Adjoint Complement to the Volume-of-Fluid Method for Immiscible Flows

Abstract The paper is concerned with an adjoint complement to the Volume-of-Fluid (VoF) method for immiscible two-phase flows, e.g. air and water, which is widely used in marine engineering due to its computational efficiency. The particular challenge of the primal and the corresponding adjoint VoF-approach refers to the sharp interface treatment featuring discontinuous physical properties. Both the continuous adjoint two-phase system (integration-by-parts) and the corresponding dual compressive convection schemes (summation-by-parts) are derived for two prominent compressive convection schemes, namely the High Resolution Interface Capturing Scheme (HRIC) and Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM). The dual scheme rigorously mirrors the primal Normalized-Variable-Diagram (NVD) stencils. Attention is restricted to steady state applications. Thus both the primal and the dual procedures are performed in pseudo-time and the backward integration of the dual approach is performed around the (pseudo-temporal) converged primal field. Therefore, the adjoint system experiences the same time step size restrictions as the primal system, is independent of the primal time horizon and forms a robust as well as an a priori stable adjoint solution process. The paper analyses the primal and adjoint equations for an engineering model problem. An analytical solution to the model problem is initially presented, which displays that the adjoint part does not offer a unique, non-trivial solution. As a remedy, an additional diffusive concentration term is introduced to the adjoint concentration equation. The imposed heuristic modification violates the dual consistency but strongly regularizes the solution of the adjoint system. The modification can be justified by reference to phase-separating diffuse-interface models and inheres a free mobility-parameter. Numerical results obtained from the modified approach are benchmarked against the analytical solution for the model problem. Supplementary, the influence of the modification on the sensitivities obtained from simulations for the two-dimensional flow around a submerged hydrofoil at Froude and Reynolds numbers of practical interest are discussed for a range of mobility-parameters. The final application refers to a shape-optimization of a generic 3D underwater vehicle and underlines a negligible influence of the free mobility parameter, even for an objective functional that directly depends on the manipulated (dual) field quantity.