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Reynolds Equation and Mixing Length Model Energy Balance in Turbulent Flow The Reynolds equation is given as
Multiplying (1) by
where
Equation (2) is the statement of balance of mean mechanical energy for the mean motion. Subtracting (1) from the Navier-Stokes equation, it follows that
Multiplying (3) by
where
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and rearranging terms, one finds
= convection,
= diffusion by turbulence,
= viscous diffusion,
= direct viscous dissipation,
= fluctuation energy production.
and taking expected value, we find the equations of balance turbulence energy fluctuation. i.e.,
is the fluctuation kinetic energy and
= convection
= turbulent diffusion
= viscous diffusion
= dissipation
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