 Download Download Presentation Chapter 2 Reynolds Transport Theorem (RTT)

Chapter 2 Reynolds Transport Theorem (RTT)

Download Presentation 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 (2)

7. 2.3The Linear Momentum Equation (3) • Special Cases

8. 2.3The Linear Momentum Equation (4)

9. 2.4 Conservation of Energy

10. 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

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

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

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

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

15. Net Rate of Mass Flux

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

17. Continuity Equation

18. 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

19. 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

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

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

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

23. 3.3 Flow Kinematics Rotation Translation y x Linear deformation z Angular deformation • Motion of a Fluid Element

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

25. Scalar component of fluid acceleration

26. Fluid acceleration in cylindrical coordinates

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

28. b y o a a' b' x

29. 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

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

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

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

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

34. b b' y o a a' x

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

36. 3.4 Momentum Equation

37. y x z

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

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

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

41. Differential Momentum Equation

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

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

44. Surface Forces

45. Momentum Equation:Navier-Stokes Equations

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

47. 3.5 Conservation of Energy

48. Summary of Basic Equations