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Using a simulated highly compressible isotropic turbulence field with turbulent Mach number around 1.0, we studied the effects of local compressibility on the statistical properties and structures of velocity gradients in order to assess salient small-scale features pertaining to highly compressible turbulence against existing theories for incompressible turbulence. A variety of statistics and local flow structures conditioned on the local dilatation – a measure of local flow compressibility – are studied. The overall enstrophy production is found to be enhanced by compression motions and suppressed by expansion motions. It is further revealed that most of the enstrophy production is generated along the directions tangential to the local density isosurface in both compression and expansion regions. The dilatational contribution to enstrophy production is isotropic and dominant in highly compressible regions. The emphasis is then directed to the complicated properties of the enstrophy production by the deviatoric strain rate at various dilatation levels. In the overall flow field, the most probable eigenvalue ratio for the strain rate tensor is found to be −3:1:2.5, quantitatively different from the preferred eigenvalue ratio of −4:1:3 reported in incompressible turbulence. Furthermore, the strain rate eigenvalue ratio tends to be −1:0:0 in high compression regions, implying the dominance of sheet-like structures. The joint probability distribution function of the invariants for the deviatoric velocity gradient tensor is used to characterize local flow structures conditioned on the local dilatation as well as the distribution of enstrophy production within these flow structures. We demonstrate that strong local compression motions enhance the enstrophy production by vortex stretching, while strong local expansion motions suppress enstrophy production by vortex stretching. Despite these complications, most statistical properties associated with the solenoidal component of the velocity field are found to be very similar to those in incompressible turbulence, and are insensitive to the change of local dilatation. Therefore, a good understanding of dynamics of the compressive component of the velocity field is key to an overall accurate description of highly compressible turbulence.

A new experimental investigation of decaying turbulence generated by a low-blockage space-filling fractal square grid is presented. We find agreement with previous works by Seoud & Vassilicos (Phys. Fluids, vol. 19, 2007, 105108) and Mazellier & Vassilicos (Phys. Fluids, vol. 22, 2010, 075101) but also extend the length of the assessed decay region and consolidate the results by repeating the experiments with different probes of increased spatial resolution. It is confirmed that this moderately high Reynolds number ${\\mathit{Re}}_{\\lambda } $ turbulence (up to ${\\mathit{Re}}_{\\lambda } \\simeq 350$ here) does not follow the classical high Reynolds number scaling of the dissipation rate $\\varepsilon \\ensuremath{\\sim} {u{}^{\\ensuremath{\\prime} } }^{3} / L$ and does not obey the equivalent proportionality between the Taylor-based Reynolds number ${\\mathit{Re}}_{\\lambda } $ and the ratio of integral scale $L$ to the Taylor microscale $\\lambda $. Instead we observe an approximate proportionality between $L$ and $\\lambda $ during decay. This non-classical behaviour is investigated by studying how the energy spectra evolve during decay and examining how well they can be described by self-preserving single-length-scale forms. A detailed study of homogeneity and isotropy is also presented which reveals the presence of transverse energy transport and pressure transport in the part of the turbulence decay region where we take data (even though previous studies found mean flow and turbulence intensity profiles to be approximately homogeneous in much of the decay region). The exceptionally fast turbulence decay observed in the part of the decay region where we take data is consistent with the non-classical behaviour of the dissipation rate. Measurements with a regular square mesh grid as well as comparisons with active-grid experiments by Mydlarski & Warhaft (J. Fluid Mech., vol. 320, 1996, pp. 331–368) and Kang, Chester & Meveneau (J. Fluid Mech., vol. 480, 2003, pp. 129–160) are also presented to highlight the similarities and differences between these turbulent flows and the turbulence generated by our fractal square grid.

A novel square-fractal-element grid was designed in order to increase the downstream measurement range of fractal grid experiments relative to the largest element of the grid. The grid consists of a series of square fractal elements mounted to a background mesh with spacing $L_0 = 100\\, {\\rm mm}$. Measurements were performed in the region $3.5 \\le x/L_0 \\le 48.5$, which represents a significant extension to the $x/L_0 < 20$ of previously reported square fractal grid measurements. For the region $x/L_0 \\gtrsim 24$ it was found that a power-law decay region following $\\langle {q}^2 \\rangle \\sim (x - x_0)^m$ exists with decay exponents of $m = -1.39$ and $-1.37$ at $\\mathit{Re}_{L_0} = 57\\, 000$ and $65\\, 000$, respectively. This agrees with decay values previously measured for regular grids ($-1 \\gtrsim m \\gtrsim -1.4$). The turbulence in the near-grid region, $x/L_0 < 20$, is shown to be inhomogeneous and anisotropic, in apparent contrast with previous fractal grid measurements. Nonetheless, power-law fits to the decay of turbulent kinetic energy in this region result in $m = -2.79$, similar to $m \\approx -2.5$ recently reported by Valente & Vassilicos (J. Fluid Mech., vol. 687, 2011, pp. 300–340) for space-filling square fractals. It was also found that $C_\\epsilon $ is approximately constant for $x/L_0 \\ge 25$, while it grows rapidly for $x/L_0 < 20$. These results reconcile previous fractal-generated turbulence measurements with classical grid turbulence measurements.

Using a conventional bi-planar turbulence-generating grid, we confirm the recent findings (Valente & Vassilicos, Phys. Rev. Lett., vol. 108, 2012, art. 214503) that show there is a turbulence decay region close to the generating grid that departs from the ‘classical’ turbulence decay (Comte-Bellot & Corrsin, J. Fluid Mech., vol. 25, 1966, pp. 657–682). In this ‘near-field’ region, the turbulence energy decays more rapidly than in the far-field and it exhibits unusual scaling properties. Based on the velocity decay laws, we show that for our conventional grid, the near-field extends from $\\def \\xmlpi #1{}\\def \\mathsfbi #1{\\boldsymbol {\\mathsf {#1}}}\\let \\le =\\leqslant \\let \\leq =\\leqslant \\let \\ge =\\geqslant \\let \\geq =\\geqslant \\def \\Pr {\\mathit {Pr}}\\def \\Fr {\\mathit {Fr}}\\def \\Rey {\\mathit {Re}}x/M \\sim 6$ to $x/M \\sim 12$ where $x$ is the downstream distance from the grid and $M$ is the mesh size. This corresponds to $1.1 \\le x/x* \\le 2.3$ where $x*$ is the wake interaction length scale (Mazellier & Vassilicos, Phys. Fluids, vol. 22, 2010, art. 075101). However, other statistics (velocity derivatives and length-scale ratios) indicate that the extent of the initial period depends on the grid mesh Reynolds number, $R_M$ , extending further for higher values of $R_M$ . In the near-field the turbulence approaches isotropy both at the large and small scales but there still is inhomogeneity in the derivative statistics. The derivative skewness also departs from values observed at comparable Reynolds numbers in the far-field decay region, and in other turbulent flows at comparable Reynolds numbers. Two values of $R_M$ were studied: $42 \\times 10^3$ and $76 \\times 10^3$ .

Understanding the non-local pressure contributions and viscous effects on the small-scale statistics remains one of the central challenges in the study of homogeneous isotropic turbulence. Here we address this issue by studying the impact of the pressure Hessian as well as viscous diffusion on the statistics of the velocity gradient tensor in the framework of an exact statistical evolution equation. This evolution equation shares similarities with earlier phenomenological models for the Lagrangian velocity gradient tensor evolution, yet constitutes the starting point for a systematic study of the unclosed pressure Hessian and viscous diffusion terms. Based on the assumption of incompressible Gaussian velocity fields, closed expressions are obtained as the results of an evaluation of the characteristic functionals. The benefits and shortcomings of this Gaussian closure are discussed, and a generalization is proposed based on results from direct numerical simulations. This enhanced Gaussian closure yields, for example, insights on how the pressure Hessian prevents the finite-time singularity induced by the local self-amplification and how its interaction with viscous effects leads to the characteristic strain skewness phenomenon.

We investigate the transfer properties of energy and helicity fluctuations in fully developed homogeneous and isotropic turbulence by changing the nature of the nonlinear Navier–Stokes terms. We perform a surgery of all possible interactions, by keeping only those triads that have sign-definite helicity content. In order to do this, we apply an exact decomposition of the velocity field in a helical Fourier basis, as first proposed by Constantin & Majda (Commun. Math. Phys, vol. 115, 1988, p. 435) and exploited in great detail by Waleffe (Phys. Fluids A, vol. 4, 1992, p. 350), and we evolve the Navier–Stokes dynamics keeping only those velocity components carrying a well-defined (positive or negative) helicity. The resulting dynamics preserves translational and rotational symmetries but not mirror invariance. We give clear evidence that this three-dimensional homogeneous and isotropic chiral turbulence is characterized by a stationary inverse energy cascade with a spectrum ${E}_{back} (k)\\sim {k}^{- 5/ 3} $ and by a direct helicity cascade with a spectrum ${E}_{forw} (k)\\sim {k}^{- 7/ 3} $ . Our results are important to highlight the dynamics and statistics of those subsets of all possible Navier–Stokes interactions responsible for reversal events in the energy-flux properties, and demonstrate that the presence of an inverse energy cascade is not necessarily connected to a two-dimensionalization of the flow. We further comment on the possible relevance of such findings to flows of geophysical interest under rotations and in thin layers. Finally we propose other innovative numerical experiments that can be achieved by using a similar decimation of degrees of freedom.

This study investigates the two-way coupling effects of finite-size solid spherical particles on decaying isotropic turbulence using direct numerical simulation with an immersed boundary method. We fully resolve all the relevant scales of turbulence around freely moving particles of the Taylor length-scale size, 1.2≤d/λ≤2.6. The particle diameter and Stokes number in terms of Kolmogorov length- and time scales are 16≤d/η≤35 and 38≤τp/τk≤178, respectively, at the time the particles are released in the flow. The particles mass fraction range is 0.026≤φm≤1.0, corresponding to a volume fraction of 0.01≤φv≤0.1 and density ratio of 2.56≤ρp/ρf≤10. The maximum number of dispersed particles is 6400 for φv=0.1. The typical particle Reynolds number is of O(10). The effects of the particles on the temporal development of turbulence kinetic energy E(t), its dissipation rate (t), its two-way coupling rate of change Ψp(t) and frequency spectra E(ω) are discussed. In contrast to particles with d < η, the effect of the particles in this study, with d > η, is that E(t) is always smaller than that of the single-phase flow. In addition, Ψp(t) is always positive for particles with d > η, whereas it can be positive or negative for particles with d < η.

The modulation of decaying isotropic turbulence by 45 000 spherical particles of Kolmogorov-length-scale size is studied using direct particle–fluid simulations, i.e. the flow field over each particle is fully resolved by direct numerical simulations of the conservation equations. A Cartesian cut-cell method is used by which the exchange of momentum and energy at the fluid–particle interfaces is strictly conserved. It is shown that the particles absorb energy from the large scales of the carrier flow while the small-scale turbulent motion is determined by the inertial particle dynamics. Whereas the viscous dissipation rate of the bulk flow is attenuated, the particles locally increase the level of dissipation due to the intense strain rate generated near the particle surfaces due to the crossing-trajectory effect. Analogously, the rotational motion of the particles decouples from the local fluid vorticity and strain-rate field at increasing particle inertia. The high level of dissipation is partially compensated by the transfer of momentum to the fluid via forces acting at the particle surfaces. The spectral analysis of the kinetic energy budget is supported by the average flow pattern about the particles showing a nearly universal strain-rate distribution. An analytical expression for the instantaneous rate of viscous dissipation induced by each particle is derived and subsequently verified numerically. Using this equation, the local balance of fluid kinetic energy around a particle of arbitrary shape can be precisely determined. It follows that two-way coupled point-particle models implicitly account for the particle-induced dissipation rate via the momentum-coupling terms; however, they disregard the actual length scales of the interaction. Finally, an analysis of the small-scale flow topology shows that the strength of vortex stretching in the bulk flow is mitigated due to the presence of the particles. This effect is associated with the energy conversion at small wavenumbers and the reduced level of dissipation at intermediate wavenumbers. Consequently, it damps the spectral flux of energy to the small scales.

A large database from direct numerical simulations of isotropic turbulence, including recent simulations for box sizes up to 40963 and the Taylor–Reynolds number Rλ ≈ 1000, is used to investigate the bottleneck effect in the three-dimensional energy spectrum and second-order structure functions, and to determine the Kolmogorov constant, CK. The difficulties in estimating CK at any finite Reynolds number, introduced by intermittency and the bottleneck, are assessed. The data conclusively show that the bottleneck effect decreases with the Reynolds number. On this basis, an alternative to the usual procedure for determining CK is suggested; this proposal does not depend on the particular choices of fitting ranges or power-law behaviour in the inertial range. Within the resolution of the numerical data, CK thus determined is a Reynolds-number-independent constant of ≈1.58 in the three-dimensional spectrum. A simple model including non-local transfer is proposed to reproduce the observed scaling features of the bottleneck.

We present numerical evidence of how three-dimensionalization occurs at small scale in rotating turbulence with Beltrami ($\\mathit{ABC}$) forcing, creating helical flow. The Zeman scale ${\\ell }_{\\Omega } $ at which the inertial and eddy turn-over times are equal is more than one order of magnitude larger than the dissipation scale, with the relevant domains (large-scale inverse cascade of energy, dual regime in the direct cascade of energy $E$ and helicity $H$, and dissipation) each moderately resolved. These results stem from the analysis of a large direct numerical simulation on a grid of $307{2}^{3} $ points, with Rossby and Reynolds numbers, respectively, equal to $0. 07$ and $2. 7\\ensuremath{\\times} 1{0}^{4} $. At scales smaller than the forcing, a helical wave-modulated inertial law for the energy and helicity spectra is followed beyond ${\\ell }_{\\Omega } $ by Kolmogorov spectra for $E$ and $H$. Looking at the two-dimensional slow manifold, we also show that the helicity spectrum breaks down at ${\\ell }_{\\Omega } $, a clear sign of recovery of three-dimensionality in the small scales.