Electromagnetically controlled multi-scale flows

Abstract

We generate a class of multi-scale quasi-steady laminar flows in the laboratory by controlling a quasi-two-dimensional shallow-layer brine flow by multi-scale Lorentz body forcing. The flows' multi-scale topology is invariant over a broad range of Reynolds numbers, $\hbox{\it Re}_{2D}$ from 600 to 9900. The key multi-scale aspects of this flow associated with its multi-scale hyperbolic stagnation-point structure are highlighted. Our multi-scale flows are laboratory simulations of quasi-two-dimensional turbulent-like flows, and they have a power-law energy spectrum $E(k)\,{\sim}\,k^{-p}$ over a range $2\pi/L\,{<}\,k\,{<}\,2\pi/\eta$ where $p$ lies between the values 5/3 and 3 which are obtained in a two-dimensional turbulence that is forced at the small scale $\eta$ or at the large scale $L$, respectively. In fact, in the present set-up, $p\,{+}\,D_{s}\,{=}\,3$ in agreement with a previously established formula; $D_s\,{\approx}\,0.5$ is the fractal dimension of the set of stagnation points and $p\,{\approx}\,2.5$. The two exponents $D_s$ and $p$ are controlled by the multi-scale electromagnetic forcing over the entire range of scales between $L$ and $\eta$ for a broad range of Reynolds numbers with separate control over $L/\eta$ and Reynolds number. The pair dispersion properties of our multi-scale laminar flows are also controlled by their multi-scale hyperbolic stagnation-point topology which generates a sequence of exponential separation processes starting from the smaller-scale hyperbolic points and ending with the larger ones. The average mean square separation $\overline{\Delta^{2}}$ has an approximate power law behaviour ${\sim}t^{\gamma}$ with ‘Richardson exponent’ $\gamma\,{\approx}\,2.45$ in the range of time scales controlled by the hyperbolic stagnation-points. This exponent is itself controlled by the multi-scale quasi-steady hyperbolic stagnation-point topology of the flow.