CFD COURSE AE 5011 Computational Fluid Dynamics 1 FINITE

EDUNEX ITB

1

Dr.-Ing. Mochammad Agoes Moelyadi

FINITE VOLUME

METHOD

06 October 2021

CFD COURSE

AE 5011 Computational Fluid Dynamics 1

Fakultas Teknik Mesin dan Dirgantara

EDUNEX ITB

2

10/6/2021

Real Words Physics Numerical Simulation

Flow Models

Dynamic

approximation

Spatial

approximation

Steadiness

approximation Space

discretizationMesh definition

Equation

discretizationDefinition of

Numerical schemes

Mathematical

Model

Discretization

Approach

Resolution of

discrete system

of Equations.

Governing

Equations

Boundary

and initial

condition

Mathematical

Behavior

REVIEW : COMPUTATIONAL MODELLING

EDUNEX ITB

3

10/6/2021

WHAT IS DISCRETIZATION

Discretization

▪ A process by which a closed-form mathematical expression, such as

a function or a differential or integral equation involving functions, is

approximated by expression which prescribe values at only a finite

number of discrete points or volumes in the domain. [Equation

discretization]

▪ A process (technique) to decompose a continuum space

/computational domain to become smaller volumes or a finite number

of points where numerical values of fluid property variables will have

to be determined [Space discretization]

Grids /

Mesh

Volum

e

Element

EDUNEX ITB

4

10/6/2021

WHAT IS DISCRETIZATION

Equation Discretization

The following steps have to be defined in the process of setting up a numerical scheme:

1. Selection of a discretization method of the equations.

This implies the selection between finite difference, finite volume or finite element

methods as well as the selection of the order of accuracy of the spatial and eventually

time discretization.

2. Analysis of the selected numerical algorithm.

This step concerns the analysis of the ‘qualities’ of the scheme in terms of stability

and convergence properties as well as the investigation of the generated errors

3. Selection of a resolution method for the system of ordinary differential equations

in time, for the algebraic system of equations and for the iterative treatment of

eventual nonlinearities

EDUNEX ITB

5

10/6/2021

Discretization Approaches

Finite

Difference

Finite

Volume

Finite

Element

X

Governing Eq. Differential. Integral. Integral.

Domain decomposition. Node based. Cell based. Element based.

DISCRETIZATION APPROACH

EDUNEX ITB

6

10/6/2021

REVIEW : MATHEMATICAL MODEL

Linear equations Non-Linear equation System equations

1. Linear convection

3. Transport (unsteady

convection-diffusion)

2. Linear diffusion

(heat conduction)

4. Laplace

1. Inviscid Burgers

2. Burgers

1. Unsteady Inviscid

compressible flow

Where p is pressure and E

is total energy per unit

volume given by

and g is ratio of specific

heats

𝜕

𝜕𝑡න𝑇 𝑑𝑉 + 𝑢න𝑛. 𝑇𝑑𝑠 = 0

𝜕

𝜕𝑡න𝑢 𝑑𝑉 +

1

2න𝑛. 𝑢2𝑑𝑠 = 0

𝜕

𝜕𝑡න𝑢 𝑑𝑉 +

1

2න𝑛. 𝑢2𝑑𝑠 − 𝜈න𝛻𝑢, 𝑑𝑠 = 0

EDUNEX ITB

7

10/6/2021

REVIEW : BOUNDARY CONDITIONS

1. Direchlet BC > a given value at boundary

Example : BC for viscous flow -> no slip condition (zero velocity)

Wall Temperature

2. Newmann BC > gradient of variable at boundary

Example : BC on the temperature gradient at wall

3. Robin BC > mixed BC’s

EDUNEX ITB

8

10/6/2021

FINITE VOLUME METHOD

(FVM)

EDUNEX ITB

9

10/6/2021

❑ the FVM is based on cell-averaged values, which appear as a most

fundamental quantity in CFD. This distinguishes the FVM from the finite

difference and finite element methods, where the main numerical quantities are

the local function values at the mesh points.

❑ Once a grid has been generated, the FVM consists in associating a local finite

volume, also called control volume, to each mesh point and applying the

integral conservation law to this local volume. This is a first major distinction

from the finite difference approach, where the discretized space is considered

as a set of points, while in the FVM the discretized space is formed by a set of

small cells, one cell being associated to one mesh point.

❑ An essential advantage of the FVM is connected to the very important concept

of conservative discretization.

Finite Volume method

EDUNEX ITB

10

10/6/2021

Finite Volume method is a discretization method based on a volume or cell solving Partial Differential equation in integral form.

Independent variables are integrated directly on physical domain.

The concept of control volume for fixed volume (Eulerian approach) is used in Finite volume describing the change of fluid properties occurred inside and on boundary of the control volume.

Fixed Control volume


EDUNEX ITB

11

10/6/2021


For unsteady flow, the time rate change of fluid properties inside of the control volume is equal to the flux across the boundaryFor steady flow, the time rate change of fluid properties inside the control volume is equal to zero

Fixed Control volume

FVM offers two major advantages :

1. It preserves the property of conservation (of mass,

momentum, etc.) very well

2. It allows complicated geometries to be dealt easy.

EDUNEX ITB

12

10/6/2021


Two groups of Finite Volume Schemes

▪ Cell Center :

Fluid properties are stored in centroid volume/cell and the

face/line of the cell associated to the grid line

▪ Cell Vertex

Fluid properties are stored in a vertex/ grid node and the

face/line of the cell associated to a line between the centroid

EDUNEX ITB

13


Cell Center Cell Vertex

Control volume are identical with

the grids, the flow variables are

associated with their centroid

Control volume are generated by

connecting the mid points of the

cells, the flow variables are stored

at the grid vertex

EDUNEX ITB

14


2 D Finite Volume method

Using Green-divergence Theorem,

it is applied to the second term

Changing differential form to Integral form

EDUNEX ITB

15


2D Integral Equation

EDUNEX ITB

16


Cell Center

Discretization in Space

The flux integral across cell edges

( ) )( kk

DA

ABk

kk

abcd

xFyEFdxEdy −− =

A

BC

D

1

5

2

4

3

An arbitrary contro volume has unlimited edge number, we

need to define simple domain quadrilateral, hexagonal, etc

which has few number of edges

EDUNEX ITB

17


Cell Center

Discretization in time

The variable Q is stored in the center of

the cell and it represent s the averaged

value of Q inside the control volume.

=

jiSji

ji QdSS

Q

,,

,

1

=

++ −+D

Ak

kkkkji yyxxS ))((2

111,

A

BC

D

1

5

2

4

3

EDUNEX ITB

18

Finite Volume Method : Flux Formulation

Moelyadi - 215.11.2016 Page 18

Cell Center

Discretization in Space

The flux integral across cell edges

( ) )( kk

DA

ABk

kk

abcd

xFyEFdxEdy −− =

A

BC

D

1

5

2

4

3

An arbitrary control volume has unlimited edge number,

we need to define simple domain quadrilateral, hexagonal,

etc which has few number of edges

EDUNEX ITB

19

Finite Volume Method

=

jiSji

ji QdSS

Q

,,

,

1


Cell Center

Discretization in time

The variable Q is stored in the center of

the cell and it represent s the averaged

value of Q inside the control volume.

=

++ −+D

Ak

kkkkji yyxxS ))((2

111,

A

BC

D

1

5

2

4

3

EDUNEX ITB

20


)(2

1,,1 jijiBA FFF += +


Cell Center

A

BC

D

1

5

2

4

3

)(2

1,1, jijiAD FFF += −

)(2

1,1, jijiCB FFF += +

)(2

1,,1 jijiDC FFF += −

kkk xxx −= +1 kkk yyy −= +1

EDUNEX ITB

21Finite Volume Method


Cell Center

A

BC

D

1

5

2

4

3

0)()( ,, =−+

=

kk

DA

ABk

kkjiji xFyEQt

S

BAjijiBAjiji

n

ji

n

ji

ji xFFyEEt

QQS +−++

−++

+

)(2

1)(

2

1)( ,,1,,1

,

1

,

,

CBjijiCBjiji xFFyEE +−++ ++ )(2

1)(

2

1,1,,1,

DCjijiDCjiji xFFyEE +−++ −− )(2

1)(

2

1,,1,,1

0)(2

1)(

2

1,1,,1, =+−++ −− ADjijiADjiji xFFyEE

EDUNEX ITB

22



Cell Vertex

A

BC

D

0)()( ,, =−+

=

kk

DA

ABk

kkjiji xFyEQt

S

BAjijiBAjiji

n

ji

n

ji

ji xFFyEEt

QQS +−++

−++

+

)(2

1)(

2

1)( ,,1,,1

,

1

,

,

CBjijiCBjiji xFFyEE +−++ ++ )(2

1)(

2

1,1,,1,

DCjijiDCjiji xFFyEE +−++ −− )(2

1)(

2

1,,1,,1

0)(2

1)(

2

1,1,,1, =+−++ −− ADjijiADjiji xFFyEE

EDUNEX ITB

23

Finite Volume Method : Cell Connectivity


Finite volume require cell connectivity for calculation of

interaction of flow between cells

Cell Forming PointsNeighbouring Cells

V3

V4 V3 V6 V5

P3 P4 P9

P9 P4 P7

Cell type

3

3

EDUNEX ITB

24

Finite Volume Method : Data structure


Data structure contains mesh connectivity, coordinate points and boundary

conditions

Cell Related PointsNeighbouring Cells

V3

V4 V3 V6 V5

P3 P4 P9

P9 P4 P7

3

3

Cell type

mesh connectivityc

Point coordinates

Boundary condition Type and numbering

EDUNEX ITB

25

Thank You

Documents

CFD COURSE AE 5011 Computational Fluid Dynamics 1 FINITE