Global Integrals

Let Q = <fC (V. n)dl and Г = <fC (V. dl), where dl represents a small oriented element of the contour and dl = |dl| is the length of the element. The first integral represents the net volume flow rate out of contour C and the second, the circulation along the same contour, taken in the clockwise direction. It is easy to show that if the contour C contains sources and sinks of intensity Q1, Q2,… the result will be

Q = Q1 + Q2 +—— the algebraic sum of the sources and sinks inside the contour.

If the contour C contains potential vortices of circulation Г1, Г2,… the result will be Г = Г1 + Г2 + ■■■ the algebraic sum of vortices inside the contour.

2.3.1 Example of Superposition: Semi-infinite Obstacle

The superposition of the uniform flow and a source (Q > 0) at the origin produces a semi-infinite body, called Rankine body. Stagnation points and stagnation streamlines play an important role in the description of the flow topology. Here, the velocity field is given by

Q 1

V = U cos в + , Ve = – U sin в (2.26)

2n r

Fig. 2.8 Half-body and flow vector field

The main flow features can be studied with the streamfunction

Q

& = Ur sin в + в (2.28)

2п

A remarkable streamline is that which corresponds to the half-body. It corresponds to & = ■§. It is also the stagnation streamline. It is made of several pieces, the negative x-axis and the half-body itself. The flow is sketched in Fig. 2.8.

Note that any streamline can be materialized as a solid surface. The flow inside the half body is also represented, however it is singular at the origin and cannot represent a realistic flow near that point. The maximum thickness of the half-body is Q. One can calculate the force on a solid surface coinciding with the separation streamline using the momentum theorem and Bernoulli’s equation applied to a contour composed of the half-body and a large circle of radius R to close the contour. The lift vanishes due to symmetry, but the axial force on the half-body is given by D = —pU Q, a thrust.