The scaling of straining motions in homogeneous isotropic turbulence

Abstract

The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from Re λ = 34.6 up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, η. The vorticity stretching motions scale with the Taylor length scale, λ T , while the flow outside the shear layer scales with the integral length scale, L. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is 120η in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of 4λ T shows that transitions in flow structure occur where Re λ ≈ 45 and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is 4λ T in width and height, which is consistent with observations in high Reynolds number flow of a 4λ T wide instantaneous shear layer with many η-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895-929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain † Email address for correspondence: g.e.elsinga@tudelft.nl ‡ Present address: Graduate School of Environmental and Life Science, Okayama University, Okayama 700-8530, Japan. 32 G. E. Elsinga, T. Ishihara, M. V. Goudar, C. B. da Silva and J. C. R. Hunt at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

Triple decomposition of velocity gradient tensor in homogeneous isotropic turbulence

“…The DNS was performed with a classical pseudo-spectral method that solves the incompressible Navier-Stokes equations, and the details of the DNS have been described in previous studies. 18,19,20,21 The statistically stationary turbulence is simulated with an artificial forcing that is delta correlated in time and uncorrelated with the velocity field. 22 Herein, the velocity components in the x, y, and z directions are denoted by u, v, and w, respectively.…”

“…The velocity jump and the thickness of the detected shear layer can be related to the extreme events in turbulence whose characteristic length and velocity scales are η and u rms , respectively. 25,28 An average velocity field extracted in the strain eigenframe also exhibited a shear layer whose thickness scales with the Kolmogorov scale, 21…”

Tracer particle dispersion around elementary flow patterns

“…More recent work on flow structures has highlighted the importance of shear-layer structures with coincident vortices (Elsinga & Marusic 2010). These shear layers are associated with intense vorticity and dissipation, and their existence appears furthermore consistent with the k −5/3 scaling of the kinetic energy spectrum (Elsinga & Marusic 2010;Ishihara, Kaneda & Hunt 2013;Hunt et al 2014;Elsinga & Marusic 2016;Elsinga et al 2017). Hence, we examine the particle dispersion in a generic vortex structure, i.e.…”

“…For example, figure 7(d) shows that at t/τ η = 4 some particles have approached the origin, where the velocity goes to zero, while others have reached distances as large as 2η away from the origin, thereby significantly increasing their velocity. Up to 5η distance from the origin of the SLS and the NST, the velocity varies approximately linearly with distance (see Elsinga et al 2017), which means that at 2η distance from the origin the particle velocity has increased approximately fourfold from its initial value when released at 0.5η from the origin. Therefore, the individual particles are subjected to strong changes in their velocity.…”

“…The corresponding particle distributions in figure 7 show that nearly all particles remain within 5η distance from the origin up to t = 10τ η , which means r < 10η. The 10η length scale is the typical size for the core of small-scale flow structures, such as the vortices (Jimenez et al 1993) and the straining motions (Elsinga et al 2017), where the velocity varies linearly in space. Therefore, the particle pairs remain within the core of the same small-scale flow structure, while still displaying approximate t 3 scaling.…”