Finite Volume Discretization
MMVN05ROBERT SZASZ
Goal & Outline
β’ 1D Cartesian
β Diffusion
β Source
β Convection
β Time dependent
β’ 2D Cartesian
β’ 2D Unstructured
Partial Differential
Equation(s)
Set of Algebraic
Equations
The Finite Volume Method
β’ Generic transport equation
β’ Integrate over a control volume
πππ
ππ‘+ πππ£ ππ’ π = πππ£ Ξππππ π + π
Time
evolutionConvection Diffusion Source
term
π
πππ
ππ‘ππ +
π
πππ£ ππ’ π ππ = π
πππ£ Ξππππ π ππ + π
π ππ
Discretization in 1D
Control volume boundaries
Control volume
P EWew
Dx
dxPe
dxPEdxWP
Diffusion problems in 1D
πππ£ Ξππππ π + π = 0
π
πππ£ Ξππππ π ππ + π
π ππ = 0
P EWew
DxdxPe
dxPEdxWP
π΄
π. Ξππππ π ππ΄ + π
π ππ = 0
Ξπ΄ππ
ππ₯π
β Ξπ΄ππ
ππ₯π€
+ ππ = 0
Ξππ΄πππ
ππ₯π
β Ξπ€π΄π€ππ
ππ₯π€
+ Su + SpπP = 0
Linear approximation
Diffusion problems in 1D
P EWew
DxdxPe
dxPEdxWP
Ξππ΄πππ
ππ₯π
β Ξπ€π΄π€ππ
ππ₯π€
+ Su + SpπP = 0
ππΞππ΄ππΏπ₯ππΈ
+Ξπ€π΄π€πΏπ₯ππ
β ππ = ππΞπ€π΄π€πΏπ₯ππ
+ ππΈΞππ΄ππΏπ₯ππΈ
+ ππ’
Ξπ =Ξπ+Ξπ
2, Ξπ€ =
ΞπΈ+Ξπ
2, ππ
ππ₯ π=ππΈβππ
πΏπ₯ππΈ, ππ
ππ₯ π€=ππβππ
πΏπ₯ππ
These are usually stored
ππππ = ππππ + ππΈππΈ + ππ’
Boundaries
P Eew
DxdxPe
dxPE
Case 1: is knownππ€ = πΉπ€
Ξππ΄πππ
ππ₯π
β Ξπ€π΄π€ππ
ππ₯π€
+ Su + SpπP = 0
ππ
ππ₯π€
=ππ β πΉπ€πΏπ₯π€π
Case 2: is known
Eastern boundary similarly
ππ
ππ₯π€
= πΊπ€
ππππ = ππΈππΈ + ππ’
Building the system of equations
1 2 N
ππππ = ππππ + ππΈππΈ + ππ’
π1π1,1 = π2π2,1 + π1
3
Building the system of equations
1 2 N
ππππ = ππππ + ππΈππΈ + ππ’
π1π1,1 = π2π2,1 + π1
π2π2,2 = π1π1,2 + π3π3,2 + π2
3
Building the system of equations
1 2 N
ππππ = ππππ + ππΈππΈ + ππ’
π1π1,1 = π2π2,1 + π1
π2π2,2 = π1π1,2 + π3π3,2 + π2
3
π3π3,3 = π2π2,3 + π4π4,3 + π3
Building the system of equations
1 2 N
ππππ = ππππ + ππΈππΈ + ππ’
π1π1,1 = π2π2,1 + π1
π2π2,2 = π1π1,2 + π3π3,2 + π2
3
π3π3,3 = π2π2,3 + π4π4,3 + π3
ππππ,π = ππβ1ππβ1,π + ππ
Convection-Diffusion problems in 1D
πππ£ ππ’ π = πππ£ Ξππππ π
π
πππ£ ππ’ π ππ = π
πππ£ Ξππππ π ππ
π΄
π. πππ’ ππ΄ = π΄
π. Ξππππ π ππ΄
P EWew
DxdxPe
dxPEdxWP
π’ππ’π€
(source terms not considered for simplicity)
ππ’π΄π π β ππ’π΄π π€ = Ξπ΄ππ
ππ₯π
β Ξπ΄ππ
ππ₯π€
Assume Ae=Aw=A and denote fluxes as:
πΉπ€ = ππ’ π€ πΉπ = ππ’ π π·π€ =Ξπ€πΏπ₯ππ
π·π=ΞππΏπ₯ππΈ
πΉπππ β πΉπ€ππ€ = π·π ππΈ β ππ β π·π€(ππ β ππ)
Convection-Diffusion problems in 1D
π(ππ’)
ππ₯= 0
The continuity equation must be also fullfilled:
P EWew
DxdxPe
dxPEdxWP
π’ππ’π€
ππ’π΄ π β ππ’π΄ π€ = 0
πΉπ β πΉπ€ = 0
Convection-Diffusion problems in 1D
P EWew
DxdxPe
dxPEdxWP
π’ππ’π€
πΉπππ β πΉπ€ππ€= π·π ππΈ β ππ β π·π€(ππ β ππ)
How to estimate fe, fw?
Central difference scheme:ππ =
ππ + ππΈ2
ππ€ =ππ + ππ
2
π·π€ +πΉπ€2+ π·π β
πΉπ2ππ = π·π€ +
πΉπ€2ππ + π·π β
πΉπ2ππΈ
ππππ = ππππ + ππΈππΈ
Fig. 5.4. N=5, u=0.1 m/s, F=0.1, D=0.5 Fig.5.5. N=5, u=2.5 m/s, F=2.5, D=0.5
Fig.5.6. N=20, u=2.5 m/s, F=2.5, D=2.0
WHY?
Properties of discretization schemes
β’ Conservativeness
β Estimate the fluxes in a consistent manner!
Fig.5.7
Fig.5.8
Properties of discretization schemes
β’ Boundedness
β In the absence of sources, the value of a property
should be bounded by its boundary values
β Requirements:
Β» All coefficients the same sign
Β»β πππ
ππβ²β€ 1 ππ‘ πππ πππππ , < 1 ππ‘ πππ ππππ ππ‘ ππππ π‘
Properties of discretization schemes
β’ Transportiveness
β Where is the information transported?
Fig.5.9
Assessment of the central differencing
scheme
β’ Conservativeness: OK
β’ Boundedness:
ae<0 for Pee=Fe/De>2 !!!
β’ Transportiveness: Not OK!
β’ Accuracy: 2nd order
π·π€ +πΉπ€2+ π·π β
πΉπ2ππ = π·π€ +
πΉπ€2ππ + π·π β
πΉπ2ππΈ
Upwind differencing scheme
β’ Compute the convective term depending on the flow
direction:
P EWew
DxdxPe
dxPEdxWP
π’ππ’π€
ππ = ππ ππ€= ππ
P EWew
DxdxPe
dxPEdxWP
π’ππ’π€
ππ = ππΈ ππ€= ππ
Upwind differencing scheme
β’ Assessment:
β Conservativeness: OK
β Boundedness: OK
β Transportiveness: OK
β Accuracy: 1st order
Fig.5.15
Hybrid differencing scheme
β’ Combine schemes:
β Central for Pe=F/D < 2
β Upwind for Pe > 2
β’ Assessment:
β Conservativeness: OK
β Boundedness: OK
β Transportiveness: OK
β Accuracy: 1st order
Power-law scheme
β’ More accurate than hybrid
β’ Diffusion set to 0 for Pe>10
β’ For Pe < 10 polynomial expression is used to evaluate
the fluxes
The QUICK scheme
β’ Quadratic Upstream Interpolation for Convective Kinetics
β’ Higher order & Upwind
β’ Face values of f obtained from quadratic functions
β’ Diffusion terms can be evaluated from the gradient
of the parabola
β’ For uw>0
β’ For ue>0
ππ€ =6
8ππ +
3
8ππ β
1
8πππ
ππ =6
8ππ +
3
8ππΈ β
1
8ππ
Fig.5.17
The QUICK scheme
β’ Generic form:
β’ Issues at boundaries: no second neighbours
β Create virtual βmirror nodesβ by linear extrapolation
ππππ = ππππ + ππΈππΈ + ππππππ + ππΈπΈππΈπΈ
Fig.5.18
The QUICK scheme
β’ Assessment
β Conservativeness: OK
β Boundedness:
Β» Only conditionnaly stable!
β Reformulated versions
to improve stability
β Transportiveness: OK
β Accuracy: better formal
accuracy than upwind
β Possible over/undershoots
Fig.5.20
General upwind-biased schemes
β’ βPureβ upwind:
β’ Linear upwind differencing:
β’ QUICK:
β’ Central differencing:
β’ General:
ππ = ππ
ππ = ππ +1
2(ππ β ππ)
ππ = ππ +1
8(3ππΈ β 2ππ β ππ)
ππ = ππ +1
2(ππΈ β ππ)
ππ = ππ +1
2π(ππΈ β ππ)
General upwind-biased schemes
β’ UD
β’ CD
β’ LUD
β’ QUICKππ = ππ +
1
2π(ππΈ β ππ)
π = π(π)
π =ππ β ππππΈ β ππ
π π = 0
π π = 1
π π = π
π π = (3 + π)/4
Fig.5.21
TVD schemes
β’ Total Variation Diminishing
β’ Criteria:
β Upper limit for TVD:
f1
f2
f3
f4
f5
ππ π= π2 β π1 + π3 β π2+ π4 β π3 + |π5 β π4|
Fig.5.23
π π β€ 2π π β€ 1
π π β€ 2 π > 1
TVD
β’ For second order:
β Must go through (1,1)
β Limited by CD and LUD
β’ To treat forward and
backward differencing
consistently: symmetry
property:
β’ Flux limiters
π π
r= π(1/π)
Fig.5.24
Fig.5.25
Unsteady flows
πππ
ππ‘+ πππ£ ππ’ π = πππ£ Ξππππ π + π
π
πππ
ππ‘ππ +
π
πππ£ ππ’ π ππ = π
πππ£ Ξππππ π ππ + π
π ππ
π‘
π‘+Ξπ‘
π
πππ
ππ‘ππππ‘ +
π‘
π‘+Ξπ‘
π
πππ£ ππ’ π ππ ππ‘
= π‘
π‘+Ξπ‘
π
πππ£ Ξππππ π ππ ππ‘ + π‘
π‘+Ξπ‘
π
π ππππ‘
Unsteady 1D heat conduction
β’ c β specific heat [J/(kg K)]
ππππ
ππ‘=π
ππ₯πππ
ππ₯+ π
P EWew
DxdxPe
dxPEdxWP
π‘
π‘+Ξπ‘
π
ππππ
ππ‘ππππ‘ =
π‘
π‘+Ξπ‘
π
π
ππ₯πππ
ππ₯ππππ‘ +
π‘
π‘+Ξπ‘
π
πππππ‘
π
π‘
π‘+Ξπ‘
ππππ
ππ‘ππ‘ππ =
π‘
π‘+Ξπ‘
ππ΄ππ
ππ₯π
β ππ΄ππ
ππ₯π€
ππ‘ + π‘
π‘+Ξπ‘
πΞπππ‘
Unsteady 1D heat conduction
π
π‘
π‘+Ξπ‘
ππππ
ππ‘ππ‘ππ =
π‘
π‘+Ξπ‘
ππ΄ππ
ππ₯π
β ππ΄ππ
ππ₯π€
ππ‘ + π‘
π‘+Ξπ‘
πΞπππ‘
ππ ππ β ππ0 ΞV =
π‘
π‘+Ξπ‘
ππ΄ππΈ β πππΏπ₯ππΈ
β ππ΄ππ β πππΏπ₯ππ
ππ‘ + π‘
π‘+Ξπ‘
πΞπππ‘
Assume uniform T
In control volume
and use backward
differencing
Use central
differencing
How do TE,TW and TP vary in time? Assume
a linear function:@ t+Dt @ t
πΌπ = π‘
π‘+Ξπ‘
πππ‘ = ππ + 1 β π π0 Ξπ‘
Unsteady 1D heat conduction
ππ ππ β ππ0 ΞV =
π‘
π‘+Ξπ‘
ππ΄ππΈ β πππΏπ₯ππΈ
β ππ΄ππ β πππΏπ₯ππ
ππ‘ + π‘
π‘+Ξπ‘
πΞπππ‘
Divide by A and Dt:
ππ ππ β ππ0Ξπ₯
Ξπ‘
= π πππΈ β πππΏπ₯ππΈ
β πππ β πππΏπ₯ππ
+ (1 β π) πππΈ0 β ππ
0
πΏπ₯ππΈβ π
ππ0 β ππ
0
πΏπ₯ππ+ πΞπ₯
ππππ= ππ πππ + 1 β π ππ
0 + ππΈ πππΈ + 1 β π ππΈ0
+ ππ0 β 1 β π ππ β 1 β π ππΈ ππ
0 + π
Explicit scheme
π = 0
ππππ = ππππ0 + ππΈππΈ
0 + ππ0 β ππ β ππΈ + ππ ππ
0 + ππ’
TP can be directly computed
First order
Can be negative! To assure stability:
Ξπ‘ < ππΞπ₯ 2
2π
Very small timesteps when the grid is refined!
Implicit scheme
π = 1
ππππ = ππππ + ππΈππΈ + ππ0ππ0 + ππ’
Several unknowns -> System of equations
Unconditionally stable
First order
Crank-Nicolson scheme
π =1
2
ππππ = ππππ + ππ
0
2+ ππΈ
ππΈ + ππΈ0
2+ ππ
0 βππ2βππΈ2+ππ2ππ0 + ππ’
System of equations
To assure stability:
Ξπ‘ < ππΞπ₯ 2
π
Very small timesteps when the grid is refined!
Second order
Unsteady 1D convection-diffusion
β’ Similarly to unsteady diffusion
β’ Additional stability limits due to convection
πππ
ππ‘+πππ’π
ππ₯=π
ππ₯Ξππ
ππ₯+ π
Finite volume method for 2D Cartesian
grids
β’ E.g. diffusion problem:
Fig.4.12
π
ππ₯Ξππ
ππ₯+π
ππ¦Ξππ
ππ¦+ π = 0
Ξπ
π
ππ₯Ξππ
ππ₯ππ +
Ξπ
π
ππ¦Ξππ
ππ¦ππ
+ Ξπ
πππ = 0
Ξππ΄πππΈ β πππΏπ₯ππΈ
β Ξπ€π΄π€ππ β πππΏπ₯ππ
+ Ξππ΄πππ β πππΏπ¦ππ
β Ξπ π΄π ππ β πππΏπ¦ππ
+ πΞπ = 0
ππππ = ππππ + ππΈππΈ + ππππ + ππππ + ππ’ = 0
Finite Volume Method on unstructured
grids
P
ni
DA
π
πππ
ππ‘ππ +
π
πππ£ ππ’ π ππ
= π
πππ£ Ξππππ π ππ + π
π ππ
π
ππ‘
π
ππ ππ + π΄
π. ππ’ π ππ΄ = π΄
π. Ξππππ π ππ΄ + π
π ππ
π
ππ‘
π
ππππ + β Ξπ΄π
ππ . ππ’ π ππ΄ = β Ξπ΄π
ππ . Ξππππ π ππ΄ + π
π ππ
β’ ni can be computed based on grid topology
Diffusion term
β’ Central differencing
β’ Only true if the grid is
orthogonal
Ξπ΄π
ππ . Ξππππ π ππ΄ β
ππ . Ξππππ π Ξπ΄π β
Ξππ΄ β ππΞπ
Ξπ΄π
Fig.11.15
Non-orthogonal grids
β’ Can be computed as:
n. gradπΞπ΄π
=π. πΞπ΄ππ. ππ
ππ΄ β ππΞπ
βππ . ππ Ξπ΄π
π. ππ
ππ β ππΞπ
Fig.11.17
Non-orthogonal grids
β’ Compare to:
n. gradπ Ξπ΄π =π. πΞπ΄ππ. ππ
ππ΄ β ππΞπ
βππ . ππ Ξπ΄π
π. ππ
ππ β ππΞπ
Direct gradient Cross-diffusion
Known Interpolated
Can be (pre)computed (and stored) based on grid topology.
n. gradπ Ξπ΄π =ππ΄ β ππΞπ
Ξπ΄π
Convective term
β’ Approximate as:
Ξπ΄π
ππ . πππ’ ππ΄ β
ππ Ξπ΄π
ππ . ππ’ ππ΄ β πππΉπ
Fi = Convective flux
normal to the surface
element.
Usually stored.Fig.11.15
fi
β’ Upwind differencing
β Fi>0 fi=fP
β Fi<0 fi=fA
β’ Linear upwind differencing
β’ QUICK
β’ TVD
β’ β¦
ππ = ππ + π»ππ . Ξπ
Needs to be computed!