Question
A flat plate with a length of 1m is placed in a wind tunnel at sea level conditions. The windtunnel is set to a speed of 25 m/s . The dynamic viscosity of air is 1.850.10^-5 Pa. s . As briefly mentioned in the course, you may assume transition to occur at Re= 500000 in these conditions.
At what distance from the leading edge in meters do you expect to find a trasition to a turbulent boundary layer?
At what distance from the leading edge in meters do you expect to find a trasition to a turbulent boundary layer?
Answers
GPT-4o mini
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.
\[
Re = \frac{U L}{\nu}
\]
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.