[mjavrincon]

Dear community,

I have been trying to calculate the turbulence kinetic energy (TKE) of large eddy simulations (LES) for a while.
I have found that using the fieldAverage utility in OpenFOAM (the standard approach as I have seen) there is a big problem in the calculation of UPrime2Mean. Let me explain by having this code in controlDict:

functions
{
    fieldAverage
    {
        type            fieldAverage;
        libs            ("libfieldFunctionObjects.so");
        writeControl    writeTime;

        fields
        (
            U
            {
                mean        on;
                prime2Mean  on;
                base        time;
            }
        );
    }
}

The problem is that using directly this approach and starting the simulation, the UMean value used to calculate UPrime2Mean is not converged, thus yielding incorrect data unless a huge simulation is run to compensate for the iterations collected in which UMean is not converged.

As a solution, I have been trying to run first only UMean averaging, and when this field is converged, start the UPrime2Mean calculation, hoping that the convergence of the 2nd order statistics does not take that long.

However, I have not succeeded at doing so since the calculation of UPrime2Mean takes into account the number of time steps already used to calculate UMean and the UPrime2Mean field barely changes from the results obtained in the first iteration.

Have someone managed to fix or use a previous UMean field and use it to calculate UPrime2Mean?

Thank you in advance!

P.S. I have attached a figure of velocity probes in function of simulation time with the instantaneous values and 1st order statistics to show what I mean.

[Lanius]

Hi Mario, I have noticed the same issue. Have you figured out a solution?
I'm thinking using the umean from the last timestep and u from every timestep to calculate the uprime2mean manually, but it will only consider "written" timestep instead of "run" timestep, so I'm not sure if that's a good idea.

Edit: I have done some more tests and found that I was wrong.
The prime2mean value is calculated by the instantaneous value of every timestep, and mean value of the "LAST" step, instead of directly using the prime2mean value from previous timesteps which are based on "PREVIOUS" mean value.

[Tobermory]

Originally Posted by mjavrincon

The problem is that using directly this approach and starting the simulation, the UMean value used to calculate UPrime2Mean is not converged, thus yielding incorrect data unless a huge simulation is run to compensate for the iterations collected in which UMean is not converged.

You haven't understood the functionObject - it's actually rather clever and extremely efficient in storage (harks back to the days when supercomputer memory was expensive, and very limiting on your DNS/LES simulations, so you had to work hared not to waste it). In a nutshell, the averages (mean and variance) BOTH get better, the longer you average for and the variance is constantly using the latest value of the mean field.

Don't believe me? Here's the derivation. Start with the mean field, and write the mean based on n samples as:



Now substitute and you have a compact way of computing the mean based on just the current sample value and the previous value of the mean (i.e. you don't have to store all the previous sample values).

Now buckle up, and let's do the variance. Start with the definition of the perturbation from which it is easy to show that . Then:



Now substitute for the summation term using the earlier identity to get:


and once again you have a method for calculating the latest, best estimate of the variance using the previous value of the variance and the current sample value. The keen-eyed amongst you will have spotted though that the above requires both the previous and current values of the mean. To save having to store another field (the old mean), the functionObject does some fancy footwork - it calls function addMeanSqrToPrime2Mean() that adds the old mean square to the old prime2 field before updating the mean and then updating the prime2 fields. A masterwork of coding.

So the long and short of it is that you just have to keep on sampling, and both the mean and the variance fields will improve.