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Transient CFD - Computational Fluid Dynamics - Lecture Slides, Slides of Fluid Dynamics

Biju Patnaik University of TechnologyFluid Dynamics

These lectures slides are from course Computational Fluid Dynamics. This lecture was delivered by Larry Caretto. Some points from the lecture are: Transient Cfd, Conduction Equation, Ftcs, Crank-Nicholson Equations, Dufort Frankel, Von Neumann Stability, Convection Equation, Transient Convection Diffusion

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Transient CFD April 21, 2010
ME 692 – Computational Fluid Dynamics 1
Transient CFD
Transient CFD
Larry Caretto
Mechanical Engineering 692
Computational Fluid Dynamics
April 21, 2010
2
Introduction
• Schedule for last two weeks of
semester
• Review transient conduction
• Density-based approaches to transient
CFD
• Pressure-based approaches to transient
CFD
• Fluent options for transient CFD
3
Remainder of Course
• April 26-28: Non-uniform grids
• May 3-5: Student presentations
– Have a total of 150 minutes of class time
for 14 students
– Will accept volunteers for May 3 or draw
names out of a hat to balance the
presentations
4
Review Conduction Equation
• Apply difference formulas derived
for ordinary derivatives to partial
derivatives
•Grids x
i= x0+ iΔt and tn= t0+ nΔt
• Try finite difference expressions
below to get explicit finite-
difference equation
n
i
x
T
t
T⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
=
∂
∂
2
2
α
])[(
)(
2
)( 2
2
11
2
21 xO
x
TTT
x
T
andtO
t
TT
t
Tn
i
n
i
n
i
n
i
n
i
n
i
n
i
Δ+
Δ
−+
=
∂
∂
Δ+
Δ
−
=
∂
∂−+
+
5
Explicit (FTCS) Method
() ()
()
n
i
n
i
n
i
n
i
n
i
n
i
n
iTfTTfT
x
t
TT
x
t
T21
)(
2
1
)( 11
2
11
2
1−++=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
Δ
−++
Δ
Δ
=−+−+
+
αα
• Method just derived is called explicit
method; can solve one equation at a time
Tni-1 TinTni+1
●----------●----------●
●Tin+1
•T
in+1 does not depend on other T values
at the new time step (n+1)
2
)( x
t
fΔ
Δ
≡
α
ff
1-2f
6
Stability of Explicit Method
• If the values of Ti+1 and Ti-1 are fixed an
increase in Tinshould increase Tin+1
• If f is greater than 0.5, an increase in Tin
will cause a decrease in Tin+1
• We can avoid this incorrect result by
keeping f = αΔt/(Δx)2≤0.5
• This imposes a time step limit that may
be less than the limit required for
accuracy in the solution
(
)
()
n
i
n
i
n
i
n
iTfTTfT 21
11
1−++= −+
+
pf3
pf4
pf5

Partial preview of the text

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Transient CFDTransient CFD

Larry Caretto

Mechanical Engineering 692

Computational Fluid Dynamics

April 21, 2010

2

Introduction

  • Schedule for last two weeks of

semester

  • Review transient conduction
  • Density-based approaches to transient

CFD

  • Pressure-based approaches to transient

CFD

  • Fluent options for transient CFD

3

Remainder of Course

  • April 26-28: Non-uniform grids
  • May 3-5: Student presentations
    • Have a total of 150 minutes of class time for 14 students
    • Will accept volunteers for May 3 or draw names out of a hat to balance the

presentations

4

Review Conduction Equation

  • Apply difference formulas derived

for ordinary derivatives to partial

derivatives

  • Grids xi = x 0 + iΔt and t (^) n = t 0 + nΔt
  • Try finite difference expressions

below to get explicit finite-

difference equation

n

x i

T

t

T

2

2

[( ) ] ( )

2 ( ) 2 1 12 2

1 2 O x x

T T T x

T O t and t

T T t

T

n i

n i

n i

n

i

n i

n i

n

i

  • Δ Δ

  • − = ∂

∂

  • Δ Δ

−

∂

∂ (^) + −

5

Explicit (FTCS) Method

( ) ( ) ( )

n i

n i

n i

n i

n i

n i

n i T fT T fT x

t T T x

t T 1 2 ( )

2 1 ( )

2 1 1 2 1 1

1 ⎟⎟ = + + − ⎠

⎞ ⎜⎜ ⎝

⎛

Δ

Δ

    • − Δ

Δ = (^) + − + −

  • α α
  • Method just derived is called explicit

method; can solve one equation at a time

Tni-1 Tin^ Tni+

● Tin+

  • Tin+1^ does not depend on other T values

at the new time step (n+1)

2

( x )

t

f

α

f f

1-2f

6

Stability of Explicit Method

  • If the values of Ti+1 and Ti-1 are fixed an

increase in Tin^ should increase Tin+

  • If f is greater than 0.5, an increase in Tin

will cause a decrease in Tin+

  • We can avoid this incorrect result by

keeping f = αΔt/(Δx) 2 ≤ 0.

  • This imposes a time step limit that may

be less than the limit required for

accuracy in the solution

( ) ( )

n i

n i

n i

n

Ti fT 1 T 1 1 2 fT

1

7

Crank-Nicholson Method

  • Seek more accurate time derivative
  • Provides implicit method
    • Value of T (^) in+1^ depends on T (^) in+1^ and T (^) in-
    • More work per step, but can take longer

time steps with this method

  • Apply to diffusion equation at time n + 1/

2

1

2

2 2

1 2 2 1

1

[( )] [( )]

2

2

    • +^ +

∂

∂

  • Δ = Δ

−

  • Δ = Δ

−

∂

∂

n

i

n i

n i

n i

n i

n

i x

T O t t

T T O t t

T T

t

T α

8

Crank-Nicholson Equations

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

− −

− −

− −

− − −

− − −

− −

−

−

−

−

n N

n N

n N

n

n

n n

n N

n N

n

n

n

R fT

R

R

R

R fT

T

T

T

T

T

f f

f f

f f

f f f

f f f

f f

1

2

3

2

1 0

1 1

1 2

1 3

1 2

1 1

0 0 0 0 21

0 0 0 0 21

0 0 21 0 0

0 21 0 0

21 0 0 0

21 0 0 0 0

M

M M

M

L

L

M M M M O M M

L

L

L

  • Consider case where boundary

temperatures T 0 and TN are specified

  • Rewrite equations in matrix form to

show tridiagonal structure

[ ]

n i

n i

n i

n Ri = fT + 1 + T − 1 + 2 ( 1 − f ) T

f = αΔt/(Δx) 2

f ≥ 1

9

Crank Nicholson Results

  • Results for α = 1, L = 1, Δx = 0.01, Δt =

0.0005, f = αΔt/(Δx) 2 = 5

n = 6 t = 0.003 0 141.46 177.47 298.2 397.

n = 5 t = 0.0025 0 56.79 252.91 334.12 422.

n = 4 t = 0.002 0 203.86 209.57 347.52 473.

n = 3 t = 0.0015 0 25.7 320.81 439.19 533.

n = 2 t = 0.001 0 352.75 305.27 440.73 599.

n = 1 t = 0.0005 0 -73.35 423.96 690.85 834.

n = 0 t = 0+ 0 1000 1000 1000 1000

t = 0 1000 1000 1000 1000 1000

x = 0 x = .01 x = .02 x = .03 x =.

i = 0 i = 1 i = 2 i = 3 i = 4

10

Crank Nicholson Results III

Error t = 0.0125 0 0.216 0.272 0.212 0.

Exact t = 0.0125 0 50.43 100.66 150.48 199.

n = 25 t = 0.0125 0 50.21 100.93 150.27 199.

n = 24 t = 0.012 0 51.73 102.36 153.78 203.

n = 23 t = 0.0115 0 52.22 105.35 156.49 208.

n = 22 t = 0.011 0 54.19 106.68 160.64 212.

n = 21 t = 0.0105 0 54.43 110.47 163.53 217.

n = 20 t = 0.01 0 57.1 111.53 168.52 222.

n = 19 t = 0.0095 0 56.86 116.5 171.59 228.

n = 18 t = 0.009 0 60.65 117 177.71 234.

x = 0 x = .01 x = .02 x = .03 x =.

i = 0 i = 1 i = 2 i = 3 i = 4

11

Fully Implicit Method

  • Discretize diffusion equation at t (^) n+

[( ) ] ( )

2 ( ) 2 2

1 1 1

1 1

1

2

(^112) O x x

T T T x

T O t and t

T T t

T (^) in in in

n

i

n i

n i

n

i

  • Δ Δ

  • − = ∂

∂

  • Δ Δ

−

∂

∂ (^) ++ −+ +

[( ),( )] 0 ( )

(^2 ) 2

1 1 1

1 1

(^11)

2

(^12)

  • Δ Δ = Δ

  • − − Δ

−

∂

∂ − ∂

∂

−

O t x x

T T T

t

T T

x

T t

T

n i

n i

n i

n i

n i

n

i

n

i

α α

n i

n i

n i

n

− fTi + + fT − fT = T

−

1 1

1 1

1 (^12 )

  • Tridiagonal system of equations
  • Almost same work as CN and no

spurious oscillations, but less accuracy

12

DuFort Frankel

  • Rearrange and introduce f = αΔt/(Δx) 2

(^1 12) ( 1 1 1 1 ) 2 ( 1 1 1 1 ) ( )

(^2) + −

  • −

  • −

  • −

  • − + − − = + − − Δ

Δ i n −^ in = Tin Tin Tin Tin fTin Tin Tin Tin x

t T T

α

( 1 + 2 f ) T (^) i n +^1 = Tin −^1 ( 1 − 2 f ) + 2 f ( T (^) in + 1 + Tin − 1 )

  • Result is explicit for values at time n+
  • Explicit start required to get first set of

values at time n-

19

von Neumann Results

  • Explicit (FTCS) method
    • G ≤ 1 if f ≤ 0.
  • Crank-Nicholson method

1 4 sin

2

Δ x

G e f

at^ β m

1 2 sin

1 2 sin

2

2

Δ

x

f

x

f

G e

m

m

at

Unconditionally stable

20

Convection Equation

  • Lax’s Method is stable if the Courant

number, NC = cΔx/Δt ≤ 1

[ ]

[ ]

n i

n i

n i

n n i

i u u

x

u u c t

u 1 1

1 1 1

  • −

    • −

[ 1 ( 1 )sin( )] 1

2 2

Δ

G e NC mx

at

x

u c t

u

21

Transient Convection Diffusion

  • Stability of FTCS algorithm

2

2

x x

c t ∂

∂ φ α

φ φ

n k

n k

n k

n k

n k

n k

n k x

t

x

ct φ φ φ

α φ φ φ φ 2 2

1 1 2 1 1

1

  • − Δ

( ) x

x

ct x i x

t G β m β m

α 1 2 2 1 cos sin Δ

  • Stability requires cΔt/Δx ≤ 1, αΔt/(Δx) 2 ≤

0.5 and Pe cell = cΔx/α ≤ 2

  • Last equation is limit for central difference 22

Density-based Solvers

  • Typically used for compressible flows in

aerodynamics calculations

  • Split stress term, σij , into sum of pressure

and viscous stress, τij = σij + Pδij

ij i

j

j

i ij x

u

x

u τ μ +κ− μΔ δ ⎥

j i

ij

i j

j i j B x x

p

x

uu

t

u ρ

ρ ρ τ

∂

∂

∂

∂ =− ∂

∂

∂

∂

23

Density-based Solvers II

  • Equation without summation convention

x

xj yj zj

j

j j j j

B

x x y z

p

z

wu

y

vu

x

uu

t

u

x

yx zx

xx

B

z

wu

y

vu

x

uu p

t

u

  • General

direction j

equation

  • x-direction

equation

24

Density-based Solvers III

  • Cast continuity, momentum, total

energy, and species balance into form

of vector equation

H

U E F G

t x y z

  • Each conservation equation is one

component of the vector equation

  • Get components by reviewing equations

25

Density-based Solvers IV

H

U E F G

t x y z

6

5

4

3

2

1

() 6

5

4

3

2

1

()

2 ( )

h

h

h

h

h

h

r

uB vB wB

B
B
B
U
U
U
U
U
U
W

e V

w

v

u

K

x y z

z

y

x

K

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

U H

26

Density-based Solvers V

H

U E F G

t x y z

6

5

4

3

2

1

() ()

2 [ ( / 2 ) ]

e

e

e

e

e

e

uW j

u e V p u v w q

wu

uv

uu p

u

K x

K

xx xy xz x

xz

xy

xx

ρ

ρ τ τ τ

ρ τ

ρ τ

ρ τ

ρ

E

27

Density-based Solvers VI

H

U E F G

t x y z

6

5

4

3

2

1

() ()

2

[ ( / 2 ) ]

f

f

f

f

f

f

vW j

e V p u v w q

wv

v

vv p

uv

v

K y

K

yx yy yz y

yz

yy

yx

F

28

Density-based Solvers VII

H

U E F G

t x y z

6

5

4

3

2

1

() ()

2 [ ( / 2 ) ]

g

g

g

g

g

g

wW j

w e V p u v w q

ww p

vw

uw

w

K z

K

zx zy zz z

zz

zy

zx

G

29

Density-based Solvers VIII

H

U E F G

t x y z

  • Compute vectors E , F , G , and H from

flow variables and use numerical

integration over time step to get U

  • Update flow variables from components

of U vector, Uk.

  • These are not velocity components
  • Details next chart

30

Density-based Solvers IX

( )

1

() 6

2 4

2 3

2 2 2 1 1

5

1

4

1

3

1

2

1

U
U
W
U U U
U U
U

e

U
U

w

U
U

v

U
U

u

U

K

37

Pressure-based Solvers VI

  • Use steady-state results

[ ( ) ( ) ]

∫

∫

Δ

Δ

+Δ

V

V

Pt t Pt

S dV x x y y

dV y

v

x

u

t

V

( )

, ,

φ φ μ

φ μ

ρφ ρφ ρφ ρφ

[( ) ( ) ]

( )

, ,

φ

a a a a a S

t

V

N N S S E E W W P P

Pt t Pt

38

Pressure-based Solvers VII

  • Transient finite-volume equation

[ ( ) ( ) ]

( )

, ,

φ

a a a a a S

t

V

N N S S E E W W P P

Pt t Pt

()

      • − , + =

φ aN φ (^) N aS φ S aE φ E aW φ W aPtransient φ P Stransient

t

V
S S

t

V

a a

Pt transient

Pt t P transient P Δ

,+ Δ () () , ,

ρ φ φ ρφ

39

What is Time Average?

  • Have same choices used for conduction

equation

  • Explicit – use values at old time step
  • Implicit – use values at new time step
  • Crank-Nicholson – use average of values at old and new time steps
  • Can also use more accurate time

derivatives

  • Fluent has various options

40

41

Explicit or Implicit?

  • Explicit stability limits on time step (set

by the local Courant number, uΔx/α)

  • The Δt required for stability is usually

much lower than the Δt for accuracy

  • Implicit algorithms will generally take

less computer time

  • Moving waves ( e. g. shock waves)

require small time steps so that explicit

algorithms are preferred here

  • Available in Fluent only with density solver

42

Other Fluent Options

  • Non-iterative time advancement –

simplifies iterations to reduce computer

time for solution

  • Does not do “outer” iteration
  • Frozen-flux formulation uses aK

coefficients from previous time step

  • Does not update during iterations
  • Another item to save computer time