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The large-scale structure of many turbulent flows encountered in practical situations such as aeronautics, industry, meteorology is nowadays successfully computed using the Kolmogorov–Kármán–Howarth energy cascade picture. This theory appears increasingly inaccurate when going down the energy cascade that terminates through intermittent spots of energy dissipation, at variance with the assumed homogeneity. This is problematic for the modelling of all processes that depend on small scales of turbulence, such as combustion instabilities or droplet atomization in industrial burners or cloud formation. This paper explores a paradigm shift where the homogeneity hypothesis is replaced by the assumption that turbulence contains singularities, as suggested by Onsager. This paradigm leads to a weak formulation of the Kolmogorov–Kármán–Howarth–Monin equation (WKHE) that allows taking into account explicitly the presence of singularities and their impact on the energy transfer and dissipation. It provides a local in scale, space and time description of energy transfers and dissipation, valid for any inhomogeneous, anisotropic flow, under any type of boundary conditions. The goal of this article is to discuss WKHE as a tool to get a new description of energy cascades and dissipation that goes beyond Kolmogorov and allows the description of small-scale intermittency. It puts the problem of intermittency and dissipation in turbulence into a modern framework, compatible with recent mathematical advances on the proof of Onsager’s conjecture.
In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier–Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $\unicode[STIX]{x1D714}_{max}$ as a function of vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}$ in the range $2000\leqslant R_{\unicode[STIX]{x1D6E4}}\leqslant 3400$, and deduce a compatible behaviour $\unicode[STIX]{x1D714}_{max}\sim \unicode[STIX]{x1D714}_{0}\exp [1+220(\log [R_{\unicode[STIX]{x1D6E4}}/2000])^{2}]$ as $R_{\unicode[STIX]{x1D6E4}}\rightarrow \infty$. This may be described as a physical (although not strictly mathematical) singularity, for all $R_{\unicode[STIX]{x1D6E4}}\gtrsim 4000$.
We have conducted an extensive study of the scaling properties of small scale turbulence using both numerical and experimental data of a flow in the same von Kármán geometry. We have computed the wavelet structure functions, and the structure functions of the vortical part of the flow and of the local energy transfers. We find that the latter obey a generalized extended scaling, similar to that already observed for the wavelet structure functions. We compute the multi-fractal spectra of all the structure functions and show that they all coincide with each other, providing a local refined hypothesis. We find that both areas of strong vorticity and strong local energy transfer are highly intermittent and are correlated. For most cases, the location of local maximum of the energy transfer is shifted with respect to the location of local maximum of the vorticity. We, however, observe a much stronger correlation between vorticity and local energy transfer in the shear layer, that may be an indication of a self-similar quasi-singular structure that may dominate the scaling properties of large order structure functions.
Recognizing the fact that the finite-time singularity of the Navier–Stokes equations is widely accepted as a key issue in fundamental fluid mechanics, and motivated by the recent model of Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019a, pp. 930–967; J. Fluid Mech., vol. 870, 2019b, R1) on this issue, we have performed direct numerical simulation (DNS) for two colliding slender vortex rings of radius $R$. The separation between the two tipping points $2s_{0}$ and the scale of the core cross-section $\unicode[STIX]{x1D6FF}_{0}$ are chosen as $\unicode[STIX]{x1D6FF}_{0}=0.1s_{0}=0.01R$; the vortex Reynolds number ($Re=\text{circulation/viscosity}$) ranges from 1000 to 4000. In contrast to the claim that the core remains compact and circular, there is notable core flattening and stripping, which further increases with $Re$ – akin to our previous finding in the standard anti-parallel vortex reconnection. Furthermore, the induced motion of bridges arrests the curvature growth and vortex stretching at the tipping points; consequently, the maximum vorticity grows with $Re$ substantially slower than the exponential scaling predicted by the model – implying that, for this configuration, even physical singularity is unlikely. Our simulations not only shed light on the longstanding question of finite-time singularities, but also further delineate the detailed mechanisms of reconnection. In particular, we show for the first time that the separation distance $s(\unicode[STIX]{x1D70F})$ before reconnection follows 1/2 scaling exactly – a significant DNS result.
Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance $\delta$ between interacting vortices follows a $\delta(t) \sim t^{1/2}$ scaling. Due to the complex nature of the dynamics (e.g. the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using direct numerical simulation of the Navier–Stokes equations, we study viscous reconnection of slender vortices, whose core size is much smaller than the radius of the vortex curvature. For separations that are large compared to the vortex core size, we discover that $\delta (t)$ between the two interacting viscous vortices surprisingly also follows the 1/2-power scaling for both pre- and post-reconnection events. The prefactors in this 1/2-power law are found to depend not only on the initial configuration but also on the vortex Reynolds number (or viscosity). Our finding in viscous reconnection, complementing numerous works on quantum vortex reconnection, suggests that there is indeed a universal route for reconnection – an essential result for understanding the various facets of the vortex reconnection phenomena and their potential modelling, as well as possibly explaining turbulence cascade physics.
We study the three-dimensional structure of turbulent velocity fields around extreme events of local energy transfer in the dissipative range. Velocity fields are measured by tomographic particle velocimetry at the centre of a von Kármán flow with resolution reaching the Kolmogorov scale. The characterization is performed through both direct observation and an analysis of the velocity gradient tensor invariants at the extremes. The conditional average of local energy transfer on the second and third invariants seems to be the largest in the sheet zone, but the most extreme events of local energy transfer mostly correspond to the vortex stretching topology. The direct observation of the velocity fields allows for identification of three different structures: the screw and roll vortices, and the U-turn. They may correspond to a single structure seen at different times or in different frames of reference. The extreme events of local energy transfer come along with large velocity and vorticity norms, and the structure of the vorticity field around these events agrees with previous observations of numerical works at similar Reynolds numbers.
All previous experiments in open turbulent flows (e.g. downstream of grids, jets and the atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions (Anselmet et al., J. Fluid Mech., vol. 140, 1984, pp. 63–89; Stolovitzky et al., Phys. Rev. E, vol. 48 (5), 1993, R3217; Arneodo et al., Europhys. Lett., vol. 34 (6), 1996, p. 411). The only measurement of scaling exponents at high order (${>}6$) in closed turbulent flow (von Kármán swirling flow) using Taylor’s frozen flow hypothesis, however, produced scaling exponents that are significantly smaller, suggesting that the universality of these exponents is broken with respect to change of large scale geometry of the flow. Here, we report measurements of longitudinal structure functions of velocity in a von Kármán set-up without the use of the Taylor hypothesis. The measurements are made using stereo particle image velocimetry at four different ranges of spatial scales, in order to observe a combined inertial subrange spanning approximately one and a half orders of magnitude. We found scaling exponents (up to ninth order) that are consistent with values from open turbulent flows, suggesting that they might be in fact universal.
The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ is studied, starting from a finite-energy configuration of two vortex rings of circulation $\pm \unicode[STIX]{x1D6E4}$ and radius $R$, symmetrically placed on two planes at angles $\pm \unicode[STIX]{x1D6FC}$ to a plane of symmetry $x=0$. The minimum separation of the vortices, $2s$, and the scale of the core cross-section, $\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities $\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$ is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$ at the tipping points and by $s(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$ is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when $\unicode[STIX]{x1D705}s$ and $\unicode[STIX]{x1D6FF}/\!s$ become of order unity. The dynamical system admits ‘partial Leray scaling’ of just $s$ and $\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of $s,\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter $\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.