On helicity fluctuations and the energy cascade in turbulence

Recent conjectures concerning the correlation between regions of high local helicity and low dissipation are examined from a rigorous theoretical standpoint based on the Navier-Stokes equations. It is proven that only the solenoidal part of the Lamb vector omega x u (which is directly tied to the nonlocal convection and stretching of vortex lines) contributes to the energy cascade in turbulence. Consequently, it is shown that regions of low dissipation can be associated with either low or high helicity, a result which disproves earlier speculations concerning this direct connection between helicity and the energy cascade. Some brief examples are given along with a discussion of the consistency of these results with the most recent computations of helicity fluctuations in incompressible turbulent flows

On the advantages of the vorticity-velocity formulation of the equations of fluid dynamics

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

On the freestream matching condition for stagnation point turbulent flows

The problem of plane stagnation point flow with freestream turbulence is examined from a basic theoretical standpoint. It is argued that the singularity which arises from the standard kappa-epsilon model is not due to a defect in the model but results from the use of an inconsistent freestream boundary condition. The inconsistency lies in the implementation of a production equals dissipation equilibrium hypothesis in conjunction with a freestream mean velocity field that corresponds to homogeneous plane strain - a turbulent flow which does not reach such a simple equilibrium. Consequently, the adjustment that has been made in the constants of the epsilon-transport equation to eliminate this singularity is not self-consistent since it is tantamount to artificially imposing an equilibrium structure on a turbulent flow which is known not to have one

Coherent structures

In order to develop more quantitative measures of coherent structures that would have comparative value over a range of experiments, it is essential that such measures be independent of the observer. It is only through such a general framework that theories with a fundamental predictive value can be developed. The triple decomposition phi = bar-phi + phi(c) + phi(r) (where bar-phi is the mean, phi(c) is the coherent part, and phi(r) is the random part of any turbulent field phi) serves this purpose. The equations of motion for the mean and coherent flow fields, based on the triple decomposition, are presented and modeling methods for the time-averaged and phase-averaged Reynolds stress are discussed

Discussion of turbulence modelling: Past and future

The full text of a paper presented at the Whither Turbulence Workshop (Cornell University, March 22-24, 1989) on past and future trends in turbulence modeling is provided. It is argued that Reynolds stress models are likely to remain the preferred approach for technological applications for at least the next few decades. In general agreement with the Launder position paper, it is further argued that among the variety of Reynolds stress models in use, second-order closures constitute by far the most promising approach. However, some needed improvements in the specification of the turbulent length scale are emphasized. The central points of the paper are illustrated by examples from homogeneous turbulence

Second-order closure models for supersonic turbulent flows

Recent work on the development of a second-order closure model for high-speed compressible flows is reviewed. This turbulent closure is based on the solution of modeled transport equations for the Favre-averaged Reynolds stress tensor and the solenoidal part of the turbulent dissipation rate. A new model for the compressible dissipation is used along with traditional gradient transport models for the Reynolds heat flux and mass flux terms. Consistent with simple asymptotic analyses, the deviatoric part of the remaining higher-order correlations in the Reynolds stress transport equations are modeled by a variable density extension of the newest incompressible models. The resulting second-order closure model is tested in a variety of compressible turbulent flows which include the decay of isotropic turbulence, homogeneous shear flow, the supersonic mixing layer, and the supersonic flat-plate turbulent boundary layer. Comparisons between the model predictions and the results of physical and numerical experiments are quite encouraging

Analysis of an RNG based turbulence model for separated flows

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

A simple nonlinear model for the return to isotropy in turbulence

A quadratic nonlinear generalization of the linear Rotta model for the slow pressure-strain correlation of turbulence is developed. The model is shown to satisfy realizability and to give rise to no stable non-trivial equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. The absence of stable non-trivial equilibrium solutions is a necessary condition to ensure that the model predicts a return to isotropy for all relaxational turbulent flows. 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. Direct comparisons are also made with the predictions of the Rotta model and the Lumley model

Predicting equilibrium states with Reynolds stress closures in channel flow and homogeneous shear flow

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

Turbulent separated flow past a backward-facing step: A critical evaluation of two-equation turbulence models

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