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Continuity Equation Net outflow in x direction Continuity Equation net out flow in y direction,

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Presentation on theme: "Continuity Equation Net outflow in x direction Continuity Equation net out flow in y direction,"— Presentation transcript:

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2 Continuity Equation

3 Net outflow in x direction

4 Continuity Equation net out flow in y direction,

5 Continuity Equation Net out flow in z direction

6 Net mass flow out of the element

7 Time rate of mass decrease in the element Net mass flow out of the element = Time rate of mass decrease in the control volume Continuity Equation

8 The above equation is a partial differential equation form of the continuity equation. Since the element is fixed in space, this form of equation is called conservation form.

9 If the density is constant

10 This is the continuity equation for incompressible fluid

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12 Momentum equation is derived from the fundamental physical principle of Newton second law F x = m a = F g + F p + F v F g is the gravity force F p is the pressure force F v is the viscous force Since force is a vectar, all these forces will have three components. First we will go one component by next component than we will assemble all the components to get full Navier – Stokes Equation. MOMENTUM EQUATION [NAVIER STOKES EQUATION]

13 F x – Inertial Force Inertial Force = Mass X Acceleration derivative. Inertial Force in x direction = m X represents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.

14 Inertial force per unit volume in x direction= Is called Material derivative or Substantial derivative or Acceleration derivative ‘u’ is variable

15 Inertial force / volume in y direction Inertial force / volume in z direction Inertial force / volume in x direction

16 Body forces act directly on the volumetric mass of the fluid element. The examples for the body forces are Eg: gravitational Electric Magnetic forces. Body force = Body force in y direction Body force in z direction Body force per unit volume

17 Pressure on left hand face of the element Pressure on right hand face of the element Net pressure force in X direction is Net pressure force per unit volume in X direction Pressure forces per unit volume

18 Net pressure force per unit volume in X direction Net pressure force per unit volume in Y direction Net pressure force per unit volume in Z direction Net pressure force in all direction Net pressure force in 3 direction

19 Viscous forces

20 Resolving in the X direction Net viscous forces

21 Net viscous force per unit volume in X direction Net viscous force per unit volume in Y direction Net viscous force per unit volume in Z direction

22 UNDERSTANDING VISCOUS STRESSES

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31 LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

32 Linear strain in X direction Volumetric strain

33 Three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality. 1. dynamic viscosity. 2. relate stresses to volumetric deformation.

34 [ Effect of viscosity ‘ ’ is small in practice. For gases a good working approximation can be obtained taking Liquids are incompressible. div V = 0] In this the second component is negligible

35 SHEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

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38 Having derived equations for inertial force per unit volume, pressure force per unit volume body force per unit volume, and viscous force per unit volume now it is time to assemble together the subcomponents.

39 Assembly of all the components X direction:- Y direction:- Z direction:-

40 X direction:-

41 Y direction:-

42 Z direction:-

43 +

44 CONVERTING NON CONSERVATION FORM ON N-S EQUATION TO CONSERVATION FORM Navier-stokes equation in the X direction is given by Divergence of the product of scalar times a vector.

45 Taking RHS of N-S Equation we have

46 since Is equal to zero

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48 CONSERVATION FORM:-

49 SIMPLICATION OF NAVIER STOKES EQUATION If is constant

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52 For Incompressible flow

53 Energy Equation Energy is not a vector So we will be having only one energy equation which includes the energy in all the direction. The rate of Energy = Force X velocity Energy equation can be got by multiplying the momentum equation with the corresponding component of velocity

54 dQ= dE + dW dE = dQ - dW = dQ + dW [Work done is negative] because work is done on the system. Work done is given by dot product of viscous force and velocity vector. for Xdirection

55 for Y direction for Z direction

56 Body force is given by

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58 Total work done Net Heat flux into element = Volumetric Heating + Heat transfer across surface. Volumetric heating

59 Heat transfer in X direction = Heating of fluid element

60 dQ = B = dQ = B

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62 Energy Equation Nonconservation form

63 Non conservation:-

64 Conservation:-

65 Momentum Equation Non conservation form X direction Y direction Z direction

66 Momentum Equation Conservation form X direction Y direction Z direction

67 Energy Equation Non conservation form

68 Energy equation Conservation form

69 FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD

70 Solution vectar

71 Variation in x direction

72 Variation in y direction

73 Variation in z direction

74 Source vectar

75 Time marching Types of time marching 1. Implicite time marching 2. Explicite time marching

76 Explicit FDM

77 Implicit FDM

78 Crank-Nicolson FDM

79 Space marching

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