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The dynamics of a boiling bubble before and after detachment

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Abstract

An overview is given of prediction methods for motion and deformation of a bubble that is created by boiling at a wall, at times before and after detachment, with a focus on added mass forces in the vicinity of the wall. The possibility to apply added mass coefficients derived in potential flows also to flows with vorticity is examined. An introduction to Lagrangian methods is given. Added mass tensors are derived for deforming bubbles at and away from a plane wall. Expressions for induced hydrodynamic lift forces are given, and validation experiments are briefly discussed.

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Notes

  1. In the case of instantaneous motion with velocity U in the direction \({{\varvec e}_1, \phi= U x_1 + {{\frac{1}{2}}} U (d_b/2)^3x_1 r^{-3}}\) and Eq. (2) yields the familiar result \({2U \mu (2/d_b) \int ({\varvec{\nabla}} \phi)^2 {\rm d}S = 6 \pi \mu U d_b.}\)

  2. It is noted that if a cylindrical coordinate system is employed, a contour has to be chosen that excludes the axis where r = 0 and the inverse mapping not defined. If a different type of contour would be used, an erroneous minus sign would appear. In the case of a truncated sphere this necessitates a limiting procedure in order to cover A bL entirely.

Abbreviations

A :

area (m2)

A 11 :

added mass coefficient (–)

\({\tilde{A}_{rs}}\) :

added mass coefficient (–)

A c :

acceleration number (–)

b n :

generalized coordinate n (–)

c :

coefficient (–)

c AM :

added mass coefficient (–)

d b :

diameter of a bubble (m)

dS :

element of area (m2)

\({{\rm d}{{{\mathcal{V}} \hspace*{-0.5em}\rule[0.7ex]{0.4em}{.4pt}\,\,}}}\) :

element of volume (m3)

F :

force (N)

g :

acceleration of gravity (m2/s)

h :

distance of center to the wall (m)

I :

added mass tensor (–)

M :

mass (kg)

n :

normal (m)

U :

velocity (m/s)

q :

generalized coordinate

\({\dot{q}}\) :

Generalized velocity

Q :

generalized force

Q W :

generalized drag force (N)

r :

radius (m)

R :

radius (m)

Sr :

dimensionless vorticity (–)

t :

time (s)

T :

kinetic energy (J)

tr(β):

added mass coefficient (–)

v :

velocity (m/s)

V :

velocity parallel to the wall (m/s)

v b :

velocity of a bubble (m/s)

v L :

velocity of the liquid (m/s)

v rel :

velocity of object relative to fluid (m/s)

\({{{{\mathcal{V}} \hspace*{-0.5em}\rule[0.7ex]{0.4em}{.4pt}\,\,}}}\) :

volume (m3)

x :

coordinate (m)

x :

position vector (m)

x CM :

position of the center of mass (m)

y :

coordinate (m)

\({\dot{x}}\) :

velocity component in x-direction (m/s)

α:

added mass coefficient to U2 (–)

α2 :

added mass coefficient to V2 (–)

\({\partial {{{\mathcal{V}} \hspace*{-0.5em}\rule[0.7ex]{0.4em}{.4pt}\,\,}}}\) :

boundary of volume (m2)

γ m :

added mass coefficient m (–)

θ:

contact angle (o)

μ:

dynamic viscosity (kg/m/s)

ρ:

mass density (kg/m3)

σ:

surface tension coefficient (N/m)

ϕ:

velocity potential (m2/s)

\({\dot{\Phi}}\) :

energy dissipation rate (W)

ψ:

added mass coefficient (–)

ψ ij :

added mass coefficient i,j (–)

ω:

vorticity (1/s)

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Acknowledgments

This research was funded by the EC project AD-700-2. I thank prof. J. Passos for providing a stimulating place to study, and Dr. J. Kuerten for useful comments to the manuscript.

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  1. Faculty of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    C. W. M. van der Geld

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van der Geld, C.W.M. The dynamics of a boiling bubble before and after detachment. Heat Mass Transfer 45, 831–846 (2009). https://doi.org/10.1007/s00231-007-0254-7

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