# Effect of the Direction of Uniform Horizontal Magnetic Field on the Linear Stability of Natural Convection in a Long Vertical Rectangular Enclosure

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Analyses of Basic Flows

_{X}, B

_{Y}) = (1,0), it represents the X-directional magnetic field, and if (B

_{X}, B

_{Y}) = (0,1), it represents the Y-directional magnetic field. Furthermore, if this condition is satisfied, an arbitrary horizontal magnetic field can be applied.

## 4. Linear Stability Analyses

#### 4.1. Linearized Disturbance Equations

#### 4.2. Numerical Methodology

_{R}and the imaginary part S

_{I}of the eigenvalue S. The two values can be obtained by applying it to one of Equations (16)–(19). The following explanation is the case where Equation (16) is applied. It is divided into a real part and an imaginary part and is regarded as a function of only S

_{R}and S

_{I}, respectively.

_{1}and C

_{2}are constants for adjusting the convergence speed of Newton’s method. By imposing Equation (24) at an arbitrarily selected grid point during the iterative calculation, the linear growth rate S

_{R}and the angular frequency S

_{I}can be simultaneously obtained.

_{R}of the eigenvalue obtained is not always zero (and the imaginary part S

_{I}is not always zero either). For the purpose of drawing a neutral stability curve, one may want to find a combination of Pr, Ha, Gr, k, and S

_{I}such that S

_{R}becomes zero. Therefore, Pr, Ha, and k should be fixed first, while Gr is allowed to be slightly shifted from the preobtained solution, so that S

_{R}becomes zero. At this time, S

_{I}is also modified to some extent. Newton’s method can also be applied to the iterative correction. For example, using the growth rate S

_{R}, it is given as follows.

_{3}is a positive constant for adjusting the convergence speed of Newton’s method. When the growth rate S

_{R}is positive, it means that Gr is decreased. This is effective in finding the lower branch in the neutral stability curve. As can be seen from Equation (25), when S

_{R}approaches zero, the value of Gr does not change and it converges. In this way, a neutral Grashof number can be obtained. On the other hand, in order to find the upper branch, the sign of the constant should be inverted. In the region connecting the upper branch and the lower branch, the gradient of the curve with respect to k is large, so it is more efficient to fix Gr and find k in the neutral stable state in order to keep computational accuracy. The more detailed numerical method on the linear stability analysis can be found in the recent paper [33].

#### 4.3. Verification of the Code Developed

_{c}= 1.347 and Gr

_{c}= 487.97 and k

_{c}= 1.405 and Gr

_{c}= 502.59, respectively. Figure 3 shows the visualization of the velocity and temperature under such conditions. Since periodicity is assumed in the vertical direction, only about one wave length is shown here. Convection cells (vortices) rotate in different directions between adjacent vortices, as observed in Bénard convection. However, the convection cell is not like a rectangle, but rather a shape close to a parallelogram. A clockwise vortex has the effect of strengthening the basic flow, so the pressure drops, while a counterclockwise vortex weakens the basic flow, so the pressure rises. It can be seen that the temperature is high in the flow part that separates from the heating surface on the left side, and the temperature is low in the flow part that separates from the cooling surface on the right side. Even if the Prandtl number is changed, the value of the critical Grashof number does not vary significantly as shown in Figure 4. This result was obtained with the number of grids 201 using the discretization of a fourth-order central difference method.

_{R}and the angular frequency S

_{I}on the wavenumber for Pr = 15 and Gr = 400. Since the angular frequency increases almost linearly, it can be seen that the phase velocity of the wave hardly depends on the wavenumber. Figure 5b shows the visualization of the travelling wave mode for k = 0.5. The temperature of the higher and lower parts near the right cold wall moves downward at a constant phase velocity.

#### 4.4. Results for MHD Natural Convection

_{I}always converged to zero, that is, the traveling wave mode did not appear in this study.

## 5. Discussion

## 6. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

b_{i} | magnetic flux density = (b_{x}, b_{y}, 0) (T) |

B_{i} | dimensionless magnetic flux density = (B_{X}, B_{Y}, 0) (-) |

b_{0} | absolute value of magnetic flux density imposed (T) |

C | constant (-) |

g | gravitational acceleration (m/s^{2}) |

Gr | Grashof number (-) |

Ha | Hartmann number (-) |

i | imaginary unit (-) |

j_{i} | electric current density = (j_{x}, j_{y}, j_{z}) (A/m^{2}) |

J_{i} | dimensionless electric current density = (J_{X}, J_{Y}, J_{Z}) (-) |

k | dimensionless wavenumber (-) |

l | characteristic length (m) |

p | pressure (Pa) |

P | dimensionless pressure (-) |

Pr | Prandtl number (-) |

S | complex eigenvalue (rad/s) |

S_{I} | angular frequency (rad/s) |

S_{R} | linear growth rate (rad/s) |

t | time (s) |

T | temperature (K) |

T_{c} | temperature at cold wall (K) |

T_{h} | temperature at hot wall (K) |

T_{0} | reference temperature = (T_{h} + T_{c})/2 (K) |

ΔT | temperature difference between hot and cold walls = (T_{h} - T_{c}) (K) |

u_{i} | velocity vector = (u, v, w) (m/s) |

U_{i} | dimensionless velocity vector = (U, V, W) (-) |

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_{i} | Cartesian coordinate (m) |

X_{i} | dimensionless Cartesian coordinate (-) |

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 (m^{2}/s) |

β | volumetric coefficient of thermal expansion at T_{0} (1/K) |

δ_{ij} | Kronecker delta (-) |

ε_{ijk} | Levi-Civita symbol (-) |

Θ | dimensionless temperature (-) |

θ | angle between X-axis and magnetic field (rad) |

ν | kinematic viscosity (m^{2}/s) |

ρ | density (kg/m^{3}) |

ρ_{0} | density at T_{0} (kg/m^{3}) |

σ | electric conductivity (1/(Ω·m)) |

τ | dimensionless time (-) |

φ | electric potential (V) |

Φ | dimensionless electric potential (-) |

Subscripts or superscripts | |

${}^{\prime}$ | infinitesimal disturbance |

$\overline{}$ | basic state |

$\tilde{}$ | amplitude function |

I | imaginary part |

R | real part |

n | number of iterative steps |

## References

- Moreau, R.J. Fluid Mechanics and Its Application. In Magnetohydrodynamics; Kluwer Academic Publishers: Dordrecht, The Netherlands; Norwell, MA, USA, 1990; Volume 3. [Google Scholar]
- Molokov, S.; Moreau, R.; Moffatt, H.K. Fluid Mechanics and Its Application. In Magnetohydrodynamics: Historical Evolution and Trends; Springer: Berlin/Heidelberg, Germany, 2007; Volume 80. [Google Scholar]
- Ozoe, H. Magnetic Convection; Imperial College Press: London, UK, 2005. [Google Scholar]
- Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Dover Publication: Mineola, NY, USA, 1961. [Google Scholar]
- Burr, U.; Müller, U. Rayleigh–Bénard convection in liquid metal layers under the influence of a vertical magnetic field. Phys. Fluids
**2001**, 13, 3247–3257. [Google Scholar] [CrossRef] - Burr, U.; Müller, U. Rayleigh-Bénard convection in liquid metal layers under the influence of a horizontal magnetic field. J. Fluid Mech.
**2002**, 453, 345. [Google Scholar] [CrossRef] - Mistrangelo, C.; Bühler, L. Magneto-convective instabilities in horizontal cavities. Phys. Fluids
**2016**, 28, 024104. [Google Scholar] [CrossRef] - Vest, C.M.; Arpaci, V.S. Stability of natural convection in a vertical slot. J. Fluid Mech.
**1969**, 36, 1–15. [Google Scholar] [CrossRef] - Hart, J.E. Stability of the flow in a differentially heated inclined box. J. Fluid Mech.
**1971**, 47, 547–576. [Google Scholar] [CrossRef] - Bergholz, R.F. Instability of steady natural convection in a vertical fluid layer. J. Fluid Mech.
**1978**, 84, 743–768. [Google Scholar] [CrossRef] - Choi, I.; Korpela, S.A. Stability of the conduction regime of natural convection in a tall vertical annulus. J. Fluid Mech.
**1980**, 99, 725–738. [Google Scholar] [CrossRef] - Lee, Y.; Korpela, S. Multicellular natural convection in a vertical slot. J. Fluid Mech.
**1983**, 126, 91–121. [Google Scholar] [CrossRef] - Chen, Y.M.; Pearlstein, A.J. Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots. J. Fluid Mech.
**1989**, 198, 513–541. [Google Scholar] [CrossRef] - Fujimura, K.; Mizushima, J. Nonlinear equilibrium solutions for travelling waves in a free convection between vertical parallel plates. Eur. J. Mech. B Fluids
**1991**, 10, 25–30. [Google Scholar] - McAllister, A.; Steinolfson, R.; Tajima, T. Vertical Slot Convection: A Linear Study (No. DOE/ET/53088-584; IFSR--584); Institute for Fusion Studies, Texas University: Austin, TX, USA, 1992. [Google Scholar]
- Lartigue, B.; Lorente, S.; Bourret, B. Multicellular natural convection in a high aspect ratio cavity: Experimental and numerical results. Int. J. Heat Mass Transf.
**2000**, 43, 3157–3170. [Google Scholar] [CrossRef] - Bratsun, D.A.; Zyuzgin, A.V.; Putin, G.F. Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side. Int. J. Heat Fluid Flow
**2003**, 24, 835–852. [Google Scholar] [CrossRef] - Wright, J.L.; Jin, H.; Hollands, K.G.T.; Naylor, D. Flow visualization of natural convection in a tall, air-filled vertical cavity. Int. J. Heat Mass Transf.
**2006**, 49, 889–904. [Google Scholar] [CrossRef] [Green Version] - Ganguli, A.A.; Pandit, A.B.; Joshi, J.B. Numerical predictions of flow patterns due to natural convection in a vertical slot. Chem. Eng. Sci.
**2007**, 62, 4479–4495. [Google Scholar] [CrossRef] - Fogaing, M.T.; Nana, L.; Crumeyrolle, O.; Mutabazi, I. Wall effects on the stability of convection in an infinite vertical layer. Int. J. Therm. Sci.
**2016**, 100, 240–247. [Google Scholar] [CrossRef] - Suslov, S.A.; Paolucci, S. Stability of natural convection flow in a tall vertical enclosure under non-Boussinesq conditions. Int. J. Heat Mass Transf.
**1995**, 38, 2143–2157. [Google Scholar] [CrossRef] - Kitada, T.; Kato, Y.; Fujimura, K. Non-Boussinesq effects on the linear stability of thermal convection in an inclined slot. Trans. Jpn. Soc. Mech. Eng. Ser. B
**2002**, 68, 1002–1007. [Google Scholar] [CrossRef] - Takashima, M. The stability of natural convection in a vertical layer of electrically conducting fluid in the presence of a transverse magnetic field. Fluid Dyn. Res.
**1994**, 14, 121. [Google Scholar] [CrossRef] - Nagata, M. Nonlinear analysis on the natural convection between vertical plates in the presence of a horizontal magnetic field. Eur. J. Mech. B Fluids
**1998**, 17, 33–50. [Google Scholar] [CrossRef] - Burr, U.; Barleon, L.; Jochmann, P.; Tsinober, A. Magnetohydrodynamic convection in a vertical slot with horizontal magnetic field. J. Fluid Mech.
**2003**, 475, 21. [Google Scholar] [CrossRef] - Hudoba, A.; Molokov, S. Linear stability of buoyant convective flow in a vertical channel with internal heat sources and a transverse magnetic field. Phys. Fluids
**2016**, 28, 114103. [Google Scholar] [CrossRef] [Green Version] - Wong, C.P.C.; Malang, S.; Sawan, M.; Dagher, M.; Smolentsev, S.; Merrill, B.; Youssef, M.; Reyes, S.; Sze, D.; Morley, N.; et al. An overview of dual coolant Pb–17Li breeder first wall and blanket concept development for the US ITER-TBM design. Fusion Eng. Des.
**2006**, 81, 461–467. [Google Scholar] [CrossRef] [Green Version] - Soto, C.; Smolentsev, S.; García-Rosales, C. Mitigation of MHD phenomena in DCLL blankets by Flow Channel Inserts based on a SiC-sandwich material concept. Fusion Eng. Des.
**2020**, 151, 111381. [Google Scholar] [CrossRef] - Tagawa, T.; Authié, G.; Moreau, R. Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 1. Fully-established flow. Eur. J. Mech. B Fluids
**2002**, 21, 383–398. [Google Scholar] [CrossRef] - Authié, G.; Tagawa, T.; Moreau, R. Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures. Eur. J. Mech. B Fluids
**2003**, 22, 203–220. [Google Scholar] [CrossRef] - Lyubimov, D.V.; Lyubimova, T.P.; Perminov, A.B.; Henry, D.; Hadid, H.B. Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 2. Effect of a magnetic field. J. Fluid Mech.
**2009**, 635, 297–319. [Google Scholar] [CrossRef] [Green Version] - Kitaura, T.; Tagawa, T. Linear stability analysis of thermal convection in an infinitely long vertical rectangular enclosure in the presence of a uniform horizontal magnetic field. J. Fluids
**2014**, 2014, 642042. [Google Scholar] [CrossRef] [Green Version] - Tagawa, T. Linear stability analysis of liquid metal flow in an insulating rectangular duct under external uniform magnetic field. Fluids
**2019**, 4, 177. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Geometrical configuration of the problem: (

**a**) the rectangular enclosure is sufficiently long in the vertical direction. The basic temperature is in heat conduction state and the basic flow is assumed to have only vertical component of velocity. (

**b**) A uniform horizontal magnetic field is applied in either the X-direction, the Y-direction, or the oblique direction.

**Figure 2.**Basic states within a cross-section for the case of an infinitely long vertical enclosure. The upper ones indicate the case of X-directional magnetic field at Ha = 2, and the lower the case of Y-directional magnetic field at Ha = 10: (

**a**) contour map of the vertical component of velocity, (

**b**) electric current density vectors, and (

**c**) contour map of the electric potential.

**Figure 3.**Velocity vectors, temperature and pressure at the onset of instability: (

**a**) Pr = 0.01, k

_{c}= 1.347 and Gr

_{c}= 487.97; (

**b**) Pr = 0.7, k

_{c}= 1.405 and Gr

_{c}= 502.59.

**Figure 4.**Variations of the critical Grashof number and the critical wavenumber for the wide range of the Prandtl number.

**Figure 5.**(

**a**) Linear growth rate S

_{R}and angular frequency S

_{I}as a function of wavenumber k at Pr = 15 and Gr = 400 and (

**b**) contour map of temperature and velocity vectors (visualized by compressing twice in the vertical direction) at Pr = 15, Gr = 400, k = 0.5, S

_{I}= 0.0297, and S

_{R}= 1.06 × 10

^{−4}.

**Figure 7.**Effect of the Hartmann number on the critical Grashof number and the wavenumber at Pr = 0.025: (

**a**) X-directional magnetic field and (

**b**) Y-directional magnetic field.

**Figure 8.**The mode at the onset of instability for the case of X-directional magnetic field at Pr = 0.025, Ha = 2, and k = 0.5566. Each figure is visualized by two isosurfaces as well as contour maps on the walls: (

**a**) X-directional velocity, (

**b**) Y-directional velocity, (

**c**) Z-directional velocity, (

**d**) temperature, (

**e**) pressure, (

**f**) X-directional current density, (

**g**) Y-directional current density, (

**h**) Z-directional current density, and (

**i**) electric potential.

**Figure 9.**The mode at the onset of instability for the case of Y-directional magnetic field at Pr = 0.025, Ha = 10, and k = 0.8196. Each figure is visualized by two isosurfaces as well as contour maps on the walls: (

**a**) X-directional velocity, (

**b**) Y-directional velocity, (

**c**) Z-directional velocity, (

**d**) temperature, (

**e**) pressure, (

**f**) X-directional current density, (

**g**) Y-directional current density, (

**h**) Z-directional current density, and (

**i**) electric potential.

**Figure 10.**Effect of the direction of the magnetic field on the basic state at Ha = 3: (

**a**) vertical component of velocity, (

**b**) electric current density vectors, and (

**c**) electric potential.

**Table 1.**The critical wavenumbers and Grashof numbers obtained for various Hartmann numbers with the mesh systems of 50 × 50 and 100 × 100 at Pr = 0.025.

Ha (Direction) | k_{c} (50 × 50) | Gr_{c} (50 × 50) | Gr_{n} (100 × 100) |
---|---|---|---|

0 | 0.7085 | 2501.77 | 2497.79 |

0.1 (X) | 0.7082 | 2506.12 | 2502.12 |

0.5 (X) | 0.6986 | 2612.84 | 2608.45 |

1.0 (X) | 0.6689 | 2977.45 | 2971.56 |

1.5 (X) | 0.6207 | 3708.12 | 3698.37 |

2.0 (X) | 0.5566 | 5072.35 | 5052.06 |

2.5 (X) | 0.4838 | 7705.94 | 7650.83 |

3.0 (X) | 0.4172 | 13,193.0 | 12,989.7 |

3.5 (X) | 0.3562 | 26,057.7 | 25,019.8 |

4.0 (X) | 0.3485 | 57,759.7 | 52,146.2 |

1 (Y) | 0.7212 | 2442.15 | 2438.47 |

2 (Y) | 0.7524 | 2310.72 | 2307.64 |

3 (Y) | 0.7887 | 2189.21 | 2186.67 |

4 (Y) | 0.8198 | 2124.65 | 2122.49 |

5 (Y) | 0.8413 | 2129.52 | 2127.56 |

7 (Y) | 0.8548 | 2338.08 | 2336.37 |

10 (Y) | 0.8196 | 3121.67 | 3121.36 |

15 (Y) | 0.6904 | 6149.12 | 6180.34 |

20 (Y) | 0.5687 | 14,081.9 | 14,435.9 |

25 (Y) | 0.4987 | 36,162.0 | 38,663.4 |

**Table 2.**The effect of the angle of the magnetic field on the critical wavenumber and Grashof number at Pr = 0.025 and Ha = 3 when the mesh system of 50 × 50 was employed.

Angle, θ rad | k_{c} | Gr_{c} |
---|---|---|

0 (X-mag.) | 0.4172 | 13,193 |

π/12 | 0.4424 | 11,964 |

π/6 | 0.5155 | 8240.4 |

π/4 | 0.6082 | 4813.4 |

π/3 | 0.7015 | 3112.7 |

5π/12 | 0.7662 | 2389.4 |

π (Y-mag.) | 0.7887 | 2189.2 |

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**MDPI and ACS Style**

Tagawa, T.
Effect of the Direction of Uniform Horizontal Magnetic Field on the Linear Stability of Natural Convection in a Long Vertical Rectangular Enclosure. *Symmetry* **2020**, *12*, 1689.
https://doi.org/10.3390/sym12101689

**AMA Style**

Tagawa T.
Effect of the Direction of Uniform Horizontal Magnetic Field on the Linear Stability of Natural Convection in a Long Vertical Rectangular Enclosure. *Symmetry*. 2020; 12(10):1689.
https://doi.org/10.3390/sym12101689

**Chicago/Turabian Style**

Tagawa, Toshio.
2020. "Effect of the Direction of Uniform Horizontal Magnetic Field on the Linear Stability of Natural Convection in a Long Vertical Rectangular Enclosure" *Symmetry* 12, no. 10: 1689.
https://doi.org/10.3390/sym12101689