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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).

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which $E(k\ensuremath{\rightarrow} 0)\ensuremath{\sim} {k}^{2} $, where $E$ is the energy spectrum and $k$ the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor ${\Phi }_{ij} (\mathbi{k})$ at small $k$. These are ${M}_{\parallel } = {\Phi }_{\parallel } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$ and ${M}_{\perp } = 2{\Phi }_{\perp } ({k}_{\parallel } = 0, {k}_{\perp } \ensuremath{\rightarrow} 0)$, where the subscripts $\parallel $ and $\perp $ indicate quantities parallel and perpendicular to the axis of symmetry. Since ${M}_{\parallel } \ensuremath{\sim} { u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $ and ${M}_{\perp } \ensuremath{\sim} { u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } $, $u$ and $\ell $ being integral scales, self-similarity of the large scales (when it applies) demands ${ u}_{\parallel }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $ and ${ u}_{\perp }^{2} { \ell }_{\perp }^{2} {\ell }_{\parallel } = \text{constant} $. This, in turn, requires that ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that ${M}_{\parallel } $ and ${M}_{\perp } $ are indeed invariants and that, in the fully developed state, ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} = \text{constant} $. Somewhat surprisingly, the same simulations also show that ${\ell }_{\parallel } / {\ell }_{\perp } $ is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when ${ u}_{\parallel }^{2} / { u}_{\perp }^{2} $ and ${\ell }_{\parallel } / {\ell }_{\perp } $ are both constant, the integral scales should evolve as ${ u}_{\perp }^{2} \ensuremath{\sim} { u}_{\parallel }^{2} \ensuremath{\sim} {t}^{\ensuremath{-} 6/ 5} $ and ${\ell }_{\perp } \ensuremath{\sim} {\ell }_{\parallel } \ensuremath{\sim} {t}^{2/ 5} $, irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.

We measured the correlation spectrum ${\hat {Q} }_{p} (\mathbi{k})$ of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number ${R}_{\lambda } $ up to ${\simeq }700$ 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 ${S}_{ij} $ 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 ${\hat {Q} }_{p} (\mathbi{k})$ as ${\hat {Q} }_{p} (\mathbi{k})= { \hat {Q} }_{p}^{(0)} (\mathbi{k})+ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$, where ${ \hat {Q} }_{p}^{(0)} (\mathbi{k})$ is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small ${S}_{ij} $ the deviation $ \mrm{\Delta} {\hat {Q} }_{p} (\mathbi{k})$ due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to ${S}_{ij} $, $k$ 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 ${S}_{ij} $ at large ${R}_{\lambda } $.

The roles of the Coriolis force and the convection associated with the fluid motion in the formation of columnar eddies in freely decaying homogeneous rotating turbulence at a moderate Rossby number are studied by direct numerical simulation of the Navier–Stokes equations in a periodic box. The simulated field is compared with a series of artificial fields generated by switching off the nonlinear and viscous terms in the Navier–Stokes equation at given instants. The comparison shows that, without the nonlinear convection effect, the Coriolis force cannot sustain the substantial growth in the direction parallel to the rotational axis of the length scale defined on the basis of the two-point correlation of the square of the vorticity, i.e. cannot sustain the formation of the columnar eddies. The length scale characterizes well the intuitive impression from visualization of flow obeying the dynamics with or without the nonlinear effect. It is shown that the lack of substantial growth is insensitive to the scale of the eddies, the box size and viscosity, at least in the case studied here.

We examine the integral properties of freely decaying homogeneous magnetohydrodynamic (MHD) turbulence subject to an imposed magnetic field B0 at low-magnetic Reynolds number. We confirm that, like conventional isotropic turbulence, the fully developed state possesses a Loitsyansky-like integral invariant, in this case I// = − ∫ r⊥2 〈 u⊥·u′⊥〉 dr, where 〈u(x) ·u(x + rc)〉 = 〈u·u′〉 is the usual two-point velocity correlation and the subscript ⊥ indicates components perpendicular to the imposed field. The conservation of I// for fully developed turbulence places a fundamental restriction on the way in which the integral scales can develop, i.e. it implies u⊥2 ℓ⊥4 ℓ// ≈ constant where u⊥, ℓ⊥ and ℓ// are integral scales. This constraint can be used to estimate the evolution of u⊥(t; B0), ℓ⊥(t; B0) and ℓ//(t; B0), and these theoretical decay laws are shown to be in good agreement with numerical simulations.

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 40963 grid points. The DNS consist of two series of runs; one is with kmaxη ~ 1 (Series 1) and the other is with kmaxη ~ 2 (Series 2), where kmax is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number Rλ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on Rλ of the skewness and flatness factors (S and F). The DNS data suggest that the Rλ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S ~ −0.32Rλ0.11 and F ~ 1.14Rλ0.34, in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with Rλ similarly to each other at Rλ > 100, in contrast to Rλ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at Rλ ≈ 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with Rλ approximately proportional to RλαE and RλαL, respectively, where αE ≈ 0.5 and αL ≈ 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to RλβE and RλβL, respectively, where βE ≈ 1.5 and βL ≈ 3.0.

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.

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

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.

We investigate the decay of freely-evolving, isotropic turbulence whose spectrum takes the form E(k→0)∼Ik4, I being Loitsyansky's integral. We report numerical simulations in a periodic domain whose dimensions, lbox, are much larger than the integral scale of the turbulence, l. We find that, provided lbox≫l and Re≫1, the turbulence evolves to a state in which Loitsyansky's integral is approximately constant and Kolmogorov's decay law, u2∼t−10/7, holds true. The approximate conservation of I in fully-developed turbulence implies that the long-range interactions between remote eddies, as measured by the triple correlations, are very weak.

The 2.5-dimensional model of the turbulent field near a wall, proposed by Hatziavramidis & Hanratty (1979) and modified by Chapman & Kuhn (1981), has been used to test the subgrid models of Schumann (1973, 1975) and Moin & Kim (1982). The results are disquieting, both trends and orders of magnitude sometimes being seriously in error. It also appears that the contribution of the subgrid energy to the pseudopressure calculated in large-eddy simulations can be large, although this contribution is usually neglected. On the positive side, Leonard's model for the Leonard stress is extremely good, and Schumann's synthetic boundary condition is also found to be reliable.These results must be taken with a grain of salt, since the tests reported in §5 show that the 2.5-dimensional model cannot reproduce important characteristics of the turbulence in the neighbourhood of y+ = 40.

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.