Page 233 - Aerodynamics for Engineering Students

21 6 Aerodynamics for Engineering Students

Fig. 5.6

5.2.2 The Biot-Savart law
The original application of this law was in electromagnetism, where it relates the
intensity of the magnetic field in the vicinity of a conductor carrying an electric
current to the magnitude of the current. In the present application velocity and
vortex strength (circulation) are analogous to the magnetic field strength and electric
current respectively, and a vortex filament replaces the electrical conductor. Thus the
Biot-Savart law can also be interpreted as the relationship between the velocity
induced by a vortex tube and the strength (circulation) of the vortex tube. Only the
fluid motion aspects will be further pursued here, except to remark that the term
induced velocity, used to describe the velocity generated at a distance by the vortex
tube, was borrowed from electromagnetism.
Allow a vortex tube of strength I?, consisting of an infinite number of vortex
filaments, to terminate in some point P. The total strength of the vortex filaments
will be spread over the surface of a spherical boundary of radius R (Fig. 5.7) as the
filaments diverge from the point P in all directions. The vorticity in the spherical
surface will thus have the total strength I?.
Owing to symmetry the velocity of flow in the surface of the sphere will be
tangential to the circular line of intersection of the sphere with a plane normal to
the axis of the vortex. Moreover, the direction will be in the sense of the circulation
about the vortex. Figure 5.8 shows such a circle ABC of radius I subtending a conical
angle of 28 at P. If the velocity on the sphere at R, 8 from P is v, then the circulation
round the circuit ABC is I?’ where
I?’ = 21rR sin 8v (5.1)

Spherical boundary
surrounding ‘free’
end at point P

Fig. 5.7