# Viscous Drag

### Skin Friction

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

The shear stress is then related to the viscosity by:

C_{f} is related to the drag coefficient by C_{D} (skin friction) = C_{f}*S_{wetted}/S_{ref}.

where S_{wetted} is the area "wetted" by the air and S_{ref} 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: C_{D} = k * C_{f} * S_{wetted} / S_{ref} 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, U_{e} is normalized by the freestream
U. H_{te} is the shape factor of the boundary layer at the trailing edge. (See
the section on Boundary Layer Theory)

Note that when U_{e} is 1.0, the drag is just twice the momentum thickness
on upper and lower surfaces. When H = 2 and U_{e} = 0.9 (C_{p} = .2),
C_{Du} = 1.38 θ