Questions and Answers of Fundamentals Of Aerodynamics

In Probs. 5.20 and 5.21, we noted that the reservoir temperature required for a continuous flow air Mach 20 hypersonic wind tunnel was beyond the capabilities of heaters in the reservoir. On the
Consider a uniform flow_straight parallel streamlines with no gradients of any flow properties, i.e., constant velocity. Prove that this flow is irrotational.
Consider a flow that is made up of straight streamlines emanating from a central point \(\mathrm{O}\), and where the flow velocity along each of the streamlines varies inversely with distance from
Consider a subsonic flow with an upstream Mach number of \(M_{\infty}\). This flow moves over a wavy wall with a contour given by \(y_{w}=h \cos (2 \pi x / l)\), where \(y_{w}\) is the ordinate of
Consider a supersonic flow with an upstream Mach number of \(M_{\infty}\). This flow moves over the same wavy wall as first shown in Fig. 9.5 and as given in Example 9.1. For small \(h\), use linear
Show that this nonlinear equation is valid for transonic flow with small perturbations:\[\left(1-M_{\infty}^{2}\right) \frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial
The low-speed lift coefficient for an NACA 2412 airfoil at an angle of attack of \(4^{\circ}\) is 0.65 . Using the Prandtl-Glauert rule, calculate the lift coefficient for \(M_{\infty}=0.7\).
In low-speed flow, the pressure coefficient at a point on an airfoil is -0.9 . Calculate the value of \(C_{p}\) at the same point for \(M_{\infty}=0.6\) by means ofa. The Prandtl-Glauert ruleb.
Consider a flat plate with chord length \(c\) at an angle of attack \(\alpha\) to a supersonic free stream of Mach number \(M_{\infty}\). Let \(L\) and \(D\) be the lift and drag per unit span,
For the flat plate in problem 9.4, the quarter-chord point is located, by definition, at a distance equal to \(c / 4\) from the leading edge. Using linearized theory, derive the following expression
Consider a flat plate at an angle of attack of \(4^{\circ}\).a. Calculate \(C_{L}\) and \(C_{M_{c / 4}}\) for \(M_{\infty}=0.03\) (essentially incompressible flow). (Hint: Consult a book, such as
Consider a diamond-shaped airfoil such as that sketched in Fig. 4.35. The half-angle is \(\varepsilon\), thickness is \(t\), and chord is \(c\). For supersonic flow, use linearized theory to derive
Supersonic linearized theory predicts that, for a thin airfoil of arbitrary shape and thickness at angle of attack \(\alpha, C_{L}=4 \alpha / \sqrt{M_{\infty}^{2}-1}\), independent of the shape and
Repeat problem 4.17, except using linearized theory. Plot the linearized results on top of the same graphs produced for problem 4.17 in order to assess the differences between linear theory (which is
Linear supersonic theory predicts that the curve of wave drag versus Mach number has a minimum point at a certain value of \(M_{\infty}>1\).a. Calculate this value of \(M_{\infty}\).b. Does it make
At \(\alpha=0^{\circ}\), the minimum pressure coefficient for an NACA 0009 airfoil in low-speed flow is -0.25 . Calculate the critical Mach number for this airfoil usinga. The Prandtl-Glauert ruleb.
Consider a \(15^{\circ}\) half-angle cone at \(0^{\circ}\) angle of attack in a free stream at standard sea level conditions with \(M_{\infty}=2.0\). Obtain:a. The shock wave angleb. \(\quad p, T,
For the cone in Prob. 10.1, below what value of \(M_{\infty}\) will the shock wave be detached? Compare this with the analogous value for a wedge.Data From Problem 10.1:Consider a \(15^{\circ}\)
The drag coefficient for a cone can be defined as \(C_{D}=D / q_{\infty} A_{b}\), where \(A_{b}\) is the area of the base of the cone. For a \(15^{\circ}\) half-angle cone, plot the variation of
Using the method of characteristics, compute and graph the contour of a two-dimensional minimum-length nozzle for the expansion of air to a design exit Mach number of 2.
Repeat problem 11.1, except consider a nozzle with a finite expansion section which is a circular arc with an equal to three throat heights. Compare this nozzle contour and total length with the
Consider the external supersonic flow over the pointed body sketched in Fig. 11.22. Outline in detail how you would set up a method-of-characteristics solution for this flow. Poc Po Tee M Initial
Consider a convergent-divergent nozzle of length \(L\) with an area-ratio variation given by \(A / A^{*}=1+10|x / L|\), where \(-0.5 \leq x / L \leq 0.5\). Assume quasi-one-dimensional flow and a
Consider the two-dimensional, subsonic-supersonic flow in a convergent divergent nozzle.a. If the sonic line is straight, sketch the limiting characteristics.b. If the sonic line is curved, sketch
Consider a \(15^{\circ}\) half-angle right-circular cone. Using newtonian theory, calculate the drag coefficient for \(1.5 \leq M_{\infty} \leq 7\), assuming the base pressure is equal to
Consider a blunt axisymmetric body at an angle of attack \(\alpha\) in a supersonic stream. Assume a calorically perfect gas. Outline in detail how you would carry out a time-dependent,
Consider a hemisphere with a flat base in a hypersonic flow at \(0^{\circ}\) angle of attack (the hemispherical portion faces into the flow). Assuming that the base pressure is equal to free-stream
In the discussion of Computational Fluid Dynamics, the time-dependent (time-marching) solution of isentropic subsonic-supersonic quasi-one-dimensional flow is given, albeit under rather controlled
In the discussion of Computational Fluid Dynamics, Using the computer program and nozzle shape from problem 12.1, calculate the purely subsonic isentropic flow through the nozzle for the case when
In the discussion of Computational Fluid Dynamics, Using your computer program from problem 12.1, involving a normal shock wave inside the nozzle. (Again, do not be surprised if you have difficulty,
A uniform supersonic stream with \(M_{1}=3.0, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=288 \mathrm{~K}\) encounters a compression corner (see Fig. 4.4a) which deflects the stream by an angle
In Example 4.1, the deflection angle is increased to \(\theta=30^{\circ}\). Calculate the pressure and Mach number behind the wave, and compare these results with those of Example 4.1.Data From
In Example 4.1, the free-stream Mach number is increased to 5. Calculate the pressure and Mach number behind the wave, and compare these results with those of Example 4.1.Data From Example 4.1:A
Consider a Mach 2.8 supersonic flow over a compression corner with a deflection angle of \(15^{\circ}\). If the deflection angle is doubled to \(30^{\circ}\), what is the increase in shock strength?
Consider a compression corner with a deflection angle of \(28^{\circ}\). Calculate the shock strengths when \(M_{1}=3\) and when \(M_{1}\) is doubled to 6 . Is the shock strength also doubled?
Consider a Mach 4 flow over a compression corner with a deflection angle of \(32^{\circ}\). Calculate the oblique shock wave angle for the weak shock case using (a) Fig. 4.8 and (b) the
A \(10^{\circ}\) half-angle wedge is placed in a "mystery flow" of unknown Mach number. Using a Schlieren system, the shock wave angle is measured as \(44^{\circ}\). What is the free-stream Mach
Consider a \(15^{\circ}\) half-angle wedge at zero angle of attack. Calculate the pressure coefficient on the wedge surface in a Mach 3 flow of air.
Consider a \(15^{\circ}\) half-angle wedge at zero angle of attack in a Mach 3 flow of air. Calculate the drag coefficient. Assume that the pressure exerted over the base of the wedge, the base
Consider a horizontal supersonic flow at Mach 2.8 with a static pressure and temperature of \(1 \mathrm{~atm}\) and \(519^{\circ} \mathrm{R}\), respectively. This flow passes over a compression
Consider the geometry shown in Fig. 4.19. Here a supersonic flow with Mach number, pressure, and temperature \(M_{1}, p_{1}\), and \(T_{1}\), respectively, is deflected through an angle
a. Consider the supersonic flow described in Example 4.10, where \(M_{1}=2.8, p_{1}=1 \mathrm{~atm}\), and \(M_{3}=1.45\). This flow is shown in Fig. 4.20a. Calculate the total pressure in region 3
A uniform supersonic stream with \(M_{1}=1.5, p_{1}=1700 \mathrm{lb} / \mathrm{ft}^{2}\), and \(T_{1}=460^{\circ} \mathrm{R}\) encounters an expansion corner (see Fig. 4.32) which deflects the stream
Consider the arrangement shown in Fig. 4.34. A \(15^{\circ}\) half-angle diamond wedge airfoil is in a supersonic flow at zero angle of attack. A Pitot tube is inserted into the flow at the location
Consider an infinitely thin flat plate at a \(5^{\circ}\) angle of attack in a Mach 2.6 free stream. Calculate the lift and drag coefficients.
Consider an infinitely thin flat plate at an angle of attack of \(20^{\circ}\) in a Mach 3 free stream. Calculate the magnitude of the flow direction angle \(\Phi\) downstream of the trailing edge,
Consider the \(15^{\circ}\) half-angle wedge shown in Fig. 4.40. This is the same flow problem sketched in Fig. 4.12, with the added feature of the expansion waves at the corners of the base. We make
Consider an oblique shock wave with a wave angle equal to \(35^{\circ}\). Upstream of the wave, \(p_{1}=2000 \mathrm{lb} / \mathrm{ft}^{2}, T_{1}=520^{\circ} \mathrm{R}\), and \(V_{1}=3355
Consider a wedge with a half-angle of \(10^{\circ}\) flying at Mach 2. Calculate the ratio of total pressures across the shock wave emanating from the leading edge of the wedge.
Calculate the maximum surface pressure (in newtons per square meter) that can be achieved on the forward face of a wedge flying at Mach 3 at standard sea level conditions ( \(\left.p_{1}=1.01 \times
In the flow past a compression corner, the upstream Mach number and pressure are 3.5 and \(1 \mathrm{~atm}\), respectively. Downstream of the corner, the pressure is \(5.48 \mathrm{~atm}\). Calculate
Consider a \(20^{\circ}\) half-angle wedge in a supersonic flow at Mach 3 at standard sea level conditions ( \(p_{1}=2116 \mathrm{lb} / \mathrm{ft}^{2}\) and \(\left.T_{1}=519^{\circ}
A supersonic stream at \(M_{1}=3.6\) flows past a compression corner with a deflection angle of \(20^{\circ}\). The incident shock wave is reflected from an opposite wall which is parallel to the
An incident shock wave with wave angle \(=30^{\circ}\) impinges on a straight wall. If the upstream flow properties are \(M_{1}=2.8, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=300 \mathrm{~K}\), calculate
Consider a streamline with the properties \(M_{1}=4.0\) and \(p_{1}=1 \mathrm{~atm}\). Consider also the following two different shock structures encountered by such a streamline: (a) a single normal
Consider the flow past a \(30^{\circ}\) expansion corner, as sketched in Fig. 4.32. The upstream conditions are \(M_{1}=2, p_{1}=3 \mathrm{~atm}\), and \(T_{1}=400 \mathrm{~K}\). Calculate the
For a given Prandtl-Meyer expansion, the upstream Mach number is 3 and the pressure ratio across the wave is \(p_{2} / p_{1}=0.4\). Calculate the angles of the forward and rearward Mach lines of the
Consider a supersonic flow with an upstream Mach number of 4 and pressure of \(1 \mathrm{~atm}\). This flow is first expanded around an expansion corner with \(\theta=15^{\circ}\), and then
Consider the incident and reflected shock waves as sketched in Fig. 4.17. Show by means of sketches how you would use shock polars to solve for the reflected wave properties.Figure 4.17: +7 0.41 My M
Consider a supersonic flow past a compression corner with \(\theta=20^{\circ}\). The upstream properties are \(M_{1}=3\) and \(p_{1}=2116 \mathrm{lb} / \mathrm{ft}^{2}\). A Pitot tube is inserted in
Can shock polars be used to solve the intersection of shocks of opposite families, as sketched in Fig. 4.23? Explain.Figure 4.23: Slip line Horizontal
Using shock-expansion theory, calculate the lift and drag (in pounds) on a symmetrical diamond airfoil of semiangle \(\varepsilon=15^{\circ}\) (see Fig. 4.35) at an angle of attack to the free stream
Consider a flat plate with a chord length (from leading to trailing edge) of \(1 \mathrm{~m}\). The free-stream flow properties are \(M_{1}=3, p_{1}=1 \mathrm{~atm}\), and \(T_{1}=270 \mathrm{~K}\).
A flat plate is immersed in a Mach 2 flow at standard sea level conditions at an angle of attack of \(2^{\circ}\). Assuming the same shear stress distribution given in Example 1.8, calculate, per
Calculate the drag coefficient for a wedge with a \(20^{\circ}\) half-angle at Mach 4. Assume the base pressure is free-stream pressure.
The flow of a chemically reacting gas is sometimes approximated by the use of relations obtained assuming a calorically perfect gas, such as in this chapter, but using an "effective gamma," a ratio
For the two cases treated in Problem 4.20, calculate and compare the pressure ratio (shock strength) across the oblique shock wave. What can you conclude about the effect of a chemically reacting gas
Consider the isentropic subsonic-supersonic flow through a convergent-divergent nozzle. The reservoir pressure and temperature are \(10 \mathrm{~atm}\) and \(300 \mathrm{~K}\), respectively. There
A supersonic wind tunnel is designed to produce Mach 2.5 flow in the test section with standard sea level conditions. Calculate the exit area ratio and reservoir conditions necessary to achieve these
Consider a rocket engine burning hydrogen and oxygen; the combustion chamber temperature and pressure are \(3517 \mathrm{~K}\) and \(25 \mathrm{~atm}\), respectively. The molecular weight of the
Consider the flow through a convergent-divergent duct with an exit-to-throat area ratio of 2. The reservoir pressure is \(1 \mathrm{~atm}\), and the exit pressure is \(0.95 \mathrm{~atm}\). Calculate
Consider a convergent-divergent duct with an exit-to-throat area ratio of 1.6. Calculate the exit-to-reservoir pressure ratio required to achieve sonic flow at the throat, but subsonic flow
A supersonic wind tunnel is designed to produce flow in the test section at Mach 2.4 at standard atmospheric conditions. Calculate:a. The exit-to-throat area ratio of the nozzleb. Reservoir pressure
The reservoir pressure of a supersonic wind tunnel is \(10 \mathrm{~atm}\). A Pitot tube inserted in the test section measures a pressure of \(0.627 \mathrm{~atm}\). Calculate the test section Mach
The reservoir pressure of a supersonic wind tunnel is \(5 \mathrm{~atm}\). A static pressure probe is moved along the centerline of the nozzle, taking measurements at various stations. For these
Consider the purely subsonic flow in a convergent-divergent duct. The inlet, throat, and exit area are \(1.00 \mathrm{~m}^{2}, 0.70 \mathrm{~m}^{2}\), and \(0.85 \mathrm{~m}^{2}\), respectively. If
Consider the subsonic flow through a divergent duct with area ratio \(A_{2} / A_{1}=\) 1.7. If the inlet conditions are \(T_{1}=300 \mathrm{~K}\) and \(u_{1}=250 \mathrm{~m} / \mathrm{s}\), and the
The mass flow of a calorically perfect gas through a choked nozzle is given by\[\dot{m}=\frac{p_{o} A^{*}}{\sqrt{T_{o}}} \sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)
When the reservoir pressure and temperature of a supersonic wind tunnel are \(15 \mathrm{~atm}\) and \(750 \mathrm{~K}\), respectively, the mass flow is \(1.5 \mathrm{~kg} / \mathrm{s}\). If the
A blunt-nosed aerodynamic model is mounted in the test section of a supersonic wind tunnel. If the tunnel reservoir pressure and temperature are \(10 \mathrm{~atm}\) and \(800^{\circ} \mathrm{R}\),
Consider a flat plate mounted in the test section of a supersonic wind tunnel. The plate is at an angle of attack of \(10^{\circ}\) and the static pressure on the top surface of the plate is \(1.0
Consider a supersonic nozzle with a Pitot tube mounted at the exit. The reservoir pressure and temperature are \(10 \mathrm{~atm}\) and \(500 \mathrm{~K}\), respectively. The pressure measured by the
Consider a convergent-divergent duct with exit and throat areas of \(0.50 \mathrm{~m}^{2}\) and \(0.25 \mathrm{~m}^{2}\), respectively. The inlet reservoir pressure is \(1.0 \mathrm{~atm}\) and the
Consider a supersonic wind tunnel where the nozzle area ratio is \(A_{e} / A_{t_{1}}=104.1\). The throat area of the nozzle is \(A_{t_{1}}=1.0 \mathrm{~cm}^{2}\). Calculate the minimum area of the
At the exit of the diffuser of a supersonic wind tunnel which exhausts directly to the atmosphere, the Mach number is very low \((\approx 0.1)\). The reservoir pressure is \(1.8 \mathrm{~atm}\), and
In a supersonic nozzle flow, the exit-to-throat area ratio is \(10, p_{o}=10.00 \mathrm{~atm}\), and the backpressure \(p_{B}=0.04 \mathrm{~atm}\). Calculate the angle \(\theta\) through which the
Consider an oblique shock wave with \(M_{1}=4.0\) and \(\beta=50^{\circ}\). This shock wave is incident on a constant-pressure boundary, as sketched in Fig. 5.26. For the flow downstream of the
Consider a rocket engine burning hydrogen and oxygen. The combustion chamber temperature and pressure are \(4000 \mathrm{~K}\) and \(15 \mathrm{~atm}\), respectively. The exit pressure is \(1.174
We wish to design a Mach 3 supersonic wind tunnel, with a static pressure and temperature in the test section of \(0.1 \mathrm{~atm}\) and \(400^{\circ} \mathrm{R}\), respectively. Calculate:a. The
Consider two hypersonic wind tunnels with the same reservoir temperature of \(3000 \mathrm{~K}\) in air.(a) One tunnel has a test-section Mach number of 10. Calculate the flow velocity in the test
Consider a hypersonic wind tunnel with a reservoir temperature of \(3000 \mathrm{~K}\) in air. Calculate the theoretical maximum velocity obtainable in the test section. Compare this result with the
As problems 5.18 and 5.19 reflect, the air temperature in the test section of conventional hypersonic wind tunnels is low. In reality, air liquefies at a temperature of about \(50 \mathrm{~K}\)
The reservoir temperature calculated in problem 5.20 is beyond the capabilities of heaters in the reservoir of continuous-flow wind tunnels using air. This is why you do not see a Mach 20
The result from problem 5.21 shows that the reservoir temperature for a Mach 20 helium tunnel can be very reasonable. This is why several very high Mach number helium hypersonic wind tunnels exist.
Write the \(y\) component of Eq. (6.7), and use it to derive Eq. (6.12).Equation 6.12:\(\frac{\partial(ho v)}{\partial t}+abla \cdot(ho v \mathbf{V})=-\frac{\partial p}{\partial y}+ho f_y\) V pfd7
Write the \(z\) component of Eq. (6.7), and use it to derive Eq. (6.13).Equation 6.13:\(\frac{\partial(ho w)}{\partial t}+abla \cdot(ho w \mathbf{V})=-\frac{\partial p}{\partial z}+ho f_z\) pf d V
Starting with Eq. (6.12), which is in conservation form, derive Eq. (6.27), which is in non-conservation form.Equation 6.12:\(\frac{\partial(ho v)}{\partial t}+abla \cdot(ho v
Starting with Eq. (6.13), which is in conservation form, derive Eq. (6.28), which is in non-conservation form.Equation 6.13:\(\frac{\partial(ho w)}{\partial t}+abla \cdot(ho w
A flat plate with a chord length of \(3 \mathrm{ft}\) and an infinite span (perpendicular to the page in Fig. 1.12) is immersed in a Mach 2 flow at standard sea level conditions at an angle of attack
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

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