Introduction to Computational
Fluid Dynamics
Adapted from notes by:
Tao Xing and Fred Stern
The University of Iowa
Outline
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What is CFD?
Why use CFD?
Where is CFD used?
Physics
Modeling
Numerics
CFD process
Resources
2
What is CFD?

What is CFD and its objective?
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Computational Fluid Dynamics
Historically Analytical Fluid Dynamics (AFD) and EFD
(Experimental Fluid Dynamics) was used. CFD has become
feasible due to the advent of high speed digital computers.
Computer simulation for prediction of fluid-flow phenomena.
The objective of CFD is to model the continuous fluids with
Partial Differential Equations (PDEs) and discretize PDEs into an
algebra problem (Taylor series), solve it, validate it and achieve
simulation based design.
3
Why use CFD?

Why use CFD?
– Analysis and Design
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Simulation-based design instead of “build & test”
– More cost effectively and more rapidly than with experiments
– CFD solution provides high-fidelity database for interrogation of
flow field
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Simulation of physical fluid phenomena that are difficult to be
measured by experiments
– Scale simulations (e.g., full-scale ships, airplanes)
– Hazards (e.g., explosions, radiation, pollution)
– Physics (e.g., weather prediction, planetary boundary layer,
stellar evolution)
– Knowledge and exploration of flow physics
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Where is CFD used? (Aerospace)
• Where is CFD used?
– Aerospace
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Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
F18 Store Separation
Wing-Body Interaction
Hypersonic Launch
Vehicle
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Where is CFD used? (Appliances)
• Where is CFD used?
– Aerospace
– Appliances
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Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
Surface-heat-flux plots of the No-Frost
refrigerator and freezer compartments helped
BOSCH-SIEMENS engineers to optimize the
location of air inlets.
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Where is CFD used? (Automotive)
• Where is CFD used?
– Aerospace
– Appliances
– Automotive
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Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
External Aerodynamics
Interior Ventilation
Undercarriage
Aerodynamics
Engine Cooling 7
Where is CFD used? (Biomedical)
• Where is CFD used?
– Aerospace
– Appliances
– Automotive
– Biomedical
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Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
Medtronic Blood Pump
Temperature and natural
convection currents in the eye
following laser heating.
Spinal Catheter
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Where is CFD used? (Chemical Processing)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
– Chemical Processing
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HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
Sports
Polymerization reactor vessel - prediction
of flow separation and residence time
effects.
Twin-screw extruder
modeling
Shear rate distribution in twinscrew extruder simulation
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Where is CFD used? (HVAC&R)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
Chemical Processing
Streamlines for workstation
ventilation
– HVAC&R
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Hydraulics
Marine
Oil & Gas
Power Generation
Sports
Mean age of air contours indicate
location of fresh supply air
Particle traces of copier VOC emissions
colored by concentration level fall
behind the copier and then circulate
through the room before exiting the
exhaust.
Flow pathlines colored by
pressure quantify head loss
in ductwork
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Where is CFD used? (Hydraulics)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
– Hydraulics
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Marine
Oil & Gas
Power Generation
Sports
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Where is CFD used? (Marine)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
– Marine
– Oil & Gas
– Power Generation
– Sports
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Where is CFD used? (Oil & Gas)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Volume fraction of gas
Flow vectors and pressure
distribution on an offshore oil rig
Volume fraction of oil
Volume fraction of water
– Oil & Gas
Analysis of multiphase
separator
– Power Generation
– Sports
Flow of lubricating
mud over drill bit
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Where is CFD used? (Power Generation)
• Where is CFD used?
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Aerospace
Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Flow around cooling
towers
Flow in a
burner
– Power Generation
– Sports
Flow pattern through a water
turbine.
Pathlines from the inlet
colored by temperature
during standard 14
operating conditions
Where is CFD used? (Sports)
• Where is CFD used?
– Aerospace
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Appliances
Automotive
Biomedical
Chemical Processing
HVAC&R
Hydraulics
Marine
Oil & Gas
Power Generation
– Sports
15
Physics
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CFD codes typically designed for representation
of specific flow phenomenon
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Viscous vs. inviscid (no viscous forces) (Re)
Turbulent vs. laminar (Re)
Incompressible vs. compressible (Ma)
Single- vs. multi-phase (Ca)
Thermal/density effects and energy equation (Pr, g, Gr,
Ec)
– Free-surface flow and surface tension (Fr, We)
– Chemical reactions, mass transfer
– etc…
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Physics
Fluid Mechanics
Inviscid
Viscous
Laminar
Compressible
(air, acoustic)
Incompressible
(water)
Turbulence
Internal
External
(pipe,valve)
(airfoil, ship)
Components of Fluid Mechanics
17
Governing Equations
(Equations based on “average” velocity)
 


  ux   u y   uz  0
t x
y
z
Continuity
 u

u
u
u 
p 


  x  u x x  u y x  u z x   
   xx   yx   zx    g x
x
y
z 
x  x
y
z 
 t
Equation of motion
18
Navier-Stokes Equations
Claude-Louis Navier
George Gabriel Stokes
C.L. M. H. Navier, Memoire sur les Lois du Mouvements des Fluides, Mem. de l’Acad. d. Sci.,6, 398 (1822)
C.G. Stokes, On the Theories of the Internal Friction of Fluids in Motion, Trans. Cambridge Phys. Soc., 8, (1845)
Navier-Stokes Equations
(constant  and m)
  2 ux  2 ux  2 ux 
 u x
u x
u x
u x   p
    m  2  2  2   g x
   u x
 uy
 uz
x
y
z 
x
y
z 
 t
 x
2
2
2


u

u

u

u

u

u

uy 
 y
p
y
y
y 
y
y
    m  2  2  2   g y
   u x
 uy
 uz
x
y
z  y
y
z 
 x
 t
 u z
u z
u z
u z   p   2 u z  2 u z  2 u z 
    m  2  2  2   g z
   u x
 uy
 uz
x
y
 z  z
y
z 
 t
 x
D

v    p  m  2v   g
Dt
20
Navier–Stokes Example
u y
u y
u y   p   2 u y  2 u y  2 u y 
 u y
   u x
 uy
 u z     m  2  2  2    g y
x
y
z  y
y
z 
 x
 t
 d 2uy 
dp
0 m  2  g
dy
 dx 

x dp
     g   C1
dx m  dy


x2  d p
Integrate  u y      g   C1 x  C2
2m  dy

Integrate 
duy
B.C.  u y  0 at x  0, u y  0 at x  L
C1 

L  dp




g

2m  dy

Final Expression  u y 
C2  0

1  dp
2




g
(
Lx
x
)


2m  dy

Fluid
L
y
x
Laminar Flow
Static Parallel Plates
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Modeling
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Mathematical representation of the physical problem
– Some problems are exact (e.g., laminar pipe flow)
– Exact solutions only exist for some simple cases. In
these cases nonlinear terms can be dropped from the NS equations which allow analytical solution.
– Most cases require models for flow behavior [e.g., K-e,
K-w, Reynolds Averaged Navier Stokes equations
(RANS) or Large Eddy Simulation (LES) for turbulent
flow]
Initial —Boundary Value Problem (IBVP), include:
governing Partial Differential Equations (PDEs), Initial
Conditions (ICs) and Boundary Conditions (BCs)
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Turbulent Flow Representation
(K-e as an example)
u i  u  u' Where : u'  deviating velocity, u  constant net veloci ty
in the direction of flow, and u i  instantane ous velocity
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Turbulent Boundary Layer
y
Bulk Stream
x
U0
Edge of boundary layer
Outer layer
d
Fully turbulent layer
Sublayer + buffer layer
Wall
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Wall Shear Stress
d U 


dy

 y 0
 w   

y 
y
d

u y

Friction Velocity
u 
w

Viscous Length Scale
d 

u
y+ is similar to a local Reynolds number.
Small y+ - Viscous effects dominate
Large y+ - Turbulence dominates
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y+ and Turbulence Models
COMSOL has many turbulent models available
Low-Re models require a y+ resolution of < 1 to guarantee
accuracy
Low-Re models are necessary to accurately estimate skin
friction and flow separation
High-Re models use wall functions to approximate averaged
turbulent flow properties
Less accurate, but more computationally efficient
In COMSOL, a minimum y+ of 11.06 is enforced. To
maintain accuracy, ensure cells meet this requirement
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Numerics / Discretization

Computational solution of the IBVP
 Method dependent upon the model equations and
physics
 Several components to formulation
– Discretization and linearization
– Assembly of system of algebraic equations
– Solve the system and get approximate solutions
27
Finite Differences
 u 
  
 x  i, j
u i 1, j  u i , j
  2u
  2
 x
x
  x    3u

  3
 i, j 2
 x
Finite difference
representation
  x 2


 i, j 6
Truncation error
Methods of Solution
Direct methods
Cramer’s Rule, Gauss elimination
LU decomposition
Iterative methods
Jacobi method, Gauss-Seidel
Method, SOR method
28
Numeric Solution
(Finite Differences)
u i 1, j  u i , j
jmax
j+1
j
j-1
o
  2u
 u 
    x   2
 x  i, j
 x
   x 2   3 u

  3
 i, j 2
 x
   x 3


 i, j 6
x
y
i-1 i i+1
Taylor’s Series Expansion
u i,j = velocity of fluid
imax x
Discrete Grid Points
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Finite Difference Truncation Error
f  x  x   f ( x ) 
f
x 
x
 2 f
 2
 x
 x 2
 n f

   n
2
i , j
 x
 x n

i , j n !
f ( x)  sin 2 x
at : x  0.2
f ( x)  0.9511
f (0.22)  ????
f  x  x   f ( x ) 
x  0.02
f
x
x
f (0.22)  f (0.2)  2 cos[ 2 (0.2)](0.02)  0.9899
Exact solution for f (0.22)  0.9823
Error  0.775 percent
30
CFD process
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Geometry description
Specification of flow conditions and properties
Selection of models
Specification of initial and boundary conditions
Grid generation and transformation
Specification of numerical parameters
Flow solution
Post processing: Analysis, and visualization
Uncertainty assessment
31
Geometry description
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Typical approaches

– Make assumptions and
simplifications
– CAD/CAE integration
– Engineering drawings
– Coordinates include Cartesian
system (x,y,z), cylindrical system (r,
θ, z), and spherical system(r, θ, Φ)
32
Flow conditions and properties

Flow conditions and properties required are
unique for each flow code and application
– FlowLab requires all variables in dimensional
form
– Because of focused application, research codes
often use non-dimensional variables.
33
Selection of models for flow field
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Direct Numerical Simulations (DNS) is to solve the N-S
equations directly without any modeling. Grid must be fine
enough to resolve all flow scales. Applied for laminar flow
and rare be used in turbulent flow.
Reynolds Averaged Navier-Stokes (NS) equations (RANS)
is to perform averaging of NS equations and establishing
turbulent models for the eddy viscosity. Too many
averaging might damping vortical structures in turbulent
flows
Large Eddy Simulation (LES), Smagorinsky’ constant
model and dynamic model. Provide more instantaneous
information than RANS did. Instability in complex
geometries
Detached Eddy Simulation (DES) is to use one single
formulation to combine the advantages of RANS and LES.
34
Initial and boundary conditions

For steady/unsteady flow
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IC should not affect final solution, only convergence path, i.e.
iteration numbers needed to get the converged solution.
Robust codes should start most problems from very crude IC, .
But more reasonable guess can speed up the convergence.
Boundary conditions
– No-slip or slip-free on the wall, periodic, inlet (velocity
inlet, mass flow rate, constant pressure, etc.), outlet
(constant pressure, velocity convective, buffer zone,
zero-gradient), and non-reflecting (compressible flows,
such as acoustics), etc.
35
Grid generation

Grids can either be structured (hexahedral)
or unstructured (tetrahedral). Depends
upon type of discretization scheme and
application
– Scheme
 Finite differences: structured
 Finite volume or finite element:
structured or unstructured
– Application
 Thin boundary layers best resolved
with highly-stretched structured
grids
 Unstructured grids useful for
complex geometries
 Unstructured grids permit automatic
adaptive refinement based on the
pressure gradient, or regions of
interest (FLUENT)
36
Grid Resolution
37
Grid generation and transformation
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Grids designed to resolve important
flow features which are dependent
upon flow parameters (e.g., Re)
Commercial codes such as Gridgen,
Gambit
For research code, grid generated by
one of several methods (algebraic vs.
PDE based, conformal mapping)
For complex geometries, body-fitted
coordinate system will have to be
applied (next slide). Grid
transformation from the physical
domain to the computational domain
will be necessary
Sample grid established by
Gambit of FLUENT
38
Grid transformation

y
o
x
Physical domain
Transformation between
physical (x,y,z)
and computational (,,z) domains,
important for body-fitted grids. The partial
derivatives at these two domains have the
relationship (2D as an example)
o

Computational domain
f f  f 
f
f


 x
 x
x  x  x


f f  f 
f
f


 y
y
y  y  y


39
Numerical parameters & flow
solution


Numerical parameters are used to control flow
solution.
– Under relaxation factor, tridiagonal or
pentadiagonal solvers
– CFD Labs using FlowLab
 Monitor residuals (change of results
between iterations)
 Number of iterations for steady flow or
number of time steps for unsteady flow
Flow solution
– Solve the momentum, pressure Poisson
equations and get flow field quantities, such as
velocity, turbulence intensity, pressure and
integral quantities (drag forces)
40
Numerical parameters & flow
solution

Typical time
history of
residuals
 The closer the
flow field to the
converged
solution, the
smaller the speed
of the residuals
decreasing.
Solution converged, residuals do
not change after more iterations
41
Post-processing

Analysis, and visualization
– Calculation of derived variables
Vorticity
 Wall shear stress
– Calculation of integral parameters: forces,
moments
– Visualization (usually with commercial software)
 Simple X-Y plots
 Simple 2D contours
 3D contour carpet plots
 Vector plots and streamlines (streamlines are
the lines whose tangent direction is the same
as the velocity vectors)
 Animations (dozens of sample pictures in a
series of time were shown continuously)

42
Post-processing (Parallel Plates)
43
Post-Processing (example)

Pressure contour and
velocity vectors .
 Note the locations of
the highest and lowest
pressure regions.
44
Uncertainty assessment

Rigorous methodology for uncertainty assessment using
statistical and engineering concepts
– Verification: process for assessing simulation numerical
uncertainty


Iterative convergence: monitoring point & integral quantities should
change within the convergence criterions
Grid independent studies: 3-grids and Richardson Extrapolation
– Validation: process for assessing simulation modeling uncertainty
by using benchmark experimental data

Certification: full Verification and Validation done for a
certain range of geometries & parameters which are well
known and then extrapolated, qualitatively as well as
quantitative
– Simulating flows for which experiments are difficult (e.g., full-
scale Reynolds numbers, hypersonic flows, off-design conditions)
– Objective: Simulation-based design
45
CFD Example
Sulzer Chemtech
250 Y Plastic
Structured Packing
46
Geometry
• CT > STL > CFD
• CT = 0.322 mm
Min Resolution
• Copy/Pasted 2x
• Surface Wrapping
• Adaptive Meshing
• Tetrahedral Mesh
• Polyhedral Mesh
47
Mess Dimensions
48
Experiment vs. Simulation
160
Simulation
N2 - July 27
N2 - July 28
150
140
130
Pressure Loss (Pa)
120
110
100
y = 23.462x1.8022
R² = 0.9998
90
80
y = 21.97910x1.76234
R² = 0.99996
70
60
50
40
30
20
10
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
F-factor (ft/s*[lb/ft3]1/2)
2.25
2.5
2.75
49
3
Velocity Map
50
Software and resources



CFD software was built upon physics, modeling, numerics.
Two types of available software
– Commercial (e.g., FLUENT, CFX, Star-CCM, COMSOL)
– Research (e.g., CFDSHIP-IOWA, U2RANS)
More information on CFD can be got on the following website:
– CFD Online: http://www.cfd-online.com/
– CFD software
 FLUENT: http://www.fluent.com/
 COMSOL http://www.comsol.com/
 CD-adapco: http://www.cd-adapco.com/
– Grid generation software
 Gridgen: http://www.pointwise.com
 GridPro: http://www.gridpro.com/
– Visualization software
 Tecplot: http://www.amtec.com/
 Fieldview: http://www.ilight.com/
51