Question

Starting from the linearized, perturbed, stability equations of a normal mode for nearly parallel viscous flows, apply Squire’s transformation and derive the stability equations.

Answer 1

Solution:-

Relation between the two-dimensional and three-dimensional solutions

ω = αc

which results in the following slightly different version of the Orr-Sommerfeld equation

(U − c)(D 2 − k 2 )˜v − U ′′v˜ − 1/ iαRe (D 2 − k 2 ) 2 v˜ = 0.

Squire’s transformation is found by comparing this equation to the two-dimensional Orr-Sommerfeld equation, i.e., the Orr-Sommerfeld equation with β = 0, given as

(U − c)(D 2 − α 2 2D)˜v − U ′′v˜ − 1/ iα2DRe2D (D 2 − α 2 2D) 2 v˜ = 0.

Comparing these two equations, it is evident that they have identical solutions if the following relations hold

α2D = k =

α2DRe2D = αRe

Squire’s theorem

Given ReL as the critical Reynolds number for the onset of linear instability for a given α, β, the Reynolds number Rec below which no exponential instabilities exist for any wave numbers satisfies

Rec ≡ min ReL(α, β) = min αReL(α, 0).

Thus parallel shear flows first become unstable to two-dimensional wavelike perturbations at a value of the Reynolds number that is smaller than any value for which unstable three-dimensional perturbations exist. The proof of this theorem follows directly from Squire’s transformation: If a three-dimensional mode is unstable, a two-dimensional mode is unstable at a lower Reynolds number.