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Random Data Analysis Part 1: Techniques for Transient Flow Field Data

CFD Analysis
June 5, 2015 By: Jonathan Dudley

A stochastic process is by definition a random process. A stochastic process is said to be ergodic if its statistical properties can be derived from only a sub-set of the entire dataset. This subset, or window of data, must represent the average statistical properties of the entire process regardless of where you extract the window from in the process. In a nutshell, this implies that we expect that the statistics of the signal would converge to a single value if given a large enough sampling window. With RANS CFD simulations, we most commonly focus on the first three statistical moments: the mean, variance, and standard-deviation. For advanced turbulence modeling with data acquired by the use of LES and DNS solvers, we may also decide to include skew and kurtosis in our CFD analysis.

For illustration purposes, let’s consider a random pressure signal as shown in Figure 1. The data is acquired at a single point in space and sampled at uniform frequency of 100,000 hertz.

Figure 1: Acquired random pressure signal

To calculate these statistics as a function of sample size Eqn. 1 is used incrementing from i=1 to i=N. Figure 2 illustrates a plot for the running average and running standard deviation , given by Eqn 1 below, of this signal.

The dotted red lines represent the mean and standard deviation to within plus or minus one percent of the calculated value when using the entire dataset. Upon inspection of Figure 2, we readably observe two key points:

1) The mean convergences much more rapidly than the standard-deviation (within approximately 500-600 samples). This is not a surprising result as the mean is utilized in the calculation of the standard-deviation.

2) The block size required for convergence of the first 3 statistical moments (to within one percent) is approximately Nc=20,000 (recall, the variance is nothing more than the standard-deviation squared).

 

Figure 2: Running mean and standard deviation

This implies we would require approximately of simulation time per block for our mean and variance to be fully converged to within one percent.

For transient analysis which is expected to be stationary and ergodic, this simple analysis can be quite powerful. The only way to ensure the initial transients have settled in your time marching simulation is by monitoring these statistics and ensuring they have converged to within a reasonable tolerance. If one wishes to only post-process mean flow field results from the transient data set, such as velocity and pressure quantities, the required amount of simulation effort is far less than acquiring statistics related to the fluctuating components; (as they relate to the higher order statistical moments) such as turbulent velocities, Reynolds stress, or block sizes used for Fast Fourier Transform analyses.

In the example above, I have only included the results from a single point in the flow field. Using ANSYS CFX you can monitor these results using the transient-statistics tab and the “Output Control” menu which writes the running mean and standard deviation for the entire computational domain as illustrated in Figure 3 and 4 for flow over a cylinder.  Using CFD-Post and ability to compare-cases (say, at iteration “i” compared to iteration “i+1000”), you could generate contour plots and monitor these statistics for multiple planes. This would allow you to rapidly gain a qualitative assessment of global transient-statistical convergence levels.
 

Fig 3. Instantaneous u-velocity contours for flow over a cylinder started from 0 velocity


Fig 4: Mean u-velocity contours for flow over a cylinder started from 0 velocity

Hopefully, this sheds some light on how you can apply random data analysis statistical techniques to monitor the convergence levels related to the initial transients for your time marching analysis. If you have further experiences or questions please feel free to add a comment.