Fluid Flow Control with Transformation Media

Abstract

We introduce a new concept for the manipulation of fluid flow around three-dimensional bodies. Inspired by transformation optics, the concept is based on a mathematical idea of coordinate transformations and physically implemented with anisotropic porous media permeable to the flow of fluids. In two situations-for an impermeable object placed either in a free-flowing fluid or in a fluid-filled porous medium-we show that the object can be coated with an inhomogeneous, anisotropic permeable medium, such as to preserve the flow that would have existed in the absence of the object. The proposed fluid flow cloak eliminates downstream wake and compensates viscous drag, hinting at the possibility of novel propulsion techniques. As a subset of the separation of variables method, conformal transformations thus offer a unique and powerful approach to the forward problems of incompressible flow, as well as electro-and magnetostatics.The utility of more general coordinate transformations in the inverse electromagnetic [3][4][5] and acoustic [6][7][8][9] problems has been demonstrated recently, most impressively by showing the possibility of electromagnetic invisibility, dubbed cloaking [3,4,[10][11][12]. The key idea that enabled progress in those areas was the combination of coordinate transformations with coordinate-dependent, and often extremely exotic [5,[13][14][15], material properties. Transformation optics [3][4][5]11,16] and transformation acoustics [6-8] offer solutions to some inverse scattering problems [17] by reducing them to an elementary one-for example, scattering off a point object in free space.There is no reason why this conceptual approach cannot be used in other areas of physics [18], and, in fact, it has already been applied to conductive heat transfer [19], linear elastodynamics [5,8,[20][21][22][23][24], surface wave [25], and quantum-mechanical matter wave [26] dynamics. The fundamental requirement for the applicability of this concept is the presence of a medium with sufficiently flexible properties, which enables manipulation of the coefficients in the equations describing the dynamical process. Here, we propose porous media as a substrate for transformation fluid dynamics and investigate the feasibility of controlling certain features of incompressible fluid flow.The electromagnetic cloak of invisibility, which inspired this approach, manipulates the field lines of electric and magnetic fields, and the streamlines of momentum flux, in such a manner that these lines are virtually indistinguishable from those in the absence of any object [3]. Because the streamlines are unperturbed in the free space outside the cloak, electromagnetic radiation avoids scattering off the structure and therefore exerts no electromagnetic pressure on it. Achieving a similar level of control over the streamlines of the fluid flow outside the structure would imply canceling the viscous drag exerted on the structure. In this Letter, we demonstrate the possibility of fluid flow cloaking using porous media whose properties...

Flow stabilization with active hydrodynamic cloaks

“…Supplemented with a macroscopic design methodology, known as transformation optics (acoustics, and so on), MMs enable new wave applications by offering extra flexibility in manipulating the dynamics of waves and matter. Recently, it was proposed [1] that hydrodynamic metamaterials (HDMMs), consisting of a fluidfilled solid matrix with a variable fluid permeability, can enable exotic flow regimes in which pressure and velocity differentials are confined to a small volume occupied by the permeable shell. These flows were demonstrated in three dimensions for a spherical object surrounded by a concentric porous shell with anisotropic and partially negative permeability.…”

“…(2), we solve the problem of two-dimensional flow cloaking [1] or wake elimination. We then proceed to demonstrate that wake elimination, indeed, results in flow stabilization at flow rates that normally result in turbulence.…”

“…When referring to normalized permeability, the normalization constant κ 0 = a 2 /4 is assumed. Previously, approximate solutions were reported for the case of flow past a spherical cloak [1]. The proposed technique was based on the introduction of the (Stokes) stream function and solving the linear equation in the regime with vanishingly small Re's.…”

“…Still, this term, due to its angular symmetry, would break down the separation of variables in the polar coordinates, thus, rendering the method of Ref. [1] much more difficult to apply. Thus, we choose to obtain the wake-free (cloaking) solutions numerically by employing a gradient-based optimization technique.…”

“…[1], it was shown that wake-free solutions exist when the permeability is spherically symmetric but locally anisotropic with the spherically uniaxial type of anisotropy. Here, we show that approximately wake-free designs can be obtained by assuming locally isotropic and cylindrically symmetric permeability κ(r).…”