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Chapter: 3 -Problem: 22 >> 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 some region of the flow field exceeds about 250 mi/h, compressibility begins to have an effect in that region. Calculate the velocity of the flow in the test section of the wind tunnel above which compr
Answer Preview: Calculation. Velocity of flow is given by; v=250 mi/h We kno…

, Chapter: 3 -Problem: 23 >> 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. Transcribed Image Text: and where u = Voo v = h 2? [1 + 127 - l 2?.? (cos 27x) e-² =) COS 2? 2?.? Voch ² (sin 27x) e l B=?1-M² –2???/? ,?2??y/1 (2.35) (2.
Answer Preview: Thorough solution with explanation. The flow field spec…

, Chapter: 10 -Problem: 17 >> 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 curve smoothly and continuously downward through the expansion fan, and emerge parallel to the downward sloping surface downstream of the tail of the wave, as shown in Figure 9.2b. Imagine a polar coor
Answer Preview: the Mach wave is perpendicular to the free stream, so the …

, Chapter: 5 -Problem: 10 >> 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 this value with that obtained in Problem 5.9. What can you conclude about the relative effect of planform shape change on the drag of the airplane at high speeds? Data from Problem 5.9:Consider the Super
Answer Preview: For low speeds, the induced drag coefficient is given by: Cdi = (CL)/(ARe) where CL is the lift coef…

, Chapter: 12 -Problem: 4 >> Equation (12.24), from linear supersonic theory, predicts that cd for a flat plate decreases as M? increases? Does this mean that the drag force itself decreases as M? increases? To answer this question, derive an equation for drag as a function of M?, and evaluate this equation. Transcribed Image Text:
Answer Preview: Yes, according to linear supersonic theory, the drag force decreases as M increases. To see …

, Chapter: 9 -Problem: 18 >> Consider a two-dimensional duct with a straight horizontal lower wall, and a straight upper wall inclined upward through the angle ? = 3?. The height of the duct entrance is 0.3 m. A uniform horizontal flow at Mach 2 enters the duct and goes through a Prandtl-Mayer expansion wave centered at the top corner of the entrance. The wave propagates to the bottom wall, where the leading edge (the forward
Answer Preview: The average flow Mach number over AB = 1/2 M(xA) + 1/2 M(xB) = 1/2 [(0 3 - xA)(3 + sin )] …

, Chapter: 4 -Problem: 16 >> 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 transitioning to a turbulent boundary layer at some point downstream. Calculate the skin-friction coefficient for the Spitfire’s airfoil described in Problem 4.15, but this time assuming a critical Rey
Answer Preview: Calculation. The skin friction coefficient is given by; Cf=(0 564/R…

, Chapter: 9 -Problem: 20 >> 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 flow back to the horizontal. Calculate the pressure of the flow downstream of the compression corner.
Answer Preview: Pressure formula: p = p + 1/2 rho v^2 (1) p is the pressure, p is the pressur…

, Chapter: 4 -Problem: 4 >> Starting with Equation (4.35), derive Equation (4.36). Transcribed Image Text: ["(dll.) = Poc Vso [^2y (4) d - V? - M?E = - * 5 (dL) = LE (4.35)
Answer Preview: Answer: Equation (4 35): $Pleft(x_{1}, x_{2}, ldots, x_{n}ight)=frac{1}{sqrt{2 pi}^{n} left|Sigmaigh…

, Chapter: 9 -Problem: 19 >> Repeat Problem 9.18, except with ? = 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 examine the effect of the much more highly two-dimensional flow generated in this case by a much larger expansion angle. Data from Problem 9.18:Consider a two-dimensional duct with a straight horizontal lowe
Answer Preview: The Prandtl-Mayer expansion wave is a shock wave that forms when a supersonic f…

, Chapter: 12 -Problem: 5 >> Consider a flat plate at an angle of attack in an inviscid supersonic flow. From linear theory, what is the value of the maximum lift-to-drag ratio, and at what angle of attack does it occur?
Answer Preview: Solution; From the linear theory the maximum lift-to-drag ratio occurs at …

, Chapter: 4 -Problem: 9 >> Starting with Equations (4.35) and (4.43), derive Equation (4.62). Transcribed Image Text: MLE = - [ 5 (dL) =-PxV?0 [°* &Y (? 0 0 == Voo 51 ??(?) d? (4.35)
Answer Preview: Answer: Starting with Equations (4 35) and (4 43), we can derive Equatio…

, Chapter: 1 -Problem: 15 >> 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 is a function of the lift coefficient CL for reasons that are given in Chapter 5; this function for the given airplane is CD = 0.025 + 0.054C2L .a. For a steady level flight at sea level, where the amb
Answer Preview: CL = 2W/(V^2S) CD = C_D0 + C_L^2/(eAR) L/D = CL/CD Explanation: Lift is a force that is perpendicula…

, Chapter: 1 -Problem: 16 >> 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 is 282 N/m2. The gas temperature at the wall is equal to standard sea level temperature. At this point, calculate the velocity gradient at the wall normal to the wall.
Answer Preview: The velocity gradient at the wall normal to …

, Chapter: 4 -Problem: 15 >> 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. The measured profile drag coefficient of the NACA 2213 airfoil is 0.006 at a Reynolds number of 9 × 106. Consider the Spitfire cruising at an altitude of 18,000 ft. (a) At what velocity is it flying for
Answer Preview: Calculation. (a) The velocity is given by; v=Re/ Where, Re is the Reynolds number, is the dynam…

, Chapter: 5 -Problem: 8 >> 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 attack of 3 degrees, corresponding to a design lift coefficient of 0.546. When they tested their gliders at Kill Devil Hills near Kitty Hawk, North Carolina, in 1900 and 1901, however, they measured only
Answer Preview: These were the first gliders the Wright brothers designed, and they were still learning the ropes. The Wright brothers probably chose three degrees be…

, Chapter: 5 -Problem: 9 >> 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 weight is 5820 lb, wing area is 242 ft2, and wing span is 36.1 ft. It is powered by a supercharged Merlin engine, which produced 1050 horsepower at 18,500 ft. (a) Calculate the induced drag coefficient of
Answer Preview: Calculation. (a) The induced drag coefficient of the Spitfire a…

, Chapter: 8 -Problem: 19 >> Prove that the total pressure is constant throughout an isentropic flow.
Answer Preview: The total pressure in a constant isentropic flow is equal to the …

, Chapter: 9 -Problem: 17 >> 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 trailing edge shock and expansion waves, is not precisely in the freestream direction. Why? Outline a method to calculate the strengths of the trailing edge shock and expansion waves, and the directio
Answer Preview: Calculations. One method to calculate the strengths of the trailing edge shock and expansion wave…

, Chapter: 19 -Problem: 7 >> 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 vehicle is 2.54 cm. Assume the Prandtl number for air at these conditions is 0.72, that cp is 1008 joules/(kg K), and that the viscosity coefficient is given by Sutherland’s law. The wall temperature at t
Answer Preview: (a) 1500 m/s, The answer is 416 . 5 W / m ^ 2 The aerodynamic heat transfer to the stagnation point can be calculated using the following equation: Q = q_w + q_s where q_w is the heat transfer from th…

, Chapter: 12 -Problem: 7 >> Using the same flight conditions and the same value of the skin-friction coefficient from Example 12.3, and the results of Problem 12.6, calculate the maximum lift-to-drag ratio of the flat plate that is used to simulate the F-104 wing and the angle of attack at which it occurs. Data from Problem 12.6:Consider a flat plate at an angle of attack in a viscous supersonic flow; i.e., there is both ski
Answer Preview: To solve this problem, you will need to use the expressions for the angle of attack at which maximu…

, Chapter: 4 -Problem: 14 >> 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 0.1 per degree. (a) Calculate the lift coefficient at an angle of attack of 5°. (b) Now imagine the same airfoil turned upside-down, but at the same 5° angle of attack as part (ii). Calculate its lift co
Answer Preview: Calculation. (a) The lift coefficient is given by; Clift=2 Where, …

, Chapter: 12 -Problem: 8 >> The result from Problem 12.6 demonstrates that maximum lift-to-drag ratio decreases as the Mach number increases. This is a fact of nature that progressively causes designers of supersonic airplanes grief as they strive toward aerodynamically efficient airplanes at higher supersonic Mach numbers. What physics is nature using against the airplane designer in this case, and how might the designer me
Answer Preview: The designer can meet this challenge by making a sharp nose. Doing that causes an aircraft to bec…

, Chapter: 10 -Problem: 16 >> Return to Problem 9.19, where the average Mach number across the two-dimensional flow in a duct was calculated, and where ? for the upper wall was 30?. Assuming quasi-one-dimensional flow, calculate the Mach number at the location AB in the duct. Data from Problem 9.19:Repeat Problem 9.18, except with ? = 30?. Again, we will use these results to compare with a quasi-one-dimensional calculation in
Answer Preview: We are given the following information: We can calculate the Mach numbers along the upper and lower …

, Chapter: 6 -Problem: 1 >> Prove that three-dimensional source flow is irrotational.
Answer Preview: Prove that three-dimensional source flow is irrotational. …

, Chapter: 7 -Problem: 12 >> Repeat Problem 7.10, considering the flow of Problem 7.11 Data from Problem 7.10:Calculate the percentage error obtained if Problem 7.9 is solved using (incorrectly) the incompressible Bernoulli equation. Data from Problem 7.9:An airfoil is in a freestream where p? = 0.61 atm, ?? = 0.819 kg/m3, and V? = 300 m/s. At a point on the airfoil surface, the pressure is 0.5 atm. Assuming isentropic flow,
Answer Preview: Problem 7 10: The percentage error would be -38 3%. Problem 7 11: The velocity at the point on the a…

, Chapter: 10 -Problem: 15 >> Return to Problem 9.18, where the average Mach number across the two-dimensional flow in a duct was calculated, and where ? for the upper wall was 3?. Assuming quasi-one dimensional flow, calculate the Mach number at the location AB in the duct. Data from Problem 9.18:Consider a two-dimensional duct with a straight horizontal lower wall, and a straight upper wall inclined upward through the angle
Answer Preview: For the two-dimensional flow in a duct, the averag…

, Chapter: 5 -Problem: 6 >> 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 straight wing, and (b) a swept wing, with a half-chord line sweep of 45 degrees.
Answer Preview: Calculation; Calculation for straight wing: Here, A = (152/6)^…

, Chapter: 8 -Problem: 17 >> 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 with the ratio of specific heats equal to 1.4, predict the gas temperature at the stagnation point of the Apollo at Mach 36 at an altitude where the freestream temperature is 300 K. Comment on the validity
Answer Preview: I predict that the gas temperature at the stagnation point of th…

, Chapter: 4 -Problem: 12 >> 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 at a velocity of 60 m/s at a standard altitude of 3 km (see Appendix D). The chord length of the airfoil is 2 m. Calculate the lift per unit span when the angle of attack is 4°. 
Answer Preview: The circulation around the airfoil can be calculated using the equation: = 2v/c Where: = …

, Chapter: 14 -Problem: 3 >> 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 the point on the surface of the nose located 20? away from the stagnation point, estimate the: (a) Pressure, (b) Temperature, (c) Mach number, (d) Velocity of the flow.
Answer Preview: Calculations. a. The pressure is given by: P = P0(1 + 0 2M) where P0 is the …

, Chapter: 1 -Problem: 2 >> Starting with Equations (1.7), (1.8), and (1.11), derive in detail Equations (1.15), (1.16), and (1.17).    Transcribed Image Text: N' == TE -TE * (P, cos 0 + T? sine) ds, + ² (p, cos 0 — , sine) ds; (1.7) - TE TE - fort (- Pu sine + T cose) dsu + f JLE A' = = (p? sine + 7, cos 0) ds; (1.8)
Answer Preview: Answer: Equation (1 7): F = ma Equation (1 8): m = F/…

, Chapter: 9 -Problem: 21 >> 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 tutorial and a quantitative calculation involving the reader. We first discuss some necessary thermodynamic background, followed by an examination of the physical nature of the flow field.
Answer Preview: The expression for the coefficient of lift is given by: Cl=41M2 Here, is the angle of attack. …

, Chapter: 1 -Problem: 19 >> 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 attack of 3 degrees, and made all their calculations of size, weight, etc., based on this design angle of attack. Why do you think they chose three degrees? Hint: From the table, calculate the ratio of lif
Answer Preview: They chose three degrees because of the Lilienthal data, but you can …

, Chapter: 3 -Problem: 3 >> 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 when the venturi is placed in an airstream. (The vacuum is defined as the pressure difference below the outside ambient pressure.) The venturi has a throat-to-inlet area ratio of 0.85. Calculate the maxim
Answer Preview: Calculations. The maximum vacuum obtainable in the reservoir is given by: Pvac = …

, Chapter: 8 -Problem: 3 >> 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?.
Answer Preview: Calculation; Here, p0= p+p= 1 2atm + 0 203atm = 1 43…

, Chapter: 2 -Problem: 14 >> In Example 2.1, the statement is made that the streamline an infinite distance above the wall is straight. Prove this statement. Transcribed Image Text: The subsonic compressible flow over a cosine-shaped (wavy) wall is illustrated in Fig- ure 2.17. The wavelength and amplitude of the wall are I and
Answer Preview: Answer: This statement can be proven using the equation for streamline curvat…

, Chapter: 1 -Problem: 17 >> 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 surface of the wing is 0.03 MW/m2. Calculate the normal temperature gradient in the air at this point on the wall, assuming the gas temperature at the wall is equal to the standard sea-level temperatu
Answer Preview: Calculations. The temperature gradient in the air is given by: T/z = …

, Chapter: 4 -Problem: 10 >> 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 coefficients are 0.65 and ?0.037, respectively. Calculate the location of the aerodynamic center.
Answer Preview: For the NACA 2412 airfoil, the lift coefficient and moment coefficient a…

, Chapter: 7 -Problem: 13 >> 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 in Bernoulli’s equation have dimensions of energy per unit volume (check it out), which prompt some argument that Bernoulli’s equation is an energy equation for incompressible flow. If this is so, then it s
Answer Preview: Answer: To derive Bernoulli's equation from the energy equation for compressible flow, we start wit…

, Chapter: 8 -Problem: 18 >> 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 with a ratio of specific heats equal to 1.4. The difference is due to chemical reactions that occur in air at these high temperatures—dissociation and ionization. The analyses in this book assuming a c
Answer Preview: Calculations. The value of the effective gamma is given by; e=Tt…

, Chapter: 12 -Problem: 6 >> Consider a flat plate at an angle of attack in a viscous supersonic flow; i.e., there is both skin friction drag and wave drag on the plate. Use linear theory for the lift and wave-drag coefficients. Denote the total skin friction drag coefficient by Cf , and assume that it does not change with angle of attack. (a) Derive the expression for the angle of attack at which maximum lift-to-drag ratio o
Answer Preview: (a)Derive the expression for the angle of attack at which maximum lif…

, Chapter: 10 -Problem: 18 >> 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 10.17: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 c
Answer Preview: Given: M1 = 1 0 and M2 = 1 6 Find: Plot the streamline that passes through the expansi…

, Chapter: 1 -Problem: 18 >> 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 length (distance from the front to the back of the wing where the wing joins the fuselage) is 14.25ft. Consider the DC-3 flying at 200 miles per hour at sea level. Calculate the Reynolds number for the f
Answer Preview: Calculation. (a) The Reynolds number is given by; Re=vc/ Where, v is the veloc…

, Chapter: 11 -Problem: 9 >> 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 Mcr for a circular cylinder using the incompressible result for Cp and the Prandtl-Glauert compressibility correction, and compare your result with the experimental value. The Prandtl-Glauert rule is
Answer Preview: First, recall that the Prandtl-Glauert compressibility correction is based on linear theory. This me…

, Chapter: 1 -Problem: 20 >> 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 answer.
Answer Preview: A forward - facing ax ial aer odynamic force can exist on a flat plate at an angle of attack in a fl…

, Chapter: 5 -Problem: 7 >> 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 magnitude of this effect is affected by aspect ratio. Data from Problem 5.6:Consider a finite wing with an aspect ratio of 6. Assume an elliptical lift distribution. The lift slope for the airfoil sec
Answer Preview: The lift slope for the straight wing is 0 1/degree, while the …

, Chapter: 3 -Problem: 21 >> 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 its stream function, ?1, ?2, and ?3. The first streamline wets the surface of the cylinder; designate ?1 = 0. The streamline above that is ?2, and the next one above that is ?3. Assume the streamlines st
Answer Preview: Calculation. The location of the point directly above the top of the cylinder throu…

, Chapter: 5 -Problem: 11 >> 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 the high-speed case in Problem 5.9. From this, what can you conclude about the relative importance of the induced drag coefficient at low speeds compared to that at high speeds? Data from Problem 5.9:C
Answer Preview: For low speeds, the induced drag coefficient is given by: Cdi = (CL)/(ARe) where CL is the lift coef…

, Chapter: 3 -Problem: 20 >> 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 two-dimensional body of arbitrary shape. Although this general result can be proven mathematically, it also can be accepted by making a physical argument as well. Make this physical argument by drawing
Answer Preview: Answer: The physical argument for the general application of the Kutta-Joukowsk…

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