Fundamental limitations in initial-state estimation using surface measurements in wall turbulence*

Estimating the initial state of turbulent channel flow from surface measurements is formulated as an adjoint variational minimization. The cost function is the difference between the true measurements and their prediction from the reconstructed state. The accuracy of the state reconstruction and the fundamental limitations in its prediction from wall measurement are examined in detail. The analysis relies on evaluation of the Hessian matrix of the cost function, in the vicinity of the true solution, and analysis of the Hessian eigen-spectra. The leading eigenmodes correspond to the highest sensitivity of measurements to the flow; these modes are concentrated in the near-wall region upstream of the sensor location. Their structure explains the fundamental difficulty of estimating the flow state from wall measurements.