For a two-dimensional incompressible flow over a flat plate, the laminar boundary layer thickness at a distance $x$ from the leading edge is $\delta$. If $Re_x$ is the Reynolds number defined based on length scale $x$, $\left(\frac{\delta}{x}\right)$ is proportional to:
Options:
(A) $Re_x^{-1/2}$
(B) $Re_x^{-1}$
(C) $Re_x^{-3/2}$
(D) $Re_x^{-2}$
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.17)**
**1. Intuitive Physical Setup & Given Parameters**:
We are asked to determine how the laminar boundary layer thickness ratio $\frac{\delta}{x}$ scales with local Reynolds number $Re_x$ for 2D incompressible flow over a flat plate.
**2. Step-by-Step Mathematical Derivation**:
1. **Blasius Boundary Layer Solution**:
For steady, 2-D, incompressible laminar flow over a flat plate with zero pressure gradient ($dp/dx = 0$), the Blasius similarity variable is:
\eta = y \sqrt{\frac{U_\infty}{\nu x}}
2. **Boundary Layer Thickness (99% Thickness)**:
The boundary layer thickness $\delta$ is defined where $u/U_\infty = 0.99$, which occurs at $\eta \approx 5.0$:
\delta = \frac{5.0 x}{\sqrt{Re_x}} = 5.0 x \, Re_x^{-1/2}
3. **Non-dimensional Thickness Ratio**:
Dividing by streamwise distance $x$:
\frac{\delta}{x} = \frac{5.0}{\sqrt{Re_x}} = 5.0 \cdot Re_x^{-1/2}
Therefore, $\frac{\delta}{x} \propto Re_x^{-1/2}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Re_x^(-1/2)**: ✅ **CORRECT**. Blasius laminar boundary layer thickness grows as $\delta \propto x/\sqrt{Re_x}$, so $\delta/x \propto Re_x^{-1/2}$.
- **(B) Re_x^(-1)**: ❌ **INCORRECT**. Corresponds to scaling of skin friction in creeping flows.
- **(C) Re_x^(-3/2)**: ❌ **INCORRECT**. Incorrect power.
- **(D) Re_x^(-2)**: ❌ **INCORRECT**. Overestimates decay rate.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: For laminar flat plate boundary layers: $\delta \propto x^{1/2}$, so $\delta/x \propto Re_x^{-1/2}$. Local skin friction coefficient scales as $C_{fx} \propto Re_x^{-1/2}$, and total drag coefficient scales as $C_D \propto Re_L^{-1/2}$.