Thermal Velocity Fluctuations Research Articles

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 at sub-Kolmogorov scales. Starting with the nonlinear Landau-Lifshitz 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, and Vanden-Eijnden [J. Stat. Mech.: Theory Exp. (2014) P04004], we derive asymptotic high Schmidt limiting equations for the concentration field, in which the thermal velocity fluctuations 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 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, either Gaussian or exponential, we obtain a ${k}^{\ensuremath{-}2}$ power-law spectrum over a couple of decades starting just below the Batchelor length. This spectrum corresponds to nonequilibrium giant concentration fluctuations, 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 wave numbers, 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 applications of our methods to experimentally accessible laminar flows.

A Thermostat‐Consistent Fully Coupled Molecular Dynamics–Generalized Fluctuating Hydrodynamics Model for Non‐Equilibrium Flows

AbstractThe thermostat‐consistent fully coupled molecular dynamics–generalized fluctuating hydrodynamics method is developed for non‐equilibrium water flow simulations. The model allows for strong coupling between the atomistic and the continuum hydrodynamics representations of water and shows an improved stability in comparison with the previous formulations of similar multiscale methods. Numerical results are demonstrated for a periodic nano‐scale Poiseuille flow problem with SPC/E water. The computed time‐averaged velocity profiles are compared with the analytical solution, and the thermal velocity fluctuations are well reproduced in comparison with the equilibrium molecular dynamics simulation. Several options to account for the long‐range electrostatics interactions available in GROMACS are incorporated in the model and compared. It is demonstrated that the suggested non‐equilibrium multiscale model is a factor of 4 to 18 faster in comparison with the standard all‐atom equilibrium molecular dynamics model for the same computational domain size.

Nonequilibrium Casimir pressures in liquids under shear.

In stationary nonequilibrium states coupling between hydrodynamic modes causes thermal fluctuations to become long ranged inducing nonequilibrium Casimir pressures. Here we consider nonequilibrium Casimir pressures induced in liquids by a velocity gradient. Specifically, we have obtained explicit expressions for the magnitude of the shear-induced pressure enhancements in a liquid layer between two horizontal plates that complete and correct results previously presented in the literature. In contrast to nonequilibrium Casimir pressures induced by a temperature or concentration gradient, we find that in shear nonequilibrium contributions from short-range fluctuations are no longer negligible. In addition, it is noted that currently available computer simulations of model fluids in shear observe effects from molecular correlations at nanoscales that have a different physical origin and do not probe shear-induced pressures resulting from coupling of long-wavelength hydrodynamic modes. Even more importantly, we find that in actual experimental conditions, shear-induced pressure enhancements are caused by viscous heating and not by thermal velocity fluctuations. Hence, isothermal computer simulations are irrelevant for the interpretation of experimental shear-induced pressure enhancements.

Droplets behavior of subcooled dispersed phase under nucleate boiling of continuous phase of liquid emulsion

The paper considers the influence of the dispersed phase of the emulsion upon the nature of heat and mass transfer in systems of immiscible liquids. In particular, is studied the interfacial interaction and breakage of subcooled drops of the dispersed phase during bubble boiling of the continuous phase of the emulsion. They predetermine the complex nature of the joint behavior of hydrodynamic and thermal processes. It is known that the controlled conditions for dispersion of emulsions in MQCL method under the minimum amount of cooling lubrication be able to improve the quality of metal processing by 10–18% (Maruda et al., 2017). Phase changes liquid generate in it a special kind of pulsation motion: «hot» turbulence of the superheated continuous phase. She is generated by process formation of nuclei and growth of steam bubbles accompanying by the pressure pulsations. Thus heat energy is converted into mechanical energy of the perturbed by the movement of vapor bubbles in evaporating liquid. The formula for the evaluating perturbations of the continuous phase of the emulsion under the influence of vapor bubbles is proposed. The behavior of these fluctuations depends on the frequency of the viable vapor phase nuclei formation related to the heat flow density. The dependence, obtained by Labuntsov (2000) for boiling homogeneous liquid, is used as a characteristic of the thermal velocity.Taking into account the character of the boiling of the continuous phase of emulsion, the generalization of the Kolmogorov-Hinze model has been performed for breakage drops in isothermal flow of emulsion under the influence of turbulent fluctuations of velocity. A correlation model of “thermal” breakup is proposed, which is complementary to the physical representations of the breakup for subcooled drops during the liquid emulsion boiling. Using dimensionless criteria was justified the form of the correlation for the maximum stable droplet size and thermal velocity fluctuations. This makes it possible to relate mechanisms hydrodynamic and thermal processes at use of usual computer tools used in practice for engineering calculations. The correlation for the maximum stable droplet size and thermal velocity fluctuations was developed in a dimensionless form. The results of the model studies were evaluated using the experimental data on the breakup of drops of oil in coolant “oil-water” (Januszkiewicz et al., 2004). They were obtained under conditions of creeping flow in a narrow channel, eliminating the creation of ordinary hydrodynamic turbulence. A good agreement with experimental data indicates that the present model would be promising for the extension of model representations about the breakup of droplets in complex processes of heat and mass transfer in liquid emulsions.

Fluctuation-enhanced electric conductivity in electrolyte solutions

We analyze the effects of an externally applied electric field on thermal fluctuations for a binary electrolyte fluid. We show that the fluctuating Poisson-Nernst-Planck (PNP) equations for charged multispecies diffusion coupled with the fluctuating fluid momentum equation result in enhanced charge transport via a mechanism distinct from the well-known enhancement of mass transport that accompanies giant fluctuations. Although the mass and charge transport occurs by advection by thermal velocity fluctuations, it can macroscopically be represented as electrodiffusion with renormalized electric conductivity and a nonzero cation-anion diffusion coefficient. Specifically, we predict a nonzero cation-anion Maxwell-Stefan coefficient proportional to the square root of the salt concentration, a prediction that agrees quantitatively with experimental measurements. The renormalized or effective macroscopic equations are different from the starting PNP equations, which contain no cross-diffusion terms, even for rather dilute binary electrolytes. At the same time, for infinitely dilute solutions the renormalized electric conductivity and renormalized diffusion coefficients are consistent and the classical PNP equations with renormalized coefficients are recovered, demonstrating the self-consistency of the fluctuating hydrodynamics equations. Our calculations show that the fluctuating hydrodynamics approach recovers the electrophoretic and relaxation corrections obtained by Debye-Huckel-Onsager theory, while elucidating the physical origins of these corrections and generalizing straightforwardly to more complex multispecies electrolytes. Finally, we show that strong applied electric fields result in anisotropically enhanced "giant" velocity fluctuations and reduced fluctuations of salt concentration.

A reversible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick’s law

We study diffusive mixing in the presence of thermal fluctuations under the assumption of large Schmidt number. In this regime we obtain a limiting equation that contains a diffusive stochastic drift term with diffusion coefficient obeying a Stokes–Einstein relation, in addition to the expected advection by a random velocity. The overdamped limit correctly reproduces both the enhanced diffusion in the ensemble-averaged mean and the long-range correlated giant fluctuations in individual realizations of the mixing process, and is amenable to efficient numerical solution. Through a combination of Eulerian and Lagrangian numerical methods we demonstrate that diffusion in liquids is not most fundamentally described by Fick’s irreversible law; rather, diffusion is better modeled as reversible random advection by thermal velocity fluctuations. We find that the diffusion coefficient is effectively renormalized to a value that depends on the scale of observation. Our work reveals somewhat unexpected connections between flows at small scales, dominated by thermal fluctuations, and flows at large scales, dominated by turbulent fluctuations.

Enhancement of diffusive transport by non-equilibrium thermal fluctuations

We study the contribution of advection by thermal velocity fluctuations to the effectivediffusion coefficient in a mixture of two identical fluids. The steady-state diffusive flux in afinite system subject to a concentration gradient is enhanced because of long-range correlationsbetween concentration fluctuations and fluctuations of the velocity parallel to theconcentration gradient. The enhancement of the diffusive transport depends on the system sizeL andgrows as ln(L/L0) in quasi-two-dimensional systems, while in three dimensions it grows asL0 − 1 − L − 1, whereL0 is a reference length. The predictions of a simple fluctuating hydrodynamics theory, closelyrelated to second-order mode–mode coupling analysis, are compared to results from particlesimulations and a finite-volume solver and excellent agreement is observed. We elucidatethe direct connection to the long-time tail of the velocity autocorrelation function in finitesystems, as well as finite-size corrections employed in molecular dynamics calculations. Ourresults conclusively demonstrate that the nonlinear advective terms need to be retained inthe equations of fluctuating hydrodynamics when modeling transport in small-scale finitesystems.

Diffusive Transport by Thermal Velocity Fluctuations

We study the contribution of advection by thermal velocity fluctuations to the effective diffusion coefficient in a mixture of two identical fluids. We find good agreement between a simple fluctuating hydrodynamics theory and particle and finite-volume simulations. The enhancement of the diffusive transport depends on the system size L and grows as ln(L/L₀) in quasi-two-dimensional systems, while in three dimensions it scales as L₀⁻¹ - L⁻¹, where L₀ is a reference length. Our results demonstrate that fluctuations play an important role in the hydrodynamics of small-scale systems.

An efficient direct simulation Monte Carlo method for low Mach number noncontinuum gas flows based on the Bhatnagar–Gross–Krook model

The direct simulation Monte Carlo (DSMC) method is the preferred approach for simulating rarefied gas flows in complex geometries. However, the standard DSMC method becomes inefficient in the limit M=U¯/⟨c⟩→0 when the thermal velocity fluctuations which scale with the speed of sound ⟨c⟩ are much larger than characteristic ensemble-averaged flow speed U¯. In this paper, we propose a modified DSMC algorithm which simulates the linearized Bhatnagar–Gross–Krook (BGK) approximation to the Boltzmann equation. The particles in this method are uniformly distributed in space and have a rest-state, equilibrium Maxwellian velocity distribution. The deviations from the equilibrium state are captured by weightings, indicating that each particle represents a noninteger expectation for finding gas molecules with the simulation particle’s velocity in the spatial cell where it is found. The BGK approximation makes the implementation of such a method simple and efficient because the BGK collision rule can be implemented by adjusting particle weightings without the need to create new particles during the simulation. The method can be incorporated into existing DSMC simulation programs with minimal changes. We apply the new algorithm to two test problems—pressure-driven flow through a nanochannel and flow around an isolated sphere. Results obtained with the new method are in excellent agreement with previous theories and simulations.

Design of chemically propelled nanodimer motors

The self-propelled motion of nanodimers fueled by a chemical reaction taking place under nonequilibrium steady state conditions is investigated. The nanodimer consists of a pair of catalytic and chemically inactive spheres, in general with different sizes, with a fixed internuclear separation. The solvent in which the dimer moves is treated at a particle-based mesoscopic level using multiparticle collision dynamics. The directed motion of the dimer can be controlled by adjusting the interaction potentials between the solvent molecules and the dimer spheres, the internuclear separation, and sphere sizes. Dimers can be designed so that the directed motion along the internuclear axis occurs in either direction and is much larger than the thermal velocity fluctuations, a condition needed for such nanodimers to perform tasks involving targeted dynamics.

Dynamics of freely cooling granular gases.

We study dynamics of freely cooling granular gases in two dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, E(t) approximately t(-1), and velocity statistics are characterized by a universal Gaussian distribution in the long time limit. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, Deltav, are small compared with the typical velocity, Deltav/v approximately t(-1/4). Furthermore, locally averaged shear modes dominate over acoustic modes. The small thermal velocity fluctuations suggest that the system can be heuristically described by Burgers-like equations.