Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall
Abstract
:1. Introduction
2. Mathematical Formulations
2.1. Schematic Model for Rotating Rayleigh–Bénard Convection
2.2. Governing Equations
3. Linear Stability Analysis (LSA)
3.1. Basic State
3.2. Disturbance Equations
3.2.1. One-Dimensional LSA
3.2.2. Linear Stability Analysis of Traveling-Wave Sidewall Mode
3.3. Numerical Methodology
3.3.1. One-Dimensional LSA with the Stationary Mode
3.3.2. Two-Dimensional LSA with the Oscillatory Mode
4. Results
4.1. One-Dimensional LSA
4.1.1. Stationary Mode
4.1.2. Oscillatory Mode (Overstability)
4.2. Two-Dimensional LSA (Traveling-Wave Sidewall Mode)
4.2.1. Verification of the Present Numerical Code
4.2.2. Effect of the Taylor Number and the Prandtl Number
4.2.3. Effect of the Bottom Heating Condition
5. Discussion
5.1. Effect of Sidewall
5.2. Centrifugal Force
5.3. Free-Surface
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
ay | wavenumber in y-direction (1/m) |
az | wavenumber in z-direction (1/m) |
C | constant (-) |
Fr | Froude number (-) |
ex | unit vector in x-direction (-) |
ey | unit vector in y-direction (-) |
ez | unit vector in z-direction (-) |
g | gravitational acceleration (m/s2) |
i | imaginary unit (-) |
k | dimensionless wavenumber (-) |
h | characteristic length (m) |
p | pressure (Pa) |
P | dimensionless pressure (-) |
Pr | Prandtl number (-) |
q | heat flux (W/m2) |
Ra | Rayleigh number (-) |
S | complex eigenvalue (rad/s) |
SI | angular frequency (rad/s) |
SR | linear growth rate (rad/s) |
t | time (s) |
T | temperature (K) |
Ta | Taylor number (-) |
Tc | temperature at cold wall (K) |
Th | temperature at hot wall (K) |
T0 | reference temperature = (Th + Tc)/2 (K) |
ΔT | temperature difference between hot and cold walls (K) |
u | velocity vector = (u1, u2, u3) = (u, v, w) (m/s) |
u | x-directional velocity component (m/s) |
U | dimensionless X-directional velocity component (-) |
v | y-directional velocity component (m/s) |
V | dimensionless Y-directional velocity component (-) |
w | z-directional velocity component (m/s) |
W | dimensionless Z-directional velocity component (-) |
x | x coordinate (m) |
X | dimensionless x coordinate (-) |
y | y coordinate (m) |
Y | dimensionless y coordinate (-) |
z | z coordinate (m) |
Z | dimensionless z coordinate (-) |
Greek symbols | |
α | thermal diffusivity (m2/s) |
β | volumetric coefficient of thermal expansion at T0 (1/K) |
λ | thermal conductivity (W/(m K)) |
Θ | dimensionless temperature (-) |
ν | kinematic viscosity (m2/s) |
ρ0 | density at T0 (kg/m3) |
τ | virtual dimensionless time (-) |
Ω | angular velocity of enclosure (rad/s) |
Subscripts or superscripts | |
infinitesimal disturbance | |
basic state | |
amplitude function | |
I | imaginary part |
R | real part |
(m) | number of iterative steps |
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Ta | k | Present Study | Chandrasekar [3] Ra | |
---|---|---|---|---|
Ra and Number of Grids | ||||
10 | 3.10 | 1.7128 × 103 | 400 | 1.7130 × 103 |
100 | 3.15 | 1.7564 × 103 | 400 | 1.7566 × 103 |
500 | 3.30 | 1.9403 × 103 | 400 | 1.9403 × 103 |
1000 | 3.50 | 2.1514 × 103 | 400 | 2.1517 × 103 |
2000 | 3.75 | 2.5301 × 103 | 400 | 2.5305 × 103 |
5000 | 4.25 | 3.4686 × 103 | 400 | 3.4686 × 103 |
10,000 | 4.80 | 4.7121 × 103 | 400 | 4.7131 × 103 |
30,000 | 5.80 | 8.3246 × 103 | 800 | 8.3264 × 103 |
105 | 7.20 | 1.6720 × 104 | 800 | 1.6721 × 104 |
106 | 10.80 | 7.1085 × 104 | 1600 | 7.1132 × 104 |
108 | 24.5 | 1.5252 × 106 | 3200 | 1.5313 × 106 |
1010 | 55.5 | 3.4498 × 107 | 6400 | 3.4574 × 107 |
Ta | k | SI | Ra | ||||
---|---|---|---|---|---|---|---|
Present Study | Chandrasekar | Present Study | Chandrasekar | ||||
200 Grids | 400 Grids | 200 Grids | 400 Grids | ||||
104 | 3.08 | 4.47036 × 101 | 4.47038 × 101 | 4.45 × 101 | 4.370328 × 103 | 4.370333 × 103 | 4.39 × 103 |
106 | 4.09 | 5.82579 × 102 | 5.82585 × 102 | 5.82 × 102 | 9.5102 × 103 | 9.5096 × 103 | 9.51 × 103 |
5 × 107 | 8.10 | 2.4258 × 103 | 2.4262 × 103 | 2.43 × 103 | 6.313 × 104 | 6.302 × 104 | 6.29 × 104 |
2 × 108 | 10.28 | 3.916 × 103 | 3.918 × 103 | 3.92 × 103 | 1.388 × 105 | 1.381 × 105 | 1.38 × 105 |
109 | 13.46 | 6.805 × 103 | 6.813 × 103 | 6.81 × 103 | 3.607 × 105 | 3.552 × 105 | 3.54 × 105 |
1010 | 19.70 | 1.489 × 104 | 1.494 × 104 | 1.50 × 104 | 1.520 × 106 | 1.436 × 106 | 1.42 × 106 |
1011 | 28.75 | 3.251 × 104 | 3.262 × 104 | 3.27 × 104 | 6.418 × 106 | 6.139 × 106 | 5.83 × 106 |
Rac | kc | ωc | |
---|---|---|---|
Expt. of Liu and Ecke [10] | 20,850 | 4.65 | −22.0 |
Theor. of Kuo and Cross [9] | 19,500 | 4.00 | −24.0 |
Theor. of Plaut [16] | 19,660 | 4.22 | −22.4 |
Present study | 19,649 | 4.225 | −22.39 |
Ta | kc | Rac | SI | |
---|---|---|---|---|
Case A | 105 | 3.82 | 1.1245 × 104 | −1.6653 × 101 |
Case B | 106 | 4.32 | 3.3086 × 104 | −2.5993 × 101 |
Case C | 107 | 4.80 | 1.0225 × 105 | −3.5321 × 101 |
Pr | kc | Rac | SI | |
---|---|---|---|---|
Case B | 1 | 4.32 | 3.3086 × 104 | −2.5993 × 101 |
Case 1 | 0.5 | 4.20 | 3.1615 × 104 | −4.9341 × 101 |
Case 2 | 0.25 | 4.00 | 2.9227 × 104 | −8.9960 × 101 |
Case 3 | 0.10 | 3.50 | 2.4580 × 104 | −1.8228 × 102 |
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Tagawa, T. Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall. Fluids 2023, 8, 96. https://doi.org/10.3390/fluids8030096
Tagawa T. Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall. Fluids. 2023; 8(3):96. https://doi.org/10.3390/fluids8030096
Chicago/Turabian StyleTagawa, Toshio. 2023. "Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall" Fluids 8, no. 3: 96. https://doi.org/10.3390/fluids8030096
APA StyleTagawa, T. (2023). Numerical Analysis of Linear Traveling Wave in Rotating Rayleigh–Bénard Convection with an Adiabatic Sidewall. Fluids, 8(3), 96. https://doi.org/10.3390/fluids8030096