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

Summary

A formalism for absolute and convective instabilities in parallel shear flows is extended to the three-dimensional case. Assuming that the dispersion relation function is given byD(k, l, ω), wherek andl are wave numbers, andω is a frequency, the analytic criterion is formulated by which a point (k ₀,l ₀,ω ₀) with Imω ₀>0 contributes to the absolute instability if and only if one of the two equivalent conditions is satisfied:

• (i) At least two roots inl of the systemD(k, l, ω)=0,D _k (k, l, ω)=0, originating on opposite sides of the reall-axis, collide on thel-plane for the parameter valuesk ₀,l ₀,ω ₀, asω is brought down toω ₀. Every point on thek-plane, that corresponds to a point on the collision paths on thel-plane, is itself a coalescence point ofk-roots for a fixedl ofD(k, l, ω)=0, that originate on opposite sides of the realk-axis.

• (ii) At least two roots ink of the systemD(k, l, ω)=0,D _l ,(k, l, ω)=0, originating on opposite sides of the realk-axis, collide on thek-plane for the parameter valuesk ₀,l ₀,ω ₀, asω is brought down toω ₀. Every point on thel-plane, that corresponds to a point on the collision paths on thek-plane, is itself a coalescence point ofl-roots for a fixedk ofD(k, l, ω)=0, that originate on opposite sides of the reall-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 thel ₁-plane, given byl ₁,=−βk+αl, of the systemD=0,−βD _k +gaD ₁=0. Thek ₁-components of these trajectories have to pass from above to below the real axis on ak ₁-plane, given byk ₁=αk+βl, asω moves down toω ₀. Hereω 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 theN-factor in thee ^N-method, which is based on the wave packet approach is developed.