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fluctuation component of velocity (Urms) calculation from RANS simulation results

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Old   January 14, 2021, 11:14
Default fluctuation component of velocity (Urms) calculation from RANS simulation results
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Hello All


While looking methods to extract Urms velocity from RANS simulation results I found two methods to calcuulate the Urms from RANS simulation results but I am not really sure, which method is more accurate.



As per RANS, instantaneous fluid velocity is decomposed into time averaged mean velocity (UMean) and fluctuating velocity (Urms or U'). RANS simulations directly provide the mean velocity (UMean) at each time step, so ultimately the RANS is steady solution. UMean doesn't change over time.



About fluctuation component of velocity (Urms or U'), In RANS, eddy viscosity approach is used to model effect of these fluctuating components and to close the Reynolds averaged NS Eqn. In this way we get Reynold stresses (rho*<u'v'>) and from there we can extract the Urms or U'. Other way, Urms can be directly calculated assuming the isotropic turbulent condition directly from calculated turbulent kinetic energy (k).


1st method based on Reynold Stress: After reading many threads, I found using function objects Uprime2Mean, we can get the Urms, As UPrime2Mean represents the Reynolds stresses.



Urms = sqrt(UPrime2Mean)
where by definition UPrime2Mean in openFoam is: UPrime2Mean = 1/N *sum(Ui-UMean)^2.



But What I understand, RANS simulation gives Umean at each time step, It means Ui=UMean for each time step for RANS simulation, which makes lUprime2Mean to be zero, thus Urms = 0.


2nd method based on calculated turbulent kinetic energy (k): By assuming the turbulent fluctuation to be isotropic (u'=v'=w'), we can get


Urms or U'=(2k/3)^0.5.


Above two methods I found in literature but Theoretically they produce different Urms. I would be grateful if someone could provide their expert comments on this.


Best Regards
Atul Jaiswal
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Old   January 14, 2021, 12:55
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Quote:
Originally Posted by atul1018 View Post
Hello All


While looking methods to extract Urms velocity from RANS simulation results I found two methods to calcuulate the Urms from RANS simulation results but I am not really sure, which method is more accurate.



As per RANS, instantaneous fluid velocity is decomposed into time averaged mean velocity (UMean) and fluctuating velocity (Urms or U'). RANS simulations directly provide the mean velocity (UMean) at each time step, so ultimately the RANS is steady solution. UMean doesn't change over time.



About fluctuation component of velocity (Urms or U'), In RANS, eddy viscosity approach is used to model effect of these fluctuating components and to close the Reynolds averaged NS Eqn. In this way we get Reynold stresses (rho*<u'v'>) and from there we can extract the Urms or U'. Other way, Urms can be directly calculated assuming the isotropic turbulent condition directly from calculated turbulent kinetic energy (k).


1st method based on Reynold Stress: After reading many threads, I found using function objects Uprime2Mean, we can get the Urms, As UPrime2Mean represents the Reynolds stresses.



Urms = sqrt(UPrime2Mean)
where by definition UPrime2Mean in openFoam is: UPrime2Mean = 1/N *sum(Ui-UMean)^2.



But What I understand, RANS simulation gives Umean at each time step, It means Ui=UMean for each time step for RANS simulation, which makes lUprime2Mean to be zero, thus Urms = 0.


2nd method based on calculated turbulent kinetic energy (k): By assuming the turbulent fluctuation to be isotropic (u'=v'=w'), we can get


Urms or U'=(2k/3)^0.5.


Above two methods I found in literature but Theoretically they produce different Urms. I would be grateful if someone could provide their expert comments on this.


Best Regards
Atul Jaiswal



I think you are confusing RANS and URANS. You are talking about "time step", thus I assume you are running URANS.
The resulting mean velocity you compute is actually still time-dependent (in a way it is largely debated but this is not the right post to discuss about).
Thus, you have to perform the statistical averaging on the URANS fields to get the steady mean RANS velocity. The residual v_RANS-v_URANS can be then extracted. Be careful that they are not the real physical fluctuations but contain the effect of the turbulence model.
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Old   January 14, 2021, 13:20
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Hello FMDenaro



Quote:
I think you are confusing RANS and URANS. You are talking about "time step", thus I assume you are running URANS.The resulting mean velocity you compute is actually still time-dependent (in a way it is largely debated but this is not the right post to discuss about).

-yes, I am doing you URANS simulations. But as per Reynold's Average NS equations, the solutions should be time independent and give mean values, So URANS and RANS should result to the same solution.




Quote:
Thus, you have to perform the statistical averaging on the URANS fields to get the steady mean RANS velocity. The residual v_RANS-v_URANS can be then extracted. Be careful that they are not the real physical fluctuations but contain the effect of the turbulence model.

-As URANS are time dependent, we won't get Urime2Mean (thus Urms) equal to zero by calculating it from method 1. I get URANS results which are time independent and and performed time averaging over URANS results to get statistically time independent velocity field. In this case the UPrime2Mean (or Urms) will not be zero. But still method 1 (based on UPrime2Mean) will results the different Urms value than Urms calculated using method 2 (assuming turbulence isotropic, directly from k). I checked d it by myself that both methods result different Urms values. My question is, which method is more accurate and represents the real physical fluctuations and why?


Best Regards
Atul
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Old   January 14, 2021, 13:25
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No one of the methods can give you the physical fluctuations when URANS/RANS are used. In statistical formulations you have only low order statistics.

The URANS solution is unsteady but the time-dependence of the computed mean velocity depends on external time-dependent forcing. That enters into the model, the only way URANS can differentiate from RANS. YOu have to accept that you have a global expression for the average of the velocity correlation expressed by the model.
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Old   January 15, 2021, 06:53
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Dear FMDenaro


Thanks for clarification.


Best Regards
Atul
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Old   February 4, 2021, 11:09
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Hello Foamers


This is a small observation. I have now understood (i guess) the both methods to calculate the Urms and their meanings. Here is what I think (correct me if i am wrong)



So the two methods to calculate Urms are:


(1) by definition UPrime2Mean in openFoam is: UPrime2Mean = 1/N *sum(Ui-UMean)^2


Urms = sqrt(UPrime2Mean)




(2) method based on calculated turbulent kinetic energy (k): By assuming the turbulent fluctuation to be isotropic (u'=v'=w'), we can get

Urms or U'=(2k/3)^0.5.

Its obvious that both methods will give different Urms values and that mainly depends on which method (URANS/LES) one is using to close the NS equation.



In RANS/URANS, the solution gives temporal mean values and all the small scales (fluctuating component of velocity) than mean values are modelled using eddy viscosity approach. So when you apply method 1, you get very small value (ideally zero) of Urms. Because method 1 calculates the resolved part of Urms and RANS doesn't resolve any TKE (almost all of TKE is modelled). This means in RANS/URANS, method 2 give better estimation of TKE (obviously the modelled TKE).


In LES, solution gives instantaneous velocity values upto certain sclaes (usually inertial scale) and dissipative eddies are modelled using some closing equations (similar to RANS using eddy viscosity approach). So the resolved part of Urms can be calculated using method 1 and modelled part of Urms can be calculated using method 2. In this way one can also estimate how much %age of TKE is resolved and how much is modelled.



To conclude, method 1 gives the resolved part of Urms (resolved TKE) and method 2 gives modelled part of Urms (modelled TKE). In RANS, almost all of TKE is modelled, so modelled Urms is near to the exact value but in LES, a part of TKE is resolve and a part is modelled, which depends on mesh refinement and schemes used. So in LES, the total Urms = Urms (method1; resolved)+Urms(method2; modelled), gives the good estimation of exact values.





This was my understanding, I hope it is correct and will help the others.


Best Regards
Atul
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