Fluid Mechanics Assignment
QUESTION 1
The sketch below shows a 2D micro-wind tunnel to test small aerodynamic devices. The length and height are, respectively, 2L and 2h, and the installation discharges in the atmosphere at a pressure pO. In order to characterize the flow in the installation, first a flow rate Q and temperature Ti is introduced in the empty installation. As L is sufficiently large, consider ?u/?x = 0,
- Use the continuity equation to determine v(x, y).
- Use the y-momentum equation to show that p = p(x).
- Use the x-momentum equation to formulate the problem and obtain u(y) as a function of dp/dx, including boundary conditions at the top and bottom walls.
- Use the last result and the known total flow rate Q to completely determine p(x). In particular, obtain the pressure at the inlet pi = p(x = ?L).
- Use a suitable control volume to evaluate the force F¯ exerted by the air on the walls.
After the empty installation is characterized, an airfoil with film-cooling is placed in the wind-tunnel, as shown in sketch 2. In this airfoil, cool air is injected, normal to the free-stream, through the airfoil. The airfoil, of length 2l << 2L, is placed in the center of the test cell, as shown in the sketch 2. The airfoil can be assumed to be infinitely thin (flat plate) and uniform flow rates Q1 = 2v1l and Q2 = 2v2l are injected perpendicularly to the free stream through its upper and lower surfaces, with a temperature Tb, to obtain an overal increment of flow rate ?Q = Q1 + Q2. Some assumptions will be made in the analysis. Because of the small size 2l of the airfoil, the perturbation can be considered as local and small. Both upstream and downstream of the airfoil the analysis previously carried out remains valid (notice the respective flow rates are Q and Q + ?Q). In particular, the pressure can be taken as linear in x for L < x>0 and for 0 < x>, and continuous at x = 0. If the force exerted on the airfoil is experimentally obtained to be F¯b,
- Determine the new pressure profiles in presence of the airfoil. In particular, obtain the new values for p(x = 0) and pi
- Find the force exerted by the air on the walls
- Take the walls as adiabatic and prove, using the differential equation, that the temperature at the exit will be uniform if the channel is long enough to ensure ?T/?x becomes 0 before reaching the outlet. Determine the temperature of the gas at the outlet section.
Note: Gravity and Rayleigh’s viscous dissipation ?v can be neglected
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QUESTION 2
The objective is to derive the PDE that governs the mean temperature field Tm(x, t) (1-dimensional, unsteady) for a liquid with constant properties that flows in a tube of cross section area A and adiabatic walls. The flow is characterized by a uniform, steady velocity field v¯ = Ume¯x.
- Apply the internal energy equation to the cylindrical C.V. defined in the figure, with length ?x and cross section area A.
- Divide the equation by the volume of the C.V. and take the limit ?x ? 0.
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