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March 24, 2006, 17:58 
Periodicity

#1 
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Hi,
I am solving an unsteady problem of flow in a tube subject to a periodic body force. The Reynold numbers are high so I have to use a turbulence model. If we denote the period of the body force as T, then although the problem is run forever, it never reaches a periodic state of period T. It may be periodic, but I am not able to determine what the period is. This problem only occurs when the turbulence model is on. When I run at a very low Reynolds number without a turbulence model (by setting the viscosity high) the results are strictly periodic with period T. Is this to be expected or do I have a bug somewhere ? How can I know that the initial transient is over and I have reached the stationary solution ? Thanks ! Sam 

March 24, 2006, 20:41 
Re: Periodicity

#2 
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The instantaneous flow variables themselves will never reach a periodic state. However, after the starting transient dies down enough, the turbulence statistics should develop into a periodic solution. If you check the mean velocities or the mean pressure or the mean reynolds stress or the mean eddy viscosity, etc., these should all settle down to a timeperiodic behavior. The "mean" for the statistics should be the time average over a time interval that is large compared to the turbulence time scales, but small compared to your body force period T (perhaps use T/10 or T/20). Also, the tolerance criterion for calling it periodic should not be too severe. For example, you should not expect that those mean quantities would repeat their values once every period to an accuracy of 10^(12)! If the values are reproduced to within, say, 1% of the variation in values over the period, you should regard that as pretty good periodicity.


March 26, 2006, 17:33 
Re: Periodicity

#3 
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Thanks for your reply !
>If you check the mean velocities or the mean pressure or >the mean reynolds stress or the mean eddy viscosity, etc., >these should all settle down to a timeperiodic behavior. >The "mean" for the statistics should be the time average >over a time interval that is large compared to the >turbulence time scales, but small compared to your body >force period T (perhaps use T/10 or T/20). Also, the >tolerance criterion for calling it periodic should not be >too severe. For example, you should not expect that those >mean quantities would repeat their values once every >period to an accuracy of 10^(12)! This is very interesting. Isnt this the same as saying that if you were to do a spectral decomposition of the instantaenous flow variables, that there would be a strong peak corresponding to T and perhaps mutiples of T and high frequency stuff corresponding to the turbulence ? Also, presumably, this is not true in the transient phase ? In other words, a well defined peak at the forcing frequency is a sign that the starting transient is over and you are in the "periodic" regime. 

March 27, 2006, 01:33 
Re: Periodicity

#4 
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Yes, I would agree with what you said. After the flow settles down to its periodic behavior, in the spectral decomposition, there should be a peak corresponding to the forcing period T. This peak will evolve during the transient phase. The amplitude of the peak will settle down as the initial transient dies down. The amplitude will be proportional to the magnitude of the effect of the periodic forcing on the particular flow variable at the sample flow location. Both my posts express my reasoned expectations. I have never actually run such calculations. So I hope someone who has run such a calculation and checked the statistics, either in the timeaveraged or spectral versions, will give you a definitive answer here.


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