In high-Reynolds-number turbulence the spatial distribution of velocity fluctuation at small scales is strongly non-uniform. In accordance with the non-uniformity, the distributions of the inertial and viscous forces are also non-uniform. According to direct numerical simulation (DNS) of forced turbulence of an incompressible fluid obeying the Navier–Stokes equation in a periodic box at the Taylor microscale Reynolds number $R_\lambda \approx 1100$, the average $\langle R_{loc}\rangle$ over the space of the ‘local Reynolds number’ $R_ {loc}$, which is defined as the ratio of inertial to viscous forces at each point in the flow, is much smaller than the conventional ‘Reynolds number’ given by $Re \equiv UL/\nu$, where $U$ and $L$ are the characteristic velocity and length of the energy-containing eddies, and $\nu$ is the kinematic viscosity. While both conditional averages of the inertial and viscous forces for a given squared vorticity $\omega ^{2}$ increase with $\omega ^{2}$ at large $\omega ^{2}$, the conditional average of $R_ {loc}$ is almost independent of $\omega ^{2}$. A comparison of the DNS field with a random structureless velocity field suggests that the increase in the conditional average of $R_ {loc}$ with $\omega ^{2}$ at large $\omega ^{2}$ is suppressed by the Navier–Stokes dynamics. Something similar is also true for the conditional averages for a given local energy dissipation rate per unit mass. Certain features of intermittency effects such as that on the $Re$ dependence of $\langle R_{loc}\rangle$ are explained by a multi-fractal model by Dubrulle (J. Fluid Mech., vol. 867, 2019, P1).

This paper presents an extension of Kolmogorov's local similarity hypotheses of turbulence to include the influence of mean shear on the statistics of the fluctuating velocity in the dissipation range of turbulent shear flow. According to the extension, the moments of the fluctuating velocity gradients are determined by the local mean rate of the turbulent energy dissipation $\left \langle \epsilon \right \rangle$ per unit mass, kinematic viscosity $\nu$ and parameter $\gamma \equiv S (\nu /\left \langle \epsilon \right \rangle )^{1/2}$, provided that $\gamma$ is small in an appropriate sense, where $S$ is an appropriate norm of the local gradients of the mean flow. The statistics of the moments are nearly isotropic for sufficiently small $\gamma$, and the anisotropy of moments decreases approximately in proportion to $\gamma$. This paper also presents a report on the second-order moments of the fluctuating velocity gradients in direct numerical simulations (DNSs) of turbulent channel flow (TCF) with the friction Reynolds number $Re_\tau$ up to $\approx 8000$. In the TCF, there is a range $y$ where $\gamma$ scales approximately $\propto y^ {-1/2}$, and the anisotropy of the moments of the gradients decreases with $y$ nearly in proportion to $y^ {-1/2}$, where $y$ is the distance from the wall. The theoretical conjectures proposed in the first part are in good agreement with the DNS results.

We measured the correlation spectrum of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number up to by a newly developed pressure probe with spatial and temporal resolutions that are sufficient to analyse inertial-subrange statistics. The influence of the mean velocity gradient tensor in the mixing layer, which is almost constant near its centreline, is studied using an idea similar to that underlying the linear response theory developed in statistical mechanics for systems at or near thermal equilibrium. If we write the spectrum as , where is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small the deviation due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to , and the mean energy dissipation rate. We also measured the pressure–velocity and velocity–velocity correlation spectra. Deviations from isotropy due to shear are shown to be approximately proportional to at large .

The evolution of a small but finite three-dimensional disturbance on a flat uniform vortex sheet is analysed on the basis of a Lagrangian representation of the motion. The sheet at time t is expanded in a double periodic Fourier series: R(λ1, λ2, t) = (λ1, λ2, 0) + Σn,mAn,m exp[i(nλ1 + δmλ2)], where λ1 and λ2 are Lagrangian parameters in the streamwise and spanwise directions, respectively, and δ is the aspect ratio of the periodic domain of the disturbance. By generalizing Moore's analysis for two-dimensional motion to three dimensions, we derive evolution equations for the Fourier coefficients An,m. The behaviour of An,m is investigated by both numerical integration of a set of truncated equations and a leading-order asymptotic analysis valid at large t. Both the numerical integration and the asymptotic analysis show that a singularity appears at a finite time tc = O(lnε−1) where ε is the amplitude of the initial disturbance. The singularity is such that An,0 = O(tc−1) behaves like n−5/2, while An,±1 = O(εtc) behaves like n−3/2 for large n. The evolution of A0,m(spanwise mode) is also studied by an asymptotic analysis valid at large t. The analysis shows that a singularity appears at a finite time t = O(ε−1) and the singularity is characterized by A0,2k ∝ k−5/2 for large k.

The drag on a sphere in a random array of fixed spheres of volume concentration c ([Lt ] 1) at Reynolds number R ([Lt ] 1) is discussed. In the case when c and R2 are of the same order of magnitude, the drag is determined up to terms of O(R).

A method of renormalized expansions in theory of turbulence is developed with the use of the Lagrangian position function. The introduction of this function makes it easy to express the Lagrangian development of the velocity field. A simple truncation of a set of renormalized expansions is shown to lead to an approximation which is compatible with Kolmogorov's inertial range energy spectrum.

The forces on two spherical particles moving in a fluid are investigated by the method of matched asymptotic expansions in the small Reynolds number, for the case when the particles are within each other's inner region of expansion. The particular case in which the distance l between the sphere centres is very much larger than the sphere radii a and b is studied in detail. The asymptotic expansion of the force on one of the spheres for small a/l and b/l is obtained. Some properties of the force, not to be expected from the Stokes equation, are revealed.

A spectral approximation for diffusion of passive scalar in stably and strongly stratified turbulence is presented. The approximation is based on a linearized approximation for the Eulerian two-time correlation and Corrsin's conjecture for the Lagrangian two-time correlation. For strongly stratified turbulence, the vertical component of the turbulent velocity field is well approximated by a collection of Fourier modes (waves) each of which oscillates with a frequency depending on the direction of the wavevector. The proposed approximation suggests that the phase mixing among the Fourier modes having different frequencies causes the decay of the Lagrangian two-time vertical velocity autocorrelation, and the highly oscillatory nature of these modes results in the suppression of single-particle dispersion in the vertical direction. The approximation is free from any ad hoc adjusting parameter and shows that the suppression depends on the spectra of the velocity and fluctuating density fields. It is in good agreement with direct numerical simulations for strongly stratified turbulence.