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Estimation curvature in PLIC-VOF method for interface advection

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Abstract

Heat and mass transfers’ equations of hydrocarbon liquid droplet are solved numerically by using a semi-implicit method based on the finite volume scheme (VOF). The Piece-wise Linear Interface Calculation (PLIC) is used in our modeling. A curvature estimation technique based on PLIC-VOF method is developed. The determination of the normal vector based on the calculation of the curvature is presented. Then, our numerical calculation is performed for hydrocarbon droplets by realizing several static and dynamic tests. These tests are compared to other techniques. Good agreement between the results of our method and the ones of the other techniques is observed. The deformation of the liquid droplet interface, the ascension of the liquid droplet and the liquid droplet surface regression are observed by taking into consideration the effect of the surface tension force and the drag force. Finally, instead of using smoothed color function, our curvature estimation hugely decreases the spurious current and reconstructs smoothed shape.

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Abbreviations

A :

Area of the triangle, m2

c :

Difference of the volume fractions of two-consecutive cells

Cp :

Specific heat, J.kg−1.K−1

d :

Sphere diameter, m

F :

Color function

g :

Terrestrial acceleration, m.s−2

k :

Curvature, m−1

l :

Normal vector impact point

L :

Latent heat, J.kg−1

N :

Total number of cells

n :

Normal

\( \overrightarrow{n} \) :

Normal vector

\( \left|\overrightarrow{n}\right| \) :

Normal modulus

\( \left\Vert \overrightarrow{n}\right\Vert \) :

Norm of the normal vector

P :

Pressure, atm

r :

Sphere radius, m

R :

Curvature radius, m

t :

Time, s

T :

Temperature, K

u :

Velocity component in ordinate direction, m.s−1

v :

Velocity component in coordinate direction, m.s−1

α :

Thermal diffusivity, m2.s−1

β :

Expansion number

δ :

Angle between two adjacent triangles, °

∇:

Gradient, m−1

∂:

Differential

Δ :

Difference

η :

Dimensionless radius

λ :

Thermal conductivity, W.m−1.K−1

μ :

Dynamic viscosity, Kg.m−1.s−1

ϑ :

Kinematic viscosity, m2.s−1

ρ :

Density, kg.m−3

Σ :

Summation

σ :

Surface tension, N.m−1

θ :

Polar angle, °

c :

Curvature

calc:

Calculated

drop:

Droplet

Err:

Error

g :

Gas

init:

Initial

i :

Incremental step in ordinate direction

j :

Incremental step in coordinate direction

k :

Fluids index

l :

Liquid

M :

Mass

T :

Thermal

η :

Radius axis

θ :

Polar axis

0:

Initial

∞:

Ambient medium

′:

Dimensionless

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Authors and Affiliations

  1. Laboratory of Applied Physics (LPA), Group of Mechanical, Thermal & Renewable Energies (GMTER), Lebanese University, Faculty of Sciences, Fanar, Lebanon

    Jean Nahed & Joseph Dgheim

  2. Lebanese University, Fanar, P.B.: 90656, Jdeidet El-Metn, Lebanon

    Joseph Dgheim

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  1. Jean Nahed
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  2. Joseph Dgheim
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Correspondence to Jean Nahed.

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Nahed, J., Dgheim, J. Estimation curvature in PLIC-VOF method for interface advection. Heat Mass Transfer 56, 773–787 (2020). https://doi.org/10.1007/s00231-019-02737-4

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