Introduction and problem formulation History, background, and rationale Initial-value concepts and stability bases Classical treatment: modal expansions Transient dynamics Asymptotic behavior Role of viscosity Geometries of relevance Spatial stability bases Temporal stability of inviscid incompressible flows General equations Nondimensionalization Mean plus fluctuating components Linearized disturbance equations Recourse to complex functions Three-dimensionality Squire transformation Kelvin-Helmholtz theory Interface conditions Piecewise linear profile Unconfined shear layer Confined shear layer Inviscid temporal theory Critical layer concept pagex xix xxi
Abstract. A study of instabilities in incompressible boundary-layer ow on a at plate is conducted by spatial direct numerical simulation DNS of the Navier-Stokes equations. Here, the DNS results are used to critically evaluate the results obtained using parabolized stability equations PSE theory and to study mechanisms associated with breakdown from laminar to turbulent o w. Three test cases are considered: two-dimensional TollmienSchlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-ow distortion component of the 2-D test problem. This discrepancy is attributed to far-eld boundarycondition di erences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic ow and a variation of the disturbance frequency result in better agreement with the experiments.
Measurements of the velocity field created by a shallow bump on a wall revealed that an energy peak in the spanwise spectrum associated with the driver decays and an initially small-amplitude secondary mode rapidly grows with distance downstream of the bump. Linear theories could not provide an explanation for this growing mode. The present Navier-Stokes simulation replicates and confirms the experimental results. Insight into the structure of the flow was obtained from a study of the results of the calculations and is presented. 0 199.5 American Institute of Physics.
This paper describes a self-contained, automated methodology for active o w control which couples the time-dependent N a vier-Stokes system with an adjoint N a vier-Stokes system and optimality conditions from which optimal states, i.e., unsteady ow elds and controls e.g., actuators, may be determined. The problem of boundary layer instability suppression through wave cancellation is used as the initial validation case to test the methodology. Here, the objective of control is to match the stress vector along a portion of the boundary to a given vector; instability suppression is achieved by c hoosing the given vector to be that of a steady base ow. Control is e ected through the injection or suction of uid through a single ori ce on the boundary. The results demonstrate that instability suppression can be achieved without any a priori knowledge of the disturbance, which is signi cant because other control techniques have required some knowledge of the ow unsteadiness such a s frequencies, instability t ype, etc. The present methodology has been extended to three dimensions and may potentially be applied to separation control, re-laminarization, and turbulence control applications using one to many sensors and actuators.
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