Local topology via the invariants of the velocity gradient tensor within vortex clusters and intense Reynolds stress structures in turbulent channel flow

The invariants of the velocity gradient tensor,$R$and$Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results are analysed at different scales. We show that the$R$–$Q$plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the instantaneous behaviour of the flow since they are the result of an averaging process where the mean is 3–5 times smaller than the corresponding standard deviation. The orbital periods in the$R$–$Q$plane are shown to be independent of the intensity of the events, and of the same order of magnitude as those in the enstrophy/enstrophy-production and strain/strain-production planes. Our final goal is to test whether the dynamics of the flow is self-similar in the inertial range, and the answer turns out to be that it is not. The mean shear is found to be responsible for the absence of self-similarity and progressively controls the dynamics of the eddies observed as the filter width increases. However, a self-similar behaviour emerges when the calculations are repeated for the fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms of vortex stretching is considered by computing the alignment of the vorticity at a given scale with the strain at a different one. These results generally support a non-negligible role of the phenomenological energy-cascade model formulated in terms of vortex stretching.

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Comparative study of Reynolds stress budgets of thermally and calorically perfect gases for high-temperature supersonic turbulent channel flow

Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering ◽

10.1177/0954410018818340 ◽

2018 ◽

Vol 233 (11) ◽

pp. 4222-4234 ◽

Cited By ~ 2

Author(s):

Xiaoping Chen ◽

Hua-Shu Dou ◽

Qi Liu ◽

Zuchao Zhu ◽

Wei Zhang

Keyword(s):

High Temperature ◽

Reynolds Stress ◽

Channel Flow ◽

Velocity Gradient ◽

Vibrational Energy ◽

Turbulent Channel Flow ◽

Mean Flow ◽

Perfect Gas ◽

Critical Position ◽

Stress Budgets

To study the Reynolds stress budgets, direct numerical simulations of high-temperature supersonic turbulent channel flow for thermally perfect gas and calorically perfect gas are conducted at Mach number 3.0 and Reynolds number 4800 combined with a dimensional wall temperature of 596.30 K. The reliability of the direct numerical simulation data is verified by comparison with previous results ( J Fluid Mech 1995, vol. 305, pp.159–183). The effects of variable specific heat are important because the vibrational energy excited degree exceeds 0.1. The viscous diffusion, pressure–velocity gradient correlation, and dissipation terms in the Reynolds stress budgets for TPG, except the streamwise component, are larger than those for calorically perfect gas close to the wall. Compressibility-related term decreases when thermally perfect gas is considered. The major difference for both gas models is mainly due to variations in mean flow properties. Inter-component transfer related to pressure–velocity gradient correlation term can be distinguished into inner and outer regions, whose critical position is approximately 16 for both gas models.

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Dynamics of a low Reynolds number turbulent boundary layer

Journal of Fluid Mechanics ◽

10.1017/s002211209900720x ◽

2000 ◽

Vol 404 ◽

pp. 87-115 ◽

Cited By ~ 63

Author(s):

JUAN M. CHACIN ◽

BRIAN J. CANTWELL

Keyword(s):

Boundary Layer ◽

Kinetic Energy ◽

Turbulent Boundary Layer ◽

Turbulent Kinetic Energy ◽

Reynolds Stress ◽

Velocity Gradient ◽

Rapid Change ◽

Joint Probability ◽

Gradient Tensor ◽

Velocity Gradient Tensor

The generation of Reynolds stress, turbulent kinetic energy and dissipation in the turbulent boundary layer simulation of Spalart (1988) is studied using the invariants of the velocity gradient tensor. This technique enables the study of the whole range of scales in the flow using a single unified approach. In addition, it also provides a rational basis for relating the flow structure in physical space to an appropriate statistical measure in the space of invariants. The general characteristics of the turbulent motion are analysed using a combination of computer-based visualization of flow variables together with joint probability distributions of the invariants. The quantities studied are of direct interest in the development of turbulence models. The cubic discriminant of the velocity gradient tensor provides a useful marker for distinguishing regions of active and passive turbulence. It is found that the strongest Reynolds-stress and turbulent-kinetic-energy generating events occur where the discriminant has a rapid change of sign. Finally, the time evolution of the invariants is studied by computing along particle paths in a Lagrangian frame of reference. It is found that the invariants tend to evolve toward two distinct asymptotes in the plane of invariants. Several simplified models for the evolution of the velocity gradient tensor are described. These models compare well with several of the important features observed in the Lagrangian computation. The picture of the turbulent boundary layer which emerges is consistent with the ideas of Townsend (1956) and with the physical picture of turbulent structure set forth by Theodorsen (1955).

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Topology of fine-scale motions in turbulent channel flow

Journal of Fluid Mechanics ◽

10.1017/s0022112096001802 ◽

1996 ◽

Vol 310 ◽

pp. 269-292 ◽

Cited By ~ 171

Author(s):

Hugh M. Blackburn ◽

Nagi N. Mansour ◽

Brian J. Cantwell

Keyword(s):

Strain Rate ◽

Channel Flow ◽

Velocity Gradient ◽

Outer Region ◽

Turbulent Channel Flow ◽

Principal Strain ◽

Wall Region ◽

Fine Scale ◽

Stable Focus ◽

Topological Features

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

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