Influence of vertical magnetic field

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Journal of Turbulence

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Rotating magneto-convection: Influence of vertical magnetic field Hirdesh Varshneya; Mirza Faisal Baiga a Department of Mechanical Engineering, AMU, Aligarh, India First published on: 01 January 2008

To cite this Article Varshney, Hirdesh and Baig, Mirza Faisal(2008) 'Rotating magneto-convection: Influence of vertical

magnetic field', Journal of Turbulence, Volume 9, Art. No. N 33,, First published on: 01 January 2008 (iFirst) To link to this Article: DOI: 10.1080/14685240802392451 URL: http://dx.doi.org/10.1080/14685240802392451

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Journal of Turbulence Vol. 9, No. 33, 2008, 1–20

Rotating magneto-convection: Influence of vertical magnetic field Hirdesh Varshney and Mirza Faisal Baig∗ Department of Mechanical Engineering, AMU, Aligarh, India

Downloaded By: [University of Warwick] At: 18:58 12 August 2010

(Received 12 December 2007; final version received 5 June 2008) In the present numerical study the effect of a vertically applied magnetic field on convection of low Prandtl number liquid metal, rotating in a cubical cavity of aspect-ratio 8:8:1 has been investigated. The bottom wall is heated while the top wall is cooled and all the other walls are kept thermally insulated. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Pr = 0.01, Rayleigh number Ra = 107 and magnetic Prandtl number Pm = 4.0 × 10−4 , while the Chandrasekhar number Q varies from 5.0625 × 104 to 4.9 × 105 and the Taylor number Ta is varied from 0 to 1010 . It is found that the magnetic field generates a strong damping effect on flow velocities and heat transfer at low rotation rates corresponding to Ta = 106 and Ta = 2.5 × 107 , though at higher rotation rates Ta = 1010 the Coriolis forces become comparable to the Lorentz forces thus generating conditions conducive for incipience of dynamo. At the same Ta, with an increase in magnetic fields corresponding to Qh = 4.9 × 105 there is a sharp fall in rms velocities near the top and bottom walls due to the formation of thin Hartmann layers. At low rotation rates, vertically stretched multi-cellular rolls perpendicular to the gravity axis are found. As the rotation increases, the rolls change their spatial orientation and align themselves parallel to the rotation axis. At the highest rotation rate Ta = 1010 , a cylindrical vortical column forms at the centre of the cavity. Keywords: rotating magneto-convection; Lorentz force; Coriolis force; statistical analysis

1.

Introduction

Magneto-convection is the study of convective motion of electro-conducting fluids such as liquid metal or plasma in the presence of external heating and applied electro-magnetic field. These electro-conducting fluids are influenced by Lorentz force generated by an externally applied magnetic field when the fluid convects in a thermally buoyant field due to the application of differential or Rayléigh–Benard thermal boundary conditions. Rotation with the magnetic field makes the flow more complex and can exhibit profound influences on the convection when either of the different forces, namely Lorentz, gravitational buoyancy, Coriolis and centrifugal, are of comparable magnitude. This phenomenon has wide industrial applications in weakening of the buoyancy-driven fluctuations which in turn modify the interface shape and rate of solidification especially in crystal growth. Besides this the phenomenon is significant for convection in planetary cores and stellar interiors especially in the presence of strong rotational and magnetic constraints. The most significant non-dimensional parameters in rotating magneto-convection are: 1. The Rayleigh number Ra = αgHνκT which is the ratio of gravitational buoyancy forces to viscous forces. Here α is the thermal expansion coefficient, H is the depth of fluid, 3

∗

Corresponding author. Email: [email protected]

ISSN: 1468-5248 online only C 2008 Taylor & Francis DOI: 10.1080/14685240802392451 http://www.tandf.co.uk/journals

2

H. Varshney and M.F. Baig g is the magnitude of the acceleration due to gravity, ν is the kinematic viscosity, T is the temperature difference between the top and bottom walls and thermal diffusivity κ = K/ρcp involving the thermal conductivity K, the density ρ and the specific heat capacity cp . σ H 2B2 2. The Chandrasekhar number Q = Ha2 = ρ◦ ν ◦ is the square of the Hartmann number H a and represents the ratio of Lorentz forces FL = j × B, that are produced by the interaction of the current density j with the magnetic field B, to the viscous forces. Here, σ is the electrical conductivity and B◦ is the magnitude of external magnetic field. 2 H 4 3. The Taylor number Ta = Dν 2 is the squared ratio of Coriolis forces to viscous forces. Here D is the dimensional angular velocity about the vertical axis. Another nondimensional parameter that is significant for rotational flows is the Rossby number Ro . and is expressed as Ro = PRa rT a


(2 H )αH 3 T

which is the ratio of rotational 4. The Rotational Rayliegh number Raw = D νκ buoyancy (centrifugal force) to viscous force. 5. The Prandtl number Pr = κν is the ratio of the viscous to thermal diffusion and characterizes the diffusive properties of the fluid. 6. The magnetic Prandtl number Pm = ννh is the ratio of the viscous to the magnetic diffusion. The induced magnetic field will be small for lower value of Pm and larger for larger value of Pm. , where U is the induced dimensional con7. The magnetic Reynolds number Rem = UνH h vective velocity. If Rem 1, then the magnetic diffusion is the dominant process and the magnetic field distortion will be negligible. For this study, the velocity scale is κ/H and ν = Un PPmr , where Un is the non-dimensional convective velocity. hence Rem = UHn κH νh ν L 8. Thermal convection in bounded regions also depends on the aspect ratio = LHx = Hy . To date many numerical and experimental works have been done to study the rotating or non-rotating magneto-convection in rectangular or cylindrical enclosures. Chandrasekhar [1] theoretically studied the instability of a layer of fluid heated from below and subject to the simultaneous action of the magnetic field and rotation. He found that the dynamics of convection of an electrically conducting low Prandtl number fluid is simultaneously affected both by the contradictory and reinforcing effects of the external magnetic field and rotation. Lehnert and Little [2] experimentally studied the effect of inhomogeneity and obliquity of a magnetic field in inhibiting convection. In their results they stated that when the direction of imposed magnetic field is different from the vertical, only the component of the magnetic field in the direction of the vertical is effective. Also for the case of the horizontal magnetic field, there is no discernible effect in inhibiting convection even though the field is five times that necessary to suppress convection, if acting in the vertical direction. Nakagawa [3] in his experimental study on the instability of a layer of mercury heated from below in a cylinder and subjected to the simultaneous action of a magnetic field and rotation, found discontinuous change in the size of the cells with increasing field strength (Ha = 125–3000) at constant rotation rate (Ta ∼ 8 × 107 ). Ozoe and Okada [4] numerically studied the effect of the direction of the external magnetic field on natural convection in a cubical enclosure. Three-dimensional conservation equations for natural convection of molten silicon in a cubical enclosure heated from one vertical side-wall and cooled from an opposing wall are numerically solved under three different external magnetic fields either in the x-, y- or z-directions for Ra = 106 and 107 , Ha = 0–500 at a fixed Pr = 0.054.



3

They found that the external magnetic field is the most effective in suppressing convection when applied perpendicular to the heated vertical wall. It is least effective when the magnetic field is horizontal and parallel to the heated vertical wall. Baumgartl et al. [5] both experimentally and numerically studied buoyant flow of a conducting fluid in a vertical cylinder with aspect ratio h/d = 1.9 under the influence of a hydrodynamically destabilizing vertical temperature difference with an axially symmetric magnetic field. For Ra = 6.678 × 104 and Ha = 5.8, they found a temporal variation of the temperature fluctuations to be nearly sinusoidal with typical amplitudes of about 4% of the mean. For Ha ≥ 20, the flow fluctuations are suppressed and the resulting steady flow has essentially the same features as the unsteady one: a single inclined roll with respect to the cylinder axis with small counter-rotating rolls near the top and bottom walls. Ben and Henry [6] numerically studied the convection in a cubical enclosure under the action of imposed vertical and transverse magnetic fields in a 4 × 1 × 1 rectangular cavity with constant horizontal temperature gradient. They found that in the case of the vertical magnetic field and relatively small Ha ≥ 10, the magnetic field is primarily associated with breaking of the average flow in the melt, but at sufficiently large Ha ≥ 100 the flow becomes unidirectional over a large part of the cavity with a quiescent core region surrounded at top and bottom by Hartmann boundary layers and by parallel layers at the vertical side walls. Aurnou and Olson [7] experimentally measured heat transfer for liquid gallium (Pr = 0.023), subject to the combined action of vertical rotation and a uniform vertical magnetic field in a rectangular cavity of size 15.2 cm × 15.2 cm × 3.8 cm. They found that for rotating magnetoconvection, the convective heat transfer is inhibited by rotation for supercritical Taylor number Ta > 104 . The previous studies were mainly concerned with the effect of the magnetic field on convection and velocity fields without rotation, with the exception of Aurnou and Olson [7]. The main aim of our study is to analyse the effect of the magnetic field and rotation, when applied parallel to each other, on the flow dynamics and on convection. Basically this study aims to explore the six-dimensional parametric space by varying two independent parameters, namely Ta and Q. The study also investigates the formation of coherent structures in rotating magneto-convection and how they influence the heat and momentum transport. Besides this we aim to determine the effect of rotation and magnetic field on the statistics of dynamical variables. The present work involves a numerical study of the three-dimensional rotating magnetoconvection in a large aspect-ratio (8:8:1) enclosure rotating about a vertical axis passing through its centre of gravity (see Figure 1). The bottom and top walls are isothermally heated and cooled, respectively, while the other walls are kept thermally insulated. All the walls are electrically insulated. The simulations have been carried out for liquid metal flows having a fixed Prandtl number, Pr = 0.01, Rayleigh number, Ra = 107 and magnetic Prandtl number, Pm = 4.0 × 10−4 , while the Chandrasekhar number, Q, and Taylor number, Ta, are varied by changing the strength of the applied magnetic field and rotation rate. 2. The mathematical model The basic equations used in the simulation of rotating flow subjected to a magnetic field are the incompressible 3D Navier–Stokes equations with the inclusion of Lorentz, Coriolis and centrifugal forces, energy equation and Faraday’s law of induction equation. The Boussinesq approximation, i.e. linear variation of density with small temperature difference has been considered for both the gravitational and centrifugal force terms [8, 9]. The conservation equations are made dimensionless using length scale (H ), time scale (H 2 /κ), velocity scale (κ/H ), pressure scale (ρo κ 2 /H 2 ), temperature scale ( = (T − Tc )/T ) and (ν/H 2 ) scale for rotation rate. The timescale for

4

H. Varshney and M.F. Baig

√ rotation is based on kinematic viscosity ν in order to set non-dimensional rotation rate = T a. Using these reference scales, the governing equations may be written as conservation of mass ∇ · V = 0;

(1)

conservation of momentum in rotating coordinate frame ∂V + V · ∇V = −∇p + P r∇ 2 V + (2T a 0.5 P r v − Raw xP r )iˆ + ∂t


P r2 (−2T a 0.5 P r u − Raw yP r )jˆ + RaP r kˆ + Q [∇ × B × B] ; Pm conservation of energy Q P r 2 αgH ∂ 2 + V · ∇ = ∇ + (∇ × B)2 ; ∂t Ra P m cp

(2)

(3)

Faraday’s law of induction ∂B 1 + V · ∇B = (B · ∇)V + ∇ 2 B. ∂t Rem

(4)

Initially the fluid is supposed to be in a quiescent state (u = 0, v = 0, w = 0) with respect to the rotating frame of reference at isothermal conditions T = Tc , i.e. = 0. No-slip conditions for velocity components at the solid boundaries i.e. u = v = w = 0 are enforced. The thermal boundary conditions are ∂ /∂x = 0 at x = ±L/2, ∂ /∂y = 0 at y = ±B/2, = 1 at z = −H /2 and = 0 at z = H /2. The magneto-hydrodynamic initial conditions assume homogeneous vertical magnetic field Bz = 1.0, Bx = By = 0, while the boundary conditions are ∂Bn /∂n = 0 i.e. on top and bottom walls ∂Bz = 0, Bx = By = 0 ∂z are used in order to maintain continuity of the normal components of the magnetic field as well as to enforce electrically insulated boundary conditions at the walls, i.e. jn = 0 [1]. The space-averaged Nusselt number at both the cold and hot walls has been calculated using the relation 1 Nu = − Lx × Ly

3.

Lx /2 −Lx /2

Ly /2

−Ly /2

∂ dxdy. ∂z

(5)

Numerical scheme and validation

Equations (1)–(4) are solved in time using the second-order explicit Adam–Bashforth integration scheme. The convective nonlinear terms are discretized using Taylor-series based upwind schemes. The first-order accurate upwind scheme is used at points adjacent to the domain boundaries, while the third-order accurate upwind scheme is used in the interior domain. Viscous and thermal diffusion terms as well as pressure terms present in the momentum and energy


5

Table 1. Comparison of the maximum local velocities at Ra = 102 , T a = 102 , and Raw = 106 .

Time (t)


0.05 0.10 0.15 0.20 S.S.

|umax | Lee and Lin [11]

|umax | Present work

|vmax | Lee and Lin [11]

|vmax | Present work

|wmax | Lee and Lin [11]

|wmax | Present work

130.54 122.08 119.48 118.74 118.56

127.96 122.25 120.57 120.24 120.15

94.24 90.42 89.15 89.15 89.08

91.19 88.65 88.21 88.09 88.06

22.08 21.84 21.75 21.73 21.73

21.23 20.77 20.91 20.89 20.89

equations are discretized using the second-order accurate central differencing scheme. The details of the numerical scheme used to solve the equations have been explained in Nadeem and Baig [10]. We verify our numerical scheme with Lee and Lin [11] for differentially heated rotating convection initially, without considering magnetic field. The results obtained are quite close to that obtained by Lee and Lin [11] as shown in Tables 1 and 2. We feel that our results are more accurate as the pressure Poisson solver used in Lee and Lin [11] had a high-residual norm of 10−4 compared to our residual norm of 10−7 . The above verification was to check the efficiency of the numerical scheme for rotating convection. To check the accuracy of the scheme for solving Faraday’s law of induction equations, we also validate our results with Aurnou and Olson [7] for Rayleigh–Benard convection with a vertical magnetic field. For a cubical cavity of aspect ratio 6:6:1 having the same Ra = 4 × 104 , P r = 0.023, P m = 1.5 × 10−6 and Chandrasekhar number Q = 1210, we found the mean Nusselt number on both the walls to be N u = 1.26 instead of Nu = 1.25. Regarding grid independence, we simulated rotating magneto-convection on a slightly finer grid of 141 × 141 × 61. We found that integral parameters such as mean Nusselt number N u changed by less than 4% compared to the coarse grid. Moreover, the global maximum velocities u, v and w changed by less than 4%. Hence, in order to cut down the computational time, we ran all our simulation cases on a much coarser grid of 71 × 71 × 31.

4. Result and discussion The study focuses on eight numerical simulations that were performed by varying the Chandrasekhar number Q from 5.0625 × 104 to 4.9 × 105 for four Taylor numbers Ta namely Table 2. Comparison of the average Nusselt number at Ra = 102 , T a = 102 , and Raw = 106 at the bottom heated wall. Time (t) 0.05 0.10 0.15 0.20 S.S.

Nu at x = −0.5 Lee and Lin [11]

Nu at x = −0.5 Present work

4.773 4.182 4.020 3.971 3.956

4.525 4.026 3.955 3.934 3.92

6



Table 3. Operating parameters for the eight simulation cases. Cases

Q

Ta

Rac

Ra/Rac

Ro

Re

Rem

λ = Q/T a 1/2

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8

5.0625 × 104 4.9 × 105 5.0625 × 104 4.9 × 105 5.0625 × 104 4.9 × 105 5.0625 × 104 4.9 × 105

0 0 106 106 2.5 × 107 2.5 × 107 1010 1010

5.0625 × 104 4.9 × 105 5.0625 × 104 4.9 × 105 5.0625 × 104 4.9 × 105 105 105

197.53 20.41 197.53 20.41 197.53 20.41 100 100

∞ ∞ 31.62 31.62 6.325 6.325 0.316 0.316

4500 2200 4500 2200 4200 2300 17000 8000

1.8 0.88 1.8 0.88 1.68 0.92 6.8 3.2

∞ ∞ 50.525 490 10.125 98 0.506 25 4.9

0, 106 , 2.5 × 107 and 1010 , keeping Ra = 107 , Pr = 0.01 and Pm = 4 × 10−4 as constants. The rotational Rayliegh number Raw is computed such that the ratio of Raw /Ta remains constant as 10−3 . This condition stringently satisfies the Boussinesq approximation for all the simulation cases. The Reynolds number Re has been defined based on the maximum root-mean-square dimensional velocity and is given as Re = Vrmsν H = VPrmsr . For the cases 1–6, Vrms is taken as wrms , while for the last two cases it is urms . The operating parameters, supercritical ratios Ra/Rac , Re and Rem = Re× Pm for all the eight simulation cases are listed in Table 3. The last quantity depicted in the table is the Elsasser number λ = Q/T a 1/2 and is a ratio of the Lorentz force to the Coriolis force. In theoretical studies, Eltayeb [12, 13] found that, for asymptotically large T a and Q, the magnetic scaling law ‘Rac ∼ Q holds when Q T a 1/2 ’, while the rotational scaling law ‘Rac ∼ T a 2/3 is followed when T a 1/2 Q3/2 ’ and in the range where the Lorentz and Coriolis forces are comparable, Q < T a 1/2 < Q3/2 , the critical Rayleigh number is reduced and varies as ‘Rac ∼ T a 1/2 ∼ T a/Q’. The results obtained from these simulations have been classified on the basis of either dominance of Lorentz force or Coriolis force. Broadly this yields three cases based on Elsasser number, namely λ 1, λ O(1) and λ < 1. Thus simulation cases 1–6 (see Table 3) pertain to λ O(1), the last case 8 pertains to λ O(1) while the penultimate case 7 pertains to λ < 1. For the cases Q T a 1/2 , the critical Rayleigh number comes out to be Rac 5.0625 × 104 and 4.9 × 105 corresponding to lower Ql = 5.0625 × 104 and higher values of Qh = 4.9 × 105 for all rotation speeds below = 5 × 103 . For the cases Q < T a 1/2 < Q3/2 , the critical Rayleigh number comes out to be Rac 105 for both the lower and higher magnetic field strengths at T a = 1010 . Hence the value of the Rayleigh number taken in our study is well above the critical Rayleigh number, both for non-rotating and rotating magneto-convection cases to ensure unstable convection.

4.1.

Flow structures

The 2D and 3D spatial visualization of velocity and thermal structures has been performed using stream lines and isotherms. For the first case corresponding to non-rotating magnetoconvection, Ql = 5.0625 × 104 , T a = 0 as λ = ∞, the three-dimensional streamlines exhibit multi-cellular convective vortices which are randomly oriented in space. The axis of these vortices is perpendicular to the applied vertical magnetic field as well as the rotation axis. Moreover, these vortices are stretched in the direction of the magnetic field so that their wavelength in horizontal directions is reduced, hence a large number of rolls are present. The equal number of vortices in x and y directions is due to the length and breadth of the cavity being equal and hence there is no



7

Figure 1. Schematic diagram of geometry of the domain.

preferred direction for rolls to align themselves as shown in Figures 2(a)–(c). These findings are in agreement with the results of Buhler and Oertel [14] who found that vortices basically align along the shorter dimension of the cavity in order to minimize viscous losses. On increasing the strength of the applied magnetic field for the non-rotating case, we find that there is increase in the number of rolls and this is consistent with the findings of Ozoe and Okada [4], who found that with increasing strength of the magnetic field the horizontal dimensions of the convective cells get smaller and they become more uniaxially stretched in the direction of the magnetic field. Increasing the rotation rate to T a = 2.5 × 107 , keeping the strength of the applied magnetic field as Ql = 5.0625 × 104 , i.e. Elsasser number λ = 10.125, skews the spatial alignment of the vortices as can be seen from Figures 3(a)–(c) and this is mainly brought about by the enhanced magnitude of Coriolis forces. The 3D visualization of thermal flowfield (see Figures 2 and 3(d)) shows the thermal structures for non-rotating and rotating magneto-convection (cases 1 and 5) with the lower cutoff value of isotherms set at 0.5. The thermal flow field shows a number of randomly rising plumes suggestive of unsteady heat transfer from lower hot wall to the upper cold wall. For the non-magnetic cases, Rossby [15], Yamanaka et al. [16] and Husain et al. [17] also inferred the convection to be non-stationary in the supercritical regime, on the basis of irregular fluctuations in the temperature. The induced Lorentz force j × B behaves unevenly in bulk flow as compared to the side-wall regions which leads to the formation of full-depth thermal plumes at the side-wall regions, as shown in Figures 2 and 3(d). This is due to the fact that as the applied magnetic field is in the vertical direction, the induced Lorentz force is in x and y directions but near the walls these horizontal components become small due to the insulating condition for the electric current density j at the walls [1] and this makes the Lorentz force less effective in side-wall regions. When the applied magnetic field (Lorentz force) and Coriolis force are of comparable magnitude i.e. λ O(1), corresponding to case 8 where λ = 4.9 (Qh = 4.9 × 105 , T a = 1010 ) (see Figures 4(a)–(c)), a large central columnar roll (with small end-corner vortices) aligned parallel



8

Figure 2. Streamline plots of (a) 3D flow structures for λ = ∞, Ql = 5.0625 × 104 and T a = 0, (b) 2D structures in the xz plane at y = 0, (c) 2D structures in the yz plane at x = 0 and (d) 3D thermal structures using isotherms with lower cutoff value set at 0.5.

9



Figure 3. Streamline plots of (a) 3D flow structures for λ 1, Ql = 5.0625 × 104 and T a = 2.5 × 107 , (b) 2D flow structures in the central xz plane at y = 0, (c) 2D flow structures in the central yz plane at x = 0 and (d) 3D thermal structures using isotherms.

to the applied magnetic field is found. The formation of such large-scale structures is explained on the basis that the Coriolis force tries to form two-dimensional structures in the direction perpendicular to the rotation axis, i.e. in the horizontal plane while the vertical component of the Lorentz force stretches them in the direction of the magnetic field. From the visualization of the thermal flow-field it is apparent that there is significant suppression of isolated thermal plumes



10

Figure 4. Streamline plots of (a) 3D flow structures for λ O(1), Qh = 4.9 × 105 and T a = 1010 , (b) 2D streamlines in the central xy plane at z = 0 and (c) 3D thermal structures using isotherms.

in the bulk flow as well as in the side-wall regions (see Figure 4(d)) as compared to the previous case. On further increasing the rotation rate, the Coriolis and rotational buoyancy forces get more dominant for case 7 corresponding to λ = 0.506 25 (Ql = 5.0625 × 104 , T a = 1010 and



11

Raw = 107 ), and this generates a strong large-scale circulation to form a large axisymmetric vortical column with accompanying small corner vortices. The rolls (both thermal and velocity) align themselves parallel to the rotation axis in agreement with the Proudman–Taylor theorem. The pattern of flow is characterised by azimuthal swirl and meridional circulation driven by the rotational forces and is in agreement with Hart and Ohlsen [19] and Hart et al. [20], who reported their findings regarding rotating turbulent convection for higher ranges of Rayleigh numbers Ra = 2.8 × 109 – 8 × 1011 and Taylor numbers T a = 1.2 × 108 – 5 × 1012 . Moreover, for the centrifugal buoyancy-driven circulation they reported vertical asymmetry of the mean temperature profile about the midplane and this is also in accordance with our findings as discussed in the section below. The nature of the plot remains quite similar to the previous case, as is evident by the 3D and 2D spatial flow structures in Figures 5(a) and (b). The thermal flow field shows a large rising axisymmetric column aligned with the rotation axis formed due to the accumulation of random thermal plumes supported by comparatively weak Ekman pumping (see Figure 5(c)). Due to higher rotation rates for the cases 7 and 8, the centrifugal buoyancy (Raw = 107 ) forces also become large and these forces, in the absence of Coriolis forces, have a tendency to radially direct the buoyant rising fluid towards the side walls of the enclosure, while they drive the descending colder fluid near the bottom wall towards the centre and thus generate the phenomenon of Ekman pumping accompanied with formation of side-wall aligned large horizontal rolls. Since for all our simulations Raw /T a = 10−3 , the Coriolis forces are more dominant vis-a-vis centrifugal buoyancy forces and hence the flow structures that form are more influenced by the Coriolis forces though there is weak Ekman pumping also present in the flow. 4.2.

Statistical flow analysis

Turbulent flows are not reproducible but the flow dynamics is assumed to be reproducible for a statistically stationary flow. Statistical analysis has been done by computing mean and rms values of velocity and temperature after spatial-averaging over horizontal planes and then time averaging till a statistically stationary state is achieved. The mean velocities are represented by u and v in horizontal x and y directions, while w is the vertical direction. A statistical time-averaging has been performed of volume-averaged kinetic and magnetic energy for 300 uncorrelated realizations, each realization separated from each other by a constant time-step t. It is found that with the increase of rotation, i.e. for Elsasser number λ > 1, the ratio of kinetic/magnetic energy increases by a factor of 78% compared to the non-rotating case. With further increase of rotation (case 8) pertaining to λ O(1), the ratio of kinetic/magnetic energy decreases and is only 32% greater than the non-rotating case, thus signifying that there is either conversion of kinetic energy to magnetic energy or enhanced viscous dissipation of kinetic energy. At the highest rotation rate for case 7 corresponding to λ < 1, the ratio of kinetic/magnetic energy plummets by 83% compared to the non-rotating case. This large decrease is a strong indication that there is incipience of dynamo due to which a large fraction of kinetic energy gets converted into magnetic energy. Figures 6(a) and (b) show the variation of mean velocity for the cases 1 and 2 (λ = ∞) with increasing value of Q at constant T a. At the lower magnetic field (Ql = 5.0625 × 104 ) the mean values of u, v and w turn out to be small, while the profiles are antisymmetric to each other. At the higher magnetic field (Qh = 4.9 × 105 ) the velocity components still retain an antisymmetric nature. The w values are negligibly small and their magnitudes with respect to instantaneous values are less than 1.5%. On further increasing T a corresponding to cases 5 and 6, the mean-velocity components still retain small variation about a zero mean, as shown in Figures 6(c) and (d). For cases 7 and 8, pertaining to λ < 1 and λ O(1), respectively, the mean-velocity components show small variation about the zero mean.



12

Figure 5. Streamline plots of (a) 3D flow structures for λ < 1, Qh = 5.0625 × 104 and T a = 1010 , (b) 2D streamlines in the central xy plane at z = 0 and (c) 3D thermal structures using isotherms.

Figure 7 shows the variation of rms velocities in the inhomogeneous direction between the top and bottom walls of the enclosure. As the applied magnetic field is in the vertical direction, the components of the Lorentz force are larger in the horizontal directions and have a similar damping effect in the x- and y-directions. Thus a similar variation of horizontal velocity

Journal of Turbulence 10

10

Ta = 0

0

-5

-10 -0.25

0

0.25

Ta = 0

0

-5

0.5

-0.5

-0.25

0

z

z

(a)

(b)

10

0.25

0.5

10

Q = 5.0625 E 04 Ta = 2.5 E 07

Q = 4.9 E 05

, ,

, ,


5

-10 -0.5

5

Q = 4.9 E 05

, ,

, ,

Q = 5.0625 E 04

5

13

0

-5

-10

5

Ta = 2.5 E 07

0

-5

-10 -0.5

-0.25

0

0.25

0.5

-0.5

-0.25

0

z

z

(c)

(d)

0.25

0.5

Figure 6. Variation of the mean velocities u, v and w at Ra = 107 , (a) and (b) for λ = ∞, with (a) at Ql = 5.0625 × 104 and T a = 0 and (b) at Qh = 4.9 × 105 and T a = 0, and (c) and (d) for λ > 1, with (c) at Ql = 5.0625 × 104 and T a = 2.5 × 107 and (d) at Qh = 4.9 × 105 and T a = 2.5 × 107 .

components urms and vrms is observed in the wall-normal direction. For the first simulation corresponding to case 1, i.e. Ql = 5.0625 × 104 and T a = 0 (see Figure 7(a)), urms and vrms have their maxima near the top and bottom walls and minima at the central plane (at z = 0). The wrms profile is also symmetric but has its maxima at the central x–y plane. On increasing the strength of the magnetic field (Qh = 4.9 × 105 ) at the same T a, the increase in the Lorentz force components suppresses the horizontal rms velocities comparatively more than the vertical rms velocity. Moreover, there is formation of thin Hartmann boundary layers near the top and bottom walls which results in large near-wall gradient of horizontal velocity components (see Figure 7(b)). For simulations corresponding to cases 5 and 6, i.e. for λ > 1 and Ql = 5.0625 × 104 and T a = 2.5 × 107 , the variation of rms velocities is similar as in previous cases (see Figure 7(c)). On increasing the magnetic field at the same T a, i.e. Qh = 4.9 × 105 , the rms velocities, especially in horizontal planes get appreciably reduced (see Figure 7(d)). This suggests that the spatial flow

14

H. Varshney and M.F. Baig 60 u(rms) v(rms) w(rms)

Q = 5.0625 E 04

50

Ta = 0

40 30 20 10

u(rms), v(rms), w(rms)


60

0 0

0.25

30 20 10

-0.5

-0.25

0

z

z

(a)

(b)

0.25

0.5

40 u(rms) v(rms) w(rms)

Q = 5.0625 E 04 Ta = 2.5 E 07

40 30 20 10



40

0.5

0

u(rms) v(rms) w(rms)

Q = 4.9 E 05 Ta = 2.5 E 07

30

20

10

0 -0.5

-0.25

0

0.25

0.5

-0.5

-0.25

0

z

z

(c)

(d)

0.25

0.5

250 u(rms) v(rms) w(rms)

Q = 5.0625 E 04 Ta = 1.0 E 10

150

100

50

0


250



-0.25

60

200

Ta = 0

0 -0.5

50


Q = 4.9 E 05

50


Q = 4.9 E 05

200

Ta = 1.0 E 10

150

100

50

0 -0.5

-0.25

0

z

(e)

0.25

0.5

-0.5

-0.25

0

0.25

0.5

z

(f)

Figure 7. Variation of the root-mean-square velocities urms , vrms and wrms at Ra = 107 , (a) and (b) for λ = ∞, with (a) at Ql = 5.0625 × 104 and T a = 0 and (b) at Qh = 4.9 × 105 and T a = 0, (c) and (d) for λ > 1, Qh = 5.0625 × 104 and T a = 2.5 × 107 , and (d) at Ql = 4.9 × 105 and T a = 2.5 × 107 and (e) and (f) at λ O(1) and λ < 1 (e) Qh = 4.9 × 105 and T a = 1010 and (f) Ql = 5.0625 × 104 and T a = 1010 .



15

structures generate large horizontal fluctuating velocities near the top and bottom walls, while in the bulk flow the structures generate large fluctuating vertical velocities (see Stellmach and Hansen [21]). This anisotropic nature of turbulence is responsible for the formation of rolls which are elongated in the direction of the magnetic field. Moreover, this phenomenon of anisotropy of rotating magneto-convection is in agreement with similar findings of Braginsky and Meytlis [23] and Buffett [24]. Further increase of rotation leads to cases 8 and 7, where the Elsasser number becomes either of order O(1) or even smaller. This dramatically alters the wall-normal symmetric variation of all the velocity components (see Figures 7(e) and (f)) leading to smaller horizontal rms velocities in the lower domain of the enclosure compared to the upper domain. There is even asymmetry created in the wall-normal variation of vertical rms velocities. The figure implies that wall-normal velocity fluctuations are significantly smaller than horizontal velocity fluctuations and this redistribution of kinetic energy fluctuations is brought about by the Coriolis forces. So there is conversion of buoyancy-induced vertical kinetic energy into horizontal kinetic energy by the Coriolis forces and this creates a swirling column of fluid which leads to twisting of the applied vertical magnetic field by the enhanced strain-rate of the fluid. This interaction between the Lorentz forces and the strained fluid leads to conversion of kinetic energy into magnetic energy creating conditions conducive to incipience of dynamo [25]. Figure 8 shows the variation of mean and rms rms temperature profile in the wallnormal direction. The mean-temperature profile shown in Figure 8(a) at λ 1 depicts a thermal boundary layer of equal thickness at the top and bottom walls. With increase in the strength of the magnetic field, the thickness of the thermal boundary layer increases at both the hot and cold walls. This result suggests that the vertical heat transfer is significantly inhibited due to reduced thermal gradients in the boundary layers. On increasing rotation corresponding to case 4, the thermal boundary layer thickness remains almost same as in the previous case. Further increase of rotation corresponding to case 5, generates thinner thermal boundary layers which significantly increase thermal gradients at both the upper and lower walls and hence there is augmentation of convective vertical heat transport. But with the increase in the strength of the magnetic field at same rotation rate (case 6), the thickness of the thermal boundary layers again increases leading to a decrease of thermal gradients and consequently heat transfer. For cases 7 and 8 pertaining to highest rotation rates (i.e. for λ < 1 and λ O(1), respectively), the mean-temperature profiles show a marked vertical asymmetry about the z = 0 midplane due to the dominating effect of centrifugal buoyancy forces which induce meridional circulation and this finding is in accordance with the results of Hart and Ohlsen [19] (see Figure 1 of their paper). The rms temperature ( rms ) versus z profile shows how thermal fluctuations vary in the inhomogeneous direction. For cases pertaining to λ 1, increase in the magnetic field at constant T a causes fluctuation gradients to decrease especially near the top and bottom walls. On increasing rotation rate corresponding to case 5, the thermal fluctuations increase significantly. For the cases pertaining to the highest rotation rate (cases 7 and 8, i.e. λ < 1 and λ O(1), respectively) there is significant rise in thermal fluctuations and this is due to the stronger fluctuating horizontal velocity components (u and v) due to rotational buoyancy-induced axisymmetric swirl (see Hart and Ohlsen [19] and Hart et al. [20]). Moreover, the strong centrifugal forces break the mid-plane symmetry of the rms temperature profile as can be seen from the figure. 4.3.

Heat transfer analysis

To show the effect of rotation and magnetic field on heat transfer, space-averaged Nusselt number versus time (Figures 9 and 10) is shown at both the hot and cold walls. For cases 1 and 2,

16

H. Varshney and M.F. Baig 1

λ >> 1 λ >> 1 λ > 1 λ >> 1 λ > 1 λ < 1 λ≈ 1

0.9 0.8 0.7

0.6 0.5 0.4 0.3 0.2

0 -0.5

-0.25

0

0.25

0.5

z (a) 0.5

λ >> 1 λ >> 1 λ > 1 λ >> 1 λ > 1 λ < 1 λ≈ 1

0.45 0.4

θ (rms)


0.1

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.5

-0.25

0

0.25

0.5

z (b) Figure 8. Variation of (a) mean temperatures and (b) root-mean-square temperature rms profile in the wall-normal direction.

with increasing magnetic field strength Q not only is the heat transfer inhibited as shown in Figures 9(a) and (b), but also the turbulent fluctuations get suppressed. For case 4, Figure 10(b) shows that convection is further reduced because of enhanced Coriolis forces which suppress vertical heat transfer. On further increasing T a (corresponding to cases 5 and 6) such that Ql = 5.0625 × 104 and T a = 2.5 × 107 , there is an increase in the value of Nusselt number but with further increase in the value of the magnetic field to Qh at same T a there is an

17



Figure 9. Variation of Nusselt number, (a) and (b) for λ = ∞, with (a) at Ql = 5.0625 × 104 and T a = 0 and (b) at Qh = 4.9 × 105 and T a = 0.

appreciable decrease in heat transfer, as shown in Figures 10(c) and (d). On further increasing rotation rate to T a = 1010 , the increase in Coriolis force further suppresses convective heat transfer, compared to previous cases, as shown in Figures 10(e) and (f). Further, it can be seen that case (f) pertaining to the higher magnetic field has higher N u value compared to case (e). This suggests that cases (e) and (f) are in the asymptotic regime described by Chandrasekhar where strong Coriolis and Lorentz forces partially cancel each other, and this regime as suggested by Stevenson [22] appears to be relevant for the efficient operation of planetary dynamos.



18

Figure 10. Variation of Nusselt number, (a) for λ 1, with Ql = 5.0625 × 104 and T a = 103 and (b) for λ 1 at Qh = 4.9 × 105 and T a = 106 , (c) for λ > 1, Ql = 5.0625 × 104 and T a = 2.5 × 107 , (d) for λ > 1, Qh = 4.9 × 105 and T a = 2.5 × 107 and (e) and (f) for λ < 1, with (e) at Ql = 5.0625 × 104 and T a = 1010 and for λ O(1), (f) at Qh = 4.9 × 105 and T a = 1010 .



5.

19

Conclusion

The study of the combined effects of vertical magnetic field and rotation on convective flows of an electrically conducting liquid metal shows that the dynamics of flow field is highly influenced by the strength of the magnetic field as well as Coriolis forces besides thermal buoyancy. For non-rotating magneto-convection, the increase in the magnetic field generates a larger number of thin rolls which are stretched in the direction of the magnetic field. As the rotation rate increases, for λ > 1, the rolls distort in shape due to the process of realignment. At the highest rotation rate T a = 1010 (case 8, i.e. λ O(1)), a non-axisymmetric cylindrical column aligned parallel to the rotation axis forms at the centre accompanied by small rolls near the corners of the cavity. Keeping the same T a and on decreasing Q (case 7), the Elsasser number λ becomes less than 1, yet the flow structures are slightly changed as compared to the case where λ O(1). The magnitude of rms velocities and accompanying fluctuations are greatly suppressed by higher magnetic field (Qh = 4.9 × 105 ) for all the cases and this is more prominent near the top and bottom walls due to the formation of intense Hartmann boundary layers at the top and bottom walls. Increasing the strength of the magnetic field leads to a significant rise in thickness of thermal boundary layers near the top and bottom walls which reduces the thermal gradients and hence inhibits the convective heat transfer.

References [1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York, 1981. [2] L. Lehnert and N.C. Little, Experiments on the effect of inhomogeneity and obliquity of a magnetic field in inhibiting convection, Tellus 9 (1957), pp. 97–103. [3] Y. Nakagawa, Experiments on the instability of a layer of mercury heated from below and subject to the simultaneous action of a magnetic field and rotation, I Proc. R. Soc. Lond. A 242 (1957), pp. 81–88. [4] H. Ozoe and K. Okada, The effect of the direction of the external magnetic field on the three-dimensional natural convection in a cubical enclosure, Int. J. Heat Mass transfer 32 (1989), pp. 1939–1954. [5] J. Baumgartl, A. Hubert, and G. Muller, The use of magnetohydrodynamic effects to investigate fluid flow in electrically conducting melts, Phys. Fluids A5 (1993), pp. 3280–3289. [6] H. B793 en Hadid and D. Henry, Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field: Part 2. Three-dimensional flow, J. Fluid Mech. 333 (1997), pp. 57–83. [7] J.M. Aurnou and P.L. Olson, Experiments on Rayleigh–Bénard convection, magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech. 430 (2001), pp. 283–307. [8] T.E. Faber, Fluid Dynamics for Physicists, Cambridge University Press, Cambridge, 1995, pp. 378–383 (chapter 9). [9] D.J. Tritton, Physical Fluid dynamics, Van Nostrand Reinhold, UK, 1982. [10] N. Hasan, and M.F. Baig, Evolution to aperiodic penetrative convection in odd shaped rectangular enclosures, Int. J. Numer. Methods Heat Fluid Flow 12 (2002), pp. 895–915. [11] T.L. Lee and T.F. Lin, Transient three-dimensional convection of air in a differentially heated rotating cubic cavity, Int. J. Heat Mass transfer 39 (1996), pp. 1243–1255. [12] I.A. Eltayeb, Hydromagnetic convection in a rapidly rotating fluid layer, Proc. R. Soc. Lond. A 326 (1972), pp. 229–254. [13] ———, Overstable hydromagnetic convection in a rotating fluid layer, J. Fluid Mech. 71 (1975), pp. 161–179. [14] K. Buhler and H. Oertel, Thermal cellular convection in rotating rectangular boxes, J. Fluid Mech. 114 (1982), pp. 261–282. [15] H.T. Rossby, A study of Benard convection with and without rotation, J. Fluid Mech. 36 (1969), pp. 309–335. [16] Y. Yamanaka et al., Rayleigh–Benard oscillatory natural convection of liquid gallium heated from below, Chem. Eng. J. 71 (1998), pp. 201–205. [17] A. Husain, M.F. Baig, and H. Varshney, Investigation of coherent structures in rotating Rayleigh– Benard convection, Phys. Fluids 18 (2006).

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[18] A. Giesecke, Anisotropic turbulence in weakly stratified rotating magnetoconvection, Geophys. J. Int. 171 (2007), pp. 1017–1028. [19] J.E. Hart and D.R. Ohlsen, On the thermal offset in turbulent rotating convection, Phys. Fluids 11 (1999), pp. 2101–2107. [20] J.E. Hart, S. Kittelman, and D.R. Ohlsen, Mean flow precession and temperature probability density functions in turbulent rotating convection, Phys. Fluids 14 (2002), pp. 955–962. [21] S. Stellmach and U. Hansen, Cartesian convection driven dynamos at low Ekman number, Phys. Rev. E 70 (2004), 056312. [22] D.J. Stevenson, Planetary magnetic fields, Earth Planet. Sci. Lett. 208 (2003), pp. 1–11. [23] S.I. Braginsky and V.P. Meytlis, Local turbulence in the earth’s core, Geophys. Astrophys. Fluid Dyn. 55 (1990), pp. 71–87. [24] B.A. Buffett, A comparison of subgrid-scale models for large-eddy simulations of convection in the Earth’s core, Geophys. J. Int. 153 (2003), pp. 753–765. [25] B.A. Buffett and J. Bloxham, Energetics of numerical geodynamo models, Geophys. J. Int. 149 (2002), pp. 211–224.