Control and sensitivity reduction for a viscous Burgers´ equation

The problem of designing and numerically implementing a controller for a fluid flow system at the boundary of the flow domain is complicated by the facts that the basic model is truly nonlinear and the flow is usually highly sensitive to boundary conditions. High sensitivity to small changes in the boundary such as wall roughness or dynamic excitations can trigger transition to undesirable states. One approach to preventing or delaying a transition is to introduce a simple control loop along the boundary to reduce the sensitivity. In this work we demonstrate the aforementioned idea for a system governed by the one dimensional Burgers´ equation. In particular, we focus on the initial boundary value problem for a viscous Burgers´ equation which is known to be is extremely sensitive with respect to a small perturbation in a Neumann boundary condition. We use this model to illustrate how this sensitivity can generate erroneous numerical solutions and “transitions” to these states. In particular, for a fixed viscosity and certain initial data the numerical solution z(x, t) of Burgers´ equation with a Neumann boundary condition z_x(0, t) = 0 converges to a (large) solution that satisfies z_x(0, t) = -α for a number α >; 0 less than machine precision zero. Thus, the solution of the Burgers´ equations for this problem exhibits an extreme sensitivity to the boundary perturbation α and this sensitivity can produce unexpected and undesired dynamic behavior. We use a continuous sensitivity equation method to compute these sensitivities and show that the sensitivity can be eliminated by introducing a very simple proportional error boundary feedback mechanism.