Presentation is loading. Please wait.

Presentation is loading. Please wait.

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.

Similar presentations


Presentation on theme: "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."— Presentation transcript:

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 2.1 The Reynolds Transport Theorem (1)

3 2.1 The Reynolds Transport Theorem (2)

4 2.1 The 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 2.3 The Linear Momentum Equation (1) ..

7 2.3 The Linear Momentum Equation (2)

8 2.3 The Linear Momentum Equation (3)  Special Cases

9 2.3 The 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 3.1 Conservation of mass Rectangular coordinate system x y z dx dy dz o u v w

13 x y z dx dy dz o u v w

14 x y z dx dy dz o u v w

15 dx dy dz o u v w x y z

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

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

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

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

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

26  Scalar component of fluid acceleration

27  Fluid acceleration in cylindrical coordinates

28  Fluid Rotation x y a a' b b' o xx yy

29 a a' b b' o xx yy

30 a a' b b' o xx yy 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  Fluid Circulation,  c y x o b a Circulation around a close contour =Total vorticity enclosed Around the close contour oacb,

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

34  Fluid Linear Deformation x y a a' b b' o xx yy

35 a a' b b' o xx yy

36

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

38 3.4 Momentum Equation

39 x y z

40 Forces acting on a fluid particle x y z 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


Download ppt "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."

Similar presentations


Ads by Google