ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence September 29, 2016, Axel Brandenburg

Handout 8: Inflection point instability II

• Experimental studies by Reynolds (1883). Two hypotheses: viscosity acts either as to stabilize or to destabilize. No physical mechanism suggested. Unaware of earlier inviscid studies.

• Helmholtz (1868), Kelvin (1871), and Rayleigh (1880) considered the stability of inviscid incompressible flow of constant density.

• Orr (1907) and Sommerfeld (1908) considered the stability of viscous flows and confirmed that Reynolds’ first hypothesis is valid.

• Fjørtoft (1950) finds a stricter necessary condition for instability. [He was part of a Princeton team that in 1950 performed the first successful numerical weather prediction using the ENIAC electronic computer.]

1 Fjørtoft’s theorem

Consider again Rayleigh’s equation in integral form

∫ (|ψ′|^{2} + k^{2}|ψ|^{2}) dx + ∫ \frac{U′′}{U − c}|ψ|^{2} dx = 0,  (1)

and write c = c_{r} + ic_{i}. Expand the second term with (U − c)^{∗} = (U − c_{r}) − ic_{i}. Instead of considering just the imaginary part of this equation,

c_{i} ∫ \frac{U′′}{|U − c|^{2}} |ψ|^{2} dx = 0,  (2)

which shows that U′′ must change sign at least once in the interval, he considered the real part,

∫ \frac{U′′(U − c_{r})}{|U − c|^{2}}|ψ|^{2} dx = − ∫ (|ψ′|^{2} + k^{2}|ψ|^{2}) dx < 0.  (3)

Given that

∫ \frac{U′′}{|U − c|^{2}} |ψ|^{2} dx = 0 (for instability),  (4)

we also have

(c_{r} − U_{s}) ∫ \frac{U′′}{|U − c|^{2}} |ψ|^{2} dx = 0 (for instability),  (5)

where U_{s} = U(x = x_{s}), and x_{s} is the inflection point where U′′ is zero, i.e., U′′(x = x_{s}) = 0. Adding Equation (5) to Equation (3) yields

∫ \frac{U′′(U − U_{s})}{|U − c|^{2}}|ψ|^{2} dx = − ∫ (|ψ′|^{2} + k^{2}|ψ|^{2}) dx < 0 (for instability).  (6)

This shows that, for instability, not only has U′′ to vanish at least once within the domain, but the product U′′(U − U_{s}) must be negative; see Figure 1 for two examples of which only one obeys U′′(U − U_{s}) < 0.

2 Adjoint problem

Rayleigh’s instability equation,

(U − c) (∂_{x}^{2} − k^{2}) ψ̂ − U′′ψ̂ = 0,  (7)

is not self-adjoint. The adjoint problem is given by

(∂_{x}^{2} − k^{2}) (U − c)ψ̂^{†} − U′′ψ̂^{†} = 0.  (8)

Figure 1: Sketch of two shear flow profiles. Both are Rayleigh unstable, but only one is Fjørtoft’s unstable.

Differentiating through, we obtain first

∂_{x}U′ψ̂^{†} + ∂_{x}(U − c)(ψ̂^{†})′ − k^{2}(U − c)ψ̂^{†} − U′′ψ̂^{†} = 0.  (9)

and then

U′′ψ̂^{†} + U′(ψ̂^{†})′ + U′(ψ̂^{†})′ + (U − c)(ψ̂^{†})′′ − k^{2}(U − c)ψ̂^{†} − U′′ψ̂^{†} = 0.  (10)

or

+2U′(ψ̂^{†})′ + (U − c)(ψ̂^{†})′′ − k^{2}(U − c)ψ̂^{†} = 0.  (11)

Multiplying this by U − c yields

+2(U − c)U′(ψ̂^{†})′ + (U − c)^{2}(ψ̂^{†})′′ − k^{2}(U − c)^{2}ψ̂^{†} = 0,  (12)

but since 2(U − c)U′(ψ̂^{†})′ + (U − c)^{2}(ψ̂^{†})′′ = ∂_{x} [(U − c)^{2}(ψ^{†})′], Equation (12) can also be written as

∂_{x} [(U − c)^{2}∂_{x}ψ^{†}] − k^{2}(U − c)^{2}ψ̂^{†} = 0  (13)

which is manifestly self-adjoint!

References

Fjørtoft, R., “Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex,” Geofys. Publ. Oslo 17, 1-52 (1950).

Helmholtz, H. von, “On discontinuous movements of fluids,” Phil. Mag. 36, 337-346 (1868).

Kelvin, Lord, “Hydrokinetic solutions and observations,” Phil. Mag. 42, 362-377 (1871).

Orr, W. M’F., “The stability or instability of the steady motions of a perfect liquid and a viscous liquid,” Proc. Roy. Irish Acad. A 27, 9-68 (1907).

Rayleigh, Lord, “On the stability, or instability, of certain fluid motions,” Proc. Lond. Math. Soc. 11, 57-70 (1980).

Reynolds, O., “An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels,” Phil. Trans. Roy. Soc. 174, 935-982 (1883).

Sommerfeld, A., “Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen,” Proc. 4th Int. Cong. Math. III, 116-124 (1908).

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