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Chapter 2 Reynolds Transport Theorem (RTT) PowerPoint Presentation
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Chapter 2 Reynolds Transport Theorem (RTT)

Chapter 2 Reynolds Transport Theorem (RTT)

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Chapter 2 Reynolds Transport Theorem (RTT)

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  1. Chapter 2 Reynolds Transport Theorem (RTT) 2.1 The Reynolds Transport Theorem 2.2 Continuity Equation 2.3 The Linear Momentum Equation 2.4 Conservation of Energy

  2. 2.1The Reynolds Transport Theorem (1)

  3. 2.1The Reynolds Transport Theorem (2)

  4. 2.1The Reynolds Transport Theorem (3) • Special Case 1: Steady Flow • Special Case 2: One-Dimensional Flow

  5. 2.2 Continuity Equation (1) • An Application: The Continuity Equation

  6. 2.3The Linear Momentum Equation (1) • ..

  7. 2.3The Linear Momentum Equation (2)

  8. 2.3The Linear Momentum Equation (3) • Special Cases

  9. 2.3The Linear Momentum Equation (4)

  10. 2.4 Conservation of Energy

  11. Chapter 3 Flow Kinematics 3.1Conservation of Mass 3.2 Stream Function for Two-Dimensional Incompressible Flow 3.3 Fluid Kinematics 3.4 Momentum Equation

  12. y dy dz dx x v o u z w 3.1 Conservation of mass • Rectangular coordinate system

  13. y dy dz dx x v o u z w

  14. y dy dz dx x v o u z w

  15. y dy dz dx x v o u z w

  16. Net Rate of Mass Flux

  17. Net Rate of Mass Flux Rate of mass change inside the control volume

  18. Continuity Equation

  19. 3.2 Stream Function for Two-Dimensional Incompressible Flow • A single mathematical function (x,y,t) to represent the two velocity components, u(x,y,t) and (x,y,t). • A continuous function (x,y,t) is defined such that The continuity equation is satisfied exactly

  20. Equation of Streamline • Lines drawn in the flow field at a given instant that are tangent to the flow direction at every point in the flow field. Along a streamline

  21. y v u x • Volume flow rate between streamlines Flow across AB Along AB, x = constant, and

  22. y v u x • Volume flow rate between streamlines Flow across BC, Along BC, y = constant, and

  23. Stream Function for Flow in a Corner Consider a two-dimensional flow field

  24. 3.3 Flow Kinematics Rotation Translation y x Linear deformation z Angular deformation • Motion of a Fluid Element

  25. Fluid particle path At t+dt At t y x z • Fluid Translation

  26. Scalar component of fluid acceleration

  27. Fluid acceleration in cylindrical coordinates

  28. b y y o a a' b' x x • Fluid Rotation

  29. b y o a a' b' x

  30. b y o a a' b' x Similarily, considering the rotation of pairs of perpendicular line segments in yz and xz planes, one can obtain

  31. Fluid particle angular velocity Vorticity: A measure of fluid element rotation Vorticity in cylindrical coordinates

  32. y b c a o x • Fluid Circulation,  Around the close contour oacb, Circulation around a close contour =Total vorticity enclosed

  33. y b  y o a a' x b' x • Fluid Angular Deformation

  34. b y b' y o a a' x x • Fluid Linear Deformation

  35. b b' y o a a' x

  36. Rate of Strain Rate of normal strain Rate of shearing strain(Angular deformation)

  37. 3.4 Momentum Equation

  38. y x z

  39. y x z Forces acting on a fluid particle x-direction + +

  40. Forces acting on a fluid particle x-direction + +

  41. Components of Forces acting on a fluid element x-direction y-direction z-direction

  42. Differential Momentum Equation

  43. Momentum Equation:Vector form is treated as a momentum flux

  44. Stress and Strain Relation for a Newtonian Fluid Newtonian fluid  viscous stress  rate of shearing strain

  45. Surface Forces

  46. Momentum Equation:Navier-Stokes Equations

  47. Navier-Stokes Equations For flow with =constant and =constant

  48. 3.5 Conservation of Energy

  49. Summary of Basic Equations