Control of wall shear flows and the potential for “ designer turbulence ”

This talk and paper draw heavily on related published work by this author reviewing flow control and especially the use of resolvent analysis to characterize and design control approaches. In particular, a range of almost identical material was presented at the 12th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements [21]. The financial and environmental cost of turbulence is staggering: manage to quell turbulence in the thin boundary layers on the surface of a commercial airliner and you could almost halve the total aerodynamic drag on existing platforms, dramatically cutting fuel burn, emissions and cost of operation, and enabling new aircraft designs. Consequently, there have been large investments around the world in research into means to reduce and control turbulence [6]. However practical implementation, especially of active control schemes, has remained either impractical or not economically viable given relatively small net gains. A brief review of historical and current flow control efforts targeting skin friction reduction is given. Systems-level tools to model scale interactions and control turbulence are given more detailed treatment. The resolvent analysis for turbulent flow proposed by McKeon and Sharma (J. Fluid Mech, 2010) [23] provides a simple, but rigorous, approach by which to deconstruct the full turbulence field into a linear combination of modes which interact through the nonlinear term to provide self-sustaining turbulence. A review of analysis of the linear resolvent operator and the nonlinear feedback, or the inter-connection between different scales which provides the forcing required for self-sustaining turbulence, is provided, then the formulation is extended to include control through modifications to the wall boundary conditions. After considering the information given by the resolvent operator about the relative physical locations of input and corresponding response, framed in the context of flow control, the specific examples of experimental manipulation of both the linear response and the triadic scale interactions using dynamic roughness actuation, opposition control and compliant wall design for global drag reduction are considered. A brief statement on the outlook for modeling and control of turbulent flows using this approach is also presented, focusing on possibilities related to a more general shaping of flow properties, or “designer turbulence”, enabled by more detailed understanding of the mechanisms of turbulence sustenance. Introduction to control of wall shear flows Control of wall turbulence using passive or active means as a sub-topic of the broader study of general flow control has been a topic of scientific and especial practical interest for many years given the ubiquity of the phenomenon. Overviews of various approaches have been given by Gad-el-Hak [7], Kim [13, 14] and Kim and Bewley [15]. There have been various more recent efforts, as summarized by, e.g., McKeon, Jacobi and Duvvuri [22] as part of an AIAA Journal Special Issue on Flow Control. Henceforth in this contribution we focus specifically on the potential to use resolvent analysis for a system-level approach to flow control. Closed Loop Resolvent Formulation The resolvent formulation for wall turbulence has been described in detail in previous publications, e.g. [23, 24, 20]. The essential outline of the closed-loop resolvent analysis is shown in Figure 1, and the development here is for turbulent channel (plane Poiseuille) flow in a domain which is periodic in the streamwise and spanwise directions, x and z, respectively, with no-slip and no-penetration wall boundary conditions. Defining fluctuations relative to the spatio-temporal turbulent mean (which is assumed to be known from, e.g., experimental data, simulations or an eddy viscosity model), U(x,y,z, t) = U(y) + u(x,y,z, t), and Fourier transforming the Navier-Stokes equations in (x,z, t): û(kz,kz,ω;y) = ∫ ∫ ∫ ∞ −∞ u(x,y,z, t)e −i(kxx+kzz−ωt)dxdzdt Rearranging, and formulating in velocity-vorticity form, (v,η), we arrive at