The dynamical system of mixing layers J P Parker, C P Caulfield, R R - - PowerPoint PPT Presentation

The dynamical system of mixing layers

J P Parker, C P Caulfield, R R Kerswell

February 6, 2019

Dynamical systems (for fluid dynamics)

Write down a vector X that fully describes the state of the system at a given time. The system probably also depends on one or more parameters R (Re, Ra, Ri...). Then there is some function F that describes the evolution of the system: dX dt = F(X, R)

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Geometrical picture

The state vector X probably has (infinitely?) many components, but we can consider just two, and draw a diagram.

X1 X2

Lines then show how states evolve, following ‘trajectories’. Continuity means nearby trajectories are similar. We can use ideas from topology to understand the system.

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Special trajectories

Certain states have special properties: equilibrium point (steady state), periodic orbit (limit cycle), etc. These can be stable or unstable, and finding them builds up a picture

• f the system around them.

X1 X2

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Special trajectories

Certain states have special properties: equilibrium point (steady state), periodic orbit (limit cycle), etc. These can be stable or unstable, and finding them builds up a picture

• f the system around them.

X1 X2

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Special trajectories

Certain states have special properties: equilibrium point (steady state), periodic orbit (limit cycle), etc. These can be stable or unstable, and finding them builds up a picture

• f the system around them.

X1 X2

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The problem with linear stability

Linear stability analysis for an equilibrium point tells us a series of ‘eigenvalues’. If the eigenvalue is positive, the point is unstable in the corresponding eigenvector.

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The problem with linear stability

Linear stability analysis for an equilibrium point tells us a series of ‘eigenvalues’. If the eigenvalue is negative, the point is stable in the corresponding eigenvector.

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The problem with linear stability

Linear stability analysis for an equilibrium point tells us a series of ‘eigenvalues’. If the eigenvalue is negative, the point is stable in the corresponding eigenvector.

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The problem with linear stability

This is only true locally.

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The problem with linear stability

This is only true locally.

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Bifurcations

We are interested in when the system changes qualitatively as parameters are varied, so-called ‘bifurcation points’. Bifurcation diagrams aim to capture this behaviour.

X1 R

X1 X2 X1 X2

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An example: Plane Poiseuille flow

A classical system in fluid dynamics is ‘plane Poiseuille flow’: flow between two parallel plates, driven by a pressure difference. The simple analytic state is stable for Re < 5772 but turbulence is

• bserved for Re 1000.

Re Energy 5772

This is called ‘subcritical transition to turbulence’.

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Kelvin-Helmholtz Instability

movie Re = 500, Pr = 1, Rib = 0.16, 3D

Out[70]=

• 1.0
• 0.5

0.5 1.0

• 4
• 2

2 4

u b

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Kelvin-Helmholtz in the ocean

van Haren & Gostiaux (2010)

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Richardson number

The Richardson number Rib quantifies (non-dimensionally) the ratio of buoyancy forces to flow shear. The Miles-Howard theorem tells us that our system is stable if Rib > 1/4. ‘Stratification stabilises flows’.

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Methodology

◮ First we try some small perturbations to the base flow at a few

different parameter values, to get a feel for the behaviour.

◮ If any of these appear to be converging to some steady state, we

use Newton’s method (Newton 1669) to find the exact solution.

◮ Once one steady state has been found, this can be reconverged at

different parameter values using branch continuation.

◮ We can then test the stability of the states found, and notice any

bifurcation points.

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Results (Re = 4000)

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Results (Re = 4000)

'Hopf' bifurcation

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Results (Re = 4000)

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Results (Re = 4000)

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Region 1

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Region 2

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Region 3

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Region 4

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Region 5

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Summary

◮ The complex behaviour can be understood as a two-dimensional

system.

◮ Transition is possible above critical Richardson number ◮ Seems likely that transition is possible above Richardson 1/4. ◮ Possible that this leads to full turbulence in 3D flows. ◮ States only go just past 1/4.

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