Viscous Drag

Skin Friction

The shearing stresses at the surface of a body produce skin friction drag. We define the skin friction coefficient, Cf, by:

The shear stress is then related to the viscosity by:

Cf is related to the drag coefficient by CD (skin friction) = Cf*Swetted/Sref.
where Swetted is the area "wetted" by the air and Sref is the reference area used to define the drag coefficient. This expression applies to a flat plate. When the body has thickness, the local velocities on the surface may be higher than the freestream velocity and the skin friction is increased. We usually write: CD = k * Cf * Swetted / Sref where k is a "form factor" that depends on the shape of the body.

The skin friction coefficient varies with Reynolds number, Mach number, and the character of the boundary layer. The momentum transferred between the air and the body surface appears as a velocity deficit in the viscous wake behind the body.

The plot below shows how Reynolds number and the location of the transition from laminar flow to turbulent flow, affects the skin friction coefficient.

From the basic boundary layer theory combined with experimental fits, the following results are obtained:

For laminar boundary layers on flat plates:

For fully-turbulent flat plate boundary layers:

Pressure Drag

In addition to direct skin friction, the presence of the boundary layer creates a pressure or form drag on bodies. This does not appear in the flat plate results because pressure always acts perpendicular to the drag direction in this case. In an adverse pressure gradient, the skin friction drag is reduced, but pressure drag increases. This increase in pressure drag compensates for some of the reduction in skin friction. The combined drag may be estimated by a handy expression derived by Squire and Young (see Thwaites, Incompressible Aerodynamics) and gives amazingly good estimates of the total profile drag:

where θ is nondimensionalized by the chord length and the velocity outside the boundary layer at the trailing edge, Ue is normalized by the freestream U. Hte is the shape factor of the boundary layer at the trailing edge. (See the section on Boundary Layer Theory)

Note that when Ue is 1.0, the drag is just twice the momentum thickness on upper and lower surfaces. When H = 2 and Ue = 0.9 (Cp = .2), CDu = 1.38 θ