[Paper Review] Thermal noise competes with turbulent fluctuations below millimeter scales

Dmytro Bandak, Gregory L. Eyink+2 more|arXiv (Cornell University)|Jul 7, 2021

Fluid Dynamics and Turbulent Flows4 references4 citations

TL;DR

This paper challenges the long-standing assumption that thermal noise is negligible in the turbulent dissipation range by demonstrating through statistical mechanics estimates and simulations of the Sabra shell model that thermal fluctuations compete with turbulent eddies across sub-millimeter scales down to the molecular mean free path. The key finding is that thermal effects are non-negligible throughout the dissipation range, implying a fundamental interplay between thermal and turbulent fluctuations in small-scale fluid dynamics.

ABSTRACT

Turbulent flows frequently accompany physical, chemical and biological processes, such as mixing, two-phase flow, combustion and even foraging by bacteria and plankton larvae, all of which are in principle subject to thermal fluctuations already on scales of several microns. Nevertheless the large separation between the millimeter scale at which turbulent fluctuations begin to be strongly damped and the mean free path of the fluid has been generally assumed to imply that thermal fluctuations are irrelevant to the turbulent dissipation range. Here we use statistical mechanical estimates to show that thermal fluctuations are not negligible compared to turbulent eddies in the dissipation range. Simulation of the Sabra shell model shows that intermittent bursts of turbulence lead to a fluctuating length scale below which thermal fluctuations are important: over three decades of length, from sub-millimeter scales down to the mean free path, thermal fluctuations coexist with hydrodynamics. Our results imply that thermal fluctuations cannot be neglected when modeling turbulent phenomena in the far dissipation range.

Motivation & Objective

• To challenge the widespread assumption that thermal fluctuations are irrelevant in the turbulent dissipation range.
• To investigate whether thermal noise significantly affects hydrodynamic behavior at scales below the Kolmogorov dissipation scale.
• To provide theoretical and numerical evidence that thermal and turbulent fluctuations coexist and compete in the far dissipation range.
• To extend Betchov's early insights on thermal noise in turbulence to include inertial-range intermittency.
• To assess the implications for modeling high Schmidt-number mixing, droplet formation, and microorganism locomotion at small scales.

Proposed method

• Using statistical mechanical estimates to compare thermal fluctuation amplitudes with turbulent velocity fluctuations across scales.
• Simulating the stochastic Sabra shell model with a multiplicative noise term representing thermal fluctuations.
• Applying the fluctuation-dissipation theorem to define the covariance of the stochastic stress tensor τij in the Navier-Stokes equation.
• Analyzing structure functions of velocity increments to detect equipartition of energy between thermal and turbulent modes.
• Evaluating the effective Reynolds number of thermal fluctuations, Re_th^ℓ ∝ (k_B T / ρ ν² ℓ^{d-2})^{1/2}, to assess coupling strength at small scales.
• Using dimensional analysis and renormalization group insights to interpret the UV behavior of thermal fluctuations in different spatial dimensions.

Experimental results

Research questions

• RQ1 At what length scales do thermal fluctuations become comparable to turbulent velocity fluctuations in the dissipation range?
• RQ2 To what extent does inertial-range intermittency influence the crossover between turbulent and thermal dominance in small-scale fluid dynamics?
• RQ3 How does the dimensionality of space affect the relative strength and coupling of thermal and turbulent fluctuations?
• RQ4 Can the fluctuation-dissipation theorem be consistently applied to turbulent flows at scales near the Kolmogorov microscale?
• RQ5 What are the observable consequences of thermal noise on passive scalar mixing and other sub-Kolmogorov scale processes?

Key findings

• Thermal fluctuations are not negligible in the turbulent dissipation range; they compete with turbulent eddies over a three-decade range of scales from sub-millimeter down to the mean free path.
• The structure function of order p exhibits a crossover to equipartition with thermal noise at shell number N_e(p), which depends strongly on p, with stronger thermal effects for negative p.
• For negative p, thermal effects are detectable even within the traditional inertial range, indicating that thermal noise influences scales larger than the Kolmogorov scale.
• The effective thermal Reynolds number Re_th^ℓ ∝ (k_B T / ρ ν² ℓ^{d-2})^{1/2} becomes large for d > 2 at small scales, but remains weakly coupled in 3D fluids until molecular scales are reached.
• The shell model, despite its effective d=0, accurately captures the weak-coupling regime of thermal fluctuations in 3D, validating its use for studying thermal effects in turbulence.
• The results imply that thermal noise must be included in models of high Schmidt-number scalar mixing, droplet formation, and microorganism motility at sub-Kolmogorov scales.