Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field
Abstract
:1. Introduction
2. Configuration of Problem and Governing Equations
3. Basic State and Disturbance Equations
4. Numerical Methodology
5. Results and Discussion
5.1. Validation of the Code Diveloped
5.2. Hartmann Flow
5.3. Finite Aspect Ratio
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
a | half spanwise length of duct (m) |
A | aspect ratio = a/h (-) |
b | magnetic flux density = (bx, by, 0) (T) |
B | dimensionless magnetic flux density = b/ b0 = (BX, BY, 0) (-) |
b0 | strength of external magnetic flux density = (bx2+ by2)1/2 (T) |
C | coefficient of Newton—Raphson method |
E | Electric field (V/m) |
ex | unit vector in x direction (-) |
ey | unit vector in y direction (-) |
h | half length of duct height (m) |
Ha | Hartmann number = b0h (σ/μ)1/2 (-) |
i | imaginary unit (-) |
j | electric current density = (jx, jy, jz) (A/m2) |
J | dimensionless electric current density = (JX, JY, JZ) (-) |
k | wavenumber in streamwise direction (rad/m) |
p | pressure (Pa) |
P | dimensionless pressure (-) |
Prm | magnetic Prandtl number = σμmν = ν/νm (-) |
Re | Reynolds number = uref h/ν (-) |
Rem | magnetic Reynolds number = uref h/νm = PrmRe (-) |
s | complex angular frequency = sr + isi (rad/s) |
S | dimensionless complex angular frequency = SR + iSI (-) |
SI | dimensionless angular frequency (-) |
SR | dimensionless linear growth rate (-) |
t | time (s) |
u | x-directional velocity component (m/s) |
u | velocity vector = (u, v, w) (m/s) |
uref | reference velocity (m/s) |
U | dimensionless x-directional velocity component = u/uref (-) |
v | y-directional velocity component (m/s) |
V | dimensionless y-directional velocity component = v/uref (-) |
w | z-directional velocity component (m/s) |
W | dimensionless z-directional velocity component = w/uref (-) |
z | z coordinate (m) |
Z | dimensionless z coordinate = z/h (-) |
Greek symbols | |
α | dimensionless wavenumber = kh (-) |
μ | viscosity (Pa·s) |
μm | magnetic permeability (H/m) |
ν | kinematic viscosity = μ/ρ (m2/s) |
νm | magnetic viscosity = 1/(σμm) (m2/s) |
ρ | density (kg/m3) |
σ | electric conductivity (1/(Ω·m)) |
τ | dimensionless time (-) |
φ | electric potential (V) |
Φ | dimensionless electric potential (-) |
Ψ | dimensionless stream function (-) |
Subscripts or superscripts | |
arb | arbitrary point |
int | interpolation |
c | critical |
I | imaginary part |
R | real part |
i,j | grid point |
n | time step |
m | number of iteration |
* | predictive |
perturbed | |
– | basic |
~ | complex amplitude |
Appendix A
Appendix A.1. Basic State
Appendix A.2. How to Solve the Eigenvalue Problem
Appendix A.3. How to Obtain Neutral Curve
Appendix A.4. How to Determine Critical Point
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Number of Meshes | Re | −SI | Cph | Error in Re |
---|---|---|---|---|
100 | 5720.572 | 0.2699707 | 0.2645352 | 8.95 × 10−3 |
200 | 5767.318 | 0.2694774 | 0.2640518 | 8.50 × 10−4 |
400 | 5771.866 | 0.2694287 | 0.2640041 | 6.17 × 10−5 |
800 | 5772.198 | 0.2694250 | 0.2640005 | 4.16 × 10−6 |
Exact value | 5772.222 | 0.2694248 | 0.2640003 | - |
Present | Takashima [10] | ||||||
---|---|---|---|---|---|---|---|
Ha | Number of Meshes | αc | Rec | −SIc | αc | ReTak. | Remod |
0 | 200 | 1.0207 | 5767.31 | 2.695 × 10−1 | 1.0205 | 5772.22 | 5772.22 |
400 | 1.0206 | 5771.87 | 2.694 × 10−1 | ||||
1 | 200 | 0.9722 | 10,817.5 | 2.118 × 10−1 | 0.9718 | 10,016.3 | 10837.4 |
400 | 0.9718 | 10,835.9 | 2.116 × 10−1 | ||||
2 | 200 | 0.9294 | 37,274.3 | 1.363 × 10−1 | 0.9277 | 28,603.6 | 37,577.5 |
400 | 0.9279 | 37,532.8 | 1.358 × 10−1 | ||||
3 | 200 | 0.9636 | 105,614 | 9.901 × 10−2 | 0.9582 | 65,155.2 | 107,974 |
400 | 0.9588 | 107,702 | 9.788 × 10−2 | ||||
4 | 200 | 1.0476 | 224,231 | 8.180 × 10−2 | 1.0354 | 112,395 | 233,178 |
400 | 1.0372 | 231,617 | 8.008 × 10−2 | ||||
5 | 200 | 1.1527 | 399,687 | 7.230 × 10−2 | 1.1342 | 164,090 | 415,791 |
400 | 1.1386 | 410,155 | 7.060 × 10−2 |
Direction of Magnetic Field | Ha | αc | Rec | −SIc |
---|---|---|---|---|
- | 0 | 0.9255 | 9883.7 | 0.2193 |
X | 0.5 | 0.9119 | 11,783 | 0.2051 |
1.0 | 0.8765 | 19,489 | 0.1700 | |
1.5 | 0.8797 | 36,013 | 0.1446 | |
2.0 | 1.0808 | 55,378 | 0.1711 | |
Y | 0.5 | 0.9223 | 10,139 | 0.2169 |
1.0 | 0.9124 | 10,944 | 0.2098 | |
1.5 | 0.8952 | 12398 | 0.1983 | |
2.0 | 0.8710 | 14,582 | 0.1832 | |
2.5 | 0.8437 | 17,382 | 0.1672 | |
3.0 | 0.8214 | 20,570 | 0.1536 | |
3.5 | 0.8031 | 24367 | 0.1417 | |
4.0 | 0.8987 | 54,436 | 0.1518 | |
XY | 1.0 | 0.8934 | 14,670 | 0.1884 |
2.0 | 0.8658 | 36,837 | 0.1414 |
X-mag. (B = ex) | Y-mag. (B = ey) |
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Tagawa, T. Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field. Fluids 2019, 4, 177. https://doi.org/10.3390/fluids4040177
Tagawa T. Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field. Fluids. 2019; 4(4):177. https://doi.org/10.3390/fluids4040177
Chicago/Turabian StyleTagawa, Toshio. 2019. "Linear Stability Analysis of Liquid Metal Flow in an Insulating Rectangular Duct under External Uniform Magnetic Field" Fluids 4, no. 4: 177. https://doi.org/10.3390/fluids4040177