Question

Air flowing over a flat plate at \(5 \mathrm{~m} / \mathrm{s}\) has a friction coefficient given as \(C_{f}=0.664(V x / v)^{-0.5}\), where \(x\) is the location along the plate. Using appropriate software, determine the effect of the location along the plate \((x)\) on the wall shear stress \(\left(\tau_{w}\right)\). By varying \(x\) from \(0.01\) to \(1 \mathrm{~m}\), plot the wall shear stress as a function of \(x\). Evaluate the air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\).

Short answer

Answer: The wall shear stress varies with the location along the flat plate, generally decreasing as the distance from the leading edge of the plate increases. This relationship can be observed by calculating and plotting the wall shear stress as a function of the location along the plate.

Step-by-step solution

Find kinematic viscosity of air at given conditions

To evaluate air properties at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) we refer to air property tables or use an online calculator. For this temperature and pressure, the kinematic viscosity \(\left(v\right)\) is approximately \(1.51\times10^{-5} \mathrm{~m}^2 / \mathrm{s}\).

Calculate the wall shear stress formula

To calculate the wall shear stress (\(\tau_w\)), we can use the given friction coefficient formula (\(C_f\)) and the formula for wall shear stress: $$ \tau_w = \frac12 \rho V^2 C_f $$ where \(\rho\) is the air density (approximately \(1.204 \mathrm{~kg} / \mathrm{m}^3\) at \(20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\)) and \(V = 5 \mathrm{~m} / \mathrm{s}\) is the air velocity.

Calculate shear stress for different x values

Now, we can calculate the wall shear stress for different values of \(x\) ranging from \(0.01\) to \(1 \mathrm{~m}\). For each value of \(x\), we perform the following steps: 1. Calculate the friction coefficient (\(C_f = 0.664(Vx/v)^{-0.5}\)) 2. Calculate the wall shear stress using the earlier formula.

Plot the wall shear stress as a function of x

Once we have calculated the wall shear stress for different values of \(x\), we can create a plot with \(x\) on the horizontal axis and wall shear stress (\(\tau_w\)) on the vertical axis. The plot will show the effect of location along the plate on the wall shear stress. Using appropriate software, such as Excel, Python, or MATLAB, plot the calculated data to visualize the relationship between the wall shear stress and the location along the plate. This visualization will help us understand how the location on the plate affects the wall shear stress.