To determine the distance from the leading edge at which the transition from a laminar to a turbulent boundary layer occurs, we first need to calculate the Reynolds number (\(Re\)) based on the distance from the leading edge. The Reynolds number is defined as:
\[ Re = \frac{U L}{\nu} \]
where:
- \(U\) is the flow velocity (25 m/s in this case),
- \(L\) is the characteristic length or distance from the leading edge,
- \(\nu\) is the kinematic viscosity of the fluid.
The kinematic viscosity (\(\nu\)) can be calculated from the dynamic viscosity (\(\mu\)), which is given as \(1.85 \times 10^{-5} , \text{Pa.s}\), using the following relation:
\[ \nu = \frac{\mu}{\rho} \]
At sea level, the density of air (\(\rho\)) is approximately \(1.225 , \text{kg/m}^3\). Thus, we can calculate the kinematic viscosity:
\[ \nu = \frac{1.85 \times 10^{-5}}{1.225} \approx 1.51 \times 10^{-5} , \text{m}^2/\text{s} \]
Now, we can substitute \(Re = 500,000\) into the Reynolds number equation to find the distance from the leading edge (\(L\)):
\[ 500,000 = \frac{25 \cdot L}{1.51 \times 10^{-5}} \]
Solving for \(L\):
\[ L = \frac{500,000 \cdot 1.51 \times 10^{-5}}{25} \] \[ L = \frac{7500 \times 10^{-5}}{25} \] \[ L = 0.03 , \text{m} \]
Therefore, the transition to a turbulent boundary layer is expected to occur at a distance of approximately 0.03 meters from the leading edge.