Chapter 8 - FVM for Transient Problems
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Transcript of Chapter 8 - FVM for Transient Problems
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Chapter 8: The Finite Volume Method for Transient Problems Presented by: Prof. Ir. Dr. Shahrir Abdullah Dr. Wan Mohd Faizal Wan Mahmood Dept. of Mechanical & Materials Engineering Universiti Kebangsaan Malaysia
KKKJ4164 COMPUTATIONAL FLUID DYNAMICS
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FVM for Transient Problems Transient 1-D Diffusion Problems Implicit Method for 2-D and 3-D Problems Transient Convective-Diffusion Problems
Contents
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FVM for Transient Problems • The equation for the unsteady convection-diffusion problems is:
( ) ( ) ( ) φφφρρφ St
+∇Γ⋅∇=⋅∇+∂∂ u (1)
• Integrating throughout a finite volume produces:
( ) ( )
( ) dtdVSdtdA
dtdAdtdVt
tt
t
tt
t
tt
t
tt
t
∫ ∫∫ ∫
∫ ∫∫ ∫∆+∆+
∆+∆+
+
∇Γ⋅=
⋅+
∂∂
CVCS
CSCV
φφ
φρρφ
n
un
(2)
4
FVM for Transient Problems or, in alternative form as
( ) ( )
( ) dtdVSdtdA
dtdAdVdtt
tt
t
tt
t
tt
t
tt
t
∫ ∫∫ ∫
∫ ∫∫ ∫∆+∆+
∆+∆+
+
∇Γ⋅=
⋅+
∂∂
CVCS
CSCV
φφ
φρρφ
n
un
• For 1D cases without generation/source term, the equation reduces to:
( ) ( )
∂∂
Γ∂∂
=∂∂
+∂∂
xxu
xtφφρρφ
In addition, a 1D flow also has to follow the flow continuity principle:
( ) 0=∂∂
+∂∂ u
xtρρ
5
Transient 1-D Diffusion Problems
• An unsteady 1-D diffusion problem, e.g. heat conduction, may be modelled using the following equation:
SxTk
xtTc +
∂∂
∂∂
=∂∂ρ (3)
Control Volume around Node P
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Transient 1-D Diffusion Problems • Integration throughout a finite volume produces:
dtdVSdtdVxTk
xdtdV
tTc
tt
t
tt
t
tt
t∫ ∫∫ ∫∫ ∫∆+∆+∆+
+
∂∂
∂∂
=∂∂
CVCVCV
ρ (4)
or, in alternative form as
∫∫∫ ∫∆+∆+∆+
∆+
∂∂
−
∂∂
=
∂∂ tt
t
tt
t we
e
w
tt
t
dtVSdtxTkA
xTkAdVdt
tTcρ (5)
• The left-hand side can be written as:
( ) VTTcdVdttTc PP
tt
t
∆−=∂∂
∫ ∫∆+
0
CV
ρρ (6)
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Transient 1-D Diffusion Problems • Hence, Eq. (5) becomes:
( ) ∫∫∆+∆+
∆+
−−
−=∆−
tt
t
tt
t WP
WPw
PE
PEePP dtVSdt
xTTAk
xTTAkVTTc
δδρ 0 (7)
• The integration of the right hand side term is arranged to have a weighting parameter θ:
( )[ ] tTTdtTI PP
tt
tPT ∆−+== ∫
∆+01 θθ (8)
where
θ = 0 : tTI PT ∆= 0
θ = ½ : ( ) tTTI PPT ∆+= 021
θ = 1 : tTI PT ∆=
8
Transient 1-D Diffusion Problems • Eq. (7) can be rearranged to be:
( )[ ] ( )[ ]
( ) ( ) xSTxk
xk
txc
TTxkTT
xk
Txk
xk
txc
PWP
w
PE
e
WWWP
wEE
PE
e
PWP
w
PE
e
∆+
−−−−
∆∆
+
−++−+=
++
∆∆
0
00
11
11
δθ
δθρ
θθδ
θθδ
δδθρ
(9)
• Hence, the general equation is:
( )[ ] ( )[ ]
( ) ( )[ ] bTaaaTTaTTaTa
PEWP
EEEWWWPP
+−−−−+
−++−+=00
00
11
11
θθ
θθθθ (10)
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Transient 1-D Diffusion Problems
where ( ) 0PEWP aaaa ++= θ , t
xcaP ∆∆
= ρ0 and
aW aE b
WP
w
xk
δ PE
e
xk
δ xS ∆
• The type of scheme is dependent on the value of θ:
θ = 0 : Explicit scheme
0 < θ ≤ 1 : Implicit scheme
θ = ½ : Crank-Nicolson scheme
θ = 1 : Fully implicit scheme
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Transient 1-D Diffusion Problems Explicit Scheme
• The source term is linearised to become 0PPu TSSb += .
• By taking θ = 0, thus
( )[ ] uPPEWPEEWWPP STSaaaTaTaTa +−+−++= 0000 (11)
where 0PP aa = , t
xcaP ∆∆
= ρ0 and
aW aE
WP
w
xk
δ PE
e
xk
δ
11
Transient 1-D Diffusion Problems • This scheme needs the following condition to produce a stable and
oscillation-free solution as mentioned in Chapter 5, i.e.:
( )
kxct
xk
txc
2or2 2∆
<∆∆
>∆∆ ρρ (12)
Crank-Nicolson Scheme
• By taking θ = ½, thus
bTaaaTTaTTaTa PEW
PEE
EWW
WPP +
−−+
++
+= 00
00
2222 (13)
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Transient 1-D Diffusion Problems
where ( ) pPEWP Saaaa 210
21 −++= , t
xcaP ∆∆
= ρ0 and
aW aE b
WP
w
xk
δ PE
e
xk
δ 021
Ppu TSS +
• This scheme needs the following condition to produce a stable and
oscillation-free solution as mentioned in Chapter 5, i.e.:
( )
kxct
2∆<∆ ρ (14)
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Transient 1-D Diffusion Problems Fully Implicit Scheme
• By taking θ = 1, thus
uPPEEWWPP STaTaTaTa +++= 00 (15)
where pEWPP Saaaa −++= 0 , txcaP ∆
∆= ρ0 and
aW aE
WP
w
xk
δ PE
e
xk
δ
• This scheme is always stable.
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Transient 1-D Diffusion Problems Example 1
A thin plate having a thickness of L = 2 cm and a uniform initial temperature of 200°C. At time t = 0, the temperature at its left side drops to 0°C instantly, whereas the other surface is insulated. By using a grid of 5 nodes and appropriate timestep, use an explicit scheme to obtain the following times:
(a) t = 40 s, (b) t = 80 s, (c) t = 120 s.
and compare each case with the analytical solutions. Repeat the question using a timestep sufficient to fulfil the requirement for stability at t = 40 s. Given that the coefficient for heat conductance k = 10 W/m⋅K and ρc = 10 × 106 J/m3⋅K.
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Transient 1-D Diffusion Problems
The analytical solution for this problems is
( ) ( ) ( ) ( )c
kL
nxtn
txTn
nnn
n
ραπλλαλ
π=
−=−
−−
= ∑∞
=
+
212cosexp
1214
200),(
1
21
16
Transient 1-D Diffusion Problems
Solution
The governing equation for this problems is:
∂∂
∂∂
=∂∂
xTk
xtTcρ
Thus, the general equation is:
( )[ ] uPEWPEEWWPP STaaaTaTaTa +−−++= 0000
where txcaa PP ∆
∆== ρ0 and
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Transient 1-D Diffusion Problems
Nod aW aE Su
1 0 k/∆x 0
2, 3, 4 k/∆x k/∆x 0
5 k/∆x 0 ( )02BB TT
xk
−∆
Determination of time step:
( ) ( )( ) s8102
004.010102
262
=×
=∆
<∆kxct ρ
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Transient 1-D Diffusion Problems For ∆t = 2s:
200002004.010102500
004.010 6 =×=
∆∆
==∆ t
xcx
k ρ
Hence, the discreet equations are:
Nod 1: 00 17525200 PEP TTT += Nod 2, 3, 4: 000 1502525200 PEWP TTTT ++= Nod 5: 00 12525200 PWP TTT +=
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Transient 1-D Diffusion Problems
20
Transient 1-D Diffusion Problems
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Transient 1-D Diffusion Problems
22
Transient 1-D Diffusion Problems
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Transient 1-D Diffusion Problems
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Transient 1-D Diffusion Problems Example 2
Repeat Example 7.1 using the fully implicit scheme and compare it with the explicit scheme and the implicit scheme with the stime step of 8 s.
Solution
The general equation for the implicit is:
( )[ ] uPEWPEEWWPP STaaaTaTaTa +−−++= 00
where pPEWP Saaaa −++= 0 , txcaP ∆
∆= ρ0 and
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Transient 1-D Diffusion Problems
Nod aW aE Sp Su
1 0 k/∆x 0 0
2, 3, 4 k/∆x k/∆x 0 0
5 k/∆x 0 xk
∆−
2 BT
xk
∆2
For ∆t = 2s:
200002004.01010
22500
004.010 6 =×=
∆==
∆ kxc
xk ρ
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Transient 1-D Diffusion Problems Hence, the discreet equations are:
Nod 1: 020025225 PEP TTT += Nod 2, 3, 4: 02002525250 PEWP TTTT ++= Nod 5: BPWP TTTT 5020025275 0 ++=
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Transient 1-D Diffusion Problems
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Transient 1-D Diffusion Problems
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Implicit Method for 2-D and 3-D Problems • For general CFD problems, the fully implicit scheme is the best
alternative.
• The 3-D equation for diffusion problem is:
Sz
kzy
kyx
kxt
c +
∂∂
∂∂
+
∂∂
∂∂
+
∂∂
∂∂
=∂∂ φφφφρ (16)
• Hence, the general equation is:
uPPTTBBNNSSEEWWPP Saaaaaaaa +++++++= 00φφφφφφφφ (17)
where pPTBNSEWP Saaaaaaaa −++++++= 0 , txcaP ∆
∆= ρ0 ,
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Implicit Method for 2-D and 3-D Problems aW aE aS aN aB aT
1D WP
ww
xAk
δ PE
ee
xAk
δ
2D WP
ww
xAk
δ PE
ee
xAk
δ SP
ss
yAk
δ PN
nn
yAk
δ
3D WP
ww
xAk
δ PE
ee
xAk
δ SP
ss
yAk
δ PN
nn
yAk
δ BP
bb
zAk
δ PT
tt
zAk
δ
∆V Aw = Ae An = As Ab = At
1D ∆x 1
2D ∆x ∆y ∆y ∆x
3D ∆x ∆y ∆z ∆y ∆z ∆x ∆z ∆x ∆y
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Transient Convective-Diffusion Problems • The typical equation of the unsteady convection-diffusion problems is:
( ) ( ) ( ) ( )
φφφφ
φρφρφρρφ
Szzyyxx
zw
yv
xu
t
+
∂∂
Γ∂∂
+
∂∂
Γ∂∂
+
∂∂
Γ∂∂
=
∂∂
+∂
∂+
∂∂
+∂∂
(18)
• By using the implicit scheme, the general equation is:
uPPTTBBNNSSEEWWPP Saaaaaaaa +++++++= 00φφφφφφφφ (19)
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Transient Convective-Diffusion Problems where
pPTBNSEWP SFaaaaaaaa −∆+++++++= 0 ,
txa PP ∆
∆= 00 ρ , Ppu SSVS φ+=∆ .
If the hybrid differencing scheme is used:
1D 2D 3D
aW
+ 0,
2,max w
wwF
DF
+ 0,
2,max w
wwF
DF
+ 0,
2,max w
wwF
DF
aE
−− 0,
2,max e
eeF
DF
−− 0,
2,max e
eeF
DF
−− 0,
2,max e
eeF
DF
33
Transient Convective-Diffusion Problems
aS
+ 0,
2,max s
ssF
DF
+ 0,
2,max s
ssF
DF
aN
−− 0,
2,max n
nnF
DF
−− 0,
2,max n
nnF
DF
aB
+ 0,
2,max b
bbF
DF
aT
−− 0,
2,max t
ttF
DF
∆F ( )we FF − ( ) ( )snwe FFFF −+− ( ) ( )( )bt
snwe
FFFFFF
−+−+−
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Transient Convective-Diffusion Problems
And, the formula for F and D are:
Face w e s n b t
F ( ) ww Auρ ( ) ee Auρ ( ) ss Avρ ( ) nn Avρ ( ) bb Awρ ( ) tt Awρ
D WP
ww
xA
δΓ
PE
ee
xA
δΓ
SP
ss
yA
δΓ
PN
nn
yA
δΓ
BP
bb
zA
δΓ
PT
tt
zA
δΓ
35
Transient Convective-Diffusion Problems Case Study
Consider a 1-D convection-diffusion problem with the boundary condition as followed:
Other data include L = 1.5 m, u = 2 m/s, ρ = 1.0 kg/m3, and Γ = 0.03 kg/m⋅s. Given that the source term is a time-dependent function as shown below for t > 0:
36
Transient Convective-Diffusion Problems
where a = −200, b = 100, x1 = 0.6, x2 = 0.2. Obtain the temperature distribution until it reaches the steady state condition using the explicit and hybrid schemes. Use appropriate time step and number of node.
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Thank You
Questions/Comments are welcomed