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Far August 6, 2011 02:59

Ensemble averge and time average
Dear Frends

I am studying the boundary layer on the low pressure turbine. I saw the velocity profiles in literature as ensemble average and time average.

Does time average means getting the values for many time steps in constant spatial locaion and average them w.r.t time

and ensemble average means repeating the experiment for n times at the same spatial location and then getting the average w.r.t n times.

ghorrocks August 6, 2011 07:21

Yes, you are correct.

Far August 6, 2011 09:15

Thanks for reply

When running CFD (URANS), does they mean same? if they are different how to find them in CFD as one is not going to repeat CFD?

Best Regards

Far August 6, 2011 15:00

In fluid mechanics, an ensemble is an imaginary collection of notionally identical experiments.
Each member of the ensemble will have nominally identical boundary conditions and fluid properties. If the flow is turbulent, the details of the fluid motion will differ from member to member because the experimental setup will be microscopically different; and these slight differences become magnified as time progresses. Members of an ensemble are, by definition, statistically independent of one another. The concept of ensemble is useful in thought experiments and to improve theoretical understanding of turbulence.

This is the definition I have from Wikipedia

But I confused that this definition talks about both independent experiments and time at the same time? Why time is included in this definition?

Best Regards

Far August 7, 2011 02:35

In equilibrium systems, time and ensemble averages of physical quantities are equivalent due to ergodic exploration of phase space.

This is from paper published in

Phys. Rev. Lett. 98, 220602 (2007) [4 pages]
Limits of the Equivalence of Time and Ensemble Averages in Shear Flows

Far August 7, 2011 02:51

1 Attachment(s)
Here is snapshot of notes on time average vs ensemble average. Here ensemble average again shows that the both spatial and time factors are considered

Far August 7, 2011 04:29

These notes are taken from above link:

In general, the could be realizations of any random variable. The defined by the ensemle average definition defined above represents the ensemble average of it. The quantity is sometimes referred to as the expected value of the random variables , or even simple its mean.
For example, the velocity vector at a given point in space and time , in a given turbulent flow can be considered to be a random variable, say . If there were a large number of identical experiments so that the in each of them were identically distributed, then the ensemble average of would be given by

Note that this ensemble average, , will, in general, vary with independent variables and . It will be seen later, that under certain conditions the ensemble average is the same as the average which would be generated by averaging in time. Even when a time average is not meaningful, however, the ensemble average can still be defined; e.g., as in non-stationary or periodic flow. Only ensemble averages will be used in the development of the turbulence equations here unless otherwise stated.

Far August 7, 2011 04:51

simple definition
Ensemble average : It can be defined as the recorded samples at the fixed spatial location for some non dimensional time (e.g. t/T = 0.25, 0.5, 0.75 and 1.0) and this experiment is repeated for same time interval (e.g (t/T)1 = 0.25, (t/T)2 = 0.25, (t/T) = 0.25) and averaged over all the samples taken then this average is known as ensemble average. Best example can be vortex shedding over cylinder. The sample can taken for the 8 to 12 cycles after the solution becomes periodic.

On the other hand for the time average we add all times for all time intervals and average them

So with time average we get one single value at the fixed spatial location whereas the ensemble average provides the different values for each non dimensional time i.e. t/T (0.25) is not equal to t/T (0.5) and so on so forth.

It should be noted that in steady state cases both have the same meaning. whereas it has different meaning in URANS, DES, LES and DNS

I hope this discussion may be useful to others and to my believe accurate as well. However comments are most welcome and I propose to add this definition, in more refined form with mathematical definition and some examples, to cfd wiki

Best Regards

ghorrocks August 7, 2011 08:45

To comment on your question is ensemble average the same as time average - the answer is no. Imagine a lot of blades in a rotor disk rotating - when each blade passes a specific point, if these flow fields were averaged you would have an ensemble average. But a time average requires an averaging time scale, something larger than the turbulent time scales but smaller than the bulk flow time scales. Then the RANS approximation is valid and the flow can be separated into bulk components (ie the RA velocities and pressure) and turbulent components (k, e, omega, Reynolds Stresses).

Note that often machinery does not allow a conveniant segregation of flow time scales into bulk and turbulent. For instance if the blade passing frequency is higher than the lowest turbulent time scale this would be the case. Then the RANS approximation is of reduced validity, and possibly can be made totally invalid.

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