The Growth of Localized Disturbances in Unstable Flows

The development of three-dimensional localized disturbances in unstable flows was recently studied by Craik [1] using a model dispersion relation. The adoption of such an approximate formula for the linear dispersion relation allows a dramatic reduction in computational effort, in comparison with more precise calculations (e.g., Gaster [3], [5]), yet may still yield quite accurate results. Craik [1] gives simple analytical solutions for various limiting cases of his chosen model. Here, this model is further investigated. Numerical results are given which are free of previous scaling assumptions and the accuracy of the proposed model is assessed by comparison with known exact computations for plane Poiseuille flow. Certain improvements are made by including further terms in the model dispersion relation and the influence of these additional terms is determined. A further model is investigated which yields “splitting” of the wave packet into two regions of greatest amplitude, one on either side of the axis of symmetry. Such behavior may be characteristic of many flows at sufficiently large Reynolds numbers. Extension of this work to three-dimensional and slowly varying flows seems a practical possibility.