Computational Fluids Mechanics -...
Transcript of Computational Fluids Mechanics -...
Computational Fluids Mechanics
Part 1: Numerical methodsPart 2: Turbulence modelsPart 3: Practice of numerical simulation
N.Oshima A5-32 [email protected] A5-33 [email protected] of Mechanical and Space EngineeringComputational Fluid Mechanics Laboratoryhttp://www.eng.hokudai.ac.jp/labo/fluid/index_e.html
Computational Fluid MechanicsPart1: Numerical Methods
Objective:Numerical methods for fluids mechanics and their
accuracy
1. Introduction2. Numerical methods for fluid mechanics (1) ~ Basic equations3. Numerical methods for fluid mechanics (2) ~Discretizing schemes4. Numerical methods for fluid mechanics (3) ~Coupling algorism 5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulationSummary
Prof. Oshima A5-32 [email protected]
Computational Fluid Mechanics
Part1: Numerical Methods (3a)1. Introduction2. Numerical methods for fluid mechanics (1) ~ governing equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes• Fundamentals for discretizing methods• Schemes for convection and diffusion equation• Analysis of accuracy and instability
4. Numerical methods for fluid mechanics (3) ~Coupling algorism5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulation
St
ju
SxJ
xu
t j
j
jj
jj x
J
j
SzJ
yJ
xJ
zw
xv
xu
tzyx
zzx
JJJ zyxt j
Fundamentals for discretizing methodsGeneralized conservation eq.
xf 0' xf
0x 1x 2x
(1) (2)
3x
(3)
xx2
x3
xgdxdf
Differential eq.
x
xfxfx
xfxfx
xfxfxx
xfxfxfxx
3
2
lim'
03
02
01
0
00
0
001
01 xgxx
xfxf
(1)
(2)
(3)
Polynomial eq.
Fundamentals for discretizing methods
Finite Different Method
Taylor series expansionTaylor series expansion
iiiiii fxfxfxxfff ""241'''
61"
21' 432
1
iiiiik
kk fxfxfxxff
xfx
kf ""
241'''
61"
21'
!1 432
1st order forward scheme
iiiiii fxfxxff
xff ""
241'''
61"
21' 321
truncation error
iiiiii fxfxxff
xff ""
241'''
61"
21' 321
1st order backward scheme
iiiiii fxfxfxxfff ""241'''
61"
21' 432
1
'''31 '2 3
11 iiii fxxfff
iiiiii fxfxfxxfff ""241'''
61"
21' 432
1
'''
61'
2211
iiii fxf
xff
1st derivative approx. truncation error
iiiiii fxfxfxxfff ""241'''
61"
21' 432
1
iiiii fxfxfff ""121 " 2 42
11
iiiiii fxfxfxxfff ""241'''
61"
21' 432
1 +
iiiii fxf
xfff ""
121 "2 2
211
2nd order central scheme
2nd derivative approx. truncation error
iiiii
iiiii
ii
fxfxfxff
fxfxfxff
ff
'''261"2
21'2
'''61"
21'
322
321
Exercisex
ffff iiii 2
43' 21
x
x
x
21
24
23
ii
ii
ii
iiiii
fxfxx
xx
fxfxx
xx
ffxx
xx
ffxxxx
fff
'"31'"
62
21
624
"0"2
221
224
'1'221
24
021
24
23
243
233
22
21
Approx.
Truncationerror
Look here!
2nd orderaccuracy
Taylor series expansion
Polynomialfactors
Fundamentals for discretizing methodsNumerical integration
(for Finite Element Method / Time marching scheme)
2x 1x 0x 1x
Differential eq.
Numerical integration of (g)
(1)
(2)
(3)
xfxgdxdf ,
x
x
xx dxxgxfxff
00''0
xg
Integral eq.
xxgdxxg
xfxfx
x
0
01
1
0
''
xxgxg
210 (1)
xxgxg
23 10
(3)(2)
6"
2'
3
0
2
000
xgxgxggdxx
+
62
""22
''2
301
210
10xggxggxgg
X1/2
X1/2(2)
(1)
dxgxgxxgggdx iiii
'''
61"
21' 32
6"
2'
3
1
2
11
0 xgxgxggdxx
Error for averaged value
12"
6"
4"
333 xgxgxg
0x 1x
1
0
x
xgdx(1) (2)
x
Trapezoidal scheme
21 101
0
gggdxx
gx
x
12
"2xg
estimated error
1st order approx.
2nd order approx.
estimated error
0x 1xx
gdx0
(1)
1x x
x
xgdx
2(2)
6"
2'
3
0
2
000
xgxgxggdxx
68"
23'
3
1
2
11
2 xgxgxggdxx
x
(1)
(2)
Adams-Bashforth scheme
310210
10 12"8"3
4''3
21
23 xggxggxgg
X (3/2)
X (-1/2)+
Estimated error3
"12
5"4
3"333 xgxgxg
Exercise
Flux type differential eq Integral eq.
Backwardapproximation
212
101
00
xJxJxJ
xJxJxJ
xJxJ
xxgxJxJxxgxJxJ
111
000
xJ
0x 1x
xg
2x
0x 0x1x 1x
2x
x
xdxxgxJxJ '' xgxJ
dxd
Midway-roleintegration
Fundamentals for discretizing methodsFinite Volume Method
Central interpolation is also available !
Finite Volume Method for generalized conservation eq.
xfxgxfxJx
,,
St
ju
dvSdndvt
ju
xxJ g
t
Sg
x
juxJ
(1D case)
x
xdxxgxJxJ ''
t
Sxxg
juxxJ xx
,
,
Sjuxt xx
Differential form
Integration form
+
"41
2210 fxfff
X1/2
X1/2
"
221'
2
2
0 fxfxff
"
221'
2
2
1 fxfxff
+
"
43
23 210 fxfff
X3/2
X-1/2
"
221'
2
2
0 fxfxff
・・・2nd orderaccuracy
"
23
21'
23 2
1 fxfxff
central interpolation scheme
backward approximation scheme
extrapolation scheme
・・・2nd orderaccuracy
Estimated error
Estimated error
Estimation of accuracy for Finite Volume Method
24"
3
00
2/
2/
xgxxfggdxx
x
""42
121 2
0110 JJxJJJJJJ
xxfgJJJJ 00110 21
21
''24
'"4
33
gxJx
Central interpolation
Midway roleintegration
Discretized eq. Truncation error
Exercise
For averaged value of volume x estimated error in order of (x)2 !!
t
Sgdxgx
xx1
Computational Fluid Mechanics
Part1: Numerical Methods (3b)1. Introduction2. Numerical methods for fluid mechanics (1) ~ governing equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes• Fundamentals for discretizing methods• Schemes for convection and diffusion equation• Analysis of accuracy and instability
4. Numerical methods for fluid mechanics (3) ~Coupling algorism5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulation
• Convection term
xu
xu ii
i
211
2nd order central
difference scheme
Schemes for convection and diffusion equation Finite Deference Method
1st order UPWINDdifference scheme
)0(
)0(
1
1
iii
i
iii
i
ux
u
ux
u
xu
St
ju
xi-2 xi-1 xi xi+1 xi+2
yj-1
yj
Yj+1
02
34
02
43
1
21
iiii
i
iiii
i
ux
u
ux
u
xu
2nd order UPWINDdifference scheme
yv
yv jj
j
211
)0(
1
j
jjj
vy
vy
v
• Diffusion term
yJJ
xJJ
yJ
xJ jjiiyx
2nd order centraldifference scheme
St
ju
xi-2 xi-1 xi xi+1 xi+2
yj-1
yj
Yj+1
j
xJ
xJ ii
iii
i
11 ,
211
211 22
yxjjjiii
211
211
2
2
2
22 22
yxyxjjjiii
Schemes for convection and diffusion equationFinite Deference Method (continued)
Schemes for convection and diffusion equationFinite Volume (control volume) Method
xi-1 xi xi+1
W P Ew e
xJuJuyJuJuyxSt sssnnnwwweee
wdAdVS
tnju
dxdyyx
n
s
xJ ii
w
1
211
211 22
yxdxJJdyJJ jjjiii
snwe
yxt
dxdyt
yxt
w xw dyJyJ
SdxdySS
Diffusion term ~ 2nd order accuracy
Source term ~ 2nd order accuracy
4
2
22
1 081
21 x
xxiiw
3
2
22
21 0241 63
81 x
xuxiiiw QUICK scheme
Schemes for convection and diffusion equationFinite Volume (control volume) Method
xi-1 xi xi+1
W P E
xJuJuyJuJuyxSt sssnnnwwweee
wdAdVS
tnju
w xw dyJ
yJ
1
dxdyyx
n
s
2nd Central scheme
w e
)0()0(1
ii
iiw u
u
1st Upwid scheme Convection term
Exercise Analyze the accuracy of FVM schemes by following way
www fxxff "2
'2
www
x
ixxxdxx "6
'2
32
0
2x -x xw=0 x
i-2 i-1 i
n
s
ww w e
wwwxixxxdxx "6
'2
320
1
www
x
xixxxdxx "
67'
23 32
22
(1)(2)(3)(1)
(2)
(3)
Time marching schemes
• (2nd order) leap-frog
• 2nd order backward scheme
ftu
kkk
ftuu
2
11
111
243
kkkk
ft
uuu
(u k)uk+1
f k
f k+1
u kuk+1
kff 1 kff (implicit scheme)
(explicit scheme)
u k-1
u k-1
Time marching schemes
• 2nd order Adams-Bashforth
• (2nd order) Crank-Nicolson
ftu
ft
uu kk
1
11
321
kk
kk
fft
uu
kkkk
fft
uu
1
1
21
f k-1
u kuk+1
f k
f k+1
u kuk+1
f k
•mid-point rule
• predictor (Euler) -corrector (Crank-Nicolson)
• 4th order Runge-Kutta
kk
uftuu
*
*1
ˆ uft
uu kk
u ku*
f k
u kuk+1
f *
***1*****1 226
ufufufuftuu kkkkk
kk uftuu2
*
****
2uftuu k
****** uftuu k
1)
4)
3)
2)
u*
f k
u**
f *
u*** ukuk+1
f k+1
1)2)
2) 1)
3)4)
4) 3)
Computational Fluid Mechanics
Part1: Numerical Methods (3c)1. Introduction2. Numerical methods for fluid mechanics (1) ~ governing equations
3. Numerical methods for fluid mechanics (2) ~Discretizing schemes• Fundamentals for discretizing methods• Schemes for convection and diffusion equation• Analysis of accuracy and instability
4. Numerical methods for fluid mechanics (3) ~Coupling algorism5. Numerical methods for fluid mechanics (4) ~Additional problems6. Reliability of numerical simulation
Numerical accuracy
• Steady C-D problem
2
2
xu
xuc
21111 2
2 xuuu
xuuc iiiii
•2nd central
2111 2
xuuu
xuuc iiiii
•1st upwind
xcPe
•2nd central •1st upwind
Numerical accuracy• Comparison between 1st Upwind and 2nd central schemes
2111 02
211 xuuu
xuu
x iiiii
2
2
2
1 02
1 xxuxuu
xxu
ii 1st upwind
2nd central
2
2st
2upwind) (1
xuxu
tu
eff
Upwind=Numerical diffusion
Numerical accuracy• Fourier expansion
• effective wave number
ikxkeau
ikx
k
ikx
k ikeax
eaxu
ikxxxikxxikikx
ex
xkixee
xe
sin
2
6
23 xkkkeff
Numerical instability
• von Neuman’s condition
) i.c.( ikxt eun
xuu
tu
xuuc
tuu n
ini
ni
ni
211
1
nni uxk
xtciu
sin11
2nd central
i+1 i i-1
tn+1
tn
c>0
xkc 222 sin1
(const.)
2
1 1max
2
22
xtif
tctc
(const.)
11max 22
xtif
c
ikx
t eun
Numerical instability
• von Neuman’s condition
) i.c.( ikxt eun
xuu
tu
xuuc
tuu n
ini
ni
ni
1
1
nxikni ue
xtc
xtcu
11
1st upwind
i+1 i i-1
tn+1
tn
c>0
CFL condition 1xtc
1
i
0
Neumann’s condition of diffusion step
2
2
xt
tinxn exp
211
1 2x
uuut
uu ni
nni
ni
ni i
•Euler (1st explicit)
•Crank-Ncolsin (2nd implicit)
... stable ?
2
111
111
11
222
xuuuuuu
tuu n
inn
ini
nni
ni
ni ii
?
Exercise