Taylor Hypothesis in high turbulence flows

lucamirtanini · October 7, 2021, 16:49

As you can see here, the Taylor hypothesis can be used only for low turbulence flow.
But, what really mean:
$\frac{u}{U}<<1$

How can I quantify how much should be lower than one. Lin explored this limit, and looking his paper., seems to have resolved it in eq. 6.3, but I cannot understand what does it mean concretely.
Is there a limit at which we can say that a tubulence intensity is high? Here there is a classification. Can be this in accordance with the limit of the Taylor hypothesis?

LuckyTran · October 7, 2021, 17:04

6.3 basically says the limit is 1/5 (or 0.2, or 20%)

The high, med, low turbulence intensities on the wiki are not related to this Taylor's hypothesis. That being said, you can see that the a turbulence intensity of 20% or higher is rather uncommon and you see it only in special places.

Having also said that, it should not be a surprise that the condition that turbulence intensity be lower than 20% covers just about all flows, since it is an order of magnitude estimate. It's kind of hard to guess the wrong order of magnitude for something that is usually between 0 and 1. But the paper shows the theoretical basis for such estimates.

Furthermore, 6.3 isn't an exact result either since it is based on the assumption in section 5 that the quadruple correlation can be approximated by the cross-correlations (two correlations). In other words, section 5 tries to show that you can ignore the higher correlations past two, which is itself implicitly assuming that taylor's hypothesis is already true.

If you are looking for an a priori prediction of whether Taylor's hypothesis should hold in an arbitrary flow, we can look at other literature. You can of course always directly calculate the two point correlations in time and space (using DNS-like data for example) to show that taylor's hypothesis does or does not hold if you have already run the simulation.

lucamirtanini · October 7, 2021, 17:34

Thank you for your answer! Really!

Quote:

Originally Posted by LuckyTran

6.3 basically says the limit is 1/5 (or 0.2, or 20%)

Why can the left member of Eq. (6.3) be considered 1?
Since $\frac{u'^{2}}{U^{2}}$ is the square of turbulence intensity. Does it mean that the limit in term of turbulence intensity is $\sqrt{(1/5)}$ , i.e. 0.44 ? It is a very high turbulence with regards to this and in my experience.

Quote:

Originally Posted by LuckyTran

The high, med, low turbulence intensities on the wiki are not related to this Taylor's hypothesis.

I also tried to look at the reference in the wiki, but I didn't find this classification. Do you know where it comes from?

Quote:

Originally Posted by LuckyTran

If you are looking for an a priori prediction of whether Taylor's hypothesis should hold in an arbitrary flow, we can look at other literature. You can of course always directly calculate the two point correlations in time and space (using DNS-like data for example) to show that taylor's hypothesis does or does not hold if you have already run the simulation.

Yes, I was searching for an a priori prediction. I will look also for other literature. Do you have suggestion?
I have found a paper that says generally that is commonly accepted that to apply the Taylor's hypothesis , the turbulence intensity should be less than 0.1, but no references are shown

LuckyTran · October 8, 2021, 13:58

I didn't notice the square, so yeah the turbulence intensity is 0.44 and not 0.2 like I said. I honestly haven't figured out yet precisely where the 5 comes from yet.

The low,med, & high classification from the wiki are very subjective. Those are just typical conventions. Basically, pipe flows are average or medium. Low speed wind tunnels with well conditioned flows being low. And complex devices being high. Don't worry about where the classification comes from, it's just a convention.

Without knowing your specific motivation, two things make make continuing this conversation a bit difficult. One is that turbulence intensity is known (and you see this already in Lin's work from way back when) to not be a sufficient criteria. Certain (very common) flow scenarios it breaks down easily.

The other thing about Taylor's hypothesis is that specific correlations break down in a way that isn't easily summarized as "Taylor's hypothesis doesn't work." Here's a recent work (from a friend of a friend) on this topic for example. That is, we're looking for specific two-point correlations in specific directions.

lucamirtanini · October 8, 2021, 14:23

Quote:

Originally Posted by LuckyTran

Without knowing your specific motivation, two things make make continuing this conversation a bit difficult. One is that turbulence intensity is known (and you see this already in Lin's work from way back when) to not be a sufficient criteria. Certain (very common) flow scenarios it breaks down easily.

I am working with a jet flow having a turbulence intensity higher than 20% and that arrives also to a peak of 100%. The Taylor hypothesis does not work, but I was searching for a theorical basis such as the one of Lin with a more specific criteria about the turbulence intensity. What seems strange to me is that 0.44 does not seems to me so far lower than 1.

I still didn't understand why, from your calculation, the left member of 6.3 is 1.

LuckyTran · October 8, 2021, 16:10

You want 6.2

6.3 tells you the ratio of those terms is $5\frac{<u^2>}{U^2}$

So you take 6.3 and put it back into 6.2 which then gives you $\frac{<u^2>}{U^2}<<1$

Why is it important to you whether or not Taylor's hypothesis holds or not? You don't even need it unless you're trying to infer spatial correlation using temporal correlations. And you only need to do this usually if you are an experimentalist who is meauring flows using hotwires. CFD'ers can just do the spatial calculation.

lucamirtanini · October 9, 2021, 07:39

If you use a LES formulation, and your mesh is coarse, than you can have a "low resolution" wavenumber spectra (not enough point), but a "good resolution" frequency spectra.

LuckyTran · October 9, 2021, 14:22

So you just end up with low resolution in your results.

The same happens if you run LES on any grid and super duper small timesteps, e.g. picosecond time steps. You can't resolve more temporal scales than the timesales already linked to the length scales resolved by your coarse grid. You don't gain more physics by going to smaller time-steps, you just gain more samples for your statistical averaging. You can't use a coarse grid for example and try to resolve Kolmogorov timescales by using a smaller timestep.

October 7, 2021, 16:49	Taylor Hypothesis in high turbulence flows	#1
lucamirtanini Senior Member luca mirtanini Join Date: Apr 2018 Posts: 165 Rep Power: 8	As you can see here, the Taylor hypothesis can be used only for low turbulence flow. But, what really mean: $\frac{u}{U}<<1$ How can I quantify how much should be lower than one. Lin explored this limit, and looking his paper., seems to have resolved it in eq. 6.3, but I cannot understand what does it mean concretely. Is there a limit at which we can say that a tubulence intensity is high? Here there is a classification. Can be this in accordance with the limit of the Taylor hypothesis? Bilbo88 likes this.

October 7, 2021, 17:04		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	6.3 basically says the limit is 1/5 (or 0.2, or 20%) The high, med, low turbulence intensities on the wiki are not related to this Taylor's hypothesis. That being said, you can see that the a turbulence intensity of 20% or higher is rather uncommon and you see it only in special places. Having also said that, it should not be a surprise that the condition that turbulence intensity be lower than 20% covers just about all flows, since it is an order of magnitude estimate. It's kind of hard to guess the wrong order of magnitude for something that is usually between 0 and 1. But the paper shows the theoretical basis for such estimates. Furthermore, 6.3 isn't an exact result either since it is based on the assumption in section 5 that the quadruple correlation can be approximated by the cross-correlations (two correlations). In other words, section 5 tries to show that you can ignore the higher correlations past two, which is itself implicitly assuming that taylor's hypothesis is already true. If you are looking for an a priori prediction of whether Taylor's hypothesis should hold in an arbitrary flow, we can look at other literature. You can of course always directly calculate the two point correlations in time and space (using DNS-like data for example) to show that taylor's hypothesis does or does not hold if you have already run the simulation. lucamirtanini likes this.

October 8, 2021, 16:10		#6
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	You want 6.2 6.3 tells you the ratio of those terms is $5\frac{<u^2>}{U^2}$ So you take 6.3 and put it back into 6.2 which then gives you $\frac{<u^2>}{U^2}<<1$ Why is it important to you whether or not Taylor's hypothesis holds or not? You don't even need it unless you're trying to infer spatial correlation using temporal correlations. And you only need to do this usually if you are an experimentalist who is meauring flows using hotwires. CFD'ers can just do the spatial calculation. Last edited by LuckyTran; October 8, 2021 at 19:44.

October 9, 2021, 14:22		#8
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	So you just end up with low resolution in your results. The same happens if you run LES on any grid and super duper small timesteps, e.g. picosecond time steps. You can't resolve more temporal scales than the timesales already linked to the length scales resolved by your coarse grid. You don't gain more physics by going to smaller time-steps, you just gain more samples for your statistical averaging. You can't use a coarse grid for example and try to resolve Kolmogorov timescales by using a smaller timestep. FMDenaro likes this.

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October 8, 2021, 13:58		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,747 Rep Power: 66	I didn't notice the square, so yeah the turbulence intensity is 0.44 and not 0.2 like I said. I honestly haven't figured out yet precisely where the 5 comes from yet. The low,med, & high classification from the wiki are very subjective. Those are just typical conventions. Basically, pipe flows are average or medium. Low speed wind tunnels with well conditioned flows being low. And complex devices being high. Don't worry about where the classification comes from, it's just a convention. Without knowing your specific motivation, two things make make continuing this conversation a bit difficult. One is that turbulence intensity is known (and you see this already in Lin's work from way back when) to not be a sufficient criteria. Certain (very common) flow scenarios it breaks down easily. The other thing about Taylor's hypothesis is that specific correlations break down in a way that isn't easily summarized as "Taylor's hypothesis doesn't work." Here's a recent work (from a friend of a friend) on this topic for example. That is, we're looking for specific two-point correlations in specific directions.

October 9, 2021, 07:39		#7
lucamirtanini Senior Member luca mirtanini Join Date: Apr 2018 Posts: 165 Rep Power: 8	If you use a LES formulation, and your mesh is coarse, than you can have a "low resolution" wavenumber spectra (not enough point), but a "good resolution" frequency spectra.