the higher turbulence correlations based on an ensemble mean are frame‐indifferent while the Reynolds stress transport equations are frame dependent. As a result of this incompatibility, second moment closure cannot form a general foundation for the study of turbulence. Alternative approaches that are properly invariant are discussed.

The effect of superimposed rigid body motions on the structure of two‐dimensional turbulence is examined. It is found that with regard to the fluctuation dynamics of the flow, the rotational behavior of two‐dimensional turbulence is quite different from its three‐dimensional counterpart. The implications that this has on turbulence modeling are discussed briefly.

The modelling of the subgrid-scale stresses in the large-eddy simulation of turbulence is examined from a theoretical standpoint. While there are a variety of approaches that have been proposed, it is demonstrated that one of the more recent models gives rise to equations of motion for the large eddies of turbulence which are not Galilean-invariant. Consequently, this model cannot be of any general applicability, since it is inconsistent with the basic physics of the problem, which requires that the description of the turbulence be the same in all inertial frames of reference. Alternative models that have been proposed which are properly invariant are discussed and compared.