Instabilities of MHD flows driven by traveling magnetic fields

The flow of an electrically conducting fluid driven by a traveling magnetic field imposed at the end caps of a cylindrical annulus is numerically studied. At sufficiently large magnetic Reynolds number, the system undergoes a transition from synchronism with the traveling field to a stalled flow, similar to the one observed in electromagnetic pumps. An unusual type of boundary layer is identified for such electromagnetically driven flows that can be understood as a combination of Hartmann and Shercliff layers generated by the spatiotemporal variations of the magnetic field imposed at the boundaries. An energy budget calculation shows that energy dissipation mostly occurs within these boundary layers and we observe that the ohmic dissipation $D_{\eta }$ always overcomes the viscous dissipation $D_{\nu }$, suggesting the existence of an upper bound for the efficiency of electromagnetic pumps. Finally, we show that the destabilization of the flow occurs when both dissipations are nearly equal, $D_{\nu }\sim D_{\eta }$.

Figure 1

(a) Geometry of the channel. The fluid is confined between two fixed cylinders and submitted to a sinusoidal magnetic field imposed at the end caps and traveling in the azimuthal direction. (b) Azimutal component of the $\theta$-averaged velocity field $U_{\theta }$ in the ($r,z$) plane for $\text{Ha}=600$, $\text{Rm}=30$, and $\text{Re}=4000$. (c) Corresponding magnitude of the $z$ component of the magnetic field ($B_z$) in the ($z,\theta$) plane at $r=(r_i+r_o)/2$. A mean flow is therefore generated along the azimuthal direction under the influence of the TMF which is nearly homogeneous in the axial direction.

Figure 2

Structure of the boundary layer close to the bottom boundary, in the synchronous regime, for $\text{Re}=4000$, ${\text{Rm}}_s=15$, and three different values of the Hartmann number: $\text{Ha}=100$ (top), $\text{Ha}=2000$ (middle), and $\text{Ha}={10}^4$ (bottom). The figure shows color plots of the velocity component $U_{\theta }$ and magnetic field lines in the $(z,\theta )$ plane. Note how the local thickness depends on the geometry of the local magnetic field.

Figure 3

(a) Axial evolution of the azimuthal velocity for $\theta$ corresponding to normal field (left) or tangential field (right), for $\text{Re}=4000$, ${\text{Rm}}_s=15$, and various values of the Hartmann number (in the synchronous regime). (b) Evolution of the boundary layer thickness $\delta$ as a function of the Hartmann number in both cases. Solid black line indicates the Hartmann scaling $\delta \sim 1/\text{Ha}$, while the dashed red line is the Shercliff scaling $\delta \sim 1/\sqrt{\text{H}a}$. Note the two different behaviors, illustrating the existence of a complex “Schercliff-Hartmann” boundary layer.

Figure 4

Normalized flow rate $Q=\int \frac{U_{\theta }}{c}dS$ developed in the channel as a function of $\text{Rm}$, for different values of $\text{Ha}$, $\text{Re,}$ and the wave number $m$. Note the transition from synchronous ($Q>0.5$) to stalled ($Q<0.5$) flows as $\text{Rm}$ is increased.

Figure 5

Structure of the magnetic field for $\text{Ha}=600$, $\text{Rm}=300,$ and $\text{Re}=4000$. (a) Magnitude of the $z$ component of the magnetic field ($B_z$) in the ($z,\theta$) plane at $r=(r_i+r_o)/2$. (b) Skin depth of the magnetic field measured in the simulations as a function of the slip magnetic Reynolds number, for various values of $\text{Ha}$ and $\text{Re.}$

Figure 6

(a) Time evolution of energy fluxes for $\text{Re}=4000$, $\text{Ha}=600,$ and ${\text{Rm}}_s=10$ (synchronous regime). (b) Same as panel (a), normalized by the injected power.

Figure 7

(a) Evolution of energy fluxes for $\text{Re}=4000$, $\text{Ha}=600,$ as a function of $\text{Rm}$. (b) Same thing but normalized by the injected power.

Figure 8

Evolution of the dimensionless flow rate $Q$ as a function of $R$, for various values of $H$ and $\kappa$, obtained by numerical integration of the reduced model [Eqs. (14) and (15)]. Inset: all the curves collapse when plotted as a function of the dimensionless number $N$ (see text).

Figure 9

(a) Total flow rate $Q$ developed in the channel as a function of $N$, for different values of $\text{Ha}$, $\text{Re,}$ and $\text{Pm}$. All the numerical points are rescaled on a single curve, corresponding to the reduced model (14) and (15) (black curve). (b) Evolution of the dissipation ratio $\epsilon =D_{\nu }/D_{\eta }$ as a function of $N$. Note the bound on the dissipation ratio, maximum for $N\sim 1$.