Computational Fluids Mechanics - 北海道大学cfml.eng.hokudai.ac.jp/wp-content/uploads/2017/09/...Computational Fluids Mechanics Part 1：Numerical methods Part 2：Turbulence models

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


61"

21' 432

1

'''31 '2 3

11 iiii fxxfff


61"

21' 432

1

'''

61'

2211

iiii fxf

xff

1st derivative approx. truncation error


61"

21' 432

1

iiiii fxfxfff ""121 " 2 42

11


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


Part1： Numerical Methods (3b)1. Introduction2. Numerical methods for fluid mechanics (1) ~ governing equations



• 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)


Part1： Numerical Methods (3ｃ)1. Introduction2. Numerical methods for fluid mechanics (1) ~ governing equations



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

Documents

Computational Fluids Mechanics - 北海道大学cfml.eng.hokudai.ac.jp/wp-content/uploads/2017/09/...Computational Fluids Mechanics Part 1：Numerical methods Part 2：Turbulence models