Control of hydrothermal waves in a thermocapillary flow using a gradient‐based control strategy

Abstract

SUMMARYThe problem of controlling the hydrothermal waves in a thermocapillary flow is addressed using a gradient‐based control strategy. The state equations are the two‐dimensional unsteady incompressible Navier–Stokes and energy equations under the Boussinesq approximation. The modeled problem is the ‘open boat’ process of crystal growth, the flow which is driven by Marangoni and buoyancy effects. The control is a spatially and temporally varying heat flux boundary condition at the free surface. The control that minimizes the hydrothermal waves is found using a conjugate gradient method, where the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations. The effectiveness of choices of the parameters governing the control algorithm is examined. Almost complete suppression of the hydrothermal waves is obtained for certain choices of the parameters governing the control algorithm. The numerical issues involved with finding the control using the optimizer are discussed, and the features of the resulting control are analyzed with the goal of understanding how it affects the flow.Copyright © 2012 John Wiley & Sons, Ltd.

Objective function choice for control of a thermocapillary flow using an adjoint-based control strategy

“…In contrast to previous investigations we make use of the full information provided by the free surface temperature and control is applied at the whole free surface. In a work closely related to the present work Muldoon (2013) demonstrated control of flow oscillations in this geometry. The mathematical model, the two-dimensional unsteady incompressible Navier-Stokes equations and energy equations, was the same as in the present work.…”

“…The gradient used in the present work is obtained by this method which provides a means to obtain the gradient of the objective function with respect to the control variables, the computational cost, as measured by the number of arithmetical operations, of which is independent of the number of control variables. The means by which the adjoint equations are derived and solved in the present work are described in Muldoon (2013). Derivations of the adjoint equations can be found in Muldoon (2008), Giles et al (2003), Nielsen (1998).…”

“…Unless otherwise mentioned, all results were obtained using a 266 Â 50 grid in the x and y-directions respectively. In the interests of brevity and to avoid duplication, we refer the reader to Muldoon (2013) for the details of the method used to discretize the governing equations and solve the resulting algebraic system.…”

“…A literature review concerning the combination of optimization/control with Computational Fluid Dynamics in general exists in Muldoon (2013). To avoid repetition, we briefly mention here only the work on thermocapillary flows.…”

“…It was found that the length of the ''event horizon'' (i.e., the distance ahead in time over which optimal control was found), played a significant role in the success of finding control that suppressed the flow oscillations. The present work represents a continuation of Muldoon (2013) focusing on investigation of different objective functions and issues involved with solving the optimization problems required to determine the optimal control.…”