Taylor-Couette flow

Building a Couette-Taylor system

Taylor-Couette flow describes the flow of an incompressible viscous liquid that is located in the space between two coaxial cylinders rotating relative to one another. The flow between the cylinders is not only dependent on the speed of rotation, but also on whether the inner or outer cylinder rotates.

If the relative speed of the cylinders is low (see below) and the gap between them is small compared to their diameter, the flow can be treated as a plane laminar flow (Couette flow). The velocity profile is similar to the idealized case of a plane flow between two plates, one of which is moved slowly relative to the other. One plate can be viewed as stationary and the other as moving.

The flow was named after Maurice Couette , who constructed the first functioning rotational viscometer at the end of the 19th century and used the laminar basic flow (Couette flow) for this purpose, and after Geoffrey Ingram Taylor , who examined and theoretically explained the instabilities at higher rotational speeds. Couette tried to avoid vortices and just turned the outer cylinder. The fact that eddies arise when the inner cylinder rotates, however, was already suspected by George Stokes in 1880, was found experimentally by Henry RA Mallock (1888) , among others , and Rayleigh , to whom Lord Kelvin communicated the phenomenon, published the basic explanation for this in 1916. They were analyzed in detail by Taylor whose work was a fundamental work on hydrodynamics in several respects (confirmation of boundary conditions without slip in viscous liquids, confirmation of the validity of the Navier-Stokes equations, one of the first examples of linear stability analysis in hydrodynamics).

Taylor vortices in the Taylor-Couette flow, Reynolds number 950, radial coordinates horizontal, axial coordinates along the axis of rotation upwards in millimeters

Formation and vortex formation

If only the outer cylinder rotates, the flow behaves according to the naive expectation: a uniform laminar flow is formed between the two cylinders. This intuitively accessible image is retained even when the inner cylinder rotates slowly.

At higher speeds of the inner cylinder, however, the flow breaks up into strips, as the liquid accelerated by the centrifugal force pushes outwards on the inner cylinder; this creates Taylor vortices that are perpendicular to the axis of rotation. Typical vortex wavelengths are 2 d (with d being the length of the gap between the cylinders). The formation conditions of the eddies are characterized by the Taylor number . According to Taylor (1923), eddies arise if the Reynolds number

${\displaystyle {\mathit {Re}}={\frac {v\cdot d}{\nu }}>41,3{\sqrt {\frac {r}{d}}}}$

where v is the peripheral speed, the kinematic viscosity, d is the gap distance and r is the mean radius. Turbulence only occurs at significantly higher speeds (factor 50). Typically the inner cylinder rotates. The vortices have a torus shape and are arranged along the axis of rotation, with the direction of rotation alternating. ${\displaystyle \nu }$

A further increase in the speed of rotation of the inner cylinder results in more complex wave formulas that eventually turn into turbulence. The Taylor-Couette current was therefore also used to investigate transition scenarios into the chaotic regime , for example in classical experiments by Harry Swinney and Jerry Gollub .

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  Taylor instabilities with increasing Reynolds number

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  Taylor vortices between cylinder walls, made visible with the addition of color

Application example

This article or the following section is not adequately provided with supporting documents ( e.g. individual evidence ). Information without sufficient evidence could be removed soon. Please help Wikipedia by researching the information and including good evidence.

Taylor-Couette system in the sewage treatment plant

Sludge dewatering is a process step in industry and municipal sewage treatment plants . For optimal dewatering in the dewatering units, flocculation of the sludge constituents through the addition of flocculation aids is a precondition. The flakes that form are not particularly shear stable. To improve the stability, a method is used whose physical basis is the Taylor-Couette flow. The sludge mixed with flocculants passes through an assembly known as a “mechanical flake former”.

The solid phase in a suspension basically deviates in the direction of a decreasing gradient of the shear flow, i.e. in a Taylor-Couette flow into the interior of the Taylor vortex. There only a steady rolling of the coarse flakes that arise takes place until they are released undestroyed at the end of the vortex path (see picture). Due to the settling movement of the coarser particles into the vortex, very fine suspended matter is also bound in the pellets and then goes off with the solid.

The downstream dewatering unit can then carry out the separation process faster and with better separation accuracy. The supply of flocculants can thus be carried out more economically.

literature

• Pascal Chossat, Gérard Iooss : The Couette – Taylor Problem (= Applied Mathematical Sciences. 102). Springer, New York NY et al. 1994, ISBN 0-387-94154-1 .
• Russell J. Donnelly : Taylor-Couette flow: The early days. In: Physics Today . Vol. 44, No. 11, 1991, pp. 32-39, doi : 10.1063 / 1.881296 .
• EL Koschmieder: Bénard Cells and Taylor Vortices. Cambridge University Press, Cambridge et al. 1993, ISBN 0-521-40204-2 .
• Richard Lueptow: Taylor-Couette Flow. In: Scholarpedia. Vol. 4, No. 11, 2009, 6389, doi : 10.4249 / scholarpedia.6389 .
• Herbert Oertel (ed.): Prandtl guide through fluid mechanics. 10th, completely revised edition. Vieweg, Braunschweig et al. 2002, ISBN 3-528-38209-0 .

Web links

• Hydrodynamic instabilities in Taylor-Couette flow
• Nonlinear dynamics and structure formation in continuous systems (PDF; 1.5 MB)

Individual evidence

1. ^ Geoffrey I. Taylor : Stability of a Viscous Liquid contained between Two Rotating Cylinders. In: Philosophical Transactions of the Royal Society . Series A: Mathematical, Physical and Engineering Sciences. Vol. 223, 1923, pp. 289-343, doi : 10.1098 / rsta.1923.0008 .
2. ^ Maurice Couette: Études sur le frottement des liquides. In: Annales de Chimie et de Physique . Série 6, Vol. 21, 1890, pp. 433-510 .
3. ^ Lord Rayleigh : On the dynamics of revolving fluids In: Proceedings of the Royal Society of London . Series A: Mathematical, Physical and Engineering Sciences. Vol. 93, No. 648, 1916, pp. 148-154, JSTOR 93794 .
4. ^ Ludwig Prandtl , Klaus Oswatitsch , Karl Wieghardt : Guide through the fluid dynamics. 9th, improved and enlarged edition. Vieweg, Braunschweig 1990, ISBN 3-528-28209-6 , p. 189.
5. ↑ Jerry P. Gollub , Harry L. Swinney : Onset of Turbulence in a Rotating Fluid. In: Physical Review Letters . Vol. 35, No. 14, 1975, 927-930, doi : 10.1103 / PhysRevLett.35.927 .
6. ↑ An application of the Taylor-Couette flow principle in sludge dewatering ( memento of the original from February 21, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 688 kB).