fluid_mechanics

6.6 Superposition of Basic, Plane Potential Flows 301
U
Ψ = 0
2U
r
θ
a
F I G U R E 6.26
circular cylinder.
The flow around a
The pressure distribution
on the
cylinder surface is
obtained from the
Bernoulli equation.
V6.8 Circular
cylinder with separation
The pressure distribution on the cylinder surface is obtained from the Bernoulli equation
written from a point far from the cylinder where the pressure is and the velocity is U so that
p 0 1 2rU 2 p s 1 2
2rv us
where p s is the surface pressure. Elevation changes are neglected. Since v us 2U sin u, the
surface pressure can be expressed as
p s p 0 1 2rU 2 11 4 sin 2 u2
(6.116)
A comparison of this theoretical, symmetrical pressure distribution expressed in dimensionless form
with a typical measured distribution is shown in Fig. 6.27. This figure clearly reveals that only on
the upstream part of the cylinder is there approximate agreement between the potential flow and
the experimental results. Because of the viscous boundary layer that develops on the cylinder, the
main flow separates from the surface of the cylinder, leading to the large difference between
the theoretical, frictionless fluid solution and the experimental results on the downstream side of
the cylinder 1see Chapter 92.
The resultant force 1per unit length2 developed on the cylinder can be determined by integrating
the pressure over the surface. From Fig. 6.28 it can be seen that
2p
F x p s cos u a du
0
p 0
(6.117)
3
U
2
β
1
_______ p s – p 0
U 2
0
1_
2 ρ Experimental
–1
–2
–3
0 30 60 90 120 150 180
β (deg)
Theoretical
(inviscid)
F I G U R E 6.27 A comparison
of theoretical (inviscid) pressure distribution
on the surface of a circular cylinder
with typical experimental distribution.