On spatial and temporal stability in three dimensional fluid flows

An analytical examination of the relation between the three-dimensional spatial and temporal stabilities, and between the spatial stabilities along two directions of three-dimensional flows over a rotating disk is presented. A method of multiple scales is used to define partial differential equations for the spatial and temporal modulations of the amplitude and phase of a three-dimensional wave in three-dimensional incompressible nonparallel mean flows. Relationships are found for the spatial and temporal growth rates as well as between the wavenumber and frequency. The feasibility of transforming any spatial growth into a temporal growth is demonstrated, as is the converse, in regards to a particular direction. Spatial growth rates are shown to be connected to spatial growth rates in any other direction.