Jignesh Ph D Credit Seminar 1 Presentation

7/28/2019 Jignesh Ph D Credit Seminar 1 Presentation

http://slidepdf.com/reader/full/jignesh-ph-d-credit-seminar-1-presentation 1/63

Presented By:- Jignesh P. Thaker (D12ME008)

Ph.D. Student, MED, SVNIT

Guided By:- Dr. Jyotirmay Banerjee

Associate Professor, MED, SVNIT 1

DEVELOPMENT OF NAVIER-STOKES SOLVER FOR SINGLE AND T WO-PHASE FLOW



To develop Navier-Stokes solver for singleand two phase flow and validate with theavailable benchmark results.

First develop Navier-Stokes solver for singlephase flow using derived variable approach.

To develop Navier-Stokes solver for single

phase flow using primitive variableapproach.

To solve the volume fraction equation for

two phase flow. 2



Navier-Stokes Solver forSingle Phase Flow(Derived Variable Approach)

3



4

Stream function-vorticity formulation is a powerful andaccurate approach of solving 2D Navier-Stokes equations.

Discretized equations using finite volume method forstream function and vorticity are solved for isothermalflow and validate with Ghia’s results.

Lid-driven cavity problem is solved for Reynolds numbers(Re) in the range of 100 ≤ Re ≤ 3200 using explicit scheme.

Streamline pattern and vorticity contours are plotted fordifferent Reynolds numbers.

For Non-isothermal flow with the discretized energyequation is used to calculate the temperature in the flowfield.

Streamlines pattern and isotherms contours at different

Rayleigh number Ra = 10

3

, 10

4

, 10

5

, 10

6

and Prandtlnumber Pr = 0.1 and 1 are analyzed.



5

The lid driven cavity flow is most probably one of the most studied fluid problems in computational fluid dynamics field.

The simplicity of the geometry of the cavity flow makes the problem

easy to code and apply boundary conditions.

Driven cavity flow serve as a benchmark problem for numericalmethods in terms of accuracy and numerical efficiency.

Isothermal Flow Non-Isothermal Flow



6

The law of conservation of mass Newton’s second law of motion ( Law of conservation of

momentum)

The law of conservation of energy

Continuity EquationMomentum Equations

X- Momentum

Y- Momentum

Z- Momentum

Energy Equation

0 = V.∇

( ) ( ) y f v y pV v

t v ρ

ρ µ

ρ +∇∇+∂∂−=∇+∂∂.1.

( ) ( )z

f w z

pV w

t

w ρ

ρ

µ

ρ +∇∇+

∂

∂−=∇+

∂

∂.

1.

( ) ( )T k V T

t

T C P ∇∇=

∇+∂

∂.. ρ

( ) ( )x f u

x

pV u

t

u ρ

ρ

µ

ρ +∇∇+

∂

∂−=∇+

∂

∂.

1.



2D, Incompressible, Unsteady, Newtonian Fluid FlowContinuity Equation

Momentum Equations

X- Momentum

Y- Momentum

Stream Function Equation Vorticity Equation

0=∂

∂+

∂

∂

y

v

x

u

∂

∂

+∂

∂

+∂

∂

−=∂

∂

+∂

∂

+∂

∂2

2

2

21

y

u

x

u

x

p

y

u

v x

u

ut

u

ρ

µ

ρ

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂2

2

2

21

y

v

x

v

y

p

y

vv

x

vu

t

v

ρ

µ

ρ

v x

u y

−=∂

∂=

∂

∂ ψ ψ ,

∂

∂+

∂

∂−=

∂

∂−

∂

∂=

2

2

2

2

y x y

u

x

v ψ ψ ω

∂

∂

+∂

∂

=∂

∂

+∂

∂

+∂

∂2

2

2

2

y x yv xut

ω ω

ρ

µ ω ω ω

y∂

∂ x∂

∂



Dimensionless Form of Isothermal Flow Equations

Stream Function Equation Vorticity Equation

Vorticity Equation becomes,

*

*

*

*

*

*,

xv

yu

∂

∂=

∂

∂=

ψ ψ

ULU

LU

L

t t

U

vv

U

uu

L

y y

L

x x

ψ ψ

ω ω =======

*******,,,,,,

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂2*

*2

2*

*2

*

**

*

**

*

*

Re

1

y x yv

xu

t

ω ω ω ω ω

∂

∂+

∂

∂−=

∂

∂−

∂

∂=

2*

*2

2*

*2

*

*

*

**

y x y

u

x

v ψ ψ ω



Solve Vorticity Equation using Explicit Time Marching SchemeVorticity Equation

Face Velocities

;2

;2

;2

;2

PS

s

P N

n

PW

w

P E

e

vvv

vvv

uuu

uuu

+=

+=

+=

+=

( ) ( )ω ω ω 2

Re

1. ∇=∇+

∂

∂V

t

nPPnS S n N N nW W n E E nPP aaaaaa ω ω ω ω ω ω 01 ++++=+

;Re

4

22221

;2Re

;2Re

;2Re

;2Re

;1

2

22220

−+−+−+=

+=

−=

+=

−==

dx

dt

dx

dt v

dx

dt v

dx

dt u

dx

dt u

a

dx

dt v

dx

dt a

dx

dt v

dx

dt a

dx

dt u

dx

dt a

dx

dt u

dx

dt aa

snwe

P

s

S

n

N

w

W

e

E P



Solve Stream Function Equation using Vorticity ValueStream Function Equation

Cell Center Velocities

∂

∂+

∂

∂−=

∂

∂−

∂

∂=

2

2

2

2

y x y

u

x

v ψ ψ ω

ψ ω 2−∇=1

0

11111 ++++++ ++++= n

PP

n

S S

n

N N

n

W W

n

E E

n

PP aaaaaa ψ ψ ψ ψ ψ ω

;4;1;1 202dx

adx

aaaaa PS N W E P =−=====

x

v

y

u∂

∂−=

∂

∂=

ψ ψ ,



At t = 0, Initialize with u = 0, v = 0,ψ = 0 and ω = 0

Obtain the face velocities from the cell center velocities.

Calculate the coefficients for the vorticity equation.

Solve the vorticity equation by explicit method and obtainvorticity.

Using this value of vorticity on the RHS solve the streamfunction equation to obtain stream function.

Calculate u and v from the stream function equation. Go to step (2) and replace u, v, ψ and ω from nth level to n+1th

level values and repeat till steady state is obtained by thefollowing condition.

Isothermal Flow

( )6

,

1,1

21

10

−

==

==

+

<×

−=

∑nm RMS

m jni

ji

n

P

n

P ω ω



Boundary ConditionsSquare Cavity X=0 to L and Y=0 to B. (L=B=1)

Variable Left Wall Right Wall Top Wall Bottom Wall

VelocityComponents

u=v=0 u=v=0 u=U(1),v=0 u=v=0

StreamFunction

0 0 0 0

Vorticity(ω) 2

2

y∂

∂

−

ψ 2

2

y∂

∂

−

ψ 2

2

x∂

∂

−

ψ 2

2

x∂

∂

−

ψ

( )ψ



Code Validation

Contour Number c d e f g h i j k l

Value of ψ -1e-05 -1e-04 -0.01 -0.03 -0.05 -0.07 -0.09 -0.1 -0.11 -0.115

Contour Number 1 2 3 4 5 6 7 8 9 m

Value of ψ 1e-07 1e-06 1e-05 5e-05 1e-04 2.5e-04 5e-04 1e-03 1.5e-03 -0.1175

Streamline Plot for Re=100

Result of Ghia et al.[1] Result of Code

Values of Streamlines Contours for Square Cavity



Vorticity Plot for Re=100


The figures on the right is result obtained by code and those on theleft is from Ghia et al. [1]. The values of the vorticity obtained by code

are exactly comparable to result obtained by Ghia et al. [1].

Code Validation



The velocity components are plotted along horizontal and vertical lines

through the center of the cavity

Code Validation



Re=400 Re=1000 Re=3200

Streamlines Contours for Square Cavity at Different Re

It is observed that as Re increases from 100 to 3200, the secondaryvortices start developing and getting larger in magnitude.The center of the primary vortex is towards the top right corner for

lower values of Reynolds number. It moves towards the geometriccenter of the cavity with the increase in Re.

Results and Discussion



Re=400 Re=1000 Re=3200

Vorticity Contours for Square Cavity at Different Re

As Re increases, several regions of high velocity gradients, indicated byconcentration of the vorticity contours, appear within the cavity.

These regions are not aligned with the geometric boundaries of the

cavity.




As the Reynolds number increases, the maximum magnitude of both velocitycomponents increases.

Velocity components along the horizontal and vertical lines through

the geometric center of the cavity




2D, Incompressible, Unsteady, Newtonian Fluid FlowContinuity Equation

Momentum Equations

X- Momentum

Y- Momentum

Energy Equation

0=∂

∂+

∂

∂

y

v

x

u

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂2

2

2

21

y

u

x

u

x

p

y

u

v x

u

ut

u

ρ

µ

ρ

( ) β ρ

µ

ρ TcT g

y

v

x

v

y

p

y

vv

x

vu

t

v−+

∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

+∂∂

2

2

2

21

∂∂+

∂∂=

∂∂+

∂∂+

∂∂

2

2

2

2

y

T

x

T

C

k

y

T v x

T ut

T

P ρ

x

T

g y x yv xut ∂

∂

+

∂

∂

+∂

∂

=∂

∂

+∂

∂

+∂

∂

β

ω ω

ρ

µ ω ω ω

2

2

2

2 x∂

∂

y∂

∂



Dimensionless Form of Non-Isothermal Flow Equations

Vorticity Equation becomes,

Energy Equation becomes,

Rayleigh Number Prandtl Number

C H

C

T T

T T T

L L

t t

L

vv

L

uu

L

y y

L

x x

−

−========

**

2

*

2

*****,,,,,,,

α

ψ ψ

α

ω ω

α

α α

∂

∂+

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂*

*

2*

*2

2*

*2

*

**

*

**

*

*

Pr Pr x

T Ra

y x yv

xu

t

ω ω ω ω ω

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂2*

*2

2*

*2

*

**

*

*

*

*

*

y

T

x

T

y

T v

x

T u

t

T

( )αν

β 3 LT T g

Ra

C H −

=k

C Pµ =Pr



Solve Temperature Equation using Explicit Time Marching Scheme

Energy Equation

( ) ( )T V T t

T 2. ∇=∇+

∂

∂

n

PP

n

S S

n

N N

n

W W

n

E E

n

PP T aT aT aT aT aT a0

1++++=

+

;4

22221

;

2

;

2

;

2

;

2

;1

2

22220

−+−+−+=

+=

−=

+=

−==

dx

dt

dx

dt v

dx

dt v

dx

dt u

dx

dt ua

dx

dt v

dx

dt a

dx

dt v

dx

dt a

dx

dt u

dx

dt a

dx

dt u

dx

dt aa

snwe

P

s

S

n

N

w

W

e

E P



Solve Vorticity Equation using Explicit Time Marching SchemeVorticity Equation

Face Velocities

( ) ( )

∂

∂+∇=∇+

∂

∂

x

T RaV

t Pr Pr .

2ω ω

ω

dxT T Raaaaaaa

n

W

n

E nPP

nS S

n N N

nW W

n E E

nPP

−+++++=

++

+

2Pr

11

01 ω ω ω ω ω ω

;Pr 42222

1

;2

Pr ;

2

Pr ;

2

Pr ;

2

Pr ;1

2

22220

−+−+−+=

+=

−=

+=

−==

dxdt

dxdt v

dxdt v

dxdt u

dxdt ua

dx

dt v

dx

dt a

dx

dt v

dx

dt a

dx

dt u

dx

dt a

dx

dt u

dx

dt aa

snweP

sS

n N

wW

e E P

;

2

;

2

;

2

;

2

PS s

P N n

PW w

P E e

vvv

vvv

uuu

uuu

+=

+=

+=

+=



23

Solve Stream Function Equation using Vorticity ValuesStream Function Equation

Cell Center Velocities

∂∂

+∂∂

−=

∂∂

−∂∂

=2

2

2

2

y x y

u

x

v ψ ψ ω

ψ ω 2−∇=1

0

11111 ++++++ ++++= n

PP

n

S S

n

N N

n

W W

n

E E

n

PP aaaaaa ψ ψ ψ ψ ψ ω

;4

;1

;1 202dx

adx

aaaaa PS N W E P =−=====

x

v

y

u∂

∂−=

∂

∂=

ψ ψ ,



24

At t = 0, Initialize with u = 0, v = 0,ψ = 0, ω = 0 and T=0

Obtain the face velocities from the cell center velocities.

Calculate the coefficients for the temperature equation.

Solve the temperature equation by explicit method and obtain new

temperature. By using this value of temperature calculate the coefficients for the

vorticity equation. Solve the vorticity equation by explicit method andobtain vorticity.

Using this value of vorticity on the RHS solve the stream function

equation to obtain stream function. Calculate u and v from the stream function equations.

Go to step (2) and replace u, v, ψ, ω and T from nth level to n+1th levelvalues and repeat till steady state is obtained by the following condition.

Non-Isothermal Flow

( )6

,

1,1

21

101 −

==

==

+

<×

−

= ∑ nm RMS

m jni

ji

n

P

n

P

ω ω

( ) 6

,

1,1

21

102 −

==

==

+

<×

−

= ∑ nm

T T RMS

m jni

ji

n

P

n

P



25

Heated Square Cavity X=0 to L and Y=0 to B. (L=B=1) Boundary Conditions


VelocityComponents u=v=0 u=v=0 u=v=0 u=v=0

Temperature T=TH = 1 T=TC = 0

Stream Function 0 0 0 0

Vorticity (ω) 2

2

y∂

∂

−

ψ

2

2

y∂

∂

−

ψ

2

2

x ∂

∂

−

ψ

2

2

x ∂

∂

−

ψ

( )ψ

0=∂

∂

y

T 0=

∂

∂

y

T



26

Results of Bubnovich

Results of Code

Ra = 104 Ra = 105Ra = 103

Non-dimensional velocity v along the horizontal center line of the cavity at diff. time( Pr =1 )

Code Validation



27

Results of Bubnovich

Results of Code

Ra = 104 Ra = 105Ra = 103

Non-dimensional Temperature T along the horizontal center line of the cavity at diff. time

( Pr =1 )

Code Validation



28

Streamlines Contours

Ra = 104 Ra = 105Ra = 103

Pr=0.1

Pr=1




29

Isotherm Contours

Ra = 104 Ra = 105Ra = 103

Pr=0.1

Pr=1




30

Isotherm Contours

Ra = 106

Pr=0.1

Pr=1

Streamlines Contours Results and Discussion



31

u-velocity and v-velocity along the vertical and horizontal geometric center line of cavity

Pr = 0.1

Pr = 1




Navier-Stokes Single Phase Solver

(Primitive Variable Approach)

32



33

The semi-explicit method has been used to solve the unsteadyNavier-Stokes equations in which the momentum equations arediscretized in an explicit manner while the pressure gradientterms are treated implicitly.

Two steps are implemented at each time level:

First, a predicted velocity is obtained from thediscretized momentum equation using the previous time-levelpressure field;

Second, corrected step consists of iterative solution ofthe pressure correction equation and in obtaining the

corresponding velocity corrections such that the final velocityfield satisfies the continuity equation to the prescribed limit.

Lid-driven cavity problem is solved for Reynolds numbers (Re)in the range of 100 ≤ Re ≤ 7500 using Semi-explicit method.

Streamline patterns are plotted for Re = 100, 400, 1000, 3200,

5000 and 7500.



34

Predictor Step (For Uniform Grid Size dx=dy)

X- Momentum

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂2

2

2

21

y

u

x

u

x

p

y

uv

x

uu

t

u

ρ

µ

ρ

Non-dimensional X- Momentum Equation

∂

∂+∂

∂+∂

∂−=∂

∂+∂

∂+∂

∂2

2

2

2

Re

1

y

u

x

u

x

p

y

uv x

uut

u

DX CX F x

pF

t

u+

∂

∂−=+

∂

∂

∫∫∫∫∫ =∇= dAV udV V uF CX ..

[ ] [ ] [ ] [ ]n

ss

n

nn

n

ww

n

eeCX dxvudxvudyuudyuuF −+−=

( ) ( )∫∫∫∫∫ ∇=∇= dAudV uF DX

.Re

1 2

[ ]n

P

n

S

n

N

n

W

n

E DX uuuuuF 4

Re

1−+++=

Similarly from Y- Momentum equation

[ ] [ ] [ ] [ ]n

ss

n

nn

n

ww

n

eeCY dxvvdxvvdyuvdyuvF −+−= [ ]n

P

n

S

n

N

n

W

n

E DY vvvvvF 4

Re

1−+++=



35

Predictor Step (For Uniform Grid Size dx=dy)Discretized X- Momentum Equation

DX

n

w

n

e

CX

n

PP F dxdydx

p pF dxdy

dt

uu+

−−=+

−*Predicted X- Velocity

( ) dt dx

p p

dxdy

dt F F uu

n

w

n

e

CX DX

n

PP

−−−+=

*

DX

n

w

n

e

CX

n

P

n

P F dxdydx

p pF dxdy

dt

uu+

−−=+

−+++ 111

( )*

Pu

( ) ( )dxdydx

p p p pdxdy

dt

uu n

w

n

w

n

e

n

eP

n

P −−−−=

− +++ 11*1

Subtraction

Similarly for Y- Momentum Equation

( ) ( )dxdydy

p p p p

dxdydt

vvn

s

n

s

n

n

n

nP

n

P −−−

−=

−+++ 11*1



36

Velocity and Pressure Correction'

1 p p p

nn+=

+ where p’= pressure correction

'*1

uuun

+=+

'*1

vvvn

+=+

( ) ( )dxdy

dx

p p p pdxdy

dt

uun

w

n

w

n

e

n

eP

n

P −−−−=

−+++ 11*1

( )dx

dt p pu weP ''' −−=

( )dy

dt p pv

snP ''' −−=

All face velocity corrections are found from the above two equations



37

Pressure Correction EquationThere is no explicit equation for pressure but the continuity equation

can be converted into an equation for the pressure corrections.

0 = V.∇

0 =dA.∫∫V 0

1111=−+−

++++ n

s

n

n

n

w

n

e vvuu

K pa pa pa pa pa S S N N E E ww pP=++++ '''''

dx

dt aaaa

dx

dt a

S N E W P−===== ;

4

****''''

snwesnwe vvuuvvuu −+−=−+−

The above equation is known as pressure correction equation

****

snwe vvuuK −+−=where,



38

( ) dt dx

p p

dxdy

dt F F uu

n

w

n

e

CX DX

n

P

n

P

11

1

++

+ −−−+=

This New pressure value is useful to calculate the velocity

components for new time level.

Pressure Correction Equation

( ) dt dy

p p

dxdy

dt F F vv

n

s

n

n

CY DY

n

P

n

P

11

1

++

+ −−−+=

New n+1th level x velocity is given by

New n+1th level y velocity is given by



39

1) Set the initial conditions and boundary conditions forvelocity and pressure.

2) Compute the face center mass flux for all real cells using

initial values.3) Compute convection and diffusion terms.

4) Compute predicted velocity at the cell centers andcalculate cell face predicted velocity from the linear

interpolation of the cell center values.5) Set pressure correction for all real cells.

6) Impose pressure correction BC.

7) Solve the pressure correction equation by GSSOR to

calculate the pressure correction.

Semi-Explicit Method



40

8) Convergence criterion for pressure correction is given byfollowing equation.

where R is the residual of the pressure correction equation9) If the above condition is satisfied then the pressure

correction obtained is the final pressure correction,otherwise go to step 7 and recalculate the same till the

convergence is reached.10) Calculate the correct pressure from the definition of the

pressure correction

11) Using the correct pressure (n+1th level pressure) calculate

the final velocity.


( )6

,

1,1

2

10−

==

== <×

=∑

nm

R RMS

m jni

ji



41

12) Convergence criterion for velocities are as follows

13) If the above criterion is satisfied then velocity obtained isthe final velocity components otherwise go to step 2 withthe new velocity and pressure as the initial conditions andrepeat the process until the convergence criteria isattained.


( )8

,

1,1

21

10−

==

==

+

<×

−=

∑nm

uu RMSu

m jni

ji

n

P

n

P

( )8

,

1,1

21

10−

==

==

+

<×

−=

∑nm

vv RMSv

m jni

ji

n

P

n

P



42

Boundary Conditions

Square Cavity X=0 to L and Y=0 to B. (L=B=1)

An iterative computer code is developed which has flexibility tochoose various inputs like square cavity size, grid size, Reynolds number

(Re), time step (Δt)

The output files are being generated as the data files to plot various

flow field parameters like u, v, ψ using tecplot360.


VelocityComponents u=v=0 u=v=0 u=U(1),v=0 u=v=0

Pressure0

',0 =∂

∂=

∂

∂

x

p

x

p0

',0 =∂

∂=

∂

∂

x

p

x

p0

',0 =∂

∂=

∂

∂

y

p

y

p0

',0 =∂

∂=

∂

∂

y

p

y

p



43

Contour Number c d e f g h i j k l

Value of ψ -1e-05 -1e-04 -0.01 -0.03 -0.05 -0.07 -0.09 -0.1 -0.11 -0.115

Contour Number 1 2 3 4 5 6 7 8 9 m

Value of ψ 1e-07 1e-06 1e-05 5e-05 1e-04 2.5e-04 5e-04 1e-03 1.5e-03 -0.1175

Streamline Plot for Re=100


Values of Streamlines Contours for Square Cavity

Code Validation



44

The velocity components are plotted along horizontal and vertical lines

through the center of the cavity

Code Validation



45

Re=3200 Re=5000 Re=7500

Streamlines Contours for Square Cavity at Different Re

It is observed that as Re increases from 3200 to 7500, the secondaryvortices start developing and getting larger in magnitude.The centre of the primary vortex is towards the top right corner forlower values of Reynolds number. It moves towards the geometriccentre of the cavity with the increase in Re.




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Re=400 (200 × 200)

Influence of Grid Size on the Flow Pattern Distribution for Re=400

Re=400 (129 × 129)




Navier-Stokes Two Phase Solver

(Advection Test only)

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48

A flow situation where two immiscible, non reactingfluids flow together is known as two phase flows.

Analysis of such flows occurring in engineeringsystems is required to be done.

One thing that separates the analysis of two-phaseflows from single fluid flow is existence of interface(boundary which separates the two fluids) acrosswhich step change in properties exists.

Some examples are flow of molten metal duringcasting, flow in pipe carrying steam and water, liquidsloshing in tankers, liquid jet issuing intoenvironments, bubble formation, breakup andpropagation in the surrounding fluid.



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There are so many Two-Phase flow simulationmethodologies available such as

Marker and Cell (MAC) method ,

Volume of Fluid method given by Hirt et al. Level set method given by Sussman et al.

In all methods a flag or some marker particles are

used to represent region of a fluid or the interface. Governing equation of that flag or any marker

particle is derived from physical principles and isa pure advection equation.



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VOF is popular because of its characteristic ofmaintaining the conservation of mass.

Present work manly focuses on development of

solution algorithm for tracking the interface usingVolume of Fluid Method.

The main issues involved in the interface tracking:

Representation of interface in the computational

domain.

Evolution of interface in space and time.

Application of the boundary conditions at the interface.



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Actual Interface Partition Discretized domain with

VOF function values Cells which contain primary fluid are labelled as full of fluid

means volume fraction value 1.

The cells which contain secondary fluid are labelled as empty of

fluid means volume fraction value 0.

Representation of Interface in VOF Method



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When the interface information is required out of thevolume fraction field, it is done so geometrically.

In a discrete space the interface has to be represented by a

curve in a cell. The collection of curves in domain represents the

interface.

Hirt et al. approximated the interface as a line segment in

the cell which can be either aligned to horizontal orvertical axis.

This type of interface reconstruction is known as SimpleLine Interface Calculation (SLIC).




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Simple Line Interface Calculation (SLIC): possible cases of interface orientation and

relative position of fluid




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In the original VOF method of Hirt et al. , the dynamics oflighter fluid in comparison to heavier fluid is neglected.

Navier-Stokes equations are solved only in the region

occupied by heavier fluid and the interface is treated ascomputational boundary.

Governing equation for VOF or Volume Fraction function as given by

Hirt et al. ( ) 0. =∇+

∂

∂C V

t

C

The above equation integrated over a cell than, the flux of C at the cell

face needs to be calculated. A convection scheme is used to interpolate the value of the C at

the cell face.

Volume fraction equation is a hyperbolic equation.

Evolution of VOF Function in Space and Time



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First Order Upwind as an advection scheme then due tolarge numerical diffusion it will smear the interface.

Second Order Upwind, QUICK then it will produceovershoots and undershoots in the solution due tonumerical dispersive nature.

It is necessary to develop some method to track theinterface without smearing and obtaining the exactvolume fraction field in the domain.

Following are methods used to solve the VOF equation.

Interface Reconstruction methods (Original VOF,Geometric PLIC, Moment of Fluid)

High resolution schemes for advection. (Flux CorrectedTrans ort Al orithm, CICSAM)

Evolution of VOF Function in Space and Time



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The Original VOF is developed by Hirt et al.

They have used the Donor-Acceptor method.

It is the combination of upwind and downwind

differencing schemes to find the flux of fluid at cell facesin partially filled cells.

The time advanced value of VOF function is given by,

( ) f snwe f

P

nnut VOFfluxedo

V

C C ∑ =

+

∆

−=,,,

1 1

( ) ( )f D D f AD f f

S xC AF t uC MIN usignut VOFfluxedo ∆∆+∆= ,

( ) ( ) 0.0,11 D D f AD xC t uC MAX AF ∆−−∆−=

Sign of velocity at face is 1 for positive flow direction and -1 for

negative flow direction and ΔSf is the cell face area vector.

Original Volume of Fluid Method



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In the previous equation subscript A meanacceptor/downwind cell, D means donor/upwind celland AD means either donor or acceptor cell.

The choice of donor or acceptor cell is made based on thefollowing criteria,

CAD = CA, if interface moves normal to itself

= CD, Otherwise

Donor-acceptor method is a 1-D method and it can be extended to

2-D by operator splitting that is 1D form of equation is solved inone direction and intermediate value of volume fraction field is

calculated.

Based on those intermediate values final volume fractions are

obtained by solving the other 1D form of equation in the remaining

direction.

Original Volume of Fluid Method



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1) Initialize velocities and volume fraction in all cells.

2) All the partially filled cells are marked as the boundary cells.

3) The VOF equation is solved explicitly for X-direction.

4) This new value of VOF is used to solve the VOF equationexplicitly for Y-direction.

5) Convergence criterion for VOF function is as follows

6) If the above criterion is satisfied then VOF function obtained isthe final VOF function otherwise go to step 3 with the new valueof VOF function as the initial conditions and repeat the process

until the convergence criteria is attained.

VOF SOLUTION ALGORITHM FOR ADVECTION TEST

( )6

,

1,1

21

101−

==

==

+

<×

−

=

∑nm

C C

RMS

m jni

ji

n

P

n

P



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Flow equations namely momentum and conservation ofmass are not solved rather direct velocity field is appliedfor advection test (Translational Test).

Advection Test (Translational Test)

Domain Description and Velocity Field

The computational domain is a square

of size 1×1 and in this domain a square

body of length of each side 0.2 is placed

at (0.20, 0.20)The velocity components for translation

test are u = 1 unit and v = 1 unit. Under

this velocity field the square fluid body

translates at 45 deg. across the mesh.



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Simple Translation

Test using DA Method

Advection Test (Translational Test)

80×

8040×40

160×160

20×

20 Interface plots at different

times for simple translation

test for DA method on 20×20,

40×40, 80×80 and 160×160

grid sizes



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[1] U. Ghia, K. N. Ghia and C. T. Shin, “High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multi-Grid Method”, Journal of Computational Physics, 1982, 48,

pp.387-411.

[2] V. Bubnovich, C. Rosas, R. Santander and G. Caceres, ”Computation of Transient Natural

Convection in a Square Cavity by an Implicit Finite-Difference Scheme in terms of the Stream

function and temperature”, Numerical Heat transfer, Part A, 2002, 42, pp.401-425.

[3] V. V. Chudanov, A. G. Popkov, A. G. Churbanov, P. N. Vabishchevich and M. M. Makarov,

“Operator-Spitting Schemes for the Stream function-Vorticity Formulation”, Computers & Fluids,

1995, 26 (7), pp.771-786.

[4] O. Bottela and R. Peyret, “Benchmark Spectral Results on The Lid-Driven Cavity Flow”,

Computers & Fluids, 1998, 27(4), pp.421-433.

[5] H. N. Dixit and V. Babu “Simulation of High Rayleigh Number Natural Convection in a Square

Cavity Using the Lattice Boltzmann Method”, International Journal of Heat and Mass transfer,

2006, 49, pp.727-739.

[6] D. C. Wan, B. S. V. Patnaik and G. W. Wei, “A new Benchmark quality solution for the Buoyancy-

Driven cavity by Discrete Singular Convolution”, Numerical Heat transfer, Part B, 2001, 40, pp.

199-228.

[7] K. Murlidhar and T. Sundararajan, “Computational Fluid Flow and Heat Transfer”, Narosa

Publishing House, 1995.



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[8] J. M. Hyman, “Numerical Methods for Tracking Interfaces”, 1984, Physica, D (12), pp. 396-407.[9] C. W. Hirt and B. D. Nichols, “Volume of Fluid (VOF) Method for the Dynamics of Free

Boundaries” Journal of Computational Physics, 1981, 39 (1), pp.201-225.

[10] M. Sussman, “A Level Set Approach for Computing Solutions to Incompressible Two-Phase

Flow”, Journal of Computational Physics, 1994, 114 (1), pp.146-159.

[11] W. J. Rider, D. B. Kothe, “Reconstructing Volume Tracking”, Journal of Computational Physics ,

1998, 141, pp. 112-152.[12] W. J. Rider, D. B. Kothe, S. J. Mosso, J. H. Cerutti, J. I. Hochstein, “Accurate solution algorithms

for incompressible multiphase flows”, 1995, AIAA Paper No. 95-0699, 33rd Aerospace Sciences

Meeting, Reno, NV.

Jignesh Ph D Credit Seminar 1 Presentation

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Transcript of Jignesh Ph D Credit Seminar 1 Presentation