A positive line vortex K is trapped in a corner, as in Fig. P8.74. Compute the total

Prove that the streamlines (r, ) in polar coordinates from Eqs. (8.10) are orthogonal to the potential lines (r, ).

Prove that the streamlines (r, ) in polar coordinates from Eqs. (8.10) are orthogonal to the potential lines (r, ).

Using cartesian coordinates, show that each velocity component (u, , w) of a potential fl ow satisfi es Laplaces equation separately

Is the function 1/r a legitimate velocity potential in plane polar coordinates? If so, what is the associated stream function (r, )? P8.5 A proposed harmonic function F(x, y, z) is given by F 5 2x2 1 y3 2 4xz 1 f(y) (a) If possible, fi nd a function f (y) for which the laplacian of F is zero. If you do indeed solve part (a), can your fi nal function F serve as (b) a velocity potential or (c) a stream function?

A proposed harmonic function F(x, y, z) is given by F 5 2x2 1 y3 2 4xz 1 f(y) (a) If possible, fi nd a function f (y) for which the laplacian of F is zero. If you do indeed solve part (a), can your fi nal function F serve as (b) a velocity potential or (c) a stream function?

An incompressible plane fl ow has the velocity potential 5 2Kxy, where B is a constant. Find the stream function of this fl ow, sketch a few streamlines, and interpret the fl ow pattern.

Consider a fl ow with constant density and viscosity. If the fl ow possesses a velocity potential as defi ned by Eq. (8.1), show that it exactly satisfi es the full Navier-Stokes equations (4.38). If this is so, why for inviscid theory do we back away from the full Navier-Stokes equations?

For the velocity distribution u 5 2By, 5 1Bx, w 5 0, evaluate the circulation about the rectangular closed curve defi ned by (x, y) 5 (1,1), (3,1), (3,2), and (1,2). Interpret your result, especially vis- -vis the velocity potential.

Consider the two-dimensional fl ow u 5 2Ax, 5 Ay, where A is a constant. Evaluate the circulation around the rectangular closed curve defi ned by (x, y) 5 (1, 1), (4, 1), (4, 3), and (1, 3). Interpret your result, especially vis- -vis the velocity potential

A two-dimensional Rankine half-body, 8 cm thick, is placed in a water tunnel at 208C. The water pressure far upstream along the body centerline is 105 kPa. What is the nose radius of the half-body? At what tunnel fl ow velocity will cavitation bubbles begin to form on the surface of the body?

A power plant discharges cooling water through the manifold in Fig. P8.11, which is 55 cm in diameter and 8 m high and is perforated with 25,000 holes 1 cm in diameter. Does this manifold simulate a line source? If so, what is the equivalent source strength m?

Consider the fl ow due to a vortex of strength K at the origin. Evaluate the circulation from Eq. (8.23) about the clockwise path from (r, ) 5 (a, 0) to (2a, 0) to (2a, 3/2) to (a, 3/2) and back to (a, 0). Interpret the result.

Starting at the stagnation point in Fig. 8.6, the fl uid acceleration along the half-body surface rises to a maximum and eventually drops off to zero far downstream. (a) Does this maximum occur at the point in Fig. 8.6 where Umax 5 1.26U? (b) If not, does the maximum acceleration occur before or after that point? Explain

A tornado may be modeled as the circulating fl ow shown in Fig. P8.14, with r 5 z 5 0 and (r) such that 5 r r # R R2 r r . R Determine whether this fl ow pattern is irrotational in either the inner or outer region. Using the r-momentum equation (D.5) of App. D, determine the pressure distribution p(r) in the tornado, assuming p 5 p as r . Find the location and magnitude of the lowest pressure. R (r)

Hurricane Sandy, which hit the New Jersey coast on Oct. 29, 2012, was extremely broad, with wind velocities of 40 mi/h at 400 miles from its center. Its maximum velocity was 90 mi/h. Using the model of Fig. P8.14, at 208C with a pressure of 100 kPa far from the center, estimate (a) the radius R of maximum velocity, in mi; and (b) the pressure at r 5 R.

Hurricane Sandy, which hit the New Jersey coast on Oct. 29, 2012, was extremely broad, with wind velocities of 40 mi/h at 400 miles from its center. Its maximum velocity was 90 mi/h. Using the model of Fig. P8.14, at 208C with a pressure of 100 kPa far from the center, estimate (a) the radius R of maximum velocity, in mi; and (b) the pressure at r 5 R.

Find the position (x, y) on the upper surface of the half-body in Fig. 8.9a for which the local velocity equals the uniform stream velocity. What should be the pressure at this point?

Plot the streamlines and potential lines of the fl ow due to a line source of strength m at (a, 0) plus a source 3m at (2a, 0). What is the fl ow pattern viewed from afar?

Plot the streamlines and potential lines of the fl ow due to a line source of strength 3m at (a, 0) plus a sink 2m at (2a, 0). What is the pattern viewed from afar?

Plot the streamlines of the fl ow due to a line vortex 1K at (0, 1a) and a vortex 2K at (0, 2a). What is the pattern viewed from afar?

At point A in Fig. P8.21 is a clockwise line vortex of strength K 5 12 m2 /s. At point B is a line source of strength m 5 25 m2 /s. Determine the resultant velocity induced by these two at point C. x y 3 m B C A 4 m

Consider inviscid stagnation fl ow, 5 Kxy (see Fig. 8.19b), superimposed with a source at the origin of strength m. Plot the resulting streamlines in the upper half-plane, using the length scale a 5 (m/K) 1/2. Give a physical interpretation of the fl ow pattern.

Sources of strength m 5 10 m2 /s are placed at points A and B in Fig. P8.23. At what height h should source B be placed so that the net induced horizontal velocity component at the origin is 8 m/s to the left? P8.23 2 m A y B h x

Line sources of equal strength m 5 Ua, where U is a reference velocity, are placed at (x, y) 5 (0, a) and (0, 2a). Sketch the stream and potential lines in the upper half plane. Is y 5 0 a wall? If so, sketch the pressure coeffi cient Cp 5 p 2 p0 1 2U2 along the wall, where p0 is the pressure at (0, 0). Find the minimum pressure point and indicate where fl ow separation might occur in the boundary layer.

Let the vortex/sink fl ow of Eq. (8.16) simulate a tornado as in Fig. P8.25. Suppose that the circulation about the tornado is 5 8500 m2 /s and that the pressure at r 5 40 m is 2200 Pa less than the far-fi eld pressure. Assuming inviscid fl ow at sealevel density, estimate (a) the appropriate sink strength 2m, (b) the pressure at r 5 15 m, and (c) the angle at which the streamlines cross the circle at r 5 40 m (see Fig. P8.25). P8.25 40 m

580 Chapter 8 Potential Flow and Computational Fluid Dynamics P8.26 25 cm/s 8 in 110 m3/s Manifold

Water at 208C fl ows 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 velocity and its uncertainty

Sources of equal strength m are placed at the four symmetric positions (x, y) 5 (a, a), (2a, a), (2a, 2a), and (a, 2a). Sketch the streamline and potential line patterns. Do any plane walls appear? P8.27 U m A B

A uniform water stream, U 5 20 m/s and 5 998 kg/m3 , combines with a source at the origin to form a half-body. At (x, y) 5 (0, 1.2 m), the pressure is 12.5 kPa less than p. (a) Is this point outside the body? Estimate (b) the appropriate source strength m and (c) the pressure at the nose of the body.

A tornado is simulated by a line sink m 5 21000 m2 /s plus a line vortex K 5 1600 m2 /s. Find the angle between any streamline and a radial line, and show that it is independent of both r and . If this tornado forms in sea-level standard air, at what radius will the local pressure be equivalent to 29 inHg?

A Rankine half-body is formed as shown in Fig. P8.31. For the stream velocity and body dimension 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 A. (0, 3 m) 7 m/s (4m, 0) A x y h +m Source

Line sources m1 and m2 are near point A, as in Fig. P8.32. If m1 5 30 m2 /2, fi nd the value of m2 for which the resultant velocity at point A is exactly vertical. P8.32 m1 m2 3 m 4 m 4 m 3 m

Sketch the streamlines, especially the body shape, due to equal line sources 1m at (0, 1a) and (0, 2a) plus a uniform stream U 5 ma

Consider three equally spaced sources of strength m placed at (x, y) 5 (1a, 0), (0, 0), and (2a, 0). Sketch the resulting streamlines, noting the position of any stagnation points. What would the pattern look like from afar?

A uniform stream, U 5 4 m/s, approaches a Rankine oval as in Fig. 8.13, with a 5 50 cm. Find the strength m of the sourcesink pair, in m2 /s, which will cause the total length of the oval to be 250 cm. What is the maximum width of this oval?

When a line sourcesink pair with m 5 2 m2 /s is combined with a uniform stream, it forms a Rankine oval whose minimum dimension is 40 cm. If a 5 15 cm, what are the stream velocity and the velocity at the shoulder? What is the maximum dimension?

A Rankine oval 2 m long and 1 m high is immersed in a stream U 5 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 stagnation point achieves its maximum deceleration. A B? 10 m/s 2 m 1 m

Consider potential fl ow of a uniform stream in the x direction plus two equal sources, one at (x, y) 5 (0, 1a) and the other at (x, y) 5 (0, 2a). Sketch your ideas of the body contours that would arise if the sources were (a) very weak and (b) very strong.

A large Rankine oval, with a 5 1 m and h 5 1 m, is immersed in 208C water fl owing at 10 m/s. The upstream pressure on the oval centerline is 200 kPa. Calculate (a) the value of m; and (b) the pressure on the top of the oval (analogous to point A in Fig. P8.37).

Modify the Rankine oval in Fig. P8.37 so that the stream velocity and body length are the same but the thickness is unknown (not 1 m). The fl uid is water at 308C and the pressure far upstream along the body centerline is 108 kPa. Find the body thickness for which cavitation will occur at point A.

A Kelvin oval is formed by a linevortex pair with K 5 9 m2 /s, a 5 1 m, and U 5 10 m/s. What are the height, width, and shoulder velocity of this oval?

The vertical keel of a sailboat approximates a Rankine oval 125 cm long and 30 cm thick. The boat sails in seawater in standard atmosphere at 14 knots, parallel to the keel. At a section 2 m below the surface, estimate the lowest pressure on the surface of the keel.

Water at 208C fl ows past a 1-m-diameter circular cylinder. The upstream centerline pressure is 128,500 Pa. If the lowest pressure on the cylinder surface is exactly the vapor pressure, estimate, by potential theory, the stream velocity

Suppose that circulation is added to the cylinder fl ow of Prob. P8.43 suffi cient to place the stagnation points at equal to 358 and 1458. What is the required vortex strength K in m2 /s? Compute the resulting pressure and surface velocity at (a) the stagnation points and (b) the upper and lower shoulders. What will the lift per meter of cylinder width be?

If circulation K is added to the cylinder fl ow in Prob. P8.43, (a) for what value of K will the fl ow begin to cavitate at the surface? (b) Where on the surface will cavitation begin? (c) For this condition, where will the stagnation points lie?

A cylinder is formed by bolting two semicylindrical 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 (gage). Using potential theory for the outside pressure, compute the tension force in each bolt if the fl uid outside is sea-level air. P8.46 U = 25 m/s D = 2 m p = 50 kPa (gage)

A circular cylinder is fi tted with two surface-mounted pressure sensors, to measure pa at 5 1808 and pb at 5 1058. The intention is to use the cylinder as a stream velocimeter. Using inviscid theory, derive a formula for estimating U in terms of pa, pb, , and the cylinder radius a.

Wind at U and p fl ows past a Quonset hut which is a half-cylinder of radius a and length L (Fig. P8.48). The internal pressure is pi. Using inviscid theory, derive an expression for the upward force on the hut due to the difference between pi and ps. P8.48 U, p A a pi ps( )

In strong winds the force in Prob. P8.48 can be quite large. Suppose that a hole is introduced in the hut roof at point A to make pi equal to the surface pressure there. At what angle should hole A be placed to make the net wind force zero?

It is desired to simulate fl ow past a two-dimensional ridge or bump by using a streamline that passes above the fl ow over a cylinder, as in Fig. P8.50. The bump is to be a/2 high, where a is the cylinder radius. What is the elevation h of this streamline? What is Umax on the bump compared with stream velocity U? U a/2 Umax? Bump a U

A hole is placed in the front of a cylinder to measure the stream velocity of sea-level fresh water. The measured pressure at the hole is 2840 lbf/ft2 . If the hole is misaligned by 128 from the stream, and misinterpreted as stagnation pressure, what is the error in velocity

The Flettner rotor sailboat in Fig. E8.3 has a water drag coeffi cient of 0.006 based on a wetted area of 45 ft2 . If the rotor spins at 220 r/min, fi nd the maximum boat velocity that can be achieved in a 15-mi/h wind. What is the optimum angle between the boat and the wind?

Modify Prob. P8.52 as follows. For the same sailboat data, fi nd the wind velocity, in mi/h, that 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 rotating at 750 r/min, which is far outside the range of Fig. 8.15. The measured lift and drag coeffi cients for each rotor were about 10 and 4, respectively. If the ship is moored and subjected to a crosswind of 25 ft/s, as in Fig. P8.54, what will the wind force parallel and normal to the ship centerline be? Estimate the power required to drive the rotors.

Assume that the Flettner rotor ship of Fig. P8.54 has a water resistance coeffi cient of 0.005. How fast will the ship sail in seawater at 208C in a 20-ft/s wind if the keel aligns itself with the resultant force on the rotors? [Hint: This is a problem in relative velocities.] P8.54 U

A proposed free-stream velocimeter would use a cylinder with pressure taps at 5 1808 and at 1508. The pressure difference would be a measure of stream velocity U. However, the cylinder must be aligned so that one tap exactly faces the free stream. Let the misalignment angle be ; that is, the two taps are at (1808 1 ) and (1508 1 ). Make a plot of the percentage error in velocity measurement in the range 2208 , , 1208 and comment on the idea.

In principle, it is possible to use rotating cylinders as aircraft wings. Consider a cylinder 30 cm in diameter, rotating at 2400 r/min. It is to lift a 55-kN airplane cruising at 100 m/s. What should the cylinder length be? How much power is required to maintain this speed? Neglect end effects on the rotating wing

Plot the streamlines due to the combined fl ow of a line sink 2m at the origin plus line sources 1m at (a, 0) and (4a, 0). [Hint: A cylinder of radius 2a will appear

In principle, it is possible to use rotating cylinders as aircraft wings. Consider a cylinder 30 cm in diameter, rotating at 2400 r/min. It is to lift a 55-kN airplane cruising at 100 m/s. What should the cylinder length be? How much power is required to maintain this speed? Neglect end effects on the rotating wing.

One of the corner fl ow patterns of Fig. 8.18 is given by the cartesian stream function 5 A(3yx2 2 y3 ). Which one? Can the correspondence be proved from Eq. (8.53)?

Plot the streamlines of Eq. (8.53) in the upper right quadrant for n 5 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 this increase be linear in x? For what corner angle and n would the increase be as x5 ?

Combine stagnation fl ow, Fig. 8.19b, with a source at the origin: f(z) 5 Az2 1 m ln z Plot the streamlines for m 5 AL2 , where L is a length scale. Interpret.

The superposition in Prob. P8.62 leads to stagnation fl ow near a curved bump, in contrast to the fl at wall of Fig. 8.19b. Determine the maximum height H of the bump as a function of the constants A and m.

Consider the polar-coordinate stream function 5 Br1.2 sin(1.2 ), with B equal, for convenience, to 1.0 ft0.8/s. (a) Plot the streamline 5 0 in the upper half plane. (b) Plot the streamline 5 1.0 and interpret the fl ow pattern. (c) Find the locus of points above 5 0 for which the resultant velocity 5 1.2 ft/s.

Potential fl ow past a wedge of half-angle leads to an important application of laminar boundary layer theory called the Falkner-Skan fl ows [15, pp. 239245]. Let x denote distance along the wedge wall, as in Fig. P8.65, and let 5 108. Use Eq. (8.53) to fi nd the variation of surface velocity U(x) along the wall. Is the pressure gradient adverse or favorable? P8.65 x U(x)

The inviscid velocity along the wedge in Prob. P8.65 has the analytic form U(x) 5 Cxm, where m 5 n 2 1 and n is the exponent in Eq. (8.53). Show that, for any C and n, computation of the boundary layer by Thwaitess method, Eqs. (7.53) and (7.54), leads to a unique value of the Thwaites parameter . Thus wedge fl ows are called similar [15, p. 241].

Investigate the complex potential function f(z) 5 U(z 1 a2 /z) and interpret the fl ow pattern.

Investigate the complex potential function f(z) 5 U z 1 m ln [(z 1 a)/(z2a)] and interpret the fl ow pattern.

Investigate the complex potential f(z) 5 A cosh [(z/a)], and plot the streamlines inside the region shown in Fig. P8.69. What hyphenated word (originally French) might describe such a fl ow pattern? P8.69 y x y = a ( = 0) Plot the streamlines inside this region

Show that the complex potential f 5 U5z 1 1 4a coth [(z/a)]} represents fl ow past an oval shape placed midway between two parallel walls y 5 ;1 2a. What is a practical application?

Show that the complex potential f 5 U5z 1 1 4a coth [(z/a)]} represents fl ow past an oval shape placed midway between two parallel walls y 5 ;1 2a. What is a practical application?

Use the method of images to construct the fl ow pattern for a source 1m near two walls, as shown in Fig. P8.72. Sketch the velocity distribution along the lower wall (y 5 0). Is there any danger of fl ow separation along this wall? P8.72 0 a a y x +m

Set up an image system to compute the fl ow of a source at unequal distances from two walls, as in Fig. P8.73. Find the point of maximum velocity on the y axis. P8.73 a x y +m 2a

A positive line vortex K is trapped in a corner, as in Fig. P8.74. Compute the total induced velocity vector at point B, (x, y) 5 (2a, a), and compare with the induced velocity when no walls are present. P8.74 B 2a a 0 a 2a x y K V?

Using the four-source image pattern needed to construct the fl ow near a corner in Fig. P8.72, fi nd the value of the source strength m that will induce a wall velocity of 4.0 m/s at the point (x, y) 5 (a, 0) just below the source shown, if a 5 50 cm.

Use the method of images to approximate the fl ow pattern past a cylinder a distance 4a from a single wall, as in Fig. P8.76. To illustrate the effect of the wall, compute the velocities at corresponding points A, B, C, and D, comparing with a cylinder fl ow in an infi nite expanse of fl uid. P8.76 2a D U B A C 4a 4a

Discuss how the fl ow pattern of Prob. P8.58 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 fl ow of a uniform stream past a Rankine half-body constrained between two parallel walls, as in Fig. P8.78. For the particular dimensions shown in this fi gure, estimate the position of the nose of the resulting half-body. P8.78 2a a y a x U

Explain the system of images needed to simulate the fl ow of a line source placed unsymmetrically between two parallel walls as in Fig. P8.79. Compute the velocity on the lower wall at x 5 a. How many images are needed to estimate this velocity within 1 percent? P8.79 +m 2a a x y 0

The beautiful expression for lift of a two-dimensional airfoil, Eq. (8.59), arose from applying the Joukowski transformation, 5 z 1 a2 /z, where z 5x 1 iy and 5 1 i. The constant a is a length scale. The theory transforms a certain circle in the z plane into an airfoil in the plane. Taking a 5 1 unit for convenience, show that (a) a circle with center at the origin and radius . 1 will become an ellipse in the plane and (b) a circle with center at x 5 2 ! 1, y 5 0, and radius (1 1 ) will become an airfoil shape in the plane. [Hint: The Excel spreadsheet is excellent for solving this problem.]

Given an airplane of weight W, wing area A, aspect ratio AR, and fl ying at an altitude where the density is . Assume all drag and lift is due to the wing, which has an infi nite-span drag coeffi cient CD. Further assume suffi cient thrust to balance whatever drag is calculated. (a) Find an algebraic expression for the best cruise velocity Vb, which occurs when the ratio of drag to speed is a minimum. (b) Apply your formula to the data in Prob. P7.119 for which a laborious graphing procedure gave an answer Vb < 180 m/s.

The ultralight plane Gossamer Condor in 1977 was the fi rst to complete the Kremer Prize fi gure-eight course under human power. Its wingspan was 29 m, with Cav 5 2.3 m and a total mass of 95 kg. The drag coeffi cient was approximately 0.05. The pilot was able to deliver 1 4 hp to propel the plane. Assuming two-dimensional fl ow at sea level, estimate (a) the cruise speed attained, (b) the lift coeffi cient, and (c) the horsepower required to achieve a speed of 15 kn.

The worlds largest airplane, the Airbus A380, has a maximum weight of 1,200,000 lbf, wing area of 9100 ft2 , wingspan of 262 ft, and CDo 5 0.026. When cruising at maximum weight at 35,000 ft, the four engines each provide 70,000 lbf of thrust. Assuming all lift and drag are due to the wing, estimate the cruise velocity, in mi/h.

Reference 12 contains inviscid theory calculations for the upper and lower surface velocity distributions V(x) over an airfoil, where x is the chordwise coordinate. A typical result for small angle of attack is as follows: x/c V/U(upper) V/U(lower) 0.0 0.0 0.0 0.025 0.97 0.82 0.05 1.23 0.98 0.1 1.28 1.05 0.2 1.29 1.13 0.3 1.29 1.16 0.4 1.24 1.16 0.6 1.14 1.08 0.8 0.99 0.95 1.0 0.82 0.82 Use these data, plus Bernoullis equation, to estimate (a) the lift coeffi cient and (b) the angle of attack if the airfoil is symmetric.

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 sealevel standard air at 200 ft/s and is found to have lift of 30 lbf and drag of 1.5 lbf. Estimate from wing theory (a) the angle of attack, (b) the minimum drag of the wing and the angle of attack at which it occurs, and (c) the maximum lift-to-drag ratio

An airplane has a mass of 20,000 kg and fl ies at 175 m/s at 5000-m standard altitude. Its rectangular wing has a 3-m chord Problems 585 and a symmetric airfoil at 2.58 angle of attack. Estimate (a) the wing span, (b) the aspect ratio, and (c) the induced drag.

A freshwater boat of mass 400 kg is supported by a rectangular hydrofoil of aspect ratio 8, 2 percent camber, and 12 percent thickness. If the boat travels at 7 m/s and 5 2.58, estimate (a) the chord length, (b) the power required if CD 5 0.01, and (c) the top speed if the boat is refi tted with an engine that delivers 20 hp to the water.

The Boeing 787-8 Dreamliner has a maximum weight of 502,500 lbf, a wingspan of 197 ft, a wing area of 3501 ft2 , and cruises at 567 mi/h at 35,000 ft altitude. When cruising, its overall drag coeffi cient is about 0.027. Estimate (a) the aspect ratio, (b) the lift coeffi cient, (c) the cruise Mach number, and (d) the engine thrust needed when cruising

The Beechcraft T-34C aircraft has a gross weight of 5500 lbf and a wing area of 60 ft2 and fl ies at 322 mi/h at 10,000-ft standard altitude. It is driven by a propeller that delivers 300 hp to the air. Assume for this problem that its airfoil is the NACA 2412 section described in Figs. 8.23 and 8.24, and neglect all drag except the wing. What is the appropriate aspect ratio for the wing?

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 advantage for each of these wing positions. If you cant think of one, read the article [37] and report to the class. P8.90 a b c

If (r, ) in axisymmetric fl ow is defi ned by Eq. (8.72) and the coordinates are given in Fig. 8.28, determine what partial differential equation is satisfi ed by .

A point source with volume fl ow Q 5 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 source to

The Rankine half-body of revolution (Fig. 8.30) could simulate the shape of a pitot-static tube (Fig. 6.30). According to inviscid theory, how far downstream from the nose should the static pressure holes be placed so that the local velocity is within 60.5 percent of U? Compare your answer with the recommendation x < 8D in Fig. 6.30.

Determine whether the Stokes streamlines from Eq. (8.73) are everywhere orthogonal to the Stokes potential lines from Eq. (8.74), as is the case for Cartesian and plane polar coordinates.

Show that the axisymmetric potential fl ow formed by superposition of a point source 1m at (x, y) 5 (2a, 0), a point sink 2m at (1a, 0), and a stream U in the x direction forms a Rankine body of revolution as in Fig. P8.95. Find analytic expressions for determining the length 2L and maximum diameter 2R of the body in terms of m, U, and a. P8.95 y x a a +m m r

Consider inviscid fl ow along the streamline approaching the front stagnation point of a sphere, as in Fig. 8.31. Find (a) the maximum fl uid deceleration along this streamline and (b) its position.

The Rankine body of revolution in Fig. P8.97 is 60 cm long and 30 cm in diameter. When it is immersed in the lowpressure water tunnel as shown, cavitation may appear at point A. Compute the stream velocity U, neglecting surface wave formation, for which cavitation occurs. U p a = 40 kPa Water at 20C A 80 cm Rankine ovoid

We have studied the point source (sink) and the line source (sink) of infi nite depth into the paper. Does it make any sense to defi ne a fi nite-length line sink (source) as in 586 Chapter 8 Potential Flow and Computational Fluid Dynamics Fig. P8.98? If so, how would you establish the mathematical properties of such a fi nite line sink? When combined with a uniform stream and a point source of equivalent strength as in Fig. P8.98, should a closed-body shape be formed? Make a guess and sketch some of these possible shapes for various values of the dimensionless parameter m/(U L2 ). P8.98 U +m m y x 0 Point source Line sink of total strength L

Consider air fl owing past a hemisphere resting on a fl at surface, as in Fig. P8.99. If the internal pressure is pi, fi nd an expression for the pressure force on the hemisphere. By analogy with Prob. P8.49, at what point A on the hemisphere should a hole be cut so that the pressure force will be zero according to inviscid theory? P8.99 U, p pi 2

A 1-m-diameter sphere is being towed at speed V in fresh water at 208C as shown in Fig. P8.100. Assuming inviscid theory with an undistorted free surface, estimate the speed V in m/s at which cavitation will fi rst appear on the sphere surface. Where will cavitation appear? For this condition, what will be the pressure at point A on the sphere, which is 458 up from the direction of travel? P8.100 pa = 101.35 kPa D = 1 m A V 3

Consider a steel sphere (SG 5 7.85) of diameter 2 cm, dropped from rest in water at 208C. Assume a constant drag coeffi cient CD 5 0.47. Accounting for the spheres hydrodynamic mass, estimate (a) its terminal velocity and (b) the time to reach 99 percent of terminal velocity. Compare these to the results when hydrodynamic mass is neglected, Vterminal < 1.95 m/s and t99% < 0.605 s, and discuss.

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 208, and a backspin (front of the ball rotating upward). Assume that the lift coeffi cient on the ball (based on frontal area) follows Fig. P7.108. If the ground is level and drag is neglected, make a simple analysis to predict the impact point (a) without spin and (b) with backspin of 7500 r/min.

Consider inviscid fl ow past a sphere, as in Fig. 8.31. Find (a) the point on the front surface where the fl uid acceleration amax is maximum and (b) the magnitude of amax. (c) If the stream velocity is 1 m/s, fi nd the sphere diameter for which amax is 10 times the acceleration of gravity. Comment.

Consider a cylinder of radius a moving at speed U through a still fl uid, as in Fig. P8.104. Plot the streamlines relative to the cylinder by modifying Eq. (8.32) to give the relative fl ow with K 5 0. Integrate to fi nd the total relative kinetic energy, and verify the hydrodynamic mass of a cylinder from Eq. (8.91). P8.104 Still fluid U a

A 22-cm-diameter solid aluminum sphere (SG 5 2.7) is accelerating at 12 m/s2 in water at 208C. (a) According to potential theory, what is the hydrodynamic mass of the sphere? (b) Estimate the force being applied to the sphere at this instant.

Laplaces equation in plane polar coordinates, Eq. (8.11), is complicated by the variable radius. Consider the fi nite difference mesh in Fig. P8.106, with nodes (i, j) equally spaced D and Dr apart. Derive a fi nite difference model for Eq. (8.11) similar to the cartesian expression (8.96). P8.106 r r   i 1, j i + 1, j i, j + 1 i, j 1 rj + 1 rj 1 i, j r

SAE 10W30 oil at 208C is at rest near a wall when the wall suddenly begins moving at a constant 1 m/s. (a) Use Dy 5 1 cm and Dt 5 0.2 s and check the stability criterion (8.101). (b) Carry out Eq. (8.100) to t 5 2 s and report the velocity u at y 5 4 cm.

Consider two-dimensional potential fl ow into a step contraction as in Fig. P8.108. The inlet velocity U1 5 7 m/s, and the outlet velocity U2 is uniform. The nodes (i, j) are labeled in the fi gure. Set up the complete fi nite difference algebraic relations for all nodes. Solve, if possible, on a computer and plot the streamlines in the fl ow. i = 1 j = 1 2 3 4 5 6 7 8 U1 U2 2 3 4 5 6 7 8 9 10 P8.108

Consider inviscid fl ow through a two-dimensional 908 bend with a contraction, as in Fig. P8.109. Assume uniform fl ow at the entrance and exit. Make a fi nite difference computer analysis for small grid size (at least 150 nodes), determine the dimensionless pressure distribution along the walls, and sketch the streamlines. (You may use either square or rectangular grids.) V1 = 10 m/s V2 5 m 6 m 10 m 10 m 15 m 16 m

For fully developed laminar incompressible fl ow through a straight noncircular duct, as in Sec. 6.8, the Navier-Stokes equations (4.38) reduce to 0 2 u 0y2 1 0 2 u 0z 2 5 1 dp dx 5 const , 0 where (y, z) is the plane of the duct cross section and x is along the duct axis. Gravity is neglected. Using a nonsquare rectangular grid (Dx, Dy), develop a fi nite difference model for this equation, and indicate how it may be applied to solve for fl ow in a rectangular duct of side lengths a and b.

Solve Prob. P8.110 numerically for a rectangular duct of side length b by 2b, using at least 100 nodal points. Evaluate the volume fl ow rate and the friction factor, and compare with the results in Table 6.4: Q < 0.1143 b4 a2dp dx b f ReDh < 62.19 where Dh 5 4A/P 5 4b/3 for this case. Comment on the possible truncation errors of your model.

In CFD textbooks [5, 2327], one often replaces the lefthand sides of Eqs. (8.102b and c) with the following two expressions, respectively: 0 0x (u2 ) 1 0 0y (u) and 0 0x (u ) 1 0 0y ( 2 ) Are these equivalent expressions, or are they merely simplifi ed approximations? Either way, why might these forms be better for fi nite difference purposes?

Formulate a numerical model for Eq. (8.99), which has no instability, by evaluating the second derivative at the next time step, j 1 1. Solve for the center velocity at the next time step and comment on the result. This is called an implicit model and requires iteration.

If your institution has an online potential fl ow boundary element computer code, consider fl ow past a symmetric airfoil, as in Fig. P8.114. The basic shape of an NACA symmetric airfoil is defi ned by the function [12] 2y tmax < 1.48451/2 2 0.63 2 1.7582 1 1.42153 2 0.50754 where 5 x/C and the maximum thickness tmax occurs at 5 0.3. Use this shape as part of the lower boundary for zero angle of attack. Let the thickness be fairly large, say, tmax 5 0.12, 0.15, or 0.18. Choose a generous number of nodes ($60), and calculate and plot the velocity distribution V/U along the airfoil surface. Compare with the

Use the explicit method of Eq. (8.100) to solve Prob. P4.85 numerically for SAE 30 oil at 208C with U0 5 1 m/s and 5 M rad/s, where M is the number of letters in your surname. (This author will solve the problem for M 5 5.) When steady oscillation is reached, plot the oil velocity versus time at y 5 2 cm.