Finite Difference Method and Finite Volume Method

1. Introduction to CFD
Mr.S.Velmurugan.
Assistant Professor,
Aeronautical Engineering,
KIT-kalaignar karunanidhi Institute of
Technology, Coimbatore-641 407.

2. 1

3. What is computational fluid dynamics?
• Computational fluid dynamics (CFD) is a branch of continuum
mechanics which deals with numerical simulation of fluid flow and
heat transfer
• The result of CFD analyses is relevant engineering data used in:
– Conceptual studies of new designs.
– Detailed product development.
– Troubleshooting.
– Redesign.
• CFD analysis complements testing and experimentation.
– Reduces the total effort required in the laboratory.
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4. Knowledge Prequisite
• Fluid dynamics
• Numerical Methods
• Heat transfer
Knowledge on Specific topics are required for specific applications
• Combustion
• Mass transfer with multispecies and multiphase
• Melting and solidification
• Rotodynamics
• Heat exchangers
• Aerospace
• Automotive
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5. Fluid dynamics
• Fluid dynamics is the science of fluid motion.
• Fluid flow is commonly studied in one of three ways:
– Experimental fluid dynamics (EFD).
– Theoretical fluid dynamics (TFD).
– Numerically: computational fluid dynamics (CFD).
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6. Numerical Vs Analytical Vs Experimental
• Experimental Investigations:
- full scale
• expensive and often impossible
• measurement errors
- on a scaled model
• simplified
• difficult to extrapolate results
• measurement errors
• Theoretical calculation:
- analytical solutions
• exist only for a few cases (example)
• sometimes complex
- numerical solutions
• for almost any problem 5

7. Modelling
• Advantages of modelling:
- cheaper
- more complete information
- capable of solving any complex problem
• Disadvantages of modelling:
- deals with a mathematical description not with reality
- multiple solutions can exist
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8. Why use CFD?
• Relatively low cost.
– Using physical experiments and tests to get essential engineering
data for design can be expensive.
– CFD simulations are relatively inexpensive, and costs are likely to
decrease as computers become more powerful.
• Speed.
– CFD simulations can be executed in a short period of time.
– Quick turnaround means engineering data can be introduced early in
the design process.
• Ability to simulate real conditions.
– Many flow and heat transfer processes can not be (easily) tested,
e.g. hypersonic flow.
– CFD provides the ability to theoretically simulate any physical
condition.
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9. Why use CFD?
• Ability to simulate ideal conditions.
– CFD allows great control over the physical process, and provides the
ability to isolate specific phenomena for study.
– Example: a heat transfer process can be idealized with adiabatic,
constant heat flux, or constant temperature boundaries.
• Comprehensive information.
– Experiments only permit data to be extracted at a limited number of
locations in the system (e.g. pressure and temperature probes, heat
flux gauges, LDV, etc.).
– CFD allows the analyst to examine a large number of locations in the
region of interest, and yields a comprehensive set of flow parameters
for examination.
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10. Boundary conditions
Prescribed Temperature BC (First kind) Dirichlet conditions
Prescribed Heat Flux BC (Second kind) Neumann conditions
Convection BC (Third kind) Robbins cinditions
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11. Boundary conditions cont…
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12. Boundary conditions cont…
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13. Boundary conditions cont…
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14. Where is CFD used?
• Where is CFD
used?
• Aerospace
• Automotive
• Biomedical
• Chemical
Processing
• HVAC
• Hydraulics
• Marine
• Turbomachine
• Power Generation
• Sports
Temperature and natural
convection currents in the eye
following laser heating. 13

15. Chemical Processing
Streamlines for workstation
ventilation
Where is CFD used?
• Where is CFD used?
• Aerospacee
• Automotive
• Biomedical
• Chemical
Processing
• HVAC
• Hydraulics
• Marine
• Turbomachine
• Power Generation
• Sports
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16. Where is CFD used?
• Where is CFD used?
• Aerospace
• Automotive
• Biomedical
• Chemical Processing
• HVAC
• Hydraulics
• Marine
• Turbomachine
• Power Generation
• Sports
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Power Generation
Turbomachine
Marine (wave pattern)
Sports

17. Post processor
•X-Y graph
•Contour
•Velocity vectors
•others
Pre processor
• Creation of geometry
• Mesh generation
• Material properties
• Boundary conditions
Solver settings
•Initialization
•Solution control
•Monitoring solution
•Convergence criteria
Physical Model
•Turbulence
•Combustion
•Radiation
•Other processes
Transport equation
•Mass
•Momentum
•energy
•Equation of state
•Supporting models
Three main elements of CFD Software
Solver
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18. CFD - how it works (2)
• CFD applies numerical methods (called
discretization) to develop approximations of
the governing equations of fluid mechanics in
the fluid region of interest.
– Governing differential equations: algebraic.
– The collection of cells is called the grid.
– The set of algebraic equations are solved
numerically (on a computer) for the flow field
variables at each node or cell.
– System of equations are solved
simultaneously to provide solution.
• The solution is post-processed to extract
quantities of interest (e.g. lift, drag, torque,
heat transfer, separation, pressure loss, etc.).
Mesh for bottle filling
problem.
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19. 
Discretization
• Domain is discretized into a finite set of control volumes
or cells. The discretized domain is called the “grid” or the “mesh.”
• General conservation (transport) equations for mass, momentum,
energy, etc., are discretized into algebraic equations.
• All equations are solved to render flow field.
 
t
div u  div grad  S

t V
dV  VdA 
A
 dA  A V
S dV Fluid region of
pipe flow
unsteady convection diffusion generation
control
volume
discretized into
finite set of
control volumes
(mesh).
Other Discretization methods:
FDM, FEM 18
Eqn. 
continuity 1
x-mom. u
y-mom. v
energy h

20. Discretization Methods
Classification of PDE
Elliptic, Parabolic & Hyperbolic Eqns
2 2 2  
cont…
a
x2
 b
xy

c
y2

d
x

e
y
 f   g 0
where coefficients are constants or fn's of the independent variables
b2  4ac 
b2  4ac 
b2  4ac 
0 elliptic
0 parabolic
0 hyperbolic
(a) Elliptic PDE
2

x2
2
y2
 0  Laplace eqn &
2

x2
2
y2
 g(x,y)  Poisson's eqn
where, b=0, a=1 and c=1 b
2
 4ac  4  0

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22. Discretization Methods
Classification of PDE
(b) Parabolic PDE
cont…

 
2
t x2
(heat conduction or diffusion equation)
where  is thermal diffusivity and b=0, c=0, a=
b2  4ac  0
(c) Hyperbolic PDE
2

t2
 2
2

x2
(wave equation)
where 2 is a constant and a=2 , b=0, c=-1
b
2
 4ac  42 which is >0
Example:

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24. Discretization Methods
Classification of PDE
cont…
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25. Discretization Method
Finite Difference Method
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26. Finite Difference Method cont…
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27. Finite Difference Method cont…
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28. Finite Difference Method cont…
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29. Finite Difference Method cont…
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30. Finite Difference Method cont…
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31. Finite Difference Method cont…
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32. Finite Difference Method cont…
4 1 1 0  T1  100 1 4 0 1  T   30 
   2    
 1 0 4 1  T3
 100
0 1 1 4
 
T
 
30

   4    29

33. Finite Difference Method
Nume rical Er ror s: Numerical errors arise during computations due
to round-off errors and truncation errors.
(i) Round-off error
It is the difference between an approximation of a number used in
computation and its exact (correct) value.
e.g. 4.756 is rounded off to 4.76
(ii) Truncation error
It is the difference between an exact expression and the corresponding
truncated form used in the numerical solution.
Note: The difference between exact solution and numerical solution
with no round-off error is called Discretization error.
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34. Grid Independence Test
It is nothing but the testing of how numerical solution has become
independent of grid spacing
Variation in the heat transfer rate with respect to the mesh count.
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600
500
400
Selectedmeshcount
300
(520000)
200
100
0 2 4 6 8 10
Meshcountx105
HeatTransferRate(W)

35. Discretization Method
Finite Volume Method
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36. Finite Volume Method cont…
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37. Finite Volume Method cont…
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38. Finite Volume Method cont…
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39. Finite Volume Method cont…
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40. Finite Volume Method cont…
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41. Finite Volume Method cont…
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42. Finite Volume Method cont…
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43. Finite Volume Method cont…
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44. Useful Web links
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45. Useful Text books
1] “An Introduction to CFD – The Finite Volume Method” by
H K Versteeg & W Malalasekara , Publication:: Pearson Education Ltd
2] “Numerical Heat Transfer and Fluid Flow” by Suhas V. Patankar,
Publication:: Taylor & Francis
3] “Computer simulation of Flow and Heat Transfer” by P.S. Ghoshdastidar,
Publication:: TMH
4] “CFD A Practical Approach” by Jiyuan Tu, Guan Yeoh & ChaoqunLiu,
Publication:: Elsevier
4] “The FEM for Fluid Dynamics” by O.C. Zienkiewicz, R.L. Taylor &
P. Nithiarasu, Publication:: Elsevier
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