Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field
Abstract
:1. Introduction
2. Mathematical Formulations
2.1. Schematic Model for Spin-Up
2.2. Governing Equations
2.3. Dimensionless Equations
2.4. Numerical Methodology
3. Results
3.1. Grid Dependency
3.2. Effect of Surface Tension
3.3. Effect of the Ekman Number
3.4. Spin-Up in the Axial Uniform Magnetic Field
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
a | radius of cylindrical enclosure (m) |
Ar | aspect ratio = a/h (-) |
b0 | absolute value of magnetic flux density (T) |
b | magnetic flux density (T) |
E | Ekman number (-) |
eR | unit vector in radial direction (-) |
eZ | unit vector in axial direction (-) |
eθ | unit vector in azimuthal direction (-) |
fst | surface normal force (N/m3) |
Fr | Froude number (-) |
g | gravitational acceleration (m/s2) |
g | absolute value of gravitational acceleration (m/s2) |
h | initial height of liquid (m) |
Ha | Hartmann number (-) |
Hε(φ) | smoothed Heaviside step function (-) |
j | electric current density = (jr, jθ, jz) (A/m2) |
jr | radial component of electric current density (A/m2) |
jz | axial component of electric current density (A/m2) |
jθ | azimuthal component of electric current density (A/m2) |
J | dimensionless electric current density (-) |
JR | dimensionless radial component of electric current density (-) |
JZ | dimensionless axial component of electric current density (-) |
Jθ | dimensionless azimuthal component of electric current density (-) |
p | pressure (Pa) |
P | dimensionless pressure (-) |
r | radial coordinate (m) |
R | dimensionless r coordinate (-) |
t | time (s) |
u | velocity = (u, v, w) (m/s) |
u | radial velocity component (m/s) |
U | dimensionless radial velocity component (-) |
v | azimuthal velocity component (m/s) |
V | dimensionless azimuthal velocity component (-) |
w | axial velocity component (m/s) |
W | dimensionless axial velocity component (-) |
We | Weber number (-) |
z | axial coordinate (m) |
Z | dimensionless z-coordinate (-) |
Greek symbols | |
γ | surface tension (N/m) |
δε(φ) | smoothed Dirac delta function (1/m) |
κ | curvature of interface (1/m) |
μ | viscosity (Pa·s) |
ν | kinematic viscosity = μ/ρ (m2/s) |
ρ | density (kg/m3) |
σ | electric conductivity (1/(Ω·m)) |
τ | dimensionless time (-) |
φ | level-set function (m) |
Φ | dimensionless level-set function (-) |
ψe | electric potential (V) |
Ψe | dimensionless electric potential (-) |
ΨJ | Stokes stream function for electric current density (-) |
ΨS | Stokes stream function for velocity (-) |
Ω | angular velocity (rad/s) |
Subscripts or superscripts | |
G | gas |
i,k | grid point |
L | liquid |
n | time step |
m | iteration number |
Φ | dependence of level-set function |
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Tagawa, T.; Song, K. Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field. Fluids 2021, 6, 438. https://doi.org/10.3390/fluids6120438
Tagawa T, Song K. Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field. Fluids. 2021; 6(12):438. https://doi.org/10.3390/fluids6120438
Chicago/Turabian StyleTagawa, Toshio, and Kewei Song. 2021. "Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field" Fluids 6, no. 12: 438. https://doi.org/10.3390/fluids6120438
APA StyleTagawa, T., & Song, K. (2021). Spin-Up from Rest of a Liquid Metal with Deformable Free Surface in a Cylinder under the Influence of a Uniform Axial Magnetic Field. Fluids, 6(12), 438. https://doi.org/10.3390/fluids6120438