Session T27: Turbulence: Theory I

High-order inertial range scaling exponents in incompressible turbulence using generalized extended self-similarity

Inertial range (IR) scaling exponents of velocity structure functions for incompressible turbulent flows can be measured very reliably at high-Reynolds numbers (Re). Extended self-similarity (ESS) has also allowed for measurement of these exponents at lower-Re. However, the measurements are limited to a range of orders where multiple theories provide similar predictions. Direct numerical simulations (DNS) of higher-Re flows for very long times, to guarantee statistical reliability, is computationally prohibitive on current generation of computers. Recent theoretical advances have however shown that scaling in certain turbulent quantities emerges in very low-Re flows, at least an order magnitude lower than needed for observing IR scaling. DNS in this regime, even with fine small-scale resolution, is currently feasible. In this talk, we measure the IR scaling exponents using generalized extended self-similarity at orders larger than reported in literature using highly resolved DNS at low to moderate-Re numbers. The particular focus is to improve reliability of IR scaling exponents in low-Re flows and determine the Re range where IR scaling is first observed. Measured scaling exponents are compared to different theories in order to enable discrimination between them.   

Non-locality and scaling of extreme events in fluid turbulence

Intense velocity gradient fluctuations spontaneously develop in turbulent flows at very high Reynolds numbers. The intense fluctuations of vorticity are amplified via the nonlinear coupling with the rate of strain tensor, known as vortex stretching. The relation between strain and vorticity, however, is highly nonlocal. An important observation is that the averaged value of strain, conditioned on vorticity, behaves as a power law.

We investigate this nonlocal coupling with highly resolved direct numerical simulations of isotropic turbulence in periodic domains of up to 12,2883 grid points and Taylor-scale Reynolds number R_λ in the range 140–1300. Specifically, we decompose the strain-rate tensor into local and nonlocal contributions obtained through Biot-Savart integration of vorticity in a sphere of radius R. We find that vorticity is predominantly amplified by the nonlocal strain due to scales beyond a characteristic scale size, which varies as a simple power law of vorticity magnitude. The underlying dynamics preferentially align vorticity with the most extensive eigenvector of nonlocal strain. The remaining local strain aligns vorticity with the intermediate eigenvector and does not contribute significantly to amplification; instead it surprisingly attenuates intense vorticity, leading to breakdown of the observed power law and ultimately also the scale invariance of vorticity amplification.

Based on elementary fluid mechanical considerations, and on the empirical relation between strain and vorticity, we propose a phenomenological description of the intense events in turbulence. Our approach provides an excellent description of the numerical data, which captures very well the observed variations as a function of R_λ.

    

Probability density functions of dissipation rate and enstrophy in turbulence

Variation of the probability density functions (PDFs) for the dissipation rate and enstophy from very law to high Reynolds numbers in the isotropic steady turbulence is studied numerically and theoretically. It is shown that the asymptotic PDFs at very high Reynolds numbers are both the stretched gamma distribution. The left tails of the PDFs are of the power law with the exponents 3/2 for the dissipation rate and 1/2 for the enstrophy, on the other hand, the right tails are stretched exponential with the same exponents which are smaller than unity but with different decay rates. It is theretically shown that the right PDF tail of the enstrophy is longer than that of the dissipation at all Reynolds number. Implications of these asymptotic PDFs are also discussed.   

Analyzing energy cascade of filtered vortices using a novel turbulence database framework

The Karman-Howarth-Monin-Hill (KHMH) equation is a generalization of the Karman-Howarth equation and is valid for non-homogeneous, non-isotropic flow conditions. It is here applied to spatial subsets of DNS data from homogeneous isotropic turbulence at R_λ=1,300 to explore possible correlations between the rate of energy cascade and features of large-scale motions (such as filtered vorticity). The cascade rate is identified and computed from DNS using a spherical surface integration in length-scale space on the triple velocity difference term of the KHMH equation. On global average, this term is related to the mean rate of dissipation by a factor of -4/5 as in the classic formulation of the 4/5th law. Locally, we find anticorrelations between the cascade rate and the filtered vorticity magnitude at the same length scale, confirming that locally the cascade rate is reduced when large-scale rotation is strong. To explore connections with the filtering approach used in the LES literature, we compare the KHMH cascade rate with the subfilter-scale dissipation rate. Qualitative similarity and positive correlation are observed, but the two quantities differ in detail, and are therefore not equivalent.  The high-resolution isotropic DNS data are accessible via novel cyberinfrastructure tools built upon the data housed in the Johns Hopkins Turbulence Database (JHTDB). This new suite of database access tools is based on python notebooks and provides fast and stable operation on the existing turbulence data sets while enabling user-programmable, server-side computations.   

Statistical equilibrium of large scales in three-dimensional hydrodynamic turbulence

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimeter-scale magnetic stirrers immersed in a large fluid reservoir. By measuring the statistics of the fluid velocity field, we experimentally evidence that the large scales in 3D turbulence are in statistical equilibrium, a regime predicted seventy years ago, but had not been reported experimentally so far. This equipartition regime of large scales can then be described with an effective temperature, although it is not isolated from the turbulent Kolmogorov cascade that develops towards small scales. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. These findings pave the way to use classical concepts of equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.

J.-B. Gorce and E. Falcon, Statistical Equilibrium of Large Scales in Three-Dimensional Hydrodynamic Turbulence, in press in Physical Review Letters (2022).   

Author not Attending

Despite significant experimental, computational and theoretical advances, a complete general mathematical framework for the turbulent motion of fluids has yet to be put forth. Here we propose such a modelling framework, by establishing a set of coupled equations of motion corresponding to the fluid's turbulent kinetic energy density, which are derived from the Navier-Stokes equations for an incompressible fluid. This approach most notably involves an amplitude-projection of the whole turbulent flow field onto a complete set of basis functions spanning the domain volume, followed by a volume average to simplify the representation. The result is a detailed account of the kinetic energy exchange between the projected amplitudes across all length scales of turbulent fluctuation. The nonlinear convective term in the Navier-Stokes equation specifically determines the selection rules for amplitude exchange, and yields the Kolmogorov energy cascade.   

Lagrangian curvature statistics from Gaussian sub-ensembles in turbulent von-Kármán flow

A salient feature of fully turbulent flows far from onset is the intermittent occurrence of extreme fluctuations at small spatial and temporal scales. Here, we derive an expression for the curvature probability density function (pdf) for the ensemble of tracer particle trajectories in isotropic turbulence that includes effects of spatio-temporal intermittency. We obtain a master curve for the pdf for near-Gaussian sub-ensembles, generated by conditioning on the squared acceleration coarse-grained over a few viscous time units (Bentkamp, Lalescu, Wilczek, Nat. Comm., 10, 3550 (2019)), where an analytic form of the pdf is known (Xu, Ouellette, Bodenschatz, Phys. Rev. Lett., 98, 050201 (2007)). Using this expression, we calculate the pdf for the full ensemble resulting in a re-scaled version the master curve. The scaling factor is related to moments of the coarse-grained acceleration, and thus includes the effect of spatio-temporal intermittency. The derived pdf agrees qualitatively and quantitatively with the curvature pdf sampled from tracer particle data in von-Kármán flow obtained by Shake-The-Box processing (Schanz, Gesemann, Schröder, Exp. Fluids 57:70 (2016)).   

Scaling of Lagrangian acceleration in isotropic turbulence at high Reynolds numbers

The acceleration of a fluid element in a turbulent flow, given by the Lagrangian derivative of the velocity, and resulting from balance of forces acting on it, is the simplest description of its motion. Hence, the statistics of acceleration are of paramount importance from both a fundamental viewpoint and for modeling purposes. Here, we examine the scaling of acceleration moments by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our work highlights the urgent need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.   

Dynamic Phase Alignment in Navier-Stokes Turbulence

In Navier-Stokes turbulence, energy and helicity injected at large scales are subject to a joint direct cascade, with both quantities exhibiting a spectral scaling $\sim k^{-5/3}$. A ``na\"ive", dimensional estimate of the spectrum of helicity would, however, yield the scaling $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \sim k^{-2/3}$, violating conservation of the helicity flux in the inertial range. We demonstrate via direct numerical simulations that this apparent contradiction is revolved because of the existence of a strong scale-dependent Fourier phase alignment between velocity and vorticity fluctuations, with the phase alignment angle scaling as $\cos\alpha_k\propto k^{-1}$ [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Navier-Stokes Turbulence", Physical Review Letters, 2021]. This strong dependence on scale of $\cos\alpha_k$, termed \textit{dynamic phase alignment}, underpins the spectral scaling of helicity, i.e., $\mathcal{H}(k) \sim k v_{\lambda} \omega_{\lambda} \cos\alpha_k \sim k^{-5/3}$. Dynamic phase alignment plays a role in the turbulent dynamics in the presence of two invariants beyond Navier-Stokes, and it has been shown to underpin the joint direct cascade of energy and (generalized) helicity in a variety of turbulent plasma environments [L. M. Milanese \textit{et al.}, ``Dynamic Phase Alignment in Inertial Alfv\'en Turbulence", Physical Review Letters, 2020].   

Characteristics of significant and insignificant regions in isotropic turbulence

Significant and insignificant regions of isotropic turbulence are classified based on their sensitivity to localized perturbations.

The growth of the perturbations is assessed by running a massive amount of simulations $(\sim\mkern-5mu 10^6)$ with perturbed initial conditions at different locations.

The regions of the flow where perturbations lie in the top 5\% of amplification after a given time are labeled as `significant' and those in the bottom 5\% as `insignificant'.

The properties of both regions are studied, uncovering several differences between both regions. First, significant regions are found to have intense kinetic energy, enstrophy and dissipation. The converse applies for insignificant regions which are weak compared to the background flow. Studying their topology shows that in significant regions, the rate of strain dominates the vorticity, which results in the invariants of the velocity gradient tensor, $Q$ and $R$, leaning towards the Vieillefosse tail. Insignificant regions are dominated by vorticity, and thus dominated by positive $Q$ and neutral $R$. In conclusion, it is shown that straining motions are more efficient at propagating perturbations than vortical ones.   

Isotropy, super-isotropy, and the extension of von Karman-Howarth equation: a Lundgren-equation based probability density function approach and its solution to homogeneous isotropic turbulence

We analyze homogeneous isotropic turbulence (HIT) considering Lundgren's (1967), infinite probability density functions (PDF) integro-differential equation hierarchy (IDE), which is a complete description of turbulence statistics. Since there is no mean velocity, the one-point PDF equation vanishes for HIT and the ensuing two-point equation, allows a spherical dimensional reduction. For further dimensional reduction of the higher multi-point equation, we introduce the new concept of super-isotropy. This leads to another significant dimensional reduction and each of the infinite equations then depends only on the spherical radius and the spherical velocity as sample-space variables. The corresponding side conditions of the PDF hierarchy are also derived. To solve the linear system, we formally introduce (i) a product ansatz and (ii) the superposition principle. Using the product ansatz together with the permutation side condition, we derive a new scalar IDE for the PDF that is truly closed, but non-linear. Further side conditions necessitate the superposition of solutions of the latter IDE. With the above approach, we have significantly generalized the Karman-Howarth equation.