1. Estimation of Prandtls Mixing Length
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Identification of Macro Mean Free Path....

2. Prandtl Mixing Length Model
Thus, the component of the Reynolds stress tensor becomes
The turbulent shear stress component becomes
This is the Prandtl mixing length hypothesis.
Prandtl deduced that the eddy viscosity can be expressed as

3. Fully Developed Duct Flow
For x > Le, the velocity becomes purely axial and varies only with the lateral coordinates.
V= W = 0 and U = U(y,z).
The flow is then called fully developed flow.
For fully developed flow, the Reynolds Averaged continuity and momentum equations for incompressible flow are simplified as:
With

4. Turbulent Viscosity is a Flow Property
The true Reynolds Averaged momentum equations for incompressible fully developed flow is:

5. Fully Developed Turbulent flow in a Circular Pipe: Modified Hagen-Poiseuille Flow
The single variable is r.
The equation reduces to an ODE:
The solution of above Equation is: ?????
Engineering Conditions:
The velocity cannot be infinite at the centerline.
Is this condition useful???

6. Estimation of Mixing Length
To find an algebraic expression for the mixing length lm, several empirical correlations were suggested in literature.
The mixing length lm does not have a universally valid character and changes from case to case.
Therefore it is not appropriate for three-dimensional flow applications.
However, it is successfully applied to boundary layer flow, fully developed duct flow and particularly to free turbulent flows.
Prandtl and many others started with analysis of the two-dimensional boundary layer infected by disturbance.
For wall flows, the main source of infection is wall.
The wall roughness contains many cavities and troughs, which infect the flow and introduce disturbances.

7. Quantification of Infection by seeing the Effect
Develop simple experimental test rigs.
Measure wall shear stress.
Define wall friction velocity using the wall shear stress by the relation
Define non-dimensional boundary layer coordinates.

8. Approximation of velocity distribution for a fully turbulent 2D Boundary Layer

9. Approximation of velocity distribution for a fully turbulent 2D Boundary Layer
For a fully developed turbulent flow, the constants are experimentally found to be =0.41 and C=5.0.

10. Measures for Mixing Length
Outside the viscous sublayer marked as the logarithmic layer, the mixing length is approximated by a simple linear function.
Accounting for viscous damping, the mixing length for the viscous sublayer is modeled by introducing a damping function D.
As a result, the mixing length in viscous sublayer:
The damping function D proposed by van Driest
with the constant A+ = 26 for a boundary layer at zero-pressure gradient.

11. Based on experimental evaluation of a large number of velocity profiles, Kays and Moffat developed an empirical correlation for that accounts for different pressure gradients and boundary layer suction/blowing.
For zero suction/blowing this correlation reduces to:
With

12. Van Driest damping function

13. Distribution of Mixing length in near-wall region

14. Mixing length in lateral wall-direction

15. Conclusions on Algebraic Models
Few other algebraic models are:
Cebeci-Smith Model
Baldwin-Lomax Algebraic Model
Mahendra R. Doshl And William N. Gill (2004)
Gives good results for simple flows, flat plate, jets and simple shear layers
Typically the algebraic models are fast and robust
Needs to be calibrated for each flow type, they are not very general
They are not well suited for computing flow separation
Typically they need information about boundary layer properties, and are difficult to incorporate in modern flow solvers.