Small-scale universality in fluid turbulence

Abstract

Turbulent flows in nature and technology possess a range of scales. The largest scales carry the memory of the physical system in which a flow is embedded. One challenge is to unravel the universal statistical properties that all turbulent flows share despite their different large-scale driving mechanisms or their particular flow geometries. In the present work, we study three turbulent flows of systematically increasing complexity. These are homogeneous and isotropic turbulence in a periodic box, turbulent shear flow between two parallel walls, and thermal convection in a closed cylindrical container. They are computed by highly resolved direct numerical simulations of the governing dynamical equations. We use these simulation data to establish two fundamental results: (i) at Reynolds numbers Re ∼ 10(2) the fluctuations of the velocity derivatives pass through a transition from nearly Gaussian (or slightly sub-Gaussian) to intermittent behavior that is characteristic of fully developed high Reynolds number turbulence, and (ii) beyond the transition point, the statistics of the rate of energy dissipation in all three flows obey the same Reynolds number power laws derived for homogeneous turbulence. These results allow us to claim universality of small scales even at low Reynolds numbers. Our results shed new light on the notion of when the turbulence is fully developed at the small scales without relying on the existence of an extended inertial range.

Keywords: energy dissipation rate; fluid dynamics.

Fig. 1.

Three turbulent flows and structure of energy dissipation field. (A) Homogeneous and isotropic turbulence in a cube of length L on the side with periodic boundary conditions in all three space directions. Statistical homogeneity is present in all directions. (B) Turbulent shear flow in a channel of height 2H with solid walls at the top and bottom and periodic boundaries in the horizontal directions. The unidirectional mean flow in the downstream direction is indicated and homogeneity is sustained in both horizontal directions. (C) Cylindrical convection cell of height H and radius R with isothermal hot bottom and cold top plate. The cell of unit aspect ratio is given by V = {(r, ϕ, z):0 ≤ r/R ≤ 0.5; 0 ≤ z/H ≤ 1}. The mean flow, a 3D large-scale circulation, is shown schematically (see description of Fig. 2B). The convective flow in the cylindrical cell is statistically homogeneous in the azimuthal direction only. (D) A slice (parallel to one of the walls) of the instantaneous kinetic energy dissipation rate field for Re = 5,587 in box turbulence (10). The box with a side length L = 2π was resolved with 2,048³ equidistant grid points. (E) Kinetic energy dissipation rate in the midplane of a channel flow for Re = 1,160. The simulation required 2,048 × 2,049 × 1,024 points for a channel with spatial extent 2πH × 2H × πH. (F) Same quantity in the midplane of a convection cell at Re = 4,638. The convection cell of unit aspect ratio is covered by 875,520 spectral elements, each containing 12³ Gauss–Lobatto–Legendre collocation points.

Fig. 2.

Global flow conditions in inhomogeneous convective turbulence. (A) Reynolds number vs. Rayleigh number. Reynolds numbers are calculated using the full velocity field, u_i, and velocity fluctuations, v_i. The corresponding power law fits are shown as dashed lines. (B) Visualization of the 3D large-scale mean flow ${\overline{u}}_i\left(x_j\right)$ that is obtained by time averaging at Ra = 10⁷.

Fig. 3.

Transition of velocity gradient statistics from Gaussian to super-Gaussian. Data are for homogeneous isotropic turbulence (HIT) and Rayleigh–Bénard convection (RBC). (A) Probability density functions of the longitudinal velocity derivative ∂v_x/∂x normalized by the corresponding root-mean-square at four different Reynolds numbers are displayed, z = ∂v_x/∂x/(∂v_x/∂x)_rms. The Gaussian distribution is added as a dashed line for comparison. (B) Skewness $\overline{z^3}/{\overline{z}}^3$ of the longitudinal derivative z vs. Reynolds number Re (Eq. 3). (C) Same as A. Now four examples for the convection case are shown. (D) Flatness $\overline{z^4}/{\overline{z}}^4$ of the longitudinal derivative. In B and D, dashed lines are added, which indicate the skewness of zero and the flatness of three, respectively, which would hold for a Gaussian field. Velocity derivatives for the RBC analysis have been obtained in a bulk volume V_b ⊂ V with V_b = {(r, ϕ, z):0 ≤ r/R ≤ 0.3; 0.2 ≤ z/H ≤ 0.8}. Velocity derivatives in isotropic turbulence have been collected in the whole volume.

Fig. 4.

Universality of the energy dissipation statistics. The Reynolds number dependence of normalized moments of orders n = 2, 3, and 4 of the energy dissipation rate, $\overline{{\mathit{ϵ}}^n}/{\overline{\mathit{ϵ}}}^n$, is compared for three turbulent flows. Black asterisks and white open triangles are for homogeneous isotropic turbulence data of refs. and , respectively. Blue open circles are for turbulent channel flow (13) and red solid triangles are for turbulent Rayleigh–Bénard convection. Dashed lines correspond to a theoretical prediction by Yakhot (6) for the case of homogeneous isotropic turbulence (Fig. 1A). Datasets for different flows have been shifted vertically by constant factors to collapse the data records for the different orders. Data for RBC have been collected in subvolume V_b inside the cell.