Boundary Layer Theory and Definitions

Boundary Layer Thickness

The boundary layer thickness, d, is defined as the distance required for the flow to nearly reach U_e. We might take an arbitrary number (say 99%) to define what we mean by "nearly", but certain other definitions are used most frequently. The displacement thickness and momentum thickness are alternative measures of boundary layer thickness and are used in the calculation of various boundary layer properties.

The displacement thickness is defined by considering the total mass flow through the boundary layer. This mass flow is the same as if the boundary layer were completely at rest, with a thickness, d*:

For laminar boundary layers d* is about one third of the distance to the edge of the boundary layer, d.

The momentum thickness, q, is defined similarly, using the momentum flux rather than the mass flux:

For laminar boundary layers, d tends to be about an order of magnitude greater than q.
The ratio of d* to q is termed the shape factor, H:

The Boundary Layer Equations

Newton's law applied to a fluid element in 2-D is:

and, in the x direction:

This leads, directly to:

This boundary layer equation is combined with the continuity equation and dp/dy = 0 to obtain the 3 equations in the unknowns: p, u, and v.

One of the basic results (assumptions?) in this development of boundary layer theory is that the static pressure is constant through the boundary layer (dp/dy = 0). This is a rather important result, but it is not exactly true*.

We start with the 2-D boundary layer equation shown above. For steady flow, this reduces to:

The approach to the solution of this equation is to assume that the pressure does not vary with y, so it is specified by the external velocity distribution. v is computed from the continuity equation, leaving a PDE in u to be integrated. However, this holds only for laminar flow since turbulent boundary layers are inherently unsteady. Subsequent sections deal with the solution of these equations in more detail.

*If one considers the balance of normal forces on a fluid element we can see that dp/dn = r V²/R
where R is the radius of curvature. This holds even for viscous fluids since we expect viscosity to produce shear stresses and not normal stresses.

Now, if C_p = (p-p₀)/(.5 r U₀²)
then, dCp/dn = 2 (V/U₀)² / R
Thus, the change in C_p through the boundary layer is: C_p ~ d/R * (V/U₀)²

So, as long as the radius of curvature is much larger than the boundary layer thickness, and the local velocities are not too large, this is true. Such conditions are not always met, though!