Continuous adjoint approach to the Spalart–Allmaras turbulence model for incompressible flows

Abstract

A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart–Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.