Taylor–Couette flow
Encyclopedia
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
.
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. Different flow regimes have been categorized over the years including twisted Taylor vortices, wavy outflow boundaries, etc. It has been a well researched and documented flow in fluid dynamics.
) are vortices
formed in rotating Taylor–Couette flow when the Taylor number
(
) of the flow exceeds a critical value 
.
For flow in which
instabilities
in the flow are not present, i.e. perturbations to the flow are damped out by viscous forces, and the flow is steady. But, as the
exceeds 
, axisymmetric instabilities appear. The nature of these instabilities is that of an exchange of stabilities (rather than an overstability), and the result is not turbulence but rather a stable secondary flow pattern that emerges in which large toroidal vortices form in flow, stacked one on top of the other. These are the Taylor vortices. While the fluid mechanics
of the original flow are unsteady when
, the new flow, called Taylor–Couette flow, with the Taylor vortices present, is actually steady until the flow reaches a large Reynolds number, at which point the flow transitions to unsteady "wavy vortex" flow, presumably indicating the presence of non-axisymmetric instabilities.
Rotating Couette flow is characterized geometrically by the two parameters
and
where the subscript "1" refers to the inner cylinder and the subscript "2" refers to the outer cylinder. The idealized mathematical problem is posed by choosing a particular value of
, 
, and 
. As 
and 
from below, the critical Taylor number is 
.
Many of the flow regimes have been observed in multiple experiments and have thus acquired a standard naming convention. For instance:
as well as a number of others. "Wavy" in this sense refers to the progression of changes to the flow in the angular direction. The entire map of flow regimes is incomplete; experiments are sometimes conducted to elucidate a particular region of interest, but gaps in understanding remain. E.g., a potentially distinct regime called "soft turbulence" has been identified.
Taylor-Couette experiments may sometimes include additional system features, such as an imposed axial flow, pulsating flow, etc. designed to better understand certain transitions.
published a paper on the onset of turbulence in rotating fluid. In a Taylor–Couette flow system, they observed that, as the rotation rate inreases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent. Their study helped establish the Ruelle-Takens scenario in turbulence.
This type of experiment studies the motion of a liquid contained between two cylinders. The outer cylinder is kept fixed while the inner cylinder rotates. At low rotation rates the fluid flows uniformly. As the rotation rate increases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, 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
Laminar flow
Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...
is known as circular Couette flow
Couette flow
In fluid dynamics, Couette flow refers to 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...
, after Maurice Marie Alfred Couette who used this experimental device as a means to measure viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
. Sir Geoffrey Ingram Taylor
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM was a British physicist, mathematician and expert on fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this century".-Biography:Taylor was born in St. John's Wood, London...
investigated the stability of the Couette flow
Couette flow
In fluid dynamics, Couette flow refers to 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...
in a ground-breaking paper which has been a cornerstone in the development of hydrodynamic stability
Hydrodynamic stability
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. Instabilities may develop further into turbulence....
theory.
Taylor showed that when the angular velocity of the inner cylinder is increased above a certain threshold, Couette flow
Couette flow
In fluid dynamics, Couette flow refers to 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...
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
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
.
Circular Couette flow has wide applications ranging from desalination to magnetohydrodynamics
Magnetohydrodynamics
Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...
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. Different flow regimes have been categorized over the years including twisted Taylor vortices, wavy outflow boundaries, etc. It has been a well researched and documented flow in fluid dynamics.
Taylor vortex
Taylor vortices (also named after Sir Geoffrey Ingram TaylorGeoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM was a British physicist, mathematician and expert on fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this century".-Biography:Taylor was born in St. John's Wood, London...
) are vortices
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...
formed in rotating Taylor–Couette flow when the Taylor number
Taylor number
In fluid dynamics, the Taylor number is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about an axis, relative to viscous forces. The typical context of the Taylor number is in characterization of the...
(
) of the flow exceeds a critical value .For flow in which
instabilities
Instability
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds...
in the flow are not present, i.e. perturbations to the flow are damped out by viscous forces, and the flow is steady. But, as the
exceeds , axisymmetric instabilities appear. The nature of these instabilities is that of an exchange of stabilities (rather than an overstability), and the result is not turbulence but rather a stable secondary flow pattern that emerges in which large toroidal vortices form in flow, stacked one on top of the other. These are the Taylor vortices. While the fluid mechanicsFluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
of the original flow are unsteady when
, the new flow, called Taylor–Couette flow, with the Taylor vortices present, is actually steady until the flow reaches a large Reynolds number, at which point the flow transitions to unsteady "wavy vortex" flow, presumably indicating the presence of non-axisymmetric instabilities.Rotating Couette flow is characterized geometrically by the two parameters
and
where the subscript "1" refers to the inner cylinder and the subscript "2" refers to the outer cylinder. The idealized mathematical problem is posed by choosing a particular value of
, , and . As and from below, the critical Taylor number is .Flow regimes
One significance of Taylor–Couette flow is due to the changes in flow regimes which eventually lead to turbulence. It is hoped that by studying these systems a more general understanding of transitions to turbulence will emerge.Many of the flow regimes have been observed in multiple experiments and have thus acquired a standard naming convention. For instance:
- TVF - Taylor vortex flow
- WVF - wavy vortex flow
- MWV - modulated wavy vortices
- TTV - turbulent Taylor vortices
- TUR - featureless turbulent flow
as well as a number of others. "Wavy" in this sense refers to the progression of changes to the flow in the angular direction. The entire map of flow regimes is incomplete; experiments are sometimes conducted to elucidate a particular region of interest, but gaps in understanding remain. E.g., a potentially distinct regime called "soft turbulence" has been identified.
Taylor-Couette experiments may sometimes include additional system features, such as an imposed axial flow, pulsating flow, etc. designed to better understand certain transitions.
Gollub-Swinney circular Couette experiment
In 1975, J. P. Gollub and H. L. SwinneyHarry Swinney
Harry L. Swinney is an American physicist noted for his contributions to the field of nonlinear dynamics.- Biography :Swinney graduated from Rhodes College in 1961 with a Bachelor degree and obtained his Ph.D...
published a paper on the onset of turbulence in rotating fluid. In a Taylor–Couette flow system, they observed that, as the rotation rate inreases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent. Their study helped establish the Ruelle-Takens scenario in turbulence.
This type of experiment studies the motion of a liquid contained between two cylinders. The outer cylinder is kept fixed while the inner cylinder rotates. At low rotation rates the fluid flows uniformly. As the rotation rate increases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent.