A continuous adjoint approach to design optimization in multiphase flow

Continuous adjoint methods are developed for design optimization in multiphase flow based on two homogeneous multiphase mixture models for cavitating flow—a barotropic model and a transport-equation-based model.;The barotropic model consists of variable-density mass and momentum equations and uses an equation of state for the density that depends on only the local static pressure and the vapor pressure of the liquid. In that case, the primary equations are homogeneous, and a conventional hybrid multistage explicit method based on central differencing and second- and fourth-order scalar artificial dissipation can be applied to solve both the primary and adjoint systems. Results are presented for both surface- and volume-based vapor minimization cost functions for a two-dimensional cavitating hydrofoil in which the geometry is parameterized using B-splines. The cost function gradients computed using the adjoint method are shown to compare well with gradients computed using standard and complex-step finite-difference methods, and several examples serve to demonstrate that these gradients can be used to inform a fixed-step method of steepest descent for shape optimization.;For the transport-equation-based model, the governing flow equations include source terms that model the mass transfer between liquid and vapor, and the fact that these source terms are not continuously differentiable with respect to the state variables gives rise to a discontinuity in the adjoint solution. A high-resolution fluctuation-splitting method with wave limiters is used to discretize the non-conservative, variable-coefficient linear system of equations, where the preconditioning matrix from the primary algorithm is included in the adjoint system to render it hyperbolic. Results are presented for several cost functions applied to quasi-one-dimensional flow through a converging-diverging nozzle. The geometry is once again parameterized using B-splines, and once again, the cost function gradients computed using the adjoint method are shown to compare well with gradients calculated using standard and complex-step finite-difference methods.;Exploration of several cost functions for the transport-equation-based model in multiphase flow reveals that the corresponding minimization surfaces can be poorly conditioned and non-convex. Nevertheless, with careful cost function design and the use of a low-memory quasi-Newton descent method, the cost functions are minimized or significantly decreased in every case. Thus, this work establishes that the continuous adjoint method can serve as the basis for a new generation of hydroelectric design tools that provide detailed design guidance from high-fidelity multiphase flow simulations.