arXiv subjects

Gregory Eyink

Publications and source records attributed to Gregory Eyink.

At least 19 recordsLinked to original sources

arXiv2026

Data-driven impeller model for efficient large eddy simulations of metastable von K\'arm\'an flows

The von K\'arm\'an turbulent swirling flow exhibits intriguing large-scale metastable dynamics, including low-frequency state switching. The study of state switching demands long-duration high-fidelity simulations at high Reynolds numbers that capture the flow generated by the impellers. Blade-resolved Large Eddy Simulations (LES) are computationally prohibitive, limiting access to these slow dynamics. Here, we develop a model for the action of the impellers on the flow using experimental data from Particle Image Velocimetry (PIV) and torque measurements of the von K\'arm\'an flow. The impeller-region velocity is parametrized via B-splines and coupled to the LES through momentum forcing. An initial set of B-spline coefficients is inferred using the Optimizing a DIscrete Loss (ODIL) framework constrained by the Reynolds-Averaged Navier--Stokes (RANS) equations, PIV measurements in the optically accessible portion of the device, and impeller torque measurements. The coefficients are then refined by the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), which minimizes the discrepancy between the LES time-averaged velocity and torque and their experimental counterparts. Using the data-driven impeller model, we perform long-duration LES of the von K\'arm\'an flow. We find that the simulation reproduces the mean flow in the bulk and displays metastable state-switching dynamics. We further show that these metastable states are not axisymmetric and consist of an alternating four-cell flow pattern that slowly rotates around the axis of the cylindrical vessel. The proposed approach provides a practical and computationally efficient route to investigating large-scale dynamics in impeller-driven turbulent flows.

physics.flu-dyn↗

arXiv2026

On the importance of stochasticity in closures of turbulence

Deterministic closures for coarse-grained turbulence models help reproduce mean statistics, but often fail to capture the finite-time growth of uncertainty. Using the framework of shell models as a quantitative multi-scale testbed, we compare fully resolved simulations with large-eddy simulations using either stochastic or deterministic subgrid closures. While in the fully resolved system a single microscopic perturbation is rapidly amplified by strongly chaotic dynamics, truncation produces a strong delay and suppression of variance growth when uncertainty is introduced through initial condition perturbations only. We show that a data-driven Langevin-type stochastic closure restores the correct timing and magnitude of variance growth across scales, demonstrating that sustained stochasticity is essential for predictability in reduced turbulent dynamics.

physics.flu-dyn↗

arXiv2025

Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equations

We present an operator learning framework for solving non-perturbative functional renormalization group equations, which are integro-differential equations defined on functionals. Our proposed approach uses Gaussian process operator learning to construct a flexible functional representation formulated directly on function space, making it independent of a particular equation or discretization. Our method is flexible, and can apply to a broad range of functional differential equations while still allowing for the incorporation of physical priors in either the prior mean or the kernel design. We demonstrate the performance of our method on several relevant equations, such as the Wetterich and Wilson--Polchinski equations, showing that it achieves equal or better performance than existing approximations such as the local-potential approximation, while being significantly more flexible. In particular, our method can handle non-constant fields, making it promising for the study of more complex field configurations, such as instantons.

cs.LG↗

arXiv2025

Lighthill's mechanism and vorticity cascade in the logarithmic layer of wall turbulence

We investigate Lighthill's proposed turbulent mechanism for near-wall concentration of spanwise vorticity by calculating mean flows conditioned on motion away from or toward the wall in an $Re_\tau=1000$ database of plane-parallel channel flow. Our results corroborate Lighthill's proposal throughout the entire logarithmic layer, but extended by counterflows that help explain anti-correlation of vorticity transport by advection and by stretching/tilting. We present evidence also for Lighthill's hypothesis that the vorticity transport in the log-layer is a ``cascade process'' through a scale-hierarchy of eddies, with intense competition between transport outward from and inward to the wall. Townsend's model of attached eddies of hairpin-vortex type accounts for half of the vorticity cascade, whereas we identify necklace-type or ``shawl vortices'' that envelop turbulent sweeps as supplying the other half.

physics.flu-dyn↗

arXiv2024

A Josephson-Anderson relation for drag in classical channel flows with streamwise periodicity: Effects of wall roughness

The detailed Josephson-Anderson relation equates instantaneous work by pressure drop over any streamwise segment of a general channel and wall-normal flux of spanwise vorticity spatially integrated over that section. This relation was first derived by Huggins for quantum superfluids, but it holds also for internal flows of classical fluids and for external flows around solid bodies, corresponding there to relations of Burgers, Lighthill, Kambe, Howe and others. All of these prior results employ a background potential Euler flow with the same inflow/outflow as the physical flow, just as in Kelvin's minimum energy theorem, so that the reference potential incorporates information about flow geometry. We here generalize the detailed Josephson-Anderson relation to streamwise periodic channels appropriate for numerical simulation of classical fluid turbulence. We show that the original Neumann b.c. used by Huggins for the background potential create an unphysical vortex sheet in a periodic channel, so that we substitute instead Dirichlet b.c. We show that the minimum energy theorem still holds and our new Josephson-Anderson relation again equates work by pressure drop instantaneously to integrated flux of spanwise vorticity. The result holds for both Newtonian and non-Newtonian fluids and for general curvilinear walls. We illustrate our new formula with numerical results in a periodic channel flow with a single smooth bump, which reveals how vortex separation from the roughness element creates drag at each time instant. Drag and dissipation are thus related to vorticity structure and dynamics locally in space and time, with important applications to drag-reduction and to explanation of anomalous dissipation at high Reynolds numbers.

physics.flu-dyn↗

arXiv2024

Onsager's "Ideal Turbulence" Theory

Lars Onsager in 1945-1949 made an exact analysis of the high Reynolds-number limit for individual turbulent flow realizations modeled by incompressible Navier-Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea, that such flows are well-described by distributional or "weak" solutions of ideal Euler equations. 1/3 H\"older singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov 4/5th law. The approach is closely related to renormalization group methods in physics and envisages "conservation-law anomalies", as discovered later in quantum field theory. There are also deep connections with Large-Eddy Simulation modeling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his 1/3 H\"older singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or "convex integration" techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting "spontaneously stochastic" behavior of high-Reynolds number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space-time local phenomena. This theory thus makes testable predictions and offers new perspectives on Large-Eddy Simulation in presence of solid walls.

physics.flu-dyn↗

arXiv2023

Vorticity cascade and turbulent drag in wall-bounded flows: plane Poiseuille flow

Drag for wall-bounded flows is directly related to flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution arises from cross-stream "vorticity cascade" by nonlinear advection and stretching of vorticity. We study this process using numerical simulation data of turbulent channel flow at $Re_τ=1000$. The net transfer from the wall of fresh spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is "down-gradient" or outward from the wall, where most vorticity concentrates, and the other which is "up-gradient" or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient transport occurs by a mechanism of correlated inflow and spanwise vortex stretching that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra that identify eddies involved in nonlinear vorticity transport in the two opposing directions, and (iii) visualizations of coherent vortex structures which contribute dominantly to the transport. The "D-type" vortices contributing dominantly to down-gradient transport in the log-layer are found to be attached, hairpin-type vortices. However, the "U-type" vortices contributing dominantly to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.

physics.flu-dyn↗

arXiv2022

The Kraichnan Model and Non-Equilibrium Statistical Physics of Diffusive Mixing

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.

cond-mat.stat-mech↗

arXiv2022

High Schmidt-Number Turbulent Advection and Giant Concentration Fluctuations

We consider the effects of thermal noise on the Batchelor-Kraichnan theory of high Schmidt-number mixing in the viscous-dissipation range of turbulent flows. Using fluctuating hydrodynamics for a binary fluid mixture at low Mach numbers, we justify linearization around the deterministic Navier-Stokes solution in the dissipation range. For the latter solution we adopt the standard Kraichnan model and derive asymptotic high-Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations are exactly represented by a Gaussian random velocity which is white in time. We obtain the exact solution for the concentration spectrum in this high-Schmidt limiting model, showing that the Batchelor prediction in the viscous-convective range is unaltered. Thermal noise dramatically renormalizes the bare diffusivity in this range, but the effect is the same as in laminar flow and thus hidden phenomenologically. However, in the viscous-diffusive range at scales below the Batchelor length (typically micron scales) the predictions based on deterministic Navier-Stokes equations are drastically altered by thermal noise. Whereas the classical theories predict rapidly decaying spectra in the viscous-diffusive range, we obtain a $k^{-2}$ power-law spectrum starting just below the Batchelor length. This spectrum corresponds to non-equilibrium giant concentration fluctuations, due to the imposed concentration variations advected by thermal velocity fluctuations which are experimentally well-observed in quiescent fluids. At higher wavenumbers, the concentration spectrum instead goes to a $k^2$ equipartition spectrum due to equilibrium molecular fluctuations. We work out detailed predictions for water-glycerol and water-fluorescein mixtures. Finally, we discuss broad implications for turbulent flows and novel applications of our methods to experimentally accessible laminar flows.

physics.flu-dyn↗

arXiv2021

Dissipation-Range Fluid Turbulence and Thermal Noise

We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equation of Landau and Lifschitz is more appropriate. In particular, our analysis implies that both the exponentially decaying energy spectrum and the far-dissipation range intermittency predicted by Kraichnan for deterministic Navier-Stokes will be generally replaced by Gaussian thermal equipartition at scales just below the Kolmogorov length. Stochastic shell model simulations at high Reynolds numbers verify our theoretical predictions and reveal furthermore that inertial-range intermittency can propagate deep into the dissipation range, leading to large fluctuations in the equipartition length scale. We explain the failure of previous scaling arguments for the validity of deterministic Navier-Stokes equations at any Reynolds number and we provide a mathematical interpretation and physical justification of the fluctuating Navier-Stokes equation as an ``effective field-theory'' valid below some high-wavenumber cutoff $Λ$, rather than as a continuum stochastic partial differential equation. At Reynolds number around a million the strongest turbulent excitations observed in our simulation penetrate down to a length-scale of microns. However, for longer observation times or higher Reynolds numbers, more extreme turbulent events could lead to a local breakdown of fluctuating hydrodynamics.

physics.flu-dyn↗

arXiv2021

Thermal Fluctuations in the Dissipation Range of Homogeneous Isotropic Turbulence

Using fluctuating hydrodynamics we investigate the effect of thermal fluctuations in the dissipation range of homogeneous, isotropic turbulence. Simulations confirm theoretical predictions that the energy spectrum is dominated by these fluctuations at length scales comparable to the Kolmogorov length. We also find that the extreme intermittency in the far-dissipation range predicted by Kraichnan is replaced by Gaussian thermal equipartition.

physics.flu-dyn↗

arXiv2021

Spatio-temporal correlations in 3D homogeneous isotropic turbulence

We use Direct Numerical Simulations (DNS) of the forced Navier-Stokes equation for a 3-dimensional incompressible fluid in order to test recent theoretical predictions. We study the two- and three-point spatio-temporal correlation functions of the velocity field in stationary, isotropic and homogeneous turbulence. We compare our numerical results to the predictions from the Functional Renormalization Group (FRG) which were obtained in the large wavenumber limit. DNS are performed at various Reynolds numbers and the correlations are analyzed in different time regimes focusing on the large wavenumbers. At small time delays, we find that the two-point correlation function decays as a Gaussian in the variable $kt$ where $k$ is the wavenumber and $t$ the time delay. The three-point correlation function, determined from the time-dependent advection-velocity correlations, also follows a Gaussian decay at small $t$ with the same prefactor as the one of the two-point function. These behaviors are in precise agreement with the FRG results, and can be simply understood as a consequence of sweeping. At large time delays, the FRG predicts a crossover to an exponential in $k^2 t$, which we were not able to resolve in our simulations. However, we analyze the two-point spatio-temporal correlations of the modulus of the velocity, and show that they exhibit this crossover from a Gaussian to an exponential decay, although we lack of a theoretical understanding in this case. This intriguing phenomenon calls for further theoretical investigation.

physics.flu-dyn↗

arXiv2015

Resonance Van Hove Singularities in Wave Kinetics

Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space $D=(N-2)d$ ($d$ physical space dimension, $N$ the number of waves in resonance) and the degree of degeneracy $δ$ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for $D>2$ but produce divergences when $D\leq 2$ and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when $D-δ\leq 2,$ as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell \& Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.

physics.flu-dyn↗

arXiv2010

Robert H. Kraichnan

Robert Harry Kraichnan {(1928-2008)} was one of the leaders in the theory of turbulence for a span of about forty years (mid-fifties to mid-nineties). Among his many contributions, he is perhaps best known for his work on the inverse energy cascade (i.e. from small to large scales) for forced two-dimensional turbulence. This is a review of Kraichnan's main scientific contributions.

physics.hist-ph↗

arXiv2009

Matrix Exponential-Based Closures for the Turbulent Subgrid-Scale Stress Tensor

Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the `Recent Fluid Deformation' (RFD) approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the product of the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in Large Eddy Simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other `nonlinear scrambling' terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy.

physics.flu-dyn↗

arXiv2008

A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence

A public database system archiving a direct numerical simulation (DNS) data set of isotropic, forced turbulence is described in this paper. The data set consists of the DNS output on $1024^3$ spatial points and 1024 time-samples spanning about one large-scale turn-over timescale. This complete $1024^4$ space-time history of turbulence is accessible to users remotely through an interface that is based on the Web-services model. Users may write and execute analysis programs on their host computers, while the programs make subroutine-like calls that request desired parts of the data over the network. The users are thus able to perform numerical experiments by accessing the 27 Terabytes of DNS data using regular platforms such as laptops. The architecture of the database is explained, as are some of the locally defined functions, such as differentiation and interpolation. Test calculations are performed to illustrate the usage of the system and to verify the accuracy of the methods. The database is then used to analyze a dynamical model for small-scale intermittency in turbulence. Specifically, the dynamical effects of pressure and viscous terms on the Lagrangian evolution of velocity increments are evaluated using conditional averages calculated from the DNS data in the database. It is shown that these effects differ considerably among themselves and thus require different modeling strategies in Lagrangian models of velocity increments and intermittency.

physics.flu-dyn↗

arXiv2002

Statistical Analysis of a Semilinear Hyperbolic System Advected by a White in Time Random Velocity Field

We study a system of semilinear hyperbolic equations passively advected by smooth white noise in time random velocity fields. Such a system arises in modeling non-premixed isothermal turbulent flames under single-step kinetics of fuel and oxidizer. We derive closed equations for one-point and multi-point probability distribution functions (PDFs) and closed form analytical formulas for the one point PDF function, as well as the two-point PDF function under homogeneity and isotropy. Exact solution formulas allows us to analyze the ensemble averaged fuel/oxidizer concentrations and the motion of their level curves. We recover the empirical formulas of combustion in the thin reaction zone limit and show that these approximate formulas can either underestimate or overestimate average concentrations when reaction zone is not tending to zero. We show that the averaged reaction rate slows down locally in space due to random advection induced diffusion; and that the level curves of ensemble averaged concentration undergo diffusion about mean locations.

nlin.SI↗

arXiv1996

Universality of the Inertial-Convective Range in Kraichnan's Model of a Passive Scalar

We establish by exact, nonperturbative methods a universality for the correlation functions in Kraichnan's ``rapid-change'' model of a passively advected scalar field. We show that the solutions for separated points in the convective range of scales are unique and independent of the particular mechanism of the scalar dissipation. Any non-universal dependences therefore must arise from the large length-scale features. The main step in the proof is to show that solutions of the model equations are unique even in the idealized case of zero diffusivity, under a very modest regularity requirement (square-integrability). Within this regularity class the only zero-modes of the global many-body operators are shown to be trivial ones (i.e. constants). In a bounded domain of size $L$, with physical boundary conditions, the ``ground-state energy'' is strictly positive and scales as $L^{-γ}$ with an exponent $γ>0$.

chao-dyn↗