Turbulent Boundary Layer Theory

The equations for turbulent flow are the full Navier-Stokes equations. We can simplify the equations by using empirical models for the more complex aspects of the turbulence and making the assumption of a thin layer. The method described here is used to compute the characteristics of a "steady" turbulent boundary layer with a pressure gradient.

In this case we solve two coupled first order ODE's. The first is the von Karman integral equation:

where q and Ue are made dimensionless with the chord length and freestream velocity and H is the shape factor, d*/q.

The second is an expression describing the entrainment of flow into the boundary layer:


Here, H1 is the mass flow shape factor:

and F is an entrainment parameter.

These variables are related to each other by the following expressions.
The skin friction coefficient is related to H and q by the Ludwieg-Tillman skin friction law:


The mass flow shape factor, H1 can be related to H by the following fits:


Finally, the entrainment parameter, F, is approximated by:


We then just integrate this system of equation numerically to obtain the variation of q and H with position along the airfoil.


These expressions must be evaluated numerically, but some useful results for flat plates are given below. (Re = 0.5 to 10 million):



The formulas involving Re.2 are based on the assumption of a 1/7th power law shape for the boundary layer profile u/Ue = (y/d)1/7. Results agree with experiments up to Re = 20 million. Above this, the skin friction is underestimated. The results based on the logarithmic distribution agree well with limited test data out to Re = 500 million.

In many cases, the flow starts laminar, undergoes transition, and continues as a turbulent boundary layer. The details of the transition process are still a matter of current research, but we may estimate the gross features of the flow by matching assuming a virtual start of the turbulent layer upstream of transition so that the momentum thicknesses of the laminar and turbulent layers match at the transition "point".