Direct assessment of Kolmogorov's first refined similarity hypothesis

Using volumetric velocity data from a turbulent laboratory water flow and numerical simulations of homogeneous, isotropic turbulence, we present a direct experimental and numerical assessment of Kolmogorov's first refined similarity hypothesis based on three-dimensional measurements of the local energy dissipation rate ${\epsilon }_r$ measured at dissipative scales $r$. We focus on the properties of the stochastic variables $V_L=\mathrm{\Delta }u\left(r\right)/{\left(r{\epsilon }_r\right)}^{1/3}$ and $V_T=\mathrm{\Delta }v\left(r\right)/{\left(r{\epsilon }_r\right)}^{1/3}$, where $\mathrm{\Delta }u\left(r\right)$ and $\mathrm{\Delta }v\left(r\right)$ are longitudinal and transverse velocity increments. Over one order of magnitude of scales $r$ within the dissipative range, the distributions of $V_L$ and $V_T$ from both experiment and simulation collapse when parametrized by a suitably defined local Reynolds number, providing conclusive experimental evidence in support of the first refined similarity hypothesis and its universality.

Figure 1

Scaling of (a) longitudinal and (b) transverse velocity increment magnitude, for fixed scale $r/\eta$, given the local Reynolds number ${\text{Re}}_r$. Symbols $◃, ▹, \times , +, \star , \square , \diamond , ▿, ▵, ○$ show experimental data at ten scales $r$ logarithmically spaced between $1.5\eta$ and $36.2\eta$. Solid lines show data from DNS at comparable scales. Inset: Conditional average magnitudes of $\langle \left|V_L\right||{\text{Re}}_r\rangle$ and $\langle \left|V_T\right||{\text{Re}}_r\rangle$. (a) Longitudinal velocity increment. (b) Transverse velocity increment.

Figure 2

Conditional distribution of dimensionless longitudinal increments $V_L$ (left panels) and transverse increments $V_T$ (right panels) at different scales $r/\eta$ given a fixed local Reynolds number. In each pair of panels, the local Reynolds number, number of experimental curves, and minimum and maximum scales $r/\eta$ are respectively (a), (b) 14.6, 5, 4.3, and 17.8, (c), (d) 38.1, 4, 8.8, and 25.4, and (e), (f) 99.5, 3, 17.8, and 36.2. The local Reynolds number at each scale is matched to within $5\%$ of the nominal value. Lines show (- -) experimental and (—) DNS data. Symbols and color denote scale $r/\eta$. Markers are as in Fig. 1.