Thermal convection in a toroidal duct of a liquid metal blanket. Part I. Effect of poloidal magnetic field

Fusion Engineering and Design 116 (2017) 52–60

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Thermal convection in a toroidal duct of a liquid metal blanket. Part I. Effect of poloidal magnetic field Xuan Zhang, Oleg Zikanov ∗ Department of Mechanical Engineering, University of Michigan, Dearborn, MI 48128-1491, USA

h i g h l i g h t s • • • •

2D convection flow develops with internal heating and strong axial magnetic field. Poloidal magnetic field suppresses turbulence at high Hartmann number. Flow structure is dominated by large-scale counter-rotation vortices. Effective heat transfer is maintained by surviving convection structures.

a r t i c l e

i n f o

Article history: Received 9 May 2016 Received in revised form 27 September 2016 Accepted 17 January 2017 Available online 14 February 2017 Keywords: Magnetohydrodynamics Liquid metal blanket Thermal convection

a b s t r a c t We explore the effect of poloidal magnetic field on the thermal convection flow in a toroidal duct of a generic liquid metal blanket. Non-uniform strong heating (the Grashof number up to 1011 ) arising from the interaction of high-speed neutrons with the liquid breeder, and strong magnetic field (the Hartmann number up to 104 ) corresponding to the realistic reactor conditions are considered. The study continues our earlier work [1], where the problem was solved for a purely toroidal magnetic field and the convection was found to result in two-dimensional turbulence and strong mixing within the duct. Here, we find that the poloidal component of the magnetic field suppresses turbulence, reduces the flow’s kinetic energy and high-amplitude temperature fluctuations, and, at high values of Hartmann number, leads to a steadystate flow. At the same time, the intense mixing by the surviving convection structures remains able to maintain effective heat transfer between the liquid metal and the walls. © 2017 Elsevier B.V. All rights reserved.

1. Introduction The use of lithium-containing liquid metals (pure Li or LiPb alloy) as a cooling, shielding, and breeding material in fusion reactor blankets is a promising concept. Development of such blankets is, however, far from complete and faces serious challenges [2]. Two factors are particularly important to understand as they determine the flow and heat transfer in the liquid metal. One is the internal heating caused by the high-energy neutrons generated in the fusion reaction and captured by the liquid metal. The conversion of the neutrons’ kinetic energy into heat leads to very intensive (tens of MW/m3 near the wall facing the reaction chamber) steady internal heating and implies strong potential for thermal convection. The second factor is the exposure of the blanket to a very strong (up to 10–12 T) magnetic field.

∗ Corresponding author. http://dx.doi.org/10.1016/j.fusengdes.2017.01.024 0920-3796/© 2017 Elsevier B.V. All rights reserved.

Traditionally, the studies of liquid metal flow within the blanket were focused on the magnetohydrodynamic (MHD) effects, in particular, on the prediction and ways of reduction of MHD pressure drop. Recently, more concerns have been given to the effects of thermal convection. Laboratory experiments [3,4] and computations [1,5–12] carried out at various but, generally, moderately high values of Hartmann and Grashof numbers have addressed this issue. Although the typical parameters corresponding to the conditions of a real fusion reactor cannot yet be attained in such studies, the results provide convincing indications that the thermal buoyancy-driven convection profoundly affects the flow structures and transport properties and may lead to safety and efficiency problems in blanket performance. In many configurations, in particular in flows through vertical [3,4,7,10] and horizontal [12] ducts and in boxes with conducting walls [9], the convection leads to large-amplitude low-frequency fluctuations of temperature (the so-called ‘anomalous fluctuations’) or to formation of hot and cold spots. This should result in strong and possibly unsteady thermal stresses in the walls, which may cause rapid deterioration of wall

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material and even loss of structural integrity of the blanket [2]. In this sense, the convection effect is highly undesirable and may even require substantial modification of the currently pursued blanket designs, such as DCLL [13] or HCLL [14]. The studies of convection have so far concerned the models of blankets with poloidal ducts. The just mentioned undesirable effects of convection invite us to revisit the old concept of a blanket with toroidal channels [15,16]. In such a blanket, all or some of the channels are oriented parallel to the main (toroidal) component of the magnetic field. Active work on such blankets was carried out in the 1980s and 1990s [17]. It was focused on the self-cooled blanket concept and eventually abandoned due to a combination of reasons, such as the difficulty of electrical insulation of the walls, MHD pressure drop in manifolds, and inter-duct electromagnetic coupling (the Madrame effect) [18]. Remarkably, the effect of thermal convection was ignored in these studies. It was incorrectly assumed that the suppression by the magnetic field would lead to laminar steady-state flow fully determined by the balance between the pressure, Lorentz, and viscous friction forces, and with only passive heat transfer. The first attempt to analyze the role of thermal convection in toroidal ducts was made in our recent work [1]. We have considered a hypothetical flow in a toroidal duct with volumetric internal heating due to the interaction between high speed neutrons and the breeder. The system was assumed to be cooled by auxiliary cooling circuits within the walls. The mean flow through the duct was taken to be negligibly weak (just sufficient for tritium transport but insignificant in the other aspects) and the walls were assumed to be maintained at a constant temperature. The key finding of [1] is that the thermal convection causes turbulence at the Grashof and Hartmann numbers typical for the reactor conditions. Due to the strong toroidal magnetic field, the flow is either purely two-dimensional (2D) or nearly 2D with weak three-dimensional (3D) perturbations superimposed on the 2D velocity. The turbulence results in strong and nearly uniform heat transfer into the walls. The turbulent mixing also removes the strong gradient of temperature and, thus, the reason for development of high-amplitude temperature fluctuations found in poloidal ducts. This paper and the accompanying paper [19] continue the investigation. Our goal is to determine whether the promise of convection-induced turbulence detected in the idealized model of [1] is also demonstrated by more realistic systems. Toroidal duct flow with axial convective heat transfer is considered in [19]. The focus of this paper is on the validity of one model simplification used in [1], namely on taking into account only the main toroidal component of the magnetic field. In an actual blanket, there is also the weak poloidal component representing about 5% of the field strength. In the present work, we include the poloidal field and show that, at the high Hartmann numbers typical for the reactor conditions, it profoundly changes the flow behavior.

2. Physical model and numerical method 2.1. Physical model The flow configuration is illustrated in Fig. 1. A liquid metal modeled as an incompressible and electrically conducting Newtonian fluid fills a horizontal duct of square cross-section. Steady and uniform magnetic field B = Bt ex + Bp ez is imposed in the flow domain. Here, the axial component Bt represents the main toroidal component. The poloidal component Bp is vertical and has the amplitude Bp = 0.05Bt . Presence of this component is the new feature of the model in comparison with [1], where B = Bt ex is used. The duct walls are electrically perfectly insulating and maintained at the constant temperature T0 (see Fig. 1b). There is no

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mean flow along the duct. This can be considered as a conceptual simplified model of the flow and heat transfer in a toroidal duct, in which cooling is predominantly accomplished by auxiliary (e.g. pressurized He) circuits built into walls, and the mean flow of liquid metal along the duct is only needed for purification and tritium extraction and, therefore, is negligibly slow. Neutron heating load is modeled as non-uniform internal heating of volumetric rate Q0 q(y) decreasing exponentially with the distance from the wall facing the reaction zone (see Fig. 1c). We assume that the duct is long, so, as a model approximation, the effect of its ends is neglected. Both 3D flows with periodicity imposed at the x-boundaries and 2D flows independent of the xcoordinate are considered. It has been found in [1] that the strong axial magnetic field suppresses the velocity gradients in the xdirection, so the flow becomes 2D or strongly anisotropic. For each duct length Lx and Grashof number Gr, there is a critical Hartmann number Hacr such that the flow is purely 2D at Ha > Hacr . Three-dimensionality occurs at Ha < Hacr . If Ha is still large, however, the effect of three-dimensionality is not strong. The kinetic energy of 3D perturbations remains small compared to the energy of the 2D part of the flow. The conclusion of [1] is that the assumption of two-dimensionality leads to reasonably accurate results at the practically interesting values of Gr, Ha and Lx . As discussed in Section 3, a similar conclusion is reached in this study. 2 −1 We use the duct’s half-width d as the length  scale, T = Q0 d  as the temperature scale, free-fall speed U = ˇgTd as the velocity scale, d/U as the time scale, strength of toroidal magnetic field Bt as the scale of the magnetic field, dUBt as the scale of the electric potential, U2 as the pressure scale, and the maximum magnitude Q0 as the scale of the internal heating rate. Here,  and ˇ are thermal conductivity and thermal expansion coefficient of the fluid, and g is acceleration of gravity. With the Boussinesq approximation of thermal convection and the quasi-static approximation of MHD effects, the non-dimensional governing equations are:

∂u 1 + (u · ∇ )u = −∇ p + √ ∇ 2 u + T ez + F L , ∂t Gr

(1)

∇ · u = 0,

(2)





∂T 1 ∇2T + q , + u · ∇T = √ ∂t Pr Gr

(3)

where u = (u, v, w) and T are the non-dimensional velocity and temperature deviation from the wall temperature T0 . The non-dimensional internal heating rate is approximated as [20] q = exp(−y − 1),

(4)

where the y =−1 marks the wall nearest to the reaction chamber. The Lorentz force is computed as: Ha2 F L = √ j × eB , Gr

(5)

where eB = ex + 0.05ez is the non-dimensional magnetic field. The electric current j is determined by the Ohm’s law j = −∇  + u × eB ,

(6)

where the electric potential  is as a solution of the Poisson equation expressing the instantaneous electric neutrality of the fluid:

∇ 2  = ∇ · (u × eB ).

(7)

The boundary conditions at the walls are those of perfect electric insulation

∂ =0 ∂n

at

y = ±1,

z = ±1,

(8)

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Fig. 1. (a) Three-dimensional geometry of the duct, magnetic field, coordinate system and boundary conditions. (b) Transverse cross-section of the duct, in which twodimensional flow occurs. jn is the wall-normal current, and T0 is the constant wall temperature. The toroidal and poloidal components of the magnetic field and the boundary conditions at the duct’s walls are shown. (c) The non-dimensional heating rate distribution used in the model.

constant temperature T =0

y = ±1,

at

z = ±1,

(9)

and no-slip u = 0 at y = ±1, z = ±1.

(10)

In the 3D simulations, the periodic conditions at x = 0, Lx are used for the electric potential , velocity u, temperature T and pressure p. In 2D simulations, the flow’s uniformity along the x-axis and the electrically insulating walls imply that no electric currents are generated in response to the motion in the toroidal magnetic field (it can be shown in a straight forward manner that ∇ × j = 0, which, together with ∇ · j = 0 and boundary conditions, gives j = 0). Only the currents associated with the poloidal field eB = 0.05ez need to be considered. These currents are purely toroidal and do not interact with toroidal magnetic field to create a Lorentz force. We see that in the case of 2D flows, only the poloidal component of the field needs to be taken into account in (5)–(7). The non-dimensional parameters are the Prandtl number Pr= /, the Grashof number Gr =

gˇQ0 d5 , 2

(11)

where , , and  are, respectively, the kinematic viscosity, temperature diffusivity, electric conductivity, and density of the fluid. Simulations are conducted at 106 ≤ Gr ≤ 1011 , 0 ≤ Ha ≤ 104 and Pr= 0.0321 (PbLi alloy at about 570 K). 2.2. Numerical method The numerical method is based on the conservative finite difference scheme of the second order introduced in [21] and modified to include thermal convection and implicit treatment of viscosity and conduction terms in [8,11,12,1]. A detailed description of the method can be found in [28]. The computational grid in 3D simulations is uniform in the axial direction. In both 3D and 2D simulations, the grid in the y-z-plane is clustered toward the walls. Two versions of the clustering scheme are used. One, applied at moderate values of Ha, utilizes the coordinate transformation y = Ay sin( /2) + (1 − Ay ),

z = Az sin( /2) + (1 − Az ) ,

(13)

where −1 ≤ ≤ 1 and −1 ≤  ≤ 1 are the transformed coordinates, in which the grid is uniform, and the blending coefficients are Ay = Az = 0.96. In computations at high Hartmann numbers, the clustering scheme is based on the coordinate transformation

    tanh Az   ,z= ,

tanh Ay 

which is a square of the Reynolds number Re = Ud/, and the Hartmann number

y=

Ha = Bt d

where Ay , Az are the coefficients determining the degrees of clustering.

 1/2 

,

(12)

tanh Ay

tanh (Az )

(14)

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Table 1 Numerical resolution used in 2D simulations. In 3D simulations, the same grids are extended in the x-direction with the step x about 0.12. Gr

Ha

Ny

Nz

Clustering

Ay

Az

109 109 109 109 1010 1010 1011

Ha ≤ 1200 1200 ≤ Ha ≤ 2000 2000 ≤ Ha ≤ 4000 4000 ≤ Ha ≤ 104 Ha ≤ 4000 4000 ≤ Ha ≤ 104 104

128 128 128 128 128 128 128

128 128 128 128 128 128 128

(13) (14) (14) (14) (14) (14) (14)

0.96 1.5 2.0 2.0 2.0 2.0 2.0

0.96 2.5 3.0 3.5 3.0 3.5 3.5

The clustering is selected in the computations so as to secure that the Hartmann and sidewall boundary layers contain, respectively, not less than 6 and 12 grid points. As shown in the high-Ha duct flow simulations [22,12], such resolution is sufficient to accurately reproduce the effect of the boundary layers on complex turbulent and convection-dominated flows. We have also performed grid-sensitivity studies using 2D flows in our system and found that the grid parameters shown in Table 1 are sufficient for accurate simulations. For the diagnostics, the global characteristics have been computed using the integration over the duct’s cross-section (in 2D computations) or the entire volume of the computational domain (in 3D computations). This includes the Nusselt number Nu =

Q , T 

(15)



 −1

where Q = A qdA is the total heating rate and T = A A T dA is the mean deviation of the temperature from that of the wall, and the average kinetic energy 1 E= A



|u|2 dA.

(16)

A

We also compute the effective Nusselt number ˜ = Nu

Furthermore and also similarly to the results of [1], at Ha smaller than Hacr , but still sufficiently large, the deviation from two-dimensionality is small and has no significant effect on the flow characteristics. The spatial structure and time evolution of the velocity and temperature field in the 3D simulations are nearly the same as in the 2D ones. The energy of 3D fluctuations (18) remains within a few percent of the total flow energy. The values of Nu and E are also very close in the 2D and 3D simulations. In general, we conclude that at the high values of Hartmann numbers typical for the blankets, and for the model system analyzed in this paper, the effects of three-dimensionality are either absent or weak, and the flow can be accurately analyzed using the 2D model [29]. The results of 2D computations are presented in the rest of the paper.

Tcond  Nu , = T  Nucond

(17)

where Tcond  is the mean temperature deviation in the steady state of the system with zero flow velocity and purely conductive heat transfer, and Nucond = Q/Tcond . For unsteady flow regimes, the global characteristics (15)–(17) are averaged in time over long (many hundreds of time units) periods of evolution of a fully developed flow. In order to evaluate the deviation of the 3D solutions from twodimensionality, we compute the average perturbation energies E  = f  , 2

(18)

where f is the scalar field of perturbations of a velocity component or temperature with respect to the streamwise averaged value f  = f − f¯ . 3. Results 3.1. Comparison between 2D and 3D models Gr = 109 ,

1010

3D computations were carried out at and 800 ≤ Ha ≤ 7000 in the domains of length 4 ≤ Lx ≤ 30 . The results were compared with those obtained in the 2D computations. The conclusion was very similar to that of simulations [1] of flows with purely toroidal magnetic field. At Ha above the critical value Hacr (Gr, Lx ), the flow in 3D simulations always evolves into a state with practically zero streamwise gradients of velocity, i.e. into a 2D state. In the presence of the poloidal magnetic field component, Hacr is only slightly higher than in the case of purely toroidal magnetic field. For example, at Gr = 109 and Lx = 4 we find Hacr about 1100, which can be compared with 1025 found in [1].

3.2. Results of 2D simulations The results obtained at Gr ≥ 109 are summarized in Table 2, which shows the integral characteristics Nu and E and indicates the type of the flow. For comparison, the analogous data obtained in the 2D simulations with purely toroidal magnetic field [1] are shown. The results at smaller Gr are considered less relevant to the blanket and not presented here. Here and in the following discussion, we use the Hartmann number Ha associated with the toroidal magnetic field (see (12)) to indicate the strength of the MHD effect. It must be remembered that in 2D flows the toroidal magnetic field does not generate the Lorentz force. The force acting on the flow is either zero (in the results from [1]) or corresponds to the effective Hartmann number Hap = 0.05Ha based on the poloidal magnetic field. We see that the poloidal magnetic field has substantial effect on the kinetic energy. This has an obvious explanation of the MHD suppression of the flow. It is known from theory and simulations (see, e.g. [23–25]) that a constant magnetic field suppresses a turbulent flow of an electrically conducting liquid when the Stuart number N = Ha2 Re−1 is not very small and may lead to full laminarization at a sufficiently large N.1 It is also known that the magnetic field results in flow anisotropy in the sense that its structures become elongated in the direction of the magnetic field lines. While our case is different due to the two-dimensionality enforced by the toroidal field, the basic physical effect is qualitatively the same. In an attempt to quantify it, Table 2 shows the effective Stuart number based on the poloidal field Bp and the convection velocity scale U = Np =

Ha2p Re

=

(0.05Ha)2 Gr 1/2

=



ˇgTd:

Bp2 d U

.

(19)

We see that the values of Np are not small, so significant MHD effect must be anticipated.

1 The actual values of N, at which significant suppression and anisotropy, or laminarization occur, vary with the system under consideration as well as with the choice of the typical velocity and length scales.

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Table 2 Results of 2D simulations. Averaged kinetic energy and Nusselt number for flows with both poloidal and toroidal magnetic field, and for the flows with purely toroidal magnetic field (see [1]) are shown for different parameters. For unsteady flow regimes, the integral characteristics are obtained by time-averaging over long (many hundreds ˜ (see (17)) can be computed by dividing the values of Nu in the table by Nucond = 15.08. of time units) periods. The effective Nusselt number Nu Gr

Ha

Np

Bp = 0 [1] Flow state

9

10 109 109 109 109 109 109 109 1010 1010 1010 1010 1011

800 1200 1500 2500 3000 4000 7000 104 3000 5000 7000 104 104

0.05 0.11 0.18 0.49 0.71 1.26 3.87 7.91 0.23 0.63 1.23 2.50 0.78

turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent turbulent

Bp = 0.05Bt E

Nu −3

8.0 × 10 8.0 × 10−3 8.0 × 10−3 8.0 × 10−3 8.0 × 10−3 8.0 × 10−3 8.0 × 10−3 8.0 × 10−3 5.0 × 10−3 5.0 × 10−3 5.0 × 10−3 5.0 × 10−3 3.0 × 10−3

81.6 81.6 81.6 81.6 81.6 81.6 81.6 81.6 128 128 128 128 206.6

flow state turbulent oscillating oscillating oscillating oscillating steady steady steady turbulent oscillating oscillating steady turbulent

E

Nu −3

3.4 × 10 2.4 × 10−3 2.0 × 10−3 1.6 × 10−3 1.3 × 10−3 1.1 × 10−3 6.4 × 10−4 3.8 × 10−4 9.1 × 10−4 7.1 × 10−4 5.3 × 10−4 3.8 × 10−4 2.9 × 10−4

80 78 78 75 72 67 56 48 119 114 108 90 179

Fig. 2. Average kinetic energy (a), Nusselt number (b), point signals of amplitude of vertical velocity (c) and temperature (d) in 2D flows at Gr = 109 , Bp = 0 (black, dotted lines, results of [1]), Ha = 800 (green, solid lines), Ha = 1500 (blue, dashed lines) and Ha = 4000 (red, dash-dot-dot lines). Only parts of actual simulations are shown and the point signals correspond to different physical locations due to different grid clustering schemes used. Point coordinate is (−0.7,0) for Ha = 0 and Ha = 800 and is (−0.8,0) for Ha = 1500 and Ha = 4000. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

The effect is illustrated by the transformation of the flow at Gr = 109 shown in Figs. 2–4. We see that at small Np = 0.05 corresponding to Ha = 800 the flow remains turbulent. This is demonstrated by the time signals in Fig. 2 (green, solid lines) and, more convincingly, by the power energy spectrum (see Fig. 4b), which shows continuous distribution of frequencies in a wide range and the slope E(ω) ∼ ω−3 expected for 2D turbulence. The spatial

structure of velocity and temperature fields in Figs. 3d–f are quantitatively similar to those of flows with purely toroidal magnetic field shown in Figs. 3a–c. At the same time, the effect of MHD suppression is evident even at such a small Np . As one can see in Table 2, the kinetic energy of the flow is reduced more than two-fold. Increase of the poloidal magnetic field results in further suppression of the flow, as manifested by decrease of E in Table 2.

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Fig. 3. Instantaneous distributions of streamfunction  (solid lines indicate counter-clockwise motion, while dashed lines indicate clockwise motion), temperature deviation T and amplitude of vertical velocity uz in 2D flows at Gr = 109 , purely axial magnetic field Bp = 0 (a)–(c), Ha = 800 (d)–(f), Ha = 1500 (g)–(i), Ha = 4000 (j)–(l). Note that different sets of isolevels are used at different Ha.

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Fig. 4. Power energy spectra in 2D flows at Gr = 109 , Bp = 0 (a), Ha = 800 (b), Ha = 1500 (c) (in log-scale and ω is frequency).

Fig. 5. Instantaneous distributions of streamfunction  (solid lines indicate counter-clockwise motion, while dashed lines indicate clockwise motion), temperature deviation T and amplitude of velocity uz in 2D flows at Gr = 1010 , Ha = 5000 (a)–(c), Gr = 1010 , Ha = 104 (d)–(f), Gr = 1011 , Ha = 104 (g)–(i).

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Fig. 6. Point (at (−0.8,0)) signals of amplitude of vertical velocity (a), and temperature deviation (b), in 2D flows shown in Fig. 5: at Gr = 1010 , Ha = 5000 (green, dotted lines), Gr = 1010 , Ha = 104 (blue, dashed lines), Gr = 1011 , Ha = 104 (black, solid lines). Power energy spectra in 2D flows at Gr = 1010 , Ha = 5000 (c), Gr = 1011 , Ha = 104 (d) (in log-scale). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

At Np = 0.18 (Ha = 1500) turbulence is replaced by an unsteady flow, evolution of which can be described as low-amplitude oscillations with several dominating frequencies (see Fig. 4c). The circulation vortices are elongated in the vertical direction (see Fig. 3g). Smallscale structures disappear from the velocity field, and the flow acquires the spatial form of a pair of counter-rotating vortices with upward flow in the core of the duct and downward flow in the two jets near the vertical walls. The vortices are closed within the Hartmann layers near the top and bottom walls. Such flow regimes are marked as ‘oscillating’ in Table 2. Further increase of the poloidal magnetic field enhances the flow transformation. At Ha ≥ 4000 the MHD suppression results in a steady-state flow. Remarkably, the transformation of the structure of the flow does not lead to a similarly dramatic change of its ability to transport heat toward walls. Table 2 shows that at moderate Ha the Nusselt number deviates only slightly from the value found in the case of purely toroidal magnetic field. At large values of Ha, when the unsteadiness of the flow is fully suppressed, the reduction of Nu is noticeable, but not very strong. This can be explained by the fact that the flow retains sufficiently strong large-scale circulation that facilitates the convective transport of heat between the core and the boundary layers. The small-scale mixing and disruption of the boundary layers are suppressed together with turbulence, but their contribution to heat transfer is moderate at low Prandtl number.

Fig. 5 illustrates the transformation of the velocity and temperature fields at high Grashof and Hartmann numbers, i.e. in the parameter range typical for an actual operating blanket. The corresponding change of the integral flow characteristics is shown in Table 2. We see that the principal features of the transformation are the same as in the case of Gr = 109 . Stronger poloidal magnetic field leads to suppression of small-scale turbulent fluctuations, so the flow becomes dominated by two counter-rotating vortices with descending jets near the vertical walls. The kinetic energy is greatly reduced, while much weaker reduction is observed for the Nusselt number. The flow changes from turbulent to oscillating with one or several dominant frequencies, and, at even higher Ha, to steady-state. At the same time, the relation between the degree of transformation and the value of the effective Stuart number Np is not the same as at Gr = 109 . Unsteadiness of the flow and turbulence are observed at larger Np . A particularly important example is the flow at Gr = 1011 and Ha = 104 , which corresponds to Np = 0.79. As we see in Figs. 5g and 6d, the flow is unsteady and retains weak turbulence. 4. Concluding remarks We have considered the effect of the poloidal component of the magnetic field on the convection-driven flow and heat transfer in a simplified model of a toroidal duct of a generic liquid metal blanket.

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X. Zhang, O. Zikanov / Fusion Engineering and Design 116 (2017) 52–60

It is found that a sufficiently strong poloidal field suppresses the two-dimensional turbulence observed in our earlier simulations conducted with purely toroidal magnetic field [1]. The flow’s kinetic energy is strongly reduced and its spatial structure becomes dominated by two large-scale counter-rotating vortices. Interestingly, the value of the effective Stuart number Np based on the strength of the poloidal field component and the typical convection velocity (see (19)) does not fully determine the degree of the flow transformation. The flows at higher Gr tend to retain unsteadiness and turbulence at higher Np . One conclusion that can be drawn from the results presented in this paper is that even though the poloidal magnetic field is quite weak compared to the main field, it is critical to the flow behavior in the toroidal ducts and should always be considered. Another conclusion is more interesting. The benefits of the convection-induced turbulence for the blanket operation can be recognized as two-fold: intensification of the heat transfer toward the walls and removal of the high-amplitude low-frequency ‘anomalous’ temperature fluctuations anticipated in other blanket schemes on the basis of recent studies [3,4,7–10,12,26,27]. The results presented in this paper demonstrate that the benefits are retained in the presence of the poloidal field. We observe turbulence at Gr = 1011 , Ha = 104 , and fully expect it to be present at Gr = 1012 , Ha = 104 which corresponds to Np = 0.25. Even when turbulence is suppressed, the heat transfer toward the walls remains strong, many time stronger than the transport by heat conduction alone (for example, Nu = 11.87Nucond at Gr = 1011 , Ha = 104 ). Finally, as illustrated in Figs. 2 and 6 and found in all our simulations, the system experiences no anomalous temperature fluctuations. Taking, again, the case Gr = 1011 , Ha = 104 as an example and using the respective curve in Fig. 6b (black, solid line), we find that for a duct of half-width d = 10 cm the typical dimensional fluctuation amplitude would be just 6.6 K. Acknowledgments Financial support is provided by the US NSF (Grant CBET 1435269). References [1] X. Zhang, O. Zikanov, Two-dimensional turbulent convection in a toroidal duct of a liquid metal blanket of a fusion reactor, J. Fluid Mech. 779 (2015) 36–52. [2] M. Abdou, N.B. Morley, S. Smolentsev, A. Ying, S. Malang, A. Rowcliffe, M. Ulrickson, Blanket/first wall challenges and required R&D on the pathway to DEMO, Fusion Eng. Des. 100 (2015) 2–43. [3] I.A. Melnikov, E.V. Sviridov, V.G. Sviridov, N.G. Razuvanov, Heat transfer of MHD flow: experimental and numerical research, in: Proc. 9th PAMIR Conf. Fund. Appl. MHD, Riga, Latvia, Vol. 1, INP, Open Library, 2014, pp. 65–69. [4] I.R. Kirillov, D.M. Obukhov, L.G. Genin, V.G. Sviridov, N.G. Razuvanov, V.M. Batenin, I.A. Belyaev, I.I. Poddubnyi, N.Y. Pyatnitskaya, Buoyancy effects in vertical rectangular duct with coplanar magnetic field and single sided heat load, Fusion Eng. Des. 104 (2016) 1–8.

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