Search

In 1981, Frisch and Morf [1] postulated the existence of complex singularities in solutions of Navier-Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier-Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.

Statistical physics and dynamical systems theory are key tools to study high-impact geophysical events such as temperature extremes, cyclones, thunderstorms, geomagnetic storms and many more. Despite the intrinsic differences between these events, they all originate as temporary deviations from the typical trajectories of a geophysical system, resulting in well-organised, coherent structures at characteristic spatial and temporal scales. While statistical extreme value analysis techniques are capable to provide return times and probabilities of occurrence of certain geophysical events, they are not apt to account for their underlying physics. Their focus is to compute the probability of occurrence of events that are large or small with respect to some specific observable (e.g. temperature, precipitation, solar wind), rather than to relate rare or extreme phenomena to the underlying anomalous geophysical regimes. This paper outlines this knowledge gap, presenting some related challenges, new formalisms and briefly commenting on how stochastic approaches tailored to the study of extreme geophysical events can help to advance their understanding., Comment: This perspective paper has issued from discussions held during the UNDERPIN symposium held in Erice, Italy in 2022

Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase-space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set as it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise, despite the fact that the two cases exhibit very similar stochastic attractors at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems., Comment: 9 pages, 6 figures

Turbulent flows present rich dynamics originating from non-trivial energy fluxes across scales, non-stationary forcings and geometrical constraints. This complexity manifests in non-hyperbolic chaos, randomness, state-dependent persistence and unpredictability. All these features have prevented a full characterization of the underlying turbulent (stochastic) attractor, which will be the key object to unpin this complexity. Here we propose a novel formalism to trace the evolution of the structural characteristics of phase-space trajectories across scales, providing a full characterization of the attractor. We demonstrate that the properties of the dynamically invariant objects depend on the scale we are focusing on. In the case of laboratory experiments on fluids we observe the emergence of an intrinsic timescale, solely determined by nonlinear interactions, controlling the geometrical and topological properties of phase-space trajectories. Given the changing nature of such attractors in time and scales we term them chameleon attractors., Comment: 19 pages, 6 figures

We report hot-wire measurements performed in two very different, co- and counter-rotating flows, in normal and superfluid helium at 1.6 K, 2 K, and 2.3 K. As recently reported, the power spectrum of the hot-wire signal in superfluid flows exhibits a significant bump at high frequency (Diribarne et al. [1]). We confirm that the bump frequency does not depend significantly on the temperature and further extend the previous analysis of the velocity dependence of the bump, over more than one decade of velocity. The main result is that the bump frequency depends on the turbulence intensity of the flow, and that using the turbulent Reynolds number rather than the velocity as a control parameter collapses results from both co- and counter-rotating flows. The vortex shedding model previously proposed, in its current form, does not account for this observation. This suggests that the physical origin of the bump is related to the small scale turbulence properties of the flow. We finally propose some qualitative physical mechanism by which the smallest structures of the flow, at intervortex distance, could affect the heat flux of the hot-wire., Comment: 13 pages, 8 figures

We investigate the ability of 4D Particle Tracking Velocimetry measurements at high particle density to explore intermittency and irreversibility in a turbulent swirling flow at various Reynolds numbers. For this, we devise suitable tools to remove the experimental noise, and compute the statistics of both Lagrangian velocity increments and wavelet coefficients of the Lagrangian power (the time derivative of the kinetic energy along a trajectory). We show that the signature of noise is strongest on short trajectories, and results in deviations from the regularity condition at small time scales. Considering only long trajectories to get rid of such effect, we obtain scaling regimes that are compatible with a reduced intermittency, meaning that long trajectories are also associated with areas of larger regularity. The scaling laws, both in time and Reynolds number, can be described by the multifractal model, with a log-normal spectrum and an intermittency parameter that is three times smaller than in the Eulerian case, where all the areas of the flow are taken into account.

We introduce a model of interacting singularities of Navier-Stokes, named pin\,cons. They follow a Hamiltonian dynamics, obtained by the condition that the velocity field around these singularities obeys locally Navier-Stokes equations. This model can be seen of a generalization of the vorton model of Novikov, that was derived for the Euler equations. When immersed in a regular field, the pin\,cons are further transported and sheared by the regular field, while applying a stress onto the regular field, that becomes dominant at a scale that is smaller than the Kolmogorov length. We apply this model to compute the motion of a dipole of pin\,cons. When the initial relative orientation of the dipole is inside the interval (0, pi/2), a dipole made of pin\,con of same intensity exhibits a transient collapse stage, following a scaling with dipole radius tending to 0 like (tc - t) power 0.63. For long time, the dynamics of the dipole is however repulsive, with both components running away from each other to infinity., Comment: 24 pages 13 figures

We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov's 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) \geq 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h \geq (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h \geq -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed., Comment: 9 pages, 1 figure

We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano-Oboukhov picture, but evidence of a mixture of Bolgiano-Oboukhov and Kolmogorov scalings. In particular, all the terms are compatible with a Bolgiano-Oboukhov local H\"older exponent for the temperature and a Kolmogorov 41 for the velocity. This behavior may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of Bolgiano-Oboukhov local scales. The scale-by-scale analysis allows us also to compare the theoretical Bolgiano-Oboukhov length $L_{BO}$ computed from its definition with that empirically extracted through scalings obtained from weak analysis. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties., Comment: 22 pages, 7 figures

We investigate universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and the direct numerical simulations (DNS) of the Navier-Stokes equations. We show that the numerical and experimental velocity structure functions up to order 9 follow a log-universality; we find that they collapse on a universal curve, if we use units that include logarithmic dependence on the Reynolds number. We then investigate the meaning and consequences of such log-universality, and show that it is connected with the properties of a "multifractal free energy", based on an analogy between multifractal and themodynamics. We show that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Holder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.

We present a comprehensive study of the statistical features of a three-dimensional time-reversible Navier-Stokes (RNS) system, wherein the standard viscosity $\nu$ is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter $\mathcal{R}_r$, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale $\ell_{\rm f}$ and the total energy $E_0$. We find that the system exhibits a transition from a high-enstrophy phase at small $\mathcal{R}_r$, where truncation effects tend to produce partially thermalized states, to a hydrodynamical phase with low enstrophy at large $\mathcal{R}_r$. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous phase transition, where $\mathcal{R}_r$ indeed behaves as a thermodynamic control parameter, e.g., a temperature, the enstrophy plays the role of an order parameter, while the symmetry breaking parameter $h$ is (one over) the truncation scale $k_{\rm max}$. We find that the signatures of the phase transition close to the critical point $\mathcal{R}_r^\star$ can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti, Phys.Lett.A, 223, 1996 could hold true. Our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes (NS) equations with viscosity chosen so as to match the average value of the reversible viscosity., Comment: Revised version; results unchanged

We present hydrodynamic and magneto-hydrodynamic simulations of a liquid sodium flow using the compressible MHD code PLUTO to investigate the magnetic field regeneration in the Von-Karman-Sodium dynamo experiment. The aim of the study is to analyze the influence of the fluid resistivity and turbulence level on the collimation by helicoidal motions of a remnant magnetic field. We use a simplified cartesian geometry to represent the flow dynamics in the vicinity of one cavity of a multi-blades impeller inspired by those used in the Von-Karman-Sodium (VKS) experiment. We perform numerical simulations with kinetic Reynolds numbers up to 1000 for magnetic Prandtl numbers between 30 and 0.1. Our study shows that perfect ferromagnetic walls favour enhanced collimation of flow and magnetic fields even if the turbulence degree of the model increases. The location of the helicoidal coherent vortex in between the blades changes with the impinging velocity. It becomes closer to the upstream blade and impeller base if the flow incident angle is analogous to the TM73 impeller configuration rotating in the unscooping direction. This result is also obtained at higher kinetic Reynolds numbers when the helicoidal vortex undergoes a precessing motion, leading to a reinforced effect in the vortex evolution and in the magnetic field collimation when using again perfect ferromagnetic boundary conditions. We estimate the efficiency of a hypothetical dynamo loop occurring in the vicinity of the impeller and discuss the relevance of our findings in the context of mean field dynamo theory.

All previous experiments in open turbulent flows (e.g. downstream of grids, jet and atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions. The only measurement in closed turbulent flow (von K\'arm\'an swirling flow) using Taylor-hypothesis, however, produced scaling exponents that are significantly smaller, suggesting that the universality of these exponents are 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 K\'arm\'an setup without the use of Taylor-hypothesis. The measurements are made using Stereo Particle Image Velocimetry at 4 different ranges of spatial scales, in order to observe a combined inertial subrange spanning roughly one and a half order of magnitude. We found scaling exponents (up to 9th order) that are consistent with values from open turbulent flows, suggesting that they might be in fact universal., Comment: Under review in J. Fluid Mech

We present hydrodynamic and magneto-hydrodynamic simulations of liquid sodium flow with the PLUTO compressible MHD code to investigate influence of magnetic boundary conditions on the collimation of helicoidal motions. We use a simplified cartesian geometry to represent the flow dynamics in the vicinity of one cavity of a multi-blades impeller inspired by those used in the Von-K\'{a}rm\'{a}n-Sodium (VKS) experiment. We show that the impinging of the large scale flow upon the impeller generates a coherent helicoidal vortex inside the blades, located at a distance from the upstream blade piloted by the incident angle of the flow. This vortex collimates any existing magnetic field lines leading to an enhancement of the radial magnetic field that is stronger for ferromagnetic than for conducting blades. The induced magnetic field modifies locally the velocity fluctuations, resulting in an enhanced helicity. This process possibly explains why dynamo action is more easily triggered in the VKS experiment when using soft iron impellers.

We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely the number of quasi-stationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can neither be recovered using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasi-stationary states.

We introduce two new singularity detection criteria based on the work of Duchon-Robert (DR) [J. Duchon and R. Robert, Nonlinearity, 13, 249 (2000)], and Eyink [G.L. Eyink, Phys. Rev. E, 74 (2006)] which allow for the local detection of singularities with scaling exponent $h\leqslant1/2$ in experimental flows, using PIV measurements. We show that in order to detect such singularities, one does not need to have access to the whole velocity field inside a volume but can instead look for them from stereoscopic particle image velocimetry (SPIV) data on a plane. We discuss the link with the Beale-Kato-Majda (BKM) [J.T. Beale, T. Kato, A. Majda, Commun. Math. Phys., 94, 61 (1984)] criterion, based on the blowup of vorticity, which applies to singularities of Navier-Stokes equations. We illustrate our discussion using tomographic PIV data obtained inside a high Reynolds number flow generated inside the boundary layer of a wind tunnel. In such a case, BKM and DR criteria are well correlated with each other.