me 747 introduction to computational fluid dynamics ...chainarong.me.engr.tu.ac.th/documents/me747...

ME 747 Introduction to computational

fluid dynamics

By Chainarong Chaktranond

Lecture 5_2Introduction to solve engineering

problems with finite-difference method

2Chainarong Chaktranond, Thammasat University

Lecture schedule

5. Introduction to solve engineering problems with finite-difference method

- Taylor series expansion, Approximation of the second derivative, Initial condition and

Boundary conditions

6 – 7

4. Introduction to numerical methods

- Finite different method, Finite volume method, Finite element method, etc.

5

3. Overviews of governing equations for flow and heat transfer

-Elliptic, Parabolic and Hyperbolic equations

4

2. Introduction to Fortran programming

- Basic commands in Fortran programming

2 - 3

1.Overviews of computational fluid dynamics

- Overviews and importance of heat transfer in real applications1

TopicsSession


Contents

Iteration method

Gauss-Seidel Method

Successive Over Relaxation Method


Iterative method: Gauss-Seidel Method

[ ]{ } { }A X C=

1

, 100n ni i

a i sni

x xx

ε ε−−

= × <

Controlled error

11 1 12 2 13 3 1 1

21 1 22 2 23 3 2 2

1 22 2 23 3

m m

m m

m mm m m

a x a x a x a x ca x a x a x a x c

a x a x a x a x c

+ + + + =+ + + + =

+ + + + =

……

…



1 2 3

1 2 3

1 2 3

3 0.1 0.2 7.850.1 7 0.3 19.30.3 0.2 10 71.4

x x xx x xx x x

− − =+ − = −

− + =

Example

1st Step:2 3

1

1 32

1 23

7.85 0.1 0.23

19.3 0.1 0.37

71.4 0.3 0.210

x xx

x xx

x xx

+ +=

− − +=

− +=



2nd Step: Trail with x2 = x3 = 0

( ) ( )1

7.85 0.1 0 0.2 02.616666667

3x

+ += =

3rd Step: Trail with x3 = 0 and employ x1 from 1st step

( ) ( )2

19.3 0.1 2.6166667 0.3 02.79452381

7x

− − += = −

4th Step: Trail with employ x1 from 1st step and x2 from 2nd step

( ) ( )3

71.4 0.3 2.616666667 0.2 2.794523817.005609524

10x

− + −= =


5th step: Control error

1

, 100n ni i

a i sni

x xx

ε ε−−

= × <


1

0 2.616666667 100 1002.616666667xε−

= × =

( )( )2

0 2.79452381100 100

2.79452381xε− −

= × =−

2

0 7.005609524 100 1007.005609524xε−

= × =

6th step: Repeat from 1st to 5th step

program gaussseidel

parameter (n=20)

real*8 x1(0:n),x2(0:n),x3(0:n)

open(1000,file='Gauss-seidel',status='unknown')

x1(0) = 0.d0

x2(0) = 0.d0

x3(0) = 0.d0

error1 = 100.d0

error2 = 100.d0

error3 = 100.d0

write(1000,1001) i,x1(0),x2(0),x3(0),error1,error2,error3

do 10 i = 1,n

x1(i) = (7.85+0.1d0*x2(i-1)+0.2d0*x3(i-1))/3.d0

x2(i) = (-19.3d0-0.1d0*x1(i)+0.3d0*x3(i-1))/7.d0

x3(i) = (71.4d0-0.3d0*x1(i)+0.2d0*x2(i))/10.d0

error1 = dabs((x1(i)-x1(i-1))/x1(i))*100.d0



write(1000,1001) i,x1(i),x2(i),x3(i),error1,error2,error3

1001 format(i2,2x,3e15.7,2x,3e15.7)

10 continue

close(1000)

stop

0.00E+000.00E+000.00E+007.000000-2.5000003.00000011

0.00E+000.00E+000.00E+007.000000-2.5000003.00000010

0.00E+001.78E-141.48E-147.000000-2.5000003.0000009

5.08E-146.39E-133.42E-127.000000-2.5000003.0000008

5.80E-129.05E-114.01E-107.000000-2.5000003.0000007

6.97E-101.83E-086.44E-087.000000-2.5000003.0000006

1.62E-071.41E-061.18E-057.000000-2.5000003.0000005

1.01E-054.82E-041.05E-037.000000-2.5000003.0000004

4.16E-031.45E-023.16E-016.999999-2.4999883.0000323

7.60E-021.18E+011.25E+017.000291-2.4996252.9905562

1.00E+021.00E+021.00E+027.005610-2.7945242.6166671

1.00E+021.00E+021.00E+020.0000000.0000000.0000000

error3error2error1x3x2x1Step


Successive Over Relaxation method (SOR)

( )1new new oldi i ix x xλ λ= + −

1 2 3

1 2 3

1 2 3

3 0.1 0.2 7.850.1 7 0.3 19.30.3 0.2 10 71.4

x x xx x xx x x

− − =+ − = −

− + =

Example

1st Step:2 3

1

1 32

1 23

7.85 0.1 0.23

19.3 0.1 0.37

71.4 0.3 0.210

x xx

x xx

x xx

+ +=

− − +=

− +=

1 2λ< <where



2nd Step: Trail with x2 = x3 = 0

( ) ( )1

7.85 0.1 0 0.2 02.616666667

3x

+ += =

3rd Step: find new x1

( )( ) ( )( )1 1.0005 2.616666667 1 1.0005 0 2.617975x = + − =


1.0005λ =


4th Step: substitute x1 and Trail with x3 = 0

( ) ( )2

19.3 0.1 2.617975 0.3 0-2.7945425

7x

− − += =

5th Step: find new x2

( )( ) ( )( )2 1.0005 -2.7945425 1 1.0005 0 -2.79594x = + − =


1.0005λ =



6th Step: substitute x1 and x2

7th Step: find new x2

( )( ) ( )( )2 1.0005 7.00554195 1 1.0005 0 7.009044x = + − =


1.0005λ =


( ) ( )3

71.4 0.3 2.617975 0.2 -2.795947.00554195

10x

− += =

8th step: Repeat from 1st to 7th step

program sor

parameter (n=20)

real*8 x1(0:n),x2(0:n),x3(0:n)

open(1000,file='SOR-output',status='unknown')

x1(0) = 0.d0

x2(0) = 0.d0

x3(0) = 0.d0

error1 = 100.d0

error2 = 100.d0

error3 = 100.d0

ramda = 1.0005d0

write(1000,1001) i,x1(0),x2(0),x3(0),error1,error2,error3

do 10 i = 1,n

x1(i) = (7.85+0.1d0*x2(i-1)+0.2d0*x3(i-1))/3.d0

x1(i) = ramda*x1(i)+(1.d0-ramda)*x1(i-1)

x2(i) = (-19.3d0-0.1d0*x1(i)+0.3d0*x3(i-1))/7.d0


x3(i) = (71.4d0-0.3d0*x1(i)+0.2d0*x2(i))/10.d0





write(1000,1001) i,x1(i),x2(i),x3(i),error1,error2,error3

1001 format(i2,2x,3e15.7,2x,3e15.7)

10 continue

close(1000)

stop

end

0.00E+000.00E+000.00E+007.000000-2.5000003.00000010

0.00E+001.78E-140.00E+007.000000-2.5000003.0000009

5.08E-143.55E-144.50E-127.000000-2.5000003.0000008

1.67E-121.51E-109.42E-117.000000-2.5000003.0000007

1.19E-091.18E-089.46E-087.000000-2.5000003.0000006

1.22E-072.56E-068.79E-067.000000-2.5000003.0000005

1.82E-054.46E-041.51E-037.000000-2.5000003.0000004

4.04E-032.62E-023.04E-016.999999-2.4999893.0000453

1.25E-011.19E+011.25E+017.000281-2.4993352.9909252

1.00E+021.00E+021.00E+027.009044-2.7959402.6179751

1.00E+021.00E+021.00E+020.0000000.0000000.0000000

error3error2error1x3x2x1Step

0.00E+000.00E+000.00E+007.000000-2.5000003.00000010

0.00E+001.78E-141.48E-147.000000-2.5000003.0000009

Gauss-Seidel

SOR


Solving the fluid dynamic equations

2 2

2 2

u u ut x y

∂ ∂ ∂= +

∂ ∂ ∂

m

l

i,j i+1,ji-1,j

i,j+1

i,j-1


2 2

2 2

u u ut x y

∂ ∂ ∂= +

∂ ∂ ∂

1, ,

1 ,2

i j i j

i j

u uux x

+

+

−Δ=

Δ Δ

, 1,

1 ,2

i j i j

i j

u uux x

−

−

−Δ=

Δ Δ

( ) ( )21, , , 1,

2,

1 i j i j i j i j

i j

u u u uux x xx

+ −⎛ ⎞− −Δ ⎜ ⎟= −⎜ ⎟Δ Δ ΔΔ ⎝ ⎠

1, , 1,2

2i j i j i ju u ux

− +− +⎛ ⎞= ⎜ ⎟Δ⎝ ⎠

( ) ( )2, 1 , , , 1

2,

1 i j i j i j i j

i j

u u u uuy y yy

+ −⎛ ⎞− −Δ ⎜ ⎟= −⎜ ⎟Δ Δ ΔΔ ⎝ ⎠

, 1 , , 12

2i j i j i ju u uy

− +− +⎛ ⎞= ⎜ ⎟Δ⎝ ⎠

Solving the fluid dynamic equations


2 2

2 2

u u ut x y

∂ ∂ ∂= +

∂ ∂ ∂

1 1 1 1 1 1 1, , 1, , 1, 1, , , 1

2 2

2 2n n n n n n n ni j i j i j i j i j j j i j i ju u u u u u u u

t x y

+ + + + + + +− + − +⎛ ⎞ ⎛ ⎞− − + − +

= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Δ Δ Δ⎝ ⎠ ⎝ ⎠

1 1 1 1 1, 1, , 1, , 1 , 12 2 2 2 2 2

2 21n n n n n ni j i j i j i j i j i j

t t t t t tu u u u u ux x y x y y

+ + + + +− + − +

⎛ ⎞Δ Δ Δ Δ Δ Δ= − + + + − − −⎜ ⎟Δ Δ Δ Δ Δ Δ⎝ ⎠

a b c d e

Implicit Euler method

[ ]{ } { }1n ni iA X X+ =


1 1 1 1, 1, 1, , 1 , 12 2 2 2

1,

2 22 21

n n n n ni j i j i j i j i j

ni j

t t t tu u u u ux x y yu

t tx y

+ + + +− + − +

+

Δ Δ Δ Δ+ + + +Δ Δ Δ Δ

=⎛ ⎞Δ Δ+ +⎜ ⎟Δ Δ⎝ ⎠

Gauss-Seidel

{ } [ ]{ }1n ni iX A X +=


1 1 1 11

n n n nn P S S N N W W E EP

P

Q A A A AA

φ φ φ φφ

+ + + ++ − − − −=

1 1 1 1 1, 1, , 1, , 1 , 12 2 2 2 2 2



+ + + + +− + − +

⎛ ⎞Δ Δ Δ Δ Δ Δ= − + + + − − −⎜ ⎟Δ Δ Δ Δ Δ Δ⎝ ⎠


SOR

( )1 1 1 1

1 1n n n n n

n nP S S W W N N E EP P

P

u A u A u A u A uu u

Aλ λ

+ + + ++ − − − −= + −

( )1 1 1 1

1 1n n n n

n nP S S N N W W E EP P

P

Q A A A AA

φ φ φ φφ λ λ φ

+ + + ++ − − − −= + −


1 1 1 1 1, 1, , 1, , 1 , 12 2 2 2 2 2



+ + + + +− + − +

⎛ ⎞Δ Δ Δ Δ Δ Δ= − + + + − − −⎜ ⎟Δ Δ Δ Δ Δ Δ⎝ ⎠

Flow

1,1u 2,1u 3,1u 4,1u 5,1u6,1u

1,1u

1,2u

1,3u

1,4u

1,5u

1,6u

2,2u

( ),1 0u x =

( ),6 0u x =

( )( )1, 1

, 0

u y

u n y

=

=

( ), 0u x y =

Initial condition

a b c d e

0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0

0 0 0 00 0 0 0 0

0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0

b c ea b c e

a b c ea b c e

d a b c ed a b c e

d a b c ed a b c e

d a b c ed a b c e

d a b c ed a b c e

d a b cd a b c

d a b cd a b

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎦⎢

12,2 2,2

13,2 3,2

14,2 4

15,2

12,3

13,3

14,3

15,3

12,4

13,4

14,4

15,4

12,5

13,5

14,5

15,5

n n

n n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

u du du duuuuuuuuuuuuu

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥ ⎪ ⎪ =⎨ ⎬⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎪ ⎪⎥⎥ ⎪ ⎪⎩ ⎭⎥

1 12,2 1,2 2,1

13,2 3,1

1,2 4,2 4,1

15,2 5,2 5,1

2,3 2,3

3,3 3,3

4,3 4

5,3

2,4

3,4

4,4

5,4

2,5

3,5

4,5

5,5

n n n

n n

n n n

n n n

n n

n n

n

n

n

n

n

n

n

n

n

n

u au duu duu du

d u dud ud ud uddddddddd

+ +

+

+

+

⎧ ⎫ − −⎪ ⎪ −⎪ ⎪⎪ ⎪ −⎪ ⎪

−⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ =⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

,3

5,3

2,4

3,4

4,4

5,41

2,5 2,61

2,5 3,61

2,5 4,61 1

2,5 6,5 5,6

n

n

n

n

n

n

n n

n n

n n

n n n

uuuuu

u euu euu eu

u cu du

+

+

+

+ +

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪

−⎪ ⎪⎪ ⎪−⎪ ⎪⎪ ⎪−⎪ ⎪

− −⎪ ⎪⎩ ⎭

1 1 1 1 1, 1, , 1, , 1 , 12 2 2 2 2 2



+ + + + +− + − +

⎛ ⎞Δ Δ Δ Δ Δ Δ= − + + + − − −⎜ ⎟Δ Δ Δ Δ Δ Δ⎝ ⎠


SOR

( )1 1 1 1

1 1n n n n n

n nP S S W W N N E EP P

P

d A u A u A u A uu u

Aλ λ

+ + + ++ − − − −= + −

( )1 1 1 1

1 1n n n n

n nP S S N N W W E EP P

P

Q A A A AA

φ φ φ φφ λ λ φ

+ + + ++ − − − −= + −


1 1 1 1 1, 1, , 1, , 1 , 12 2 2 2 2 2



+ + + + +− + − +

⎛ ⎞Δ Δ Δ Δ Δ Δ= − + + + − − −⎜ ⎟Δ Δ Δ Δ Δ Δ⎝ ⎠

program flowsor

parameter(n=21,m =21,ntime=100)

real*8 unew(n,m),uold(n,m),a,b,c,d,e

open(1000,file='FlowSOR',status='unknown')

do 10 j = 1,m

do 20 i = 1,n

unew(i,j) = 0.d0

uold(i,j) = 0.d0

unew(1,j) = 1.d0

uold(1,j) = 1.d0

unew(i,1) = 0.d0

uold(i,m) = 0.d0

unew(n,j) = 0.d0

20 continue

10 continue

dx = 10.d0/dble(n-1)

dy = 10.d0/dble(m-1)

dt = 0.001d0

a = -1.d0*dt/dx/dx

b = 1.d0+2.d0/dx/dx+2.d0/dy/dy

c = -1.d0*dt/dx/dx

d = -1.d0*dt/dy/dy

e = -1.d0*dt/dy/dy

ramda = 1.5d0

do 100 nt = 1,ntime

do 110 j = 2,m-1

do 120 i = 2,n-1

unew(1,j) = 1.d0

uold(i,j) = unew(i,j)

unew(n,j) = 0.d0

unew(i,1) = 0.d0

unew(i,m) = 0.d0

unew(i,j) = ramda*(uold(i,j)-d*unew(i,j-1)-e*unew(i,j+1)

# -a*unew(i-1,j)-c*unew(i+1,j))/b+(1.d0-ramda)*uold(i,j)

120 continue

110 continue

100 continue

do 200 i = 1,n

do 210 j = 1,m

unew(i,1) = 0.d0

unew(i,m) = 0.d0

write(1000,1001) i,j,unew(i,j)

1001 format(2i2,2x,e15.8)

210 continue

200 continue

close(1000)

stop

end


1.0

0.8

0.6

0.4

0.2

0.0

Y

2015105

X

x = 1 x = 5 x = 10 x = 15 x = 21

Y

u

me 747 introduction to computational fluid dynamics ...chainarong.me.engr.tu.ac.th/documents/me747...

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