Session A16: Flow Control: Theory

Real-time Estimation of the Gaseous Plume Using a Formation of Unmanned Aerial Vehicles

This work proposes an approach for the real-time estimation of gaseous plume caused by a source moving along an unknown trajectory using a formation of seven unmanned aerial vehicles (UAVs) with concentration sensors onboard. The process of gas release is modeled with unsteady advection-diffusion equation and is solved numerically using a finite volume method (FVM) with total variation diminishing (TVD) scheme. The concentration estimator is based on the Luenberger observer. The UAVs are assumed to maintain a rigid flying formation throughout the process. The UAVs dynamics is described by the point-mass model of a fixed wing aircraft. The guidance of the leader UAV is coupled to the performance of the estimator through Lyapunov redesign methods. An appropriate choice of the Lyapunov functional results in the desired direction for the leader UAV, which is expressed in terms of the concentration estimation error and the error gradients at the sensors locations. For computational efficiency in the real-time applications the computational grid for the estimator is adapted dynamically to provide higher resolution near the flying formation. Numerical tests are implemented to illustrate the performance of the proposed approach.   

The importance of being fractional in mixing: optimal choice of the index $s$ in $H^{-s}$ norm

A natural measure of homogeneity of a mixture is the variance of the concentration field, which in the case of a zero-mean field is the $L^2$-norm. Mathew \emph{et.al.} (Physica D, 2005) introduced a new multi-scale measure to quantify mixing referred to as the mix-norm, which is equivalent to the $H^{-1/2}$ norm, the Sobolev norm of negative fractional index. Unlike the $L^2$-norm, the mix-norm is not conserved by the advection equation and thus captures mixing even in the non-diffusive systems. Furthermore, the mix-norm is consistent with the ergodic definition of mixing and Lin~\emph{et al.} (JFM, \ 2011) showed that this property extends to any norm from the class $H^{-s}, \ s>0$. We consider a zero-mean passive scalar field organised into two layers of different concentrations advected by a flow field in a torus. We solve two non-linear optimisation problems. We identify the optimal \emph{initial perturbation} of the velocity field with given initial energy as well as the \emph{ optimal forcing} with given total action (the time integral of the kinetic energy of the flow) which both yield maximal mixing by a target time horizon. We analyse sensitivity of the results with respect to $s$-variation and thus address the importance of the choice of the fractional index   

Low order modelling for feedback control of fluid flows around complex geometries

The majority of goods transportation vehicles' power is consumed in overcoming aerodynamic drag. Reduction in pressure drag via feedback control could have significant economic and environmental effects on CO$_2$ emissions, and reduce fatigue on the body by suppressing vortex shedding. The difficulty in designing such controllers lies in obtaining models suited to modern control design methods, which are necessarily of much lesser complexity than typical Computational Fluid Dynamics (CFD) models, or models derived from immediate spatial discretisation of the Navier-Stokes equations. This work develops an approach for modelling fluid flows using frequency response data generated for individual computational node sub-systems that result from a CFD type spatial discretisation of the governing equations. Input-to-sensor frequency response data for the overall system are then computed by forming interconnections between adjacent nodes via a Redheffer Star Product operation, from which one typically observes low-order dynamics. With this data, a low-order model can be identified and used for controller design. This method avoids manipulating large matrices and is therefore computationally efficient and numerically well-conditioned. It can be readily applied to complex geometry flows.   

Contractive control design for Navier-Stokes systems with the incompressibility constraint relaxed

One approach to the linear stabilization of near-wall transitional channel flow is via the Orr-Sommerfeld/Squire equations. This formulation is delicate, as it reduces the three momentum equations and the divergence-free constraint of the incompressible NSE down to a highly non-normal set of two equations, one for the wall-normal velocity and one for the wall-normal vorticity, and involves inverting a Laplacian with boundary conditions embedded. A simpler formulation for the purpose of control design is given by simply dropping the divergence-free constraint from the problem considered altogether, and at the same time dropping the pressure gradient from the momentum equations, which acts to enforce this constraint. What remains is three coupled Burgers equations. In general, there is no relationship between the stability of such constrained and unconstrained systems; however, if the unconstrained system is contractive (a condition stronger than just stability), the constrained system is also contractive. We have investigated this approach to control design for NS systems. We have proved a fundamental limit: if an uncontrolled, unconstrained channel flow system is not contractive, there is no boundary control that can make it contractive.   

Control of laminar wake flows using the Sum-of-Squares approach

A novel feedback control design methodology for finite-dimensional, reduced-order models of incompressible turbulent fluid flows, aiming at reduction of long-time averages of key quantities, is presented. The key enabler is a recent advance in control design for systems with polynomial dynamics, i.e. the discovery that the Sum-of-Squares decomposition of a non-negative polynomial, or the construction of one of such functions, can be computed via semidefinite programming techniques. Firstly, the theoretical difficulties of treating long-time averages are relaxed by abstracting the analysis to upper bounds of such averages. Then, the problems of estimation and optimisation via control design of these bounds are conveniently reformulated into constructing suitable non-negative polynomial functions, using Sum-of-Squares programming techniques. To showcase the methodology, linear and nonlinear polynomial-type state-feedback controllers are designed to reduce the fluctuations kinetic energy in the wake of a circular cylinder at $Re=100$, using rotary oscillations. A compact, reduced-order Galerkin model of the actuated wake is first derived using Proper Orthogonal Decomposition. Controllers are then designed and implemented in direct numerical simulations of the flow.   

A suboptimal feedback control theory based on a quadratic sensitivity and tabulation approach

The main objective of this study is to develop a new systematic flow control approach based on the suboptimal feedback control (SFC) theory by addressing a few issues in controlling flows for a practical purpose. The Fréchet differential is applied to the governing equations to derive a systematic controller based on partial differential equations. In the previous SFC theory, a physical assumption or tuning process is necessary for a user-defined parameter to control flows successfully. In the present study, this issue is addressed by introducing an approximate optimality condition based on a quadratic control sensitivity. In order to build a practical control framework, the revised theory is reformulated as a tabulation approach using a modified Green's function method, which achieves both efficiency and accuracy. The effectiveness of the proposed method is tested using laminar and turbulent flows such as a two-dimensional Taylor vortex problem and turbulent channel flow. As a result, the proposed approach shows a similar or better control performance compared to the previous one, without a need to determine a control parameter.   

Control of Quantum Fluid Dynamics and Adaptive Phase Compensation for Laser Propagation in Turbulence

Equations of High Energy Laser propagation in a turbulent medium and the equations of quantum fluid dynamics are connected through a mathematical transformation. In this way the problem of adaptive phase compensation can be phrased as an initial velocity control problem for quantum fluid dynamics. The quantum hydrodynamics equation can be derived by applying the Madelung transformation to the time-dependent linear or nonlinear Schr\"{o}dinger equation. The resulting equations are similar to incompressible Euler equations with an additional term denoted the quantum pressure term. The quantum hydrodynamics equation can thus be a good way to understand adaptive optics and laser propagation through the atmosphere. A Riemann solver within the Clawpack framework has been developed. An initial value optimization problem will be solved using adjoint methods. The initial phase can be controlled when the beam leaves the laser appartus. The control method can also be coupled to a Navier-Stokes solver in order to study thermal blooming where the laser heats the air and changes the index of refraction. The change in refractive index will in turn affect the propagation of the Laser beam. Using optimal control techniques, it is possible to adjust the beam in order to compensate for the heating.   

Shell model of optimal passive-scalar mixing

Optimal mixing is significant to process engineering within industries such as food, chemical, pharmaceutical, and petrochemical. An important question in this field is ``How should one stir to create a homogeneous mixture while being energetically efficient?" To answer this question, we consider an initially unmixed scalar field representing some concentration within a fluid on a periodic domain. This passive-scalar field is advected by the velocity field, our control variable, constrained by a physical quantity such as energy or enstrophy. We consider two objectives: local-in-time (LIT) optimization (what will maximize the mixing rate now?) and global-in-time (GIT) optimization (what will maximize mixing at the end time?). Throughout this work we use the $H^{-1}$ mix-norm to measure mixing. To gain a better understanding, we provide a simplified mixing model by using a shell model of passive-scalar advection. LIT optimization in this shell model gives perfect mixing in finite time for the energy-constrained case and exponential decay to the perfect-mixed state for the enstrophy-constrained case. Although we only enforce that the time-average energy (or enstrophy) equals a chosen value in GIT optimization, interestingly, the optimal control keeps this value constant over time.   

Oscillatory flow past a slip cylindrical inclusion embedded in a Brinkman medium

Transient flow past a circular cylinder embedded in a porous medium is studied based on Brinkman model with Navier slip conditions. Closed form analytic solution for the stream-function describing slow oscillatory flow around a solid cylindrical inclusion is obtained in the limit of low-Reynolds-number. The key parameters such as the frequency of oscillation $\lambda$, the permeability constant $\delta$, and the slip coefficient $\xi$ dictate the flow fields and physical quantities in the entire flow domain. Asymptotic steady-state analysis when $\delta\to 0$ reveals the paradoxical behavior detected by Stokes. Local streamlines for small times demonstrate interesting flow patterns. Rapid transitions including flow separations and eddies are observed far away from the solid inclusion. Analytic expressions for the wall shear stress and the force acting on the cylinder are computed and compared with existing results. It is noted that the slip parameter in the range $0\le\xi\le0.5$ has a significant effect in reducing the stress and force. In the limit of large permeability, Darcy (potential) flow is recovered outside a boundary layer. The results are of some interest in predicting maximum wall stress and pressure drop associated with biological models in fibrous media.