Forced underwater laminar flows with active magnetohydrodynamic metamaterials

Theory and practical implementations for wake-free propulsion systems are proposed and proven with computational fluid dynamic modeling. Introduced earlier, the concept of active hydrodynamic metamaterials is advanced by introducing magnetohydrodynamic metamaterials, structures with custom-designed volumetric distribution of Lorentz forces acting on a conducting fluid. Distributions of volume forces leading to wake-free, laminar flows are designed using multivariate optimization. Theoretical indications are presented that such flows can be sustained at arbitrarily high Reynolds numbers. Moreover, it is shown that in the limit $\mathrm{Re}\gg {10}^2$, a fixed volume force distribution may lead to a forced laminar flow across a wide range of Re numbers, without the need to reconfigure the force-generating metamaterial. Power requirements for such a device are studied as a function of the fluid conductivity. Implications to the design of distributed propulsion systems underwater and in space are discussed.

Figure 1

Schematic of the annular cloak domain around a cylinder. The ideal cloak domain yields uniform flow for $r>r_e$.

Figure 2

Flow velocity (a) and pressure (b) as prescribed by a three-term polynomial ansatz for $\mathrm{Re}={10}^3$.

Figure 3

Flow velocity (a) and pressure (b) as prescribed by a four-term polynomial ansatz for $\mathrm{Re}={10}^3$.

Figure 4

Function $\mathrm{\Phi }$ for a range of aspect ratios $\gamma$.

Figure 5

(a) Flow field around an uncloaked cylinder. (b) Power dissipation for different volumetric force scaling factors, i.e., how “turned on” is the volumetric force. (c) Flow field around a cloaked cylinder using forced Stokes flow.

Figure 6

(a) LW in the flow surrounding the uncloaked cylinder at $\mathrm{Re}=100$. (b, top center) Volumetric force (normalized by $F_{V,0})$ in the $x$ direction necessary to achieve cloaking as determined by optimization. (c) Volumetric force (normalized by $F_{V,0})$ in the $y$ direction necessary to achieve cloaking as determined by optimization. (d) LW in the flow surrounding the cloaked cylinder at $\mathrm{Re}=100$.

Figure 7

(a) LW in the flow surrounding the uncloaked cylinder at $\mathrm{Re}={10}^7$. (b) Volumetric force (normalized by $F_{V,0})$ in the $x$ direction necessary to achieve cloaking as determined by optimization. (c) Volumetric force (normalized by $F_{V,0})$ in the $y$ direction necessary to achieve cloaking as determined by optimization. (d) LW in the flow surrounding the cloaked cylinder at $\mathrm{Re}={10}^7$.

Figure 8

Optimized configuration of an MHD array surrounding a cylindrical core for wake elimination. This differs from the configuration in Figs. 6 and 7 because the force production is discretized into cells (possibly by the presence of out-of-plane inductors). (a) The current lines and current density in the flow field of the cload. (b) The magnetic field magnitudes in the various cells of the array. (c) The flow velocity of the wake-minimized, cloaked field.

Figure 9

Schematic of the magnetohydrodynamic cloaking device for 2D flow around a cylinder. The only alterations are in the downstream magnetic fields. In place of a single ellipse, three circles now help to control the flow. $R_{\mathrm{ds},1}=0.75R_c$ and $R_{\mathrm{ds},2}=R_c$. The off-axis domains are displaced by $2R_c$.

Figure 10

Cloaking results using a modified MHD array to simulate the results of the general body force optimization. (a) The uncloaked 2D flow around a cylinder at Reynolds number $10000$. (b) Electrical current lines around the cylindrical core due to optimized electrode potential. (c) Optimized out-of-plane magnetic fields in tesla. (d) Resulting local wake of the cloaked configuration.

Figure 11

Body force resulting from the first MHD configuration. The body-force magnitude is normalized by $F_{B,0}$.