Dissipation-Range Fluid Turbulence and Thermal Noise

Eyink, Gregory L.; Bandak, Dmytro; Goldenfeld, Nigel; Mailybaev, Alexei A.

doi:10.48550/arxiv.2107.13954

Cited by 3 publications

(14 citation statements)

References 96 publications

(210 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted here that this structure function is related to the scalar spectrum commonly considered in turbulence theory by the relation 2 . The result which has been confirmed by experiment is a power-law scaling…”

Section: Introductionmentioning

confidence: 90%

“…The prefactor √ 2ηk B T is chosen according to the standard fluctuation-dissipation relation so that the correct Gibbs equilibrium distribution is obtained for the equaltime velocity statistics, with energy equipartition among wave-number modes. For example, see [2], Appendix A, for a careful discussion.…”

Section: A Fluid At Restmentioning

confidence: 99%

“…The second difference, the neglect of the nonlinear advection term, is justified by the renormalization group analysis of [75,76], which implies that in thermal equilibrium the nonlinearity becomes negligible at sufficiently low wavenumbers and frequencies. A quantitative estimate provided in [2] implies that the nonlinear coupling should be weak except for length scales of order the radius of the fluid molecules, where no hydrodynamic description is valid in any case. Finally, the key assumption of DFvE that tracer particles are advected by the smoothed velocity field ( 14) is intuitively plausible, since such particles can feel only a resultant velocity averaged over fluctuations at a scale smaller than their size σ.…”

Section: A Fluid At Restmentioning

confidence: 99%

“…Recent work [1][2][3][4] has sparked renewed interest in the effects of thermal noise on turbulent flow, a problem much neglected since the pioneering work of Betchov more than 60 years ago [5][6][7]. These new studies have confirmed Betchov's insight that the dissipation range of turbulent flows must be strongly affected by thermal noise.…”

Section: Introductionmentioning

confidence: 99%

“…These new studies have confirmed Betchov's insight that the dissipation range of turbulent flows must be strongly affected by thermal noise. In particular, the energy spectrum below the Kolmogorov dissipation scale [8], which has long been expected to exhibit an exponential decay [9][10][11][12][13][14][15][16], instead demonstrates an equilibrium equipartition energy spectrum in numerical simulations which incorporate molecular noise [1][2][3]. The question remains which turbulent processes at sub-Kolmogorov length scales can be essentially altered by such noise.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

High Schmidt-Number Turbulent Advection and Giant Concentration Fluctuations

Eyink¹,

Jafari²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the effects of thermal noise on the Batchelor-Kraichnan theory of high Schmidtnumber mixing in the viscous dissipation range of turbulent flows at sub-Kolmogorov scales. Starting with the nonlinear Landau-Lifschitz fluctuating hydrodynamic equations 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, a Gaussian random velocity with spatially-constant strain but white-noise in time. Then, following prior work of Donev, Fai & vanden-Eijnden, we derive asymptotic high-Schmidt limiting equations for the concentration field, in which the thermal velocity fluctations are exactly represented by a Gaussian random velocity that is likewise white in time. We obtain the exact solution for concentration spectrum in this high-Schmidt limiting model, showing that the Batchelor prediction in the viscousconvective 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, either Gaussian or exponential, we obtain a k −2 power-law spectrum over a couple of decades starting just below the Batchelor length. This spectrum corresponds to non-equilibrium giant concentration fluctuations (GCF's), which are due to the imposed concentration variations being advected by thermal velocity fluctuations and 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.

show abstract

Section: Introductionmentioning

confidence: 90%

Section: A Fluid At Restmentioning

confidence: 99%

Section: A Fluid At Restmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

High Schmidt-Number Turbulent Advection and Giant Concentration Fluctuations

Eyink¹,

Jafari²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Ab initio method of calculating invariant measure for turbulent flow

Macháček

2022

Phys. Scr.

View full text Add to dashboard Cite

We present a method of calculating the invariant measure (IM) of hydrodynamical systems describing incompressible viscous fluid flow in bounded 3D volume~$V$. Using a basis $\{\wv_I\}$ of zero-divergence vector functions on $V$ satisfying boundary conditions we transform the Navier-Stokes equations (NSE) for the velocity field $\vv(\xv,t)=\sum_I v_I(t)\,\wv_I(\xv)$ into a simple system of ordinary differential equations for $v_I(t)$. The fact that fluid consists of a finite number of molecules in thermal motion implies that the number~$N$ of variables $v_I$ is finite too and that the system of equations for $v_I(t)$ contains a white-noise term. We prove that all solutions of this system are global, the IM exists and is unique. Its density $\psi(v)$ defined on the phase space $\mathbb{R}^N$ satisfies an elliptic partial differential equation (PDE). Expanding $\psi(v)=\sum_\mu\bla \He_\mu\bra\,\He_\mu(v)\,\phe(v)$ into Hermite functions $\He_\mu(v)\,\phe(v)$ (where $\phe$ is a Gaussian function on $\mathbb{R}^N$, $\He_\mu$ are Hermite polynomials orthonormal with the weight $\phe$ and $\bla\He_\mu\bra=\int\He_\mu(v)\,\psi(v)\,d^Nv $ are IM mean values of $\He_\mu$) we transform this PDE into an infinite system of linear algebraic equations for $\bla\He_\mu\bra$ and suggest some methods for solving it numerically. From known first- and second-order $\bla\He_\mu\bra$ we could easily calculate mean turbulent velocity $\bla\vv(\xv)\bra$ and Reynolds stress at any point $\xv$. Cylindrical pipe flow is used to illustrate the method everywhere in the paper. All conclusions are derived from the mathematical model by rigorous mathematics, with no further assumptions. All approximations can be arbitrarily improved.

show abstract