Skin friction and pressure: the "footprints" of turbulence

Thomas R. Bewley, Bartosz Protas
2004 Physica D : Non-linear phenomena  
This article addresses the information available at the wall in the problem of state estimation in wall-bounded incompressible flows. It is shown that, if precise measurements are made of the two components of wall skin friction, ∂u ∂y and ∂w ∂y, and the wall pressure, p, all terms in the Taylor-series expansions of the flow state near the wall may be determined. Combining this fact with the analyticity of solutions of the Navier-Stokes equation on the attractor, in theory complete
more » ... n of a turbulent flow in a channel at time t is possible given only precise measurements of the flow at the wall in a neighborhood of time t. Implications of this result, in light of the standard framework for adjoint-based state reconstruction in turbulent flow systems, are discussed. § x 2 § x 3 coordinate system, and consider the flow in the entire channel¨0 1 Note that referring to the boundary values of ∂u ∂y and ∂w ∂y as "wall skin friction" is, admittedly, a bit sloppy notationally, as the corresponding components of the shear-stress tensor at the wall, τ xy µ ∂u ∂y ∂v ∂x and τ zy µ ∂w ∂y ∂v ∂z , both include contributions from the (prescribed) boundary values of v on the wall and are scaled by the viscosity µ. We assume the viscosity µ and the value of v at the wall are known in this work, so ∂u ∂y and ∂w ∂y may easily be determined from measurements of τ xy and τ zy at the wall. The idealized problem of a continuous distribution of both actuation and sensing on the wall is not quite physically realizable anyway; how this configuration might be approximated in a real implementation is an application-specific issue which we will not address here. We will thus use the words "streamwise and spanwise wall skin friction distributions" to refer to the distributions of ∂u ∂y and ∂w ∂y on the wall without ambiguity, with apology to the reader for this abuse of notation.
doi:10.1016/j.physd.2004.02.008 fatcat:b5f56fpakvednejgo36n4mvmwe