Page 238 - Aerodynamics for Engineering Students

Finite wing theory 221

circulating round the centre in steady motion under the influence of the force due to
the radial pressure gradient.
Considering unit axial length, the inwards force due to the pressures is:
1 1
(p+Sp)(r+Sr)SB-prS6- 2(p+-Sp)Sr-S8
2 2
which reduces to Sp(r - +Sr)Se. Ignoring 4Sr in comparison with r, this becomes
rSpS0. The volume of unit length of the element is rSrSB and therefore its mass is
pr Sr 68. Its centripetal acceleration is (velocity)2/radius, and the force required to
produce this acceleration is:

Equating this to the force produced by the pressure gradient leads to
r ~p = pq2 Sr since se # o (5.13)

Now, since the flow outside the vortex core is assumed to be inviscid, Bernoulli’s
equation for incompressible flow can be used to give, in this case,
1 1
p+pa= @+6p)+p(q+6qI2

Expanding the term in q + Sq, ignoring terms such as (Sq)2 as small, and cancelling,
leads to:
sp + pqsq = 0

i.e.
SP = -P4 (5.14)
Substituting this value for Sp in Eqn (5.13) gives

p$ Sr + pqr Sq = 0
which when divided by pq becomes

q Sr + r Sq = 0
But the left-hand side of this equation is S(qr). Thus
S(qr) = 0
qr = constant (5.15)

This shows that, in the inviscid flow round a vortex core, the velocity is inversely
proportional to the radius (see also Section 3.3.2).
When the core is small, or assumed concentrated on a line axis, it is apparent from
Eqn (5.15) that when r is small q can be very large. However, within the core the air
behaves as though it were a solid cylinder and rotates at a uniform angular velocity.
Figure 5.13 shows the variation of velocity with radius for a typical vortex.
The solid line represents the idealized case, but in reality the boundary is not so
distinct, and the velocity peak is rounded off, after the style of the dotted lines.