An Optimal Control Approach to Bounding Transport Properties of Thermal Convection.

Abstract

This dissertation explores and develops an optimal control approach to upper bounds on transport properties of fluid flows inspired by the physical phenomenon of buoyancy-driven Rayleigh-Bénard convection. This method is applied in the context of three different problems: the Lorenz equations, the Double Lorenz equations, and the Boussinesq approximation to the Navier-Stokes equations. Rather than restricting attention to flows that satisfy an equation of motion, we consider incompressible flows that satisfy suitable bulk integral constraints and boundary conditions. Bounds on transport are formulated in terms of optimal control problems where the flows are the "control" and a passive scalar tracer field is the "state". All three problems lead to non-convex optimization problems. Sharp upper bounds to the Lorenz equations are proven analytically, and it is shown that any sustained time-dependence of the control variable strictly lowers transport. For the Double Lorenz equations an upper bound is proven and saturated by steady optimizing flow fields and any time-periodic stirring protocol strictly lowers transport. In contrast to the Lorenz equations, however, the optimizing steady flow fields (solutions to the Euler-Lagrange equations for optimal transport) are not solutions to the original equations of motion. In the Boussinesq equation context the optimal control problem is rigorously formulated for steady flows, and analytic upper bounds to transport are deduced using the background method. A gradient ascent procedure for numerically solving the associated the Euler-Lagrange equations for optimal transport is developed, including optimality conditions for the domain size. The numerically computed optimizing flow fields consist of convection cells of decreasing aspect ratio as one allows for a stronger flow fields. Implications for natural convective transport in the motivating Rayleigh-Bénard problem are discussed.