Turbulence topologies predicted using large eddy simulations

In this paper, turbulence topologies related to the invariants of the resolved velocity gradient and strain rate tensors are studied based on large eddy simulation. The numerical results presented in the paper were obtained using two dynamic models, namely, the conventional dynamic model of Lilly and a recently developed dynamic nonlinear subgrid scale (SGS) model. In contrast to most of the previous research investigations which have mainly focused on isotropic turbulence, the present study examines the influence of near-wall anisotropy on the flow topologies. The SGS effect on the so-called SGS dissipation of the discriminant is examined and it is shown that the SGS stress contributes to the deviation of the flow topology of real turbulence from that of the ideal restricted Euler flow. The turbulence kinetic energy (TKE) transfer between the resolved and subgrid scales of motion is studied, and the forward and backward scatters of TKE are quantified in the invariant phase plane. Some interesting phenomenological results have also been obtained, including a wing-shaped contour pattern for the density of the resolved enstrophy generation and the near-wall dissipation shift of the peak location (mode) in the joint probability density function of the invariants of the resolved strain rate tensor. The newly observed turbulence phenomenologies are believed to be important and an effort has been made to explain them on an analytical basis.

Keywords:

• Large eddy simulation
• Subgrid-scale stress
• Turbulence topology
• Wall-bounded flow
• Vortex dynamics

Acknowledgments

Support from the National Sciences and Engineering Research Council (NSERC) and constructive comments from the anonymous reviewers are gratefully acknowledged.

Notes

* In the current literature, the quantity P _r = −τ _ij * _ij is referred to in several ways. Traditionally, it has been referred to as the SGS TKE dissipation. The meaning of this traditional term is clearly expressed in the original constant-coefficient (C) Smagorinsky model [35], in which case there is no backscatter since P _r = −τ _ij * _ij = 2ν _{sgs ij ij} = (C )²||³ ≥ 0. However, Pope [34] indicated that it is more appropriate to refer to this term as the rate of TKE production for SGS motion, since it corresponds to an inviscid and inertial TKE transfer process which is distinct from a real viscous dissipation. The rate of the SGS TKE production as represented by P _r is not exactly equal to the SGS dissipation rate. In particular, if P _r < 0, SGS TKE is backscattered to the filtered scale, and this backscattered TKE can then be further redistributed through a variety of mechanisms (such as advection and diffusion) as governed by the TKE transport equation at the filtered scale. In addition, if Pope's suggestion is adopted, the definition of P _r is then analogous to the classical definition of the production term due to the deviatoric part of the Reynolds stress tensor in a Reynolds-averaged Navier–Stokes (RANS) approach, i.e. −⟨ u _i ′ u _j ′⟩ _t *⟨ S _ij ⟩ _t . Here ⟨·⟩ _t represents an ensemble average used for the Reynolds decomposition of an instantaneous flow quantity. Considering the characteristics of P _r as mentioned above, some researchers also refer it to as the local energy flux [36, 21].

^† The state of invariants corresponding to the wall condition must occur at the origin of the R _A –Q _A phase plane (i.e., Q _A |_wall = 0 and R _A |_wall = 0). However, the converse is not necessarily true, which follows readily from the following relation between the invariants of _ij and _ij

from which it is seen that any states in which Q _S = − ²/4 and R _S = /4 are sufficient for ensuring that Q _A = 0 and R _A = 0.