Question #21
At the Al-Zaafarana wind farm, the wind turbine is placed on a plateau, as shown in Figure 2. The wind approaching the plateau has a speed U of 30 km/hr on average (The free-stream turbulence intensity is less than 0.5%). Each turbine blade is 26 m in length. The distance d = 1000 m and the average air temperature is 27°C. v = 1.55 * 10^-5 m^2/s. Assume that the surface of the plateau is smooth and that the critical Reynolds number Recr = 10^5.
A. How high should you place the centerline of the turbine if you want the blades not to come within 3 m of the developed turbulent boundary layer? Apply the von Karman's integral momentum equation using the following expressions for the velocity profile and the skin friction coefficient: Cs = 0.027/Re^(1/7) (2)
B. To your knowledge, what type of transition to turbulence does the boundary layer flow experience?
C. In a few sentences, explain what is meant by the turbulent Reynolds stress, the eddy viscosity, and the Prandtl mixing length.
Question #22
Part I: An experimentalist has measured the boundary layer velocity profile on a wing and found that a better fit to the velocity profile is one in which u = U and the velocity gradient Ou/ay = 0 at the edge of the boundary layer y = 0. If the following velocity profile is assumed: u/U = aoy/8 - a1y/82 - a2y/84
a) Apply the above boundary conditions to relate the constants a1 and a to ao.
b) What physical situation does this profile represent for the cases where ao = 2 and ao = 0? Plot the corresponding distributions of u, Ou/y, and 2u/y^2 within the boundary layer.
Question #23
In our fluid mechanics lab, the wind tunnel has a square-duct U = 1 m/s test section, as shown in Figure 3. It has a length of L = 60 cm. A uniform airflow at U = 1 m/s enters the test section through an opening D = 30 cm. Due to the developed boundary layer, it is necessary to design a variable cross-sectional area in order to maintain constant velocity throughout the test section. Determine the duct size D(x) as a function of x. Explain how the result will be affected if U = 20 m/s instead of U = 1 m/s. (v = 1.55 * 10^-5 m/s)
Figure 2: Al-Zaafarana Wind Farm, Egypt.
Figure 3: Square-Duct Wind Tunnel.