Taylor number

Physical key figure
Surname	Taylor number
Formula symbol	${\displaystyle {\mathit {Ta}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Ta}}=4\cdot {\mathit {Re}}^{2}\cdot {\frac {R_{\mathrm {a} }-R_{\mathrm {i} }}{R_{\mathrm {a} }+R_{\mathrm {i} }}}}$
${\displaystyle {\mathit {Re}}}$ Reynolds number ${\displaystyle R_{\mathrm {a} }}$ Outside radius of the cylinder ${\displaystyle R_{\mathrm {i} }}$ Inner radius of the cylinder
Named after	Geoffrey Ingram Taylor
scope of application	Taylor vortex

The Taylor number ( ), named after Geoffrey Ingram Taylor , is a dimensionless parameter to describe the tendency to form Taylor vortices . ${\displaystyle {\mathit {Ta}}}$

definition

A Taylor-Couette flow can be used to define the Taylor number . This is a laminar flow of an incompressible viscous liquid that is located in the space between two coaxial cylinders rotating relative to one another or between two infinitely long and wide plates that are moving relative to one another. The diameter of the Taylor vortex is approximately equal to the gap width , which is determined by the difference between the outer radius and the inner radius . This Taylor number is related to the Reynolds number on the inner cylinder ${\displaystyle d=(R_{\mathrm {a} }-R_{\mathrm {i} })}$ ${\displaystyle R_{\mathrm {a} }}$ ${\displaystyle R_{\mathrm {i} }}$ ${\displaystyle {\mathit {Re}}}$

${\displaystyle Re=v_{\text{i}}\cdot {\frac {d}{\nu }}=\omega R_{\text{i}}\cdot {\frac {R_{\text{a}}-R_{\text{i}}}{\nu }}}$

in the following way:

${\displaystyle {\mathit {Ta}}=4\cdot {\mathit {Re}}^{2}\cdot {\frac {R_{\mathrm {a} }-R_{\mathrm {i} }}{R_{\mathrm {a} }+R_{\mathrm {i} }}}=4\cdot \left({\frac {\omega }{\nu }}\right)^{2}\cdot R_{\mathrm {i} }^{2}\cdot {\frac {(R_{\mathrm {a} }-R_{\mathrm {i} })^{3}}{R_{\mathrm {a} }+R_{\mathrm {i} }}}}$

Here is the density, the kinematic viscosity of the fluid and the angular velocity . The Taylor number, which describes the occurrence of the Taylor vortex, depends reciprocally on the kinematic viscosity. If the viscosity gets too high, the Taylor number drops below a critical value and the Taylor vortices disappear. ${\displaystyle \rho }$ ${\displaystyle \nu }$ ${\displaystyle \omega }$

From a critical Taylor number, which depends on the ratio , Taylor vortices form. (E.g. for ("wide gap") or for ("narrow gap")). ${\displaystyle R_{\mathrm {i} }/R_{\mathrm {a} }}$ ${\displaystyle {\mathit {Ta}}_{\mathrm {c} }=6200}$ ${\displaystyle R_{\mathrm {i} }/R_{\mathrm {a} }=0{,}5}$ ${\displaystyle {\mathit {Ta}}_{\mathrm {c} }=3448}$ ${\displaystyle R_{\mathrm {i} }/R_{\mathrm {a} }=0{,}975}$

literature

• GI Taylor: Stability of a Viscous Liquid Contained between Two Rotating Cylinders . Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character Vol. 223, pp. 289-343 (57 pages), 1923.
• Ludwig Prandtl, Klaus Oswatitsch, Karl Wieghardt: Guide through fluid mechanics . 9th improved and expanded edition, 1990.