Next: Boundary layer separation and Up: Viscosity effects at high Previous: Boundary layers

## Skin friction

The boundary layer produces a drag on the plate due to the viscous stresses which are developed at the wall. The viscous stress at the surface of the plate is

Once this stress is known, we have only to integrate it over the surface of the plate to obtain the total drag force D:

To get an estimate of the velocity gradient near the wall, we note that by definition the width of the boundary layer is the distance over which the velocity returns to its free stream value, so

Performing the integral to obtain the drag, we find

Defining the coefficient of drag for the plate as CD = D/(rU2A/2), with A = lxlz, we find that . A real calculation starting from the Navier-Stokes equation yields

not far from our simple estimate. This drag is often referred to as skin friction , and is due to the viscous stresses acting on the surface of the plate. If the boundary layer remains attached to the body (which it may not; see below), then this is the sole source of aerodynamic drag on a body. At high Reynolds numbers, say , this gives a drag coefficient of , which is relatively small.

The previous analysis assumed that the flow in the boundary layer was laminar. However, in large Reynolds number flow we often encounter turbulent boundary layers , which tend to produce a larger drag. The turbulent mixing of the fluid near the surface of a solid body leads to more efficient momentum transport away from the body, increasing the gradient of the velocity profile at the surface and therefore the viscous stress on the plate. For boundary layers which remain attached to a body the drag due to skin friction can be reduced if the boundary layer can be persuaded to remain laminar.

Next: Boundary layer separation and Up: Viscosity effects at high Previous: Boundary layers

Notes by A. Dorsey, revised by V. Celli, Univ. of Virginia
Sun Sep 28 22:13:11 EDT 1997