Optimal mixing enhancement

We introduce a general-purpose method for optimising the mixing rate of advective fluid flows. An existing velocity field is perturbed in a $C^1$ neighborhood to maximize the mixing rate for flows generated by velocity fields in this neighborhood. Our numerical approach is based on the infinitesimal generator of the flow and is solved by standard linear programming methods. The perturbed flow may be easily constrained to preserve the same steady state distribution as the original flow, and various natural geometric constraints can also be simply applied. The same technique can also be used to optimize the mixing rate of advection-diffusion flow models by manipulating the drift term in a small neighborhood

Optimizing the Source Distribution in Fluid Mixing

A passive scalar is advected by a velocity field, with a nonuniform spatial source that maintains concentration inhomogeneities. For example, the scalar could be temperature with a source consisting of hot and cold spots, such that the mean temperature is constant. Which source distributions are best mixed by this velocity field? This question has a straightforward yet rich answer that is relevant to real mixing problems. We use a multiscale measure of steady-state enhancement to mixing and optimize it by a variational approach. We then solve the resulting Euler--Lagrange equation for a perturbed uniform flow and for simple cellular flows. The optimal source distributions have many broad features that are as expected: they avoid stagnation points, favor regions of fast flow, and their contours are aligned such that the flow blows hot spots onto cold and vice versa. However, the detailed structure varies widely with diffusivity and other problem parameters. Though these are model problems, the optimization procedure is simple enough to be adapted to more complex situations.Comment: 19 pages, 23 figures. RevTeX4 with psfrag macro

Closed-Loop Control of a Piezo-Fluidic Amplifier

Fluidic valves based on the Coand\u{a} effect are increasingly being considered for use in aerodynamic flow control applications. A limiting factor is their variation in switching time, which often precludes their use. The purpose of this paper is to demonstrate the closed-loop control of a recently developed, novel piezo-fluidic valve that reduces response time uncertainty at the expense of operating bandwidth. Use is made of the fact that a fluidic jet responds to a piezo tone by deflecting away from its steady state position. A control signal used to vary this deflection is amplitude modulated onto the piezo tone. Using only a pressure measurement from one of the device output channels, an output-based LQG regulator was designed to follow a desired reference deflection, achieving control of a 90 m/s jet. Finally, the controller's performance in terms of disturbance rejection and response time predictability is demonstrated.Comment: 31 pages, 23 figures. Published in AIAA Journal, 4th May 202

Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics

We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators

Deceleration of one-dimensional mixing by discontinuous mappings

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equal sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes non-monotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counter-intuitively predicts a deceleration in the asymptotic mixing rate with increasing diffusivity rate. The implication of the results on finite time mixing are discussed