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Reynolds Equation and Mixing Length Model
The probabilistic (ensemble) average is defined as
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(6)
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where  is the
probability density function of  . Ergodicity
assumption implies that the time average and ensemble average are equal. Hence,
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(7)
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Note that the ergodicity hypothesis has not been proven for turbulence; however,
it is commonly used to relate the theoretical results to the experimental data.
It is also well known that while
 ,
 ,
About a century ago, Reynolds suggested to use the
decomposition given by (2) and (3) into the Navier-Stokes equation and
average the resulting equation. Noting that
 ,
 ,
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(9)
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it follows that
 ,
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(10)
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 ,
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(11)
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,
,
.