Questions and Answers of Fundamentals Of Aerodynamics

What happens to the airflow when it encounters an oblique shock wave?
Will the pilot experience any negative compressibility effects at the critical Mach number?
What happens to the kinetic energy of the airflow when it encounters a normal shock wave?
Identify the speed at which force divergence becomes significant.
VMO is typically a ________ limitation.
MMO is typically a ________ limitation.
Assuming standard atmospheric conditions, what will happen to your true airspeed when you conduct a constant indicated airspeed climb from sea level to 20,000 feet?
What is the crossover altitude?
Explain the term “coffin corner.”
What happens to the air density, pressure, temperature, and total energy when it passes through a normal shock wave?
What happens to the center of lift as you accelerate from subsonic to supersonic speeds?
Explain why flight speeds below L/DMAX are not recommended.
Identify an advantage of a supercritical airfoil.
Consider an airfoil flying at zero angle of attack. At low speeds \(\left(M_{\infty}
An aircraft is in steady level flight at \(M_{\infty}=0.55\) with \(T_{\infty}=273^{\circ} \mathrm{K}\) and \(p_{\infty}=10,000 \mathrm{~N} / \mathrm{m}^{2}\). Performance data for the aircraft's
An F-15 is in straight and level flight at about \(36,000 \mathrm{ft}\) where \(p_{\infty}=22,700 \mathrm{~N} / \mathrm{m}^{2}\). Its wing loading \(\left(W_{S}\right)\) is about \(400 \mathrm{~kg} /
Consider a length of pipe bent into a \(U\)-shape. The inside diameter of the pipe is \(0.5 \mathrm{~m}\). Air enters one leg of the pipe at a mean velocity of \(100 \mathrm{~m} / \mathrm{s}\) and
A pressure field is defined by \(P(x, y)=\sqrt{x^{2}+y^{2}}+2\). Here, \(x\) and \(y\) are in units of meters, and the pressure \(P\) has units of \(\mathrm{Pa}\). Calculate the resulting force that
A jet engine undergoing ground testing vents its exhaust through a smokestack. Investigators set up two probes to measure the dispersion of the exhaust into the atmosphere. Probe A is set up at a
If \(\vec{V}=(3 x+2 y) \mathbf{i}+4(x+y+2 z) \mathbf{j}+(x-3 z) \mathbf{k}\) please:a. Compute the curl of \(\mathbf{V}\) and explain what the curl signifies physically.b. Compute the divergence of
Write the expression for the substantial derivative and explain its physical significance.
Show how the differential form of the \(\mathrm{x}\)-component momentum equation [Equation (2.70a)] can be expressed using the substantial derivative:\[ \frac{\partial(ho u)}{\partial t}+abla
Consider steady flow through the duct with circular cross section illustrated below.Use mass conservation to find \(U_{2}\) in terms of \(U_{1}\) assuming \(D_{2} / D_{1}=2\) and:a. The flow is
RP-1 flows steadily through the \(90^{\circ}\) reducing elbow shown in the illustration. The absolute pressure at the inlet to the elbow is \(107.3 \mathrm{kPa}\) and the cross-sectional area is
The space shuttle main engine consumes approximately \(140.3 \mathrm{~kg} / \mathrm{s}\) of liquid hydrogen and \(374.2 \mathrm{~kg} / \mathrm{s}\) of liquid oxygen at full power. The overall ratio
Consider the velocity given by:\[ \boldsymbol{V}=\left(9 x^{2}+5 y\right) \hat{i}+(5 x) \hat{j}+\hat{k} \]Does a velocity potential exist and if so, what is it?
Prove that flows that can be represented using a velocity potential are irrotational.
Calculate the stream function and angular velocity associated with the following velocity field:\[ \boldsymbol{V}=-y^{3} \hat{i}+x^{4} \hat{j} \]Can this flow have a velocity potential?
Starting with Equation (4.35), derive Equation (4.36).4.35:\(M_{\mathrm{LE}}^{\prime}=-\int_0^c \xi(d L)=-ho_{\infty} V_{\infty} \int_0^c \xi \gamma(\xi) d
We studied the case of the lifting flow over a circular cylinder. In real life, a rotating cylinder in a flow will produce lift; such real flow fields are shown in the photographs in Figure 3.34b and
Derive the 1-D energy equation for adiabatic compressible flow staring from the first law of thermodynamics and Euler's equation, that is, derive \(c_{p} T+\frac{1}{2} u^{2}=C\) from \(d e=\delta
An airplane flies at Mach 0.8 at an altitude of \(10 \mathrm{~km}\). Its engine draws in air from the free stream and raises its pressure by a factor of 80 by the time it exits the engine's
Starting with the integral form of energy conservation [Equations (2.95 and 7.43)], derive an expression for energy conservation in a steady, inviscid, nonchemically reacting flow without body forces
In a thermally perfect, but not calorically perfect gas, the specific heat at constant pressure can no longer be considered constant. How would this affect the prediction of the stagnation
Plot \(p_{2} / p_{1}, T_{2} / T_{1}, p_{02} / p_{01}, T_{02} / T_{01}\), and \(M_{2}\) from \(M_{1}=1\) to \(M_{1}=10\) assuming that the gas is air \((\gamma=1.4)\) and Krypton \((\gamma=1.68)\).
Plot \(p_{0} / p\) for a pitot-static system from \(M_{1}=0\) to \(M_{1}=5\).
The purpose of the inlet in a ramjet engine is to slow the incoming air to subsonic conditions so that it can be mixed and reacted with the fuel. Often, this is accomplished via a system of shock
Find \(M_{3}\) in the engine inlet illustrated below when \(M_{\infty}=3\). Assume that the first two shocks are weak and that \(\gamma=1.4\).
Below is a sketch of a bow shock stabilized in front of a rounded strut that protrudes into a supersonic flow of air. Find the velocity of the free stream if the angle of the bow shock far away from
The static and total pressures of air downstream of a \(30^{\circ}\) expansion corner are \(0.0155 \mathrm{~atm}\) and \(151.8 \mathrm{~atm}\), respectively. What are the Mach number and static
Consider the smooth expansion corner illustrated below where a Mach 2.55 flow at \(1 \mathrm{~atm}\) is turned away from itself by \(15^{\circ}\). Please sketch the important features of the flow and
Recognizing that the Mach waves emanating from the curved wall eventually coalesce into an oblique shock that turns the flow in the far field. What additional structure(s) must form at the
Write a computer program in the language/environment of your choice to solve the \(\theta-\beta-M\) relation [Equation (9.23)] for \(\beta\) given \(\theta\) and \(M\).9.23:\(\tan \theta=2 \cot \beta
Air at \(500 \mathrm{kPa}\) and \(573^{\circ} \mathrm{K}\) flows at \(0.1 \mathrm{~kg} / \mathrm{s}\) through a converging diverging nozzle whose exit is at \(100 \mathrm{kPa}\). Assume
Show that for an arbitrary isentropic flow through a nozzle (i.e., one that is not necessarily choked) that the mass flow rate is given by:\[ \dot{m}=A_{t} \psi \sqrt{2 p_{0} ho_{0}} \quad \text {
Starting with Equations (1.7), (1.8), and (1.11), derive in detail Equations (1.15), (1.16), and (1.17).\begin{aligned}1.15: c_n= & \frac{1}{c}\left[\int_0^c\left(C_{p, l}-C_{p, u}\right) d
Derive \(P=ho R T\) from \(P V=n R_{u} T\), where \(n\) is the number of moles.
Calculate \(\mathrm{R}\) for a 20 percent mixture of \(\mathrm{H}_{2}\) in \(\mathrm{N}_{2}\).
Calculate the lift force generated by a NACA 1408 airfoil with a chord of \(1.5 \mathrm{~m}\) and a span of \(15 \mathrm{~m}\) operating at \(50 \mathrm{~m} / \mathrm{s}\) and an angle of attack of 8
Starting with Equations (1.1) and (1.2), use the definitions of the force coefficients to show that:\[ C_{L}=C_{N} \cos \alpha-C_{A} \sin \alpha \text { and } C_{D}=C_{N} \sin \alpha+C_{A} \cos
Derive the expression for the moment coefficient about the leading edge [Equation (1.17)] from the expression for the moment about the leading edge [Equation (1.11)].\(\begin{aligned} 1.11:
Consider two airfoils tested in two different environments where Airfoil 2 is a three times scale replica of Airfoil 1.\begin{array}{lll}\hline \text { Factors } & \text { Airfoil 1 } & \text {
Consider an airplane in level flight at \(578 \mathrm{MPH}\) at an altitude of \(38,000 \mathrm{ft}\). It has a rectangular wing \(25 \mathrm{ft}\) long and \(4 \mathrm{ft}\) wide with a drag
Derive an expression for dynamic pressure in terms of pressure and the Mach number.
Consider the intersection of two shocks of opposite families, as sketched in Fig. 4.23. For \(M_{1}=3, p_{1}=1 \mathrm{~atm}, \theta_{2}=20^{\circ}\), and \(\theta_{3}=15^{\circ}\), calculate the
Following the guidance in Secs. 13.2 and 13.3, lay out in detail a plan of attack for the solution of the supersonic flow over a right-circular cone at an angle of attack low enough so that the shock
Repeat problem 13.1, except now for a cone with an elliptic cross-section. What are the differences between this problem and problem 13.1? In fact, are there any differences?Data From Problem
Consider a cone at a very high angle of attack to the flow, high enough so that the shock wave is detached from the cone vertex. Lay out in detail a plan of attack for solving this flow field.
Using equation (14.9), tabulate the entropy increase across a normal shock from Mach 1 to Mach 1.3 in Mach number increments of 0.02. What does the variation tell you about the nature of the
Consider a flat plate at an angle of attack in a hypersonic flow. As the angle of attack is increased from zero, the lift coefficient first increases, then reaches a maximum value, and then decreases
Consider a circular cylinder of infinite span in a hypersonic flow. Using newtonian theory, show that the drag coefficient based on cross-sectional area perpendicular to the flow is \(4 / 3\).
Consider a sphere in a hypersonic flow. Using newtonian theory, show that the drag coefficient based on cross-sectional area is equal to 1 .
Starting with Eq. (16.4), derive the most probable population distribution for Fermions, namely, Eq. (16.19).\[\begin{equation*}W=\prod_{j} \frac{g_{j} !}{\left(g_{j}-N_{j}\right) ! N_{j} !}
Derive Eqs. (16.34) and (16.35).\[\begin{equation*}S=N k\left(\ln \frac{Q}{N}+1\right)+N k T\left(\frac{\partial \ln Q}{\partial T}\right)_{V} \tag{16.34}\end{equation*}\]\[\begin{align*}Q= &
Starting with Eq. (16.35), derive the perfect gas equation of state, \(p=ho R T\). (This demonstrates that the perfect gas equation of state, which historically was first obtained empirically, falls
Starting with the quantum mechanical expression for the quantized translational energy levels as a function of the quantum numbers \(n_{1}, n_{2}\), and \(n_{3}\), derive in detail the translational
In a similar vein as Problem 16.4, derive in detail the rotational partition function given by Eq. (16.39)\[\begin{equation*}Q_{\mathrm{rot}}=\frac{8 \pi^{2} I k T}{h^{2}}
Consider \(1 \mathrm{~kg}\) of pure diatomic \(\mathrm{N}_{2}\) in thermodynamic equilibrium. The fundamental vibrational frequency of \(\mathrm{N}_{2}\) is \(v=7.06 \times 10^{13} / \mathrm{s}\),
Frequently in the literature, a characteristic temperature for vibration is defined as \(\theta_{\text {vib }}=h v / k\). Express \(e\) and \(c_{v}\) for a diatomic molecule [Eqs. (16.49) and
Consider an equilibrium chemically reacting mixture of three general species denoted by A, B, and AB. In detail, derive Eqs. (16.54) and (16.55) for such a
Consider an equilibrium chemically reacting mixture of oxygen at \(p=\) \(1 \mathrm{~atm}\) and \(T=3200 \mathrm{~K}\). The only species present are \(\mathrm{O}_{2}\) and O. \(K_{p,
For the conditions of problem 16.9, calculate the internal energy of the mixture in joules per kilogram, including the translational, rotational, vibrational, and electronic energies. Note the
Consider air at \(p=0.5 \mathrm{~atm}\) and \(T=4500 \mathrm{~K}\). Assume the chemical species present are \(\mathrm{O}_{2}, \mathrm{O}, \mathrm{N}_{2}\), and \(\mathrm{N}\). (Ignore NO.) Calculate
Consider a unit mass of \(\mathrm{N}_{2}\) in equilibrium at \(p=1 \mathrm{~atm}\) and \(T=300 \mathrm{~K}\). For these conditions, the vibrational relaxation time is \(190 \mathrm{~s}\). Assume
Consider a normal shock wave in pure \(\mathrm{N}_{2}\). The upstream pressure, temperature, and velocity are \(0.1 \mathrm{~atm}, 300 \mathrm{~K}\), and \(3500 \mathrm{~m} / \mathrm{s}\),
The temperature that would exist at a point in the flow if the fluid elements were brought to rest adiabatically at that point. For each of these chemically reacting flows, is \(T_{0}\) constant or
Obtain the lift and propulsive force coefficients of an airfoil given in Example 8.6, and compare the results with Problem 8.30. Assume the profile pitches about midchord.Example 8.6The NACA 0012
Find the heat transfer rate \(\mathrm{q}_{\mathrm{w}}\) at \(\mathrm{x}=10 \mathrm{~cm}\) and \(100 \mathrm{~cm}\) for the flat plate given in Problem 7.31.Problem 7.31A flat plate of \(4
What are the values of the feathering parameters for the airfoils given by Examples 8.5 and 8.6?Examples 8.5Assume an airfoil pitching about its leading edge and plunging with \(k=0.35\) as
For a chordwise flexible airfoil obtain the quasi unsteady edge velocity, Eq. 8.41, and the suction force coefficient, Eq. 8.42, formulae assuming that the parabolic camber of the airfoil, whose
Derive the equations of continuity, Eq. 8.44, and the vorticity transport, Eq. 8.45, for skewed coordinates as shown in Fig. 8.38.Eq. 8.44Eq. 8.45Fig. 8.38 de = tan -[-(h+WaLE) /U]
Obtain the time dependent but steady lift coefficient, Eq. 8.48, and the boundary layer edge velocity, Eq. 8.49 for a chordwise flexible parabolically cambered thin airfoil whose equation is given by
Obtain the quasi steady lift coefficient, Eq. 8.51, and the boundary layer edge velocity, Eq. 8.52 for a chordwise flexible and parabolically cambered thin airfoil whose equation is given by Eq.
Obtain the quasi unsteady lift coefficient using FFT and the arbitrary angle of attack change associated with the equivalent quasi steady lift given by Eq. 8.51 for the reduced frequency of 0.8 and
The wing shown in Fig. 8.60 pitching and plunging with \(3 \mathrm{~Hz}\) in a free stream of \(15 \mathrm{~m} / \mathrm{s}\). Using the strip theory, obtain the total lift and the propulsive force
Which type of spanwise flexibility is preferred for a finite wing?
Two high aspect ratio identical rectangular wings \(\left(\begin{array}{lll}2L & x & 2 b\end{array}\right)\) are separated with a distance \(h\). Establish an expression for the downwash in terms of
The Newtonian impact theory is valid at high angles of attack. The wall inclination for a blunt body gradually decreases along the free stream direction. For such cases, when this angle is less than
For the attached flows over slender delta wings, show that at low angles of attack Eqs. 1.11 and 1.33 are identical.Eq 1.11Eq 1.13 = 1 2 AR CL=
The ellipsoid given in Problem 2.3 is also undergoing a pulsative major axis change with the same period but with phase difference \(\phi\). Express the equation of surfaces.Problem 2.3An oblate
Express the components of stress tensor in generalized coordinates in terms ofvelocity gradients.
Find the sectional lift and moment coefficients taken about the midchord for the airfoil given in Problem 3.2 undergoing a simple harmonic motion \(h=\bar{h} e^{i \omega t}\).Problem 3.2An airfoil is
Solve Problem 5.13 using Doublet-Lattice Method with taking 8 points in spanwise and 8 points in chordwise directions. Compare the results.Problem 5.13A thin wing with aspect ratio 3 is oscillating
The wing given by Problem 5.25 oscillates with the reduced frequency of \(k=0.2\). Obtain the lifting pressure curve for the spanwise change. Find the total lift coefficient.Problem 5.25A delta wing
Obtain the surface pressure distribution of a spherical segment which has an apex angle of \(60^{\circ}\) and \(2 \mathrm{~m}\) radius oscillating about its nose with small amplitude.
The profile given in Problem 7.12. is in plunging motion with \(\mathrm{k}=0.3\). Determine the upper surface pressure distribution in terms of the amplitude of plunging.Problem 7.12For a 5% thick
Using the improved piston theory for the profile given in Problem 7.13 find the lifting pressure distribution. What can be the thickness ratio for the same profile at free stream of \(\mathrm{M}=3\)
Eq. 7.36a is written for the conservation of momentum in y direction. Obtain Eq. 7.36-b wherein the stream function is independent variable.Eq. 7.36(a,b) y - v v +(1-y/R)v- + momentum: u u R-y + R P
Show that the derivative of the boundary layer edge velocity is given by Eq. 7.64 for the figure given below.Eq. 7.64 M>>1 Ue dx R dy/dx = 1/R
Using Maslen method, find the approximate value of pressure and density at the junction of the sphere and the cone of Problem 7.29 at Mach number 8.Problem 7.29An empirical way to determine shock

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