# Applied Computational Fluid Dynamics 2003

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```					                          Applied Computational Fluid Dynamics
FACE8 – Applied CFD

2003

Institute of Energy Technology
Aalborg University

1
Course objectives

• To give an introduction to the possibilities of Computational Fluid Dynamics
• To provide an appreciation for the complexity of fluid mechanics and CFD methods
• To introduce important aspects related to the quality and trust in CFD
FACE8 – Applied CFD

• To give an idea about the potential of CFD in fluid mechanics

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Course outline
∙   1 MM
▫ Introduction to applied CFD
▫ Review of the finite volume method
▫ Geometry creation and CAD import
∙   2 MM
FACE8 – Applied CFD

▫   Grid generation
▫   Grid quality
▫   Grid adaption – static and dynamic
▫   Boundary conditions
∙   3 MM
▫ Solution methods
▫ Solution quality
∙   4 MM
∙   5 MM
▫ Post-processing
▫ Validation

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Hands – on (room 5)

∙   Available software (Win2K or Win-XP platforms)
▫ FLUENT currently ver. 6.1
◦ Gambit
▫ FIDAP
FACE8 – Applied CFD

◦ Gambit
▫ FIELDVIEW 7
▫ CFX4
◦ CFX-Build
◦ CFX-Analyse
▫ CFX5
◦ CFX-Visualize
∙   Simple problems
∙   CFD in projects

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Exercise 1

symmetry

=1
FACE8 – Applied CFD

Outflow

=0

symmetry

Outflow
=1

=0
Outflow

5
Useful homepages
FACE8 – Applied CFD

Course homepage

www.cfd-online.com

www.qnet-cfd.net

www.fluent.com

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What is CFD (the history)?

∙       Colorful fluid dynamics, colors for directors, ... or?
FACE8 – Applied CFD

LES and DNS simulation
Explosions
Multiphase flow
Single phase                                                       Disperse multiphase flow
Combustion                                                         Multiple frame of reference
models                              GUI                            Fluid-solid interaction
Optimization algorithms
Process coupling techniques
1960’s –
aircraft industry                                                                 Unstructured codes
2D structured, single
aerodynamic profiles                                  3D structured, single
block codes
block codes
1-2 eq. turbulence models
Multi-block codes

Timeline

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Why use CFD?

Pure/classical                      Experimental
fluid dynamics                     fluid dynamics
FACE8 – Applied CFD

Limited knowledge                                              Expensive
Unsolvable sets of equations

Few industrial type
?                Pre-prototype testing
Timeconsuming
Difficult to isolate subproblems
problems solvable even                                Virtual prototyping
through approximations                                  Prohibitive to perform
Reduced time-to-market
full-scale tests
Subproblem testing
Impossible to measure in
Agressive environments
Full-scale testing expensive

Sensitivity analysis
Computational
Agressive environments
fluid dynamics              Complex physics

...

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CFD application areas
Power generation
Aerodynamics (cars, aeroplanes etc.)
Air conditioning
Electronics
FACE8 – Applied CFD

Safety issues
Purpose                Reliability
Efficiency
Forecasts
Velocities
Leading CFD codes on the market:                        Turbulence
Pressure
- CFX (CFX4, CFX5, CFX-Tascflow         Properties      Temperature
- Fluent (Fluent6, Fluent4, FIDAP)                      Species
- STAR-CD
Heat fluxes
Shear stresses
A minimum of time should be spent on pre-processing!
- CAD import, automatic grid generation                     Trajectories

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CFD – short and sweet

Pre-processor
FACE8 – Applied CFD

CFD                   Solver

Post-processor

Wall boundary - ”no-slip” Udløb      Neumann condition
Dirichlet inlet                                          
  0  kst 
condition                                              n
  kst
p  kst

Wall boundary - ”no-slip”
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FLUENT program structure
FACE8 – Applied CFD

11
The CFD user must be an expert...

Physical models
Numerical methods              - Turbulence
- Truncation errors            - Two-phase flow
- Stability                    - Chemistry
FACE8 – Applied CFD

- Class of PDE                 - Compressibility
- ……………..                      - Heat transfer
- ……………
Boundary conditions
- Experience from similar problems
- Measurements
- Knowledge from “simple” cases
- ……………..                                 ?
The Engineer
- Physical insight (choice of models)
- Experience
- Review / evaluation of results
- Choice of approach (exp. or num)
- ……………..

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FLUENT recommendations
FACE8 – Applied CFD

13
Quality and trust

Required level of detail
- prediction of small losses (e.g. in pumps)
FACE8 – Applied CFD

- Overall flow direction (recirculation zones etc.)

Even qualitative properties can reveal weak points in a design!

Limit

Predicted                                                                 Error bands

14
Quality and trust cont’d

Model uncertainties:
- incomplete knowledge of underlaying physics
FACE8 – Applied CFD

Errors (these are the ones we can minimize):
- discretization error
- convergence error (iteration)
- round-off (numerical precision)
- user (garbage in - garbage out)
- coding

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FACE8 – Applied CFD

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Applications
Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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Applications cont’d
FACE8 – Applied CFD

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An overview
the governing equations

Quantity/
Equation
property
 
Mass                         0  divu  
FACE8 – Applied CFD

t
p             
   divgradu   S Mx
Du
x-momentum                 
Dt     x
p                 
   divgradu   S My
Dv
y-momentum                 
Dt      y
p                 
   divgradu   S Mz
Dw
z-momentum                
Dt       z

  p div u  divk grad T     Si
Di
Internal energy         
Dt
Equations of              p  p , T ,     i  i , T 
state                   p  RT ,         i  cvT
General transport                  
 divu   div grad    S
equation                t

Diffusion coefficient
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Governing Equations

Recall the general transport equation:

(  )          
FACE8 – Applied CFD

 div(  u )  div(  grad )  S
t

rate of change   convection       diffusion   source term
with time

Examples:

Continuity:         =1       =0 S=0
U-momentum:         =u        =  S = -dp/dx

25
Spatial discretization (1)
diffusion

We are beginning with the diffusion term (not only due to a concentration gradient
but in general):
(  )          
 div(  u )  div(  grad )  S
t
FACE8 – Applied CFD

As an example, consider 1-dimensional conduction of heat:

d  dT 
k   S 0
dx  dx 

where:

k = thermal conductivity
T = temperature
S = rate of heat generation per unit volume

26
Spatial discretization (2)
diffusion

Consider the 1-dimensional grid system (y and z = 1):

(x) w                 (x) e
FACE8 – Applied CFD

w                      e

W                    P                 E
x

We integrate the heat equation over this volume:

 dT   dT       e
k     k     w S dx  0
 dx  e  dx  w

27
Spatial discretization (3)
diffusion

 dT   dT       e
k     k     w S dx  0
 dx  e  dx  w
FACE8 – Applied CFD

We need to assume a profile to evaluate the gradient terms:

W     w   P   e    E                    W     w   P     e   E

dT/dx NOT defined for this profile!

28
Spatial discretization (4)
diffusion

If we use the linear profile assumption:

 dT   dT       e
k     k     w S dx  0
 dx  e  dx  w
FACE8 – Applied CFD

W       w   P   e   E

We arrive at the following discretized equation (with uniform cells):

k e (TE  TP ) k w (TP  TW )
                S V  0
(x ) e        (x ) w

(x) w                (x) e
w                    e

W                   P                      E
29                                          x
Spatial discretization (5)
diffusion

The discretized equation is written as:

aPTP  aETE  aW TW  b   anbTnb  b
FACE8 – Applied CFD

where:
ke                 kw
aE =               aW =              aP = aE + aW      b = S Dx
(dx )e             (dx )w

In 2 dimensions, this is often illustrated as an amoeba:

30
Spatial discretization (6)
diffusion and convection

We extend the discretization equation to include convection also:

(  )          
FACE8 – Applied CFD

 div(  u )  div(  grad )  S
t

We look at the 1-dimensional case for simplicity:

d            d  d 
( u )        
dx           dx  dx 

31
Spatial discretization (7)
diffusion and convection

Consider again the 1-dimensional grid system (y and z = 1):
FACE8 – Applied CFD

(x) w               (x) e
w                   e

W                   P                   E
x
We integrate the diffusion-convection equation over this volume:

 dT   dT 
(  u ) e  (  u ) w          
 dx  e  dx  w

32
Spatial discretization (8)
diffusion and convection

If we use the same linear profile assumption for:

(  u ) e  (  u ) w
FACE8 – Applied CFD

W    w   P   e   E

We find that (assuming P to be midways between W and E):

 e  ½( E   P ) and  w  ½( P  W )

(x) w             (x) e
w                     e

W                   P                 E
x
This approximation is termed central differencing
- it is termed second-order accurate since error O(x2) in terms of the Taylor series
33
Spatial discretization (9)
diffusion and convection

Now inserting these discretized diffusion and convection terms yields:

1                          1                         (   P ) w ( P  W )
(  u ) e ( E   P )  (  u ) w ( P  W )  e E        
FACE8 – Applied CFD

2                          2                           (x ) e       (x ) w

If we introduce:

F  u           D
x
we can again write:

aPP  aEE  aWW  b   anbnb  b

now with
Fe            F
a E  De       , aW  Dw  w
2              2
34
Spatial discretization (10)
diffusion and convection

If we consider the case where:
De  Dw  1 and         Fe  Fw  4
FACE8 – Applied CFD

We can calculate the value at point P from those at points E and W:

(a )  E  200 and W  100   P  50!
(b)  E  100 and W  200   P  250!

This is clearly unrealistic!

Conclusion: Central differencing does not always work for the convection term
(specifically it can be shown that -2<F/D<2 is required => low velocity + fine grids)

We need another approximation!
35
Spatial discretization (11)
diffusion and convection

The upwind scheme identifies flow direction:
FACE8 – Applied CFD

W   w   P   e   E

Flow from right to left (Fe< 0 and Fw< 0)

e   E       and      w  P

Similarly, if flow from left to right (Fe< 0 and Fw< 0)

e   P       and  w  W

36
Spatial discretization (12)
diffusion and convection

For the flux across the east face, e, this scheme leads to:

Fee  P Fe ,0  E  Fe ,0
FACE8 – Applied CFD

Again we can write:

aPP  aEE  aWW  b   anbnb  b

Now, for the upwind scheme the coefficient are:

aE  De   Fe ,0 , aW  Dw   Fw ,0

The Upwind scheme is only first-order accurate hence the error O(x)

37
Spatial discretization (13)
diffusion and convection

There is a large number of differencing schemes out there:

•the Hybrid scheme combines the Central difference and the Upwind schemes
•the Power law scheme approximates the exponential behavior (not exact in 2 or 3D)
FACE8 – Applied CFD

More advanced schemes extend the amoeba to include more neighbour cells:

parabola

W    w   P   e    E

•second-order Upwind
•QUICK
•and many others
38
Spatial discretization (14)
false diffusion

Often it is stated that the Central difference scheme is superior to the Upwind
scheme because it is second-order accurate whereas the Upwind scheme is
only first-order accurate.
FACE8 – Applied CFD

If we compare the Central difference and Upwind schemes:

 ux
Upwind   
2
solution at large Peclet number (large cells)!

39
Spatial discretization (15)
false diffusion, the proper view

False diffusion is a multidimensional phenomenon
FACE8 – Applied CFD

Consider the following case where  = 0:



=0
No false diffusion

=1
False diffusion

False diffusion occurs when the flow is NOT perpendicular to the grid lines!

We will investigate this further in the exercise

40
Spatial discretization (16)
Quality and trust

The convection terms are of primary concern

The accuracy stated may be reduced on non-uniform grids (e.g. QUICK scheme)
FACE8 – Applied CFD

Halving the cell size reduces the error with a factor of:
- 8 for third-order schemes
- 2 for first-order schemes

Higher-order schemes more unstable (perhaps due to physially unsteady flow)

Guidelines:
- use a least second-order accurate schemes
- first-order schemes may be used to start the calculation
- use grid refinement test to estimate discretization error
- look at distribution of residuals

41
Temporal discretization (1)

We have now discretized the spatial terms:

(  )             
 div(  u )  div(  grad )  S
FACE8 – Applied CFD

t

time        convection      diffusion 
dependence

For simplicity we consider time-dependent heat conduction:

T     T 
c       k   
t x  x 

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Temporal discretization (2)

We integrate in time (from t to t+t) and space:

e t  t                t  t e
T                        T 
 c            dt dx                k    dx dt
t                      t  x 
FACE8 – Applied CFD

w    t                    t   w

(x) w                  (x) e
w                       e

W                     P                       E
x

If we assume the value of T prevails over the entire volume:
e t  t
T
 c            dt dx   cx (TP  TP0 )
1

w   t
t

43
Temporal discretization (3)

For the diffusion term:
t  t
 k e (TE  TP ) k w (TP  TW ) 
 cx (TP  TP0 )                                
1
 (x )               (x ) w    dt
          e                    
FACE8 – Applied CFD

t

We need an assumption for the variation of T in time (between t and t+t):

t  t


1

TP dt  f TP  (1  f ) TP0 t  
t

where 0 < f < 1

44
Temporal discretization (4)

f =0:      Explicit scheme (first-order accurate)
f = 0.5:   Crank-Nicolcon (second-order accurate)
f = 1:     Implicit (first-order accurate)
FACE8 – Applied CFD

Explicit

T P0

Crank-Nicolson

T P1

Implicit

t                 t+dt
45
Temporal discretization (5)

For stability, the Explicit scheme requires (on uniform grid):

 ( x ) 2
t 
FACE8 – Applied CFD

2

To give realistic results, the Crank-Nicolson scheme requires:

 ( x ) 2
t 


The Imlicit scheme is always stable (but still only first-order)!

Temporal and spatial discretization are strongly coupled

46
Temporal discretization (6)

Schemes that are higher-order accurate in time exist, e.g.:
FACE8 – Applied CFD

T   1
    (3 T n 1  4 T n  T n 1 )
t 2 t

47
Temporal discretization (7)
quality and trust

Physically time-dependent flows often fail to converge using steady-state methods

The order of the scheme and the time-step size determine the:
FACE8 – Applied CFD

- phase error
- amplitude error

Guidelines:

- second-order accuracy is recommended
- check the influence from temporal discretization on frequency etc.
- check the influence from time-step size
- ensure the time-step is sufficient to capture feature of interest
- begin with small time-steps (e.g. CFL = t v/x < 1)

48

∙   Be aware of
FACE8 – Applied CFD

▫   Blueprints are up-to-date
▫   last minute design modifications
▫   imported CAD geometry is sufficiently accurate
▫   tolerances e.g. in turbomachinery
▫   changes in geometry due to wear or fouling
▫   UNITS

49

∙   Short Gambit demo
FACE8 – Applied CFD

50

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