Consider the flow over an oval modeled using the elementary potential flows as shown below. \(U\) represents uniform flow velocity and \(\Gamma\) represents circulation around an irrotational line vortex. Which of the following statements is/are TRUE for this model?
Show Hint
The oval is the closed streamline balancing the vortex-induced velocity against \(U\); it grows with \(\Gamma\), shrinks with \(U\), and needs the vortices close enough together to stay closed.
Interchanging the sense of the two vortices does not alter the oval
Moving the vortices too far apart causes the oval to break up
Show Solution
The Correct Option isB, D
Solution and Explanation
This flow is built by adding a uniform stream $U$ to two opposite-sign line vortices, $-\Gamma$ above and $+\Gamma$ below, separated by a gap $a$. The oval is the closed streamline that this combination traces out, similar in spirit to a Rankine oval but built from a vortex pair instead of a source-sink pair. Checking each statement means asking how the balance between the vortex-induced velocity and the free stream shifts.
Increasing $U$ enlarges the oval: false. A bigger free stream speed overwhelms the fixed induced velocity of the vortices more easily, so the balance point (and the oval boundary) is pushed inward, closer to the vortices, not outward. A stronger $U$ shrinks the oval.
Increasing $\Gamma$ enlarges the oval: true. A stronger vortex pair induces higher velocities further away from itself, so the location where this induced velocity matches the fixed $U$ moves outward. The oval grows with $\Gamma$.
Interchanging the sense of the two vortices does not alter the oval: false. Flipping which vortex is positive and which is negative reverses the direction of the flow each vortex induces in the region near its partner and near the free stream. That changes the balance of velocities that defines the oval boundary, so the same oval is not reproduced.
Moving the vortices too far apart causes the oval to break up: true. The single oval depends on the two vortices being close enough that their combined field, added to $U$, gives one continuous closed streamline around both of them. Stretch the spacing $a$ past a critical value and the two vortices start acting nearly independently, so the flow can no longer sustain a single closed boundary enclosing both; the oval breaks up.
Only increasing $\Gamma$ and separating the vortices too far are correctly described.
Let's summarize:
The oval size grows with vortex strength $\Gamma$ and shrinks with free stream speed $U$.
Reversing the sense of the vortices changes the induced flow pattern, so it does change the oval.
Too large a spacing between the vortices breaks the single closed oval apart.
