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The articles in this volume cover recent work in the area of flow control from the point of view of both engineers and mathematicians. These writings are especially timely, as they coincide with the emergence of the role of mathematics and systematic engineering analysis in flow control and optimization. Recently this role has significantly expanded to the point where now sophisticated mathematical and computational tools are being increasingly applied to the control and optimization of fluid flows. These articles document some important work that has gone on to influence the practical, everyday design of flows; moreover, they represent the state of the art in the formulation, analysis, and computation of flow control problems. This volume will be of interest to both applied mathematicians and to engineers.



Active Control of Acoustic Pressure Fields Using Smart Material Technologies

An overview describing the use of piezoceramic patches in reducing noise in a structural acoustics setting is presented. The passive and active contributions due to patches which are bonded to an Euler-Bernoulli beam or thin shell are briefly discussed and the results are incorporated into a 2-D structural acoustics model. In this model, an exterior noise source causes structural vibrations which in turn lead to interior noise as a result of nonlinear fluid/structure coupling mechanisms. Interior sound pressure levels. are reduced via patches bonded to the flexible boundary (a beam in this case) which generate pure bending moments when an out-of-phase voltage is applied. Wellposedness results for the infinite dimensional system are discussed and a Galerkin scheme for approximating the system dynamics is outlined. Control is implemented by using LQR optimal control theory to calculate gains for the linearized system and then feeding these gains back into the nonlinear system of interest. The effectiveness of this strategy for this problem is illustrated in an example.
H. T. Banks, R. C. Smith

On the Presence of Shocks in Domain Optimization of Euler Flows

In this paper we consider a shape optimization problem for a 1-D Euler flow. We show that for problems with shocks, the use of high order CFD schemes can produce artificial local minima in the approximate cost functional. These local minima can cause optimization algorithms to fail. We illustrate this phenomenon, show how hybrid algorithms may be constructed to overcome this problem and speculate on potential difficulties that may occur in more complex situations.
J. T. Borggaard

A Sensitivity Equation Approach to Shape Optimization in Fluid Flows

In this paper we apply a sensitivity equation method to shape optimization problems. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.
Jeff Borggaard, John Burns

Quasi-Analytical Shape Modification for Neighboring Steady-State Euler Solutions

Aerodynamic inverse design methods which are governing equation consistent are generally limited to the Full Potential equations. Consistent design methods use identical governing equations for all fluid dynamic segments of the algorithm, including shape modification. This ensures that all relevant physical information is included within each design estimate, and therefore, a minimum number of analysis/design iterations are required. This report presents a new, and consistent, shape modification method for future use within a direct-iterative inverse design algorithm. The method is simple, being developed from a truncated quasi-analytical Taylor’s series expansion of the global governing equations. The method is general, since it may use either the Euler or Navier-Stokes equations, any combination of numerical techniques, and any number of spatial dimensions. The proposed method also includes a unique iterative algorithm, and new geometry/grid constraints, to solve the over-determined design problem. An upwind, cell-centered, finite-volume formulation of the two-dimensional Euler equations is used within the present effort. The method is evaluated within a symmetric channel where the design variable is a mid-channel ramp angle which is nominally θ = 5°. Tests were conducted for three target ramp angle perturbations, Δθ = 2%, 10%, and 40%, and three inlet Mach numbers, M = 0.30, 0.85, and 2.00. For a single design estimate, using design-like test conditions, the new method is demonstrated to accurately predict geometry shape changes. This includes the transonic test case with an extreme 40% design variable perturbation where the target geometry was predicted with 95% accuracy.
J. S. Brock, W. F. Ng

Control of Steady Incompressible 2D Channel Flow

We consider steady incompressible flows in a 2D channel with flow quantities measured along some fixed, transverse sampling line. From a set of allowable flows it is desired to produce a flow that matches a given set of measurements as closely as possible. Allowable flows are completely specified by a set of control parameters which determine the shape of the inflow at the boundary and the shape of an internal bump which partially obstructs the flow. Difficulties concerning the transformation of this problem into a standard optimization problem are discussed, including the correct choice of functional and algorithm, and the existence of local minima.
John Burkardt, Janet Peterson

Optimality Conditions for Some Control Problems of Turbulent Flows

In this article, we are concerned with the control of the turbulence of viscous, incompressible flows. The control are the body forces or the heat flux through the boundary of the domain occupied by the fluid. The state is the velocity of the fluid and the turbulence is measured by some integral involving the vorticity within the flow. We consider steady and time-dependent three-dimensional flows described by the Navier-Stokes equations, sometimes coupled with the heat equation. We prove existence of optimal controls and derive some first order optimality conditions.
Eduardo Casas

On Controllability of Certain Systems Simulating a Fluid Flow

Approximate controllability of the Stokes system is established by a constructive method when control is a right-hand-side concentrated in subdomain i.e. in the case of local distributed control. Approximate uncontrollability of the Burgers equation is shown in the cases of boundary and local distributed controls. A local theorem of exact controllability for the Burgers equation with boundary control is proved. With its help it is shown that the controlled trajectory going out an arbitrary initial point can achieve the attractor of the Burgers equation during a finite time and after that belongs to attractor. The sets possessing such property we call an absorbing set of reachability. For the boundary and local distributed controls the description of absorbing points of reachability for the Burgers equation is given.
Andrei V. Fursikov, Oleg Yu Imanuvilov

A Prehistory of Flow Control and Optimization

Flow control and optimization is an ancient practice of man. For example any dam, sluice, canal, levee, irrigation ditch, valve, duct, pipe, pump, hose, vane, etc., is an exercise in flow control or optimization, i.e., and attempt to
control the mechanical state, e.g., the rate and direction of motion, and/or the thermodynamic state, e.g. the temperature, of a fluid in order to achieve a desired purpose.
Max D. Gunzburger

Mathematical Issues in Optimal Design of a Vapor Transport Reactor

In this paper the optimal design of a vertical reactor for growing crystals and epitaxial layers by physical vapor transport technique is discussed. The transport phenomena involved in the deposition process is modeled by the gasdynamics equations and chemical kinematics. The problem is formulated as a shape optimization with respect to the geometry of the reactor and an optimal control problem by controlling the wall temperature. The material and shape derivatives of solutions to the so-called Boussinesq approximation are derived. Optimality condition and a numerical optimization method based on the augmented Lagrangian method are discussed for the boundary control of the Boussinesq flow. A numerical approximation based on the Jacobi polynomials for the axi-symmetric flow is developed along with a discussion of an iterative method based on GMRES for solving the resulting system of nonlinear equations.
Kazufumi Ito, Hien T. Tran, Jeffery S. Scroggs

Mathematical Modeling and Numerical Simulation in External Flow Control

This paper presents an investigation of some active control problems for an external flow field. A series of numerical simulations are performed to investigate an unsteady viscous flow generated by a circular cylinder undergoing a combined rotary and rectilinear motion. By treating the rotation rate as a control variable, we present results of the time histories of forces acting on the cylinder surface and their time-averaged values under several types of rotations. The impact of changing rotation rate on the vortex formation, including the synchronization of cylinder and wake, is demonstrated. Based on the optimal control theory, an optimality system is formulated to determine the optimal rotation rates and the solution orbits. Though only the moving boundary mechanism is discussed, the results presented here add insight to the optimal design of control mechanism and may provide guidance to the formulation of other complex optimal flow control problems.
Yuh-Roung Ou

Optimal Feedback Control of Hydrodynamics: a Progress Report

In this article we review some of the recent results in the mathematical theory of optimal feedback control of viscous flow. Main results are existence of ordinary and chattering controls, Pontryagin maximum principle and feedback synthesis using infinite dimensional Hamilton-Jacobi equation of dynamic programming. Some preliminary results on stochastic control also presented.
S. S. Sritharan

Nonsmooth Analysis and Free Boundary Problems for Potential Flow

New approach to some Free boundary problems, is introduced. Those problems are studied first by Alt and Caffarelli [2] in the case of a potential flow. Their approach seem not to be possible to extend to the case of a Stokes flow. In this paper, the variable domain problem is relaxed so that it becomes a nonsmooth optimization problem on the fixed domain for the somewhat singular state equation. State equation is considered, and the multivalued generalized gradient of the variational functional is studied. Here, we considered Potential flow.
Srdjan Stojanovic

Computational Fluid Dynamics Analysis of the Flow in an APCVD Applicator System

Application of Atmospheric Pressure Chemical Vapor Deposition (APCVD) to the production of coated glass is addressed in this study. Several layers of thin films are deposited on the surface of the glass as it moves underneath the APCVD applicator system at high temperature. A memory effect in the form of film thickness streaks, corresponding to the location of the inlet holes located upstream in the upper manifold feed channel, is evident on the glass. This nonuniform film across the glass causes a color variation of the coating. Effective mixing of the gas streams is required to treat the hole memory problem. However, a premature reaction is to be avoided. Optimum design parameters to correct this problem include the geometry of the applicator and the sensitivity of the flow field to boundary conditions is of major interest. The Computational Fluid Dynamics (CFD) simulation and analysis package FIRE is used to predict the flow. The flow of gases involved is treated as that of a steady, viscous, incompressible fluid. Results for both two- and three-dimensional cases demonstrate that the deposition process can be improved by injecting the flow at an angle counter to the direction of glass motion, and that CFD techniques can be successfully used to predict the flow behavior of an APCVD applicator system and help optimize its design.
Gary S. Strumolo

Shape Optimization and Control of Separating Flow in Hydrodynamics

A model for computing flows with specified separation characteristics is presented. This is based on a shape optimization method for constructing a surface with a given tangential vorticity field.
Thomas Svobodny

Recent Advances in Steady Compressible Aerodynamic Sensitivity Analysis

An overview is given of some recent accomplishments by different researchers in calculating gradient information of interest from modern flow-analysis codes. Of particular interest here is advanced computational fluid dynamics (CFD) software, which solves the nonlinear multidimensional Euler and/or Navier-Stokes equations. The accurate, efficient calculation of aerodynamic sensitivity derivatives is very important in design-oriented applications of these CFD codes to single discipline and multidisciplinary problems [1, 2].
Arthur C. Taylor, Perry A. Newman, Gene J.-W. Hou, Henry E. Jones

Remarks on the Control of Turbulent Flows

Our aim in the article is to address some theoretical and computational questions related to the control of viscous incompressible flows governed by the Navier-Stokes equations or related equations.
Roger Temam
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