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Published byAddison Grahame Modified over 3 years ago

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Continuity Equation

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Continuity Equation Net outflow in x direction

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Continuity Equation net out flow in y direction,

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Continuity Equation Net out flow in z direction

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**Net mass flow out of the element**

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**Continuity Equation Time rate of mass decrease in the element**

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

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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.

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**If the density is constant**

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**This is the continuity equation for incompressible fluid**

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**[NAVIER STOKES EQUATION]**

MOMENTUM EQUATION [NAVIER STOKES EQUATION] Momentum equation is derived from the fundamental physical principle of Newton second law Fx = m a = Fg + Fp + Fv Fg is the gravity force Fp is the pressure force Fv 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.

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** Fx – Inertial Force Inertial Force = Mass X Acceleration derivative. **

Fx – 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.

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**Is called Material derivative or Substantial derivative or **

Acceleration derivative ‘u’ is variable Inertial force per unit volume in x direction =

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**Inertial force / volume in x direction**

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

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**Body force per unit volume**

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

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**Pressure forces per unit volume**

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

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**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

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Viscous forces

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**Resolving in the X direction**

Net viscous forces

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**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

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**UNDERSTANDING VISCOUS STRESSES**

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**LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE**

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**Linear strain in X direction**

Volumetric strain

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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.

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**In this the second component is negligible**

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

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**SHEAR STRESSES = ELASTIC**

CONSTANT X STRAIN RATE

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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.

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**Assembly of all the components**

X direction:- Y direction:- Z direction:-

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X direction:-

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Y direction:-

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Z direction:-

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+

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**Divergence of the product of scalar times a vector.**

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.

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**Taking RHS of N-S Equation we have**

Taking RHS of N-S Equation we have

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since Is equal to zero

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CONSERVATION FORM:-

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**SIMPLICATION OF NAVIER STOKES EQUATION**

If is constant

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**For Incompressible flow**

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**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

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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

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for Y direction for Z direction

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Body force is given by

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

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**Heat transfer in X direction**

= Heating of fluid element

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dQ = B = dQ = B

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**Energy Equation Nonconservation form**

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Non conservation:-

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Conservation:-

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**Momentum Equation Non conservation form X direction Y direction**

X direction Y direction Z direction

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**Momentum Equation Conservation form **

X direction Y direction Z direction

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Energy Equation Non conservation form

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Energy equation Conservation form

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**FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD**

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Solution vectar

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**Variation in x direction**

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**Variation in y direction**

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**Variation in z direction**

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Source vectar

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**Time marching Types of time marching 1. Implicite time marching**

2. Explicite time marching

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Explicit FDM

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Implicit FDM

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Crank-Nicolson FDM

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Space marching

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