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Reynold's Theory | Prandtl mixing length Theory

                                         Reynold's Theory

Fig: Transfer of momentum in turbulent flow



Relative velocity of layer 'A' w.r.t. layer 'B' is



So, fluctuating components in X and Y directions due to turbulence are Vx and Vy .
Now if over a surface of area 'A' perpendicular to the Y-direction and separating two adjacent fluid layers the component 'Vy ' is uniformly distributed.
 So, Mass of fluid transferred across that surface from one layer to another per second =                                                        pAVy
This mass of fluid has started moving with a relative velocity 'Vx'.
So, Transfer of momentum = pAVxVy takes place, resulting in developing tangential forces on each of the layers.
Now, the corresponding turbulent shear stress exerted on fluid layers ( 𝜏) = pVx Vy 

Since, Vx and Vy are varying, the magnitude of '𝜏' will also vary. Hence, usually the time average value of shear stress is considered. So, turbulent shear stress becomes


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Prandtl's mixing length theory


In 1925 Ludwing Prandtl, a German Engineer made an important advance in this direction by presenting mixing length hypothesis, by means of which the turbulent shear stress can be express in terms of measurable quantities related to average flow characteristics.

Fig: Prandtl's hypothesis

According to Prandtl, mixing length is that length in the transverse direction which must be covered by a lump of fluid particles travelling with its original mean velocity in order to make the difference between its velocity and the velocity of the new layer equal to the mean transverse fluctuation in turbulent flow.
So, mixing length is the distance between the two layers where momentum exchange between two layers occurs due to the displacement of fluid particles.
Now, according to Prandtl, fluctuating component ' Vx 'may be related to the mixing length 'l' as
















Which is the required prandtl's mixing length equation for turbulent shear stress. 
Again, total shear stress at any point is the sum of the viscous shear stress and turbulent shear stress and it may be expressed as, 





However, the viscous shear stress is negligible as compared with turbulent shear stress.


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