Prandtl's mixing length at the wall of a pipe is
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In analogy with the coefficient of viscosity for laminar flow, J. Boussinesq introduced a mixing coefficient  for the Reynolds stress term, defined as

Using  the shearing stresses can be written as

such that the equation

may be written as

(33.13)
The term νt is known as eddy viscosity and the model is known as eddy viscosity model .
Unfortunately the value of νt is not known. The term ν is a property of the fluid whereas νt is attributed to random fluctuations and is not a property of the fluid. However, it is necessary to find out empirical relations between νt, and the mean velocity. The following section discusses relation between the aforesaid apparent or eddy viscosity and the mean velocity components
Prandtl's Mixing Length Hypothesis
Consider a fully developed turbulent boundary layer . The stream wise mean velocity varies only from streamline to streamline. The main flow direction is assumed parallel to the x-axis (Fig. 33.4).
The time average components of velocity are given by  . The fluctuating component of transverse velocity transports mass and momentum across a plane at y1 from the wall. The shear stress due to the fluctuation is given by

(33.14)
Fluid, which comes to the layer y1 from a layer (y1- l) has a positive value of . If the lump of fluid retains its original momentum then its velocity at its current location y1 is smaller than the velocity prevailing there. The difference in velocities is then

(33.15)

Fig. 33.4 One-dimensional parallel flow and Prandtl's mixing length hypothesis
The above expression is obtained by expanding the function  in a Taylor series and neglecting all higher order terms and higher order derivatives. l is a small length scale known as Prandtl's mixing length . Prandtl proposed that the transverse displacement of any fluid particle is, on an average, 'l' .

Using  the shearing stresses can be written as

such that the equation

may be written as

(33.13)
The term νt is known as eddy viscosity and the model is known as eddy viscosity model .
Unfortunately the value of νt is not known. The term ν is a property of the fluid whereas νt is attributed to random fluctuations and is not a property of the fluid. However, it is necessary to find out empirical relations between νt, and the mean velocity. The following section discusses relation between the aforesaid apparent or eddy viscosity and the mean velocity components
Prandtl's Mixing Length Hypothesis
Consider a fully developed turbulent boundary layer . The stream wise mean velocity varies only from streamline to streamline. The main flow direction is assumed parallel to the x-axis (Fig. 33.4).
The time average components of velocity are given by  . The fluctuating component of transverse velocity transports mass and momentum across a plane at y1 from the wall. The shear stress due to the fluctuation is given by

(33.14)
Fluid, which comes to the layer y1 from a layer (y1- l) has a positive value of . If the lump of fluid retains its original momentum then its velocity at its current location y1 is smaller than the velocity prevailing there. The difference in velocities is then

(33.15)

Fig. 33.4 One-dimensional parallel flow and Prandtl's mixing length hypothesis
The above expression is obtained by expanding the function  in a Taylor series and neglecting all higher order terms and higher order derivatives. l is a small length scale known as Prandtl's mixing length . Prandtl proposed that the transverse displacement of any fluid particle is, on an average, 'l' .
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