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The paper reports on an alternative approach to modelling the turbulent scalar fluxes that arise from time averaging the transport equation for a scalar. In this approach, a functional relationship between these fluxes and various tensor quantities is constructed with guidance from the exact equations governing the transport of fluxes. Results from tensor representation theory are then used to obtain an explicit relationship between the fluxes and the terms in the assumed functional relationship. Where turbulence length– and time–scales are implied, these are determined from two scalar quantities: the turbulence kinetic energy and its rate of dissipation by viscous action. The general representation is then reduced by certain justifiable assumptions to yield a practical model for the turbulent scalar fluxes that is explicit and algebraic in these quantities and one that correctly reflects their dependence on the gradients of mean velocity and on the details of the turbulence. Examination of alternative algebraic models shows most to be subsets of the present proposal. The new model is calibrated using results from large–eddy simulations (LESs) of homogeneous turbulence with passive scalars and then assessed by reference to benchmark data from heated turbulent shear flows. The results obtained show the model to correctly predict the anisotropy of the turbulent diffusivity tensor. The asymmetric nature of this tensor is also recovered, but only qualitatively, there being significant quantitative differences between the model predictions and the LES results. Finally, comparisons with data from benchmark two–dimensional free shear flows show the new model to yield distinct improvements over other algebraic scalar–flux closures.

This paper presents a rational approach to modelling the triple velocity correlations that appear in the transport equations for the Reynolds stresses. All existing models of these correlations have largely been formulated on phenomenological grounds and are defective in one important aspect: They all neglect to allow for the dependence of these correlations on the local gradients of mean velocity. The mathematical necessity for this dependence will be domonstrated in the paper. The present contribution lies in the novel use of Group Representation Theory to determine the most general tensorial form of these correlations in terms of all the second- and third-order tensor quantities that appear in the exact equations that govern their evolution. The requisite representation did not exist in the literature and therefore had to be developed specifically for this purpose by Professor G. F. Smith. The outcome of this work is a mathematical framework for the construction of algebraic, explicit, and rational models for the triple velocity correlations that are theoretically consistent and include all the correct dependencies. Previous models are reviewed, and all are shown to be an incomplete subset of this new representation, even to lowest order.

The analysis and modelling of the structure of turbulent flow in a circular pipe subjected to an axial rotation is presented. Particular attention is paid to determining the terms in various turbulence closures that generate the two main physical features that characterize this flow: a rotationally dependent axial mean velocity and a rotationally dependent mean azimuthal or swirl velocity relative to the rotating pipe. It is shown that the first feature is well represented by two-dimensional explicit algebraic stress models but is irreproducible by traditional two-equation models. On the other hand, three-dimensional frame-dependent models are needed to predict the presence of a mean swirl velocity. The latter is argued to be a secondary effect which arises from a cubic nonlinearity in standard algebraic models with conventional near-wall treatments. Second-order closures are shown to give a more complete description of this flow and can describe both of these features fairly well. In this regard, quadratic pressure–strain models perform the best overall when extensive comparisons are made with the results of physical and numerical experiments. The physical significance of this problem and the implications for future research in turbulence are discussed in detail.

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.

An entirely new approach to the large-eddy simulation (LES) of high-speed compressible turbulent flows is presented. Subgrid scale stress models are proposed that are dimensionless functions of the computational mesh size times a Reynolds stress model. This allows a DNS to go continuously to an LES and then a Reynolds-averaged Navier–Stokes (RANS) computation as the mesh becomes successively more coarse or the Reynolds number becomes much larger. Here, the level of discretization is parameterized by the nondimensional ratio of the computational mesh size to the Kolmogorov length scale. The Reynolds stress model is based on a state-of-the-art two-equation model whose enhanced performance is documented in detail in a variety of benchmark flows. It contains many of the most recent advances in compressible turbulence modeling. Applications to the high-speed aerodynamic flows of technological importance are briefly discussed.

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 modelling of anisotropies in the dissipation rate of turbulence is considered based on an analysis of the exact transport equation for the dissipation rate tensor. An algebraic model is systematically derived using integrity bases methods and tensor symmetry properties. The new model differs notably from all previously proposed models in that it depends nonlinearly on the mean velocity gradients. This gives rise to a transport equation for the scalar dissipation rate that is of the same general form as the commonly used model with one major exception: the coefficient of the production term is dependent on the invariants of both the rotational and irrotational strain rates. The relationship between the new model and other recently proposed models is examined in detail. Some basic tests and applications of the model are also provided along with a discussion of the implications for turbulence modelling.

The linear response of turbulence to a distortion and simple rotation is investigated in this paper from a fundamental theoretical standpoint. Quadratic flows are a special case of planar flows with constant mean velocity gradients, which can be characterized by a constant rate of strain D and a constant spanwise (normal to the plane) vorticity component −W=2Ω0, with arbitrary values (here, we can take D>0 and W>0 without any loss of generality). According to the sign of D2−Ω02, streamlines are hyperbolic, rectilinear (pure shear flow) or elliptic. Since these flows can also be considered as mean flows, when superimposing a three-dimensional disturbance (or fluctuating turbulent) field which satisfies statistical homogeneity, the linearized analysis of the disturbance field is of interest both from the point of view of hydrodynamic stability (e.g., the elliptical flow instability) and from the point of view of homogeneous rapid distortion theory (RDT) including applications to the basic statistics. The case of quadratic flow in a rotating frame (with angular velocity Ω in the direction normal to the plane of the basic flow) is revisited in this paper, in order to complete—with the three parameters D, −W=2Ω0 and Ω—previous works on linear theory by Cambon et al. [J. Fluid Mech. 278, 175 (1994)] and Speziale, Abid, and Blaisdell [Phys. Fluids 8, 781 (1996)]. From a simplified “pressure-less” linear approach, a general stability criterion is derived based on the value of the modified Bradshaw number Br=[D2−(2Ω+Ω0)2]/S2 (S=D−Ω0), which coincides with the rotational Richardson number introduced by Bradshaw [J. Fluid Mech. 36, 177 (1969)] and denoted by B=2Ω(S−2Ω)/S2 (in particular, for the case of pure shear flow where D=W/2=−Ω0=S/2). It is shown that this criterion gives results identical to the “true” linear stability analysis (including the effect of the fluctuating pressure) if the absolute vorticity 2Ω+2Ω0 has a zero value. In addition, the relevance of this criterion is checked with respect to the true linear approach in distorted wave space and related RDT applications. For all of the cases, the maximum amplification for the three-dimensional disturbance field is found for zero tilting vorticity 2Ω+Ω0 and for pure spanwise modes (with wave vector normal to the plane of the quadratic flow), in accordance with the generalized Bradshaw criterion and other results in hydrodynamic stability. For other spectral directions, the agreement is not as complete except for the pure shear case, and this is particularly discussed looking at statistical RDT solutions, which involve a summation over all directions of the wave vector. Finally, the impact of the whole analysis on second-order, one-point modeling is discussed.The linear response of turbulence to a distortion and simple rotation is investigated in this paper from a fundamental theoretical standpoint. Quadratic flows are a special case of planar flows with constant mean velocity gradients, which can be characterized by a constant rate of strain D and a constant spanwise (normal to the plane) vorticity component −W=2Ω0, with arbitrary values (here, we can take D>0 and W>0 without any loss of generality). According to the sign of D2−Ω02, streamlines are hyperbolic, rectilinear (pure shear flow) or elliptic. Since these flows can also be considered as mean flows, when superimposing a three-dimensional disturbance (or fluctuating turbulent) field which satisfies statistical homogeneity, the linearized analysis of the disturbance field is of interest both from the point of view of hydrodynamic stability (e.g., the elliptical flow instability) and from the point of view of homogeneous rapid distortion theory (RDT) including applications to the basic statistics. The ca...

Data from free turbulent jets both with and without swirl are used to assess the performance of the pressure-strain model of Speziale, Sarkar and Gatski, which is quadratic in the Reynolds stresses. Comparative predictions are also obtained with the two versions of the Launder, Reece and Rodi model, which are linear in the same terms. All models are used as part of a complete second-order closure based on the solution of differential transport equations for each nonzero component of uiuj together with an equation for the scalar energy dissipation rate. For nonswirling jets, the quadratic model underestimates the measured spreading rate of the plane jet but yields a better prediction for the axisymmetric case without resolving the plane jet/round jet anomaly. For the swirling axisymmetric jet, the same model accurately reproduces the effects of swirl on both the mean flow and the turbulence structure in sharp contrast with the linear models which yield results that are in serious error. The reasons for these differences are discussed.

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.

A new approach for the development of second-order closure models suitable for nonequilibrium turbulent shear flows is presented. The central idea is based on the implementation of a relaxation time approximation to the nonequilibrium extension of an explicit algebraic stress model. Here, the algebraic stress model is systematically derived from the full Speziale et al. 1991 (SSG) second-order closure in the equilibrium limit of homogeneous turbulence. It is then extended to nonequilibrium turbulent flows by means of a Pade approximation, whereby approximate consistency with the rapid distortion theory (RDT) solution for homogeneous shear flow is established. The resulting model is tested in homogeneous shear flow turbulence under a wide range of shear rates. Substantially improved results are obtained in comparison to conventional second-order closure models that are based on the direct modeling of terms in the Reynolds stress transport equation.

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 restricted Euler equation captures many important features of the behavior of the velocity gradient tensor observed in direct numerical simulations (DNS) of isotropic turbulence. However, in slightly more complex flows the agreement is not good, especially in regions of low dissipation. In this paper, it is demonstrated that the Reynolds-averaged restricted Euler equation violates the balance of mean momentum for virtually all homogeneous turbulent flows with only two major exceptions: isotropic and homogeneously-sheared turbulence. A new model equation which overcomes this shortcoming and is more widely applicable is suggested. This model is derived from the Navier-Stokes equation with a restricted Euler type approximation made on the fluctuating velocity gradient field. Analytical solutions of the proposed modified restricted Euler equation appear to be difficult to obtain. Hence, a strategy for numerically calculating the velocity gradient tensor is developed. Preliminary calculations tend to indicate that the modified restricted Euler equation captures many important aspects of the behavior of the fluctuating velocity gradients in anisotropic homogeneous turbulence.

It is shown that realizability of second-moment turbulence closure models can be established by finding a Langevin equation for which they are exact. All closure models currently in use can be derived formally from the type of Langevin equation described herein. Under certain circumstances a coefficient in that formalism becomes imaginary. The regime in which models are realizable is, at least, that for which the coefficient is real. The present method does not imply unrealizable solutions when the coefficient is imaginary, but it does guarantee realizability when the coefficient is real; hence, this method provides sufficient, but not necessary, conditions for realizability. Illustrative computations of homogeneous shear flow are presented. It is explained how models can be modified to guarantee realizability in extreme non-equilibrium situations without altering their behaviour in the near-equilibrium regime for which they were formulated.

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.

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

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.

Reynolds stress calculations of homogeneous turbulent shear flow are conducted with a second-order closure model modified to account for non-equilibrium vortex stretching in the dissipation rate transport equation, as recently proposed by Bernard and Speziale. As with the earlier reported k-epsilon model calculations incorporating this vortex stretching effect, a production-equals-dissipation equilibrium is obtained with bounded turbulent kinetic energy and dissipation. However, this equilibrium is not achieved until the dimensionless time greater than 60, an elapsed time that is at least twice as large as any of those considered in previous numerical and physical experiments on homogeneous shear flow. Direct quantitative comparisons between the model predictions and the results of experiments are quite favorable. In particular, it is shown that the inclusion of this non-equilibrium vortex stretching effect has the capability of explaining the significant range of production to dissipation ratios observed in experiments.

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.

Equations governing the motion of a specific class of singularities of the Euler equation in the extended complex spatial domain are derived. Under some assumptions, it is shown how this motion is dictated by the smooth part of the complex velocity at a singular point in the unphysical domain. These results are used to relate the motion of complex singularities to the stability of steady solutions of the Euler equation. A sufficient condition for instability is conjectured. Several examples are presented to demonstrate the efficacy of this sufficient condition which include the class of elliptical flows and the Kelvin–Stuart cat’s eye.

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.

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 assumption of self-preservation permits an analytical determination of the energy decay in isotropic turbulence. Batchelor (1948), who was the first to carry out a detailed study of this problem, based his analysis on the assumption that the Loitsianskii integral is a dynamic invariant – a widely accepted hypothesis that was later discovered to be invalid. Nonetheless, it appears that the self-preserving isotropic decay problem has never been reinvestigated in depth subsequent to this earlier work. In the present paper such an analysis is carried out, yielding a much more complete picture of self-preserving isotropic turbulence. It is proven rigorously that complete self-preserving isotropic turbulence admits two general types of asymptotic solutions: one where the turbulent kinetic energy K ∼ t−1 and one where K ∼ t−α with an exponent α > 1 that is determined explicitly by the initial conditions. By a fixed-point analysis and numerical integration of the exact one-point equations, it is demonstrated that the K ∼ t−1 power law decay is the asymptotically consistent high-Reynolds-number solution; the K ∼ t−α decay law is only achieved in the limit as t → ∞ and the turbulence Reynolds number Rt vanishes. Arguments are provided which indicate that a t−1 power law decay is the asymptotic state toward which a complete self-preserving isotropic turbulence is driven at high Reynolds numbers in order to resolve an O(R1½) imbalance between vortex stretching and viscous diffusion. Unlike in previous studies, the asymptotic approach to a complete self-preserving state is investigated which uncovers some surprising results.

Turbulence models are developed by supplementing the renormalization group (RNG) approach of Yakhot and Orszag [J. Sci. Comput. 1, 3 (1986)] with scale expansions for the Reynolds stress and production of dissipation terms. The additional expansion parameter (η≡SK/■) is the ratio of the turbulent to mean strain time scale. While low‐order expansions appear to provide an adequate description for the Reynolds stress, no finite truncation of the expansion for the production of dissipation term in powers of η suffices−terms of all orders must be retained. Based on these ideas, a new two‐equation model and Reynolds stress transport model are developed for turbulent shear flows. The models are tested for homogeneous shear flow and flow over a backward facing step. Comparisons between the model predictions and experimental data are excellent.

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.

New subgrid-scale models for the large-eddy simulation of compressible turbulent flows are developed and tested based on the Favre-filtered equations of motion for an ideal gas. A compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model, in terms of Favre-filtered fields, is obtained for the subgrid-scale stress tensor. An analogous thermal linear combination model is also developed for the subgrid-scale heat flux vector. The two dimensionless constants associated with these subgrid-scale models are obtained by correlating with the results of direct numerical simulations of compressible isotropic turbulence performed on a963grid using Fourier collocation methods. Extensive comparisons between the direct and modelled subgrid-scale fields are provided in order to validate the models. A large-eddy simulation of the decay of compressible isotropic turbulence – conducted on a coarse323grid – is shown to yield results that are in excellent agreement with the fine-grid direct simulation. Future applications of these compressible subgrid-scale models to the large-eddy simulation of more complex supersonic flows are discussed briefly.

The equilibrium structure of homogeneous turbulent shear flow is investigated from a theoretical standpoint. Existing turbulence models, in apparent agreement with physical and numerical experiments, predict an unbounded exponential time growth of the turbulent kinetic energy and dissipation rate; only the anisotropy tensor and turbulent time scale reach a structural equilibrium. It is shown that if a residual vortex stretching term is maintained in the dissipation rate transport equation, then there can exist equilibrium solutions, with bounded energy states, where the turbulence production is balanced by its dissipation. Illustrative calculations are presented for a k–ε model modified to account for net vortex stretching. The calculations indicate an initial exponential time growth of the turbulent kinetic energy and dissipation rate for elapsed times that are as large as those considered in any of the previously conducted physical or numerical experiments on homogeneous shear flow. However, vortex stretching eventually takes over and forces a production-equals-dissipation equilibrium with bounded energy states. The plausibility of this result is further supported by independent calculations of isotropic turbulence which show that when this vortex stretching effect is accounted for, a much more complete physical description of isotropic decay is obtained. It is thus argued that the inclusion of vortex stretching as an identifiable process may have greater significance in turbulence modeling than has previously been thought and that the generally accepted structural equilibrium for homogeneous shear flow, with unbounded energy growth, could be in need of re-examination.

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 modelling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - appears to yield improved results over the Launder, Reece & Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in five independent homogeneous turbulent flows. However, some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modelling of the pressure–strain correlation. Possible alternative approaches are discussed briefly.

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.

The pressure driven, fully developed turbulent flow of an incompressible viscous fluid in curved ducts of square cross-section is studied numerically by making use of a finite volume method. A nonlinear K -1 model is used to represent the turbulence. The results for both straight and curved ducts are presented. For the case of fully developed turbulent flow in straight ducts, the secondary flow is characterized by an eight-vortex structure for which the computed flowfield is shown to be in good agreement with available experimental data. The introduction of moderate curvature is shown to cause a substantial increase in the strength of the secondary flow and to change the secondary flow pattern to either a double-vortex or a four-vortex configuration.

The structure of the subgrid scale fields in plane channel flow has been studied at various stages of the transition process to turbulence. The residual stress and subgrid scale dissipation calculated using velocity fields generated by direct numerical simulations of the Navier-Stokes equations are significantly different from their counterparts in turbulent flows. The subgrid scale dissipation changes sign over extended areas of the channel, indicating energy flow from the small scales to the large scales. This reversed energy cascade becomes less pronounced at the later stages of transition. Standard residual stress models of the Smagorinsky type are excessively dissipative. Rescaling the model constant improves the prediction of the total (integrated) subgrid scale dissipation, but not that of the local one. Despite the somewhat excessive dissipation of the rescaled Smagorinsky model, the results of a large eddy simulation of transition on a flat-plate boundary layer compare quite well with those of a direct simulation, and require only a small fraction of the computational effort. The inclusion of non-dissipative models, which could lead to further improvements, is proposed.

A quadratic nonlinear generalization of the linear Rotta model for the slow pressure‐strain correlation of turbulence is developed for high Reynolds number flows. The model is shown to satisfy realizability and to give rise to no stable nonzero equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. In order for any model to predict a return to isotropy for all relaxational flows, it is necessary to ensure that there is no nonzero stable fixed point that attracts realizable initial conditions. Both the phase space dynamics and the temporal behavior of the model are examined and compared against experimental data for the return to isotropy problem. It is demonstrated that the quadratic model successfully captures the experimental trends which clearly exhibit nonlinear behavior. Comparisons are also made with the predictions of the linear Rotta model, the quasilinear Lumley model, and the nonlinear model of Shih, Mansour, and Moin. The simple quadratic model proposed in this study does better than the Rotta model as anticipated, and also compares quite favorably with the other more complicated nonlinear models.

Some aspects of a recent paper on rotating turbulence by Canuto and Dubovikov [Phys. Fluids 9, 2132 (1997)] are examined from historical and scientific perspectives. Their claim to have discovered a new energy spectrum scaling law for rotating turbulence is examined in light of previous publications on this subject. We answer an objection raised to the consistency of this spectral scaling law with the Bardina-type dissipation rate transport equation derived from this law by two of the authors [Phys. Fluids 8, 3172 (1996)]. Finally, some difficulties with the alternative model for the dissipation rate transport equation proposed by Canuto and Dubovikov are described in both the weak and strong rotation limits.

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

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.

Traditional turbulence models invoke a range of equilibrium assumptions that renders them incapable of describing turbulent flows where the departures from equilibrium are large. Even the commonly used second-order closures have an implicit equilibrium assumption for the pivotal pressure-strain correlation that makes them incapable of describing such non-equilibrium turbulent flows. It will be shown that explicit algebraic stress models can be partially extended to non-equilibrium turbulent flows by a Pade approximation. Then, by implementing a relaxation time approximation, second-order closures are obtained where — to the lowest order — the rapid pressure-strain correlation is represented by models that depend nonlinearly on the invariants of the non-dimensional strain rates. However, unlike in many of the more recent second-order closures, linearity is maintained in the Reynolds stress anisotropy tensor consistent with the definition of the rapid pressure-strain correlation. It will be demonstrated by a variety of examples how this leads to an improved performance in non-equilibrium turbulence without compromising the predictions for the near-equilibrium case. A new approach to large-eddy simulations will also be presented that allows subgrid scale stress models to continuously go to Reynolds stress models in the coarse mesh/infinite Reynolds number limit. Furthermore, the modeling of the turbulent dissipation rate will be considered, particularly in regard to the non-equilibrium effects of vortex stretching and anisotropic dissipation. The status of these recent developments and the prospects for future research will be thoroughly discussed.