Estimation of structure of air entrained flow in a spillway chute

摘要

The structure of the air entrained flow in a spillway chute is estimated in the framework of an optimal control problem. The specific gravity of air-water mixture is supposed to be governed by a partial differential equation, which includes the diffusion and the convection coefficients. The coefficients are taken as the control variables to minimize a boundary and domain observation type functional which evaluates the error between measured specific gravity and the solution of the governing equation. The minimum principle is deduced to characterize the optimal control variables using the adjoint partial differential equation. Constraints are imposed on the control variables so that tile diffusion coefficient and the air bubble rise velocity do not become negative. Further assumption that these two components of the control variables are functions of the specific gravity is made to simplify the optimization process. An iterative numerical procedure using a finite element scheme to solve the partial differential equations is developed. Test runs of the procedure prove that the method is valid enough and can be used for analysis of air entrained flows.