Anisotropic Taylor microscale

rdemyan · January 3, 2023, 19:00

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

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

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

which is expected and is basically the first equation in this post.

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

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

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

Lalilulelo · January 25, 2023, 10:06

I gave you an answer on SE.

January 3, 2023, 19:00	Anisotropic Taylor microscale	#1
rdemyan Member Join Date: Jan 2023 Posts: 71 Rep Power: 4	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 So for isotropic turbulence, equation 3 (third equation in this text) yields: $\lambda = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{3v'_1}^2} = \sqrt{ 15 \frac{\nu \ {v'_1}^2}{\epsilon} }$ which is expected and is basically the first equation in this post. 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 $\lambda_A = \sqrt{ 5 \frac{\nu}{\epsilon} }\sqrt{{v'_1}^2}=\sqrt{ 5 \frac{\nu \ {v'_1}^2}{\epsilon} }$ where $\lambda_A$ is the anisotropic Taylor microscale. Does this seem correct? Also, does anyone know of a reference where this derivation was already done?

January 25, 2023, 10:06		#2
Lalilulelo New Member Alan Join Date: Jul 2012 Posts: 5 Rep Power: 15	I gave you an answer on SE. arjun likes this.

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