# Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows

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- 19 Citations

## Summary

*D*(

*k, l, ω*), where

*k*and

*l*are wave numbers, and

*ω*is a frequency, the analytic criterion is formulated by which a point (

*k*

_{0},

*l*

_{0},

*ω*

_{0}) with Im

*ω*

_{0}>0 contributes to the absolute instability if and only if one of the two equivalent conditions is satisfied:

- (i)
At least two roots in

*l*of the system*D*(*k, l, ω*)=0,*D*_{ k }(*k, l, ω*)=0, originating on opposite sides of the real*l*-axis, collide on the*l*-plane for the parameter values*k*_{0},*l*_{0},*ω*_{0}, as*ω*is brought down to*ω*_{0}. Every point on the*k*-plane, that corresponds to a point on the collision paths on the*l*-plane, is itself a coalescence point of*k*-roots for a fixed*l*of*D*(*k, l, ω*)=0, that originate on opposite sides of the real*k*-axis. - (ii)
At least two roots in

*k*of the system*D*(*k, l, ω*)=0,*D*_{ l },(*k, l, ω*)=0, originating on opposite sides of the real*k*-axis, collide on the*k*-plane for the parameter values*k*_{0},*l*_{0},*ω*_{0}, as*ω*is brought down to*ω*_{0}. Every point on the*l*-plane, that corresponds to a point on the collision paths on the*k*-plane, is itself a coalescence point of*l*-roots for a fixed*k*of*D*(*k, l, ω*)=0, that originate on opposite sides of the real*l*-axis.

Consequently, the causality condition for spatially amplifying 3-D waves in absolutely stable, but convectively unstable flow is derived as follows. We denote by (*α, β*) a unit vector on the (*x, y*) plane. The contributions to amplification in the direction of this vector come from the end points of the trajectories that consist of the coalescence roots on the*l*_{1}-plane, given by*l*_{1},=−*βk*+*αl*, of the system*D*=0,−*βD*_{ k }+*gaD*_{1}=0. The*k*_{1}-components of these trajectories have to pass from above to below the real axis on a*k*_{1}-plane, given by*k*_{1}=*αk*+*βl*, as*ω* moves down to*ω*_{0}. Here*ω*_{0} is the real frequency of excitation. At each point of such trajectories the group velocity vector (*D*_{ k },*D*_{ l }) is collinear with the direction vector (*α, β*). There exists a direction for which the spatial amplification rate reaches its maximum.

The formalism is illustrated with a simple model example. A procedure for computing the*N*-factor in the*e*^{ N }-method, which is based on the wave packet approach is developed.

### Keywords

Dispersion Relation Wave Packet Group Velocity Real Axis Direction Vector## Preview

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### References

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