Adjoint complement to the universal momentum Law of the wall

The paper is devoted to an adjoint complement to the universal Law of the Wall (LoW) for fluid dynamic momentum boundary layers. The latter typically follows from a strongly simplified, unidirectional shear flow under a constant stress assumption. We first derive the adjoint companion of the simplified momentum equation, while distinguishing between two strategies. Using mixing-length arguments, we demonstrate that the frozen turbulence strategy and a LoW-consistent (differentiated) approach provide virtually the same adjoint momentum equations, that differ only in a single scalar coefficient controlling the inclination in the logarithmic region. Moreover, it is seen that an adjoint LoW can be derived which resembles its primal counterpart in many aspects. The strategy is also compatible with wall-function assumptions for prominent RANS-type two-equation turbulence models, which ground on the mixing-length hypothesis. As a direct consequence of the frequently employed assumption that all primal flow properties algebraically scale with the friction velocity, it is demonstrated that a simple algebraic expression provides a consistent closure of the adjoint momentum equation in the logarithmic layer. This algebraic adjoint closure might also serve as an approximation for more general adjoint flow optimization studies using standard one- or two-equation Boussinesq-viscosity models for the primal flow. Results obtained from the suggested algebraic closure are verified against the primal/adjoint LoW formulations for both, low- and high-Re settings. Applications included in this paper refer to two- and three-dimensional shape optimizations of internal and external engineering flows. Related results indicate that the proposed adjoint algebraic turbulence closure accelerates the optimization process and provides improved optima at no computational surplus in comparison to the frozen turbulence approach.