Winter School # Finite Volume Method - I

8/11/2019 Winter School # Finite Volume Method - I

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Finite Volume Method in Heat

Transfer & Fluid Flow - I

Professor Parthapratim GuptaChemical Engineering Department

National Institute of Technology Durgapur


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ConservationEquations


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Differential Conservation Equation - 1

Differential conservation eqn represents

conservation principle of a physical quantity

like Mass, Momentum & Energy

Dependent variablespecific property, i.e.

property per unit mass, f

e.g. mass fraction of a species, velocity,

specific enthalpy

STgradkdivTUcdivt

Tcp

p

)](.[)(

)(


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Differential Conservation Equation - 2

Infinitesimal

Control Volume

Jx, Jy, Jz - Total Flux of the property

in x , y& zdirections


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Conservation Principle

[Rate of change of fin the controlvolume with respect to time]

+

[Net efflux of fdue to convection &diffusion out of the control volume]

=[Net rate of generation of fin the control

volume]


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Net Efflux

through theCV faces in xdirection =

zyxx

xJ

...

Jdivz

zJ

y

yJ

x

xJ

Conservation Equation - 1

Net Efflux fromthe CV per unitvolume =


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Mass Conservation Equation - 1

Conservation of mlis expressed as:

unsteady / accumulation

per unit volume

Efflux due to convective &diffusive flux

SlRate of generation of the species perunit volume per unit time. (+) ve forgeneration & (-) ve for consumption of l

t

ml

)(

ldll SJUmdiv

t

m

)()(

)( dl JUmdiv


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Mass Conservation Equation - 2

Diffusive flux expressed by Ficks law:

- Diffusivity of l

Resultant mass conservation equation:

Each term represents mass of lper unit volume

per unit time

l

)( lld mgradJ

lllll SmgraddivUmdiv

t

m

)]([)(

)(

ldll SJUmdivt

m

)(

)(


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Energy Conservation Equation - 1

)(. TgradkJd

STgradkdivUhdivt

h

)](.[)(

)(

SJUhdivt

hd

)(

)(

Tch p


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Energy Conservation Equation - 2

Shgradc

kdivUhdiv

t

h

p

)](.[)(

)(

)(1)( hgrad

pcTgrad

pp c

STgrad

c

kdivUTdiv

t

T

)](.[)(

)(

STgradkdivUhdivt

h

)](.[)(

)(


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Momentum Conservation Equation

Bx , By , BzBody force per unit volume inx, y & z directions

Vx,Vy,VzAdditional viscous force tems

xx VBxpugraddivUudiv

tu

)](.[)()(

yy VB

y

pvgraddivUvdiv

t

v

)](.[)(

)(

zz VBz

pwgraddivUwdiv

t

w

)](.[)(

)(


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General Conservation Equation - 1

ff ff

f

SgraddivUdivt

)]([)(

)(

Accumulation Diffusion Convection Source

Equation ff

Mass ml l

Energy h k /cp

Momentum u , v, w

0)(

Udiv

t

- Continuity Equation


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General Conservation Equation - 2

ffff

fff

ffff

Szzyyxx

wz

vy

uxt

)()()(

)()()()(

0)()()(

w

zv

yu

xt

Cartesian Coordinates (3-D)

Conservation Equation:

Continuity Equation:


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Finite Volume

Method


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Computation Grid


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Finite Volume Method (FVM) - 1

Computation domain is divided into a

number of non-overlapping controlvolumes such that there is one control

volume surrounding each grid point.

Values of thedependent variable

f

at a finite number

of locations (grid

points) in thecalculation domain

are the basic

unknowns


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Finite Volume Method - 2

The differential equation is integrated

over each control volume.

Piecewise profiles expressing variation

of fbetween grid points are used to

evaluate required integrals

Resultant

equations

involving fare

discretized


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Discretization equations are algebraic

equations correlating the unknown valuesof fat chosen grid points. These are

solved.

The solutions constitute the computed

values of fonly. The interpolating profiles

are used to discretize the differential

equation only.

Different interpolating profiles may be

used to integrate different terms


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The most attractive feature of the controlvolume is that the resulting solution

would imply that the integral

conservation of quantities such as mass,

momentum, and energy, is exactly

satisfied over any group of control

volume and the whole calculation

domain. Even a coarse-grid solutionexhibits exact integral balance.


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When the number of grid pointsbecomes very large, the solution of

discretization equations are expected to

approach the exact solution of the

corresponding differential equations.


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( ){ .( ) } { .[ ] }

t t t t

t V V t V V

dV U dV dt dV S dV dt

t f f

f f f

( ){ ( ). } { ( ). }

t t t t

t V S t S V

dV U ndA dt ndA S dV dt

t

f f

f f f

Integrating the governing differential

equation over the control volume & thetime interval

Using Gauss Divergence

theorem the volume

integrals are converted tosurface integrals


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eA

e fdAF

Flowrate through the CV boundary

denoted e (east face of CV), Fe

f is component ofconvective/ diffusive

flux in the direction

normal to face e& A eis the area of face e.

eee AfF


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The last Volume integral is

SfPis the source

term Sfat cellcenter P, and V

is volume of

control volume

VSdVS PV

ff


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( ){ ( ). } { ( ). }

t t t t

t V S t S V

dV U ndA dt ndA S dV dt

t f f

f f f

VP

Sys

Ayn

Axw

Axe

A

susAnunAwuwAeueAtV

ff

ff

ff

ff

f

ffffff

)()()()(

)()()()(]0)()[(


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Finite Volume Method10

A linear profile between the neighboring

nodes assumed. The gradient of fat facee can be written

bNANSASEAEWAPAP W fffff

Replacing all terms in the integral and

rearranging


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1-D Steady

Diffusion


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1-D Steady Diffusion - 1

0)( Sdx

d

dx

d f

Boundary values

at A & B specified.(Dirichlet

condition)

Grid Generation: A number of interior node

points (W, P, E, ) are placed between the

boundary grid points A & B. Control volume

faces are placed mid way between the

adjacent node points.

Computation Grid


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Node Points

P - CentralW - Western

E - Eastern

Control Volume Faces

eeastern

w - western

Control Volume

dxWPDistance between W & P

dxPEDistance between P & E

xWidth of the control volume


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0)( Sdx

d

dx

d f

Discretization Control Volume

Net Diffusive transfer = Generation

Balance of fover the control volume

So, even discretized equation in FVM has a

clear physical interpretation.

S - Average value of S over control volume


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Control Volume

Diffusive terms:

Source terma linear function off


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The Four Basic Rules - 1

(1) Consistency at Contro l Volume Faces

(Conservat iveness) :When a face is

common to two adjacent control volumes,

the flux across it must be represented by

the same expression in the discretizationequations for the two control volumes.


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The Four Basic Rules - 2(2) Boundedness:

( 1 at all nodes,


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The Four Basic Rules - 3

(3) Negative Slope L inearization Of TheSou rce Term:When the source term is

linearized as , the co-efficient

SP must always be less than or equal to

zero.

(4) Sum Of The Neighbour ing Co-ef fic ients :

, for situations where thedifferential equation continues to remain

satisfied after a constant is added to the

d d t i bl

uS

PPSVS f

anbap

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Winter School # Finite Volume Method - I