me106_s11 - ME-106 FLUID MECHANICS Spring 2011 O Sava s...

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ME-106 FLUID MECHANICS Spring 2011 ¨ O. Sava¸ s Department of Mechanical Engineering University of California at Berkeley c January 11, 2011
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Preface These skeletal notes are meant to be a record of what was covered in class. ¨ O. S. Cover Photos: Top A Karman Vortex Street in the wake of Beerenber volcano on Jan Mayen island, June 6, 2001 Image courtesy NASA/GSFC/LaRC/JPL, MISR Team http://earthobservatory.nasa.gov/IOTD/view.php?id=2270 Bottom Smoke wire visualization of the flow past a semi-cylinder in wind tunnel. ME-107B, Sp03-Group T4 i
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Contents 1 Lecture-1 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Constitutive relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.3 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Three vital fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Kitchen experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Lecture-2 4 2.1 Fluid statics - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Pressure at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Cauchy’s tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Fluid statics equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Lecture-3 6 3.1 Fluid statics - 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.1 Fluid statics equation - rigid body motion . . . . . . . . . . . . . . . . . . . . 6 3.1.2 U-tube manometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.3 The atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.4 Standard atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Lecture-4 8 4.1 Fluid statics - 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.1 Fluid in rigid body rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.2 Static force on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.3 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Energy equation: Bernoulli equation - 1 . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2.2 Incompressible flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ii
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5 Lecture-5 10 5.1 Energy equation: Bernoulli equation - 2 . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1.1 Compressible flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Lecture-6 11 6.0.2 Example: Punctured tire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.1 Euler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 Flow examples – 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2.1 Rigid body rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2.2 Line vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 7 Lecture-7 13 7.1 Flow examples – 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.1.1 Viscous vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.1.2 Pitot-static tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.1.3 Uniform flow: u = ( U, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8 Lecture-8 15 8.1 Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 8.2 Laminar Pipe Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 9 Lecture-9 17 9.1 Flow Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 9.1.1 Streamlines, Pathlines, Streaklines . . . . . . . . . . . . . . . . . . . . . . . . 17 9.2 Streamline/Pathline example: Stagnation point flow . . . . . . . . . . . . . . . . . . 18 10 Lecture-10 20 10.1 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 11 Lecture-11 22 11.1 Conservation Equations - I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 12 Lecture-12 24 12.1 Conservation Equations - II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 12.1.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 iii
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13 Lecture-13 26 13.1 Conservation Equations - III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 13.1.1 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 13.1.2 Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 13.1.3 Surface Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 13.1.4 Total Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 14 Lecture-14 27 14.1 Conservation Equations - IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 15 Lecture-15 28 15.0.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 15.0.2 Steady Stream Tube Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 16 Lecture-16: Mid Term Exam 1 30 17 Lecture-17 31 17.1 Exam-1 distributed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 17.2 Differential flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 18 Lecture-18 33 18.1 Mass conservation for an infinitesimal element . . . . . . . . . . . . . . . . . . . . . . 33 18.1.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 19 Lecture-19 35 19.0.2 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 19.1 Differential flow analysis - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 19.1.1 Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 19.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 20 Lecture-20 37 20.1 Differential flow analysis - 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 20.1.1 linear extension rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 20.1.2 dilatation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 20.1.3 strain rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 iv
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20.1.4 rotation - vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 20.1.5 potential function / potential flow . . . . . . . . . . . . . . . . . . . . . . . . 37 21 Lecture-21 38 21.1 Differential flow analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 21.1.1 Uniform flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 21.1.2 2D Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 21.1.3 Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 21.2 Complex potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 21.2.1 Uniform flow + Source + Vortex . . . . . . . . . . . . . . . . . . . . . . . . . 39 21.2.2 2D Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 21.2.3 Uniform flow + Dipole: flow past cylinder . . . . . . . . . . . . . . . . . . . . 39 22 Lecture-22 40 22.1 Complex potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 22.1.1 Uniform flow + Dipole: flow past cylinder . . . . . . . . . . . . . . . . . . . . 40 22.1.2 Pressure coefficient flow past cylinder . . . . . . . . . . . . . . . . . . . . . . 40 22.1.3 Uniform flow + Dipole + Vortex: flow past lifting cylinder . . . . . . . . . . 40 22.1.4 Viscous flow past a cylinder
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