Taylor number

Taylor number
Name	Taylor number
Field	Fluid dynamics, Magnetohydrodynamics
Introduced	By Geoffrey Ingram Taylor (major contributor)
Related	Reynolds number, Ekman number, Rossby number, Rayleigh number

Taylor number

The Taylor number is a dimensionless parameter that characterizes the relative importance of centrifugal or rotational forces compared to viscous forces in rotating fluid systems. It appears prominently in problems involving rotating cylinders, rotating disks, planetary atmospheres, and magnetohydrodynamic stability, and is used to predict transitions between laminar and vortical or turbulent flow regimes.

Definition and physical meaning

The Taylor number quantifies the competition between rotation-induced inertial effects and viscous diffusion in a rotating flow, analogous to how the Reynolds number measures inertial versus viscous forces and how the Ekman number measures viscous effects in rotating frames. In classical centrifugal-instability problems such as the flow between concentric rotating cylinders studied by researchers associated with Royal Society and institutions like Cambridge University, a large Taylor number indicates dominance of centrifugal forces leading to instabilities, while a small Taylor number indicates viscous suppression of disturbances. In magnetohydrodynamic contexts relevant to research at laboratories such as Princeton Plasma Physics Laboratory and Max Planck Institute for Plasma Physics, the Taylor number helps balance rotational forcing against dissipative processes alongside parameters like the Magnetic Reynolds number.

Mathematical formulation

A common nondimensional form of the Taylor number, used for flow between rotating cylinders or for rotating disks, is proportional to (Ω^2 L^4)/(ν^2), where Ω is a characteristic angular velocity, L is a characteristic length (gap or radius), and ν is the kinematic viscosity. This structure mirrors configurational forms found in the definitions of the Rayleigh number and the Grashof number in buoyancy-driven convection problems studied historically at institutions including Birkbeck, University of London and University of Chicago. Depending on geometry, prefactors and geometric ratios appear; for example, in classical Taylor–Couette flow formulations developed in the early 20th century by investigators linked to Trinity College, Cambridge and experimentalists at Imperial College London, the squared angular velocity and fourth power of gap width enter the numerator while viscosity squared appears in the denominator. Alternative forms introduce radius ratios or include density ρ when expressed with dynamic viscosity μ, analogous to treatments used by theoreticians at Harvard University and Massachusetts Institute of Technology.

Derivation and theoretical background

Derivations begin from the Navier–Stokes equations in a rotating reference frame, historically grounded in analyses by figures such as Geoffrey Ingram Taylor and contemporaries at Trinity College, Cambridge and University of Cambridge. Linear stability analysis of the base Couette flow leads to a nondimensional eigenvalue problem in which the square of angular speed times a geometric length scale to the fourth power arises naturally after nondimensionalizing time with a viscous diffusion timescale. The critical Taylor number for the onset of centrifugal instability emerges where the leading eigenvalue crosses zero, a technique shared with stability studies of phenomena like the Taylor–Couette instability, the Rayleigh–Bénard convection onset, and the Kelvin–Helmholtz instability. Mathematical methods for these derivations are standard in texts produced by scholars at Princeton University and ETH Zurich, using normal-mode decomposition, asymptotic matching, and spectral methods developed in part at Courant Institute.

Applications and examples

The Taylor number is central to characterizing transitions in Taylor–Couette flow between laminar circular Couette states and cellular vortical patterns such as Taylor vortices, wavy vortices, and turbulent states observed in experiments at facilities like Los Alamos National Laboratory and Woods Hole Oceanographic Institution. It also guides analysis of rotating machinery and turbomachinery problems researched by teams at General Electric and Siemens, informs geophysical fluid studies of planetary cores and atmospheric jets pursued at NASA and European Space Agency, and appears in magnetohydrodynamic stability criteria for dynamos and accretion disks studied at University of Cambridge and Caltech. Specific example regimes include the onset threshold for axisymmetric Taylor vortices in finite-gap systems and centrifugal instability in rotating annuli used in laboratory models of atmospheric circulation investigated at Scripps Institution of Oceanography.

Experimental measurement and observation

Experimental determination of the critical Taylor number typically involves precise control of rotation rates, gap geometries, and fluid properties, tasks carried out in dedicated facilities at institutions such as University of Oxford, Imperial College London, and Utrecht University. Measurements combine high-speed flow visualization, torque sensors, particle image velocimetry developed at labs like Stanford University, and rheometric characterization of viscosity. Observables include the appearance of axisymmetric vortices, onset of secondary instabilities, changes in torque scaling, and emergence of turbulence; these signatures were first documented in classical experiments conducted by researchers affiliated with Cambridge University and later extended in experiments at KTH Royal Institute of Technology.

Relation to other dimensionless numbers

The Taylor number relates to the Reynolds number through geometric and rotational scaling, often scaling like Ta ∼ Re^2 in canonical problems when Re is defined with rotational velocity and radial gap. It complements the Ekman number in rotating stratified flows studied in contexts such as planetary boundary layers at University of California, Los Angeles and combines with the Rossby number when Coriolis effects from planetary rotation (studied at NOAA) are significant. In magnetohydrodynamic settings it interacts with the Magnetic Reynolds number and the Hartmann number in determining stability and transport properties in experiments at facilities such as DIII-D National Fusion Facility and theoretical studies from Imperial College London.

Category:Fluid dynamics