Adjoint Navier-Stokes methods for hydrodynamic shape optimisation

Adjoint Navier-Stokes methods are presented for incompressible flow. The sensitivity derivative of scalar hydrodynamic objective functionals with respect to the shape is obtained at the cost of one flow-field computation, so that the numerical effort is practically independent of the number of shape parameters. The adjoint Navier-Stokes problem was derived on the level of partial differential equations (PDE) first. According to the frozen-turbulence assumption, variations of the turbulence field with respect to the shape control were neglected in the adjoint analysis. Second, consistent dicretisation schemes were derived for the individual terms of the adjoint PDE on the basis of the primal, unstructured finite-volume discretisation. A unified, discrete formulation for the adjoint wall boundary condition and the sensitivity equation is presented that supports both low- and high-Reynolds number boundary treatments. The segregated pressure-correction scheme used to solve the primal problem was also pursued in the adjoint code. Reusing huge portions of the flow code lead to a compact adjoint module, reduced coding effort and yielded a consistent implementation. Analytical adjoint solutions were tailored to validate the adjoint approach. Moreover, the adjoint-based sensitivity derivative was verified against the direct-differentiation method for both internal and external flow cases. The adjoint method was used for ship-hydrodynamic design optimization to compute the sensitivity derivative of a wake objective functional with respect to the hull shape. The sensitivity map yields considerable insight into the design problem from the objective point of view. The adjoint-based sensitivity analysis was carried out at model scale to support a manual redesign of the hull and led to an improved wake field. An explicit, filtering-based preconditioning of the sensitivity derivative is first-order equivalent to the concept of “Sobolev-smoothing”. Particularly the sensitivity derivative obtained in conjunction with the low-Reynolds number treatment of boundary walls needed to be filtered before applied to the design surfaces. Guided by the adjoint-based sensitivity derivative, automatic shape optimization runs were performed for 2D and 3D internal flow problems to reduce the power loss.