Chapter 2 Reynolds Transport Theorem (RTT)

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

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

38. 3.4 Momentum Equation

39. y x z

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

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

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

43. Differential Momentum Equation

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

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

46. Surface Forces

47. Momentum Equation:Navier-Stokes Equations

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

49. 3.5 Conservation of Energy

50. Summary of Basic Equations