Question: Prove that the flow field specified in

> Consider the isentropic flow over an airfoil. The freestream conditions are T∞ = 245 K and p∞ = 4.35 × 104 N/m2. At a point on the airfoil, the pressure is 3.6 × 104 N/m2. Calculate the density at this point.

> Just upstream of a shock wave, the air temperature and pressure are 288 K and 1 atm, respectively; just downstream of the wave, the air temperature and pressure are 690 K and 8.656 atm, respectively. Calculate the changes in enthalpy, internal energy, an

> Calculate cp, cv, e, and h for a. The stagnation point conditions given in Problem 7.1 b. Air at standard sea level conditions (If you do not remember what standard sea level conditions are, find them in an appropriate reference, such as Reference 2.)

> 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 volum

> Repeat Problem 7.10, considering the flow of Problem 7.11.

> Repeat Problem 7.9, considering a point on the airfoil surface where the pressure is 0.3 atm.

> Calculate the percentage error obtained if Problem 7.9 is solved using (incorrectly) the incompressible Bernoulli equation.

> The temperature and pressure at the stagnation point of a high-speed missile are 934 ◦R and 7.8 atm, respectively. Calculate the density at this point.

> A sphere and a circular cylinder (with its axis perpendicular to the flow) are mounted in the same freestream. A pressure tap exists at the top of the sphere, and this is connected via a tube to one side of a manometer. The other side of the manometer is

> Prove that three-dimensional source flow is a physically possible incompressible flow.

> Consider a circular cylinder in a hypersonic flow, with its axis perpendicular to the flow. Let φ be the angle measured between radii drawn to the leading edge (the stagnation point) and to any arbitrary point on the cylinder. The pressure coefficient di

> Prove that three-dimensional source flow is irrotational.

> 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

> 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 coeffi

> 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.

> 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

> Consider the airplane and flight conditions given in Problem 5.4. The span efficiency factor e for the complete airplane is generally much less than that for the finite wing alone. Assume e=0.64. Calculate the induced drag for the airplane in Problem 5.4

> The Piper Cherokee (a light, single-engine general aviation aircraft) has a wing area of 170 ft2 and a wing span of 32 ft. Its maximum gross weight is 2450 lb. The wing uses an NACA 65-415 airfoil, which has a lift slope of 0.1033 degree−1 and αL=0=-3◦.

> The measured lift slope for the NACA 23012 airfoil is 0.1080 degree−1, and αL=0 = −1.3◦. Consider a finite wing using this airfoil, with AR = 8 and taper ratio = 0.8. Assume that δ = τ . Calculate the lift and induced drag coefficients for this wing at a

> Consider the same vortex filament as in Problem 5.1. Consider also a straight line through the center of the loop, perpendicular to the plane of the loop. Let A be the distance along this line, measured from the plane of the loop. Obtain an expression fo

> 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

> The German Zeppelins of World War I were dirigibles with the following typical characteristics: volume =15,000 m3 and maximum diameter= 14.0 m. Consider a Zeppelin flying at a velocity of 30 m/s at a standard altitude of 1000 m (look up the corresponding

> 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 yo

> Consider a vortex filament of strength Г in the shape of a closed circular loop of radius R. Obtain an expression for the velocity induced at the center of the loop in terms of Г and R.

> Starting with Equations (4.35) and (4.43), derive Equation (4.62).

> Compare the results of Problems 4.6 and 4.7 with experimental data for the NACA 4412 airfoil, and note the percentage difference between theory and experiment. (Hint: A good source of experimental airfoil data is Reference 11.)

> For the airfoil given in Problem 4.6, calculate cm,c/4 and xcp/c when α = 3◦.

> The NACA 4412 airfoil has a mean camber line given by Using thin airfoil theory, calculate (a) αL=0 (b) cl when α = 3◦

> Consider a thin, symmetric airfoil at 1.5◦ angle of attack. From the results of thin airfoil theory, calculate the lift coefficient and the moment coefficient about the leading edge.

> Starting with Equation (4.35), derive Equation (4.36).

> Starting with the definition of circulation, derive Kelvin’s circulation theorem, Equation (4.11).

> Consider an NACA 2412 airfoil with a 2-m chord in an airstream with a velocity of 50 m/s at standard sea level conditions. If the lift per unit span is 1353 N/m, what is the angle of attack?

> A U-tube mercury manometer is used to measure the pressure at a point on the wing of a wind-tunnel model. One side of the manometer is connected to the model, and the other side is open to the atmosphere. Atmospheric pressure and the density of liquid me

> 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 poi

> 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 a

> 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 a

> In Section 3.15 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 Figures 3.34(b) and (c). Here, the viscous shear stress

> For the airfoil in Problem 4.11, calculate the value of the circulation around the airfoil.

> 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 atta

> 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 o

> Consider the data for the NACA 2412 airfoil given in Figure 4.10. Calculate the lift and moment about the quarter chord (per unit span) for this airfoil when the angle of attack is 4◦ and the freestream is at standard sea level conditions with a velocity

> Show that a source flow is a physically possible incompressible flow everywhere except at the origin. Also show that it is irrotational everywhere.

> Consider a uniform flow with velocity V∞. Show that this flow is a physically possible incompressible flow and that it is irrotational.

> Consider a Lear jet flying at a velocity of 250 m/s at an altitude of 10 km, where the density and temperature are 0.414 kg/m3 and 223 K, respectively. Consider also a one-fifth scale model of the Lear jet being tested in a wind tunnel in the laboratory.

> At a given point on the surface of the wing of the airplane in Problem 3.6, the flow velocity is 130 m/s. Calculate the pressure coefficient at this point.

> A Pitot tube on an airplane flying at standard sea level reads 1.07 *105 N/m2. What is the velocity of the airplane?

> Assume that a Pitot tube is inserted into the test-section flow of the wind tunnel in Problem 3.4. The tunnel test section is completely sealed from the outside ambient pressure. Calculate the pressure measured by the Pitot tube, assuming the static pres

> Consider a low-speed open-circuit subsonic wind tunnel with an inlet-to-throat area ratio of 12. The tunnel is turned on, and the pressure difference between the inlet (the settling chamber) and the test section is read as a height difference of 10 cm on

> Consider a venturi with a small hole drilled in the side of the throat. This hole 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

> 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, co

> 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, ψ1, ψ2, and ψ3. The first streamline

> 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

> Consider a venturi with a throat-to-inlet area ratio of 0.8, mounted on the side of an airplane fuselage. The airplane is in flight at standard sea level. If the static pressure at the throat is 2100 lb/ft2, calculate the velocity of the airplane.

> Consider two different flows over geometrically similar airfoil shapes, one airfoil being twice the size of the other. The flow over the smaller airfoil has freestream properties given by T∞ =200 K, ρ∞=1.23 kg/m3, and V∞=100 m/s. The flow over the larger

> A typical World War I biplane fighter (such as the French SPAD shown in Figure 3.50) has a number of vertical interwing struts and diagonal bracing wires. Assume for a given airplane that the total length for the vertical struts (summed together) is 25 f

> The lift on a spinning circular cylinder in a freestream with a velocity of 30 m/s and at standard sea level conditions is 6 N/m of span. Calculate the circulation around the cylinder.

> 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