The momentum integral expression for the drag of a two-dimensional body

Consider a two-dimensional control volume fixed in space (see Fig. 7.48) of unit width, with two faces (planes 0 and 2) perpendicular to the free stream, far ahead of and far behind the body respectively, the other two lying parallel to the undisturbed flow direction, and situated respectively far above and far below the body. For any stream tube (of vertical height Sy) that is contained within the wake at the downstream boundary, the mass flow per unit time is ригбуг and the velocity reduction between upstream and downstream is Ua0 — «2- The loss of momentum per unit time in the stream tube = pui{Ux — иг)6у2 and, for the whole field of flow:

r

Total loss of momentum per unit time = pu2(Ux — U2)&y2

Joe

In fact, the limits of this integration need only extend across the wake because the term Ux — U2 becomes zero outside it.

This rate of loss of momentum in the wake is brought about by the reaction on the fluid of the profile drag force per unit span D, acting on the body. Thus

D = f pu2(Uo0 – u2)dy2 (7.166)

J W

This expression enables the drag to be calculated from an experiment arranged to determine the velocity profile at some considerable distance downstream of the body, i. e. where p = p00.

For practical use it is often inconvenient, or impossible, to arrange for measurement so far away from the body, and methods that allow measure­ments to be made close behind the body (plane 1 in Fig. 7.48) have been
developed by Betz* and B. M. Jones. t The latter’s method is considerably the simpler and is reasonably accurate for most purposes.