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TECHNICAL PAPERS

A Mean-Field Pressure Formulation for Liquid-Vapor Flows

[+] Author and Article Information
Shi-Ming Li

Department of Mechanical Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, VA 24060smli@vt.edu

Danesh K. Tafti

Department of Mechanical Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, VA 24060dtafti@vt.edu

It was established that the initial distribution had no effect on the final solution, only on how quickly the solution equilibrated.

J. Fluids Eng 129(7), 894-901 (Dec 28, 2006) (8 pages) doi:10.1115/1.2742730 History: Received July 10, 2006; Revised December 28, 2006

A nonlocal pressure equation is derived from mean-field free energy theory for calculating liquid-vapor systems. The proposed equation is validated analytically by showing that it reduces to van der Waals’ square-gradient approximation under the assumption of slow density variations. The proposed nonlocal pressure is implemented in the mean-field free energy lattice Boltzmann method (LBM). The LBM is applied to simulate equilibrium liquid-vapor interface properties and interface dynamics of capillary waves and oscillating droplets in vapor. Computed results are validated with Maxwell constructions of liquid-vapor coexistence densities, theoretical relationship of variation of surface tension with temperature, theoretical planar interface density profiles, Laplace’s law of capillarity, dispersion relationship between frequency and wave number of capillary waves, and the relationship between radius and the oscillating frequency of droplets in vapor. It is shown that the nonlocal pressure formulation gives excellent agreement with theory.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Comparison between LBM simulations and analytical Maxwell constructions of liquid-vapor coexistence densities

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Figure 2

Planar interface surface tension versus normalized temperature

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Figure 3

Comparison of analytical solution and the current LBM simulation for the density profiles of the planar interfaces at KbT=0.54, 0.55, 0.56; lines: the analytical solutions, symbols: the current LBM

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Figure 4

Snapshots of the droplet iteration process

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Figure 5

Density profiles of six droplets in different directions: the north, west, south, and east, Kbt=0.55, time step=10,000

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Figure 6

Pressure difference between inside and outside for different droplet sizes: Solid lines are the linear correlations of the LBM data, which give the surface tension 0.00764 for KbT=0.55 and 0.0310 for KbT=0.52

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Figure 7

Droplet shape (rh∕rv) response to initial speed U0

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Figure 10

Snapshot of an oscillating droplet with the initial dimension of RL×RS=25×18, KbT=0.50

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Figure 11

Oscillating frequency versus droplet radius, KbT=0.50; solid circles: the current LBM results; line: the linear correlation of the LBM results, giving the slope of the line −1.458; the analytical solution gives the slope of −1.50

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Figure 12

Simulation of droplet coalescence process, KbT=0.55

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Figure 8

Snapshots of capillary waves at different time steps, KbT=0.52, case 4, ℓ=65, ℏ=2993∕2: (a)Δt=0, (b)Δt=180, (c)Δt=454, (d)Δt=580, (e)Δt=850, and (f)Δt=908

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Figure 9

Dispersion relation of capillary waves, KbT=0.52; solid circle: the current LBM, solid line: correlation for the LBM, which gives the line slope of 1.551; exact solution gives the slope of 1.50

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