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For homogeneous flows the rapid pressure-strain correlation tensor can be written in terms of the mean velocity gradient and an integral of the two-point correlation tensor. In the present study the two-point correlation tensor is modeled and analytically integrated to obtain the pressure-strain model coefficients, which can then be expressed in terms of closed integrals of the two-point correlation model coefficients. Therefore, deficiencies of the pressure-strain models can be detected and resolved on the two-point correlation level. As an example for the new approach, direct numerical simulations results for a homogeneous shear flow are analyzed in detail and the resulting coefficients are compared to existing model coefficients for linear and nonlinear pressure-strain models.

A nonlinear stress–strain model, derived from the modeled Reynolds stress transport equation, is modified to account for the near wall effects in wall-bounded turbulent flows. Since it is known that wall reflection of the turbulent pressure field modifies the pressure–strain correlation, the approach taken is to introduce a correction to the coefficients in the closure for the pressure–strain correlation purely based on ideas for full Reynolds stress closures. The stress–strain relation is implemented in the context of the k – ϵ model with a variable C μ . Results are presented for plane channel flow and both zero and adverse pressure gradient boundary layers. Favorable results for the anisotropies in the Reynolds stresses are obtained by the new model as validated by comparisons against direct numerical simulation (DNS) and experimental data.

The experimental budgets for the transport equations for heat flux and Reynolds stress are presented for a round turbulent buoyant plume. The pressure correlation terms are deduced as the closing terms and are found to constitute a substantial part of these budgets. Even though a buoyant plume is initiated by buoyancy, it is found that the production of heat flux and Reynolds stress is largely maintained by the mean flow gradients, because the buoyancy production terms are not as large. The results are used to assess the local equilibrium assumption, which implies that the production and destruction terms of the transport equations balance each other. The results are also used to investigate why the mechanical to thermal time scale ratio for a buoyant plume is different than the commonly used value. Finally, some simpler models for the pressure correlation terms, which appear in the heat flux and the Reynolds stress equations, are assessed against those deduced from the experiment.

A closure for the compressible portion of the pressure-strain covariance is developed. It is shown that, within the context of a pressure-strain closure assumption linear in the Reynolds stresses, an expression for the pressure-dilatation can be used to construct a representation for the pressure-strain. Additional closures for the unclosed terms in the Favre–Reynolds stress equations involving the mean acceleration are also constructed. The closures accommodate compressibility corrections depending on the magnitude of the turbulent Mach number, the mean density gradient, the mean pressure gradient, the mean dilatation, and, of course, the mean velocity gradients. The effects of the compressibility corrections on the Favre–Reynolds stresses are consistent with current DNS results. Using the compressible pressure-strain and mean acceleration closures in the Favre–Reynolds stress equations an algebraic closure for the Favre–Reynolds stresses is constructed. Noteworthy is the fact that, in the absence of mean velocity gradients, the mean density gradient produces Favre–Reynolds stresses in accelerating mean flows. Computations of the mixing layer using the compressible closures developed are described. Favre–Reynolds stress closure and two-equation algebraic models are compared to laboratory data for the mixing layer. Experimental data from diverse laboratories for the Favre–Reynolds stresses appears inconsistent and, as a consequence, comparison of the Reynolds stress predictions to the data is not conclusive. Reductions of the kinetic energy and the spread rate are consistent with the sizable decreases seen in these classes of flows.

Direct numerical simulation (DNS) is conducted of a homogeneous turbulent shear flow laden with mono-size particles. The dispersed phase is simulated in the Lagrangian frame and the carrier phase is considered in the Eulerian manner. The coupling between the two phases is `two-way' which allows investigation of the effects of the mass loading ratio and the particle time constant on both phases. A new Reynolds stress model (RSM) is developed based on a `two-fluid' methodology in which both the carrier phase and the dispersed phase are considered in the Eulerian frame. Closures are suggested for the unclosed terms (including the pressure–velocity gradient) which manifest the effects of two-way coupling. The results generated by DNS are used to determine the magnitudes of some of the empirical constants appearing in RSM. The final model predictions for all the components of the fluid, the particle, and fluid-particle Reynolds stresses are assessed via detailed comparisons against DNS data.

A nonlinear stress-strain relation (NLSM) for the turbulence stresses is applied to axisymmetric free shear flows with and without swirl. This relation is an explicit solution to an algebraic Reynolds stress model (ARSIVI). The stress relation is a finite sum of tensor groups depicting various interactions between the mean strain and vorticity fields. Implementation is in the context of a k −e type model. Comparisons are made between flow field predictions obtained with the full Reynolds stress model, the NLSM corresponding to improved and standard ARSMs, the k −e model and experimental data.

Experimental data of Hussein et al. (1993) for the turbulent round jet are used to evaluate individual components of Reynolds stress turbulence models. Models for terms in the Reynolds stress equations are reviewed, with particular emphasis on linear and nonlinear pressure-strain models. Improved coefficients for the Choi and Lumley return-to-isotropy expressions have been developed by the authors. These coefficients are valid for a wider range of flows than the currently used coefficients. Pressure-strain and transport model components are compared to the experimental data for the jet, and agreement is very good, indicating that the models are reasonably correct. Predictions using the linear models are generally as good as those obtained using nonlinear models, indicating that nonlinear models may not be necessary for engineering accuracy for this flow.

An improved algebraic Reynolds stress model is developed from the modeled dynamic equations for the Reynolds stress. The improved model more closely represents the original Reynolds stress model equation than the standard algebraic Reynolds stress model over the range of time scales for the turbulence and mean flow strain field. Quasi‐non‐local convective effects are also included in the formulation. Explicit solutions to the algebraic Reynolds stress model equation set, in the form of nonlinear stress‐strain models, are presented for two and three dimensions.

A theoretical study was conducted to determine the effects of adverse pressure gradient and compressibility in modeling turbulent compressible flows. The zero equation, kinetic-energy/dissipation eddy-viscosity models, and Reynolds stress model predictions are presented and compared with experimental data. It is shown that the effects of compressibility, which include the mass-averaged fluctuation term u j '', the pressure-dilatation term p'∂u l ''/∂x l , and the dilatation dissipation μ(∂u l ''/∂x l ) 2 /p, are important in modeling turbulent compressible flows

The unique problems confronting turbulence modelers and experimenters who wish to directly test closure hypotheses are reviewed. Particular attention is paid to the problems presented by systematic errors in the experimental data. Some of the sources of these errors in thermal and laser-Doppler anemometry techniques for measurement are also reviewed, as are problems common to all techniques. The discussion is illustrated by examples from the authors' own work in axisymmetric jets.

This paper discusses the computation of turbulent backstep flow, focusing on the use of Reynolds stress models. A review of existing backstep calculations shows that these calculations generally underpredict reattachment length, and some calculations produce physically unrealistic behavior. Some of these problems are shown to be related to the commonly used pressure-strain coefficients, particularly the linear return-to-isotropy coefficient, C 1 . A new expression for C 1 that is consistent with both homogeneous shear flow experiments and return-to-isotropy experiments produces reasonable results in the present backstep calculations. Some of the present backstep calculations result in unsteady periodic vortex shedding consistent with experimental evidence. This presents a dilemma in the context of Reynolds averaging, which is discussed.

In this Brief Communication, the nonlinear stress–strain model for three‐dimensional turbulent flows, as given by Taulbee [Phys. Fluids A 4, 11 (1992)], is expanded upon. That relation represents a closed form solution to the algebraic Reynolds stress model equation set which is obtained from the modeled transport equation for the Reynolds stress. The parameter values, which appear in the linear pressure–strain closure, that were suggested by Taulbee to obtain a simplified stress relation for three dimensions, are justified in this Brief Communication. A stress solution is also presented for a wider range of pressure–strain model parameter values.

In the chapter, a nonlinear stress-strain model, derived from the modeled Reynolds stress transport equation, is modified to account for the near wall effects in wall-bounded turbulent flows. Since it is known that wall reflection of the turbulent pressure field modifies the pressure-strain correlation, the approach taken is to introduce a correction to the coefficients in the closure for the pressure-strain correlation. The stress-strain relation is implemented in the context of the k – ∈ model with a variable Cμ. Results are presented for plane channel flow, and both zero and adverse pressure gradient boundary layers. Favorable results for the anisotropies in the Reynolds stresses are obtained by the new model as validated by comparisons against direct numerical simulation (DNS) and experimental data.

Two physical phenomena have been the primary subject of investigation: (1) multiphase transport in turbulence, (2) realistic chemistry in large scale numerical simulation of turbulent combustion. In addition, two other phenomena have also been considered: (3) scalar mising in turbulence, and (4) magnetohydrodynamic turbulence. This Final Report provides a summary of our accomplishments in research on each of the above four problems.

The gravitational settling of monodisperse particles from fully developed laminar flow in a horizontal curved tube is considered. The particle size is small compared with the tube radius, and their concentration is small so that they do not interact with each other or affect the flow. It was found that for sufficiently heavy particles, or for tubes with very small curvature, or for very small flow Reynolds numbers, the settling to the tube surface versus axial distance is much the same as that for a straight tube. However, for the opposite size of these parameters, the settling behavior becomes much different than that of a straight tube. Particles are maintained in the flow in helical spiral motions and do not settle to the tube surface.

The stability of a streaming layer of incompressible, inviscid, self-gravitational fluid bounded in an infinite similar medium is investigated. The critical wave numbers are found when the layer has a linear streaming velocity profile.

Finally, we should mention a problem that can arise with the heat-flux probe when determining Qconvective by means of Eq. (2). It is difficult to know whether Qcooiant actually remains constant, or whether during a test the coolant absorbs heat before it reaches the platinum-film sensor. This is minimized by keeping the exposed glass tube as short as possible on the inlet side and by carefully designing the supporting tube bundles. However, operating experience indicates that Qcooiant does vary and that this probably contributes to the errors noted in Fig. 4.

Predictions using boundary-layer theory with a low-Reynolds number kinetic-energy/dissipation turbulence model are reported for the unsteady heat-flux distributions on the rotor blade surfaces. With an assumed sinusoidal variation for the unsteady surface pressure depicting the effects of convected wake segments, unsteady inviscid-freestream velocity and total enthalpy distributions are determined. Unsteady variations in freestream turbulence due to the wake passings are also considered. The unsteady boundary-layer equations, subject to these freestream conditions, are solved numerically. It was found that most of the unsteady variations in surface heat transfer are due to unsteady pressure changes and not variations in the freestream turbulence. Calculations are made for the Garrett TFE 731-2 HP turbine and the magnitudes of the heat-transfer variations are compared with the measurements of Dunn and Seymour.