[LES] Estimating resolved TKE from instantaneous velocities

Gerry Kan · March 21, 2022, 07:01

Howdy Folks:

I am looking for a way to estimate total TKE in an LES code, since it only has the SGS TKE as a field variable, but not the resolved TKE. Since the TKE is used for a function returning a Yes / No response, a solution within 50% of what it should be is considered acceptable.

Currently I have a very simple way of making the estimate: $\displaystyle{\left<{k}\right>\approx\frac{k_\text{SGS}}{\phi_\text{SGS}}}$

where $k_\text{SGS}$ is a user-defined fraction between resolved and SGS scale TKE.

For a more sophisticated step, I am thinking of using the above resolved scale TKE as the "upper bound" of TKE, while calculating it using gradient transport theorem:

$\displaystyle{ \left<{k}\right> \approx \text{max}\left[ \frac{1}{2} \nu_\text{turb} \overline{ \left( \frac{\partial{\left<{u_i}\right>}} {\partial{x_i}} \right) }, \;\; \frac{k_\text{SGS}}{\phi_\text{SGS}} \right] }$ .

Note that the mean (overline) indicates the mean over the three components, and not the RANS average of the velocity.

While I can use a random distribution ( $\psi$ ) to estimate relationship between the instantaneous and RANS velocity component, where

$\left<{u_i}\right> \approx \psi{u_i}$ ,

I am not sure how I could approximate the eddy viscosity $\nu_\text{turb}$ because the text book definition involves both

and $\epsilon$ . At the minimum I can simply set it to a fixed value so that $\nu_\text{turb} >> \nu$ , and as an estimate it should be sufficient.

Before I do something like this, is there a less ad hoc way of doing it?

Thanks in advance, Gerry.

LuckyTran · March 21, 2022, 15:12

The resolved tke you get from the rmse of the filtered velocity. In other words, you can directly calculate it. The textbook definition is:
$k=\frac{1}{2}({u'u'}+{v'v'}+{w'w'})$

where $u'=u-\overline{u}$ and so on

Well, technically, there should be a density in there too if you want this in units of energy.

FMDenaro · March 21, 2022, 15:55

The title of your post is addressing a question but the content you wrote makes me in some confusion.

From the title, I assume you have obtained the LES velocity fields, that is u_filt, v_filt, w_filt. Now, from these variables, you can compute u_filt^2/2+ v_filt^2/2+w_filt^2/2. Is that the question in title?

The part that involves the fluctuations around the filtered fields are not solved but are in the SGS model.
I see you use an equation for the filtered k, which is a different variable.

However, you can evaluate other forms of the kinetic energy equations. You can evaluate the fluctuation fields around the RANS means by performing explicitly the statistical means of the filtered fields.

Could you better clarify the title of your post?

Gerry Kan · March 22, 2022, 02:42

Folks:

Thanks, and the terminologies could be a little confusing. To clarify, here is what I have to work with:

1) Resolved scale velocity $\left<{u}\right>$

2) SGS TKE $k_\Delta$

Since time-averaged field variables are part of postprocessing, I basically cannot calculate the TKE in the textbook fashion when the simulation is still running. So the short version of the original question is, how I could obtain an estimate of the resolved scale TKE with only the above quantities during simulation time, for which accuracy is of primary concern.

I have come up with two possible ways of doing this, as outlined in the original post. In particular, the second way (gradient transport) currently involves an ad hoc treatment of the eddy viscosity, where an a priori constant value is assigned. The specific question is, therefore, whether there is a more systematic way to represent this without resorting to calculating k and epsilon?

Gerry.

LuckyTran · March 22, 2022, 05:59

I don't know why you are obsessed with trying to calculate it using Boussinesq. Btw, calculating k is exactly what you want. Why would you calculate k to calculate a turbulent viscosity which isn't even applicable, just to calculate.... k

I recommend you look into ways for calculating resolved turbulent kinetic energy at runtime. Almost nobody that does LES can store all the data at every time-step to calculate mean statistics after the fact in a productive calculation. While you're collecting statistics for means, also get the rmse's so you can get the tke. You can even use surrogates for the mean fields by using a preexisting mean field from RANS or use a running average. What software are you using that is not collecting or unable to collect these statistics? I would like to know so I can recommend it to never be used by anyone else. Sorry to say, but if you're doing a time-resolved calclation and can't do something like retrieve mean fields from it, that kind of defeats the purpose of doing a time-resolved simulation.

Gerry Kan · March 22, 2022, 06:29

Lucky:

For the record I am not obsessed with any particular method. Gradient transport seems like a simple and robust enough idea to try out on first attempt.

I don't / can't realize your suggestions at this point because:

1) I am not the code owner for the solver core and (for this code) the calculation of turbulent statistics is optional for performance reasons. Requesting another team to modify this will take this off tangent from the overall objectives.

2) At the moment I only need an estimate. I don't need something accurate when in the end it is fed into a logistic function that gives me a yes / no response. In case you need some more information of what it is about, it is to determine whether the TKE is high enough to shed pollens off flowers. So it is definitely not something that I need immediate rigorous statistical treatment.

3) Since the TKE estimation is only one piece of the puzzle, it is at the moment economically inefficient to go through (1) to get (2) immediately. We will eventually do that, but for the time being I needed a reasonably quick feedback on the effectiveness of the whole algorithm. If it works well, I can justify investing the time and effort to improve its accuracy.

I can appreciate the disagreement that arose from this proposed approach due to its lack of correctness, something which I am very well aware of. That's why I emphasized the "ad hoc" aspect. If I have not stressed this enough, I am not looking for high accuracy, at least not at this very moment, but some way to give me values that are reasonable from which a binary decision could be made.

Gerry.

P.S. - It is a "community code" so I am not profiting financially from this.

Gerry Kan · March 22, 2022, 06:40

I realized what you meant. I think we might have something like that. Let me check. But this still doesn't alleviate points 2 and 3.

Gerry.

FMDenaro · March 22, 2022, 06:44

Quote:

Originally Posted by Gerry Kan

Howdy Folks:

Currently I have a very simple way of making the estimate: $\displaystyle{\left<{k}\right>\approx\frac{k_\text{SGS}}{\phi_\text{SGS}}}$

Thanks in advance, Gerry.

Just to clarify, the <*> operator is the time-averaged mean?

LuckyTran · March 22, 2022, 07:17

There are 0 degrees of freedom in a single sample observation. The uncertainty of any estimator you have for a statistic (the resolved k is a statistic) is at least 100%. You mathematically cannot have any confidence in any number you calculate for the resolved tke using only a snapshot of the velocity field unless it is via sheer dumb luck. And if it is luck, then you might as well flip a coin.

That means you need to resort to post-processing, which is being denied, or rely on a priori knowledge. Does this problem really have no RANS precursor or anything like that to where you have no clue what the tke ought to be? And can you really not run your simulation for just 1 eddy turnover time, get the kinetic energy from there and then keep going? You can even restart the simulation from zero time. It's 1 eddy turnover, shouldn't be a significant part of your calculation.

What about order of magnitude analysis?

Gerry Kan · March 22, 2022, 10:32

Lucky:

I talked to the owner of the core module ... averages only come after 30 minutes of simulation time at intervals of 30 minutes. It won't provide the temporal response.

Like I said ... the TKE is going to yield a yes/no solution and it is (in mine and his opinion) too much work to go through the formalism which I normally would exercise.

Fillipo:

The <> operator is the spatial filter for the resolved scale. But I don't know if you would say there is an implicit temporal average in the TKE already simply because of the way TKE is defined.

Gerry.

FMDenaro · March 22, 2022, 10:41

Quote:

Originally Posted by Gerry Kan

Lucky:

I talked to the owner of the core module ... averages only come after 30 minutes of simulation time at intervals of 30 minutes. It won't provide the temporal response.

Like I said ... the TKE is going to yield a yes/no solution and it is (in mine and his opinion) too much work to go through the formalism which I normally would exercise.

Fillipo:

The <> operator is the spatial filter for the resolved scale. But I don't know if you would say there is an implicit temporal average in the TKE already simply because of the way TKE is defined.

Gerry.

At this point I am not sure to understand what you want to compute… If you have a solved equation for <k> (filtered variabile) in your LES code and you have, pf course, <u>, <v>,<w> (filtered velocity components), you have at each time step what you need.

LuckyTran · March 22, 2022, 16:12

Quote:

Originally Posted by Gerry Kan

I talked to the owner of the core module ... averages only come after 30 minutes of simulation time at intervals of 30 minutes. It won't provide the temporal response.

You don't need temporal response. You just need any estimator for the mean so that you can put an upper bound on the kinetic energy. It doesn't even have to be from the LES. If can be from RANS or whatever. You just need an estimator. If you're choosing to withold this information, that's fine. But you need it.

The idea behind $<k>=\frac{k_{sgs}}{\phi_{sgs}}$ is that in a typical LES simulation one would resolve a certain % of the tke. So let's suppose that in this particular pollen problem the resolved tke is 95%of the total kinetic energy. Knowing the subgrid kinetic energy at runtime one could guestimate that the resolved tke is approximately 20 times the subgrid kinetic energy and there's your estimator.

The actual ratio you wouldn't know until you actually run the LES and get the statistics. Unless.... there exists some prior analysis that was used to generate the grid to ensure that such a resolution was achievable. Because who even generates an LES grid without doing this first? If you could somehow guarantee that at least 75% of the tke is resolved then this estimation would work. I presume this analysis has been done and you know something about the mean fields that you're not telling us. Whatever, fine.

Otherwise, it's crazy talk. It's like saying I measured a velocity of 3 km/s at some unknown location in the universe. Guess what the turbulent kinetic energy was at that instant in space and time? For all we know, it could have been laminar... There's no theory that will help you get any info here. You need some a priori knowledge about your specific flow problem.

Alternatively, you could also abuse Germano's identity and find the trace of the Leonard tensor. But that requires coding, which is outside the scope of this discussion.

Gerry Kan · March 23, 2022, 02:00

Lucky:

Long story short, my strategy is more or less what you have described above.

The resolved TKE is estimated using an a priori fraction (based on previous LES runs on idealized geometry I have a good idea what this should be; in my case 80 to 90 per cent). This serves as my upper bound.

As I anticipate the local resolved TKE be lower than this value due to grid resolution and geometry, I thought an ad hoc method using local filtered velocities might give methis. An obvious choice is to use gradient transport because it is pretty simple to program, so that I can quickly evaluate the soundness of my overall algorithm, which the TKE estimation is a part of.

In the end I went with fixing a turbulent viscosity value about 50x larger than the molecular viscosity of air at room temperature, though I did stress the "ad hoc" aspect of this approach. The resolved TKE seems to fall below my upper bound for the most part. So empirically this method seems to fit the bill.

I was hoping there could be another way to estimate the eddy viscosity without outright prescribing a fixed value, hence the original post, but at the moment it is "good emough".

Gerry.

LuckyTran · March 23, 2022, 03:25

If you could estimate tke using a single arbitrary snapshot of a velocity field we would've solved turbulence 50 years ago. But who knows... maybe I'll wake up tomorrow and there will be a paper that provides exactly this theory.

Gerry Kan · March 23, 2022, 07:58

Quote:

Originally Posted by LuckyTran

If you could estimate tke using a single arbitrary snapshot of a velocity field we would've solved turbulence 50 years ago. But who knows... maybe I'll wake up tomorrow and there will be a paper that provides exactly this theory.

I am not so ambitious and I make no such claim. Given that this will potentially take a another 50 years to solve, life is really too short.

March 21, 2022, 07:01	[LES] Estimating resolved TKE from instantaneous velocities	#1
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	Howdy Folks: I am looking for a way to estimate total TKE in an LES code, since it only has the SGS TKE as a field variable, but not the resolved TKE. Since the TKE is used for a function returning a Yes / No response, a solution within 50% of what it should be is considered acceptable. Currently I have a very simple way of making the estimate: $\displaystyle{\left<{k}\right>\approx\frac{k_\text{SGS}}{\phi_\text{SGS}}}$ where $k_\text{SGS}$ is a user-defined fraction between resolved and SGS scale TKE. For a more sophisticated step, I am thinking of using the above resolved scale TKE as the "upper bound" of TKE, while calculating it using gradient transport theorem: $\displaystyle{ \left<{k}\right> \approx \text{max}\left[ \frac{1}{2} \nu_\text{turb} \overline{ \left( \frac{\partial{\left<{u_i}\right>}} {\partial{x_i}} \right) }, \;\; \frac{k_\text{SGS}}{\phi_\text{SGS}} \right] }$ . Note that the mean (overline) indicates the mean over the three components, and not the RANS average of the velocity. While I can use a random distribution ( $\psi$ ) to estimate relationship between the instantaneous and RANS velocity component, where $\left<{u_i}\right> \approx \psi{u_i}$ , I am not sure how I could approximate the eddy viscosity $\nu_\text{turb}$ because the text book definition involves both and $\epsilon$ . At the minimum I can simply set it to a fixed value so that $\nu_\text{turb} >> \nu$ , and as an estimate it should be sufficient. Before I do something like this, is there a less ad hoc way of doing it? Thanks in advance, Gerry.

March 22, 2022, 02:42		#4
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	Folks: Thanks, and the terminologies could be a little confusing. To clarify, here is what I have to work with: 1) Resolved scale velocity $\left<{u}\right>$ 2) SGS TKE $k_\Delta$ Since time-averaged field variables are part of postprocessing, I basically cannot calculate the TKE in the textbook fashion when the simulation is still running. So the short version of the original question is, how I could obtain an estimate of the resolved scale TKE with only the above quantities during simulation time, for which accuracy is of primary concern. I have come up with two possible ways of doing this, as outlined in the original post. In particular, the second way (gradient transport) currently involves an ad hoc treatment of the eddy viscosity, where an a priori constant value is assigned. The specific question is, therefore, whether there is a more systematic way to represent this without resorting to calculating k and epsilon? Gerry. Last edited by Gerry Kan; March 22, 2022 at 06:03.

March 22, 2022, 05:59		#5
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	I don't know why you are obsessed with trying to calculate it using Boussinesq. Btw, calculating k is exactly what you want. Why would you calculate k to calculate a turbulent viscosity which isn't even applicable, just to calculate.... k I recommend you look into ways for calculating resolved turbulent kinetic energy at runtime. Almost nobody that does LES can store all the data at every time-step to calculate mean statistics after the fact in a productive calculation. While you're collecting statistics for means, also get the rmse's so you can get the tke. You can even use surrogates for the mean fields by using a preexisting mean field from RANS or use a running average. What software are you using that is not collecting or unable to collect these statistics? I would like to know so I can recommend it to never be used by anyone else. Sorry to say, but if you're doing a time-resolved calclation and can't do something like retrieve mean fields from it, that kind of defeats the purpose of doing a time-resolved simulation. FMDenaro likes this.

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March 21, 2022, 15:12		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	The resolved tke you get from the rmse of the filtered velocity. In other words, you can directly calculate it. The textbook definition is: $k=\frac{1}{2}({u'u'}+{v'v'}+{w'w'})$ where $u'=u-\overline{u}$ and so on Well, technically, there should be a density in there too if you want this in units of energy.

March 21, 2022, 15:55		#3
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	The title of your post is addressing a question but the content you wrote makes me in some confusion. From the title, I assume you have obtained the LES velocity fields, that is u_filt, v_filt, w_filt. Now, from these variables, you can compute u_filt^2/2+ v_filt^2/2+w_filt^2/2. Is that the question in title? The part that involves the fluctuations around the filtered fields are not solved but are in the SGS model. I see you use an equation for the filtered k, which is a different variable. However, you can evaluate other forms of the kinetic energy equations. You can evaluate the fluctuation fields around the RANS means by performing explicitly the statistical means of the filtered fields. Could you better clarify the title of your post?

March 22, 2022, 06:29		#6
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	Lucky: For the record I am not obsessed with any particular method. Gradient transport seems like a simple and robust enough idea to try out on first attempt. I don't / can't realize your suggestions at this point because: 1) I am not the code owner for the solver core and (for this code) the calculation of turbulent statistics is optional for performance reasons. Requesting another team to modify this will take this off tangent from the overall objectives. 2) At the moment I only need an estimate. I don't need something accurate when in the end it is fed into a logistic function that gives me a yes / no response. In case you need some more information of what it is about, it is to determine whether the TKE is high enough to shed pollens off flowers. So it is definitely not something that I need immediate rigorous statistical treatment. 3) Since the TKE estimation is only one piece of the puzzle, it is at the moment economically inefficient to go through (1) to get (2) immediately. We will eventually do that, but for the time being I needed a reasonably quick feedback on the effectiveness of the whole algorithm. If it works well, I can justify investing the time and effort to improve its accuracy. I can appreciate the disagreement that arose from this proposed approach due to its lack of correctness, something which I am very well aware of. That's why I emphasized the "ad hoc" aspect. If I have not stressed this enough, I am not looking for high accuracy, at least not at this very moment, but some way to give me values that are reasonable from which a binary decision could be made. Gerry. P.S. - It is a "community code" so I am not profiting financially from this.

March 22, 2022, 06:40		#7
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	I realized what you meant. I think we might have something like that. Let me check. But this still doesn't alleviate points 2 and 3. Gerry.

March 22, 2022, 07:17		#9
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	There are 0 degrees of freedom in a single sample observation. The uncertainty of any estimator you have for a statistic (the resolved k is a statistic) is at least 100%. You mathematically cannot have any confidence in any number you calculate for the resolved tke using only a snapshot of the velocity field unless it is via sheer dumb luck. And if it is luck, then you might as well flip a coin. That means you need to resort to post-processing, which is being denied, or rely on a priori knowledge. Does this problem really have no RANS precursor or anything like that to where you have no clue what the tke ought to be? And can you really not run your simulation for just 1 eddy turnover time, get the kinetic energy from there and then keep going? You can even restart the simulation from zero time. It's 1 eddy turnover, shouldn't be a significant part of your calculation. What about order of magnitude analysis?

March 22, 2022, 10:32		#10
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	Lucky: I talked to the owner of the core module ... averages only come after 30 minutes of simulation time at intervals of 30 minutes. It won't provide the temporal response. Like I said ... the TKE is going to yield a yes/no solution and it is (in mine and his opinion) too much work to go through the formalism which I normally would exercise. Fillipo: The <> operator is the spatial filter for the resolved scale. But I don't know if you would say there is an implicit temporal average in the TKE already simply because of the way TKE is defined. Gerry.

March 23, 2022, 02:00		#13
Gerry Kan Senior Member Gerry Kan Join Date: May 2016 Posts: 376 Rep Power: 11	Lucky: Long story short, my strategy is more or less what you have described above. The resolved TKE is estimated using an a priori fraction (based on previous LES runs on idealized geometry I have a good idea what this should be; in my case 80 to 90 per cent). This serves as my upper bound. As I anticipate the local resolved TKE be lower than this value due to grid resolution and geometry, I thought an ad hoc method using local filtered velocities might give methis. An obvious choice is to use gradient transport because it is pretty simple to program, so that I can quickly evaluate the soundness of my overall algorithm, which the TKE estimation is a part of. In the end I went with fixing a turbulent viscosity value about 50x larger than the molecular viscosity of air at room temperature, though I did stress the "ad hoc" aspect of this approach. The resolved TKE seems to fall below my upper bound for the most part. So empirically this method seems to fit the bill. I was hoping there could be another way to estimate the eddy viscosity without outright prescribing a fixed value, hence the original post, but at the moment it is "good emough". Gerry.

March 23, 2022, 03:25		#14
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	If you could estimate tke using a single arbitrary snapshot of a velocity field we would've solved turbulence 50 years ago. But who knows... maybe I'll wake up tomorrow and there will be a paper that provides exactly this theory.