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ABSTRACT
A coupled volume-of-fluid, level set, and smoothed physical parameter (VOF+LS+SPP) method based on FLUENT is used to simulate bubble dynamics in the gravitational and uniform electric fields. Both of the bubble and surrounding medium are assumed to be perfect dielectrics with constant but different permittivities. The effects of electric Bond number, permittivity ratio, Morton number, and Eotvos number on the deformation and rising motion of a single bubble are systematically investigated. Simulation results show that a vertical electric field elongates the bubble along the electric field direction and accelerates the bubble rising. The electric Bond number has a much greater effect on bubble deformation and rising velocity than the permittivity ratio. The bubble behaviors in the electric field are similar for different Morton numbers but totally different for various Eotvos numbers. The influence of electric field on bubble Reynolds number changes a little for different Morton numbers but decreases distinctly with the increase of Eotvos number.
Nomenclature
| A | = | dimensionless largest horizontal cross-section of bubble |
| B | = | maximum bubble width, m |
| Boe | = | electric Bond number |
| Co | = | Courant number |
| d | = | distance from interface, m |
| D | = | deformation rate |
| E | = | electric field intensity, N/C |
| Eo | = | Eotvos number |
| Fσ | = | surface tension, N/m3 |
| Fe | = | electric field force, N/m3 |
| g | = | gravitational acceleration, m/s2 |
| H(ϕ) | = | Heaviside function |
| κ(ϕ) | = | interface curvature, m−1 |
| L | = | maximum bubble length, m |
| M | = | Morton number |
| p | = | pressure, Pa |
| r | = | position vector, m |
| R | = | bubble radius, m |
| Re | = | Reynolds number |
| t | = | time, s |
| T | = | dimensionless time |
| u | = | velocity vector, m/s |
| u∞ | = | terminal bubble rising velocity, m/s |
| uz | = | velocity along the z-direction, m/s |
| U | = | dimensionless average bubble rising velocities |
| yb | = | bubble bottom position, m |
| α | = | volume fraction |
| ϕ | = | electric potential, V |
| δ(φ) | = | Dirac distribution function |
| ε | = | relative permittivity |
| ε0 | = | permittivity of vacuum, F/m |
| λε | = | permittivity ratio |
| λμ | = | viscosity ratio |
| λρ | = | density ratio |
| μ | = | fluid dynamic viscosity, Pa · s |
| ρ | = | fluid density, kg/m3 |
| σ | = | surface tension coefficient, N/m |
| φ | = | level set function, m |
| Δ | = | grid size, m |
| Subscripts | = | |
| g | = | gas phase |
| l | = | liquid phase |
Nomenclature
| A | = | dimensionless largest horizontal cross-section of bubble |
| B | = | maximum bubble width, m |
| Boe | = | electric Bond number |
| Co | = | Courant number |
| d | = | distance from interface, m |
| D | = | deformation rate |
| E | = | electric field intensity, N/C |
| Eo | = | Eotvos number |
| Fσ | = | surface tension, N/m3 |
| Fe | = | electric field force, N/m3 |
| g | = | gravitational acceleration, m/s2 |
| H(ϕ) | = | Heaviside function |
| κ(ϕ) | = | interface curvature, m−1 |
| L | = | maximum bubble length, m |
| M | = | Morton number |
| p | = | pressure, Pa |
| r | = | position vector, m |
| R | = | bubble radius, m |
| Re | = | Reynolds number |
| t | = | time, s |
| T | = | dimensionless time |
| u | = | velocity vector, m/s |
| u∞ | = | terminal bubble rising velocity, m/s |
| uz | = | velocity along the z-direction, m/s |
| U | = | dimensionless average bubble rising velocities |
| yb | = | bubble bottom position, m |
| α | = | volume fraction |
| ϕ | = | electric potential, V |
| δ(φ) | = | Dirac distribution function |
| ε | = | relative permittivity |
| ε0 | = | permittivity of vacuum, F/m |
| λε | = | permittivity ratio |
| λμ | = | viscosity ratio |
| λρ | = | density ratio |
| μ | = | fluid dynamic viscosity, Pa · s |
| ρ | = | fluid density, kg/m3 |
| σ | = | surface tension coefficient, N/m |
| φ | = | level set function, m |
| Δ | = | grid size, m |
| Subscripts | = | |
| g | = | gas phase |
| l | = | liquid phase |