Chapter 6, Compressible Flow Video Solutions, Aerodynamics for Engineering Students

Problem 1

A convergent-divergent duct has a maximum diameter of $150 \mathrm{~mm}$, and a Pitotstatic tube is placed in its throat. Neglecting the effect of the Pitot-static tube on the flow, estimate the throat diameter under the following conditions:
(a) Air at the maximum section is of standard pressure and density, and the pressure difference across the pitôt-static tube $\equiv 127$-mm water.
(b) Pressure and temperature in the maximum section are $100,300 \mathrm{Nm}^{-2}$ and $100^{\circ} \mathrm{C}$, respectively, and the pressure difference across the pitot-static tube $\equiv 127-\mathrm{mm}$ mercury.
(Answer: $123 \mathrm{~mm} ; 66.5 \mathrm{~mm}$ )

Anand Jangid

Numerade Educator

03:39

Problem 2

In the wing-flow method of transonic research, an airplane dives at a Mach number of 0.87 at a height where the pressure and temperature are 46,500 $\mathrm{N} \mathrm{m}^{-2}$ and $-24.6^{\circ} \mathrm{C}$, respectively. At the model's position, the pressure coefficient is -0.5 . Calculate the speed, Mach number, $0.7 \mathrm{p} M^2$, and kinematic viscosity of the flow past the model.
(Answer: $344 \mathrm{~m} \mathrm{~s}^{-1} ; M=1.133 ; 0.7 p M^2=30,800 \mathrm{Nm}^{-2} ; \quad v=2.64 \times 10^{-3}$ $\left.\mathrm{m}^2 \mathrm{~s}^{-1}\right)$

Satpal Satpal

Numerade Educator

01:18

Problem 3

What is the indicated air speed and the true air speed of the airplane in Exercise 6.2 assuming that the air-speed indicator is calibrated for incompressible flow in standard conditions and that there are no instrument errors?
(Answer: TAS $=274 \mathrm{~ms}^{-1}$; IAS $=219 \mathrm{~m} \mathrm{~s}^{-1}$ )

Prabhu Ramji

Numerade Educator

02:55

Problem 4

On the basis of Bernoulli's equation, discuss the assumption that the compressibility of air may be neglected for low subsonic speeds.
A symmetric airfoil at zero lift has a maximum velocity that is $10 \%$ greater than the free-stream velocity. This maximum increases at the rate of $7 \%$ of the free-stream velocity for each degree of incidence. What is the free-stream velocity at which compressibility effects begin to become important (i.e., the error in pressure coefficient exceeds $2 \%$ ) on the airfoil surface when the incidence is 5 degrees?
(Answer: Approximately $70 \mathrm{~ms}^{-1}$ )

Mayank Tripathi

Numerade Educator

01:46

Problem 5

A closed-return wind tunnel with a large contraction ratio has air at standard conditions of temperature and pressure in the settling chamber upstream of the contraction to the working section. Assuming isentropic compressible flow in the tunnel, estimate the speed in the working section where the Mach number is 0.75 . Take the ratio of specific heats for air as $\gamma=1.4$. (Answer: $242 \mathrm{~ms}^{-1}$ )

Narayan Hari

Numerade Educator

01:27

Problem 6

Derive Eq. (6.28b).

Ajay Singhal

Numerade Educator

00:58

Problem 7

Recreate Fig. 6.25 using OS_mt.m instead of OS_mw.m, as in the example in this chapter.

Nick Johnson

Numerade Educator

04:46

Problem 8

Traditionally, a second chart besides the wave angle versus turning angle chart is helpful for rapidly understanding the impact of oblique shocks on an airflow. This chart is pressure ratio across the shock $\frac{p_2}{p_1}$ versus turning angle for a constant Mach number. Use the MATLAB codes provided to create such a plot for upstream Mach numbers Mone $=[1.25: 0.25: 3]$.

Jincy M Saji

Numerade Educator

06:47

Problem 9

At what Mach number does the pressure triple through a normal shock? Solve this three ways:
(a) Using the m-file NS_unp.m.
(b) To check the numerical result, solve Eq. (6.44) for Mach number and compute.
(c) Use the Normal Shock Table (NST) for the nearest value. That is, you will not find $\frac{p_2}{p_1}=3$ in the NST, so use the row in the table (numerically) closest to it. That is the old-fashioned method, but it is still generally the in-class exam method.

Saman Zulfiqar

Numerade Educator

01:01

Problem 10

Repeat Exercise 6.9 but for tripling of density.

Raj Bala

Numerade Educator

03:27

Problem 11

Determine how downstream Mach number varies with upstream Mach number for a fixed turning angle. Specifically, for a 6-degree turning angle, plot the downstream versus the upstream Mach number from 1 to 10 . Note that there are two solutions for each upstream Mach number and that some Mach numbers may be too low to have a solution for a 6-degree turning angle. It is assumed that you will make use of the compressible flow MATLAB functions as described above. Hand in the plot and the $\mathrm{m}$-file.

Hubert Agamasu

Numerade Educator

02:09

Problem 12

Consider a Mach 2 flow that encounters a 15-degree compression corner and then a 15-degree expansion corner (Fig. Ex6.12a). What are the Mach number and pressure after the expansion corner?

Chai Santi

Numerade Educator

01:54

Problem 13

A converging-diverging nozzle is supplied with air at a total pressure of $250 \mathrm{psi}$. The area of the throat is $0.1 \mathrm{in}^2$. The exit area is $1.6 \mathrm{in}^2$. Find the following (use of the nearest values in tables is acceptable):
(a) The greatest back pressure for which the throat is choked.
(b) The one value of the back pressure for perfect supersonic expansion.
(c) The lowest back pressure for which there is a shock in the nozzle.
Use these back pressures to determine:
(a) The range of back pressures for which a shock exists in the nozzle.
(b) The range of back pressures for which the flow is overexpanded.
(c) The range of back pressures for which the flow is underexpanded.

Chai Santi

Numerade Educator

Problem 14

Consider how compressibility in subsonic flight affects the results of the liftingline theory from Chapter 5. Derive an expression for how the induced drag coefficient for a thin rectangular wing of span $b$ and chord $\mathrm{c}$ flying at a small angle of attack $\alpha$ is affected by compressibility in linearized small-perturbation subsonic compressible flow. Assume that you know whatever you need to know about the performance of the wing in incompressible flow; use the subscript $o$ to denote those values when you write them. For example, induced drag coefficient at incompressible conditions is $C_{D i o}$. The flight Mach number in the subsonic compressible regime is $M_{\infty}$. The pressure is $p_{\infty}$, and the temperature is $T_{\infty}$. There are two common answers to this problem, one is wrong. Be prepared to discuss why yours is correct.

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