(Book 23.3, 4)

Decompose the flow quantities in average and fluctuating components. Time average:

Fluctuating part:Reynolds averaged equations can be found by averaging the Navier Stokes equations:

*Continuity*

*Viscous Stress Tensor:* Assuming is constant,

*Momentum Equations:* Using the conservation form,

It is seen that the turbulent velocity fluctuations act as an additional shear stress called the Reynolds stress:

Unfortunately, the Reynolds stress is not known unless you solve the full unsteady Navier-Stokes equations. To avoid this humongous task, guessing (also known as modelling) is needed.*Molecular model of (laminar) viscosity:*

(Book 6.2)

Consider the net momentum transfer per unit area through the plane
indicated by the broken line. This momentum transfer produces
the shear stress. Molecules that cross the plane travelled
towards the plane over a distance on the order of a mean free path length
without colliding. So the molecules reaching the plane from
above have an average *x*-velocity that is roughly
higher than the average u at the plane,
decreasing the *x* momentum above the broken line. The fluid particles
that reach the plane from below have an average *x*-velocity that is roughly
lower than the average u at the plane,
also decreasing the *x* momentum above the broken line. As a result,
the fluid above the plane is slowed down at the plane. The force per unit
area follows as the part of that not cancels
between the downward and upward moving molecules. The crossing velocity
corresponding to is of the order of the mean molecular speed, which is of the order of the
speed of sound *a*. So the stress is of order
, and the kinematic viscosity
of order .

*Mixing length idea:*

Assume that the turbulent transport of momentum is similar to the
molecular one. Fluid at a given plane originates from some transverse
distance with much of its velocity difference left intact
during the trip. Then the kinematic eddy viscosity would be , where *v*' is the typical vertical velocity fluctuation. From
continuity, *u*'_{x} + *v*'_{y} = 0, so assuming that there is no strong
directionality in eddy length scales, *v*' is of the order *u*', which
was estimated as , giving an eddy
viscosity

Many things are wrong in the story: For one, turbulent motion is not
small compared to the transverse scales of the flows, so and is in fact related to the velocity at
finite distances, making the entire idea of having universal partial
differential equations (involving local quantities only) suspect.
Also, the turbulent fluctuations are not independent of the velocity
field like *a*, the turbulent shear stress would always be predicted
to be exactly zero at points of even if there is no symmetry around that point, the larger turbulence
scales are directional, etc.