Adjoint analysis for receptivity prediction

Abstract

Physical knowledge of the laminar-turbulent transition process, prediction of the transition location, as well as the ability to control transition are essential in many engineering applications. However, control of the laminar-turbulent transition depends critically on various environmental sources and their ability to excite the instability waves in the flow, which are responsible for the laminar-turbulent transition. The process by which external disturbances are converted into instability waves is called receptivity. The research described in this thesis focuses on the receptivity of two- and three-dimensional boundary layers. The main objective of this research is to formulate, validate and apply adjoint analysis in order to predict receptivity. Adjoint analysis is a powerful approach for investigating the receptivity of different flows for arbitrary environmental sources. In this work, Adjoint Navier-Stokes (ANS) equations are formulated based on the sensitivity approach, and adjoint predictions are validated against Linearized Navier-Stokes (LNS) calculations. Further, Adjoint Parabolized Stability Equations (APSE) are derived as an approximation of ANS equations and compared against the ANS results. Our studies indicate that the APSE method should be constructed as an approximation to the ANS equations, not as the formal adjoint of the PSE. When implemented in this manner, we show that APSE is a viable method for receptivity prediction, even in highly nonparallel flows. The APSE is first applied to predict receptivity of weakly nonparallel two-dimensional boundary layer flows for a variety of parameters. We find that these flows are generally more receptivity to oblique disturbances although two-dimensional disturbances are less stable. We also find that favorable pressure gradient boundary layers are more receptive then adverse pressure gradient boundary layers, although adverse pressure gradients are destabilizing. The APSE are then applied to highly nonparallel three-dimensional boundary layers where we find that for the inviscidly unstable crossflow instability, stability effects typically dominate receptivity effects. Comparison of receptivity for stationary and unsteady crossflow instabilities shows that receptivity to both localized momentum sources and streamwise wall-velocity excitations is larger for unsteady modes, but that receptivity to wall-normal excitations is larger for stationary modes. Finally, we consider receptivity for the swept parabolic cylinder and observe that convex surface curvature tends to enhance receptivity to wall roughness, in agreement with prior studies. Utilizing the efficiency of adjoint methods, we also consider other excitations for the swept parabolic cylinder and show that convex surface curvature also enhances receptivity to streamwise momentum sources, but that receptivity to both normal and tangential velocity disturbances is slightly reduced.