Reynold's decomposition
symbolically
and
to be remembered
and
symbolically
and
to be noted
The mean of the instantaneous continuity condition
Yields continuity condition for the mean flow,
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(7.7) |
Which in turn, together with Reynolds' decomposition yields
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(7.8) |
An equation of motion for the mean velocity is obtained by averaging the NS equation. The major step in the derivation is
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(7.9) |
The complete equation therefore is
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(7.10) |
Which is known as Reynold's Averaged N-S Equation. This equation may be rewritten as
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(7.11) |
or
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~ 10-3Re (considering intensity  ) |
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If Re ~ 105 , the RTV is 100
 or k (turbulent kinetic energy) = 
Turbulent stresses are invariant under arbitrary rotation and reflection of the coordinate axis
Rotation
Fig 7.1
 (rotational symmetry)
 (rotational symmetry)
 (rotational symmetry)
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Therefore
Reflection 
Figure 7.2
Reflection (mirror image symmetry)
Axis, X1
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| Axis, X2 |
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| Axis, X3 |
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