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mechanical engineering

Questions and Answers of Mechanical Engineering

A well-known exact solution to the Navier-Stokes equation (4.38) is the unsteady circulating motion [15] Where K is a constant and ν is the kinematic viscosity. Does this flow have a
A tornado may be modeled as the circulating flow shown in Fig. P8.14, with υr = υ z = 0 and υθ ( r) such that
A category-3 hurricane on the Saffir-Simpson scale (www.encyclopedia.com) has a maximum velocity of 130 mi/h. Let the match-point radius be R = 18 km (see Fig. P8.14) Assuming sea-level standard
Consider in viscid stagnation flow ψ = Kxy, superimposed with a source at the origin of strength m. Plot the resulting streamlines in the upper-half plane, using the length scale (m/K)1/2. Give
Find the position (x, y) on the upper surface of the half-body in Fig. 8.5a for which the local velocity equals the uniform stream velocity. What should the pressure be at this point?
Using the graphical method of Fig. 8.4, plot the streamlines and potential lines ofthe flow due to a line source of strength m at (a, 0) plus a line source 3m at (€“a, 0). Whatis the flow
Plot the streamlines and potential lines of the flow due to a line source of strength 3m at (a, 0) plus a line sink of strength €“m at (€“a, 0). What is the pattern viewed from
Plot the streamlines of the flow due to a line vortex of strength +K at (0, +a) plus a line vortex of strength €“K at (0, €“a). What is the pattern viewed from afar?
Plot the streamlines of the flow due to a line vortex +K at (+a, 0) and a vortex (€“2K) at (€“a, 0). What is the pattern viewed from afar?
Plot the streamlines of a uniform stream V = iU plus a clockwise line vortex –K at the origin. Are there any stagnation points?
Find the resultant velocity vector induced at point A in Fig. P8.23 due to the combination of uniform stream, vortex, and line source
Line sources of equal strength m = Ua, where U is a reference velocity, are placed at(x, y) = (0, a) and (0, €“a). Sketch the stream and potential lines in the upper half plane. Isy = 0 a
Let the vortex/sink flow of Eq. (4.134) simulate a tornado as in Fig. P8.25. Suppose that the circulation about the tornado is Γ = 8500 m2/s and that the pressure at r = 40 m is 2200 Pa less
Find the resultant velocity induced at point A in Fig. P8.26 by the uniform stream, line source, line sink, and line vortex.
Water at 20°C flows past a half body as shown in Fig. P8.27. Measured pressures at points A and B are 160 kPa and 90 kPa, respectively, with uncertainties of 3 kPa each. Estimate the stream
Sources of equal strength m are placed at the four symmetric positions (a, a), (€“a, a), (a, €“a), and (€“a, €“a). Sketch the streamline and potential-line
A uniform water stream, U∞ = 20 m/s and ρ = 998 kg/m3, combines with a source at the origin to form a half-body. At (x, y) = (0, 1.2 m), the pressure is 12.5 kPa less than(a) Is this point
Suppose that the total discharge from the manifold in Fig. P8.11 is 450 m3/s and that there is a uniform ocean current of 60 cm/s to the right. Sketch the flow pattern from above, showing the
A Rankine half-body is formed as shown in Fig. P8.31. For the conditions shown, compute (a) the source strength m in m2/s; (b) the distance a; (c) the distance h; and (d) the total velocity at point
Sketch the streamlines, especially the body shape, due to equal line sources m at (€“a, 0) and (+a, 0) plus a uniform stream U∞ = ma.
Sketch the streamlines, especially the body shape, due to equal line sources m at (0, +a) and (0, €“a) plus a uniform stream U∞ = ma.
Consider three equally spaced line sources m placed at (x, y) = (+a, 0), (0, 0), and (−a, 0). Sketch the resulting streamlines and note any stagnation points. What would the pattern look like
8.35 Consider three equal sources in a triangular configuration: one at (a/2, 0), one at (€“a/2, 0), and one at (0, a). Plot the streamlines for this flow. Are there any stagnation points?
When a line source-sink pair with m = 2 m2/s is combined with a uniform stream, it forms a Rankine oval whose minimum dimension is 40 cm, as shown. If a = 15 cm, what are the stream velocity and the
A Rankine oval 2 m long and 1 m high is immersed in a stream U∞ = 10 m/s, as in Fig. P8.37. Estimate (a) the velocity at point A and (b) the location of point B where a particle approaching the
A uniform stream U in the x direction combines with a source m at (+a, 0) and a sink €“m at (€“a, 0). Plot the resulting streamlines and note any stagnation points.
Find the value of m/(U∞a) for which the velocity in the inside center of a Rankine oval exactly equals 3U∞.
Consider a uniform stream U∞ plus line sources (+m) at (+a, 0) and (−a, 0) and a single line sink (−2m) at (0, 0). Does a closed body shape appear? If so, plot its shape for
A Kelvin oval is formed by a line vortex pair with K = 9 m2/s, a = 1 m, and U = 10 m/s. What are the height, width, and shoulder velocity of this oval?
For what value of K/ (U∞a) does the velocity at the shoulder of a Kelvin oval equal 4U∞? What is height h/a of this oval?
Consider water at 20°C flowing past a 1-m-diameter cylinder. What doublet strength in m2/s is required to simulate this flow? If the stream pressure is 200 kPa, use in viscid theory to estimate
Suppose that circulation is added to the cylinder flow of Prob. 8.43 sufficient to place the stagnation points at 35° and 145°. What is the required vortex strength
If circulation K is added to the cylinder flow in Prob. 8.43, (a) for what value of K will the flow begin to cavitate at the surface? (b) Where on the surface will cavitation begin? (c) For this
A cylinder is formed by bolting two semi cylindrical channels together on the inside, as shown in Fig. P8.46. There are 10 bolts per meter of width on each side, and the inside pressure is 50 kPa
A circular cylinder is fitted with two pressure sensors, to measure pressure at €œa€ (180°) and €œb€ (105°), as shown. The intent is to use this
Wind at U∞ and p∞ flows past a Quonset hut which is a half-cylinder of radius a and length L (Fig. P8.48). The internal pressure is pi. Using in viscid theory, derive an expression for
In strong winds, the force in Prob. 8.48 above can be quite large. Suppose that a hole is introduced in the hut roof at point A (see Fig. P8.48) to make pi equal to the surface pressure pA, at what
It is desired to simulate flow past a ridge or €œbump€ by using a streamline above the flow over a cylinder, as shown in Fig. P8.50. The bump is to be a/2 high, as shown. What is
Modify Prob. 8.50 above as follows: Let the bump be such that Umax = 1.5U∞. Find (a) The upstream elevation h; and (b) The height Z of the bump.
The Flettner-rotor sailboat in Fig E8.2 has a water drag coefficient of 0.006 based on a wetted area of 45 ft2. If the rotor spins at 220 rev/min, find the maximum boat speed that can be achieved in
Modify Prob. P8.52 as follows. For the same sailboat data, find the wind velocity, in mi/h, which will drive the boat at an optimum speed of 8 kn parallel to its keel.
The original Flettner rotor ship was approximately 100 ft long, displaced 800 tons, and had a wetted area of 3500 ft2. As sketched in Fig. P8.54, it had two rotors 50 ft high and 9 ft in diameter
Assume that the Flettner rotor ship of Fig. P8.54 has a water-resistance coefficient of 0.005. How fast will the ship sail in seawater at 20°C in a 20 ft/s wind if the keel aligns itself with the
A proposed free stream velocimeter would use a cylinder with pressure taps at θ = 180° and at 150°. The pressure difference would be a measure of stream velocity U∞. However, the
In principle, it is possible to use rotating cylinders as aircraft wings. Consider a cylinder 30 cm in diameter, rotating at 2400 rev/min. It is to lift a 55-kN airplane flying at 100 m/s. What
Plot the streamlines due to a line sink (€“m) at the origin, plus line sources (+m) at (a, 0) and (4a, 0). Hint: A cylinder of radius 2a appears
8.60 One of the corner-flow patterns of Fig. 8.15 is given by the Cartesian stream function ψ = A (3yx2 – y3). Which one? Can this correspondence be proven from Eq. (8.49)?
Combine stagnation flow, Fig. 8.14b, with a source at the origin: f(z)=Az2+mln(z) Plot the streamlines for m = AL2, where L is a length scale. Interpret.
Plot the streamlines of Eq. (8.49) in the upper right quadrant for n = 4. How does the velocity increase with x outward along the x axis from the origin? For what corner angle and value of n would
Consider the polar-coordinate velocity potential φ = Br1.2cos (1.2θ), where B is a constant. (a) Determine whether 2φ=0. If so, (b) Find the associated stream function ψ (r,
Potential flow past a wedge of half angle θ leads to an important application of laminar-boundary-layer theory called the Falkner-Skan flows [Ref. 15 of Chap. 8, pp. 242–247]. Let x denote
The in viscid velocity along the wedge in Prob. 8.65 has the form U(x) = Cxm, where m = n − 1 and n is the exponent in Eq. (8.49). Show that, for any C and n, computation of the laminar
Show that the complex potential f (z) = U∞ [z + (a/4) coth(π z /a)] represents flow past an oval shape placed midway between two parallel walls y = ±a/2. What is a practical
Investigate the complex potential function f (z) = Acosh (π z/a), where a is a constant, and plot the streamlines inside the region shown in Fig. P8.69. What hyphenated French word might describe
Figure P8.71 shows the streamlines and potential lines of flow over a thin-plate weir as computed by the complex potential method. Compare qualitatively with Fig. 10.16a. State the proper boundary
Use the method of images to construct the flow pattern for a source +m near two walls, as in Fig. P8.72. Sketch the velocity distribution along the lower wall (y = 0). Is there any danger of flow
Set up an image system to compute the flow of a source at unequal distances from two walls, as shown in Fig. P8.73 Find the point of maximum velocity on the y-axis.
A positive line vortex K is trapped in a corner, as in Fig. P8.74. Compute the total induced velocity at point B, (x, y) = (2a, a), and compare with the induced velocity when no walls are present.
Using the four-source image pattern needed to construct the flow near a corner shown in Fig P8.72, find the value of the source strength m which will induce a wall velocity of 4.0 m/s at the point
Use the method of images to approximate the flow past a cylinder at distance 4a from the wall, as in Fig. P8.76 To illustrate the effect of the wall, compute the velocities at points A, B, C, and D,
Discuss how the flow pattern of Prob. 8.58 might be interpreted to be an image-system construction for circular walls. Why are there two images instead of one?
Indicate the system of images needed to construct the flow of a uniform stream past a Rankine half-body centered between parallel walls, as in Fig. P8.78 For the particular dimensions shown, estimate
Indicate the system of images needed to simulate the flow of a line source placed unsymmetrically between two parallel walls, as in Fig. P8.79 Compute the velocity on the lower wall at x = +a. How
The beautiful expression for lift of a two-dimensional airfoil, Eq. (8.69), arose from applying the Joukowski transformation, ζ =z+a2/z, where z=x+iyand ζ = η + iβ. The constant a
A very wide NACA 4412 airfoil, with a chord of 75 cm, is tested in a sea-level wind tunnel at 45 m/s and found to have a lift of 65 lbf per foot of span. Estimate the angle of attack for this
The ultra light plane Gossamer Condor in 1977 was the first to complete the Kremer Prize figure-eight course solely under human power. Its wingspan was 29 m, with Cav 2.3 m and a
Two-dimensional lift-drag data for the NACA 2412 airfoil with 2 percent camber (from Ref. 12) may be curve-fitted accurately as follows: CL ≈ 0.178 + 0.109α – 0.00109α 2 CD
Reference 12 contains in viscid theory calculations for the surface velocity distributions V(x) over an airfoil, where x is the chord wise coordinate. A typical result for small angle of attack is
A wing of 2 percent camber, 5-in chord, and 30-in span is tested at a certain angle of attack in a wind tunnel with sea-level standard air at 200 ft/s and is found to have lift of 30 lbf and drag of
An airplane has a mass of 20,000 kg and flies at 175 m/s at 5000-m standard altitude. Its rectangular wing has a 3-m chord and a symmetric airfoil at 2.5° angle of attack. Estimate (a) The wing
A freshwater boat of mass 400 kg is supported by a rectangular hydrofoil of aspect ratio 8, 2% camber, and 12% thickness. If the boat travels at 7 m/s and α = 2.5°, estimate (a) The chord
The Boeing 727 airplane has a gross weight of 125000 lbf, a wing area of 1200 ft2, and an aspect ratio of 6. It has two turbofan engines and cruises at 532 mi/h at 30000 feet standard altitude.
The Beech craft T-34C airplane has a gross weight of 5500 lbf, a wing area of 60 ft2, and cruises at 322 mi/h at 10000 feet standard altitude. It is driven by a propeller which delivers 300 hp to the
NASA is developing a swing-wing airplane called the Bird of Prey [37]. As shown in Fig. P8.90, the wings pivot like a pocketknife blade: forward(a) Straight(b) Or backward(c) Discuss a possible
If φ (r, θ) in axisymmetric flow is defined by Eq. (8.85) and the coordinates are given in Fig. 8.24, determine what partial differential equation is satisfied by φ.
A point source with volume flow Q = 30 m3/s is immersed in a uniform stream of speed 4 m/s. A Rankine half-body of revolution results Compute (a) The distance from the source to the stagnation
The Rankine body of revolution of Fig. 8.26 could simulate the shape of a pitotstatic tube (Fig. 6.30). According to in viscid theory, how far downstream from the nose should the static-pressure
Determine whether the Stokes streamlines from Eq. (8.86) are everywhere orthogonal to the Stokes potential lines from Eq. (8.87), as is the case for Cartesian and plan polar coordinates.
Show that the axisymmetric potential flow formed by a point source +m at (–a, 0), a point sink-(−m) at (+a, 0), and a stream U in the x direction becomes a Rankine body of revolution as in
Suppose that a sphere with a single stagnation hole is to be used as a velocimeter. The pressure at this hole is used to compute the stream velocity U, but there are errors if the stream is not
The Rankine body or revolution in Fig P8.97 is 60 cm long and 30 cm in diameter. When it is immersed in the low-pressure water tunnel as shown, cavitation may appear at point A. Compute the stream
We have studied the point source (sink) and the line source (sink) of infinite depth into the paper. Does it make any sense to define a finite-length line sink (source) as in Fig P8.98? If so, how
Consider air flowing past a hemisphere resting on a flat surface, as in Fig. P8.99. If the internal pressure is pi, find an expression for the pressure force on the hemisphere. By analogy with Prob.
A 1-m-diameter sphere is being towed at speed V in fresh water at 20°C as shown in Fig. P8.100. Assuming in viscid theory with an undistorted free surface, estimate the speed V in m/s at which
Consider a steel sphere (SG = 7.85) of diameter 2 cm, dropped from rest in water at 20°C. Assume a constant drag coefficient CD = 0.47. Accounting for the sphere’s hydrodynamic mass, estimate (a)
A golf ball weighs 0.102 lbf and has a diameter of 1.7 in. A professional golfer strikes the ball at an initial velocity of 250 ft/s, an upward angle of 20°, and a backspin (front of the ball
Modify Prob. 8.102 as follows. Golf balls are dimpled, not smooth, and have higher lift and lower drag (CL ≈ 0.2 and CD ≈ 0.3 for typical backspin). Using these values, make a computer
Consider a cylinder of radius a moving at speed U∞ through a still fluid, as in Fig. P8.104. Plot the streamlines relative to the cylinder by modifying Eq. (8.32) to give the relative flow with
In Table 7.2 the drag coefficient of a 4:1 elliptical cylinder in laminar-boundary layer flow is 0.35. According to Patton [17], the hydrodynamic mass of this cylinder is πρ hb/4, where b is
Laplace€™s equation in polar coordinates, Eq. (8.11), is complicated by the variable radius r. Consider the finite difference mesh in Fig. P8.106, with nodes (i, j) at equally spaced
Set up the numerical problem of Fig. 8.30 for an expansion angle of 30°. A new grid system and non-square mesh may be needed. Give the proper nodal equation and boundary conditions. If possible,
Consider two-dimensional potential flow into a step contraction as in Fig. P8.108. The inlet velocity U1 = 7 m/s, and the outlet velocity U2 is uniform. The nodes (i, j) are labelled in the figure.
For fully developed laminar incompressible flow through a straight noncircular duct, as in Sec. 6.8, the Navier-Stokes Equation (4.38) reduce to where (y, z) is the plane of the duct cross section
Solve Prob. 8.110 numerically for a rectangular duct of side length b by 2b, using at least 100 nodal points. Evaluate the volume flow rate and the friction factor, and compare with the results in
In his CFD textbook, Patankar [Ref. 5] replaces the left-hand side of Eq. (8.119b) and (8.119c), respectively, with the following two expressions: Are these equivalent expressions, or are they merely
Repeat Example 8.7 using the implicit method of Eq. (8.118). Take Δt = 0.2 s and Δy = 0.01 m, which ensures that an explicit model would diverge. Compare your accuracy with Example 8.7.
Use the explicit method of Eq. (8.115) to solve Problem 4.85 numerically for SAE 30 oil (v = 3.25E−4 m2/s) with Uo = 1 m/s and ω = M rad/s, where M is the number of letters in your
Did you know you can solve iterative CFD problems on an Excel spreadsheet? Successive relaxation of Laplace€™s equation is easy, since each nodal value is the average of its 4 neighbors.
Use an explicit method, similar to but not identical to Eq. (8.115), to solve the case of SAE 30 oil starting from rest near a fixed wall. Far from the wall, the oil accelerates linearly, that is,
Model potential flow through the upper-half of the symmetric diffuser shown below, the expansion angle is θ = 18.5°. Use a non-square mesh and calculate and plot(a) The velocity

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