Scaling laws in axisymmetric magnetohydrodynamic duct flows

We report on a numerical study of axisymmetric flow of liquid metal in a circular duct with a rectangular cross section. The flow is forced through the combination of an axial magnetic field and a radial current. Sweeping a wide range of forcing parameters, we identify the different regimes which characterize the flows and explicate the associate scaling laws. Results from different studies in the literature are interpreted in the light of our numerical simulations.

Figure 1

Scheme of the annular duct. The axisymmetric 2D simulations presented in this study are displayed on the $(r,z)$ plane.

Figure 2

Fields $u(r,z)$, $w(r,z)$, and $b(r,z)$ for $h=2$, $r_0=1$, $r_1=5$, $\text{Pm}=5.6\times {10}^{-5}$, and $\text{Ha}=30$. (a)–(c) Equilibrium solution of the model (7)–(10) without terms proportional to $\sqrt{\text{Pm}}$. (d)–(f) Temporal average of the fields during the asymptotic state for the model (7)–(10).

Figure 3

Time evolution of $\overline{u}$ the spatial average of $u$, with the model (7)–(10). Surrounding purple curves are the spatial average of the standard deviation $\delta u$. The parameters are $\text{Pm}=5.6\times {10}^{-5}$, $h=2$, $r_0=1$, $r_1=5$, and $\text{Ha}=30$.

Figure 4

Plot of the $(r,z)$ cross section of the flow and the current (a) $u(r,z)$, (b) $b(r,z)$, (c) $w(r,z)$, and (d) $\psi (r,z)$ for the reference case $\text{Pm}=1.48\times {10}^{-7}$ and $\text{Ha}=25.9$. The parameters are $B_0=0.1$ T, $I_0=0.5$ A, $h=1$ cm, $r_0=4$ cm, and $r_1=5$ cm. The fluid is mercury $[\rho =13550$ kg/${\mathrm{m}}^3$, $\nu =1.14\times {10}^{-7}{\mathrm{m}}^2$/$s$, and $\eta ={\mu }_0(7.7\times {10}^{-1})\mathrm{\Omega }\text{m}]$.

Figure 5

Mean toroidal flow versus $B_0$ and $I_0$. The black dashed line corresponds to the critical ratio for the inertial criterion $2h^2\text{Re}/{\text{Ha}}^2{\overline{r}}^2=1$. The black solid line is the numeric estimate of $\langle \parallel (\mathbf{V}·\mathbf{\nabla })V\parallel \rangle /\langle \parallel \nu ▵V\parallel \rangle =0.7$. The red dot-dashed line is the numeric estimate of the induction ratio $\tau =0.5$ [Eq. (29)]. There are three clear regimes: the inertialess regime ILId, the inertial regime IR, and the weak regime ILR.

Figure 6

Mean toroidal flow velocity snapshots for various magnetic fields, from (a) to (e), $B_0=0.001,0.01,0.05,0.1,0.5$ T. The other parameters are reference case ones.

Figure 7

Mean toroidal flow versus imposed magnetic field. The other parameters are reference case parameters. The gray zone separates the limit of validity of the ILId and ILR regimes. The orange line shows the theoretical ILId regime and the green line the theoretical ILR regime.

Figure 8

Mean toroidal flow versus imposed current. The orange line shows the theoretical mean $V_{\theta }$ for the ILId regime [Eq. (30)] and the yellow dot-dashed line the theoretical mean $V_{\theta }$ for the IR regime [Eq. (41)]. The gray zone separates the limit of validity of the ILId and IR regimes.

Figure 9

Mean toroidal flow for the reference case with $B_0=1$ mT and with various current values, obtained from simulation (blue pluses). The scan crosses the weak regime ILR to the inertial regime IR around $I_0=20$ A. The theoretical value (36) in the ILR regime is shown by the green line and the theoretical value (41) in the IR regime is shown by the yellow dot-dashed line.

Figure 10

Plot of the $(r,z)$ cross section of the flow and the current (a) $u(r,z)$, (b) $b(r,z)$, (c) $w(r,z)$, and (d) $\psi (r,z)$ for the reference case with $B_0=1$ mT and $I_0=100$ A in the inertial regime ($\text{Pm}=1.48\times {10}^{-7}$ and $\text{Ha}=0.26$).

Figure 11

Mean toroidal flow for the reference case with $I_0=20$ A and with various magnetic field values, obtained from simulation (blue pluses). The scan crosses the inertial regime IR to the inertialess regime ILId around $B_0=1$ T. The theoretical value (30) in the ILId regime is shown by the orange line and the theoretical value (41) in the IR regime is shown by the yellow dot-dashed line.

Figure 12

Mean toroidal flow for the reference case with (a) various aspect ratio values and (b) various mean radii. Blue pluses show the results obtained from simulation and the lines are for theories. The scan crosses the identified regimes. The theoretical value (30) in the ILId regime is shown by the orange line.

Figure 13

Values of toroidal flow versus $I_0{B}_0$. The black solid line shows the scaling of Boisson et al. [6, 13]. The pluses are the maximum values of the toroidal flow and the circles are the mean values. Colors stand for the magnetic field: yellow, 10 mT; blue, 20 mT; green, 50 mT; and orange, 0.1 T. The parameters are $r_0=1$ cm, $r_1=4$ cm, $h=12$ cm, galinstan, $\eta =2.89\times {10}^{-7}\mathrm{\Omega }\text{m}$, $\rho =6440\text{kg}{\text{m}}^{-3}$, $\nu =3.73\times {10}^{-7}{\text{ms}}^{-2}$, $B_0$ from 10 mT to 0.1 T, and $I_0$ from 1 mA to 20 A. (a) Linear scale. (b) Logarithmic scale with the theoretical value of Eq. (43) (black dashed line), Eq. (30) (orange solid line for $B_0=1$ T), and Eq. (41) (black dot-dashed line). Shown in gray is $V_B$ divided by 2 to adapt the measurements from maximal values to averages.

Figure 14

Friction factor $F_{\text{Alb}}$ as a function of $R_{\text{Alb}}$. The thin black pluses show the experimental measurements (1–13 T and 10–400 A); blue pluses the simulations with $B_0=0.1$ T and $I_0=0.1$–20 A; green crosses the simulations with $B_0=0.5$ T and $I_0=0.1$–20 A; closed red circles the simulations with $B_0=1$ T and $I_0=0.1$–20 A; open yellow circles the simulations with $B_0=1$ mT and $I_0=0.1$–100 A; dashed lines the theoretical values calculated with Eq. (30) for blue, green, and red and with Eq. (36) for yellow; and dot-dashed lines the theoretical values calculated with Eq. (41). The black star denotes the reference case.

Figure 15

Same as in Fig. 5 plus extrapolation from experiment (blue dotted line) and a question mark for the limit of the ideal or resistive Ohm law (red dot-dashed line).