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HW_5 - 208 CHAPTER 5 I ntrsonucnou T0 otmnamst ANALYSIS or...

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Unformatted text preview: 208 CHAPTER 5 I ntrsonucnou T0 otmnamst ANALYSIS or FLUID M0110" 6r Navier—Stokes equations (incom- pressible. constant viscosity): REFERENCES 1. Li, W. 14.. and S. H. Lam, Principles rngluid Mechanics. Reading, MA: Addison-Wesley. 1964. 2. Daily. .l. W.. and D. R. F. Harleman. Fluid Dynamics. Reading. MA: Addison-Wesley, [966. 3. Schlichting, H.. Boundary~Layer Theory. 7th ed. New York: McGraw-Hill, I979. 4. White, F. M., Viscous Fluid Flow. 3rd ed. New York: McGraw-Hill, 2000. 2 Particle acceleration components in = 1/ 6V, + EBV, ._ E + mfg—m. + (iv. Page 177 an, r .. (5.123) cylindrical coordinates: 65 V rVagV 1" V V 3?. 8V (‘3! an,,=V—"+—"—"+ "WV. " 8r 1' 66 r 8: +5VH +81“ (5.1213) _ —V g: V9617 0V 0V. (5.12c} r36 iriiiriiiifliilill’lll'”l-‘lllllllllllllllllllljll‘ililllll?'il:llillllillllllllli'UiflllilllTil Zaz Page 189 iTEiltll-‘lllil l illii 5. Sabersky, R. H., A. J. Acosta. E. G. Hauptmann. and E. M. Gates, Fluid Flow—A First Course in Fluid Mechanics, 4th ed. New Jersey: Prentice Hall. I999. 6. Fluent. Fluent incorporated. Centerra Resources Park. 10 Cavendish Court. Lebanon, NH 03766 (wwwfluentcom). 7. STAR-CD. Adapco. 60 Broadhollow Road. Melville. NY 1 I747 (www.cd-adapco.com). PROBLEMS 5.1 Which of the following sets of equations represent possible two-dimensional incompressible flow cases? a. u : 2.x:2 + y2 — ray; 0 = x1 + x03 - 23;) b. u=2xy-.rz+y;u=2xy—yz+x2 c. u = x: + 2y; 1: = xrz — yr d. u = {x + Zytrr: v =etlx + ytyt 5.2 Which of the following sets of equations represent possible three—dimensional incompressible flow cases? a.u=y2 +2xz;u= —2yz+.rzvz'uvg:.—.r'.:2 +x3y4 b u = xyzr; v =—xyzt2: w— = (272)00‘2 - W) C. u=x 2+y+z2;v=x-y+z:w=H1xz+y2+z 5.3 The three components of Velocity in a velocity field are given byu=Ax+ By + Car) = Dx+ Ey + Faandw Gx+Htv+Ja Determine the relationship among the coefficients A through J that is necessary if this is to be a possible incompressible flow field. 5.4 For a flow in the xy plane. the x component of velocity is gi- ven byu = Arty — B), where/l =lft‘1-s_]_. B = 6 ft. andxand y are measured in feet. Find a possible y component for steady. incompressible How. Is it also valid for unsteady. incompressible flow? Why? How many y components are possible? 5.5 For a flow in the xy plane. the x component of velocity is given by u x" — 3xy2. Determine a possible y component for steady. incompressible flow. Is it also valid for unsteady. incom- pressible flow? Why? How- many possible y components are there? 5.6 The x component of velocity in a steady, incompressible flow field in the xy plane is u = A/x. where A = 2 It'll/S, and xix measured in meters. Find the simplest y component of velocity for this flow field. 5. 7 The 3: component of velocity in a steady. incompressible flow field in the 3y plane is U f ‘iixyly2 -x“) where A— “ 2 component of velocity for this flow field. 5.8 The x component of velocity in a steady incompressible flow lieldEn the xy plane is u = A?“ costar/b). where A : 10 [1115. b = 5m, and x and _v are measured in meters. Find the simplest y com- ponent of velocity for this flow field. an in the xy plane is 2.13? u = —1 (x3 + 3’2)“ iShow that the Simplest expression for the .r component of velocity is l 2 r2 (x3 +5.2)? "‘nnni 5.10 A crude approximation for the .r component of velocity in u incompressible laminar boundary layer is a linear variation 3‘; -m u = 0 at the surface (_v = 0 )to the freeSIream velocity, U. at {r boundary-layer edge [v d). The equation for the profile is .- = Uy/d, where 6 2 ex”: and c is a constant. Show that the sim- iv-st expression for the y component of velocity is u = try/4x. .. a = 0 at the surface (y = 0) to the freestream velocity. U. at edge of the boundary layer (y : 6). The equation for Ihe pro- -1: is m‘U = 20:10.) — 0785):. where d = arm and c is a constant. _:' ow that the simplest expression for the 3; component of velocity ”—i‘fl-‘fl U X 2 r5 3 r3 1.. UlU versus y/ri to find the location of the maximum value of -=' ratio ulU. Evaluate the ratio where 6 = 5 mm and x 2 0.5 m. t 7-12 A useful approximation for the .r component of velocity in l_ incompressible laminar boundary layer is a sinusoidal variation ' - u = 0 at the surface (y = O) to the freestream velocity. U. at u Id rtv av . nv —=—r .' —'—. + 7:. .‘ 4: ,1 U ax [mini (admin) l g) ulU and till] versus 3215. and find the location of the maximum - of the ratio UIU. Evaluate the ratio where x = 0.5 tn and r5mm. A useful approximation for the .r component of velocity in mcompressible laminar boundary layer is a cubic variation = u = 0 at the surface (y = 0) to the freestream velocity. U, 'illlc edge of the boundary layer 0' = :5). The equation for the j-r- e is ulU = i (yr/6) — {7016)}. where (‘i = or”2 and r is a con~ .Derive the simplest expreszsion for vIU. the y component of ity ratio. Plot u/U and MW versus _v/r5. and find the location maximum value of the ratio v/U. Evaluate the ratio where '_.'=5mm andx = 0.5 m. in—3 - s" and .r and ,v are measured in meters. Find the Simplestx 6.65.14 The y component of velocity in a steady. incompressible 5-6 suumv AND USEFUL enunnoss 209 flow field in the .ry plane is u :—B.r_v“. where B = 0.2 mTJ- s" '. and .r and ,v are measured in meters. Find the simplest 3: com- ponent of velocity for this flow field. Find the equation of the streamlines for this flow. Plot the streamlines through points ( I. 4) and (2. 4). 5.9 They component of velocity in a steady incompressible flow 365-15 For 11 flow in the xy plane. the .1' component of velocity is given by u : stray“. where A 2 0.3 m’3- s7'. and x and _v are measured in meters. Find a possible _v component for steady. in- compressible flow. is it also valid for unsteady. incompressible flow"? Why? How many possible _v components are there? Deter- mine the equation of the streamline for the simplest y component of velocity. Plot the streamlines through points ( l. 4) and (2, 4). 5. l6 Derive the differential form of conservation of mass in rec- tangular coordinates by expanding the products of density and the velocity components. on, pt). and pw. in a Taylor series about a point 0. Show that the result is identical to Eq. 5.1a. 5.17 Consider a water stream from a jet of an oscillating lawn sprinkler. Describe the corresponding pathline and strealdine. 5.18 Which of the following sets of equations represent possible incompressible flow cases? a. V, = U cos t); V" = —U sin if b. V, =-q/2m“. V" = Kertr c. Vr = Ucos 6 [1 r (an-)2]; v” : com 011 + (rt/r)2] 5.19 For an incompressible flow in the r6 plane. the r com- ponent of velocity is given as V, = —A cos Ulrz. Determine a poss- ible ll component of velocity. How many possible 0 components are there? 5.20 A viscous liquid is sheared between two parallel disks of radius R. one of which rotates while the other is fixed. The vet- ocity field is purely tangential, and the velocity varies linearly with z from V” = 0 at z = 0 (the fixed disk) to the velocity of the rotating disk at its surface it. = h). Derive an expression for the velocity field between the disks. 5.21 Evaluate V - p? in cylindrical coordinates. Use the defi- nition of V in cylindrical coordinates. Substitute the velocity vec- tor and perform the indicated operations. using the hint in footnote 1 on page 169. Collect terms and simplify; shew that the result is identical to Eq. 5.2c. 5.22 A velocity field in cylindrical coordinates is given as l7 = é,Alr+ énBlr. where A and B are constants with dimensions of mzfs. Does this represent a possible incompressible flow? Sketch the streamline that passes through the point r0 = l m. a = 90“ ifA = a 2 lmzis,ifA :1m2/s and B = 0, and if B =1m1/sandA = o. *5.23 The velocity field for the viscometric flow of Example 5.7 is l/ 2 U(ylh)i. Find the stream function for this flow. Locate the streamline that divides the total flow rate into two equal parts. *5.24 Detemiine the family of stream functions Ill that will yield the velocity field V = _v(2x + l)f+ [.r(x + l) - yzlj'. 5“5.25 The stream function for a certain incompressible flow field is given by the expression ill =—Ur sin F) + 40/21:. Obtain an expression for the velocity field. Find the stagnation poinlts) where |l7| = 0, and show that I}: = 0 there. - problems require material from sections that may be omitted without loss ol‘continuity in the text material. 210 CHAPTER 5 I imouucnon T0 DIFFEflEfl'I’lAL sumsrs or FLUID nonon ”“526 Does the velocity field of Problem 5.22 represent a possible incompressible flow case? If so. evaluate and sketch the stream function for the flow. If not. evaluate the rate of change of density in the flow field. *527 Consider a flow with velocity components ti = 0. v = y(y2 — 3:2). and w = z z2 — 3y2). a. ls this a one-. two—_. or three-dimensional flow? b. Demonstrate whether this is an incompressible or com- pressible flow. c. If possible. derive a stream function for this flow. *528 An incompressible frictionless flow field is specified by the stream function i]; =—2Ax — SAy. where A = 1 m/s. and x and y are coordinates in meters. Sketch the streamlines ti} = 0 and t!) = 5. Indicate the direction of the velocity vector at the point (0. 0) on the sketch. Determine the magnitude of the flow rate between the streamlines passing through the points (2. 2) and (4. l). *5.29 In a parallel one-dimensional flow in the positive x direc— tion. the velocity varies linearly from zero at y = 0 to 30 mls at y = [.5 m. Determine an expression for the stream function. Ill. Also determine the y coordinate above which the volume flow rate is halfthe total between y = 0 and y 2 1.5 m. *5.30 A linear velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5. IO. Derive the stream function for this flow field. Locate streamlines at one-quarter and one-half the total volume flow rate in the boundary layer. $5.31 A parabolic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5| 1. Derive the stream function for this flow field. Locate streamlines at one— quarter and one-half the total volume flow rate in the boundary layer. $5.32 Derive the stream function that represents the sinusoidal approximation used to model the x component of velocity for the boundary layer of Problem 5.12. Locate streamlines at one—quarter and one-half the total volume flow rate in the boundary layer. $5.33 A cubic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5. l3. Derive the stream function for this flow field. Locate streamlines at one-quarter and one-half the total volume flow rate in the boundary layer. $5.34 A rigid-body motion was modeled in Example 5.6 by the velocity field 17 = rote-“u. Find the stream function for this flow. Evaluate the volume flow rate per unit depth between 1'. = 0. I0 m and r3 = 0.]2 m. if m = 0.5 radfs. Sketch the velocity profile along a line of constant 0. Check the flow rate calculated from the stream function by integrating the velocity profile along this line. \9‘5.“ An incompressible liquid with negligible viscosity ll-‘_=. $5.35 Example 5.6 showed that the velocity field for a free vertex in the r0 plane is ii" = éuC/r. Find the stream function for this flow. Evaluate the volume flow rate per unit depth between r1 = 0.10 m and r2 = 0. | 2 m. if C = 0.5 mgi’s. Sketch the velocity profile along a line of constant 0. Check the flow rate calculated from the stream function by integrating the velocity profile along this line. 5.36 Consider the velocity field ‘7 =1’t(ir4 — 6r2y2 + y‘)i + A(4x_v3 —4.r3_v)f in the xy plane. where A = 0.25 m'A-s 1. and the coordinates are measured in meters. Is this a possible incom- pressible flow field? Calculate the acceleration of a fluid particle at point (.t,y) = (2. I). *These problems require material from sections that may be omitted without loss of continuity in the text material. 5.37 Consider the flow field given by t7 =2:ny — W} + iyt. r..- terrnine (a) the number of dimensions of the flow. (b) if it is a n m- - ible incompressible flow. and (c) the acceleration of a fluid panic at point (x. y. z) = {1.2. 3)- s.3s Consider the flow field given by t7 =ax3yf—byjml ' wherea = .| m‘2¢s".b = 3 s", and c = 2 ru‘I -s". Detenn' (a) the number of dimensions of the flow. (b) if it is a possible ' - compressible flow, and (c) the acceleration of a fluid particle ._- point (x, y. z) = (3. l. 2). 5.39 The velocity field within a laminar boundary layer approximated by the expression In this expression, A = 141 m‘ ”2. and U = 0.240 m/s is the -.. stream velocity. Show that this velocity field represents a possl incompressible flow. Calculate the acceleration of a fluid parti‘ at point (.r. y) = (0.5 m. 5 mm). Determine the slope of :- streamline through the point. 5.40 The x component of velocity in a steady. incompressin, flow field in the xy plane is u = A(x5—10x3y2+5xy4). where A - 2 m"4 -s’ ' and x is measured in meters. Find the simplest y i . ponent of velocity for this flow field. Evaluate the acceleration .' a fluid particle at point (x. y) : {1, 3). 5.41 Considerthe velocity field I? = Axi'(x2 + f)? + Ayiol + y‘- in the xy plane. where A = ID mails, and x and y are measured' meters. Is this an incompressible flow field? Derive an expressii.’ for the fluid acceleration. Evaluate the velocity and accelerati along the x axis. the y axis. and along a line defined by y = x. ' can you conclude about this flow field? 5.42 The y component of velocity in a two-dimensional. inconr. pressible flow field is given by U = —Axy. where v is in m/s, r and are in meters. and A is a dimensional constant. There is no velocii; component or variation in the z direction. Determine the di .ue‘. sions of the constant. A. Find the simplest 1: component of veloc' in this flow field. Calculate the acceleration of a fluid particle. point (x, y) = (l. 2)- 5.43 An incompressible liquid with negligible viscosity ll- - steadily through a horizontal pipe of constant diameter. In a n -_ ous section of length L = 0.3 m. liquid is removed at a cons -' rate per unit length. so the uniform axial velocity in the pipe 5' ri(.r) = Utl — xl2L), where U = 5 mls. Develop an expres' for the acceleration of a fluid particle along the centerline of i porous section. steadily through a horizontal pipe. The pipe diameter linearly ..- ies from a diameter of IO cm to a diameter of 2.5 cm over a lett of 2 m. Develop an expression for the acceleration of a fluid in ticle along the pipe centerline. Plot the centerline velocity and celeration versus position along the pipe. if the inlet center‘ velocity is 1 m/s. 5.45 Solve Problem 4.] 18 to show that the radial velocity in r. narrow gap is V, = QIanIi. Derive an expression for the accel tion of a fluid particle in the gap. 5.46 Consider the low-speed flow of air between parallel as. as shown. Assume that the flow is incompressible and invim ...
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