Ensemble-averaged domain of dependence of wall stress in turbulent channel flow

Estimating past events from wall measurements in turbulent channel flow, by aid of the governing equations, is a challenging and ill-posed problem. To quantify the extent to which we can determine the earlier flow state from wall data, the adjoint Navier-Stokes equations are solved backward in time, initialized by the measurements kernel. These adjoint fields define the spatio-temporal domain-of-dependence (DOD) of the measurement. In turbulent channel flow, the energy of the adjoint field amplifies exponentially in backward time at the Lyapunov exponent, while the ensemble average of adjoint realizations decays. Slow convergence of the ensemble motivates a Reynolds-averaged approach. We adopt a linear eddy-viscosity model to close the forward-adjoint correlation term in the adjoint RANS equations. As such, the averaged DOD can be computed efficiently, and compares favorably with the ensemble approach. We additionally compare the DOD to the domain of influence, the later computed from the linearized forward RANS equations. We discuss the implications of these findings for data assimilation and information propagation in wall turbulence.