Turbulent fluctuation
 

Hello,
I am working with a LES code which provides total velocity components(U and V). In order to seperate average u from fluctuating velocity, I have to use ensemble averaging, that should be on different realizations. I am a little confused, because the flow that I am working on is unsteady flow. In this case to seperate u and u' do I have to average in space?(look a little wierd to me, because it would smooth the velocity function instead)
Thank you in advance

LES formulation idea is about separting Large Eddies to DNS from small scale which will be modeled with some assumptions due to the closure problem (By separation I ment via wave number). On the other hand ensemble averaging under the assumption of ergodicity is somewhat the basics for URANS. the problem with the assumption is the basic flaw arrising from the ergodic theorem which actually makes the ensemble-average not time dependent.

Is ergodic theorem valid for k-epsilon and even when there is no turbulent model applied on NS?
and for space averaging, for example for a shock wave, if averaging would be done on space, instead of finding ensemble averaged components we would end up changing flow structure.
Thanks again

Im not certain about what you mean by "ergodic theorem valid for k-e". The ergodic theorem applied to turbulence in one context mainly states that for a stationary flow (which is said to be valid for say turbulent flows) then in the formal limit of T-->infinity which must imply in time averaging, by applying the egodic theorem one may assume that time averaging equals ensemble averaging. then by ensemble averaging NS one gets what seems to be a time dependent equation and therefore by ergodic theorem a time dependent Reynolds decompositioned NS. Only what I claimed is that the ergodic theorem states just the opposite, meaning that the solution to ensemble averaged NS can not be time dependent because of its equality to time averaged solution.
My question is what kind of decomposition is used in order to achieve k-e (Reynolds, Hilbert, LES, etc...) and folowing that what is the justification for ensemble averaging?
By principle LES decomposition comprises spatial filtering and not spatial averaging hence saperating the flow to a large wave number part which fits small scales (SGS) to be modeled, and a low wave number part for large scales which is simulated via DNS.