Which of the following statements is/are TRUE regarding critical and drag
divergence Mach numbers of a wing?
Options:
(A) Critical Mach number is the minimum freestream Mach number for which sonic
condition is attained somewhere over the wing
(B) Drag divergence Mach number is always higher than the critical Mach number
(C) Drag divergence Mach number is the local Mach number over the wing at which
the drag increases drastically
(D) Critical Mach number is independent of the angle of attack
Official Answer Key:
A, B
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** ($M_{ ext{cr}}$) and **Drag Divergence Mach number** ($M_{ ext{dd}}$) of an airfoil/wing (MSQ).
**2. Step-by-Step Aerodynamic Derivation**:
1. **Definition of Critical Mach Number ($M_{ ext{cr}}$)**:
- $M_{ ext{cr}}$ 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 $M_{ ext{cr}}$ and $M_{ ext{dd}}$**:
- When $M_\infty$ exceeds $M_{ ext{cr}}$, a local supersonic region develops, terminated by a normal shock wave.
- Initially, for $M_\infty$ slightly above $M_{ ext{cr}}$, 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 $M_{ ext{dd}}$.
- Because drag divergence requires finite, strong shock formation and separation, $M_{ ext{dd}}$ is **always strictly greater** than $M_{ ext{cr}}$ ($M_{ ext{dd}} > M_{ ext{cr}}$). Statement B is **TRUE**.
3. **Evaluation of Options C and D**:
- *Option C*: $M_{ ext{dd}}$ 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 $C_{p, ext{min}}$), causing sonic velocity to be reached at a lower freestream Mach number. Therefore, $M_{ ext{cr}}$ 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 $M_{ ext{cr}}$.
- **(B) Drag divergence Mach number is always higher than the critical Mach number**: ✅ **CORRECT**. $M_{ ext{dd}} > M_{ ext{cr}}$ always, with typically $M_{ ext{dd}} - M_{ ext{cr}} pprox 0.05 ext{--}0.10$.
- **(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 $M_{ ext{cr}}$ significantly.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B**
- **Exam Takeaway**: Order of Mach numbers: $M_{ ext{cr}} < M_{ ext{dd}} < 1.0$ (for subcritical/transonic flight). Both $M_{ ext{cr}}$ and $M_{ ext{dd}}$ decrease as angle of attack $lpha$ or airfoil thickness ratio $t/c$ increases.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.