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1 University of California, Berkeley Mechanical Engineering ME 106, Fluid Mechanics ODK/Midterm 2, Fall 2015 Last name: First name: Student ID: Discussion: Notes: You solution procedure should be legible and complete for full credit (use scratch paper as needed). Question Grade Total:

2 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 1 of 7 1. (Total 15 points) Consider the unsteady, incompressible, 2D flow described by the velocity field u = y 2 + t v = x 2 (a) Find the equation, x(y) or y(x), for the streamline passing through point (0, 0) at time t = 0. (b) Find the equations, x(t) and y(t), describing a pathline originating at point (0, 0) at time t = 0. (c) Calculate the acceleration of a fluid particle at point (0, 0) at time t = 0. (a) Streamlines at t = 0 are given by relation Integrating, x dx dy = u v = y2 x 2 x o x 2 dx = Carrying out the integral for (x 0, y 0 ) = (0, 0) gives y y o y 2 dy x = y. (5 point) (b) Pathline: NOTE: The equations are nonlinearly coupled, making a closed form solution difficult, thus solutions should be left in integral form (if you noted this you received full credit). Alternatively, if you ignored this coupling in performing the integrals (as shown below), you also still received full credit. For the x-coordinate Rewriting, x dx dt = u = y2 + t x o dx = t t o (y 2 + t)dt and carrying out the integral (ignoring y s dependence on t) for x 0 = 0 and t 0 = 0 we get x = y 2 t + t2 2. (2.5 point) For the y-coordinate Rewriting, y dy dt = v = x2 y o dy = t t o x 2 dt and carrying out the integral (ignoring x s dependence on t) for y 0 = 0 and t 0 = 0 we get y = x 2 t. (2.5 point)

3 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 2 of 7 (c) Acceleration: At (x, y) = (0, 0) and t = 0, we get Dv Dt = v t + u v x + v v y a x = u t + u u x + v u y = 1 + x2 2y a y = v t + u v x + v v y = (y2 + t)(2x) a = (1, 0). (5 point)

4 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 3 of 7 2. (Total 20 points) A sluice gate of width b into the page controls the flow of water by raising or lowering a vertical plate. The water exerts a force F on the gate. Let ρ be the water density and other variables be as shown in the diagram. Disregarding the wall shear forces at the solid surfaces, and assuming steady, uniform flow: (a) Solve for the horizontal component of the force, F x, the water imposes on the gate. Express answer in terms of (ρ, y 1, y 2, b, g and V 1 ) (b) Based on the expression you derived above, derive an expression for y 2 when F x is a maximum. Assume V 1 and y 1 remain constant. Force F x (a) Define control volume as water only. Forces F p1 and F p2 due to pressures at sections (1) and (2) are given by F p1 = P c1 A 1 = ρg y 1 2 (y 1b) (2 point) F p2 = P c2 A 2 = ρg y 2 2 (y 2b) (2 point) The momentum equation in the x direction gives F x + F 1p F p2 = (ρa 1 V 1 )V 1 + (ρa 2 V 2 )V 2. (5 point) Also from continuity V 1 y 1 = V 2 y 2. (1 point) Hence ( F x = ρby 1 V1 2 1 y ) y 2 2 ρgb ( y1 2 y2) 2. (5 point) This is the force of the gate on the water. The force on the gate is F x. (b) F x is maximum when F x = 0 = ρby2 1V1 2 ρgby 2 y 2 y 2 2 (5 point)

5 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 4 of 7

6 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 5 of 7 3. (Total 20 points) A vehicle of mass M scoops stationary water of density ρ with depth h and width b into the page, creating an upward jet with angle θ. Assume the incoming and outgoing stream of water on the scoop have the same area. Neglect air drag, wheel friction and gravity effects. (a) Determine the thrust T to maintain a constant acceleration a in terms of the variables given. (b) Assume that the thrust is removed (T = 0), hence the vehicle decelerates from initial velocity V 0 at t = 0. Based on the expression you derived above, find the expression for the velocity V (t) as a function of time (note: a = dv dt ). θ water h Mass, M Thrust, T Velocity, V (a) Choose control volume moving with the vehicle (1 point). Since the absolute fluid velocity is zero, the relative velocity entering the control volume is W = 0 ( V ) (1 point) hence the mass flux passing the control surface is ṁ = ρhbv (2 point). Due to the conservation of mass, the exit velocity is also V (1 point). The momentum equation in the x direction is T Ma = ṁv + ṁv cos(θ) (6 point). (1) Hence T = ρhbv 2 (1 cos(θ)) Ma. (b) Setting T = 0 in equation (1) and a = V yields M V = ρbhv 2 (1 cos(θ)). (1 point) (2 point) Note, we have flipped the sign on Ma since vehicle is decelerating (1 point). Let the constant C = ρbh(1 cos(θ))/m. Separating the equation V V 0 dv V 2 = Integration and simplifying for V (t) yields t 0 Cdt. (2 point) V (t) = V 0. (3 point) 1 V 0 Ct

7 ME 106, Fluid Mechanics ODK/Midterm 2-10/26/2015 Page 6 of 7

8 from or causes changes in linear momentum (velocity magnitude and/or direction) of fluid flowing through a control volume. Work and power associated with force can be involved. The moment-of-momentum equation, which involves the relationship between torque and changes in angular momentum, is obtained and used to solve flow problems dealing with turbines h04_ qxd 8/19/08 8:52 PM Page 179 (energy extracted from a fluid) and pumps (energy supplied to a fluid). The steady-state energy equation, obtained from the first law of thermodynamics (conservation of energy), is written in several forms. The first (Eq. 5.69) involves power terms. The second Summary form (Eq or of 5.84) Equations: is termed the mechanical energy equation or the extended Bernoulli equation. It consists of the Bernoulli equation with extra terms that account for energy losses due Problems to Chapter friction in the flow, 4: as well as terms accounting for the work of pumps 179 or turbines in the flow. nservation of mass Some The of the following important equations checklist in this provides chapter a are: study guide for this chapter. When your study of the entire chapter and end-of-chapter exercises has been completed you should be able to ntinuity equation ass flowrate dy Equation for streamlines (4.1) dx v near momentum write out meanings of the terms listed here u in the margin and understand each of the related equation concepts. These terms are particularly important Acceleration a 0V u 0V and are set in italic, bold, and color type 0V 0V v w (4.3) oment-ofmomentum select an appropriate control volume D1 for 2 a in the text. 0t 0x 0y 0z Material derivative 01 given 2 problem and draw an accurately labeled control volume diagram. Dt 0t 1V 21 2 (4.6) equation aft power Streamwise use the and continuity normal components equation and a control of acceleration a s V volume 0V 0s, a to n solve V 2 problems involving mass or volume flowrate. aft torque r (4.7) rst law of DB sys Reynolds transport theorem (restricted form) 0B cv use the linear momentum equation and a control r (4.15) Dt 0t 2 Avolume, 2 V 2 b 2 r 1 in A 1 Vconjunction 1 b 1 with the continuity thermodynamics equation as necessary, to solve problems involving forces related to linear momentum change. eat transfer rate DB sys ergy equation Reynolds use transport the moment-of-momentum theorem (general form) equation to 0 solve rb dv problems (4.19) Dt 0t cs rb Vinvolving nˆ da torque and related work cv ss and power due to angular momentum change. Relative and absolute velocities V W V cv (4.22) aft work head use the energy equation, in one of its appropriate forms, to solve problems involving losses ead loss due to friction (head loss) and energy input by pumps or extraction by turbines. References netic energy use the kinetic energy coefficient in the energy equation to account for nonuniform flows. coefficient 1. Streeter, V. L., and Wylie, E. B., Fluid Mechanics, 8th Ed., McGraw-Hill, New York, Chapter 2. Goldstein, R. J., Fluid 5: Mechanics Measurements, Hemisphere, New York, Some 3. Homsy, of the G. M., important et al., Multimedia equations Fluid Mechanics in this chapter CD-ROM, are 2nd Ed., given Cambridge below. University Press, New York, Magarvey, R. H., and MacLatchy, C. S., The Formation 0 and Structure of Vortex Rings, Canadian Journal of Physics, of Vol. mass 42, (5.5) Conservation 0t r dv cs rv nˆ da 0 cv Review Problems Mass flowrate m # rq rav (5.6) Guide for Fundamentals of Fluid Mechanics, by Munson et al. rv nˆ da ( 2009 John Wiley and Sons, Inc.). A Average velocity V (5.7) ra -262.qxd 9/23/08 10:19 AM Page 245 Go to Appendix G for a set of review problems with answers. Detailed solutions can be found in Student Solution Manual and Study Problems Steady flow mass conservation a m# out a m # in 0 (5.9) Note: Unless otherwise indicated, Moving use control the values volume of fluid properties found in the tables on mass the inside conservation of the front cover. Prob- flowing. Print this photo and draw in some lines to represent how 4.3 Obtain a photograph/image 0 of a situation in which a fluid is 0t r dv cs rw nˆ da 0 lems designated with an (*) are intended to be solved with the you think some streamlines cv Problems may look. Write a brief paragraph to (5.16) 245 aid of a programmable calculator or a computer. Problems describe the acceleration of a fluid particle as it flows along one designated with a ( ) are Deforming open-ended problems control and volume require of these streamlines. DM critical thinking in that to work them one must make various sys assumptions and provide mass the necessary conservation data. There is not a 4.4 The x- and y-components 0 of a velocity field are given by (5.17) unique answer to these problems. u 1V 0 2 Dtx and v 1V 0t r dv cs rw nˆ da 0 cv 0 2 y, where V 0 and are constants. Answers to the even-numbered problems are listed at the Make a sketch of the velocity field in the first quadrant end of the book. Access to Force the videos related that accompany to change problems in 1x 7 0, 0y 7 02 by drawing arrows representing the fluid velocity can be obtained through the book s web site, at representative locations. linear momentum (5.22) college/munson. The lab-type problems can also be accessed on 0t Vr dv cs VrV nˆ da a F contents of the 4.5 A two-dimensional cv velocity field is given by u 1 y and control volume this web site. v 1. Determine the equation of the streamline that passes Moving control volume force through related the origin. On a graph, plot this streamline. Section 4.1 The Velocity to Field change in linear momentum WrW nˆ da a F contents of the 4.6 The velocity field cs of a flow is given by control V volume 4.1 Obtain a photograph/image that shows a flowing fluid. Print 15z 32î 1x 42ĵ 4ykˆ ft s, where x, y, and z are in feet. Determine relative the fluid velocities speed at the origin 1x y V z 02 Wand on Uthe x this photo and write a brief Vector paragraph addition that describes of absolute the flow in and (5.29) (5.43) terms of an Eulerian description; a Lagrangian description. axis 1y z Obtain a photograph/image Shaft torque of a situation from in force which the 4.7 A flow can be visualized by plotting the velocity field as (5.45) unsteadiness of the flow is important. Print this photo and write a velocity vectors a B1r F2 contents of the R T shaft at representative control locations volume in the flow as shown in axial brief paragraph that describes the situation involved. Video V4.2 and Fig. E4.1. Consider the velocity field given in Shaft torque related to change in moment-of-momentum (angular T shaft 1 m # in21 r in V uin 2 m # out1 r out V uout 2 (5.50) momentum) eview Problems Shaft power related to change in moment-of-momentum (angular momentum) (5.53) First law of 0 thermodynamics (5.64) (Conservation of 0t er dv cs aǔ p cv r V 2 2 gzb rv nˆ da Q# net W # shaft in net in energy) Conservation of power m # c ȟ out ȟ in V 2 out V 2 in g1z out z in 2d Q # net W # shaft (5.69) 2 Conservation of p out mechanical energy (5.82) r V 2 out 2 gz out p in r V 2 in 2 gz in w shaft loss References W # shaft 1 m # in21 U in V uin 2 m # out1 U out V uout 2 1. Eck, B., Technische Stromungslehre, Springer-Verlag, Berlin, Germany, Dean, R. C., On the Necessity of Unsteady Flow in Fluid Machines, ASME Journal of Basic Engineering 81D; 24 28, March Moran, M. J., and Shapiro, H. N., Fundamentals of Engineering Thermodynamics, 6th Ed., Wiley, New York, in net in net in o to Appendix G for a set of review problems with answers. Deiled solutions can be found in Student Solution Manual and Study Guide for Fundamentals of Fluid Mechanics, by Munson et al. ( 2009 John Wiley and Sons, Inc.). roblems ote: Unless otherwise indicated, use the values of fluid roperties found in the tables on the inside of the front cover. roblems designated with an (*) are intended to be solved Section Derivation of the Continuity Equation 5.1 Explain why the mass of the contents of a system is constant

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