RMS speed from k-epsilon

obscureed · July 27, 2021, 07:41

Hi All.

This is a basic question, sorry, but I'm having a mental block.

I would like to estimate Root-Mean-Squared (RMS) flow speed from a steady-state k-epsilon CFD simulation. (Basically, I want to distinguish between regions with low time-averaged speed but plenty of fluctuations, and regions that are genuinely stagnant.)

If U is the time-averaged x-component of velocity and u is the time-varying part, and the same for V+v and W+w, and <...> indicates time-averaging, then RMS flow speed is
sqrt(<(U+u)^2 + (V+v)^2 + (W+w)^2>)
= sqrt(<(U^2+V^2+W^2 + 2(Uu + Vv + Ww) + u^2+v^2+w^2>)
= sqrt((time-averaged-speed)^2 + 2k)

Because the time-averaged <Uu> = U<u> = 0,
and k = <u^2 + v^2 + w^2> / 2.

Is this correct? I seem to remember 2k/3 instead of 2k, but I can't see where the 3 comes in and some Googling hasn't helped. Thanks in advance.
Ed

obscureed · July 27, 2021, 07:43

Oh, by the way, if anyone can point me to a decent source where this is discussed, that would be even better than the correct answer.

FMDenaro · July 27, 2021, 12:31

Quote:

Originally Posted by obscureed

Hi All.

This is a basic question, sorry, but I'm having a mental block.

I would like to estimate Root-Mean-Squared (RMS) flow speed from a steady-state k-epsilon CFD simulation. (Basically, I want to distinguish between regions with low time-averaged speed but plenty of fluctuations, and regions that are genuinely stagnant.)

If U is the time-averaged x-component of velocity and u is the time-varying part, and the same for V+v and W+w, and <...> indicates time-averaging, then RMS flow speed is
sqrt(<(U+u)^2 + (V+v)^2 + (W+w)^2>)
= sqrt(<(U^2+V^2+W^2 + 2(Uu + Vv + Ww) + u^2+v^2+w^2>)
= sqrt((time-averaged-speed)^2 + 2k)
Because the time-averaged <Uu> = U<u> = 0,

and k = <u^2 + v^2 + w^2> / 2.

Is this correct? I seem to remember 2k/3 instead of 2k, but I can't see where the 3 comes in and some Googling hasn't helped. Thanks in advance.
Ed

Sorry, I am not sure to understand what you want to obtain...

1) Using the steady state field you have only U,V,W, nothing else. The fluctuations are modelled over all the lenght scale.
2) Are you searching a general expression of the TKE? You express the RMS of the total velocity field?

3) Any time you see a term with (1/3) I would think about the trace of a tensor. See the isotropic part...

obscureed · July 27, 2021, 14:34

Yes, starting from the steady-state velocity field (U,V,W), plus k and epsilon, I would like to estimate the field of RMS flow-speed, point by point.

I think this can be estimated using sqrt(U^2+V^2+W^2+2*k), but I wanted to check (in particular the factor 2 applied to k).

I can't see anything wrong in my working. I'm sure about the factor of 2 in the definition of k (see for example https://cfd-online.com/Wiki/Introduc...kinetic_energy), so I'm going to use it.

I think my distant memory involving a 3 was of sqrt(2k/3), the estimate of the RMS value of u (or v or w, assuming isotropic turbulence) from a k-epsilon solution.

If anyone can point me at previous uses or discussions of this RMS speed, that would be useful. Thanks!

LuckyTran · July 27, 2021, 16:38

I don't know why you're looking for references for rms velocity. You can google it and get hundreds of millions of hits. It's a concept taught in classrooms throughout the world in nearly all the physical sciences.

I get that you are confused but.. I can loan you my high school algebra book if you really really need a reference.

July 27, 2021, 07:41	RMS speed from k-epsilon	#1
obscureed Senior Member Join Date: Sep 2017 Posts: 246 Rep Power: 11	Hi All. This is a basic question, sorry, but I'm having a mental block. I would like to estimate Root-Mean-Squared (RMS) flow speed from a steady-state k-epsilon CFD simulation. (Basically, I want to distinguish between regions with low time-averaged speed but plenty of fluctuations, and regions that are genuinely stagnant.) If U is the time-averaged x-component of velocity and u is the time-varying part, and the same for V+v and W+w, and <...> indicates time-averaging, then RMS flow speed is sqrt(<(U+u)^2 + (V+v)^2 + (W+w)^2>) = sqrt(<(U^2+V^2+W^2 + 2(Uu + Vv + Ww) + u^2+v^2+w^2>) = sqrt((time-averaged-speed)^2 + 2k) Because the time-averaged <Uu> = U<u> = 0, and k = <u^2 + v^2 + w^2> / 2. Is this correct? I seem to remember 2k/3 instead of 2k, but I can't see where the 3 comes in and some Googling hasn't helped. Thanks in advance. Ed

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July 27, 2021, 07:43		#2
obscureed Senior Member Join Date: Sep 2017 Posts: 246 Rep Power: 11	Oh, by the way, if anyone can point me to a decent source where this is discussed, that would be even better than the correct answer.

July 27, 2021, 14:34		#4
obscureed Senior Member Join Date: Sep 2017 Posts: 246 Rep Power: 11	Yes, starting from the steady-state velocity field (U,V,W), plus k and epsilon, I would like to estimate the field of RMS flow-speed, point by point. I think this can be estimated using sqrt(U^2+V^2+W^2+2*k), but I wanted to check (in particular the factor 2 applied to k). I can't see anything wrong in my working. I'm sure about the factor of 2 in the definition of k (see for example https://cfd-online.com/Wiki/Introduc...kinetic_energy), so I'm going to use it. I think my distant memory involving a 3 was of sqrt(2k/3), the estimate of the RMS value of u (or v or w, assuming isotropic turbulence) from a k-epsilon solution. If anyone can point me at previous uses or discussions of this RMS speed, that would be useful. Thanks!

July 27, 2021, 16:38		#5
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,675 Rep Power: 66	I don't know why you're looking for references for rms velocity. You can google it and get hundreds of millions of hits. It's a concept taught in classrooms throughout the world in nearly all the physical sciences. I get that you are confused but.. I can loan you my high school algebra book if you really really need a reference.