Flow Past a Cylindrical Obstacle

Given that the fluid velocity field a large distance upstream of the cylinder is irrotational (because we have already seen that the flow pattern associated with uniform flow is irrotational--see Section 5.4), it follows from the Kelvin circulation theorem (see Section 4.14) that the velocity field remains irrotational as it is convected past the cylinder. Hence, according to Section 5.2, the stream function of the flow satisfies Laplace's equation,

The appropriate boundary condition at the surface of the cylinder is simply that the normal fluid velocity there be zero, because the fluid must stay in contact with the cylinder, but cannot penetrate its surface. Hence, , which implies that

because is undetermined to an arbitrary additive constant. It follows that we are searching for the most general solution of Equation (5.71) that satisfies the boundary conditions (5.70) and (5.72). Comparison with Equation (5.66) reveals that this solution takes the form

(5.73) |

where

(5.74) |

and is the circulation of the flow around the cylinder. (Note that the velocity field can be irrotational, but still possess nonzero circulation around the cylinder, because a loop that encloses the cylinder cannot be spanned by a surface lying entirely within the fluid. Thus, zero fluid vorticity does not necessarily imply zero circulation around such a loop from the curl theorem.) Let us assume that , for the sake of definiteness.

Figure 5.6-5.8 show streamlines of the flow calculated for various different values of the normalized circulation,
. For
, there exist a pair of points on the surface of the
cylinder at which the flow speed is zero. These are known as *stagnation points*, and can be located in Figures 5.6
and 5.7 as the points at which streamlines intersect the surface of the cylinder at right-angles. The
tangential fluid velocity at the surface of the cylinder is

(5.75) |

The stagnation points correspond to the points at which (because the normal velocity is automatically zero at the surface of the cylinder). Thus, the stagnation points lie at . When , the stagnation points coalesce and move off the surface of the cylinder, as illustrated in Figure 5.8 (the stagnation point corresponds to the point at which two streamlines cross at right-angles).

The irrotational form of Bernoulli's theorem, (4.97), can be combined with the boundary condition as , as well as the fact that is constant in the present case, to give

(5.76) |

where is the constant static fluid pressure a large distance from the cylinder. In particular, the fluid pressure on the surface of the cylinder is

where . The net force per unit length exerted on the cylinder by the fluid has the Cartesian coordinates

(5.78) | ||

(5.79) |

Thus, it follows from Equation (5.77) that

(5.80) | ||

(5.81) |

The component of the force that a moving fluid exerts on an obstacle, placed in its path, in a direction parallel to that of the unperturbed flow is usually called

Suppose that the cylinder is placed in a fluid which is initially at rest, and that the fluid's uniform flow velocity, , is then very slowly ramped up (in such a manner that no vorticity is induced in the upstream flow at infinity). Because the flow pattern is initially irrotational, and because the flow pattern well upstream of the cylinder is assumed to remain irrotational, the Kelvin circulation theorem indicates that the flow pattern around the cylinder also remains irrotational. Consider the time evolution of the circulation, , around some fixed curve that lies entirely within the fluid, and encloses the cylinder. We have

(5.82) |

where use has been made of Equation (4.84) (with assumed constant). However,

(5.83) |

We, thus, conclude that the rate of change of the circulation around is equal to minus the flux of the vorticity across [assuming that vorticity is convected by the flow, which follows from Equation (4.85), the fact that