Turbulent Boundary Layer Theory
Introduction: 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 θ and Ue are made dimensionless with the chord length and free stream velocity and H is the shape factor, δ*/θ.
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 θ 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 θ 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/δ)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".