Taylor microscale

Calculation of the Taylor microscale is not entirely straightforward, requiring formation of certain flow correlation function(s), then expanding in a Taylor series and using the first non-zero term to characterize an osculating parabola. The Taylor microscale is proportional to ${\displaystyle {\text{Re}}^{-1/2}}$, while the Kolmogorov microscale is proportional to ${\displaystyle {\text{Re}}^{-3/4}}$, where ${\displaystyle \text{Re}}$ is the integral scale Reynolds number. A turbulence Reynolds number calculated based on the Taylor microscale ${\displaystyle \lambda }$ is given by

${\displaystyle {\text{Re}}_{\lambda }=\frac{⟨{\mathbf{v}}^{\prime }{⟩}_{rms}\lambda }{\nu },}$

where ${\displaystyle ⟨{\mathbf{v}}^{\prime }{⟩}_{rms}=\frac{1}{\sqrt{3}}\sqrt{(v_1^{\prime }{)}^2+(v_2^{\prime }{)}^2+(v_3^{\prime }{)}^2}}$ is the root mean square of the velocity fluctuations. The Taylor microscale is given as

${\displaystyle \lambda =\sqrt{15\frac{\nu }{ϵ}}⟨{\mathbf{v}}^{\prime }{⟩}_{rms},}$

where ${\displaystyle \nu }$ is the kinematic viscosity, and ${\displaystyle ϵ}$ is the rate of energy dissipation. A relation with turbulence kinetic energy ${\displaystyle k}$ can be derived as

${\displaystyle \lambda \approx \sqrt{10\nu \frac{k}{ϵ}}.}$

The Taylor microscale gives a convenient estimation for the fluctuating strain rate field

${\displaystyle {\left(\frac{\mathrm{\partial }⟨{\mathbf{v}}^{\prime }{⟩}_{rms}}{\mathrm{\partial }\mathbf{x}}\right)}^2=\frac{⟨{\mathbf{v}}^{\prime }{⟩}_{rms}^2}{{\lambda }^2}.}$

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 ${\displaystyle \lambda }$ and the Kolmogorov microscale ${\displaystyle \eta }$. Given the length scale of the larger eddies ${\displaystyle l\propto \frac{k^{3/2}}{ϵ}}$, and the turbulence Reynolds number ${\displaystyle {\text{Re}}_l}$ referred to these eddies, the following relations can be obtained:

${\displaystyle \frac{\lambda }{l}=\sqrt{10}{\text{Re}}_l^{-1/2}}$

${\displaystyle \frac{\eta }{l}={\text{Re}}_l^{-3/4}}$

${\displaystyle \frac{\lambda }{\eta }=\sqrt{10}{\text{Re}}_l^{1/4}}$

${\displaystyle \lambda =\sqrt{10}{\eta }^{2/3}{l}^{1/3}}$