Questions and Answers of Fundamentals Of Aerodynamics

At the nose of a missile in flight, the pressure and temperature are \(5.6 \mathrm{~atm}\) and \(850^{\circ} \mathrm{R}\), respectively. Calculate the density and specific volume. \(1
In the reservoir of a supersonic wind tunnel, the pressure and temperature of air are \(10 \mathrm{~atm}\) and \(320 \mathrm{~K}\), respectively. Calculate the density, the number density, and the
For a calorically perfect gas, derive the relation \(c_{p}-c_{v}=R\). Repeat the derivation for a thermally perfect gas.
The pressure and temperature ratios across a given portion of a shock wave in air are \(p_{2} / p_{1}=4.5\) and \(T_{2} / T_{1}=1.687\), where 1 and 2 denote conditions ahead of and behind the shock
Assume that the flow of air through a given duct is isentropic. At one point in the duct, the pressure and temperature are \(p_{1}=1800 \mathrm{lb} / \mathrm{ft}^{2}\) and \(T_{1}=500^{\circ}
Consider a room that is \(20 \mathrm{ft}\) long, \(15 \mathrm{ft}\) wide, and \(8 \mathrm{ft}\) high. For standard sea level conditions, calculate the mass of air in the room in slugs. Calculate the
In the infinitesimal neighborhood surrounding a point in an inviscid flow, the small change in pressure, \(d p\), that corresponds to a small change in velocity, \(d V\), is given by the differential
Consider a turbojet-powered airplane flying at a velocity of \(300 \mathrm{~m} / \mathrm{s}\) at an altitude of \(10 \mathrm{~km}\), where the free-stream pressure and density are \(2.65 \times
Consider a liquid-fueled rocket engine burning liquid hydrogen as the fuel and liquid oxygen as the oxidizer. The hydrogen and oxygen are pumped into the combustion chamber at rates of \(11
Ask yourself in what direction does aerodynamic drag (by definition) act. Then write the appropriate scaler component of the vector momentum equation in this direction. The body force term for this
Similarly, ask yourself in what direction does the lift act. Then write the appropriate scaler component of the vector equation in this direction. The body force term for this case is the lift; solve
When the National Advisory Committee for Aeronautics (NACA) measured the lift and drag on airfoil models in the 1930s and 1940s in their specially designed airfoil wind tunnel at the Langley
In the same tests described in problem 2.3, the NACA measured the lift per unit span by measuring the pressure distribution in the flow direction on the top and bottom walls of the wind tunnel. Using
At a point in the flow over an F-15 high-performance fighter airplane, the pressure, temperature, and Mach number are \(1890 \mathrm{lb} / \mathrm{ft}^{2}, 450^{\circ} \mathrm{R}\), and 1.5 ,
Return to Example 1.6. Calculate the Mach number and velocity at the exit of the rocket nozzle.Data From Example 1.6:Consider the flow through a rocket engine nozzle. Assume that the gas flow through
Return to Example 1.1. Calculate the percentage density change between the given point on the wing and the free stream, assuming compressible flow.Data From Example 1.1:Consider the low-speed flow of
Consider again the rocket engine discussed in Examples 1.6 and 3.2. If the thrust of the engine is \(4.5 \times 10^{5} \mathrm{~N}\) at an altitude where the ambient pressure is \(0.372
A normal shock wave is standing in the test section of a supersonic wind tunnel. Upstream of the wave, \(M_{1}=3, p_{1}=0.5 \mathrm{~atm}\), and \(T_{1}=200 \mathrm{~K}\). Find \(M_{2}, p_{2},
A blunt-nosed missile is flying at Mach 2 at standard sea level. Calculate the temperature and pressure at the nose of the missile.
Consider a point in a supersonic flow where the static pressure is \(0.4 \mathrm{~atm}\). When a Pitot tube is inserted in the flow at this point, the pressure measured by the Pitot tube is \(3
For the normal shock that occurs in front of the Pitot tube in Example 3.7, calculate the entropy change across the shock.Data From Example 3.7:Consider a point in a supersonic flow where the static
Transonic flow is a mixed subsonic-supersonic flow where the local Mach number is near 1. A typical example is the flow over the wing of a high-speed subsonic transport, such as the Boeing 777
Consider two flows, one of helium and one of air, at the same Mach number of 5. Denoting the strength of a normal shock by the pressure ratio across the shock, \(p_{2} / p_{1}\), which gas will
Repeat Example 3.10, except assuming equal velocities of \(1700 \mathrm{~m} / \mathrm{s}\) and temperatures of \(288 \mathrm{~K}\) for both gas flows.Data From Problem 3.10:Consider two flows, one of
Consider the normal shock wave properties calculated in Example 3.5. Show that these properties satisfy the Hugoniot equation for a calorically perfect gas.Data From Example 3.5:A normal shock wave
Air enters a constant-area duct at \(M_{1}=0.2, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=273 \mathrm{~K}\). Inside the duct, the heat added per unit mass is \(q=1.0 \times 10^{6} \mathrm{~J} /
Air enters a constant-area duct at \(M_{1}=3, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=300 \mathrm{~K}\). Inside the duct, the heat added per unit mass is \(q=3 \times 10^{5} \mathrm{~J} / \mathrm{kg}\).
In Example 3.14, how much heat per unit mass must be added to choke the flow?Data From Example 3.14:Air enters a constant-area duct at \(M_{1}=3, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=300
Consider the supersonic inflow conditions given in Example 3.14. If an amount of heat equal to \(6 \times 10^{5} \mathrm{~J} / \mathrm{kg}\) is added to this flow, what will happen to it
Consider the flow of air through a pipe of inside diameter \(=0.15 \mathrm{~m}\) and length \(=30 \mathrm{~m}\). The inlet flow conditions are \(M_{1}=0.3, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=273
Consider the flow of air through a pipe of inside diameter \(=0.4 \mathrm{ft}\) and length \(=5 \mathrm{ft}\). The inlet flow conditions are \(M_{1}=3, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=300
In Example 3.18, what is the length of the duct required to choke the flow?Data From Example 3.18:Consider the flow of air through a pipe of inside diameter \(=0.4 \mathrm{ft}\) and length \(=5
At a given point in the high-speed flow over an airplane wing, the local Mach number, pressure, and temperature are 0.7, \(0.9 \mathrm{~atm}\), and \(250 \mathrm{~K}\), respectively. Calculate the
At a given point in a supersonic wind tunnel, the pressure and temperature are \(5 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}\) and \(200 \mathrm{~K}\), respectively. The total pressure at this point
At a point in the flow over a high-speed missile, the local velocity and temperature are \(3000 \mathrm{ft} / \mathrm{s}\) and \(500^{\circ} \mathrm{R}\), respectively. Calculate the Mach number
Consider a normal shock wave in air. The upstream conditions are given by \(M_{1}=3, p_{1}=1 \mathrm{~atm}\), and \(ho_{1}=1.23 \mathrm{~kg} / \mathrm{m}^{3}\). Calculate the downstream values of
Consider a Pitot static tube mounted on the nose of an experimental airplane. A Pitot tube measures the total pressure at the tip of the probe (hence, sometimes called the Pitot pressure), and a
Consider the compression of air by means of \((a)\) shock compression and (b) isentropic compression. Starting from the same initial conditions of \(p_{1}\) and \(v_{1}\), plot to scale the \(p v\)
During the entry of the Apollo space vehicle into the Earth's atmosphere, the Mach number at a given point on the trajectory was \(M=38\) and the atmosphere temperature was \(270 \mathrm{~K}\).
Consider air entering a heated duct at \(p_{1}=1 \mathrm{~atm}\) and \(T_{1}=288 \mathrm{~K}\). Ignore the effect of friction. Calculate the amount of heat per unit mass (in joules per kilogram)
Air enters the combustor of a jet engine at \(p_{1}=10 \mathrm{~atm}, T_{1}=1000^{\circ} \mathrm{R}\), and \(M_{1}=0.2\). Fuel is injected and burned, with a fuel-air ratio (by mass) of 0.06 . The
For the inlet conditions of problem 3.9, calculate the maximum fuel-air ratio beyond which the flow will be choked at the exit.Data from problem 3.9Air enters the combustor of a jet engine at
At the inlet to the combustor of a supersonic combustion ramjet (SCRAMjet), the flow Mach number is supersonic. For a fuel-air ratio (by mass) of 0.03 and a combustor exit temperature of
Air is flowing through a pipe of \(0.02-\mathrm{m}\) inside diameter and \(40-\mathrm{m}\) length. The conditions at the exit of the pipe are \(M_{2}=0.5, p_{2}=1 \mathrm{~atm}\), and \(T_{2}=270
Consider the adiabatic flow of air through a pipe of \(0.2-\mathrm{ft}\) inside diameter and 3-ft length. The inlet flow conditions are \(M_{1}=2.5, p_{1}=0.5 \mathrm{~atm}\), and \(T_{1}=520^{\circ}
The stagnation chamber of a wind tunnel is connected to a high-pressure air bottle farm which is outside the laboratory building. The two are connected by a long pipe of 4 -in inside diameter. If the
Starting with Eq. (3.95), derive in detail Eq. (3.96).\[\begin{equation*}d p+ho u d u=-\frac{1}{2} ho u^{2} \frac{4 f d x}{D} \tag{3.95}\end{equation*}\]\[\begin{equation*}\frac{4 f d
Consider a Mach 2.5 flow of air entering a constant-area duct. Heat is added to this flow in the duct; the amount of heat added is equal to 30 percent of the total enthalpy at the entrance to the
Consider the low-speed flow of air over an airplane wing at standard sea level conditions; the free-stream velocity far ahead of the wing is 100 mi/h. The flow accelerates over the wing, reaching a
A pressure vessel that has a volume of \(10 \mathrm{~m}^{3}\) is used to store high-pressure air for operating a supersonic wind tunnel. If the air pressure and temperature inside the vessel are \(20
Calculate the isothermal compressibility for air at a pressure of \(0.5 \mathrm{~atm}\).
For the pressure vessel in Example 1.2, calculate the total internal energy of the gas stored in the vessel.Data From Example 1.2:A pressure vessel that has a volume of \(10 \mathrm{~m}^{3}\) is used
Consider the air in the pressure vessel in Example 1.2. Let us now heat the gas in the vessel. Enough heat is added to increase the temperature to \(600 \mathrm{~K}\). Calculate the change in entropy
Consider the flow through a rocket engine nozzle. Assume that the gas flow through the nozzle is an isentropic expansion of a calorically perfect gas. In the combustion chamber, the gas which results
Calculate the isentropic compressibility for air at a pressure of \(0.5 \mathrm{~atm}\). Compare the result with that for the isothermal compressibility obtained in Example 1.3.Data From Example
Consider a hypersonic vehicle with a spherical nose flying at Mach 20 at a standard altitude of 150,000 ft, where the ambient temperature and pressure are 500◦R and 3.06 lb./ft2, respectively. At
Consider a high-speed vehicle flying at a standard altitude of 35 km, where the ambient pressure and temperature are 583.59 N/m2 and 246.1 K, respectively. The radius of the spherical nose of the
Consider a flat plate at zero angle of attack in a hypersonic flow at Mach 10 at standard sea level conditions. At a point 0.5 m downstream from the leading edge, the local shear stress at the wall
Starting with Equations (1.7), (1.8), and (1.11), derive in detail Equations (1.15), (1.16), and (1.17). N' == TE -TE * (P, cos 0 + T₁ sine) ds, + ² (p, cos 0 — , sine) ds;
Consider a light, single-engine, propeller-driven airplane similar to a Cessna Skylane. The airplane weight is 2950 lb and the wing reference area is 174 ft2. The drag coefficient of the airplane CD
The purpose of this problem is to give you a feel for the magnitude of Reynolds number appropriate to real airplanes in actual flight. a. Consider the DC-3 shown in Figure 1.1. The wing root chord
For the design of their gliders in 1900 and 1901, the Wright brothers used the Lilienthal Table given in Figure 1.65 for their aerodynamic data. Based on these data, they chose a design angle of
Consider the Space Shuttle during its atmospheric entry at the end of a mission in space. At the altitude where the Shuttle has slowed to Mach 9, the local heat transfer at a given point on the lower
In Example 2.1, the statement is made that the streamline an infinite distance above the wall is straight. Prove this statement. The subsonic compressible flow over a cosine-shaped (wavy) wall is
Consider the existence of a forward-facing axial aerodynamic force on an airfoil. Can a forward-facing axial force exist on a flat plate at an angle of attack in a flow? Thoroughly explain your
Consider a venturi with a small hole drilled in the side of the throat. This Chole is connected via a tube to a closed reservoir. The purpose of the venturi is to create a vacuum in the reservoir
The Kutta-Joukowski theorem, Equation (3.140), was derived exactly for the case of the lifting cylinder. In Section 3.16 it is stated without proof that Equation (3.140) also applies in general to a
Consider the streamlines over a circular cylinder as sketched at the right of Figure 3.26. Single out the first three streamlines flowing over the top of the cylinder. Designate each streamline by
Consider the flow field over a circular cylinder mounted perpendicular to the flow in the test section of a low-speed subsonic wind tunnel. At standard sea level conditions, if the flow velocity at
Prove that the flow field specified in Example 2.1 is not incompressible; i.e., it is a compressible flow as stated without proof in Example 2.1. and where u = Voo v = h 2π [1 + 127 - l 2π.χ (cos
Starting with Equation (4.35), derive Equation (4.36). ["(dll.) = Poc Vso [^2y (4) d - V∞ - M₁E = - * 5 (dL) = LE (4.35)
Starting with Equations (4.35) and (4.43), derive Equation (4.62). MLE = - [ 5 (dL) =-PxV₂0 [°* &Y (₁ 0 0 == Voo 51 ξγ(ξ) dξ (4.35)
For the NACA 2412 airfoil, the lift coefficient and moment coefficient about the quarter-chord at −6° angle of attack are −0.39 and −0.045, respectively. At 4° angle of attack, these
For the airfoil in Problem 4.11, calculate the value of the circulation around the airfoil.Data from Problem 4.11:Consider again the NACA 2412 airfoil discussed in Problem 4.10. The airfoil is flying
The question is often asked: Can an airfoil fly upside-down? To answer this, make the following calculation. Consider a positively cambered airfoil with a zero-lift angle of −3°. The lift slope is
The airfoil section of the wing of the British Spitfire of World War II fame (see Figure 5.19) is an NACA 2213 at the wing root, tapering to an NACA 2205 at the wing tip. The root chord is 8.33 ft.
For the conditions given in Problem 4.15, a more reasonable calculation of the skin friction coefficient would be to assume an initially laminar boundary layer starting at the leading edge, and then
Consider a finite wing with an aspect ratio of 6. Assume an elliptical lift distribution. The lift slope for the airfoil section is 0.1/degree. Calculate and compare the lift slopes for (a) a
Repeat Problem 5.6, except for a lower aspect ratio of 3. From a comparison of the results from these two problems, draw some conclusions about the effect of wing sweep on the lift slope, and how the
In Problem 1.19 we noted that the Wright brothers, in the design of their 1900 and 1901 gliders, used aerodynamic data from the Lilienthal table given in Figure 1.65. They chose a design angle of
Consider the Supermarine Spitfire shown in Figure 5.19. The first version of the Spitfire was the Mk I, which first flew in 1936. Its maximum velocity is 362 mi/h at an altitude of 18,500 ft. Its
If the elliptical wing of the Spitfire in Problem 5.9 were replaced by a tapered wing with a taper ratio of 0.4, everything else remaining the same, calculate the induced drag coefficient. Compare
Consider the Spitfire in Problem 5.9 on its landing approach at sea level with a landing velocity of 70 mi/h. Calculate the induced drag coefficient for this low-speed case. Compare your result with
Prove that three-dimensional source flow is irrotational.
Repeat Problem 7.10, considering the flow of Problem 7.11Data from Problem 7.10:Calculate the percentage error obtained if Problem 7.9 is solved using (incorrectly) the incompressible Bernoulli
Bernoulli’s equation, Equation (3.13), (3.14), or (3.15), was derived in Chapter 3 from Newton’s second law; it is fundamentally a statement that force = mass × acceleration. However, the terms
At a given point in a flow, T = 300 K, p = 1.2 atm, and V = 250 m/s. At this point, calculate the corresponding values of p0, T0, p∗, T ∗, and M∗.
When the Apollo command module returned to earth from the moon, it entered the earth’s atmosphere at a Mach number of 36. Using the results from the present chapter for a calorically perfect gas
The stagnation temperature on the Apollo vehicle at Mach 36 as it entered the atmosphere was 11,000 K, a much different value than predicted in Problem 8.17 for the case of a calorically perfect gas
Prove that the total pressure is constant throughout an isentropic flow.
Consider the supersonic flow over a flat plate at an angle of attack, as sketched in Figure 9.35. As stated in Section 9.7, the flow direction downstream of the trailing edge of the plate, behind the
Consider a two-dimensional duct with a straight horizontal lower wall, and a straight upper wall inclined upward through the angle θ = 3◦. The height of the duct entrance is 0.3 m. A uniform
Repeat Problem 9.18, except with θ = 30◦. Again, we will use these results to compare with a quasi-one-dimensional calculation in Problem 10.16. The reason for repeating this calculation is to
Consider a Mach 3 flow at 1 atm pressure initially moving over a flat horizontal surface. The flow then encounters a 20 degree expansion corner, followed by a 20 degree compression corner that turns
The purpose of this problem is to explain what causes the dramatic white cloud pattern generated in the flow field over the F/A-18C Hornet shown on the cover of this book. This problem is both a
Return to Problem 9.18, where the average Mach number across the two-dimensional flow in a duct was calculated, and where θ for the upper wall was 3◦. Assuming quasi-one dimensional flow,
Return to Problem 9.19, where the average Mach number across the two-dimensional flow in a duct was calculated, and where θ for the upper wall was 30◦. Assuming quasi-one-dimensional flow,
A horizontal flow initially at Mach 1 flows over a downward-sloping expansion corner, thus creating a centered Prandtl-Meyer expansion wave. The streamlines that enter the head of the expansion wave
Consider a centered expansion wave where M1 = 1.0 and M2 = 1.6. Using the method developed in Problem 10.17, plot to scale a streamline that passes through the expansion wave.Data from Problem
In Problem 11.8, the critical Mach number for a circular cylinder is given as Mcr = 0.404. This value is based on experimental measurements, and therefore is considered reasonably accurate. Calculate

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