Taylor-Couette flow

Post-publication activity

Richard Lueptow, Northwestern University, Evanston, IL

Dr. Richard Lueptow accepted the invitation on 22 January 2009 (self-imposed deadline: 22 August 2009).

Taylor-Couette flow is the name of a fluid flow and the related instability that occurs in the annulus between differentially rotating concentric cylinders, most often with the inner cylinder rotating and the outer cylinder fixed, when the rotation rate exceeds a critical value. The stable base flow was used in the 19th century to test the fundamental Newtonian stress assumption in the Navier-Stokes equations. It has long been used as a means to measure fluid viscosity. More importantly, the flow instabilities that arise in Taylor-Couette flow (toroidal Taylor vortices stacked in the annulus) and the related theoretical framework to describe these instabilities have provided valuable insight into the commonly used no-slip boundary condition, linear stability analysis, low dimension bifurcation phenomena, chaotic advection, absolute and convective instabilities, and a host of other fundamental physical phenomenon and analytic methods. The flow is frequently studied because it is easy to produce in small closed systems, demonstrates a fundamental fluid flow phenomenon that can be mathematically predicted from basic principles, and is simple and beautiful to observe.

History

The geometric simplicity of the flow of a fluid between differentially rotating concentric cylinders has attracted the interest of scientists for centuries. Sir Isaac Newton used it to describe the circular motion of fluids in his Principia in 1687 (Newton 1946). The pioneering theoretical fluid dynamicist George G. Stokes likewise considered this simple flow in 1848 noting the difficulty in the boundary conditions at the walls of the cylinder, now taken for granted as the no-slip boundary condition. He suggested that "eddies would be produced," if the inner cylinder "were made to revolve too fast" (Stokes 1880), a remarkable insight many years before vortices were visualized.

The development of the Navier-Stokes equations for viscous fluid flow naturally brought about debate on how to best measure fluid viscosity. Henry R. A. Mallock and M. Maurice Couette both independently sought to accomplish this using two differentially rotating concentric cylinders, now known as a Taylor-Couette cell (Mallock 1888, Couette 1890). Couette rotated the outer cylinder keeping the inner cylinder fixed, which is the basis for the modern viscometer, thus avoiding the vortical structure and obtaining an accurate measurement of the viscosity of various fluids. Mallock performed similar experiments to Couette, but also rotated the inner cylinder keeping the outer cylinder fixed. He found anomalous results in this case because Taylor vortices occurred. In fact, Mallock’s experiment prompted Lord Kelvin to write a letter to Lord Rayleigh in 1895 bringing the instability to his attention (Donnelly 1991). While Rayleigh’s eventual analysis in 1916 explained the physical origin of the vortical structure, it was not until 1923, that G. I. Taylor was able to relate theory and experiment for stability in cylindrical Couette flow (Taylor 1923). His investigation was a key development in the modern study of fluid mechanics for three reasons (Donnelly 1991):

It was taken by many as convincing proof of the no-slip boundary condition wherein the velocity of a particle in contact with a wall moves at the same velocity as the wall. Although this concept has become a fundamental tenet for the study of fluid flow, it was questioned until Taylor used it with such success in his analysis of the stability of Taylor-Couette flow.
It offered convincing proof that the Navier-Stokes equations indeed accurately describe the flow of a Newtonian fluid, not just at the base flow level, but at a level that permitted the analysis of secondary flows and instabilities.
It was the first successful application of linear stability analysis that accurately predicted experimental results, namely the transition from stable flow to vortical Taylor-Couette flow.

Fundamentals of Taylor-Couette Flow

In its simplest form, Taylor-Couette flow arises from the shear flow between a rotating inner cylinder and a concentric, fixed outer cylinder. The stable flow for this geometry is known as cylindrical Couette flow (or sometimes circular Couette flow or rotating Couette flow). As with all Couette-type flows, the flow is driven by the motion of one wall bounding a viscous liquid. Applying the Navier-Stokes equation for an incompressible Newtonian fluid, the exact solution for infinitely long cylinders is of the form [in cylindrical coordinates (\(r\ ,\) \(\theta\ ,\) \(z\))]:

\[ U=0,\quad V=Ar+\frac{B}{r},\quad W=0,\quad \frac{\partial P}{\partial r}=\rho \frac {V^2}{r} \]

where \(U\ ,\) \(V\ ,\) and \(W\) are the radial, azimuthal, and axial components of velocity, \(P\) is the pressure, and \(\rho\) is the fluid density. \(A\) and \(B\) depend on the radius ratio, \(\eta=r_i/r_o\) of the inner cylinder radius \(r_i\) and the outer cylinder radius \(r_o,\) and the rotational speed of the inner cylinder, \(\Omega_i\ ,\) as

\[ A=-\Omega_i\frac{\eta^2}{1-\eta^2}, \quad B=\Omega_i\frac{r_i^2}{1-\eta^2} \]

Cylindrical Couette flow becomes unstable as the rotational speed of the inner cylinder increases resulting in pairs of counter-rotating, axisymmetric, toroidal vortices that fill the annulus superimposed on the Couette flow ( Figure 1). Each pair of vortices has a wavelength of approximately \(2d\ ,\) where \(d = r_o-r_i\) is the gap between the cylinders. The vortices are readily visualized by adding small flakes to the fluid that align with the flow ( Figure 2). As a consequence of the vortices, high-speed fluid near the rotating inner cylinder is carried outward in the outflow regions between vortices, while low-speed fluid near the fixed outer cylinder is carried inward in the inflow regions between vortices, redistributing angular momentum of the fluid in the annulus. The axial and radial velocities related to the Taylor vortices are relatively small, typically only a few percent of the surface speed of the inner cylinder (Wereley and Lueptow 1998).

The origin of the vortical flow is a centrifugal instability. Stable cylindrical Couette flow is geostrophic. That is, the centrifugal force due to the azimuthal velocity (or equivalently, the inertia related to the centripetal acceleration) is balanced by the radial pressure gradient force set up due to the azimuthal velocity. However, if a fluid particle is perturbed (moved slightly) outward from its initial radius, it reaches a region where the local restoring force due to the pressure gradient is slightly less than the outward inertia of the particle, which is based on the particle’s initial position because its angular momentum is conserved. As a result, a fluid particle perturbed outward will continue outward. Likewise, a fluid particle moved slightly inward from its initial radius will continue inward as the local restoring force due to the pressure gradient is smaller than the inward inertia of the particle. It does not matter if the initial perturbation is inward or outward, since mass conservation guarantees a return flow, which is in the form of a toroidal vortex of Taylor-Couette flow. At low rotational speeds, the instability can be suppressed by viscosity, which damps out the perturbations. The instability comes about only when the pressure gradient force decreases (because of the decreasing azimuthal velocity) with increasing radius, as is the case for the inner cylinder rotating with the outer cylinder fixed. When only the outer cylinder is rotating, the pressure gradient force increases with increasing radius, and the flow remains stable.

Theoretical Background

Lord Rayleigh first put forth the inviscid (no viscosity) approach to the instability based on an imbalance of the centrifugal force and pressure gradient force (Rayleigh 1916). In considering meteorological problems such as cyclones having a fluid angular velocity \(\Omega(r)\ ,\) he argued that if the value for \((r^2\Omega)^2\) decreases in the radial direction, as it does for an inner rotating cylinder and a fixed outer cylinder, the flow should be unstable. While the Rayleigh stability criterion describes the underlying physics of the instability, it is not strictly correct. It predicts that regardless of the speed of the inner cylinder, as long as the inner cylinder rotates within a stationary outer cylinder, the flow should be unstable. This is not the case, since viscosity damps the perturbations for low rotational speeds, preventing the vortices from forming.

G. I. Taylor first showed how viscosity stabilizes the flow at low rotational speeds using linear stability analysis (Taylor 1923). The analysis is based on small perturbations of the velocity and pressure fields, expressed as normal modes of the form:

\[u=u(r) \cos(kz)e^{qt}, \quad v=V+v(r)\cos(kz)e^{qt}, \quad w=w(r)\sin(kz)e^{qt}, \quad p=P+p(r)\cos(kz)e^{qt}\]

These expressions include the base flow (noting \(U=W=0\)) and a perturbation including sinusoidal variation of the disturbance in the \(z\)-direction with axial wavenumber \(k\ ,\) a growth rate or amplification factor \(q\) for the disturbance, and amplitudes of the disturbance [\(u(r), v(r), w(r),\) and \(p(r)\)], which are dependent on the radial position. The wavenumber describes the axial periodicity of the perturbation. Using \(\sin(kz)\) for \(w\) and \(\cos(kz)\) for the other perturbations comes about due to the phase relationship between the velocity components for the vortex structure—\(w\) is zero where the other perturbations are extrema, and vice versa.

Substituting these expressions into the Navier-Stokes equations followed by linearizing the equations (discarding higher order terms) results in a set of ordinary differential equations. These equations can be transformed into an eigenvalue problem for which the amplification factor is set to zero, corresponding to the onset of the instability. The solution yields the critical wavenumber, \(k_{crit}\ ,\) and the critical Taylor number, \(T_{crit}\ ,\) a dimensionless number above which the instability occurs. Below the critical Taylor number the flow is stable with no vortical structure; above it is unstable with toroidal vortices shown in Figure 1.

There are various forms of the Taylor number, though all represent the ratio of centrifugal (or inertial) forces to viscous forces. Above the critical Taylor number, centrifugal forces exceed viscous forces, and the flow becomes unstable. One form for the Taylor number when the inner cylinder rotating within a fixed outer cylinder is

\[T=4 Re^2 \left[\frac{1-\eta}{1+\eta}\right]\]

Table 1: Critical Reynolds number for transition to vortical flow (Recktenwald et al. 1993).
\(\eta\ !\) ! \(Re_{crit}\ !\) ! \(k_{crit}\)
0.975	260.9	3.13
0.90	131.6	3.13
0.80	94.7	3.13
0.70	79.5	3.14
0.60	71.7	3.15
0.50	68.2	3.16

where \(Re=\Omega_ir_id/\nu\) is a Reynolds number based on the surface velocity of the inner cylinder as the velocity scale and the gap width as the length scale, with \(\nu\) being the kinematic viscosity. The critical Reynolds number, \(Re_{crit}\ ,\) corresponding to the critical Taylor number and the associated critical wavenumber, \(k_{crit}\ ,\) for the onset of vortices depends on the radius ratio as indicated in Table 1. The critical wavenumber defines the axial spacing of the vortices, or wavelength, \(\lambda=2\pi/k_{crit}\ .\) Thus, since \(k_{crit}=3.13/d\) for \(\eta=0.9\ ,\) \(\lambda\approx 2d\ ,\) indicating that a counter-rotating pair of vortices (one wavelength) has an axial wavelength that is twice the radial gap width. Thus, each vortex tends to be circular (as opposed to elliptical), filling a region that is \(d \times d\ ,\) consistent with experiments.

Both Rayleigh’s and Taylor’s analyses can be extended to both cylinders rotating, either in the same direction (co-rotating) or in opposite directions (counter-rotating). Rayleigh’s inviscid analysis indicates that the flow has potential to be unstable when \((r_i^2\Omega_i)^2 > (r_o^2\Omega_o)^2\ ,\) where \(\Omega_o\) is the rotation rate of the outer cylinder. Extending Taylor’s analysis to both cylinders rotating yields another Taylor number, one of the forms of which is

\[T=4 Re^2 \left[1-\frac{\mu}{\eta^2}\right]\left[\frac{1-\eta}{1+\eta}\right]\]

where \(\mu=\Omega_o/\Omega_i\ ,\) is the ratio of the rotation rates of the outer and inner cylinders. For \(\mu<0\ ,\) corresponding to the cylinders rotating in opposite directions, only the region near the inner cylinder is unstable, since the azimuthal velocity profile changes sign at some point in the gap between the two cylinders due to the opposite rotation of the cylinders.

Higher Order Instabilities

Figure 3: Schematic of counter-rotating wavy vortices. Note that the axial flow between wavy vortices is not represented in this idealized version of the flow (© 2000, Mike Minbiole and Richard M. Lueptow)

Figure 4: Velocity vectors measured using particle image velocimetry in a meridional (\(r\)-\(\theta\)) plane showing intra-vortex flow between counter-rotating wavy vortices. Color corresponds to the azimuthal (\(\theta\)) velocity with red corresponding to the velocity of the inner cylinder on the left and blue corresponding to the velocity of the outer cylinder on the right (for details, see Akonur and Lueptow 2003) (© 2003, Alp Akonur and Richard M. Lueptow)

Increasing the Taylor number above the critical Taylor number for the case of the inner cylinder rotating and the outer cylinder fixed results in higher order instabilities in which the vortical structure is retained, but the vortices are modified. The first transition is to wavy vortex flow, which is characterized by azimuthal waviness of the vortices as shown schematically in Figure 3. The waves travel around the annulus at a speed that is 30-50% of the surface speed of the inner cylinder, depending on the Taylor number and other conditions (King et al. 1984). The mathematical formulation of the problem for wavy vortices uses perturbations of the form

\[v=V+v(r)e^{qt}e^{i(n\theta+kz)}\]

This form of the perturbation includes azimuthal waviness, where n is an integer number of waves around the annulus. The axial dependence is included in the exponential, equivalent to a \(sin\) or \(cos\) term. In theory, the perturbation is the sum of many normal modes (many \(n\)’s and \(k\)’s), but in practice the mode that dominates is the one for which the Taylor number at the stability limit is lowest. The Taylor number for the transition from axisymmetric toroidal Taylor vortices to wavy vortices is not firmly established. For instance, the transition is theoretically predicted to occur at \(T/T_{crit}\)=1.1 for \(\eta\)=0.85 for infinitely long cylinders, whereas experiments indicate a range of higher values between 1.14 and 1.31 for \(\eta\)=0.80-0.90, depending on experimental conditions (Serre et al. 2008). The number of azimuthal waves depends on experimental conditions, though it is usually less than 6 or 7 (Coles 1965).

Regions of upward (downward) deformation of a wavy vortex correspond to regions of upward (downward) axial flow. As a result, for wavy vortex flow streamtubes are destroyed leading to chaotic particle paths with intra-vortex mixing, as shown in Figure 4. By contrast, the axisymmetric cellular structure of non-wavy Taylor vortex flow results in a set of nested streamtubes (KAM tori) for each vortex with a dividing invariant streamsurface between adjacent vortices. The only mechanism for transport within a vortex or between vortices is molecular diffusion.

At higher Taylor numbers, the wavy vortices transition to modulated wavy vortices, evident upon flow visualization as a slight flattening of the outflow boundary. The transition is most easily detected from spectral analysis of a velocity or reflected light measurement at a single point in the flow. Wavy vortex flow has a single peak at a frequency related to the passage of the azimuthal wave; modulated wavy vortex flow introduces a second spectral peak at a lower frequency related to the modulation. At still higher Taylor numbers, the waviness gives way to turbulence, which raises the spectral level at all frequencies. The vortices become axisymmetric, but the flow is turbulent at small scales. At high enough Taylor number, the turbulent vortices disappear, and the flow is fully turbulent.

The rotation of the outer cylinder in addition to the inner cylinder results in a variety of other flow regimes for long cylinders: wavy inflow and outflow, wavelets, twisted vortices, and corkscrew regimes for co-rotating cylinders; interpenetrating spirals, wavy interpenetrating spirals, intermittent turbulent spots, and spiral turbulence regimes for counter-rotating cylinders (Andereck et al. 1986). (A map showing different flow regimes as a function of the Reynolds numbers for the inner and outer cylinders, \(R_i=\Omega_ir_id/\nu\) and \(R_o=\Omega_or_od/\nu\ ,\) respectively, is given in Fig. 1 of this reference; it cannot be reproduced here due to copyright restrictions.) The addition of an axial flow in the annulus or a radial flow through porous cylinders alters the critical Taylor number as well as the wavelength and structure of the vortices. Likewise, the flow is altered by oscillating the inner or outer cylinder axially or azimuthally or by varying the gap between the inner and outer cylinders. The vortical structure is very robust--Taylor vortices can occur in other geometries including between concentric cones and spheres. For example, vortices occur at the equator between an inner rotating sphere and a concentric, stationary outer sphere ( Figure 5.).

References

A. Akonur and R. M. Lueptow, “Three-dimensional velocity field for wavy Taylor-Couette flow,” Phys. Fluids, 15:947-960, 2003.

C. D. Andereck, S. S. Lui, and H. L. Swinney, "Flow regimes in a circular Couette system with independently rotating cylinders," J. Fluid Mech. 164:155, 1986.

D. Coles, “Transition in circular Couette flow,” J. Fluid Mech., 21:385-425, 1965.

M. M. Couette, "Études sur le frottement des liquides," Ann. Chim. Phys. 6, Ser. 21, 433-510, 1890.

G. P. King, Y. Li, W. Lee, H. L. Swinney, and P. S. Marcus, “Wave speeds in wavy Taylor-vortex flow," J. Fluid Mech., 141:365-390, 1984.

A. Mallock, "Determination of the viscosity of water," Proc. Royal Soc. London, 45:126-132, 1888.

I. Newton, Principia: Vol. 1 The Motion of Bodies, F. Cajori, ed., U. Calif. Press, Berkeley, 1934, pp. 385-386.

Lord Rayleigh, “On the dynamics of revolving fluids,” Proc. Royal Soc. A, 93:148-154, 1916.

A. Recktenwald, M. Lücke, and H. W. Müller, “Taylor vortex formation in axial through-flow: Linear and weakly nonlinear analysis,” Phys. Rev. E, 48:4444-4454, 1993.

E. Serre, M. A. Sprague, and R. M. Lueptow, “Stability of Taylor-Couette flow in a finite-length cavity with radial throughflow,” Phys. Fluids, 20:034106, 2008.

G. G. Stokes, Mathematical and Physical Papers, vol. 1, Cambridge Univ. Press, Cambridge, England, 1880, pp. 102-104.

G. I. Taylor, "Stability of a viscous liquid contained between two rotating cylinders," Philos. Trans. R. Soc. London, Ser. A, 223:289-343, 1923.

S. T. Wereley and R. M. Lueptow, “Spatio-temporal character of non-wavy and wavy Taylor-Couette flow,” J. Fluid Mech., 364:59-80, 1998.

Internal references

John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.

Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Paul Manneville and Yves Pomeau (2009) Transition to turbulence. Scholarpedia, 4(3):2072.

Taylor-Couette flow

Contents

History

Fundamentals of Taylor-Couette Flow

Theoretical Background

Higher Order Instabilities

References

Recommended reading

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