Skin friction and pressure: the “footprints” of turbulence

The problems of exact state reconstruction and approximate state estimation based on wall information in a wall-bounded incompressible unsteady flow are addressed. It is shown that, if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure, all terms in the Taylor-series expansions of the unsteady flow state near the wall at time t may be determined (in the linear setting, this determination may be made based on skin-friction measurements alone). Combining this fact with the analyticity of solutions of the nonlinear Navier–Stokes equation and the unique continuation theorem for analytic functions, in theory complete reconstruction of a fully-developed turbulent flow in a channel at any Reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t, without knowledge of the initial conditions of the flow. Thus, skin-friction and pressure measurements on the wall in a neighborhood of time t provide a unique “footprint” of the entire unsteady turbulent flow state; no other flow can have the same footprint. Indeed, higher-order terms are shown to uniformly improve the correlation of truncated Taylor-series expansions with the DNS of a turbulent flow near the wall. However, such series extrapolations amplify measurement noise, as they require differentiation in both space and time of the measurements, and the radius of convergence of the Taylor series expansions is less than 10 wall units. The so-called linear stochastic estimation technique, in which the polynomials forming the basis of the series expansion are replaced by well-behaved functions (such as POD modes) on the entire flow domain also demonstrates very poor convergence. In light of these limitations on direct extrapolations from measurements in the practical setting, an adjoint-based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel-flow system based on a time history of noisy measurements at the wall. This algorithm effectively uses the unsteady nonlinear Navier–Stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements.