Second Set

Solutions should be very accurate, very neat and complete. Software you can use is on the main course web page.

1.

Plot M and p/p₀₁ along the axis of a nozzle for 20 equally spaced values of p_B/p₀₁. Take the A(x) distribution of example 11.1 in the book. Include a qualitative sketch of the flow field behind the nozzle for each case. Plots should be accurate and neat. Put as many on a page as accuracy allows so that the evolution of the flow is easy to follow. (taken)

2.

For a Mach 2 windtunnel with a second throat big enough to swallow the starting shock, plot the pressure distributions through the entire duct when the pressure ratio p_B/p₀₁ is reduced from 1 in small steps. Next plot the pressure distributions when it is again increased in small steps to 1. Comment on the hysteresis effect. Plots should be accurate and neat. Put as many on a page as accuracy allows so that the evolution of the flow is easy to follow.

3.

For a windtunnel with a throat to measurement section area designed for Mach 2, and with a second, diffuser throat 20% larger than the first, plot the pressure distributions through the entire duct when the pressure ratio p_B/p₀₁ is reduced from 1 in small steps. Next plot the pressure distributions when it is again increased in small steps to 1. Comment on the hysteresis effect. (taken)

4.

The x-position of the normal shock inside the diverging section of a converging-diverging duct is a nontrivial function of the pressure ratio p_B/p₀₁. Find the analytical form of that function when the shock is close to the throat. (I.e., what power of is proportional to?) (taken)

5.

For example 11.1 in the book, graphically compare the quasi-one-dimensional approximation for the flow with the exact solution. Go in great detail to clearly show the differences.

6.

Design a windtunnel diffusor that clamps down to a bit larger than after the tunnel has been started. What improvement in operating pressure ratio can be achieved? What practical problems are there?

7.

If a windtunnel diffusor is closed to a bit larger than after the tunnel has been started, will the shock behind the second throat ever jump upstream through the measuring section? Consider the question carefully and argue your case in detail.

8.

Consider the theoretical possibility of improving diffusor efficiency by heating the flow or cooling it. Find out what can be done to improve efficiency in this way above that of a diffuser with a second throat big enough to swallow the starting shock.

9.

Consider replacing the diffuser behind a wind tunnel by a rotating turbine both theoretically and practically.

10.

Consider the theoretical possibility of improving diffusor efficiency by providing friction or an accelerating force. Find out what can be done to improve efficiency this way above that of a diffuser with a second throat big enough to swallow the starting shock.

11.

Use the MacCormack scheme to compute the supersonic flow in a constant area duct of height h. Assume that the nominal flow in the duct is that of air expanded isentropically to a M=2 flow from standard stagnation air (i.e. M_n=2, p_0n=1 atm, T_0n= 288 K, with corresponding p_n and T_n at M_n=2, while u_n=M_n a_n and v_n=0.) Assume the actual flow at the entrance x=0 has the same entropy and total enthalpy as the nominal flow, and v=0, but the pressure is given by . Compute the flow downstream of the exit using the MacCormack scheme. Use as many mesh points as the computer will take and plot the deviations from nominal to clearly show its evolution.

To find from , first find v as v=F₃/F₁. Next, noting that F₂=F₁ u + p and, with , , we can eliminate p from these two expressions to get a quadratcic equation for u with coefficients , , and c = F₁ v²/2. The roots are, using round-off insensitive expressions, ; u₂=c/(a u₁). The root u₁ will be the one you want. Then and p=F₂-F₁ u. (Taken)

12.

Use the MacCormack scheme to compute the acoustics in a pipe of length at nominally standard air conditions. The governing equations are (7.38) and (7.39), easily converted to F_x+G_y = 0. Assume the initial condition of a shock tube, in which the density in the left half is slightly raised above the standard value. Use as many mesh points as the computer will take and plot the deviations from nominal to clearly show its evolution. (Taken)

13.

Consider active noise control in a pipe. Assume the end at x=0 vibrates back and forward to generate noise. The other end at x=l is controlled to kill the noise when it reaches that end. Discuss the control method that makes the noise disappear when it hits x=l (instead of reflecting back into the pipe.) Include considerations about practical implementation.

14.

Consider a duct with two throats. In the initial situation, the ratio of the upstream stagnation pressure to the back pressure around the exit is such that both throats are just sonic, with the remaining duct subsonic. Then we squeeze the most upstream throath down slightly. What happens? What happens if we squeeze its area way down? What happens if instead we squeeze the area of the downstream throat slightly down? Way down? Carefully consider all aspects of the flow.