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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
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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
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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
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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
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Discretization Approaches
Finite
Difference
Finite
Volume
Finite
Element
X
Governing Eq. Differential. Integral. Integral.
Domain decomposition. Node based. Cell based. Element based.
DISCRETIZATION APPROACH
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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
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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
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FINITE VOLUME METHOD
(FVM)
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❑ 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
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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
Finite Volume method
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Finite Volume method
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.
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Finite Volume method
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
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Finite Volume method
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
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Finite Volume method
2 D Finite Volume method
Using Green-divergence Theorem,
it is applied to the second term
Changing differential form to Integral form
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Finite Volume method
2D Integral Equation
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Finite Volume method
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
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Finite Volume method
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
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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
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Finite Volume Method
=
jiSji
ji QdSS
Q
,,
,
1
Moelyadi - 215.11.2016 Page 19
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
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Finite Volume Method
)(2
1,,1 jijiBA FFF += +
Moelyadi - 215.11.2016 Page 20
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
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21Finite Volume Method
Moelyadi - 215.11.2016 Page 21
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
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Finite Volume Method
Moelyadi - 215.11.2016 Page 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
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Finite Volume Method : Cell Connectivity
Moelyadi - 215.11.2016 Page 23
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
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Finite Volume Method : Data structure
Moelyadi - 215.11.2016 Page 24
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
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Thank You