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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 = pRT, where R is the specific gas constant. For air at near standard conditions, R = 287 J/(kg ∙ K) in the International System of Units and R = 1716 ft ∙ lb/(slug ∙ °R) in the English Engineering System of Units. (More details on the perfect gas equation of state are given in Chap. 7.) Using the above information, consider the following two cases:
(a) At a given point on the wing of a Boeing 727, the pressure and temperature of the air are 1.9 x 104 N/m2 and 203 K, respectively. Calculate the density at this point.
(b) At a point in the test section of a supersonic wind tunnel, the pressure and density of the air are 1058 lb/ft2 and 1.23 x 10-3 slug/ft3, respectively. Calculate the temperature at this point.
Starting with Eqs (1.7) (1.8), and (1.11), derive in detail Eqs (1.15), (1.16), and (1.17)
Consider an infinitely thin flat plate of chord c at an angle of attack a in a supersonic flow. The pressures on the upper and lower surfaces are different but constant over each surface; i.e., pu(s) = c1 and p1(s) = c2, where c1 and c2 are constants and c2 > c1. Ignoring the shear stress, calculate the location of the center of pressure.
Consider an infinitely thin flat plate with a 1-m chord at an angle of attack of 10° in a supersonic flow. The pressure and shear stress distributions on the upper surfaces are given by pu = 4x 104(x- l)2 + 5.4 x 104, p1 = 2x 104(x- 1)2 + 1.73 x 105, τu = 288x –0.2, and τ1 = 731x–0.2, respectively, where x is the distance from the leading edge in meters and p and τ are in Newton’s per square meter. Calculate the normal and axial forces, the lift and drag, moments about the leading edge, and moments about the quarter chord, all per unit span. Also, calculate the location of the center of pressure.
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 NAC A 2412 airfoil (the meaning of the number designations for standard NACA airfoil shapes is discussed in Chap. 4). The following is a tabulation of the lift, drag, and moment coefficients about the quarter chord for this airfoil, as a function of angle ofattack.
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 is D = f (p∞, V∞, c, g) where c is a length scale associated with the hull, say, the maximum width of the hull. Define the drag coefficient as CD = D/q∞c2. Also, define a similarity parameter called the Froude number, Fr= V/√gc. Using Buckingham's pi theorem, prove that CD =/(Fr).
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 CDw = Dw/q∞ S, where S is a suitable reference area for the body. In supersonic flight, the flow is governed in part by its thermodynamic properties, given by the specific heats at constant pressure, cp, and at constant volume, cv. Define the ratio cp/cv= γ. Using Buckingham's pi theorem, show that CD,w =/(M∞, γ). Neglect the influence of friction.
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 free stream properties given by T∞ = 200 K., p∞ = 1.23 kg/m3, and V∞= 100 m/s. The flow over the larger airfoil is described by T∞ = 800K, p∞ = 1.739 kg/m3, and V∞ = 200rn/s. Assume that both μ and a are proportional to T1/2. Are the two flows dynamically similar?
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. The pressure in the test section of the wind tunnel is 1atm = 1.01 x 105 N/m2. Calculate the necessary velocity, temperature, and density of the airflow in the wind-tunnel test section such that the lift a id drag coefficients are the same for the wind-tunnel model and the actual airplane in flight. Note: The relation among pressure, density, and temperature is given by the equation of state described in Prob. 1.1.
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 mercury are 1.01 x 105 N/m2 and 1.36 x 104 kg/m3, respectively. When the displacement of the two columns of mercury is 20 cm, with the high column on the model side, what is the pressure on the wing?
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 density in any standard altitude table). The Zeppelin is at a small angle of attack such that its lift coefficient is 0.05 (based on the maximum cross-sectional area). The Zeppelin is flying in straight-and-level flight with no acceleration. Calculate the total weight of the Zeppelin.
Consider a circular cylinder in a hypersonic flow, with its axis perpendicular to the flow. Let 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 distribution along the cylindrical surface is given by Cp = 2 cos2 Φ for 0 < Φ < π/2 and 3π/2 < Φ > 2π and Cp = 0 for π/2 < Φ < 3 π/2. Calculate the drag coefficient for the cylinder, based on projected frontal area of the cylinder.
Derive Archimedes' principle using a body of general shape.
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 Eq. (2.68).]
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 distributions on the walls above and below the airfoil).
Consider a velocity field where the x and y components of velocity are given by u = cx/(x2 + y2) and v = cy/(x2 + 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 = cy/(x2 + y2) and v = – cx/(x2 + y2), 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 = cx and v = – cy, where c is a constant. Obtain the equations of the streamlines.
The velocity field given in Prob. 2.3 is called source flow. For source flow, calculate:
(a) The time rate of change of the volume of a fluid element per unit volume
(b) The vorticity
The velocity field given in Prob. 2.4 is called vortex flow. For vortex flow, calculate:
(a) The time rate of change of the volume of a fluid element per unit volume
(b) The vorticity
Is the flow field given in Prob. 2.5 irrotational? Prove your answer.
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 Eqs. (2.139a and b)
Assuming the velocity field given in Prob. 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
For an irrotational flow, show that Bernoulli's equation holds between any points in the flow, not just along a streamline.
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 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 a U-tube mercury manometer. (The density of liquid mercury is 1.36 x 104 kg/m3.) Calculate the velocity of the air in the test section.
Assume that a Pitot tube is inserted into the test-section flow of the wind tunnel in Prob. 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 pressure at the tunnel inlet is atmospheric.
A Pitot tube on an airplane flying at standard sea level reads 1.07 x 105 N/m2. What is the velocity of the airplane?
At a given point on the surface of the wing of the airplane in Prob. 3.6, the flow velocity is 130 m/s. Calculate the pressure coefficient at this point.
Consider a uniform flow with velocity V^. Show that this flow is a physically possible incompressible flow and that it is irrotational.
Show that a source flow is a physically possible incompressible flow everywhere except at the origin. Also show that it is irrotational everywhere.
Prove that the velocity potential and the stream function for a uniform flow, Eqs. (3.53) and (3.55), respectively, satisfy Laplace's equation.
Prove that the velocity potential and the stream function for a source flow, Eqs. (3.67) and (3.72), respectively, satisfy Laplace's equation.
Consider the flow over a semi-infinite body as discussed in Sec. 3.11. If Vx 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 pressure coefficient distribution over the body; i.e., plot Cp versus distance along the centerline of the body.
Derive Eq. (3.81). Make use of the symmetry of the flow field shown in Fig. 3.18; i.e., start with the knowledge that the stagnation points must lie on the axis aligned with the direction of V∞-
Derive the velocity potential for a doublet; i.e., derive Eq. (3.88).
Consider the non-lifting 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 Eq. (3.101) on the surface of the cylinder.
Consider the non-lifting flow over a circular cylinder of a given radius, where V∞ = 20ft/s. If V∞ is doubled, i.e., V∞ = 40ft/s, does the shape of the streamlines change? Explain.
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.
The lift on a spinning circular cylinder in a free stream 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.
A typical World War I biplane fighter (such as the French SPAD shown in Fig. 3.45) 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 ft, and that the struts are cylindrical with a diameter of 2 in. Assume also that the total length of the bracing wires is 80 ft, with a cylindrical diameter of 3/32 in. Calculate the drag (in pounds) contributed by these struts and bracing wires when the airplane is flying at 120 mi/h at standard sea level. Compare this component of drag with the total zero-lift drag for the airplane, for which the total wing area is 230 ft2 and the zero-lift drag coefficient is 0.036.
Consider the data for the NACA 2412 airfoil given in Fig. 4.5. Calculate the lift and moment about the quarter chord (per unit span) for this airfoil when the angle of attack is 4° and the free stream is at standard sea level conditions with a velocity of 50 ft/s. The chord of the airfoil is 2 ft.
Consider an NACA 2412 airfoil with a 2-m chord in an airstream with a velocity of 50m/s at standard sea level conditions. If the lift per unit span is 1353 N, what is the angle of attack?
Starting with the definition of circulation, derive Kelvin's circulation theorem, Eq. (4.11).
Starting with Eq. (4.35), derive Eq. (4.36).
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.
The NACA 4412 airfoil has a mean camber line given by Using thin airfoil theory, calculate
(a) αL = 0
(b) c1 when α = 3°
For the airfoil given in Prob. 4.6, calculate cm c/4 and xcp/c when α =3°.
Compare the results of Probs. 4.6 and 4.7 with experimental data for the NACA 4412 airfoil, and note the percentage difference between theory and experiment.
Starting with Eqs (4.35) and (4.43), derive Eq. (4.62).
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.
Consider the same vortex filament as in Prob. 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 for the velocity at distance A on the line, as induced by the vortex filament.
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 8 = r. Calculate the lift and induced drag coefficients for this wing at a geometric angle of attack = 7°.
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 24501b. The wing uses an NACA 65-415 airfoil, which has a lift slope of 0.1033 degree–1 and aL = 0 = – 3°. Assume t = 0.12. If the airplane is cruising at 120mi/h at standard sea level at its maximum gross weight and is in straight-and-level flight, calculate the geometric angle of attack of the wing.
Consider the airplane and flight conditions given in Prob. 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 Prob. 5.4.
Prove that three-dimensional source flow is irrotational.
Prove that three-dimensional source flow is a physically possible incompressible flow.
A sphere and a circular cylinder (with its axis perpendicular to the flow) are mounted in the same free stream. 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 connected to a pressure tap on the surface of the cylinder. This tap is located on the cylindrical surface such that no deflection of the manometer fluid takes place. Calculate the location of this tap.
The temperature and pressure at the stagnation point of a high-speed missile are 934°R and 7.8atm, respectively. Calculate the density at this point.
Calculate cp, cv, e, and h for
(a) The stagnation point conditions given in Prob. 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 Ref. 2.)
Just upstream of a shock wave, the air temperature and pressure are 288 K and 1atm, respectively; just downstream of the wave, the air temperature and pressure are 690 K and 8.656atm, respectively. Calculate the changes in enthalpy, internal energy, and entropy across the wave.
Consider the isentropic flow over an airfoil. The free stream conditions are T∞ = 245 K and p∞ = 4.35x 104 N/m2. At a point on the airfoil, the pressure is 3.6 x 104 N/m2. Calculate the density at this point.
Consider the isentropic flow through a supersonic wind-tunnel nozzle. The reservoir properties are T0 = 500 K and p0 = 10atm. If p = 1atm at the nozzle exit, calculate the exit temperature and density.
Consider air at a pressure of 0.2 atm. Calculate the values of τT and τs. Express your answer in SI units.
Consider a point in a flow where the velocity and temperature are 1300 ft/s and 480°R, respectively. Calculate the total enthalpy at this point.
In the reservoir of a supersonic wind tunnel, the velocity is negligible, and the temperature is 1000 K. The temperature at the nozzle exit is 600 K. Assuming adiabatic flows through the nozzle; calculate the velocity at the exit.
An airfoil is in a free stream where p∞ = 0.61atm, px, = 0.819 kg/m3, and Vx = 300 m/s. At a point on the airfoil surface, the pressure is 0.5atm; assuming isentropic flow calculates the velocity at that point.
Calculate the percentage error obtained if Prob. 7.9 is solved using (incorrectly) the incompressible Bernoulli equation.
Repeat Prob. 7.9, considering a point on the airfoil surface where the pressure is 0.3 atm.
Repeat Prob. 7.10, considering the flow of Prob. 7.11.
Consider air at a temperature of 230 K. Calculate the speed of sound.
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.
At a given point in a flow, T = 700oR, p = 1.6atm, 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 = 700oR, p = 1.6atm, and V = 2983 ft/s. At this point, calculate the corresponding values of p0, T0, p*, T*, and M*.
Consider the isentropic flow through a supersonic nozzle. If the test-section conditions are given by p = 1atm, T = 230 K, and M = 2, calculate the reservoir pressure and temperature.
Consider the isentropic flow over an airfoil. The free stream 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 consult a standard atmosphere table for this problem, such as given in Ref. 2. If you do not have one, you can find such tables in any good technical library.)
The flow just upstream of a normal shock wave is given by p1, = 1atm, T1 = 288 K, and M1 = 2.6. Calculate the following properties just downstream of the shock: p2, T2, P2, M2, p0,2, T0,2, and the change in entropy across the shock.
The pressure upstream of a normal shock wave is 1 atm. The pressure and temperature downstream of the wave are 10.33atm and 1390°R, respectively. Calculate the Mach number and temperature upstream of the wave and the total temperature and total pressure downstream of the wave.
The entropy increase across a normal shock wave is 199.5 J / (kg ∙ K). What is the upstream Mach number?
The flow just upstream of a normal shock wave is given by p1 = 1800 lb/ft2, T, =480°R, and M1 = 3.1. Calculate the velocity and M* behind the shock.
Consider a flow with a pressure and temperature of 1atm and 288 K. A Pitot tube is inserted into this flow and measures a pressure of 1.555 atm. What is the velocity of the flow?
Consider a flow with a pressure and temperature of 2116 lb/ft2 and 519°R, respectively. A Pitot tube is inserted into this flow and measures a pressure of 7712.8 lb/ft2. What is the velocity of this flow?
Repeat Probs. 8.11 and 8.12 using (incorrectly) Bernoulli's equation for incompressible flow. Calculate the percent error induced by using Bernoulli's equation.
Derive the Rayleigh Pitot tube formula, Eq. (8.80).
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. (Consult a standard altitude table for any additional information you need.)
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 test section.
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?
Consider an oblique shock wave with a wave angle of 30° in a Mach 4 flow. The upstream pressure and temperature are 2.65 x 104 N/m2 and 223.3 K, respectively (corresponding to a standard altitude of 10,000m). Calculate the pressure, temperature, Mach number, total pressure, and total temperature behind the wave and the entropy increase across the wave.
Equation (8.80) does not hold for an oblique shock wave, and hence the column in App. 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. On the other hand, the column labeled p0,2 / P0,1 can be used for an oblique shock wave, using Mn,1. Explain why all this is so.
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 App. B (the correct way)
(b) p0,2/P1 from App. B (the incorrect way) Compare the results.
Consider the flow over a 22.2° half-angle wedge. If M1 = 2.5, p1 = 1atm, and T1 = 300 K, calculate the wave angle and p2, T2, and M2.
Consider a flat plate at an angle of attack a to a Mach 2.4 airflow at 1atm 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 a does this occur?
A 30.2° half-angle wedge is inserted into a free stream 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 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 single normal shock wave
(b) An oblique shock with a deflection angle of 25.3°, followed by a normal shock
(c) An oblique shock with a deflection angle of 25.3°, followed by a second oblique shock of deflection angle of 20°, followed by a normal shock
From the results of (a), (b), and (c), what can you induce about the efficiency of the various shock systems?
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 Fig. 9.14. If the upstream flow has the properties M1 = 3.2, p1 = 1atm and T1, = 520°R, calculate M3, p3, and T3 behind the reflected shock from the upper wall. Also, obtain the angle Φ which the reflected shock makes with the upper wall.
Consider the supersonic flow over an expansion corner, such as given in Fig. 9.20. The deflection angle θ = 23.38°. If the flow upstream of the corner is given by M1 = 2, P1 = 0.7atm, T1, = 630°R, calculate M2, p2, T2, p2, P0,2, and T0,2 downstream of the corner. Also, obtain the angles the forward and rearward Mach lines make with respect to the upstream direction.
A supersonic flow at M1 = 1.58 and P1 = 1atm expands around a sharp corner. If the pressure downstream of the corner is 0.1306atm, calculate the deflection angle of the corner.
A supersonic flow at M1 = 3, T1 = 285 K, and p1 = 1atm is deflected upward through a compression corner with 0 = 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. Calculate M3, p3, and T3 downstream of the expansion corner. Since the resulting flow is in the same direction as the original flow, would you expect M3 = M1, p3 = P1, and T3 = T1? Explain.