This chapter deals with the effects of viscosity in two dimensions. The sections describe the basic phenomena and some simple theory that may be used to estimate boundary layer properties.

Boundary layers appear on the surface of bodies in viscous flow because the fluid seems to "stick" to the surface (*see note). Right at the surface the flow has zero relative speed and this fluid transfers momentum to adjacent layers through the action of viscosity. Thus a thin layer of fluid with lower velocity than the outer flow develops. The requirement that the flow at the surface has no relative motion is the "no slip condition."

The velocity in the boundary layer slowly increases until it reaches the outer flow velocity, U

The boundary layer thickness, δ, is defined as the distance required for the flow to nearly reach U

The boundary layer concept is attributed primarily to Ludwig Prandtl (1874-1953),
a professor at the University of Gottingen. His 1904 paper on the subject
formed the basis for future work on skin friction, heat transfer, and separation.
He subsequently made fundamental contributions to finite wing theory and
compressibility effects. (His name appears about 30 times in these notes.)
Theodore von Karman and Max Munk were among his many famous students. R.T.
Jones was a student of Max Munk and I have subsequently learned a great
deal from R.T. Jones -- which makes readers of these notes great-great grandstudents
of Prandtl.

The character of the boundary layer changes as it develops along the surface
of the airfoil. Generally starting out as a laminar flow, the boundary layer
thickens, undergoes transition to turbulent flow, and then continues to
develop along the surface of the body, possibly separating from the surface
under certain conditions.

In laminar flow, the fluid moves in smooth layers or lamina. There is relatively
little mixing and consequently the velocity gradients are small and shear
stresses are low. The thickness of the laminar boundary layer increases
with distance from the start of the boundary layer and decreases with Reynolds
number.

As the fluid is sheared across the surface of the body, instabilities develop
and eventually the flow transitions into turbulent motion.

Turbulent boundary layer flow is characterized by unsteady mixing due to
eddies at many scales. The result is higher shear stress at the wall, a
"fuller" velocity profile,and a greater boundary layer thickness.
The wall shear stress is higher because the velocity gradient near the wall
is greater. This is because of the more effective mixing associated with
turbulent flow. However, the lower velocity fluid is also transported outward
with the result that the distance to the edge of the layer is larger.

Several fundamental effects are produced by viscosity:

Drag: Skin friction drag caused by shear
stresses at the surface contribute a majority of the drag of most airplanes.

The pressure distribution is changed by
the presence of a boundary layer, even when no significant separation is
present. This changes C_{L} and C_{m}.

Flow separation: Viscosity is responsible
for flow separation which causes major changes to the flow patterns and
pressures.

To compute these characteristics some basic boundary
layer theory is described here with more detailed computational methods
for laminar and turbulent boundary layers.

*Actually, the zero slip condition at the surface arises from the roughness of the surface on a molecular scale. Fluid molecules hitting the surface impart a net momentum to the surface and the mean velocity of molecules hitting the surface is about the same as the surface velocity.

Even when the surface is extremely smooth, electrostatic forces exist between the surface and the air molecules, introducing the shear stress at the surface. If this interaction could be reduced, a reduction in skin friction would result, but no one has found a way to do this. (Yet.)