Reynolds stress and the physics of turbulent momentum transport

Abstract

The nature of the momentum transport processes responsible for the Reynolds shear stress is investigated using several ensembles of fluid particle paths obtained from a direct numerical simulation of turbulent channel flow. It is found that the Reynolds stress can be viewed as arising from two fundamentally different mechanisms. The more significant entails transport in the manner described by Prandtl in which momentum is carried unchanged from one point to another by the random displacement of fluid particles. One-point models, such as the gradient law are found to be inherently unsuitable for representing this process. However, a potentially useful non-local approximation to displacement transport, depending on the global distribution of the mean velocity gradient, may be developed as a natural consequence of its definition. A second important transport mechanism involves fluid particles experiencing systematic accelerations and decelerations. Close to the wall this results in a reduction in Reynolds stress due to the slowing of sweep-type motions. Further away Reynolds stress is produced in spiralling motions, where particles accelerate or decelerate while changing direction. Both transport mechanisms appear to be closely associated with the dynamics of vortical structures in the wall region.

About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity

“…[33,34,35,36,37,38]). As already implicitly assumed by Boussinesq, the basis of such modelling is an analogy with kinetic theory, where the viscous stress is expressed using the gradient of the mean field (see also the comments in [3] ou [5]).…”

“…An interesting direction for further research is provided by non-local models, that take into account the history of turbulence: several models have already been proposed [34,36,38,35,39], and an explicit expression for the kernel of the non-local model has recently been proposed and tested on simple flows [40]. Direct analysis of DNS data, using a Lagrangian approach to follow elements of fluid, could also be a good way to better understand the mechanism inside the formation of the turbulent stress [37,41], and improve the recent proposal of Hamba [40]. Turbulence models have important engineering applications, and thus in this framework, having models whose predictions are only partly in agreement with reality is better than having no agreement at all.…”

On the physical accuracy of scalar transport modeling in inhomogeneous turbulence

“…The time scale in (21) and (22) is indicated as T 22 since this choice is consistent with the premises upon which the random flight models are based. It should be noted, however, that neither this assumption nor the model itself are rigorously derived, so that it is conceivable that other choices for T 22 may improve the performance of the models.…”

“…y (13) However, a formal analysis of momentum transport along the same lines leading to ( 6) shows that the theoretically correct eddy viscosity for momentum transport 21 …”

“…The approach has been applied extensively in previous work to study the rates of momentum 21 and vorticity 22 -23 transport. Subsequently, this was extended 24 to include the effects of vortical structures in causing Reynolds stress.…”