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Mark Harten September 9, 1999 10:27

Derivation of reynolds equation

reading a text book about the derivation of the Reynold's equation I did not understand the following point.

The instataneous quantities of u, v, w and p are split up in an averaged part and a fluctuating part. Afterwards the equations are time-averaged. Why is the density and its fluctuation not taken into account ? Was it a special treatment of an incompressible case in my text book ? (it was not distinguished between compressible and incompressible case)

Could someone give me a reference where the Reynolds' equation are derived in detail for the general compressible case starting from the compressible N-S equations ?



M. Barbaro September 9, 1999 10:50

Re: Derivation of reynolds equation
See, for example:

Kenneth K. Kuo, "Principles of Combustion", Wiley, 1986, pp. 412-427

Hongjun Li September 9, 1999 11:23

Re: Derivation of reynolds equation
The Reynold's Averaged NS (RANS) equations were derivated (in about 1895 by Reynold) with assumption that density fluctuation was neglected. The averaging procedure for compressible flow is much more complex. If the density is split to R=Rave+R', there are time-averaged (R'u'), (R'v'), etc. terms in the equations. However, very little knowledge is available about these terms. So many investigators have neglected these terms, the resulting equation have variable mean density but additional terms due to density fluctuation are not included and hence, cannot influence turbulent transport.

An alternate approach to variable-density turbulent flow is the use of mass-weighted averaging procedure, sometimes referred to as the Favre Averaging NS equation. I had some references but I lost them when I moved (sorry for this useless information).

Most turbulence models were developed for RANS eqs. and were used for both incompressible and compressible flows. They are still valid according to the Morkovin's Hypothesis which says that the effect of density fluctuations on the turbulene structure will remain small for Mach number below 5 in boundary layers and wakes. You may try to find the following old artical:

Bradshaw, P. "Compressible turbulent share layer." Annual Review of FLuid Mechanics, Vol.9, pp.33-54, 1977.

Hope this will give you a preliminary idea to start with. Regards.


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