Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.29)**
**1. Intuitive Physical Setup & Given Parameters**:
The question asks to evaluate fundamental aerodynamic concepts concerning the **Critical Mach number** ($M_{\text{cr}}$) and **Drag Divergence Mach number** ($M_{\text{dd}}$) of an airfoil/wing (MSQ).
**2. Step-by-Step Aerodynamic Derivation**:
1. **Definition of Critical Mach Number ($M_{\text{cr}}$)**:
- $M_{\text{cr}}$ is defined as the freestream Mach number $M_\infty$ at which sonic flow ($M = 1$) is first achieved at some point on the airfoil surface (typically the point of minimum pressure / peak suction). Statement A is **TRUE**.
2. **Relationship Between $M_{\text{cr}}$ and $M_{\text{dd}}$**:
- When $M_\infty$ exceeds $M_{\text{cr}}$, a local supersonic region develops, terminated by a normal shock wave.
- Initially, for $M_\infty$ slightly above $M_{\text{cr}}$, the shock is extremely weak and wave drag is negligible.
- As $M_\infty$ increases further, the shock strengthens and induces severe boundary layer separation. The freestream Mach number at which drag begins to rise catastrophically (typically defined by $ rac{dC_D}{dM_\infty} = 0.1$ or $\Delta C_D = 0.0020$) is the drag divergence Mach number $M_{\text{dd}}$.
- Because drag divergence requires finite, strong shock formation and separation, $M_{\text{dd}}$ is **always strictly greater** than $M_{\text{cr}}$ ($M_{\text{dd}} > M_{\text{cr}}$). Statement B is **TRUE**.
3. **Evaluation of Options C and D**:
- *Option C*: $M_{\text{dd}}$ is defined with respect to the **freestream** Mach number $M_\infty$, not the local Mach number on the wing surface. Statement C is **FALSE**.
- *Option D*: As angle of attack $lpha$ increases, the suction peak becomes sharper (lower minimum pressure coefficient $C_{p,\text{min}}$), causing sonic velocity to be reached at a lower freestream Mach number. Therefore, $M_{\text{cr}}$ strongly depends on the angle of attack. Statement D is **FALSE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Critical Mach number is the minimum freestream Mach number for which sonic condition is attained somewhere over the wing**: ✅ **CORRECT**. Standard Anderson definition of $M_{\text{cr}}$.
- **(B) Drag divergence Mach number is always higher than the critical Mach number**: ✅ **CORRECT**. $M_{\text{dd}} > M_{\text{cr}}$ always, with typically $M_{\text{dd}} - M_{\text{cr}} pprox 0.05\text{--}0.10$.
- **(C) Drag divergence Mach number is the local Mach number over the wing at which the drag increases drastically**: ❌ **INCORRECT**. It is a *freestream* Mach number, not local.
- **(D) Critical Mach number is independent of the angle of attack**: ❌ **INCORRECT**. Higher $lpha$ produces higher suction peaks, lowering $M_{\text{cr}}$ significantly.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B**
- **Exam Takeaway**: Order of Mach numbers: $M_{\text{cr}} < M_{\text{dd}} < 1.0$ (for subcritical/transonic flight). Both $M_{\text{cr}}$ and $M_{\text{dd}}$ decrease as angle of attack $lpha$ or airfoil thickness ratio $t/c$ increases.