A Singular Value Analysis of Boundary Layer Control

Several approaches for boundary-layer control are analyzed from a linear system point of view. The singular value decomposition (SVD) is applied to the linearized NavierStokes system in the presence of control. The performance of control is examined in terms of the largest singular values, which represent the maximum disturbance energy growth ratio attainable in the linear system under control. It is shown that the maximum growth ratio is less in controlled systems than in the uncontrolled system only when control parameters are within a certain range of values. With opposition control, for example, when the detection plane is located too far away from the wall, the maximum energy growth ratio is larger, consistent with the results obtained in direct numerical simulations. The SVD analysis of other controls also shows a similarity between the trend observed in the SVD analysis (linear) and that observed in direct numerical simulations (nonlinear), thus reaffirming importance of linear mechanisms in the near-wall dynamics of turbulent boundary layers. The present study illustrates that the SVD analysis can be used as a guideline for designing controllers for drag reduction in turbulent boundary layers.