Turbulent kinetic energy expression

CFD_10 · May 27, 2019, 16:56

Hello

The turbulent kinetic energy is defined as follows:

$k = {1 \over 2} \left(\overline{u_x'u_x'} + \overline{u_y'u_y'} + \overline{u_z'u_z'}\right) = {1 \over 2} \left(\overline{u_x'^2} + \overline{u_y'^2} + \overline{u_z'^2}\right)$

But in the following cfd-online wiki article we can find the following expression:

$u' = \sqrt{{1\over 3}(u_x'^2+u_y'^2+u_z'^2)}=\sqrt{{2\over 3}k}$
As you can see in this formula they supposed that
$k = {1\over 2}(u_x'^2+u_y'^2+u_z'^2)$
which means that:

$u_x'^2 = \overline{u_x'^2}$ and $u_y'^2 = \overline{u_y'^2}$ and $u_z'^2 = \overline{u_z'^2}$

Question 1: Is that correct?

LuckyTran · May 27, 2019, 21:15

There is a notation clash.

Quote:

Originally Posted by CFD_10

As you can see in this formula they supposed that
$k = {1\over 2}(u_x'^2+u_y'^2+u_z'^2)$

Here

already is an rms has been time-averaged.

So to answer your question
1) It is the exact same formula, which looks different just because there are two different sets of notation being used.
2) In general no. But if the mean is 0 ( $\overline{u'}=0$ ) then yes. And that's why

is always taken as fluctuations about the mean: $u'(t)=u(t)-\overline{u}$ .
3) In general no. But here yes because there is a notation clash.

CFD_10 · May 28, 2019, 00:31

Quote:

Originally Posted by LuckyTran

There is a notation clash.
Here

already is an rms has been time-averaged.

Sorry, I can't understand this. could you please give more details?

FMDenaro · May 28, 2019, 02:58

I suggest to give always a look to the definition of the overbar. It is used for both RANS and LES but the resulting consequence on the average is different.

LuckyTran · May 28, 2019, 12:14

Quote:

Originally Posted by CFD_10

Sorry, I can't understand this. could you please give more details?

In the article you linked to, u' is defined as an rms (as a time-averaged value and that's why the overbar is dropped in the notation).

You didn't define it, but I presume that here, u' is an instantaneous velocity fluctuation about the mean and not the same u' in the article:

$k = {1 \over 2} \left(\overline{u_x'u_x'} + \overline{u_y'u_y'} + \overline{u_z'u_z'}\right) = {1 \over 2} \left(\overline{u_x'^2} + \overline{u_y'^2} + \overline{u_z'^2}\right)$

I make this presumption, because otherwise it would make no sense to introduce overbar notation for the time-average (note: the article drops the overbar notation because there's no need for it).

In any case, your confusion can be cleared up if you clearly define your variables and operators in all formulas used.

May 27, 2019, 16:56	Turbulent kinetic energy expression	#1
CFD_10 Member CFD USER Join Date: May 2019 Posts: 40 Rep Power: 6	Hello The turbulent kinetic energy is defined as follows: $k = {1 \over 2} \left(\overline{u_x'u_x'} + \overline{u_y'u_y'} + \overline{u_z'u_z'}\right) = {1 \over 2} \left(\overline{u_x'^2} + \overline{u_y'^2} + \overline{u_z'^2}\right)$ But in the following cfd-online wiki article we can find the following expression: $u' = \sqrt{{1\over 3}(u_x'^2+u_y'^2+u_z'^2)}=\sqrt{{2\over 3}k}$ As you can see in this formula they supposed that $k = {1\over 2}(u_x'^2+u_y'^2+u_z'^2)$ which means that: $u_x'^2 = \overline{u_x'^2}$ and $u_y'^2 = \overline{u_y'^2}$ and $u_z'^2 = \overline{u_z'^2}$ Question 1: Is that correct? Last edited by CFD_10; May 28, 2019 at 12:45.

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May 28, 2019, 02:58		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71	I suggest to give always a look to the definition of the overbar. It is used for both RANS and LES but the resulting consequence on the average is different.