How Does Anisotropy Affect the Calculation of Taylor Microscale in Turbulence?

[rdemyan]

TL;DR Summary

Anisotropic Taylor microscale

The Taylor microscale in isotropic turbulence is given by:

$$\lambda = \sqrt{ 15 \frac{\nu \ v'^2}{\epsilon} }$$

where v' is the root mean square of the velocity fluctuations. In general, for velocity fluctuations in three dimensions:

$$v' = \frac{1}{\sqrt{3}}\sqrt{{v'_1}^2+{v'_2}^2+{v'_3}^2}$$

So plugging this expression into the Taylor microscale equation yields:

$$\lambda = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{v'_1}^2+{v'_2}^2+{v'_3}^2}$$

Now for isotropic turbulence

$$v'_1=v'_2=v'_3$$

So for isotropic turbulence, equation 3 (third equation in this text) yields:

$$\lambda = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{3v'_1}^2} = \sqrt{ 15 \frac{\nu \ {v'_1}^2}{\epsilon} }$$

My question is: can I use equation 3 to calculate the Taylor microscale for anisotropic turbulence. For example if the injection of energy is highly anisotropic where $v'_2 = v'_3=0$

$$\lambda_A = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{v'_1}^2}=\sqrt{ 5 \frac{\nu \ {v'_1}^2}{\epsilon} }$$

where $λ_A$ is the anisotropic Taylor microscale. Does this seem correct? Also, does anyone know of a reference where this derivation was already done?

 

[member38644]

Yes, you can use equation 3 to calculate the Taylor microscale for anisotropic turbulence as long as you take into account the anisotropy in the velocity fluctuations. In your example, where $v'_2 = v'_3=0$, the Taylor microscale would be given by:

$$\lambda_A = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{v'_1}^2+0+0}=\sqrt{ 5 \frac{\nu \ {v'_1}^2}{\epsilon} }$$

This is because the anisotropy in the velocity fluctuations only affects the magnitude of the velocity fluctuations, not the overall expression for the Taylor microscale.

As for references, there are many papers and textbooks that discuss the anisotropic Taylor microscale, such as "Anisotropic Turbulence" by F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia, and "Turbulence: An Introduction for Scientists and Engineers" by P. A. Davidson. You can also find many research papers that use this equation to calculate the anisotropic Taylor microscale in various types of turbulence.