Kelvin-Helmholtz instability above Richardson number 1 / 4 J P - - PowerPoint PPT Presentation

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Kelvin-Helmholtz instability above Richardson number 1 / 4 J P Parker, C P Caulfield, R R Kerswell September 4, 2019 Miles-Howard theorem The local/gradient Richardson number is defined as / z Ri g = g ( u / z ) 2 . Theorem

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Kelvin-Helmholtz instability above Richardson number 1/4

J P Parker, C P Caulfield, R R Kerswell

September 4, 2019

SLIDE 2

Miles-Howard theorem

The local/gradient Richardson number is defined as Rig = g ρ ∂ρ/∂z (∂u/∂z)2 .

Theorem (Miles-Howard)

For a steady, one-dimensional, Boussinesq, inviscid, stratified shear flow, linear stability is guaranteed if Rig > 1/4 everywhere.

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SLIDE 3

Mixing layer models

∂tu + (U + u) ∂xu + w∂z (U + u) = −∂xp + 1 Re

∂2

xu + ∂2 zu

∂tw + (U + u) ∂xw + w∂zw = −∂zp + 1 Re

∂2

xw + ∂2 z w

+ Ribb,

∂tb + (U + u) ∂xb + w∂z (B + b) = 1 PrRe

∂2

xb + ∂2 zb

∂xu + ∂zw = 0. U = tanh z, B = tanh z U = tanh z, B = z

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SLIDE 4

Bifurcation diagram?

Re = ∞

Ri

1/4

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SLIDE 5

Bifurcation diagram?

Re = ∞: supercritical

Ri

1/4

Amplitude 5 of 15

SLIDE 6

Bifurcation diagram?

Re = ∞: subcritical

Ri

1/4

Amplitude 6 of 15

SLIDE 7

Bifurcation diagram?

Re = ∞: subcritical

Ri

1/4

Amplitude 7 of 15

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State tracking

Formally define F: (u(T), b(T)) = F(u(0), b(0)) Look for steady states F(u, b) = (u, b) Solve this using Newton iteration with GMRES, all built on top of a DNS code.

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SLIDE 9

Holmboe model

Re = 4000 0.2 0.4 0.6 0.8 0.244 0.246 0.248 0.25 X Rib

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SLIDE 10

Drazin model

Re = 4000 0.2 0.4 0.6 0.8 0.23 0.24 0.25 X Rib

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Bifurcation point tracking

At pitchfork/saddle-node bifurcation, attempt to solve F(u, b) = (u, b) J(u, b) · Y = Y Y · A = 1 for X, Y and Rib, where A is some fixed direction. Similar for Hopf bifurcation, with 3 time integrations.

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SLIDE 12

Bifurcation tracking

0.246 0.248 0.25 0.0005 0.001 Rib 1/Re

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SLIDE 13

Pr > 1

0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 1 1.5 2 2.5 3 3.5 Rib Pr

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SLIDE 14

Pr > 1

0.2 0.4 0.6 0.8 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 X Rib

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SLIDE 15

Conclusions

◮ Supercriticality for Pr > 1 ◮ Pr = 1 is a degenerate case, with very small subcriticality ◮ Finite amplitude instability for Pr = 3 is possible

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