Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through...
Transcript of Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through...
![Page 1: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/1.jpg)
Bifurcation Phenomena in the Flow through aSudden Expansion in a Pipe
Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2 andAndy Salinger2
1University of Nottingham
2Sandia National Laboratories
LMS Durham SymposiumComputational Linear Algebra for Partial Differential Equations
July 2008
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 2: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/2.jpg)
Acknowledgements
This work is supported by the EPSRC under grants EP/E013724/1and EP/F01340X/1.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 3: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/3.jpg)
Overview
Introduction
Bifurcation in the presence of O(2) symmetry
A posteriori error estimation
Numerical results
Summary and conclusions
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 4: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/4.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 5: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/5.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 6: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/6.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 7: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/7.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 8: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/8.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 9: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/9.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 10: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/10.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 11: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/11.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 12: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/12.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 13: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/13.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 14: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/14.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 15: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/15.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 16: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/16.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 17: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/17.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 18: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/18.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 19: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/19.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 20: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/20.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 21: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/21.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 22: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/22.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 23: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/23.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 24: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/24.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 25: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/25.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 26: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/26.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 27: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/27.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 28: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/28.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 29: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/29.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 30: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/30.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 31: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/31.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 32: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/32.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 33: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/33.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 34: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/34.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 35: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/35.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 36: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/36.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 37: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/37.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 38: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/38.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 39: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/39.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 40: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/40.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 41: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/41.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 42: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/42.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 43: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/43.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 44: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/44.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 45: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/45.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 46: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/46.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 47: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/47.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 48: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/48.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 49: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/49.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 50: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/50.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 51: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/51.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 52: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/52.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 53: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/53.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 54: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/54.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 55: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/55.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 56: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/56.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 57: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/57.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 58: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/58.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 59: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/59.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 60: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/60.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 61: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/61.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 62: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/62.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 63: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/63.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 64: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/64.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 65: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/65.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 66: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/66.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 67: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/67.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 68: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/68.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 69: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/69.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 70: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/70.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 71: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/71.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 72: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/72.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 73: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/73.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 74: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/74.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 75: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/75.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 76: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/76.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 77: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/77.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 78: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/78.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 79: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/79.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 80: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/80.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 81: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/81.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 82: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/82.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 83: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/83.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 84: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/84.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 85: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/85.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 86: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/86.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 87: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/87.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 88: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/88.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 89: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/89.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 90: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/90.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 91: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/91.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 92: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/92.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 93: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/93.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 94: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/94.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 95: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/95.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 96: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/96.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 97: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/97.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 98: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/98.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 99: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/99.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 100: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/100.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 101: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/101.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 102: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/102.jpg)
Flow past a cylinder
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 103: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/103.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 104: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/104.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 105: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/105.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 106: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/106.jpg)
Channel with a Sudden Expansion - Re = 20
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 107: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/107.jpg)
Channel with a Sudden Expansion - Re = 25
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 108: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/108.jpg)
Channel with a Sudden Expansion - Re = 30
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 109: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/109.jpg)
Channel with a Sudden Expansion - Re = 35
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 110: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/110.jpg)
Channel with a Sudden Expansion - Re = 40
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 111: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/111.jpg)
Channel with a Sudden Expansion - Re = 45
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 112: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/112.jpg)
Channel with a Sudden Expansion - Re = 50
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 113: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/113.jpg)
Channel with a Sudden Expansion - Re = 55
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 114: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/114.jpg)
Channel with a Sudden Expansion - Re = 60
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 115: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/115.jpg)
Channel with a Sudden Expansion - Re = 65
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 116: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/116.jpg)
Channel with a Sudden Expansion - Re = 70
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 117: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/117.jpg)
Channel with a Sudden Expansion - Re = 75
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 118: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/118.jpg)
Channel with a Sudden Expansion - Re = 80
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 119: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/119.jpg)
Channel with a Sudden Expansion - Re = 85
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 120: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/120.jpg)
Channel with a Sudden Expansion - Re = 90
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 121: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/121.jpg)
Channel with a Sudden Expansion - Re = 95
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 122: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/122.jpg)
Channel with a Sudden Expansion - Re = 100
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 123: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/123.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 124: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/124.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 125: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/125.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 126: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/126.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 127: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/127.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.
Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 128: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/128.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.
Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 129: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/129.jpg)
Introduction
Bifurcation phenomena in open systems:
Flow past a cylinder in a channel.Jackson JFM 1987; Cliffe and Tavener JFM 2004.
Z2 symmetry-breaking Hopf bifurcation.
Flow through a sudden expansion in a channel.Fearn, Mullin and Cliffe JFM 1990.
Steady, Z2 symmetry-breaking bifurcation.
Flow past a sphere in a pipe.Tavener PoF 1994; Cliffe, Spence and Tavener IJNMF 2000.
Steady, O(2) symmetry-breaking bifurcation.
Flow through a sudden expansion in a pipe.
Previous experimental studies inconclusive.Current project:Mullin and Seddon in Manchester and at the MRRC inCambridge.Cliffe, Hall and Houston in Nottingham; Phipps and Salingerat Sandia.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 130: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/130.jpg)
Introduction
Problem definition and properties:
Navier-Stokes equations.
Parameters: Reynolds number, Re, and expansion ratio, E .
O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 131: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/131.jpg)
Introduction
Problem definition and properties:
Navier-Stokes equations.
Parameters: Reynolds number, Re, and expansion ratio, E .
O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 132: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/132.jpg)
Introduction
Problem definition and properties:
Navier-Stokes equations.
Parameters: Reynolds number, Re, and expansion ratio, E .
O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 133: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/133.jpg)
Introduction
Problem definition and properties:
Navier-Stokes equations.
Parameters: Reynolds number, Re, and expansion ratio, E .
O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 134: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/134.jpg)
Introduction
Problem definition and properties:
Navier-Stokes equations.
Parameters: Reynolds number, Re, and expansion ratio, E .
O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 135: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/135.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 136: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/136.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 137: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/137.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 138: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/138.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 139: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/139.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 140: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/140.jpg)
Bifurcation in the Presence of O(2) Symmetry
Bifurcation with O(2) symmetry.Golubitsky, Stewart and Schaeffer 1988; Healey and Treacy IJNME 1991; Aston SIAM JMA 1991; Cliffe,
Spence and Tavener IJNMF 2000.
Consider a problem of the form
Myt + f (y ,R) = 0, y(t) ∈ H, R ∈ R.
H is a Hilbert space.
f : H× R 7→ H is a non-linear operator.
M : H 7→ H is a linear operator.
The problem has O(2) symmetry.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 141: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/141.jpg)
Bifurcation in the Presence of O(2) Symmetry
Action of O(2) on H
O(2)×H 7→ H,
(γ, y) 7→ ργ(y) ≡ γ · y .
The mapping ργ : H 7→ H is linear.
If γ1, γ2 ∈ O(2) then γ1 · (γ2 · y) = (γ1γ2) · y .
The mapping, ρ, that takes γ to ργ is called a representationof O(2) on H.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 142: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/142.jpg)
Bifurcation in the Presence of O(2) Symmetry
Action of O(2) on H
O(2)×H 7→ H,
(γ, y) 7→ ργ(y) ≡ γ · y .
The mapping ργ : H 7→ H is linear.
If γ1, γ2 ∈ O(2) then γ1 · (γ2 · y) = (γ1γ2) · y .
The mapping, ρ, that takes γ to ργ is called a representationof O(2) on H.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 143: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/143.jpg)
Bifurcation in the Presence of O(2) Symmetry
Action of O(2) on H
O(2)×H 7→ H,
(γ, y) 7→ ργ(y) ≡ γ · y .
The mapping ργ : H 7→ H is linear.
If γ1, γ2 ∈ O(2) then γ1 · (γ2 · y) = (γ1γ2) · y .
The mapping, ρ, that takes γ to ργ is called a representationof O(2) on H.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 144: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/144.jpg)
Bifurcation in the Presence of O(2) Symmetry
Action of O(2) on H
O(2)×H 7→ H,
(γ, y) 7→ ργ(y) ≡ γ · y .
The mapping ργ : H 7→ H is linear.
If γ1, γ2 ∈ O(2) then γ1 · (γ2 · y) = (γ1γ2) · y .
The mapping, ρ, that takes γ to ργ is called a representationof O(2) on H.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 145: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/145.jpg)
Bifurcation in the Presence of O(2) Symmetry
O(2) equivariance
ργM = Mργ , ∀γ ∈ O(2)
ργf (y ,R) = f (ργ(y),R), ∀γ ∈ O(2).
If y is a solution, then ργ(y) is also a solution.
HO(2) = y ∈ H, y = ργ(y),∀γ ∈ O(2).Differentiating gives
ργfy (y ,R) = fy (ργ(y),R)ργ ,
so that if y ∈ HO(2) and A = fy (yO(2),R) then
ργA = Aργ .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 146: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/146.jpg)
Bifurcation in the Presence of O(2) Symmetry
O(2) equivariance
ργM = Mργ , ∀γ ∈ O(2)
ργf (y ,R) = f (ργ(y),R), ∀γ ∈ O(2).
If y is a solution, then ργ(y) is also a solution.
HO(2) = y ∈ H, y = ργ(y),∀γ ∈ O(2).Differentiating gives
ργfy (y ,R) = fy (ργ(y),R)ργ ,
so that if y ∈ HO(2) and A = fy (yO(2),R) then
ργA = Aργ .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 147: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/147.jpg)
Bifurcation in the Presence of O(2) Symmetry
O(2) equivariance
ργM = Mργ , ∀γ ∈ O(2)
ργf (y ,R) = f (ργ(y),R), ∀γ ∈ O(2).
If y is a solution, then ργ(y) is also a solution.
HO(2) = y ∈ H, y = ργ(y),∀γ ∈ O(2).
Differentiating gives
ργfy (y ,R) = fy (ργ(y),R)ργ ,
so that if y ∈ HO(2) and A = fy (yO(2),R) then
ργA = Aργ .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 148: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/148.jpg)
Bifurcation in the Presence of O(2) Symmetry
O(2) equivariance
ργM = Mργ , ∀γ ∈ O(2)
ργf (y ,R) = f (ργ(y),R), ∀γ ∈ O(2).
If y is a solution, then ργ(y) is also a solution.
HO(2) = y ∈ H, y = ργ(y),∀γ ∈ O(2).Differentiating gives
ργfy (y ,R) = fy (ργ(y),R)ργ ,
so that if y ∈ HO(2) and A = fy (yO(2),R) then
ργA = Aργ .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 149: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/149.jpg)
Bifurcation in the Presence of O(2) Symmetry
O(2) equivariance
ργM = Mργ , ∀γ ∈ O(2)
ργf (y ,R) = f (ργ(y),R), ∀γ ∈ O(2).
If y is a solution, then ργ(y) is also a solution.
HO(2) = y ∈ H, y = ργ(y),∀γ ∈ O(2).Differentiating gives
ργfy (y ,R) = fy (ργ(y),R)ργ ,
so that if y ∈ HO(2) and A = fy (yO(2),R) then
ργA = Aργ .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 150: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/150.jpg)
Bifurcation in the Presence of O(2) Symmetry
Standard decomposition
H =∞∑
m=0
⊕Vm, Vm ⊥ Vl , m 6= l .
where the Vm are O(2) invariant.
Eigenvalue problem
λMφ = Aφ, φ ∈ H,
decouples into the infinite set of simpler eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 151: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/151.jpg)
Bifurcation in the Presence of O(2) Symmetry
Standard decomposition
H =∞∑
m=0
⊕Vm, Vm ⊥ Vl , m 6= l .
where the Vm are O(2) invariant.
Eigenvalue problem
λMφ = Aφ, φ ∈ H,
decouples into the infinite set of simpler eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 152: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/152.jpg)
Bifurcation in the Presence of O(2) Symmetry
Standard decomposition
H =∞∑
m=0
⊕Vm, Vm ⊥ Vl , m 6= l .
where the Vm are O(2) invariant.
Eigenvalue problem
λMφ = Aφ, φ ∈ H,
decouples into the infinite set of simpler eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 153: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/153.jpg)
Bifurcation in the Presence of O(2) Symmetry
Navier-Stokes in cylindrical coordinates
H = W 1,2(Ω)3 × L2(Ω),
y =
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
.
Action of O(2) on H
Rα
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
=
ur (r , θ + α, z)uθ(r , θ + α, z)uz(r , θ + α, z)p(r , θ + α, z)
,
S
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
=
ur (r ,−θ, z)−uθ(r ,−θ, z)uz(r ,−θ, z)p(r ,−θ, z)
.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 154: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/154.jpg)
Bifurcation in the Presence of O(2) Symmetry
Navier-Stokes in cylindrical coordinates
H = W 1,2(Ω)3 × L2(Ω),
y =
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
.
Action of O(2) on H
Rα
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
=
ur (r , θ + α, z)uθ(r , θ + α, z)uz(r , θ + α, z)p(r , θ + α, z)
,
S
ur (r , θ, z)uθ(r , θ, z)uz(r , θ, z)p(r , θ, z)
=
ur (r ,−θ, z)−uθ(r ,−θ, z)uz(r ,−θ, z)p(r ,−θ, z)
.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 155: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/155.jpg)
Bifurcation in the Presence of O(2) Symmetry
Subspaces Vm
Vm = Span
umr (r , z) cos(mθ)uθ(r , z) sin(mθ)uz(r , z) cos(mθ)p(r , z) cos(mθ)
,
umr (r , z) sin(mθ)
uθ(r , z) cos(mθ)uz(r , z) sin(mθ)p(r , z) sin(mθ)
.
Note that the eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
only have to be discretised in (r , z).
Can study stability to three dimensional disturbances using asequence of two dimensional problems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 156: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/156.jpg)
Bifurcation in the Presence of O(2) Symmetry
Subspaces Vm
Vm = Span
umr (r , z) cos(mθ)uθ(r , z) sin(mθ)uz(r , z) cos(mθ)p(r , z) cos(mθ)
,
umr (r , z) sin(mθ)
uθ(r , z) cos(mθ)uz(r , z) sin(mθ)p(r , z) sin(mθ)
.
Note that the eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
only have to be discretised in (r , z).
Can study stability to three dimensional disturbances using asequence of two dimensional problems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 157: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/157.jpg)
Bifurcation in the Presence of O(2) Symmetry
Subspaces Vm
Vm = Span
umr (r , z) cos(mθ)uθ(r , z) sin(mθ)uz(r , z) cos(mθ)p(r , z) cos(mθ)
,
umr (r , z) sin(mθ)
uθ(r , z) cos(mθ)uz(r , z) sin(mθ)p(r , z) sin(mθ)
.
Note that the eigenvalue problems
λMmφ = Amφ, φ ∈ Vm, m = 0, 1, 2, . . .
only have to be discretised in (r , z).
Can study stability to three dimensional disturbances using asequence of two dimensional problems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 158: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/158.jpg)
Discretisation
Discontinuous Galerkin.
Quadrilateral elements with tensor product polynomials bases.
Need to find most dangerous eigenvalue.
Solve eigenvalue problem using modified Cayley transform andARPACK (cf Alastair Spence’s lectures).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 159: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/159.jpg)
Discretisation
Discontinuous Galerkin.
Quadrilateral elements with tensor product polynomials bases.
Need to find most dangerous eigenvalue.
Solve eigenvalue problem using modified Cayley transform andARPACK (cf Alastair Spence’s lectures).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 160: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/160.jpg)
Discretisation
Discontinuous Galerkin.
Quadrilateral elements with tensor product polynomials bases.
Need to find most dangerous eigenvalue.
Solve eigenvalue problem using modified Cayley transform andARPACK (cf Alastair Spence’s lectures).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 161: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/161.jpg)
Discretisation
Discontinuous Galerkin.
Quadrilateral elements with tensor product polynomials bases.
Need to find most dangerous eigenvalue.
Solve eigenvalue problem using modified Cayley transform andARPACK (cf Alastair Spence’s lectures).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 162: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/162.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 163: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/163.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 164: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/164.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 165: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/165.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 166: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/166.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 167: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/167.jpg)
A posteriori error estimation
Simple illustration of basic ideas:Bangerth & Rannacher 2003
Suppose A,Ah ∈ Rn×n,b,bh ∈ Rn and x, xh ∈ Rn satisfy
Ax = b, Ahxh = bh.
Suppose Ah → A as h → 0.
Approximation error e = x− xh.
Truncation error τ = Ahx− bh.
Residual ρ = b− Axh.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 168: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/168.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 169: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/169.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh
= Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 170: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/170.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh
= τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 171: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/171.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 172: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/172.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 173: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/173.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 174: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/174.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh
= b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 175: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/175.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh
= ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 176: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/176.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 177: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/177.jpg)
A posteriori error estimation
A priori error analysis:
Ahe = Ahx− Ahxh = Ahx− bh = τ.
This leads to‖e‖ ≤ ‖A−1
h ‖‖τ‖.
A posteriori error analysis:
Ae = Ax− Axh = b− Axh = ρ.
This leads to‖e‖ ≤ ‖A−1‖‖ρ‖.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 178: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/178.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 179: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/179.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 180: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/180.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh)
= (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 181: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/181.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 182: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/182.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 183: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/183.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 184: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/184.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j)
= (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 185: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/185.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz)
= (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 186: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/186.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z)
= (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 187: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/187.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 188: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/188.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi |
≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 189: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/189.jpg)
A posteriori error estimation
Goal-oriented a priori error analysis:
Estimate and control the error in some linear functional of thesolution, J.
J(e) = J(x)− J(xh) = (e, j).
Consider the solution, z, of the dual problem
ATz = j.
The error can be written
J(e) = (e, j) = (e,ATz) = (Ae, z) = (ρ, z).
This gives the weighted a posteriori error estimate
|J(e)| = |n∑
i=1
ρizi | ≤n∑
i=1
|ρi ||zi |.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 190: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/190.jpg)
A posteriori error estimation
Apply to extended systems for eigenvalue problem or forbifurcation point.
Take the linear functional to be the eigenvalue or thebifurcation parameter.
Solve the dual problems using higher order interpolation.
Details (which are complicated) are omitted:
λ− λh =∑κ∈Th
ηκ ≡∑κ∈Th
η0κ + ηm
k ,
Iterative solves of bordered systems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 191: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/191.jpg)
A posteriori error estimation
Apply to extended systems for eigenvalue problem or forbifurcation point.
Take the linear functional to be the eigenvalue or thebifurcation parameter.
Solve the dual problems using higher order interpolation.
Details (which are complicated) are omitted:
λ− λh =∑κ∈Th
ηκ ≡∑κ∈Th
η0κ + ηm
k ,
Iterative solves of bordered systems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 192: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/192.jpg)
A posteriori error estimation
Apply to extended systems for eigenvalue problem or forbifurcation point.
Take the linear functional to be the eigenvalue or thebifurcation parameter.
Solve the dual problems using higher order interpolation.
Details (which are complicated) are omitted:
λ− λh =∑κ∈Th
ηκ ≡∑κ∈Th
η0κ + ηm
k ,
Iterative solves of bordered systems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 193: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/193.jpg)
A posteriori error estimation
Apply to extended systems for eigenvalue problem or forbifurcation point.
Take the linear functional to be the eigenvalue or thebifurcation parameter.
Solve the dual problems using higher order interpolation.
Details (which are complicated) are omitted:
λ− λh =∑κ∈Th
ηκ ≡∑κ∈Th
η0κ + ηm
k ,
Iterative solves of bordered systems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 194: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/194.jpg)
A posteriori error estimation
Apply to extended systems for eigenvalue problem or forbifurcation point.
Take the linear functional to be the eigenvalue or thebifurcation parameter.
Solve the dual problems using higher order interpolation.
Details (which are complicated) are omitted:
λ− λh =∑κ∈Th
ηκ ≡∑κ∈Th
η0κ + ηm
k ,
Iterative solves of bordered systems.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 195: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/195.jpg)
Solution of bordered systems
Need to solve (A BCT D
) (xy
)=
(fg
),
where
A ∈ Rn×n,
B,C ∈ Rn×m,
D ∈ Rm×m.
Rewrite as (D CT
B A
) (yx
)=
(gf
).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 196: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/196.jpg)
Solution of bordered systems
Need to solve (A BCT D
) (xy
)=
(fg
),
where
A ∈ Rn×n,
B,C ∈ Rn×m,
D ∈ Rm×m.
Rewrite as (D CT
B A
) (yx
)=
(gf
).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 197: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/197.jpg)
Solution of bordered systems
Use Householder QR algorithm(DT
C
)= Q
(R0
),
whereQ = I + UTUT,
and
Q, I ∈ R(n+m)×(n+m), Q− orthogonal, I− identity
R,T ∈ Rm×m, upper triangular
0 ∈ Rn×m,
U ∈ R(n+m)×m.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 198: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/198.jpg)
Solution of bordered systems
It follows that (D CT
B A
)Q =
(RT 0
B A
),
where
B = B + (BU1 + AU2)TUT1 ,
A = A + (BU1 + AU2)TUT2 ,
and
U =
(U1
U2
), U1 ∈ Rm×m, U2 ∈ Rn×m.
Note: A is a rank m modification of A that is non-singular.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 199: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/199.jpg)
Solution of bordered systems
It follows that (D CT
B A
)Q =
(RT 0
B A
),
where
B = B + (BU1 + AU2)TUT1 ,
A = A + (BU1 + AU2)TUT2 ,
and
U =
(U1
U2
), U1 ∈ Rm×m, U2 ∈ Rn×m.
Note: A is a rank m modification of A that is non-singular.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 200: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/200.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 201: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/201.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 202: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/202.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 203: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/203.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 204: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/204.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 205: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/205.jpg)
Solution of bordered systems
Solve the block triangular system(RT 0
B A
) (vu
)=
(gf
).
Solve small triangular system for v.
Use (P)GMRES forAu = f − Bv.
Use the preconditioner for A to precondition A.
Finally (yx
)= Q
(vu
).
Large scale, distributed memory parallel implementation(LOCA, Trilinos).
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 206: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/206.jpg)
Sudden Expansion in a Channel: Error Effectivities
r : R = 3 : 1Re = 35Eigenvalue = 0.00613553131999
Mesh No No. Eles Eig. Dof Error
∣∣∣∑κ∈Thηκ
∣∣∣Error
∣∣∣∑κ∈Thηm
κ
∣∣∣Error
1 760 16720 6.027E-05 1.92 0.14
2 1387 30514 1.540E-05 2.47 0.96
3 2479 54538 9.795E-06 1.98 1.16
4 4387 96514 6.327E-06 1.58 0.98
5 7645 168190 3.845E-06 1.33 0.80
6 13243 291346 2.231E-06 1.16 0.67
7 22585 496870 1.281E-06 1.00 0.56
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 207: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/207.jpg)
Sudden Expansion in a Channel: Mesh under Refinement
Mesh after 5 refinement steps
Contour plot of zmx
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 208: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/208.jpg)
Sudden Expansion in a Channel: Mesh Detail underRefinement
Mesh detail near expansion Contour plot of z0y near expansion
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 209: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/209.jpg)
Sudden Expansion in a Channel: Error Convergence
105
106
10−6
10−5
Uniform RefinementAdaptive Refinement
Degrees of Freedom
|λ−
λh|
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 210: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/210.jpg)
Cylindrical Blockage in a Channel: Error Effectivities
r : R = 1 : 2Re = 100Eigenvalue = 0.114789963956350 + 2.116719676204527i
Mesh No No. Eles Eig. Dof Error
∣∣∣∑κ∈Thηκ
∣∣∣Error
∣∣∣∑κ∈Thηm
κ
∣∣∣Error
1 816 17952 8.966E-02 1.08 4.51E-02
2 1443 31746 2.229E-03 1.54 0.55
3 2577 56694 1.455E-04 1.31 0.68
4 4590 100980 4.089E-05 0.980 0.53
5 8190 180180 1.033E-05 1.01 0.81
6 14400 316800 3.870E-06 0.946 0.51
7 24843 546546 1.060E-06 1.00 0.97
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 211: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/211.jpg)
Cylindrical Blockage in a Channel: Mesh under Refinement
Full Mesh
Mesh Detail near Blockage Contour plot of z0y near blockage
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 212: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/212.jpg)
Cylindrical Blockage in a Channel: Error Convergence
105
106
10−6
10−5
10−4
10−3
10−2
10−1
Uniform RefinementAdaptive Refinement
Degrees of Freedom
|λ−
λh|
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 213: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/213.jpg)
Spherical Blockage in a Pipe: Error Effectivities
r : R = 1 : 2Re = 350Eigenvalue = 0.015358133759879
Mesh No No. Eles Eig. Dof Error|∑
κ∈Thηκ|
Error
|∑
κ∈Thηm
κ |Error
1 1016 31496 1.384E-01 1.68 1.14E-02
2 1793 55583 2.552E-03 2.01 2.70
3 3158 97898 4.877E-04 0.94 0.67
4 5624 174344 2.467E-05 1.02 1.14
5 10301 319331 2.111E-06 1.08 2.69
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 214: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/214.jpg)
Spherical Blockage in a Pipe: Mesh under Refinement
Full Mesh
Mesh Detail near Blockage Contour plot of z0r near blockage
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 215: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/215.jpg)
Spherical Blockage in a Pipe: Error Convergence
105
10−6
10−5
10−4
10−3
10−2
10−1
Uniform RefinementAdaptive Refinement
Degrees of Freedom
|λ−
λh|
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 216: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/216.jpg)
Pipe with a Sudden Expansion: Experimental Results
Tom Mullin and James Seddon, University of Manchester.
New MRI flow visualisation techniques - MRRC in Cambridge.
Preliminary experiments indicate presence of steadybifurcation at Re ≈ 1100.
Onset of time dependence at Re ≈ 1500.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 217: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/217.jpg)
Pipe with a Sudden Expansion: Experimental Results
Tom Mullin and James Seddon, University of Manchester.
New MRI flow visualisation techniques - MRRC in Cambridge.
Preliminary experiments indicate presence of steadybifurcation at Re ≈ 1100.
Onset of time dependence at Re ≈ 1500.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 218: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/218.jpg)
Pipe with a Sudden Expansion: Experimental Results
Tom Mullin and James Seddon, University of Manchester.
New MRI flow visualisation techniques - MRRC in Cambridge.
Preliminary experiments indicate presence of steadybifurcation at Re ≈ 1100.
Onset of time dependence at Re ≈ 1500.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 219: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/219.jpg)
Pipe with a Sudden Expansion: Experimental Results
Tom Mullin and James Seddon, University of Manchester.
New MRI flow visualisation techniques - MRRC in Cambridge.
Preliminary experiments indicate presence of steadybifurcation at Re ≈ 1100.
Onset of time dependence at Re ≈ 1500.
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 220: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/220.jpg)
Pipe with a Sudden Expansion: Eigenvalues with Re
600 700 800 900 1000 1100 1200 1300 14000
0.002
0.004
0.006
0.008
0.01
0.012
m = 0m = 1m = 2m = 3
Re
Sm
alles
tE
igen
valu
e
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 221: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/221.jpg)
Pipe with a Sudden Expansion: Eigenvalue Errors
Re = 1300
Mesh No. No. Eles Eig. Dofs Eigenvalue |∑
κ∈Thηκ|
1 20000 420000 0.167241E-02 1.741E-06
2 34565 725865 0.167194E-02 1.914E-06
3 65909 1384089 0.167218E-02 9.771E-07
4 111956 2351076 0.167243E-02 5.765E-07
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 222: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/222.jpg)
Pipe with a Sudden Expansion: Eigenvalues with Re
0 1000 2000 3000 4000 5000 6000−2
0
2
4
6
8
10
12
14
16
Reynolds Number
Eig
enva
lue
m=
1
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 223: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/223.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 224: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/224.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 225: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/225.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 226: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/226.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 227: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/227.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 228: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/228.jpg)
Summary and Conclusions
Successfully applied DG and goal-oriented a posteriori errorestimation to stability analysis of incompressible Navier-Stokesequations.
There is a steady, supercritical, O(2)-symmetry-breakingbifurcation at Reynolds number approximately 5000.
Sadly, this has nothing to do with what is seen in theexperiments!
Next steps:
Apply goal-oriented a posteriori error estimation directly tocritical Reynolds number.
Investigate effect of perturbations that destroy the O(2)symmetry.
Conclusion:
In fluid mechanics we still need theory, computation andexperiment!
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 229: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/229.jpg)
Channel with a Sudden Expansion - Re = 35
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 230: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/230.jpg)
Channel with a Sudden Expansion - Re = 40
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 231: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/231.jpg)
Channel with a Sudden Expansion - Re = 45
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 232: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/232.jpg)
Channel with a Sudden Expansion - Re = 50
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 233: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/233.jpg)
Channel with a Sudden Expansion - Re = 55
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 234: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/234.jpg)
Channel with a Sudden Expansion - Re = 60
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 235: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/235.jpg)
Channel with a Sudden Expansion - Re = 65
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 236: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/236.jpg)
Channel with a Sudden Expansion - Re = 70
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 237: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/237.jpg)
Channel with a Sudden Expansion - Re = 75
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 238: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/238.jpg)
Channel with a Sudden Expansion - Re = 80
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 239: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/239.jpg)
Channel with a Sudden Expansion - Re = 85
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 240: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/240.jpg)
Channel with a Sudden Expansion - Re = 90
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 241: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/241.jpg)
Channel with a Sudden Expansion - Re = 95
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion
![Page 242: Bifurcation Phenomena in the Flow through a Sudden ... · Bifurcation Phenomena in the Flow through a Sudden Expansion in a Pipe Andrew Cliffe1, Ed Hall1, Paul Houston1, Eric Phipps2](https://reader034.fdocuments.in/reader034/viewer/2022042104/5e81be6f9a6ee3482f2863c9/html5/thumbnails/242.jpg)
Channel with a Sudden Expansion - Re = 100
Andrew Cliffe Bifurcation Phenomena in a Pipe Expansion