RANS equations
Good morning
looking at the RANS eq pag 61 solver theory.... I can't understand why if we do a time average we can still see some time derivative in the equations. I suppose that even if I do this averaging, the variables are still time dependent but why? Moreover: why I can't see the density fluctuations in the equations? Is the density a mean value like the others variables ? Can anybody spend few words about the Rans equations? Thank you all Alex 
Re: RANS equations
Hi,
Why is there a derivative WRT time in the RANS equations? This is because RANS assumes you can seperate turbulence timescales and bulk flow timescales. RANS averages out the turbulence timescales and puts their effect on the flow via a turbulence model, whereas the bulk flow timescale is due to changes in the bulk or Reynolds/Fauve averaged flows. That is, it is possible to have transient RANS turbulent flows. Why is density fluctuations not in the equations? Have a look at the derivation of the RANS equations in a fluids textbook, such as "Turbulence for CFD" by Wilcox. Wilcox also discusses the timescale issues mentioned above in some detail. Regards, Glenn 
Re: RANS equations
Hi Glen
Thanks for the answer. Now I understand why I can still see a time derivative in the RANS. Unfortunately I haven't the book you talk about. About the absence of density fluctuations: I read that for a slightly compressible fluid, we can suppose these fluctuations as zero. Is it rigth? Thank again for your attention Alex 
Re: RANS equations
Hi Alex,
No, the density fluctuations are included in the derivation. For the compressible Navier Stokes equations some properties are Fauve (or mass) averaged such as velocity components and enthalpy; and some are Reynolds averaged such as density and pressure. This means the density terms in the compressible Fauve averaged Navier Stokes equations are Reynolds averaged, but a number of other terms such as the velocity are Fauve averaged. The density fluctuation term appears in the Reynolds stress tensor, and this is modelled by the turbulence model. Regards, Glenn 
Thanks Glenn

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