2ª aula
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Transcript of 2ª aula
2ª aulaEvolution Equation.
The Finite Volume Method.
Objective of the lecture1. The Students “mise à zero” as francophone say.2. To show that the conservation principle can be written on:
– Words (is a concept)– Integral equation form,– Differential form (can have analytical solution),– Algebraic form (used by mathematical models).
3. To recall methods to solve advection. Temporal discretisation, stability and numerical diffusion.
4. To tell about the need for initial and boundary conditions and about the difficulties to get them.
• Fluxes:– Advective: (Why the sign “-”)?
– Diffusive:
The Integral Equation
• Accumulation Rate:
vol
volcdtt
MonRateAccumulati
dAnucA
.
dAncA
.
)(. iosurfaceCV
SSdAnAnvdVt
Differential Evolution Equation
SinksSourcesoutFlowinFlowoRateAccumulati
)( kkjj
k SiSoxc
xdtdc
)( kkj
kjj
k PFxccu
xtc
The rate of accumulation is “minus” the divergence of the fluxes + (Souces-Sinks)
)( kkjjj
kjk SiSoxc
xxcu
tc
Or:
Finite Volume
)(. iosurfaceCV
SSdAnAnvdVt
The Finite (Control) Volume :1) isolates a portion of the space,2) systematises budgets’ computation
across its faces,3) Computes the rate of accumulation,4) Permits the computation of a property
rate of change.
SinksSourcesoutFlowinFlowoRateAccumulati
Computational grid
Assuming uniform concentration inside the volume one gets
txx
txx
ttxx
ttxx VolcVolc
t
xxt
xxttxx
ttxx VolcVolc
tx
tx
ttx
ttx VolcVolc
tcVolcVolcd
ttMonRateAccumulati
ttt
volvol
Finite Volume in a 1D case
tx
tx
ttx
ttx VolcVolc t
xxt
xxttxx
ttxx VolcVolc
t
xxt
xxttxx
ttxx VolcVolc
* 2/*
2/ xxxx cQUcdA ***
*2/
dA
xcc xxx
xx
* 2/*
2/. xxxxAcQUcdAdAnuc
***
*2/.
dA
xccdAnc xxx
xxA
In 1D case properties can change along one direction only.
Summing up
txx
txx
ttxx
ttxx VolcVolc
t
xxt
xxttxx
ttxx VolcVolc
tx
tx
ttx
ttx VolcVolc
xcc
Axcc
ACQCQtVolcVolc xxxxx
xxxxxxxxxxxx
tx
tx
ttx
ttx 2/2/2/2//
Shrinking the volume to zero
)( kkjjj
kjk PFxc
xxcu
tc
That is the 1D advection-diffusion equation for one property. In a 3D case, for a generic property “k” one would get:
That represents the conservation principle in one point
xcc
Axcc
ACQCQtVolcVolc xxxxx
xxxxxxxxxxxx
tx
tx
ttx
ttx 2/2/2/2//
xAVol And assuming that the volume is a parallelepiped that doesn’t change in time:
xc
xxuc
tc
Knowing that:
Algebraic form Requires hypothesis. The upwind formulation with Q>0
xcc
Axcc
AQcQctVolcVolc xxxxx
xxxxxxxxx
tx
tx
ttx
ttx 2/2/2/2//
0:
0:
0:
0:
*2/
**2/
*2/
**2/
*2/
**2/
*2/
**2/
xxxxxx
xxxxx
xxxxx
xxxxxx
ucc
ucc
ucc
ucc
xcc
Axcc
ACQCQtVolcVolc xxxxx
xxxxxxxxxxxx
tx
tx
ttx
ttx 2/2/2/2//
tx
tx
ttx
ttx VolcVolc t
xxt
xxttxx
ttxx VolcVolc
t
xxt
xxttxx
ttxx VolcVolc
Hypothesis impose stability conditions
tttttx
xxxxxxxxtx
ttx
xxxxx
xxxxxxxx
tx
ttx
xxxxxC
xtC
xt
xtuC
xt
xtuc
xccc
xCCu
tcc
xccA
xccACCAuxAcc
222
2
2/2/
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2
xcc
Axcc
ACQCQtVolcVolc xxxxx
xxxxxxxxxxxx
tx
tx
ttx
ttx 2/2/2/2//
Stability condition:
021 2
xt
xtu
ti
tii
ti
tti xxi
CfCeCdc
11
Average values on faces => Central Differences
*
*2/
**
2/
2
2
xxxxx
xxxxx
ccc
ccc
**2/2/
222
xcc
xcc
xcc
xcc xxxxxxxxxxt
xxttxx
Explicit Central Differences
tttttx
txxx
xx
txxx
xxt
xxxxtx
ttx
xxxxxC
xt
xtuC
xtC
xt
xtuc
xccA
xccACCAuxAcc
222
2/2/
21
Stability conditions:
021 2
xt
02
xt
xtu
Understanding the Central Differences
• Why are CD instable without diffusion? – Resp: They violate the transportive property of advection.
The computing point learns about the downstream property value through advection, which is physically impossible.
• Why can be stable with diffusion• Resp: Because diffusion transports information in any
direction. If the diffusive transport is stronger than advective, the process becomes physically correct.
More Questions• Can explicit central differrences be used on advection is dominant?
– Resp: No. In that case difusion transports upstream much less than advection transport downstream (Grid Reynolds number is large).
• If diffusion is dominant is better to use centra differences or upwind?– Central differences are better they have second order accuracy and introduce
less numerical diffusion.• What about an implicit algorithm? Would it be stable without diffusion?
– Resp: Yes. In implicit algorithms fluxes are computed using the new concentrations. If Advection would generate negative concentrations the leaving flux would become positive. Thos means that it is impossible to generate negative concentrations.
• Even in upwind? – Resp: In upwind case the concentration can become negative only if we remove
from a volume more than its content. But since what is leaving the volume is computed at the end of the time step negative values can not be generated.
Other methods for advection• Upwind: Assumes that the concentration at the face is equal
to the upstream concentration.• Central Differences: Assume the average between both sides. • What if a 2nd order polynom was considered (using 3
points)? One would get the QUICK: (Quadratic Upstream Interpolation for Convective Kinematics):
• It has 3rd order accuracy. It has increased stability problems next to boundaries.
• What is the best method?
18
118
386
281
83
186
21
21
0
0
iiiii
iiiii
CCCCQ
CCCCQ
Algorithm for implicit methods
tttxx
ttx
tt
ttxxxxx
ttxxx
tx
ttx
ttxxx
xx
ttxxx
xxtt
xxxtx
ttx
xxxCc
xtc
xt
xtuC
xt
xtu
xccc
xCCu
tcc
xccA
xccACCAuxAcc
222
2
2/2/
21
2
ti
tti
ttii
tti CCfCeCd
ii
111
itt
itt
iitt
i TICfCeCdii
11
1
The Semi-implicit method (Crank – Nicholson)
ti
tii
ti
tti
ttii
tti iiii
CfCeCdCfCeCd1111 2
1211
21
21
211
21
Initial and Boundary Conditions
• Initial conditions are less important in dissipative systems (high sinks).
• Boundary conditions are less important when Sources and/or sinks are important?
Ci
Ci-1
Ci+1
ti
tti
ttii
tti CCfCeCd
ii
111
Boundary conditions
• Diffusion:– Requires the knowledge of concentrations outside
the domain. If not known zero gradient is usually the best option
• Advection– When the flow enters the domain carries
information from outside that must be known.
How to know the external boundary condition?
• Gradients can be considered null if sources and sinks are important.
• Otherwise nested models are required!
)( kkjjj
kjk PFxc
xxcu
tc
23
¼°
Downscaling
I n v e s t i n g i n o u r c o m m o n f u t u r e .
Summary• The finite control volume helps us to think about fluxes. That is usually
good..• Explicit methods get instable when the amount removed from a volume is
higher than the volume contained inside. • Central differences assume linear concentration evolution between adjacent
finite volumes. They cannot respect the transportive property of advection.• The QUICK method assume a quadratic evolution between adjacent centres.
Requires three points and consequently cannot be used next to boundaries. It does not respect completely the transportive property and can generate instabilities. It is better if combined with upwind.
• The finite volume method puts into evidence the advantage of combining methods for advection.
• Initial and boundary conditions choice determine the quality of the results.