Mitigating adjoint chaos in wall turbulence

Estimating past events in wall turbulence based solely on surface measurements and first principles is an ill-posed problem that is complicated by chaos. The sensitivity of a measurement to the earlier flow state is described by the adjoint Navier–Stokes equations, which are solved in reverse time starting from the measurement kernel at the sensing position and time. The resulting adjoint field is the spatio-temporal domain of dependence (DOD) of the sensor, which is a dual to the concept of the domain of influence (DOI) of an actuator in the linearised forward equations. In channel turbulence, the energy of each adjoint realisation grows exponentially in backward time according to the Lyapunov exponent, even though the energy of the ensemble average should decay. We introduce a linear eddy-viscosity closure model in the ensemble-averaged adjoint equations, and directly compute the mean DOD and compare our prediction with the ensemble average. Furthermore, we demonstrate that the DOD of a wall-stress measurement and the DOI resulting from a wall-stress perturbation exhibit respective universal behaviours across Reynolds numbers. However, their spatio-temporal structures differ qualitatively, due to the time asymmetry of the governing equations. The DOD field has a two-part structure: one component is associated with the Orr mechanism, characterised by rapid reorientation under mean shear, and the other is related to self-similar expanding streaky structures. These two components jointly define the sensitivity of the wall-stress measurement to past flow events.