On Spatial and Temporal Stability in Three Dimensional Fluid Flows

A three-dimensional stability analysis is presented for an incompressible viscous non-parallel boundary layer flow over a rotating disk. The method of multiple scales is used to derive partial differential equations that describe the axial and cross-flow variations of the disturbance amplitude, phase and wave numbers. These equations are then employed to derive the expressions that relate the spatial and temporal stabilities. These equations are also used to obtain the expression that relates the spatial growth rate along a given direction to a spatial growth rate along any other direction. These relations are verified by numerical results, obtained for three-dimensional disturbances. The numerical results are presented in order to relate the spatial and temporal stability for different values of Reynolds number, wave number and frequency. It is shown that the predicted results are in excellent agreement with those of the present numerical calculations. Further, it is predicted for a quasiparallel flow that any spatial growth rate can be transformed into a temporal growth rate, while a temporal growth rate can be transformed into a spatial growth rate along any specified direction.