Consider the flow over an oval modeled using the elementary potential flows as
shown below. U represents uniform flow velocity and 𝛤 represents circulation
around an irrotational line vortex.
Which of the following statements is/are TRUE for this model?
Options:
(A) Increasing U enlarges the oval
(B) Increasing 𝛤 enlarges the oval
(C) Interchanging the sense of the two vortices does not alter the oval
(D) Moving the vortices too far apart causes the oval to break up
Official Answer Key:
B, D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.47)**
**1. Intuitive Physical Setup & Given Parameters**:
An oval body (Rankine-like oval) is synthesized in 2D potential flow using elementary flows:
- Uniform freestream of velocity $U$.
- A vortex pair consisting of two counter-rotating irrotational line vortices of circulation $\pm \Gamma$ separated by distance $2d$.
We are asked to identify which statements are **TRUE** regarding this flow model (MSQ).
**2. Step-by-Step Potential Flow Derivation**:
1. **Governing Stream Function & Stagnation Points**:
- The stream function is obtained by superposing uniform flow and the vortex pair:
$\psi = U y + \frac{\Gamma}{2\pi} \ln(r_1) - \frac{\Gamma}{2\pi} \ln(r_2)$
- The closed dividing streamline $\psi = 0$ defines the boundary of the closed oval body.
- Stagnation points occur where velocity $\vec{V} = \nabla\phi = 0$.
2. **Effect of Parameter Variations**:
- **Increasing Freestream Velocity $U$**: Increasing $U$ compresses the streamlines closer to the vortex centers, shrinking the closed dividing streamline. Statement A is **FALSE**.
- **Increasing Circulation $\Gamma$**: Increasing vortex strength $\Gamma$ induces stronger opposing velocities further out, pushing the stagnation points further away and **enlarging the oval**. Statement B is **TRUE**.
- **Interchanging Sense of Vortices**: Reversing the signs of $\Gamma$ reverses the induced velocity direction (from opposing freestream to assisting it, or flipping stagnation points), which alters the stagnation streamline and shape. Statement C is **FALSE**.
- **Vortex Separation Distance**: If the vortices are moved too far apart, the induced flow between them can no longer maintain a single closed dividing streamline enclosing both vortices; instead, the dividing streamline pinches in and splits into two isolated closed bodies around each individual vortex (the oval breaks up). Statement D is **TRUE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Increasing U enlarges the oval**: ❌ **INCORRECT**. Higher freestream speed overpowers the induced velocities and shrinks the oval.
- **(B) Increasing 𝛤 enlarges the oval**: ✅ **CORRECT**. Stronger vortices extend their influence further into the freestream, expanding the closed dividing streamline.
- **(C) Interchanging the sense of the two vortices does not alter the oval**: ❌ **INCORRECT**. Alters the sign of the induced velocities and the resulting stagnation streamline shape.
- **(D) Moving the vortices too far apart causes the oval to break up**: ✅ **CORRECT**. Beyond a critical separation distance, the single oval bifurcates into two separate closed bubbles.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B, D**
- **Exam Takeaway**: In superposition of potential flows, singularity strength ($\Gamma, \Lambda$) expands body dimensions, while freestream velocity $U$ compresses them.
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