Elevator mode convection in flows with strong magnetic fields Li Liu and Oleg Zikanov∗ Department of Mechanical Engineering, University of Michigan-Dearborn, 48128-1491 MI, USA (Dated: March 24, 2015)
Abstract Instability modes in the form of axially uniform vertical jets, also called ‘elevator modes’, are known to be solutions of thermal convection problems for vertically unbounded systems. Typically, their relevance to the actual flow state is limited by three-dimensional breakdown caused by rapid growth of secondary instabilities. We consider a flow of a liquid metal in a vertical duct with a heated wall and strong transverse magnetic field and find elevator modes that are stable and, thus, not just relevant, but a dominant feature of the flow. We then explore the hypothesis suggested by recent experimental data that an analogous instability to modes of slow axial variation develops in finite-length ducts, where it causes large-amplitude fluctuations of temperature. The implications for liquid metal blankets for tokamak fusion reactors that potentially invalidate some of the currently pursued design concepts are discussed. PACS numbers: 47.65.-d,47.60.Dx,47.20.Bp
1
I.
INTRODUCTION
Elevator modes appear in thermal convection flows in idealized systems, in which top and bottom walls are replaced by periodicity planes. Convection in such a system can be driven by a vertical gradient of mean temperature dTm /dx either produced by wall heating (if there are lateral walls) or imposed (if a periodic box is considered). It is easy to see by direct substitution into the governing equations that, if dTm /dx = const, such systems allow solutions u = u(y, z, t)ex , θ(y, z, t),
(1)
where u and θ are vertical velocity and perturbations of temperature and (y, z) are the horizontal coordinates. The governing equations become linear for such solutions. At the Grashof numbers Gr , which we define later, above a typically small critical value, the solutions show exponential growth u, θ ∼ eγt , γ > 0
(2)
and have the form of a pair of vertically uniform ascending and descending jets. Detailed analysis of elevator modes in various contexts can be found, e.g., in Refs. 1–6. There are, typically, multiple eigenmodes corresponding to different values of γ. The role of the modes in turbulent convection appears to be limited to initial stages of instability. As soon as they grow to a significant amplitude, the jets experience secondary instabilities and break down into three-dimensional fluctuations.5,6 The situation changes if there is a sufficiently strong mechanism of suppression of secondary instabilities. This can be viscous friction, but only at small Gr and in narrow gaps between walls.5 Another, potentially much more effective mechanism first noticed for convection in a periodic box7,8 and further considered in this work for a wall-bounded system is the action of an imposed steady magnetic field on flows of electrically conducting fluids (e.g., liquid metals). The basic physics of this effect is well understood (see, e.g., Ref. 9). The electric currents induced by motion of a liquid conductor in a magnetic field cause Joule dissipation and, thus, conversion of flow’s kinetic energy into heat. The rate of conversion is proportional to the square of velocity gradient in the magnetic field direction, so the flow becomes anisotropic or even two-dimensional (uniform along the magnetic field lines). Furthermore, magnetohydrodynamic (MHD) boundary layers develop at solid walls (see, e.g., Ref. 10). 2
z y
B
mean flow
x
q
FIG. 1: Flow geometry and coordinate system.
Our work is motivated by development of liquid metal (Li or PbLi) blankets - essential components of future tokamak fusion reactors. The blankets, ideally, serve the triple goal of shielding magnets from the neutrons generated in the reaction zone, converting the neutrons’ energy into heat, and breeding tritium fuel. Several currently developed blankets have poloidal (vertical or close to vertical) ducts, through which the liquid metal circulates at a significant rate.11 The flow is affected by a very strong (up to 10-12 T) almost toroidal steady magnetic field and by heat generation concentrated near the wall facing the reaction chamber. The first effect has been studied for long time, attention being paid to poor transport properties caused by flow laminarization and to strong resistance due to Lorentz force and friction in MHD boundary layers. Curiously, the second effect, more specifically the thermal convection caused by non-uniform heating, has been largely ignored. Only recently it was realized that convection may change the nature of the flow, making it unsteady and creating strong fluctuations of temperature and velocity.12–14 A new mechanism of that type leading to spectacular consequences is presented in this paper. We consider mixed (combined forced and natural) convection of a liquid metal in a vertical duct of square cross-section (see Fig. 1) with electrically insulating walls. A constant mean flow directed downwards is maintained. A steady uniform transverse magnetic field B = Bey is imposed. Constant heat flux q is applied to one wall, while the other walls are thermally insulated. This is different from the situation in an actual blanket, where the heating is
3
a combination of the radiative heating of the first (facing the reaction chamber) wall and internal volumetric heating by absorbed neutrons. Use of our model allows us to avoid uncertainties associated with selection of heating parameters and is partially justified by the fact that the volumetric heating occurs primarily near one wall and subsides exponentially with the distance from it.15
II.
PHYSICAL MODEL
The fluid is assumed Newtonian, incompressible, and electrically conducting. The flow is described using the Boussinesq and quasi-static MHD9 approximations. The governing equations and boundary conditions non-dimensionalized using the duct half-width d, imposed mean velocity U , qd/κ, B, and σU B as respective typical scales are 1 2 Gr Ha 2 ∂u + (u · ∇)u = −∇p − ∇ˆ p+ ∇ u+ e θ + j × ey , x ∂t Re Re Re 2 ∂θ 1 2 dTm + u · ∇θ = ∇ θ − ux , ∂t Pe dx ∇ · u = 0,
(3) (4) (5)
j = −∇φ + u × ey ,
(6)
∇2 φ = ∇ · (u × ey ), ∂φ = u = 0 at y = ±1, z = ±1 ∂n ∂θ = −1 at z = −1, ∂z ∂θ = 0 at y = ±1, z = 1, ∂n
(7) (8) (9) (10)
where u, j, and φ are velocity, electric current, and electric potential. We consider a fully developed flow in an infinitely long duct and present pressure and temperature as pˆ(x) + p(x, t) and Tm (x) + θ(x, t), where pˆ(x) is the imposed linear distribution adjusted to maintain constant mean velocity and 1 Tm (x) ≡ A
Z T dA = − A
2x Pe
(11)
is the steady-state mean temperature established in a balance between the wall heat flux and the heat transport by the streamwise mean velocity. Periodic inlet-exit conditions are imposed on p, θ, φ, and u. 4
The parameters are the Reynolds Re ≡ U dν −1 , Prandtl Pr ≡ νχ−1 , Hartmann Ha ≡ Bd [σ(ρν)−1 ]
1/2
and Grashof Gr ≡ gβqd4 (ν 2 κ)−1 numbers. We also use the Peclet
number Pe ≡ RePr . Here, κ, σ, ν, ρ, β, and χ are, respectively, the thermal and electrical conductivities, kinematic viscosity, density, thermal expansion coefficient, and temperature diffusivity of the fluid. The typical values for a fusion reactor blanket are Pr = 0.032 (PbLi at 570 K and the value used in this paper), Ha up to ∼ 104 , Gr up to ∼ 1012 , and Re varying from nearly zero to up to ∼ 105 depending on the blanket’s design. It should be stressed that no a-priori assumptions about the flow’s dimensionality, timedependency, and the number of non-zero velocity components are made. Solutions with three-dimensional and unsteady fields u, j, φ, θ, and p are allowed.
III. A.
RESULTS Direct numerical simulations
Our investigation started as an attempt to observe the Kelvin-Helmholtz instability associated with inflection points in steady-state profiles of streamwise velocity distorted by the thermal buoyancy effect. Such instability was recently identified in configurations identical to ours, but with the mean flow directed upwards.12,16 We used the DNS approach based the second-order conservative finite difference scheme developed in Refs. 13,17 to compute solutions of (3)–(10). The computational grids, whose sufficiency was verified in grid-sensitivity studies, consisted of up to 1282 points in the transverse plane clustered at the walls using the tanh coordinate transformation.17 In the streamwise direction, the grid points were distributed uniformly with the step 0.1. The Kelvin-Helmholtz instability was found to develop near the inflection point between the upward (near the heated wall) and downward jets, but only at small Gr (less than 107 at 50 ≤ Ha ≤ 400 and Re = 5000).16 At higher Gr , the solutions quickly evolved into the form of the elevator modes (1), (2), even though the simulations were conducted for arbitrary three-dimensional flows in long (up to 100) computational domains and initial conditions consisting of three-dimensional random perturbations of velocity were used. The existence of the elevator modes is expected for our system and can be explained from the physical viewpoint if one considers the mean temperature distribution (11). In a
5
1
(a)
(b) 0.5
0
y
y
0.5
0
q -0.5
-0.5
-1 -1
-0.5
0
0.5
B -1
-1
1
-0.5
(c)
(d)
1
1
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1
1
u/umax
θ/θmax
0.5
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0
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-0.5
-1 -1
0
z
z
-0.5
0
0.5
-1 -1
1
-0.5
0
z
z
FIG. 2: Instantaneous distributions of temperature θ ((a) and (c) ) and streamwise velocity u ((b) and (d) ) computed in the three-dimensional DNS at Re = 5000, Ha = 400, Gr = 108 . Distributions in the cross-section plane x = const and along the line x = const, y = 0 are shown, respectively, in (a), (b) and (c), (d). The solution is dominated by an exponentially growing elevator mode (1), (2). Solid and dashed lines in (a), (b) are for, respectively, positive and negative values. Computations are conducted in a duct of length 4π on a grid with ∆x = 0.1 and 96 × 96 points in the transverse plane clustered toward walls to accurately resolve MHD boundary layers.
flow with downward mean velocity, mean temperature grows downwards, which creates an unstable density stratification. Rather unexpected, however, is the fact that such modes are found in a DNS solution, which means that the modes are stable to three-dimensional perturbations at moderately large Gr and Ha. A typical elevator mode obtained in the three-dimensional DNS is presented in Fig. 2. Velocity field consists of an upward jet near the heated wall and downward jet near the opposite wall. We also see that, as for many (but not all) flows at high Ha, velocity and temperature fields are nearly uniform along the magnetic field lines outside of the thin (of thickness ∼ Ha −1 ) Hartmann layers10 near the walls at y = ±1.
6
B.
Quasi-2D analysis of elevator modes
The DNS results discussed in the previous section demonstrate that the elevator mode solutions describe the actually realized state of our flow at high Ha and Gr . We now focus on these states and analyze them in the framework of a model that uses their nearly yindependent form illustrated in Fig. 2. The quasi-two-dimensional (Q2D) approximation18 rigorously valid at Ha 1, Ha 2 /Re 1 is applied. In this approximation, the flow variables are averaged in the direction of B and the electromagnetic effects are reduced to linear friction in the Hartman layers. For solutions of the form (1), the Q2D approximation of (3)–(10) is ∂u ∂t ∂θ ∂t ∂θ ∂z ∂θ ∂z
Gr 1 ∂ 2 u Ha θ + − u, Re ∂z 2 Re Re 2 1 1 ∂ 2θ = u+ , 2Pe Pe ∂z 2 =C+
(12) (13)
= −1, u = 0 at z = −1,
(14)
= u = 0 at z = 1,
(15)
where u(z, t) and θ(z, t) are now y-averaged streamwise velocity, and temperature and C stands for −∂ pˆ/∂x. The constraints Z 1 udz = 2,
Z
1
θdz = 0.
(16)
−1
−1
have to be satisfied. At this point, one can search for exponential solutions u(z, t) = U (z)eγt ,
θ(z, t) = Θ(z)eγt
and obtain the eigenvalue problem for γ, which can be written as Gr ReC iv 00 2 Θ + Θ (−Reγ − Peγ − Ha) + Θ RePeγ + HaPeγ − = , 2Re 2 Θ0 = −1, 2PeγΘ − 2Θ00 = 0 at z = −1, Θ0 = 0, 2PeγΘ − 2Θ00 = 0 at z = 1
(17)
(18) (19) (20)
and allows multiple eigenmode solutions of the form 4
X C Θ= + Aj eλj (z+1) , 2Peγ 2 + 2HaγPr − Gr Re −2 j=1 7
(21)
where the constants Aj and C are determined from the boundary conditions and the condition (16). For example, at γ = 0, we obtain a steady-state solution 4 CRe 2 X + Θ0 = − Aj eλj (z+1) , U0 = −2Θ000 Gr j=1 " #1/2 r Ha 1 2Gr λ1,2 = ± + Ha 2 + 2 2 Re " #1/2 r Ha 1 2Gr λ3,4 = ±ı − Ha 2 + + . 2 2 Re
(22)
(23)
(24)
We do not attempt to conduct a full analysis of the problem, but limit our attention to the physically relevant eigenmodes, i.e. those experiencing the fastest growth with time. Such modes can be found by direct numerical solution of the system (12)–(16), which we conduct using the second-order finite difference scheme on a uniform grid. For convenience of dealing with rapidly growing modes, u and θ are decomposed into steady-state components Θ0 (z), U0 (z), and perturbations u∗ , θ∗ , for which (12) does not have the term C and (14)-(16) have zero right-hand sides. This allows us to re-scale a solution when its amplitude becomes too large for computer arithmetics. The exponential growth rate is determined as Z 1 Z 1 1 dE ∗2 , where E = u dz or E = θ∗ 2 dz. γ= 2E dt −1 −1
(25)
We compared the growth rates found for the DNS solutions of full three-dimensional equations (3)–(10) and the Q2D equations (12)–(16) at Re = 5000, Gr = 108 , 1010 , and Ha = 100, 200, 400. The results differed by less than 2%. This shows that the Q2D model is accurate at moderately high Ha and, thus, provides accurate results at much higher Ha typical for fusion reactor blankets. The rest of the paper presents a study conducted using the model. We note that an analogous study based on solution of full equations (3)–(10) would be not only unnecessary, but also very difficult due to the high computational cost of resolving Hartmann layers at Ha 1. The growth rates γ computed at Re = 2 × 104 are shown in Fig. 3. The main conclusions are that exponentially growing elevator modes appear at all, but the smallest values of Gr and that at Gr and Ha typical for fusion reactor blankets, γ is very high (∼ 1 or even higher in our units). The results at other Re are qualitatively similar. The only significant effect is reduction of γ at higher Re, which can be attributed to weaker unstable temperature stratification (11). 8
(a) 0.3
(b)
10000
γ
4000
Ha
6000
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0.5
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2000 0
0.9 1
0.7
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1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
8000
0.1 0.2 0.3 0.4
Ha
8000
0.8
1.1
10000
5E+11
0 7
1E+12
8
9
10
11
12
lg(Gr)
Gr
FIG. 3: γ(Gr , Ha) at Re = 2 × 104 . Blanked areas in (b) indicate values of Gr and Ha, at which there are no growing elevator modes and the solution converges to the steady state (22)–(24) .
a
u, θ
b
0
-1
-0.5
0
0.5
1
z
FIG. 4: Exponentially growing elevator modes at Re = 2 × 104 , Gr = 1011 , Ha = 104 (a) and at Re = 105 , Gr = 1012 , Ha = 104 (b). Solid and dashed curves show, respectively, u∗ and 2Peγθ∗ .
The typical distributions of u∗ and θ∗ shown in Fig. 4 demonstrate that the elevator mode is asymmetric with respect to the midplane z = 0 and consists of one upward and one downward jet. This form is consistent with the DNS results (see Fig. 2) and, generally, expected since instability modes of such form have been observed in other configurations of convection with heated vertical walls (see, e.g., Ref. 19). We also see in Fig. 4 that the profile of θ∗ multiplied by 2Peγ nearly coincides with the profile of u∗ outside the boundary layers. This suggests that the elevator modes can be approximated by solutions of the non-viscous, non-conductive problem, for which 2Peγθ∗ = 9
(b)
0.1 0.2
(a)
(c)14
100
0.9
0.8
10000
12 10 8
Gr=10 12
8000 11
γ
6000 4000
Gr=10
γ
1.2
4
Gr=10 11
2
10-1
0.6
Ha
Gr=10 12
6
Gr=10 10
Gr=10 10
1.1
0
1
0
0.5
0.3
2000
0
5E+11
1E+12
5000
10000
0
5000
10000
Ha
Ha
Gr
FIG. 5: Approximation of γ(Gr , Ha) at Re = 2 × 104 by (27) (a) and comparison between values of γ computed in solution of (12)-(16) (solid curves) and approximated by (27) (dashed curves) at Re = 20000 (b) and Re = 5000 (c).
u∗ and (18) written for the growing part of the elevator mode becomes Gr ∗ 2 θ PeReγ + HaPeγ − = 0. 2Re
(26)
The larger root of this equation for γ is 1 γ= 2Re
r
2Gr Ha + − Ha Pe 2
! .
(27)
We see in Fig. 5 that (27) provides a reasonably accurate approximation of γ and its variation with Gr , Ha and Re.
IV.
QUASI-ELEVATOR MODES IN DUCTS OF FINITE LENGTH
We have found that elevator modes exist in a fully developed flow in an infinitely long duct if Gr is sufficiently high. Moreover, our DNS of three-dimensional flows show that the modes are stabilized by the magnetic field against secondary instabilities at 100 ≤ Ha ≤ 800 and, therefore, must be stable at higher Ha typical for fusion reactor blankets. One important question still remains, namely whether counterparts of these modes exist in real finite-length ducts. One can plausibly assume that in the case of a downward flow in a finite but long duct, there exist ‘quasi-elevator modes’ in the form of growing upward and downward jets, which, while not exactly axially uniform, have typical axial length scale of the order of the 10
50
q
B gU
θ [K]
25
0
-25 0
2
4
t [s]
6
8
10
FIG. 6: Experimental data:20 temperature signals at the insulated part of the wall (solid line) and at the axis (dashed line) in a downward flow in a vertical pipe with radius-based Re = 6000, Ha = 150, Gr = 3.75 × 106 . Data are courtesy of I. A. Melnikov.
duct’s length. To verify this hypothesis, we will need extensive computations, in which the Q2D model may or may not be applicable. Leaving this for future research, we provide a brief discussion of experimental evidence and implications for fusion reactor blankets. We start with the observation that if the quasi-elevator modes exist in a duct of length L < ∞, they lead to rapidly growing perturbations of temperature and velocity. The growth stops when the perturbation amplitude ∆θ becomes of the same order of magnitude as the drop of mean temperature along the duct ∆Tm ≈ LdTm /dx. At this stage, the unstable stratification by Tm is compensated by perturbations, thus removing the mechanism of the instability, and the perturbations are suppressed by friction and conduction. After the suppression, the perturbation amplitude remains low until ∆Tm is re-established and the growth is repeated. We should, therefore, expect flow evolution consisting of cycles in each of which temperature fluctuations grow to the maximum amplitude about ∆Tm and then decay. Experimental data presenting such an evolution and, thus, consistent with the just described hypothetical picture were recently reported in Refs. 20,21 (see Fig. 6). Downward flows of mercury in a vertical pipe and a duct of aspect ratio 17:56 with stainless steel walls were studied. The walls were partially heated and partially insulated, so that the net heat flux was perpendicular to the imposed transverse uniform magnetic field. Long test
11
section (tens of hydraulic diameters) were used. The analogy with our case was incomplete in terms of the cross-sectional shape of the duct and in that walls had non-negligible electrical and thermal conductivities, so only qualitative comparison can be made. The main result of the experiments was that, while turbulence was suppressed by the magnetic field, high-amplitude, low-frequency, quasi-periodic fluctuations of temperature appeared at Gr above a moderate threshold and not too high Re. Moreover, in agreement with our hypothesis, the fluctuations had amplitude about the mean temperature drop along the heated test section. For example, for the pipe flow in Fig. 6, the drop can be estimated as ∆Tm = 80qR2 κ−1 Pe −1 ≈ 33 K.
V.
CONCLUDING REMARKS
We have presented a new and, in our view, interesting phenomenon of elevator convection modes in vertical ducts with imposed magnetic field and mean flow directed downwards. The analysis is far from complete and needs to be continued in the future with two effects deserving particular attention: of finite length of the duct and of realistic internal heating. As a final remark, we consider implications of our results for liquid metal blankets of fusion reactors. We accept the hypothesis that quasi-elevator modes develop in poloidal ducts of such blankets and that their growth results in temperature fluctuations reaching the maximum amplitude of the order of the mean temperature drop. As an example, we consider a square duct of a DCLL blanket with half-width 5 cm and length 2 m and assume B = 10 T and the heat flux q varying between 104 Wm−2 and 106 Wm−2 depending on the duct’s location with respect to the wall facing the reaction chamber.22 With the physical properties of PbLi at 570 K,23 we find Re = 21750, Pe = 700, Ha = 9500, and Gr between 1.1×108 and 1.1×1010 . According to our results, elevator modes develop in such ducts at Gr > 1.03×109 , i.e. at q higher than about 105 Wm−2 . The dimensional drop of mean temperature varies between 11 and 110 K at 105 < q < 106 Wm−2 . If temperature fluctuations of such amplitude and pattern similar to that seen in Fig. 6 develop in components of a blanket, the resulting unsteady thermal stresses will lead to rapid deterioration of wall material and loss of structural integrity of the blanket. The effect will be more severe in future generations of fusion reactors, where higher heat fluxes and larger ducts are expected.
12
Financial support was provided by the US NSF (Grant CBET 1232851).
∗
Electronic address:
[email protected]
1
P. G. Baines and A. E. Gill, On thermohaline convection with linear gradients, J. Fluid Mech. 37, 289 (1969).
2
G. K. Batchelor and J. M. Nitsche, Instability of stationary unbounded stratified fluid, J. Fluid Mech. 227, 357 (1991).
3
G. K. Batchelor and J. M. Nitsche, Instability of stratified fluid in a vertical cylinder, J. Fluid Mech. 252, 419 (1993).
4
E. Calzavarini, C. R. Doering, J. D. Gibbon, D. Lohse, A. Tanabe, and F. Toschi, Exponentially growing solutions of homogeneous Rayleigh-B´enard flow, Phys. Rev. E 73, R035301 (2006).
5
L. E. Schmidt, E. Calzavarini, D. Lohse, F. Toschi, and R. Verzicco, Axially homogeneous Rayleigh-B´enard convection in a cylindrical cell, J. Fluid Mech. 691, 52 (2012).
6
T. Radko and D. P. Smith, Equilibrium transport in double-diffusive convection, J. Fluid Mech. 692, 5 (2012).
7
O. Zikanov, A. Thess, and J. Sommeria, Turbulent convection driven by an imposed temperature gradient in the presence of a constant vertical magnetic field, (1998), vol. 66 of Notes on Num. Fluid Mech., pp. 187–199.
8
O. Zikanov and A. Thess, Direct numerical simulation as a tool for understanding MHD liquid metal turbulence, Appl. Math. Mod. 28, 1 (2004).
9
P. A. Davidson, An Introduction to Magnetohydrodynamics (Cambridge University Press, 2001).
10
U. M¨ uller and L. B¨ uhler, Magnetohydrodynamics in channels and containers (Springer, Berlin, 2001).
11
S. Smolentsev, R. Moreau, L. B¨ uhler, and C. Mistrangelo, MHD thermofluid issues of liquidmetal blankets: Phenomena and advances, Fusion Eng. Design 85, 1196 (2010).
12
N. Vetcha, S. Smolentsev, M. Abdou, and R. Moreau, Study of instabilities and quasi-twodimensional turbulence in volumetrically heated magnetohydrodynamic flows in a vertical rectangular duct, Phys. Fluids 25, 024102 (2013).
13
O. Zikanov, Y. Listratov, and V. G. Sviridov, Natural convection in horizontal pipe flow with strong transverse magnetic field, J. Fluid Mech. 720, 486 (2013).
13
14
X. Zhang, O. Zikanov, Mixed convection in a horizontal duct with bottom heating and strong transverse magnetic field, J. Fluid Mech. 757, 33 (2014).
15
S. Smolentsev, M. Morley, and M. Abdou, MHD and thermal issues of the SiCf/SiC flowchannel insert, Fusion Sci. Technol. 50, 107 (2006).
16
L. Liu, A. Schigelone, X. Zhang, and O. Zikanov, in Proc. 9th PAMIR Conf. Fund. Appl. MHD, Riga, Latvia (INP, Open Library, 2014), vol. 1, pp. 12–16.
17
D. Krasnov, O. Zikanov, and T. Boeck, Comparative study of finite difference approaches to simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number, Comp. Fluids 50, 46 (2011).
18
J. Sommeria and R. Moreau, Why, how and when MHD-turbulence becomes two-dimensional, J. Fluid Mech. 118, 507 (1982).
19
R. F. Bergholz, Instability of steady natural convection in a vertical fluid layer, J. Fluid Mech. 84, 743 (1978).
20
I. A. Melnikov, E. V. Sviridov, V. G. Sviridov, and N. G. Razuvanov, in Proc. 9th PAMIR Conf. Fund. Appl. MHD, Riga, Latvia (INP, Open Library, 2014), vol. 1, pp. 65–69.
21
I. Poddubnyi, N. Razuvanov, V. Sviridov, and Y. Ivochkin, in Proc. 9th PAMIR Conf. Fund. Appl. MHD, Riga, Latvia (INP, Open Library, 2014), vol. 1, pp. 330–334.
22
S. Smolentsev, R. Moreau, and M. Abdou, Characterization of key magnetohydrodynamic phenomena for PbLi flows for the US DCLL blanket, Fusion Eng. Design 83, 771 (2008).
23
B. Schulz, Thermophysical properties of the Li(17)Pb(83) alloy, Fusion Eng. Des. 14, 199 (1991).
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