Periodic orbits - Revision history

Apwillis: /* Periodic orbits in pipe flow */

2020-01-09T09:35:04Z

Periodic orbits in pipe flow

← Older revision		Revision as of 02:35, 9 January 2020
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	=== Periodic orbits in pipe flow ===		=== Periodic orbits in pipe flow ===
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point in space we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many points, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-~~sheet~~ within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point in space we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many points, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-surface within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	'''The following video shows a selection of periodic orbits in pipe flow''', visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		'''The following video shows a selection of periodic orbits in pipe flow''', visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):
	{{#ev:youtube\|http://youtu.be/X_xKq1yt1UQ}}		{{#ev:youtube\|http://youtu.be/X_xKq1yt1UQ}}

	The setup ~~for the previous video was~~ pipe flow with flow rate measured by Re=DU/nu=2500 (D=diameter, U=mean axial speed, nu=kinematic viscosity). '''The next video shows turbulent flow in the lab frame.''' Regions of flow slower than the mean flow are indicated in blue, called 'streaks'. Rotating 'vortex structures' are indicated in yellow. To simplify analysis a 4-fold rotational symmetry has been imposed. This does not significantly affect statistical properties of the flow, e.g. the turbulent friction factor.		(The setup is pipe flow with flow rate measured by Re=DU/nu=2500; D=diameter, U=mean axial speed, nu=kinematic viscosity.)

			'''The next video shows turbulent flow in the lab frame.''' Regions of flow slower than the mean flow are indicated in blue, called 'streaks'. Rotating 'vortex structures' are indicated in yellow. To simplify analysis a 4-fold rotational symmetry has been imposed. This does not significantly affect statistical properties of the flow, e.g. the turbulent friction factor.
	{{#ev:youtube\|http://youtu.be/hSj0VnmSQro}}		{{#ev:youtube\|http://youtu.be/hSj0VnmSQro}}


	'''The next video shows the same flow viewed in a moving frame''' where it is possible to see recurring patterns, i.e. periodic orbits. The speed at which the frame moves is different for every periodic orbit. To determine this automatically we use the [[Method of slices]].		'''The next video shows the same flow viewed in a moving frame''' where it is possible to see recurring patterns, i.e. periodic orbits. The speed at which the frame moves is different for every periodic orbit. To determine this speed automatically we use the [[Method of slices]]. The periodic orbit itself was found using the [[Newton-Krylov method]].
	{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}		{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}

	Periodic orbits exist that encompass the broad range of observed flow patterns. '''The following video shows an example of a periodic orbit that is representative of a spatially localised patch of turbulence''', often called a turbulent 'puff':		Periodic orbits exist that encompass the broad range of observed flow patterns. '''The following video shows an example of a periodic orbit that is representative of a spatially localised patch of turbulence''', often called a turbulent 'puff':
	{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}		{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}

Apwillis: /* Periodic orbits in pipe flow */

2019-11-04T18:00:47Z

Periodic orbits in pipe flow

← Older revision		Revision as of 11:00, 4 November 2019
Line 14:		Line 14:

	=== Periodic orbits in pipe flow ===		=== Periodic orbits in pipe flow ===
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many ~~positions~~, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point in space we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many points, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	'''The following video shows a selection of periodic orbits in pipe flow''', visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		'''The following video shows a selection of periodic orbits in pipe flow''', visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):

Apwillis at 12:55, 6 November 2018

2018-11-06T12:55:00Z

← Older revision		Revision as of 05:55, 6 November 2018
Line 16:		Line 16:
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		'''The following video shows a selection of periodic orbits in pipe flow''', visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):
	{{#ev:youtube\|http://youtu.be/X_xKq1yt1UQ}}		{{#ev:youtube\|http://youtu.be/X_xKq1yt1UQ}}

	The setup for the previous video was pipe flow with flow rate measured by Re=DU/nu=2500 (D=diameter, U=mean axial speed, nu=kinematic viscosity). The next video shows turbulent flow in the lab frame. Regions of flow slower than the mean flow are indicated in blue, called 'streaks'. Rotating 'vortex structures' are indicated in yellow. To simplify analysis a 4-fold rotational symmetry has been imposed. This does not significantly affect statistical properties of the flow, e.g. the turbulent friction factor.		The setup for the previous video was pipe flow with flow rate measured by Re=DU/nu=2500 (D=diameter, U=mean axial speed, nu=kinematic viscosity). '''The next video shows turbulent flow in the lab frame.''' Regions of flow slower than the mean flow are indicated in blue, called 'streaks'. Rotating 'vortex structures' are indicated in yellow. To simplify analysis a 4-fold rotational symmetry has been imposed. This does not significantly affect statistical properties of the flow, e.g. the turbulent friction factor.
	{{#ev:youtube\|http://youtu.be/hSj0VnmSQro}}		{{#ev:youtube\|http://youtu.be/hSj0VnmSQro}}


	~~When~~ viewed in a moving frame it is possible to see recurring patterns, i.e. periodic orbits ~~(next video)~~. The speed at which the frame moves~~, however,~~ is different for every periodic orbit. To determine this automatically we use the [[Method of slices]].		'''The next video shows the same flow viewed in a moving frame''' where it is possible to see recurring patterns, i.e. periodic orbits. The speed at which the frame moves is different for every periodic orbit. To determine this automatically we use the [[Method of slices]].
	{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}		{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}

	Periodic orbits exist that encompass the broad range of observed flow patterns. The following video shows an example of a periodic orbit that is representative of a spatially localised patch of turbulence, often called a turbulent 'puff':		Periodic orbits exist that encompass the broad range of observed flow patterns. '''The following video shows an example of a periodic orbit that is representative of a spatially localised patch of turbulence''', often called a turbulent 'puff':
	{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}		{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}

Apwillis at 12:50, 6 November 2018

2018-11-06T12:50:14Z

← Older revision		Revision as of 05:50, 6 November 2018
Line 1:		Line 1:

	[[File:DblPendShadowing2.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]		[[File:DblPendShadowing2.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]
	Apparently complex dynamics can originate from ~~just~~ a small set of dynamically important recurring patterns.		'''Apparently complex dynamics can originate from a small set of dynamically important recurring patterns.'''
	The image to the right shows two recurring patterns (green,red) that exist for the 'double pendulum', where a second pendulum is attached to the weight of a first pendulum.		The image to the right shows two recurring patterns (green,red) that exist for the 'double pendulum', where a second pendulum is attached to the weight of a first pendulum.
	(A ~~simple android application and screenshots~~ of the double-pendulum can be found [https://play.google.com/store/apps/details?id=org.openpipeflow.doubledoublependulum HERE];)		('''A video showing periodic orbits of the double-pendulum can be found [https://play.google.com/store/apps/details?id=org.openpipeflow.doubledoublependulum HERE].''')
	As time progresses, the position of the lower pendulum traces out the lines shown, (x,y)=(position of lower pendulum). As the line is a closed loop, it traces out a repeating pattern - a 'periodic orbit'.		As time progresses, the position of the lower pendulum traces out the lines shown, (x,y)=(position of lower pendulum). As the line is a closed loop, it traces out a repeating pattern - a 'periodic orbit'.

Apwillis: /* Periodic orbits in pipe flow */

2018-03-20T13:05:53Z

Periodic orbits in pipe flow

← Older revision		Revision as of 06:05, 20 March 2018
Line 14:		Line 14:

	=== Periodic orbits in pipe flow ===		=== Periodic orbits in pipe flow ===
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic ~~trajectories~~, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic pathways, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):

Apwillis: /* Periodic orbits in pipe flow */

2018-03-20T13:05:19Z

Periodic orbits in pipe flow

← Older revision		Revision as of 06:05, 20 March 2018
Line 14:		Line 14:

	=== Periodic orbits in pipe flow ===		=== Periodic orbits in pipe flow ===
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to ~~the~~ more typical subset of states. To visualise the ~~attracting subset of states in the full space~~, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to a more typical subset of states. To visualise the typical dynamic trajectories, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):

Apwillis: /* Periodic orbits in pipe flow */

2018-03-20T13:02:50Z

Periodic orbits in pipe flow

← Older revision		Revision as of 06:02, 20 March 2018
Line 14:		Line 14:

	=== Periodic orbits in pipe flow ===		=== Periodic orbits in pipe flow ===
	In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for dissipative systems, the dynamics typically lives on a space of much smaller dimension (like a 2D-sheet within 3D space~~), because only realistic states (those on the sheet) occur~~. ~~States initially off~~ the ~~sheet are~~ attracted to ~~it as time progresses~~. To visualise ~~this~~ space, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).		In this case, the 'state' of the system corresponds to the speed and direction of travel of the fluid at every position in the flow. By listing the velocity at every point we form a state-vector, '''a'''=(a1,a2,a3,a4,...). As there are infinitely many positions, the dimension (length) of '''a''' is formally infinite. However, for 'dissipative' systems, the dynamics typically lives on a space of much smaller dimension, like a 2D-sheet within 3D space. This is because the dynamics avoids unrealistic states, and is attracted to the more typical subset of states. To visualise the attracting subset of states in the full space, it is necessary to project the state onto a shorter, more meaningful vector, e.g. (x,y)=(energy,dissipation).

	The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):		The following video shows a selection of periodic orbits in pipe flow, visualised in terms input energy driving the flow (I), dissipation (D) and kinetic energy (E) of the motion (relative to their laminar (non-turbulent) values):

Apwillis at 11:19, 23 February 2017

2017-02-23T11:19:13Z

← Older revision		Revision as of 04:19, 23 February 2017
Line 26:		Line 26:
	{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}		{{#ev:youtube\|http://youtu.be/SzDi_vORuMs}}

	Periodic orbits exist that encompass the broad range of observed flow patterns. The following video shows an example of a periodic orbit that is representative a spatially localised patch of turbulence:		Periodic orbits exist that encompass the broad range of observed flow patterns. The following video shows an example of a periodic orbit that is representative of a spatially localised patch of turbulence, often called a turbulent 'puff':
	{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}		{{#ev:youtube\|http://youtu.be/gDYdD8MBlPU}}

Apwillis at 18:28, 17 January 2017

2017-01-17T18:28:21Z

← Older revision		Revision as of 11:28, 17 January 2017
Line 1:		Line 1:

	[[File:~~DblPendShadowing~~.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]		[[File:DblPendShadowing2.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]
	Apparently complex dynamics can originate from just a small set of dynamically important recurring patterns.		Apparently complex dynamics can originate from just a small set of dynamically important recurring patterns.
	The image to the right shows two recurring patterns (green,red) that exist for the 'double pendulum', where a second pendulum is attached to the weight of a first pendulum.		The image to the right shows two recurring patterns (green,red) that exist for the 'double pendulum', where a second pendulum is attached to the weight of a first pendulum.

Apwillis at 14:32, 13 December 2016

2016-12-13T14:32:16Z

← Older revision		Revision as of 07:32, 13 December 2016
Line 2:		Line 2:
	[[File:DblPendShadowing.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]		[[File:DblPendShadowing.png\|400px\|thumb\|right\|Shadowing of two periodic orbits (red and green) leads to a longer one (blue).]]
	Apparently complex dynamics can originate from just a small set of dynamically important recurring patterns.		Apparently complex dynamics can originate from just a small set of dynamically important recurring patterns.
	The image to the right shows two recurring patterns (green,red) that exist for the double pendulum.		The image to the right shows two recurring patterns (green,red) that exist for the 'double pendulum', where a second pendulum is attached to the weight of a first pendulum.
	(A simple android application and screenshots of the double-pendulum can be found [https://play.google.com/store/apps/details?id=org.openpipeflow.doubledoublependulum HERE];)		(A simple android application and screenshots of the double-pendulum can be found [https://play.google.com/store/apps/details?id=org.openpipeflow.doubledoublependulum HERE];)
	As time progresses, the position of the lower pendulum traces out the lines shown, (x,y)=(position of lower pendulum). As the line is a closed loop, it traces out a repeating pattern - a 'periodic orbit'.		As time progresses, the position of the lower pendulum traces out the lines shown, (x,y)=(position of lower pendulum). As the line is a closed loop, it traces out a repeating pattern - a 'periodic orbit'.