  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 in comparison to the frozen turbulence approach.
