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 examine the effect of the much more highly two-dimensional flow generated in this case by a much larger expansion angle.
> 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 a
> 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 a
> 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
> 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?
> 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∞, a
> Consider a diamond-wedge airfoil such as shown in Figure 9.37, with a half-angle ε=10◦. The airfoil is at an angle of attack α=15◦ to a Mach 3 freestream. Using linear theory, calculate the lift and wave-drag coefficients for the airfoil. Compare these a
> For the conditions of Problem 12.1, calculate the pressures (in the form of p/ p∞) on the top and bottom surfaces of the flat plate, using linearized theory. Compare these approximate results with those obtained from exact shock-expansion theory in Probl
> Using the results of linearized theory, calculate the lift and wave-drag coefficients for an infinitely thin flat plate in a Mach 2.6 freestream at angles of attack of (a) α=5◦ (b) α=15◦ (c) α=30◦ Compare these approximate results with those from the e
> 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 res
> Consider the flow over a circular cylinder; the incompressible flow over such a cylinder is discussed in Section 3.13. Consider also the flow over a sphere; the incompressible flow over a sphere is described in Section 6.4. The subsonic compressible flow
> Consider a flow field in polar coordinates, where the stream function is given as ψ =ψ(r,θ). Starting with the concept of mass flow between two streamlines, derive Equations (2.148a and b).
> Figure 11.5 shows four cases for the flow over the same airfoil wherein M∞ is progressively increased from 0.3 to Mcr=0.61. Have you wondered where the numbers on Figure 11.5 came from? Here is your chance to find out. Point A on the airfoil is the point
> Consider an airfoil in a Mach 0.5 freestream. At a given point on the airfoil, the local Mach number is 0.86. Using the compressible flow tables at the back of this book, calculate the pressure coefficient at that point. Check your answer using the appro
> For a given airfoil, the critical Mach number is 0.8. Calculate the value of p/ p∞ at the minimum pressure point when M∞ = 0.8.
> In low-speed incompressible flow, the peak pressure coefficient (at the minimum pressure point) on an airfoil is-0.41. Estimate the critical Mach number for this airfoil, using the Prandtl-Glauert rule.
> Under low-speed incompressible flow conditions, the pressure coefficient at a given point on an airfoil is -0.54. Calculate Cp at this point when the freestream Mach number is 0.58, using a. The Prandtl-Glauert rule b. The Karman-Tsien rule c. Laitone’s
> Using the Prandtl-Glauert rule, calculate the lift coefficient for an NACA 2412 airfoil at 5◦ angle of attack in a Mach 0.6 freestream. (Refer to Figure 4.5 for the original airfoil data.)
> Consider a subsonic compressible flow in cartesian coordinates where the velocity potential is given by If the freestream properties are given by V∞ = 700 ft/s, p∞ = 1 atm, and T∞=519â—&brv
> Consider a convergent-divergent nozzle with an exit-to-throat area ratio of 1.53. The reservoir pressure is 1 atm. Assuming isentropic flow, except for the possibility of a normal shock wave inside the nozzle, calculate the exit Mach number when the exit
> For the flow in Problem 10.7, calculate the mass flow through the nozzle, assuming that the reservoir temperature is 288 K and the throat area is 0.3 m2.
> A convergent-divergent nozzle with an exit-to-throat area ratio of 1.616 has exit and reservoir pressures equal to 0.947 and 1.0 atm, respectively. Assuming isentropic flow through the nozzle, calculate the Mach number and pressure at the throat.
> Consider a body of arbitrary shape. If the pressure distribution over the surface of the body is constant, prove that the resultant pressure force on the body is zero. [Recall that this fact was used in Equation (2.77).]
> Repeat Problem 10.4, using the formula derived in Problem 10.5, and check your answer from Problem 10.4.
> A closed-form expression for the mass flow through a choked nozzle is Derive this expression.
> For the nozzle flow given in Problem 10.1, the throat area is 4 in2. Calculate the mass flow through the nozzle.
> A Pitot tube inserted at the exit of a supersonic nozzle reads 8.92 *104 N/m2. If the reservoir pressure is 2.02 *105 N/m2, calculate the area ratio Ae/ A∗ of the nozzle.
> A flow is isentropically expanded to supersonic speeds in a convergent-divergent nozzle. The reservoir and exit pressures are 1 and 0.3143 atm, respectively. What is the value of Ae/ A∗?
> 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.
> 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 ex
> 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, calculate the Mach number at the location AB in the duct.
> 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, calculate the Mach number at the location AB in the duct.
> For supersonic and hypersonic wind tunnels, a diffuser efficiency, ηD, can be defined as the ratio of the total pressures at the diffuser exit and nozzle reservoir, divided by the total pressure ratio across a normal shock at the test-section Mach number
> 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 siz
> Consider a rocket engine burning hydrogen and oxygen. The total mass flow of the propellant plus oxidizer into the combustion chamber is 287.2 kg/s. The combustion chamber temperature is 3600 K. Assume that the combustion chamber is a low-velocity reserv
> We wish to design a supersonic wind tunnel that produces a Mach 2.8 flow at standard sea level conditions in the test section and has a mass flow of air equal to 1 slug/s. Calculate the necessary reservoir pressure and temperature, the nozzle throat and
> The nozzle of a supersonic wind tunnel has an exit-to-throat area ratio of 6.79. When the tunnel is running, a Pitot tube mounted in the test section measures 1.448 atm. What is the reservoir pressure for the tunnel?
> A 20◦ half-angle wedge is mounted at 0◦ angle of attack in the test section of a supersonic wind tunnel. When the tunnel is operating, the wave angle from the wedge leading edge is measured to be 41.8◦. What is the exit-to-throat area ratio of the tunnel
> The reservoir pressure and temperature for a convergent-divergent nozzle are 5 atm and 520◦R, respectively. The flow is expanded isentropically to supersonic speed at the nozzle exit. If the exit-to-throat area ratio is 2.193, calculate the following pro
> Consider an oblique shock generated at a compression corner with a deflection angle θ=18.2◦. A straight horizontal wall is present above the corner, as shown in Figure 9.19. If the upstream flow has the properties M1 = 3.2, p1=1 atm and T1=520◦R, calcula
> Consider a Mach 4 airflow at a pressure of 1 atm. We wish to slow this flow to subsonic speed through a system of shock waves with as small a loss in total pressure as possible. Compare the loss in total pressure for the following three shock systems: a.
> A 30.2◦ half-angle wedge is inserted into a freestream with M∞ = 3.5 and p∞=0.5 atm. A Pitot tube is located above the wedge surface and behind the shock wave. Calculate the magnitude of the pressure sensed by the Pitot tube.
> Consider a flat plate at an angle of attack α to a Mach 2.4 airflow at 1 atm pressure. What is the maximum pressure that can occur on the plate surface and still have an attached shock wave at the leading edge? At what value of α does this occur?
> Consider the flow over a 22.2◦ half-angle wedge. If M1 = 2.5, p1 = 1 atm, and T1 = 300 K, calculate the wave angle and p2, T2, and M2.
> 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 w
> Consider an oblique shock wave with a wave angle of 36.87◦. The upstream flow is given by M1=3 and p1=1 atm. Calculate the total pressure behind the shock using a. p0,2/ p0,1 from Appendix B (the correct way) b. p0,2/ p1 from Appendix B (the incorrect wa
> Equation (8.80) does not hold for an oblique shock wave, and hence the column in Appendix B labeled p0,2/ p1 cannot be used, in conjunction with the normal component of the upstream Mach number, to obtain the total pressure behind an oblique shock wave.
> 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 reade
> 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
> Consider an oblique shock wave with a wave angle of 30◦ in a Mach 4 flow. The upstream pressure and temperature are 2.65*104 N/m2 and 223.3 K, respectively (corresponding to a standard altitude of 10,000 m). Calculate the pressure, temperature, Mach numb
> 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 horizontal flow at Mach 2 enters the duct and goes through a Pran
> 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 precis
> Consider a circular cylinder (oriented with its axis perpendicular to the flow) and a symmetric diamond-wedge airfoil with a half-angle of 5◦ at zero angle of attack; both bodies are in the same Mach 5 freestream. The thickness of the airfoil and the dia
> Consider sonic flow. Calculate the maximum deflection angle through which this flow can be expanded via a centered expansion wave.
> 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 norma
> Consider a diamond-wedge airfoil such as shown in Figure 9.36, with a half-angle ε=10◦. The airfoil is at an angle of attack α=15◦ to a Mach 3 freestream. Calculate the lift and wave-drag coefficients for the airfoil.
> Consider an infinitely thin flat plate at an angle of attack α in a Mach 2.6 flow. Calculate the lift and wave-drag coefficients for (a) α=5◦ (b) α=15◦ (c) α=30◦ (Note: Save the results of this problem for use in Chapter 12.)
> A supersonic flow at M1=3, T1=285 K, and p1=1 atm is deflected upward through a compression corner with θ=30.6◦ and then is subsequently expanded around a corner of the same angle such that the flow direction is the same as its original direction. Calcul
> A supersonic flow at M1=1.58 and p1=1 atm expands around a sharp corner. If the pressure downstream of the corner is 0.1306 atm, calculate the deflection angle of the corner.
> Consider the supersonic flow over an expansion corner, such as given in Figure 9.25. The deflection angle θ=23.38◦. If the flow upstream of the corner is given by M1=2, p1=0.7 atm, T1=630◦R, calculate M2, p2, T2, ρ2, p0,2, and T0,2 downstream of the corn
> A slender missile is flying at Mach 1.5 at low altitude. Assume the wave generated by the nose of the missile is a Mach wave. This wave intersects the ground 559 ft behind the nose. At what altitude is the missile flying?
> The entropy increase across a normal shock wave is 199.5 J/(kg. K). What is the upstream Mach number?
> The pressure upstream of a normal shock wave is 1 atm. The pressure and temperature downstream of the wave are 10.33 atm and 1390 ◦R, respectively. Calculate the Mach number and temperature upstream of the wave and the total temperature and total pressur
> The flow just upstream of a normal shock wave is given by p1=1 atm, T1=288 K, and M1=2.6. Calculate the following properties just downstream of the shock: p2, T2, ρ2, M2, p0,2, T0,2, and the change in entropy across the shock.
> Consider the isentropic flow over an airfoil. The freestream conditions correspond to a standard altitude of 10,000 ft and M∞=0.82. At a given point on the airfoil, M=1.0. Calculate p and T at this point. (Note: You will have to use the standard atmosphe
> 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
> Consider the isentropic flow through a supersonic nozzle. If the test-section conditions are given by p=1 atm, T=230 K, and M=2, calculate the reservoir pressure and temperature.
> At a given point in a flow, T=700 ◦R, p=1.6 atm, and V=2983 ft/s. At this point, calculate the corresponding values of p0, T0, p∗, T∗, and M∗.
> 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 ∗.
> The temperature in the reservoir of a supersonic wind tunnel is 519 ◦R. In the test section, the flow velocity is 1385 ft/s. Calculate the test-section Mach number. Assume the tunnel flow is adiabatic.
> Prove that the total pressure is constant throughout an isentropic flow.
> 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 differ
> 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
> In the test section of a supersonic wind tunnel, a Pitot tube in the flow reads a pressure of 1.13 atm. A static pressure measurement (from a pressure tap on the sidewall of the test section) yields 0.1 atm. Calculate the Mach number of the flow in the t
> On March 16, 1990, an Air Force SR-71 set a new continental speed record, averaging a velocity of 2112 mi/h at an altitude of 80,000 ft. Calculate the temperature (in degrees Fahrenheit) at a stagnation point on the vehicle.
> Derive the Rayleigh Pitot tube formula, Equation (8.80).
> 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 th
> Consider an NACA 2412 airfoil (the meaning of the number designations for standard NACA airfoil shapes is discussed in Chapter 4). The following is a tabulation of the lift, drag, and moment coefficients about the quarter chord for this airfoil, as a fun
> (a) A point charge q is inside a cavity in an uncharged conductor (Fig. 2.45). Is the force on q necessarily zero?11 (b) Is the force between a point charge and a nearby uncharged conductor always attractive?12
> Find the electric field a distance z above the center of a square loop (side a) carrying uniform line charge λ (Fig. 2.8). [Hint: Use the result of Ex. 2.2.]
> Two spherical cavities, of radii a and b, are hollowed out from the interior of a (neutral) conducting sphere of radius R (Fig. 2.49). At the center of each cavity a point charge is placed—call these charges qa and qb. (a) Find the surf
> A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b, as in Fig. 2.48). The shell carries no net charge. (a) Find the surface charge density σ at R, at a, and at b. (b)
> Find the interaction energy / for two point charges, q1 and q2, a distance a apart. [Hint: Put q1 at the origin and q2 on the z axis; use spherical coordinates, and do the r integral first.]
> Consider two concentric spherical shells, of radii a and b. Suppose the inner one carries a charge q, and the outer one a charge -q (both of them uniformly distributed over the surface). Calculate the energy of this configuration, (a) using Eq. 2.45, and
> Here is a fourth way of computing the energy of a uniformly charged solid sphere: Assemble it like a snowball, layer by layer, each time bringing in an infinitesimal charge dq from far away and smearing it uniformly over the surface, thereby increasing th
> Find the energy stored in a uniformly charged solid sphere of radius R and charge q. Do it three different ways: (a) Use Eq. 2.43. You found the potential in Prob. 2.21. (b) Use Eq. 2.45. Don’t forget to integrate over all space. (c) Use Eq. 2.44. Take a
> Consider an infinite chain of point charges, ±q (with alternating signs), strung out along the x axis, each a distance from its nearest neighbors. Find the work per particle required to assemble this system. [Partial Answer: / for some dimensionless nu
> Two positive point charges, qA and qB (masses m A and m B) are at rest, held together by a massless string of length a. Now the string is cut, and the particles fly off in opposite directions. How fast is each one going, when they are far apart?
> Find the transformation matrix R that describes a rotation by 120◦ about an axis from the origin through the point (1, 1, 1). The rotation is clockwise as you look down the axis toward the origin.
> (a) Three charges are situated at the corners of a square (side a), as shown in Fig. 2.41. How much work does it take to bring in another charge, +q, from far away and place it in the fourth corner? (b) How much work does it take to assemble the whole co
> (a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33. (b) Use Gauss’s law to find the field inside and outside a long hollow cylindrical tube, which carries a uniform surface charge σ. Check that your result is consi
> Find the electric field a distance z above one end of a straight line segment of length L (Fig. 2.7) that carries a uniform line charge λ. Check that your formula is consistent with what you would expect for the case z (( L.
> Check that Eq. 2.29 satisfies Poisson’s equation, by applying the Laplacian and using Eq. 1.102.
> Use Eq. 2.29 to calculate the potential inside a uniformly charged solid sphere of radius R and total charge q. Compare your answer to Prob. 2.21.
> Find the potential on the axis of a uniformly charged solid cylinder, a distance z from the center. The length of the cylinder is L, its radius is R, and the charge density is ρ. Use your result to calculate the electric field at this point. (Assume that
> A conical surface (an empty ice-cream cone) carries a uniform surface charge σ. The height of the cone is h, as is the radius of the top. Find the potential difference between points a (the vertex) and b (the center of the top).
> Using Eqs. 2.27 and 2.30, find the potential at a distance z above the center of the charge distributions in Fig. 2.34. In each case, compute E=-∇V, and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectiv
> For the configuration of Prob. 2.16, find the potential difference between a point on the axis and a point on the outer cylinder. Note that it is not necessary to commit yourself to a particular reference point, if you use Eq. 2.22.
> For the charge configuration of Prob. 2.15, find the potential at the center, using infinity as your reference point.
> (a) Prove that the two-dimensional rotation matrix (Eq. 1.29) preserves dot products. (That is, show that (b) What constraints must the elements (Rij ) of the three-dimensional rotation matrix (Eq. 1.30) satisfy, in order to preserve the length of A (fo
> Find the potential a distance s from an infinitely long straight wire that carries a uniform line charge λ. Compute the gradient of your potential, and check that it yields the correct field.
