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. Note: The Prandtl-Glauert rule is based on linear theory assuming small perturbations, and therefore we would not expect that it would be valid for the case of flow over a circular cylinder. Nevertheless, when you use it to make this calculation of Mcr , you will find your calculated result to be within 3.5 percent of the experimental value. Interesting.
> Consider the lifting flow over a circular cylinder of a given radius and with a given circulation. If V∞ is doubled, keeping the circulation the same, does the shape of the streamlines change? Explain.
> Consider the nonlifting flow over a circular cylinder of a given radius, where V∞=20 ft/s. If V∞ is doubled, that is, V∞=40 ft/s, does the shape of the streamlines change? Explain.
> Consider the nonlifting flow over a circular cylinder. Derive an expression for the pressure coefficient at an arbitrary point (r, θ) in this flow, and show that it reduces to Equation (3.101) on the surface of the cylinder.
> Derive the velocity potential for a doublet; that is, derive Equation (3.88). Hint: The easiest method is to start with Equation (3.87) for the stream function and extract the velocity potential.
> Derive Equation (3.81). Hint: Make use of the symmetry of the flow field shown in Figure 3.18; that is, start with the knowledge that the stagnation points must lie on the axis aligned with the direction of V∞.
> Consider the flow over a semi-infinite body as discussed in Section 3.11. If V∞ is the velocity of the uniform stream, and the stagnation point is 1 ft upstream of the source: a. Draw the resulting semi-infinite body to scale on graph paper. b. Plot the
> Prove that the velocity potential and the stream function for a source flow, Equations (3.67) and (3.72), respectively, satisfy Laplace’s equation.
> Prove that the velocity potential and the stream function for a uniform flow, Equations (3.53) and (3.55), respectively, satisfy Laplace’s equation.
> The shock waves on a vehicle in supersonic flight cause a component of drag called supersonic wave drag Dw. Define the wave-drag coefficient as CD,w=Dw/q∞ S, where S is a suitable reference area for the body. In supersonic flight, the flow is governed in
> For an irrotational flow, show that Bernoulli’s equation holds between any points in the flow, not just along a streamline.
> Is the flow field given in Problem 2.5 irrotational? Prove your answer.
> The velocity field given in Problem 2.4 is called vortex flow, which will be discussed in Chapter 3. For vortex flow, calculate: a. The time rate of change of the volume of a fluid element per unit volume. b. The vorticity. Hint: Again, for convenience u
> The velocity field given in Problem 2.3 is called source flow, which will be discussed in Chapter 3. For source flow, calculate: a. The time rate of change of the volume of a fluid element per unit volume. b. The vorticity. Hint: It is simpler to convert
> Consider a velocity field where the x and y components of velocity are given by u =cx and v=- cy, where c is a constant. Obtain the equations of the streamlines.
> Consider a velocity field where the radial and tangential components of velocity are Vr=0 and Vθ=cr , respectively, where c is a constant. Obtain the equations of the streamlines.
> Consider a velocity field where the x and y components of velocity are given by u= cy/(x2+ y2) and v=cx/(x 2+y2), where c is a constant. Obtain the equations of the streamlines.
> Consider a velocity field where the x and y components of velocity are given by u =cx/(x2+y2) and v=cy/(x 2+y2) where c is a constant. Obtain the equations of the streamlines.
> Consider an airfoil in a wind tunnel (i.e., a wing that spans the entire test section). Prove that the lift per unit span can be obtained from the pressure distributions on the top and bottom walls of the wind tunnel (i.e., from the pressure distribution
> In Example 2.1, the statement is made that the streamline an infinite distance above the wall is straight. Prove this statement.
> The drag on the hull of a ship depends in part on the height of the water waves produced by the hull. The potential energy associated with these waves therefore depends on the acceleration of gravity g. Hence, we can state that the wave drag on the hull
> Consider the subsonic compressible flow over the wavy wall treated in Example 2.1. Derive the equation for the velocity potential for this flow as a function of x and y.
> Consider an airfoil at 12◦ angle of attack. The normal and axial force coefficients are 1.2 and 0.03, respectively. Calculate the lift and drag coefficients.
> Consider an infinitely thin flat plate witha1m chord at an angle of attack of 10â—¦ in a supersonic flow. The pressure and shear stress distributions on in meters and p and Ï„ are in newtons per square meter. Calculate the no
> Consider an infinitely thin flat plate of chord c at an angle of attack α in a supersonic flow. The pressures on the upper and lower surfaces are different but constant over each surface; that is, pu(s) = c1 and pl (s) c2, where c1 and c2 are constants a
> Starting with Equations (1.7), (1.8), and (1.11), derive in detail Equations (1.15), (1.16), and (1.17).
> For most gases at standard or near standard conditions, the relationship among pressure, density, and temperature is given by the perfect gas equation of state: p=ρ RT, where R is the specific gas constant. For air at near standard conditions, R =287 J/(
> 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 t
> Consider a compressible, laminar boundary layer over a flat plate. Assuming Pr=1 and a calorically perfect gas, show that the profile of total temperature through the boundary layer is a function of the velocity profile via where Tw=wall temperature and
> Repeat Problem 19.4 for the case of all turbulent flow.
> Consider a length of pipe bent into a U-shape. The inside diameter of the pipe is 0.5 m. Air enters one leg of the pipe at a mean velocity of 100 m/s and exits the other leg at the same magnitude of velocity, but moving in the opposite direction. The pre
> Consider Mach 4 flow at standard sea level conditions over a flat plate of chord 5 in. Assuming all laminar flow and adiabatic wall conditions, calculate the skin-friction drag on the plate per unit span.
> For the case in Problem 19.1, calculate the skin-friction drag accounting for transition. Assume the transition Reynolds number = 5 × 105.
> For the case in Problem 19.1, calculate the boundary-layer thickness at the trailing edge for a. Completely laminar flow b. Completely turbulent flow
> The wing on a Piper Cherokee general aviation aircraft is rectangular, with a span of 9.75 m and a chord of 1.6 m. The aircraft is flying at cruising speed (141 mi/h) at sea level. Assume that the skin-friction drag on the wing can be approximated by the
> Assume that the two parallel plates in Problem 15.1 are both stationary but that a constant pressure gradient exists in the flow direction (i.e., dp/dx = constant). a. Obtain an expression for the variation of velocity between the plates. b. Obtain an ex
> Consider the incompressible viscous flow of air between two infinitely long parallel plates separated by a distance h. The bottom plate is stationary, and the top plate is moving at the constant velocity ue in the direction of the plate. Assume that no p
> 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
> Consider a flat plate at α=20◦ in a Mach 20 freestream. Using straight newtonian theory, calculate the lift- and wave-drag coefficients. Compare these results with exact shock-expansion theory.
> Repeat Problem 9.13 using a. Newtonian theory b. Modified newtonian theory Compare these results with those obtained from exact shock-expansion theory (Problem 9.13). From this comparison, what comments can you make about the accuracy of newtonian and mo
> Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at (x1, y1) = (0, 0.0684) and (x2, y2) = (0.0121, 0), where the units are meters. At point (x1, y1): u1 = 639 m/s, v1 = 232.6 m/s, p1 = 1 atm, T1 = 288 K.
> Assuming the velocity field given in Problem 2.6 pertains to an incompressible flow, calculate the stream function and velocity potential. Using your results, show that lines of constant φ are perpendicular to lines of constant ψ .
> 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
> 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
> 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
> 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
