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: d X d t = F ( X , R ) 2 of 27

Geometrical picture The state vector X probably has (infinitely?) many components, but we can consider just two, and draw a diagram. X 2 X 1 Lines then show how states evolve, following ‘trajectories’. Continuity means nearby trajectories are similar. We can use ideas from topology to understand the system. 3 of 27

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 of the system around them. X 2 X 1 4 of 27

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 of the system around them. X 2 X 1 5 of 27

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 of the system around them. X 2 X 1 6 of 27

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. 7 of 27

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. 8 of 27

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. 9 of 27

The problem with linear stability This is only true locally. 10 of 27

The problem with linear stability This is only true locally. 11 of 27

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. X 1 R X 2 X 2 X 1 X 1 12 of 27

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 observed for Re � 1000. Energy Re 5772 This is called ‘subcritical transition to turbulence’. 13 of 27

Kelvin-Helmholtz Instability Re = 500, Pr = 1, Ri b = 0 . 16, 3D 4 2 u Out[70]= - 1.0 - 0.5 0.5 1.0 b - 2 - 4 movie 14 of 27

Kelvin-Helmholtz in the ocean van Haren & Gostiaux (2010) 15 of 27

Richardson number The Richardson number Ri b quantifies (non-dimensionally) the ratio of buoyancy forces to flow shear. The Miles-Howard theorem tells us that our system is stable if Ri b > 1 / 4. ‘Stratification stabilises flows’. 16 of 27

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. 17 of 27

Results ( Re = 4000) 18 of 27

Results ( Re = 4000) 'Hopf' bifurcation 19 of 27

Results ( Re = 4000) 20 of 27

Results ( Re = 4000) 21 of 27

Region 1 22 of 27

Region 2 23 of 27

Region 3 24 of 27

Region 4 25 of 27

Region 5 26 of 27

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. 27 of 27