skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures

Abstract

We develop a low Mach number formulation of the hydrodynamic equations describing transport of mass and momentum in a multispecies mixture of incompressible miscible liquids at specified temperature and pressure, which generalizes our prior work on ideal mixtures of ideal gases [Balakrishnan et al., “Fluctuating hydrodynamics of multispecies nonreactive mixtures,” Phys. Rev. E 89 013017 (2014)] and binary liquid mixtures [Donev et al., “Low mach number fluctuating hydrodynamics of diffusively mixing fluids,” Commun. Appl. Math. Comput. Sci. 9(1), 47-105 (2014)]. In this formulation, we combine and extend a number of existing descriptions of multispecies transport available in the literature. The formulation applies to non-ideal mixtures of arbitrary number of species, without the need to single out a “solvent” species, and includes contributions to the diffusive mass flux due to gradients of composition, temperature, and pressure. Momentum transport and advective mass transport are handled using a low Mach number approach that eliminates fast sound waves (pressure fluctuations) from the full compressible system of equations and leads to a quasi-incompressible formulation. Thermal fluctuations are included in our fluctuating hydrodynamics description following the principles of nonequilibrium thermodynamics. We extend the semi-implicit staggered-grid finite-volume numerical method developed in our prior work on binary liquidmore » mixtures [Nonaka et al., “Low mach number fluctuating hydrodynamics of binary liquid mixtures,” http://arxiv.org/abs/1410.2300 (2015)] and use it to study the development of giant nonequilibrium concentration fluctuations in a ternary mixture subjected to a steady concentration gradient. We also numerically study the development of diffusion-driven gravitational instabilities in a ternary mixture and compare our numerical results to recent experimental measurements [Carballido-Landeira et al., “Mixed-mode instability of a miscible interface due to coupling between Rayleigh–Taylor and double-diffusive convective modes,” Phys. Fluids 25, 024107 (2013)] in a Hele-Shaw cell. We find that giant nonequilibrium fluctuations can trigger the instability but are eventually dominated by the deterministic growth of the unstable mode, in both quasi-two-dimensional (Hele-Shaw) and fully three-dimensional geometries used in typical shadowgraph experiments.« less

Authors:
;  [1]; ;  [2];  [3]
  1. Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 (United States)
  2. Center for Computational Science and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California 94720 (United States)
  3. Department of Physics and Astronomy, San Jose State University, San Jose, California 95192 (United States)
Publication Date:
OSTI Identifier:
22403221
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Fluids (1994); Journal Volume: 27; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 42 ENGINEERING; ABUNDANCE; CONCENTRATION RATIO; DIFFUSION; FLUCTUATIONS; GASES; GRAVITATIONAL INSTABILITY; HYDRODYNAMICS; LIQUIDS; MACH NUMBER; MASS TRANSFER; MIXING; MIXTURES; MOMENTUM TRANSFER; NUMERICAL ANALYSIS; SOUND WAVES; THERMODYNAMICS; THREE-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS

Citation Formats

Donev, Aleksandar, E-mail: donev@courant.nyu.edu, Bhattacharjee, Amit Kumar, Nonaka, Andy, Bell, John B., and Garcia, Alejandro L. Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures. United States: N. p., 2015. Web. doi:10.1063/1.4913571.
Donev, Aleksandar, E-mail: donev@courant.nyu.edu, Bhattacharjee, Amit Kumar, Nonaka, Andy, Bell, John B., & Garcia, Alejandro L. Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures. United States. doi:10.1063/1.4913571.
Donev, Aleksandar, E-mail: donev@courant.nyu.edu, Bhattacharjee, Amit Kumar, Nonaka, Andy, Bell, John B., and Garcia, Alejandro L. 2015. "Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures". United States. doi:10.1063/1.4913571.
@article{osti_22403221,
title = {Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures},
author = {Donev, Aleksandar, E-mail: donev@courant.nyu.edu and Bhattacharjee, Amit Kumar and Nonaka, Andy and Bell, John B. and Garcia, Alejandro L.},
abstractNote = {We develop a low Mach number formulation of the hydrodynamic equations describing transport of mass and momentum in a multispecies mixture of incompressible miscible liquids at specified temperature and pressure, which generalizes our prior work on ideal mixtures of ideal gases [Balakrishnan et al., “Fluctuating hydrodynamics of multispecies nonreactive mixtures,” Phys. Rev. E 89 013017 (2014)] and binary liquid mixtures [Donev et al., “Low mach number fluctuating hydrodynamics of diffusively mixing fluids,” Commun. Appl. Math. Comput. Sci. 9(1), 47-105 (2014)]. In this formulation, we combine and extend a number of existing descriptions of multispecies transport available in the literature. The formulation applies to non-ideal mixtures of arbitrary number of species, without the need to single out a “solvent” species, and includes contributions to the diffusive mass flux due to gradients of composition, temperature, and pressure. Momentum transport and advective mass transport are handled using a low Mach number approach that eliminates fast sound waves (pressure fluctuations) from the full compressible system of equations and leads to a quasi-incompressible formulation. Thermal fluctuations are included in our fluctuating hydrodynamics description following the principles of nonequilibrium thermodynamics. We extend the semi-implicit staggered-grid finite-volume numerical method developed in our prior work on binary liquid mixtures [Nonaka et al., “Low mach number fluctuating hydrodynamics of binary liquid mixtures,” http://arxiv.org/abs/1410.2300 (2015)] and use it to study the development of giant nonequilibrium concentration fluctuations in a ternary mixture subjected to a steady concentration gradient. We also numerically study the development of diffusion-driven gravitational instabilities in a ternary mixture and compare our numerical results to recent experimental measurements [Carballido-Landeira et al., “Mixed-mode instability of a miscible interface due to coupling between Rayleigh–Taylor and double-diffusive convective modes,” Phys. Fluids 25, 024107 (2013)] in a Hele-Shaw cell. We find that giant nonequilibrium fluctuations can trigger the instability but are eventually dominated by the deterministic growth of the unstable mode, in both quasi-two-dimensional (Hele-Shaw) and fully three-dimensional geometries used in typical shadowgraph experiments.},
doi = {10.1063/1.4913571},
journal = {Physics of Fluids (1994)},
number = 3,
volume = 27,
place = {United States},
year = 2015,
month = 3
}
  • Cited by 7
  • In this study we discuss the formulation of the fluctuating Navier-Stokes equations for multispecies, nonreactive fluids. In particular, we establish a form suitable for numerical solution of the resulting stochastic partial differential equations. An accurate and efficient numerical scheme, based on our previous methods for single species and binary mixtures, is presented and tested at equilibrium as well as for a variety of nonequilibrium problems. These include the study of giant nonequilibrium concentration fluctuations in a ternary mixture in the presence of a diffusion barrier, the triggering of a Rayleigh-Taylor instability by diffusion in a four-species mixture, as well asmore » reverse diffusion in a ternary mixture. Finally, good agreement with theory and experiment demonstrates that the formulation is robust and can serve as a useful tool in the study of thermal fluctuations for multispecies fluids.« less
  • Here, we formulate and study computationally the low Mach number fluctuating hydrodynamic equations for electrolyte solutions. We are also interested in studying transport in mixtures of charged species at the mesoscale, down to scales below the Debye length, where thermal fluctuations have a significant impact on the dynamics. Continuing our previous work on fluctuating hydrodynamics of multicomponent mixtures of incompressible isothermal miscible liquids (A. Donev, et al., Physics of Fluids, 27, 3, 2015), we now include the effect of charged species using a quasielectrostatic approximation. Localized charges create an electric field, which in turn provides additional forcing in the massmore » and momentum equations. Our low Mach number formulation eliminates sound waves from the fully compressible formulation and leads to a more computationally efficient quasi-incompressible formulation. Furthermore, we demonstrate our ability to model saltwater (NaCl) solutions in both equilibrium and nonequilibrium settings. We show that our algorithm is second-order in the deterministic setting, and for length scales much greater than the Debye length gives results consistent with an electroneutral/ambipolar approximation. In the stochastic setting, our model captures the predicted dynamics of equilibrium and nonequilibrium fluctuations. We also identify and model an instability that appears when diffusive mixing occurs in the presence of an applied electric field.« less
  • We present a consistent numerical model for coupling radiation to hydrodynamics at low Mach number. The hydrodynamical model is based on a low-Mach asymptotic in the compressible flow that removes acoustic wave propagation while retaining the compressibility effects resulting from combustion. Radiative transfer is modelled by the M{sub 1} entropy equations that can be viewed as a moment method. The radiation model possesses the capability to accurately approximate solution of radiative transfer at low computational cost while retaining the main physical properties of radiative energy. Consistent numerical approaches are developed for space and time discretizations in both hydrodynamics and radiation.more » A modified projection method is used for hydrodynamics, whereas an HLL-type discretization is implemented for radiation transport. The combined methods permit time steps that are controlled by the advective time scales resulting in a substantial improvement in computational efficiency compared to a compressible formulation. Numerical results are presented for the natural convection in a squared cavity with large temperature difference and also for a diffusion methane/air flame with four-step reduced chemical reactions in non-gray participating media. The present approach has been found to be feasible and satisfactory.« less