The Taylor microscale, which is sometimes called the turbulence length scale, is a
length scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot ...
used to characterize a
turbulent
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. This microscale is named after
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
. The Taylor microscale is the intermediate length scale at which fluid
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
significantly affects the
dynamics of turbulent
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid ...
in the flow. This length scale is traditionally applied to turbulent flow which can be characterized by a
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
spectrum of velocity fluctuations. In such a flow, length scales which are larger than the Taylor microscale are not strongly affected by viscosity. These larger length scales in the flow are generally referred to as the
inertial range
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleratio ...
. Below the Taylor microscale the turbulent motions are subject to strong viscous forces and
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
is
dissipated
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
into heat. These shorter length scale motions are generally termed the
dissipation range
In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy (internal, bulk flow kinetic, or system potential) transforms from an initial form ...
.
Calculation of the Taylor microscale is not entirely straightforward, requiring formation of certain flow correlation function(s), then expanding in a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
and using the first non-zero term to characterize an osculating parabola. The Taylor microscale is proportional to
, while the
Kolmogorov microscale is proportional to
, where
is the integral scale Reynolds number. A turbulence Reynolds number calculated based on the Taylor microscale
is given by
:
where
is the
root mean square
In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
of the velocity fluctuations.
The Taylor microscale is given as
:
where
is the
kinematic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
, and
is the rate of energy dissipation. A relation with
turbulence kinetic energy
In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuat ...
can be derived as
:
The Taylor microscale gives a convenient estimation for the fluctuating strain rate field
:
Other relations
The Taylor microscale falls in between the large-scale eddies and the small-scale eddies, which can be seen by calculating the ratios between
and the Kolmogorov microscale
. Given the length scale of the larger eddies
, and the turbulence Reynolds number
referred to these eddies, the following relations can be obtained:
:
:
:
:
Notes
References
*{{citation , last1=Tennekes , first1=H. , authorlink1=Hendrik Tennekes , first2=J.L. , last2=Lumley , authorlink2=John L. Lumley , title=A First Course in Turbulence , publisher=MIT Press , location=Cambridge, MA , year=1972 , isbn=978-0-262-20019-6
Fluid dynamics
Turbulence