- 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
- 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)
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Consider a circular cylinder in a hypersonic flow, with its axis perpendicular to the flow. Let <j> be the angle measured between radii drawn to the leading edge (the stagnation point) and to
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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?
- 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
- 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
- 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) ?L = 0 (b) c1 when ? = 3?
- For the airfoil given in Prob. 4.6, calculate cm c/4 and xcp/c when α =3°.Data from 4.6The NACA 4412 airfoil has a mean camber line given byUsing thin airfoil theory, calculate(a) αL=0 (b) c1 when
- 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,
- 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.
- 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,
- 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.
- 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
- 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
- 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.
- 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
- 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
- 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;
- 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
- 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
- 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*. Discuss.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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,
- 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.
- 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

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