Session P13: Turbulence: General (3:10pm - 3:55pm CST)

turbulent intermittency as consequence of stationary constraints

In its seminal work on turbulence, Kolmogorov made use of the stationary hypothesis to determine the Power Density Spectra of velocity field in turbulent flow. However to our knowledge, the constraints that stationary processes impose on the fluctuations of power have never been used in the context of turbulence. Here we first recall how the Power Density Spectra of the fluctuations of the injected power, the dissipated power and the energy flux have to converge at vanishing frequency. Then we show that these constraints cannot be satisfied by using naive scaling argument in the no-intermittent framework of the 1941 Kolmogorov theory (the K41 theory). Yet the intermittent GOY--shell model, fulfills these constraints on the power fluctuations. Hence, they might induce some intermittency. Indeed, we show that the constraints on the power fluctuations implies a relation between scaling exponents which is consistent with our GOY--shell model and agrees the She-Leveque formula. It also fixes the intermittent parameter of the log-normal model to a realistic value. The relevance of these results for real turbulence is drawn in the concluding remarks.   

A Renormalization Group Theory of Spontaneous Stochasticity

Spontaneous stochasticity is persistent randomness in solutions of singular deterministic dynamics (ODE's, PDE's) for fixed initial data, as regularizations and stochastic perturbations are both taken to vanish. First identified in Lagrangian fluid particles undergoing turbulent Richardson dispersion$^{\mathrm{a}}$, the effect was shown to be due to non-unique solutions of the singular initial-value problem and to be necessary for anomalous dissipation of both passive$^{\mathrm{a}}$ and active$^{\mathrm{b\thinspace }}$scalars. Even earlier, Lorenz$^{\mathrm{c}}$ pointed out similar spontaneous stochasticity in turbulent solutions of Eulerian fluid equations, associated to intrinsic unpredictability, and the effect has been verified in numerical simulations of a self-similar turbulent mixing layer$^{\mathrm{d}}$. Here we describe a renormalization group method$^{\mathrm{e}}$ that can determine whether spontaneous stochasticity occurs, calculate the universal statistics obtained at long times, and obtain finite Reynolds-number corrections. The method is illustrated on some simple models. $^{\mathrm{a}}$D. Bernard, K. Gawedzki, and A. Kupiainen, J. Stat. Phys. 90, 519--569 (1998) $^{\mathrm{b}}$T. D. Drivas and G. L. Eyink, J. Fluid Mech. 829, 153--189 (2017) $^{\mathrm{c}}$E. N. Lorenz, Tellus 21, 289--307 (1969) $^{\mathrm{d}}$S. Thalabard, J. Bec and A. Mailybaev, Commun. Phys. 3, 122 (2020) $^{\mathrm{e}}$https://arxiv.org/abs/2007.01333   

Experimental Results on Turbulence at Extreme Reynolds Numbers

High-quality measurements of the velocity increment statistics of turbulence at high Reynolds numbers (Re) provide insights into the dynamics of the inertial range. Recently, for decaying turbulence in the Max Planck Variable Density Turbulence Tunnel, we have shown (arXiv:2006.10993) that in the inertial range, the functional dependence of the 2nd order velocity increments on spatial scales becomes independent of the Reynolds number for sufficiently large Re. While the functional dependence reaches a universal form, effects of large-scale inhomogeneity and viscous dissipation remain important across all scales. We review these second-order results up to Taylor-scale Reynolds numbers $R_\lambda \approx 6000$, extend them to higher orders, and also report on ongoing Lagrangian particle tracking experiments.   

Stirring anisotropic turbulence with an active grid

The turbulent cascade restores isotropy, but may do that very slowly, so that even at large Reynolds numbers anisotropic stirring may be remembered at the small scales. Our active grid consists of a grid of rods with attached vanes that are rotated by servo motors. Homogeneous shear in a wind tunnel is created with a judicious choice of the axes motion protocol. We characterize turbulence using an array of 10 two-component hot wires. By changing the orientation of the array and selecting the components $\alpha$, $\beta$ of the velocity, structure functions $G_{\alpha^n, \beta^n}(\vec{r}) = \langle u_\alpha^n(\vec{x} + \vec{r})\: u_\beta^n(\vec{x}) \rangle$ that vanish in isotropic turbulence could be measured. Their scaling anomaly appears to be surprisingly large. In a second experiment we drastically change the integral length scales and drastically change the stirring anisotropy by tuning the coherence time of the signals that drive the grid. Although the spectra remain turbulence-like, the large-scale anisotropy is imprinted on small scales. \\[2ex] H. E. Cekli and W. van de Water, Phys. Fluids \textbf{32}, 075119 (2020)   

An open source Matlab\textsuperscript{\textregistered}package for solving Fokker-Plank Equation and validation of Integral Fluctuation Theorem

We present a user-friendly open-source Matlab package developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg. This package enables to perform a standard analysis of given turbulent data and extracts the stochastic equations describing the cascade process in turbulent flows through Fokker-Planck equations. As the analysis of the scale-dependent cascade process through a hierarchy of spatial and temporal scales is an integral part of turbulence theory, this stochastic treatment of the cascade process has the potential for a new way to link to fluctuation theorems of non-equilibrium stochastic thermodynamics. In particular, entropy production can be determined for local turbulent flow structures, based on this the validity of the integral fluctuation theorem can be verified.\\ The development of this package greatly enhances the practicability and availability of this method, which allows a comprehensive statistical description in terms of the complexity of turbulent velocity time series. It can also be used by researchers outside of the field of turbulence for the analysis of data with turbulent like complexity. Support is available: \\ \mbox{https://github.com/andre-fuchs-uni-oldenburg/OPEN\_FPE\_IFT}   

small scale structures of turbulence in terms of entropy and fluctuation theorems

We present experimental evidence that, together with the integral fluctuation theorem, a detailed-like fluctuation theorem holds for large entropy values in cascade processes in turbulent flows. Based on experimental data, we estimate the stochastic equations describing the scale-dependent cascade process in a turbulent flow through Fokker-Planck equations, and from the individual cascade trajectories an entropy term can be determined. Since the statistical fluctuation theorems set the occurrence of positive and negative entropy events in strict relation, we are able to verify how cascade trajectories, defined by entropy consumption or entropy production, are linked to turbulent structures: Trajectories with entropy production start from large velocity increments at large scale and converge to zero velocity increments at small scales; trajectories with entropy consumption end at small scale with finite size increments. A lower bound at small scale of these negative entropy trajectories increases linearly with the magnitude of the negative entropy value. This indicates a tendency to local discontinuities in the velocity field. Our findings show no lower bound of negative entropy values and thus for the piling up velocity differences on small scales. (Phys. Rev. F 5, 034602 (2020))   

Novel Identification of Gravity Waves in an Experimental Gravity Current

The body of gravity currents has typically been assumed to be two-dimensional, however the instantaneous three-dimensional flow structure has never been observed experimentally. This key aspect of an extensively studied flow remains poorly understood. Fully three-dimensional particle tracking velocimetry measurements (Shake-the-Box) of constant-influx solute-based gravity current flows are presented for the first time. These measurements call into question some standard assumptions made about the body of gravity current flows. Cross-stream velocity is typically neglected, yet in this work cross-stream and vertical velocities are shown to be equivalent in magnitude. Dynamic mode decomposition is used to identify the presence of three-dimensional internal waves within the body centered at the height of the velocity maximum. Estimation of the Doppler-shifted Brunt-Vaisala buoyancy frequency demonstrates that these are internal gravity waves, and that they may form a critical layer within the flow. As the body often forms by far the largest part of the gravity current, these observations suggest that a new model of the gravity current body is needed to understand these often seen and critical flows.   

Schmidt number effects in turbulence with active scalars

Active scalar turbulence refers to problems characterized by two-way couplings between the velocity field and one or more diffusing scalars, typically via small changes in the fluid density leading to buoyancy forces which may either suppress or amplify motions in the vertical. Although the most common examples are of temperature fluctuations in air or water, of Schmidt (Prandtl) numbers ($Sc$) 0.72 and 7 respectively, important applications also arise where $Sc\gg 1$ (such as salinity in the ocean) and $Sc\ll 1$ (such as in conducting fluids of high diffusivity). The physics is most intriguing when two scalars with different molecular diffusivities having opposing stabilizing versus de-stabilizing influences are present, where differential diffusion clearly plays a pivotal role. We have performed direct numerical simulations with one or two active scalars treated with the well-known Boussinesq approximation, with a focus Schmidt numbers substantially below unity. We use results on Reynolds stress budget and spectral transfer to draw attention to contrasts in particular between flows where the stabilizing influence is provided by a scalar of low versus moderate Schmidt number, with or without another scalar providing a de-stabilizing effect.