Development and assessment of adjoint sensitivity analysis method for transient two-phase flow simulations

The adjoint method is efficient for calculating sensitivities of few responses to a large number of parameters. The cost of solving an adjoint equation is comparable to the cost of solving the forward governing equation. Once the adjoint solution is available, the sensitivities to any number of parameters can be calculated with little effort. There are two methods to develop the adjoint equations: continuous method and discrete method. In the continuous method, the control theory is applied to the forward governing equation and produces an analytical partial differential equation for solving the adjoint variable; in the discrete method, the control theory is applied to the discrete form of the forward governing equation and produces a linear system of equations for solving the adjoint variable. In this article, an adjoint sensitivity analysis framework is developed using both the continuous and discrete methods. These two methods are assessed with one transient test case. Adjoint sensitivities from both methods are verified by sensitivities calculated with a perturbation method. Adjoint sensitivities from both methods are physically reasonable and match each other. The sensitivities obtained with discrete method is found to be more accurate than the sensitivities from the continuous method. Though continuous method is computationally more efficient than the discrete method, difficulties are observed in solving the continuous adjoint equation for cases where the adjoint equation contains sharp discontinuities in the source terms. In such cases, the continuous method is not as robust as the discrete adjoint method.