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2/5/13
1
ControlVolumeFiniteDifferenceMethods
BasicConcept:
1. DevelopmatrixequaBonsbasedonVolume/MassFluxesandVolume/MassconservaBonexpressionsratherthansecondorderPDEs.Forexample:
2.Youdon’tuseTaylorseriesperse.UsesimplefinitedifferencerepresentaBonsofthefluxtermsacrossthefacenormaltothecell.
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ddt
ρφdVV∫ = n ⋅ qρdA
A∫
2DExample
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δei, j =Δxi, j + Δxi, j+1
2δwi, j =
Δxi, j + Δxi, j−12
Ke =Ki,kΔxi, j + Ki, j+1Δxi, j+1
Δxi, j + Δxi, j+1Kw =
Ki,kΔxi, j + Ki, j−1Δxi, j−1Δxi, j + Δxi, j−1
€
ne = (1,0) nw = (−1,0)€
Δxi, jΔyi, jSs∂h∂t
=k= e,w,n,s∑ AkKk∇hk ⋅ nk
qwAw ⋅ nw = −KwΔyi, jhi, j − hi, j−1
δw
qeAe ⋅ ne = KeΔyi, jhi, j+1 − hi, j
δe
€
Vi, j = Δxi, jΔyi, j
Area=Δy
NodeseparaBondistance
nenw
2/5/13
2
2DExample,X‐DirecBon
€
Ss∂h∂t
=
Ke
hi, j+1 − hi, jΔxi, j + Δxi, j+1
2
−Kw
hi, j − hi, j−1Δxi, j + Δxi, j−1
2Δxi, j
€
SsΔt
hi, jk+1 − hi, j
k( ) =2Ki, j+1/ 2
Δxi, j Δxi, j + Δxi, j+1( )
h i , j+1
k+1
+2Ki, j+1/ 2
Δxi, jΔxi, j + Δxi, j+1+
2Ki, j−1/ 2
Δxi Δxi, j + Δxi,i−1( )
h
i , j
k+1 +2Ki, j−1/ 2
Δxi, j Δxi, j + Δxi,i−1( )
h
i , j−1
k+1