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
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The Correct Option isB, D
Solution and Explanation
Step 1: Identify the flow model.
The picture combines a uniform stream of speed \(U\) moving in the positive x-direction with two irrotational line vortices of equal and opposite strength, placed a fixed distance \(a\) apart: the upper vortex has circulation \(-\Gamma\) (clockwise) and the lower vortex has circulation \(+\Gamma\) (counterclockwise). Superposing these three simple flows produces one closed streamline that encloses both vortices, an oval shaped body, in the same spirit as a source and a sink placed in a uniform stream producing a Rankine oval.
Step 2: Effect of increasing the free stream speed \(U\).
The oval boundary is the streamline on which the velocity induced by the two vortices exactly balances the free stream at the stagnation points. If \(U\) is made larger while \(\Gamma\) and \(a\) are kept fixed, the vortices have to reach that balance closer to themselves, since their own induced velocity field has not changed strength. So the stagnation points, and the oval boundary they define, move inward. Increasing \(U\) shrinks the oval, it does not enlarge it. Statement (A) is FALSE.
Step 3: Effect of increasing the circulation \(\Gamma\).
Now hold \(U\) and \(a\) fixed and increase \(\Gamma\). A stronger vortex pair induces a larger velocity at any given distance from the vortices, so the point where this induced velocity matches the fixed free stream \(U\) moves further out. The closed streamline that separates the rotational region from the free stream therefore grows. Increasing \(\Gamma\) enlarges the oval. Statement (B) is TRUE.
Step 4: Effect of interchanging the sense of the two vortices.
Swapping which vortex is \(+\Gamma\) and which is \(-\Gamma\) reverses the direction of the velocity that each vortex induces on the flow near the other, and on the free stream in the gap between them. Since the shape and location of the closed oval streamline depends on how the induced velocities from the two vortices add to the uniform stream, reversing this sense changes that balance and does not reproduce the same oval. Statement (C), which claims the oval is unchanged, is FALSE.
Step 5: Effect of moving the vortices far apart.
The single closed oval exists only because the two vortices are close enough that their combined induced flow, together with the uniform stream, creates one continuous ring of closed streamlines enclosing both of them. If the spacing \(a\) is increased too much, each vortex starts to behave more like an isolated vortex in a uniform stream sitting far from its partner, and the induced velocities are no longer strong enough, at the right location, to keep a single closed boundary around both vortices together. Past a critical spacing, the single oval breaks up. Statement (D) is TRUE.
Final Answer:
Increasing \(\Gamma\) enlarges the oval, and separating the vortices too far apart breaks the single oval up.
\[ \boxed{\text{B, D}} \]
