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Maxwell models for the flow of viscoelastic fluids are not usually computable unless the dimensionless relaxation time is small. Problems with numerical instabilities arise. A new approach is presented that alleviates this problem. In this new approach, a viscous term is added which has the effect of regularizing the model. By an appeal to the kinetic theory of gases and turbulence, it is shown that this addition can be formally justified. The regularization comes from changing the constitutive equation type from purely hyperbolic to a mix that is largely parabolic and makes the model more computable. This can also alleviate the problem with specifying boundary conditions. The prospects for future research in viscoelastic fluids are thoroughly discussed.

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 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.

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.

A two-equation turbulence model of the K-epsilon type was recently derived by using Renormalization Group (RNG) methods. It was later reported that this RNG based model yields substantially better predictions than the standard K-epsilon model for turbulent flow over a backward facing step - a standard test case used to benchmark the performance of turbulence models in separated flows. The improvements obtained from the RNG K-epsilon model were attributed to the better treatment of near wall turbulence effects. In contrast to these earlier claims, it is shown in this paper that the original version of the RNG K-epsilon model substantially underpredicts the reattachment point in the backstep problem. This is a deficiency that is traced to the modeling of the production of dissipation term. However, with the most recent improvements in the RNG K-epsilon model, excellent results for the backstep problem are now obtained.

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 invariance of constitutive equations in continuum mechanics is examined from a basic theoretical standpoint. It is demonstrated the constitutive equations which are not form invariant under arbitrary translational accelerations of the reference frame are in violation of the Einstein equivalane principle. Furthermore, by making use of an analysis based on statistical mechanics, it is argued that any frame-dependent terms in constitutive equations must arise from the intrinsic spin tensor and are negligible provided that the ratio of microscopic to macroscopic time scales is extremely small. The consistency of these results with existing constitutive theories is discussed in detail along with possible avenues of future research.

A surface functional approximation to the general theory of nonlocal Stokesian fluids is obtained. The relationship between this theory and the existing phenomenological theories of turbulence is demonstrated. The theory is then applied to the problem of turbulent channel flow. An exact solution is obtained for the mean velocity and turbulent stresses which is shown to he consistent with existing experimental data.

The usefulness of the method of quadrature by differentiation for solving eigenvalue problems in hydrodynamic stability is demonstrated. A numerical example, the Taylor problem, is presented in order to show the efficacy of the method and provide a basis of comparison with other approximate methods (e.g. the Galerkin method). The results were found to be in good agreement with experimental data and it was demonstrated that the method of quadrature by differentiation in comparison with other analytical methods requires no trial and error, no extensive mathematical investigation, and no lengthy computer computations.

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 mathematical properties of the pressure-velocity and vorticity-velocity formulations of the equations of viscous flow are compared. It is shown that a vorticity-velocity formulation exists which has the interesting property that non-inertial effects only enter the problem through the implementation of initial and boundary conditions. This valuable characteristic, along with other advantages of the vorticity-velocity approach, are discussed in detail.

The frame-dependence of higher order approximations to the response functions in the kinetic theory of gases is examined. It is shown that this frame-dependence is in no way a characteristic of the kinetic theory but results from defects in the Chapman-Enskog and Maxwellian iterative procedures. A concise proof is given to show that the kinetic theory of gases is consistent with the Principle of Material Frame-Indifference of modern continuum mechanics.

The problem of turbulent flow past a backward facing step is important in many technological applications and has been used as a standard test case to evaluate the performance of turbulence models in the prediction of separated flows. It is well known that the commonly used kappa-epsilon (and K-l) models of turbulence yield inaccurate predictions for the reattachment points in this problem. By an analysis of the mean vorticity transport equation, it will be argued that the intrinsically inaccurate prediction of normal Reynolds stress differences by the kappa-epsilon and K-l models is a major contributor to this problem. Computations using a new nonlinear Kappa-epsilon model (which alleviates this deficiency) are made with the TEACH program. Comparisons are made between the improved results predicted by this nonlinear kappa-epsilon model and those obtained from the linear kappa-epsilon model as well as from second-order closure models.