M.Sc Thesis

M.Sc StudentIlia Malkine
SubjectPrediction of the Nonlinear Interaction between a 2D
Instability Wave and the Maen Flow of 2D Turbulent
DepartmentDepartment of Aerospace Engineering
Supervisor Full Professors Cohen Jacob



The long term goal of this research is to predict and understand the change in incompressible velocity field over a stalled wing due to the addition of a two-dimensional (2D) instability wave. Such an understanding will provide a simple tool for choosing optimal parameters for AFC (frequency, amplitude, angle of attack and etc.).

In the present work the nonlinear interaction between the mean flow and a 2D coherent disturbance in a 2D turbulent wake is investigated. Relative to the wing problem, this is a simpler one due to a lack of pressure gradients. Additionally, the theoretical predictions can be compared with available experimental data. For this purpose a theoretical model is developed. The proposed model is based on the triple decomposition method where it is assumed that the flow field consists of a quasiparallel mean flow, coherent sinusoidal perturbation and incoherent turbulent fluctuation which are taken into account using a constant eddy viscosity model.

However, quasiparallel approach suffers from an ambiguity of the linear solution normalization. The normalization of the eigenfunction obtained from the solution of the Orr-Sommerfeld equation influences the resulting level of the Reynolds stresses. This uncertainty can be resolved in two ways. The first one, by taking into account the divergence of the mean flow using the multiple scale method. The second one, using the procedure made in Parabolized Stability Equations (PSE) where the eigenfunction of the wave and its derivatives are assumed to vary slowly in the streamwise direction. Accordingly, centerline normalization of the linear eigenfunction can be used.

Both examined models provide close results. However, multiple scale theory is more complex and requires longer calculation time due to the adjoint problem and the iterative computation method. The most significant discrepancy between these two models is observed in the cross-stream velocity fluctuation approximation, where the multiple scale method “lags” the experimental and quasiparallel results. However, the multiple scale approach produces results which are not affected by an adopted normalization.

The results obtained for a wide range of Reynold number, depict that the spreading of the mean flow is due to two conflicting effects: the first is associated with the eddy viscosity of the unforced flow, and the second with the amplitude of the coherent wave. During the initial stage of development the effect of the eddy viscosity is dominant, and consequently, as the Reynolds number is increased the spreading rate is decreased. However, beyond a certain downstream distance where the disturbance amplitude reaches a significant magnitude the effect of Reynolds number is reversed, and now when the Reynolds number is increased so is the spreading rate of the mean flow. The results for different levels of the disturbance show that the spreading of the mean flow grows with an increase level of the forcing.