Couette Flow 4: Taylor Vortices

Couette Flow - Taylor Vortices

Settled Science generally hides away inconvenient science in the dark corners of their academic edifices.

Atmospheric Corotation is one of those “dark corners” of science where mainstream scientists “fear to tread”.
Physics and the Earth sciences seem to [currently] avoid the subject “like the plague”.

Planetary Rotation 1: Atmospheric Corotation

These dark corners of Settled Science are [figuratively] located in a “locked filing cabinet stuck in a disused lavatory with a sign on the door saying ‘Beware of the Leopard’.”

“But Mr Dent, the plans have been available in the local planning office for the last nine months.”

“Oh yes, well as soon as I heard I went straight round to see them, yesterday afternoon. You hadn’t exactly gone out of your way to call attention to them, had you? I mean, like actually telling anybody or anything.”

“But the plans were on display …”

“On display? I eventually had to go down to the cellar to find them.”

“That’s the display department.”

“With a flashlight.”

“Ah, well the lights had probably gone.”

“So had the stairs.”

“But look, you found the notice didn’t you?”

“Yes,” said Arthur, “yes I did. It was on display in the bottom of a locked filing cabinet stuck in a disused lavatory with a sign on the door saying ‘Beware of the Leopard’.”

The Hitchhiker’s Guide to the Galaxy – Douglas Adams – 1978

Couette Flow is [figuratively] locked away “in a disused lavatory with a sign on the door saying” Beware of the Tiger.

In fluid dynamics, Couette flow is the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other.

The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates.

This type of flow is named in honor of Maurice Marie Alfred Couette, a Professor of Physics at the French university of Angers in the late 19th century.

Couette flow is frequently used in undergraduate physics and engineering courses to illustrate shear-driven fluid motion.

Couette Flow

The Beware of the Tiger sign has been [figuratively] placed on the door of this disused lavatory because the “laminar flow” [Wikipedia] between “two parallel plates” where one “is moving relative to the other” can produce turbulent Tiger Stripes.

How the Turbulence Got His Stripes

Turbulent-laminar band formation in plane Couette flow.
These patterns can be observed in the transition to turbulence in shear flows.

If you ignore the Beware of the Tiger sign then Settled Science provides you with many reminders that it’s pointless rummaging about in this particular disused lavatory.

The simplest conceptual configuration finds two infinite, parallel plates separated by a distance h.

Taylor’s idealized model
The configuration shown in the figure cannot actually be realized, as two plates cannot extend infinitely in the flow direction.

This model is incomplete in that it does not account for near-wall effects in finite-width cylinders, although it is a reasonable approximation if the width is large compared to the space between the cylinders.
Generalizations of Taylor’s basic model have also been examined.

Taylor’s solution accounts for the curvature inherent in the cylindrical devices typically used to create Couette flows, but not the finite nature of the width.

A complementary idealization accounts for finiteness, but not curvature.

A mathematical result that accounts for both of these aspects was given only recently by Michael Wendl.

Taylor–Couette Flow has been [figuratively] locked away in another “disused lavatory with a sign on the door saying” Beware of the Zebra.

In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders.

For low angular velocities, measured by the Reynolds number Re, the flow is steady and purely azimuthal.

This basic state is known as circular Couette flow, after Maurice Marie Alfred Couette who used this experimental device as a means to measure viscosity.

Sir Geoffrey Ingram Taylor investigated the stability of the Couette flow in a ground-breaking paper which has been a cornerstone in the development of hydrodynamic stability theory.

Taylor showed that when the angular velocity of the inner cylinder is increased above a certain threshold, Couette flow becomes unstable and a secondary steady state characterized by axisymmetric toroidal vortices, known as Taylor vortex flow, emerges.

Subsequently increasing the angular speed of the cylinder the system undergoes a progression of instabilities which lead to states with greater spatio-temporal complexity, with the next state being called as wavy vortex flow.

If the two cylinders rotate in opposite sense then spiral vortex flow arises.

Beyond a certain Reynolds number there is the onset of turbulence.

Taylor–Couette Flow examples

T. T. Lim
Fluid Mechanics Group – Department of Mechanical Engineering
National University of Singapore

Taylor–Couette Flow schematic

Playing with Taylor Couette

Wikipedia concedes that Taylor–Couette Flow has “wide applications”.

Circular Couette flow has wide applications ranging from desalination to magnetohydrodynamics and also in viscosimetric analysis.

Furthermore, when the liquid is allowed to flow in the annular space of two rotating cylinders along with the application of a pressure gradient then a flow called Taylor–Dean flow arises.

But they play down the diversity of patterns associated with Taylor–Couette Flow.

Taylor–Couette Flow patterns

Taylor-Couette Flow – Christoph Roick
Fakultät für Physik und Astronomie – Universität Heidelberg

Spiral Poiseuille flow

Spiral Poiseuille flow
The effect of applying a constant axial pressure gradient to the Taylor-Couette system, results in a steady spiral flow which is a combination of a rotation due to the azimuthal Couette flow and an axial parabolic profile, due to the pressure gradient.

This flow is termed spiral Poiseuille flow.

The axial through flow, measured by the axial Reynolds number Ra, has a stabilizing effect, i.e. the bifurcation to Taylor vortex flow is shifted to larger rotation rates.

The resulting vortices emerge via a Hopf bifurcation; they are a travelling wave which propagates with the imposed through flow.

When the strength of the through flow is increased, the propagating Taylor vortices are superseded by a spiral barber-pole pattern.

The vortices become non-axisymmetric and not only travel axially in the direction of the imposed through flow, but also rotate, advected by the inner cylinder rotation.

Hence, spiral vortices are simultaneously travelling and rotating waves.

Taylor-Couette Flow – Computational Fluid Dynamics Lab – Marc Avila Research Group

And NASA simply slams the door [labelled Beware of the Zebra] in your face if you enter the disused lavatory in the basement of astrophysics.

Zebra Stripes

NASA is good at concocting catchy and distracting headlines for their press releases.

This week was no exception when they delivered the “zebra stripes” and “zebra patterns” distractions for the gormless and gullible.

The elephant in the room they are trying to wiggle round [aka model] is the phenomena called atmospheric corotation whereby the Earth’s atmosphere corotates with the planet.

The settled science regarding atmospheric corotation is that the surface of the rotating Earth causes the atmosphere to corotate with the planetary surface.

Unfortunately, for the mainstream, the science of viscous fluids dictates that atmospheric corotation must be driven externally.

Inner Van Allan Zebra Stripes

Therefore, it’s probably worth taking a closer look at Earth’s atmosphere.

Gallery | This entry was posted in Astrophysics, Atmospheric Science, Earth, Fluid Mechanics, Science, Vortices. Bookmark the permalink.

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