Dr Edmondo Minisci EngAn3-CFD

Engineering Analysis 3 (16363)

2012-2013 / 2nd semester

Introduction to Computational Fluid

Dynamics ( CFD )
Dr Edmondo MINISCI
edmondo.minisci@strath.ac.uk
Twitter: https://twitter.com/edmondo_minisci https://twitter.com/MAE_Strath
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Dr Edmondo Minisci EngAn3-CFD

Lecture 5

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CFD in engineering designs and …

CFD is a computer based technique to predict fluid flows and heat transfer.

Now a must during design processes and forecasting within a wide range of
industries
Aerospace industry:
Automotive industry:

Meteorology:

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CFD in engineering designs

The CFD analysis focuses on distributed properties.

The objective is the determination of entire fields such as temperature T(x,

t ) velocity v(x, t ), density ρ(x, t ), etc.

Even when an integral characteristic, such as the friction coefficient,

aerodynamic coefficient, or the net rate of heat transfer, is needed, it is
derived from distributed fields.

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Introduction to CFD [1]

Except for few strongly simplified models, the equations for distributed
properties
are partial differential equations (PDEs), often nonlinear, which are solved
analytically (exact solutions, which are only possible for a very limited
class of problems, typically formulated in an artificial, idealized way.)

or

using numerical methods (approximate solutions)

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Formulation of a PDE problem [1]

A complete PDE problem consists of
- an equation
- a domain of solution
- boundary and initial conditions.

The problem has to be solved in a spatial domain, Ω, and, in the case of

time-dependency, in a time interval between t0 and tend .

Domains can finite of infinite.

Boundary conditions have to be imposed at the boundaries of the spatial

domain ∂Ω.
Only a problem with properly set boundary conditions is well-posed (i.e.,
consistent and having a unique solution).

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Formulation of a PDE problem [2]

Physical boundary conditions are usually expressed in terms of the
boundary values of
- the unknown field u (Dirichlet boundary condition) or
- its normal derivative (Neumann boundary condition), or, again,
- a mix of both (Robin, mixed, boundary condition).

In general, all fluid flow and heat transfer processes evolve with time, but
depending on their behaviour it is advisable to consider two different
groups:
- time-independent and
- time-evolving.

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Steps of numerical approach

1- Modeling: definition of the mathematical model describing the phenomenon

2- Discretisation of the domain and the equations

3- Solution of the discretised equations

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Discretisation [4]
 we also discretise the fluid dynamic equations (PDEs) and transform them
into linear algebraic equations

PDEs are valid over continuum domain: i.e, at any position

Discretisation: transform a Partial Differential Equation into

a set of (linear) algebraic equations

Discretized equations are valid on discrete domain or grid

 the most common PDEs discretisation methods are

1. the finite difference method (FD)
2. the finite element method (FE)
3. the finite volume method (FV)

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A Taylor series expansion, which gives the value of a function f at position x+Dx
located close to position x , reads:

Dx 2 Dx 3 ''' Dx 4 '''' Dx 5 '''''

f ( x  Dx)  f ( x)  Dxf ' ( x)  f ' ' ( x)  f ( x)  f ( x)  f ( x)...
2! 3! 4! 5!

The following expression is a finite difference which represents an approximation

to a certain partial derivative at the node labelled i.

Af i 3  Bf i  2  Cf i 1  Df i
Dx 2
where the following notations have been used :

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

1. Identify the corresponding derivative order

2. Find the values of coefficients A to D

3. Identify the order of the approximation.

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Af i 3  Bf i  2  Cf i 1  Df i
 f ' '  xi   O??
Dx 2

f ( xi  3Dx)  f ( x )  3Dxf ' ( x ) 

 3Dx 
2
f ''(x ) 
 3Dx 
3
f '''
(x ) 
 3Dx 
4
f ''''
(x ) 
 3Dx 
5
f ''''' ( xi )  ...
i i i i i
2! 3! 4! 5!

f ( xi  2Dx)  f ( xi )  2Dxf ' ( xi ) 

 2Dx 
2
f ' ' ( xi ) 
  2Dx  '''
3
f ( xi ) 
 2Dx  ''''
4
f ( xi ) 
 2Dx  '''''
5
f ( xi )  ...
2! 3! 4! 5!

f ( xi  Dx)  f ( xi )  Dxf ' ( xi ) 

 Dx 
2
f ' ' ( xi ) 
  Dx  '''
3
f ( xi ) 
 Dx  ''''
4
f ( xi ) 
 Dx  '''''
5
f ( xi )  ...
2! 3! 4! 5!
f ( xi )  f ( xi )

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

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Af i 3  Bf i  2  Cf i 1  Df i
 f ' '  xi   O??
Dx 2

Af ( xi  3Dx)  Af ( x )  A3Dxf ' ( x )  A

 3Dx 
2
f ''(x )  A
 3Dx 
3
f '''
(x )  A
 3Dx 
4
f ''''
(x )  A
 3Dx 
5
f ''''' ( xi )  ...
i i i i i
2! 3! 4! 5!

Bf ( xi  2Dx)  Bf ( xi )  B 2Dxf ' ( xi )  B

 2Dx 
2
f ' ' ( xi )  B
  2Dx  '''
3
f ( xi )  B
 2Dx  ''''
4
f ( xi )  B
  2Dx  '''''
5
f ( xi )  ..
2! 3! 4! 5!

Cf ( xi  Dx)  Cf ( xi )  CDxf ' ( xi )  C

 Dx 
2
f ' ' ( xi )  C
 Dx  '''
3
f ( xi )  C
 Dx  ''''
4
f ( xi )  C
 Dx  '''''
5
f ( xi )  ...
2! 3! 4! 5!
Df ( xi )  Df ( xi )

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

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Af i 3  Bf i  2  Cf i 1  Df i
 f ' '  xi   O??
Dx 2

Af ( xi  3Dx)  Af ( x )  A3Dxf ' ( x )  A

 3Dx 
2
f ''(x )  A
 3Dx 
3
f '''
(x )  A
 3Dx 
4
f ''''
(x )  A
 3Dx 
5
f ''''' ( xi )  ...
i i i i i
2! 3! 4! 5!

Bf ( xi  2Dx)  Bf ( xi )  B 2Dxf ' ( xi )  B

 2Dx 
2
f ' ' ( xi )  B
  2Dx  '''
3
f ( xi )  B
 2Dx  ''''
4
f ( xi )  B
  2Dx  '''''
5
f ( xi )  ..
2! 3! 4! 5!

Cf ( xi  Dx)  Cf ( xi )  CDxf ' ( xi )  C

 Dx 
2
f ' ' ( xi )  C
 Dx  '''
3
f ( xi )  C
 Dx  ''''
4
f ( xi )  C
 Dx  '''''
5
f ( xi )  ...
2! 3! 4! 5!
Df ( xi )  Df ( xi )

Af i 3  Bf i  2  Cf i 1  Df i  A  B  C  D   3 A  2 B  C  f ' ( x )  9 / 2 A  4 / 2 B  1 / 2C  f ' ' ( x ) 

 f ( x ) 
Dx 2 Dx 2 Dx
i i i

  27 / 6 A  8 / 6 B  1 / 6C  f ''' ( xi ) 
 81 / 24 A  16 / 24 B  1 / 24C  f '''' ( xi )  ...

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

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Af i 3  Bf i  2  Cf i 1  Df i
 f ' '  xi   O??
Dx 2

Af i 3  Bf i  2  Cf i 1  Df i  A  B  C  D   3 A  2 B  C  f ' ( x )  9 / 2 A  4 / 2 B  1 / 2C  f ' ' ( x ) 

 f (x ) 
Dx 2 Dx 2 Dx
i i i

  27 / 6 A  8 / 6 B  1 / 6C Dxf ''' ( xi ) 
 81 / 24 A  16 / 24 B  1 / 24C Dx 2 f '''' ( xi )  ...

 A  B  C  D   0 A  1
 3 A  2 B  C   0 B4


9 / 2 A  4 / 2 B  1 / 2C   1 C  5
 27 / 6 A  8 / 6 B  1 / 6C   0
D2

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

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Af i 3  Bf i  2  Cf i 1  Df i
Dx 2
 f ' '  xi   O Dx
?? 2
 
Second
0 order

Af i 3  Bf i  2  Cf i 1  Df i  A  B  C  D 
 f ( xi ) 
Dx 2 Dx 2 0


 3 A  2 B  C  f ' ( x ) 
1
A  1 Dx
i

B4  9 / 2 A  4 / 2 B  1 / 2C  f ' ' ( xi )  0

C  5
D2   27 / 6 A  8 / 6 B  1 / 6C Dxf ''' ( xi ) 
≠0

 81 / 24 A  16 / 24 B  1 / 24C Dx 2 f '''' ( xi )  ...

fi 3  f xi  3Dx  fi 2  f xi  2Dx  f i 1  f xi  Dx 

 Dr Edmondo Minisci EngAn3-CFD

Finite Difference Method for unsteady state heat conduction [1]

 in 1D with constant heat conductivity it is

k is the thermal
T  T 2  diffusivity
 Cv
t x 2
k is the heat
conductivity

Cv is the specific
heat at constant
volume

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 Dr Edmondo Minisci EngAn3-CFD

Finite Difference Method for unsteady state heat conduction [1]

t The time variable

T  T 2

t x 2 x The position variable

 position and time variables are independent

and therefore are discretised separately
Using forward difference to
discretise the time variation
n n 1
 Ti
n is the running index for time
 T   Ti
n
   Dt 
 t i Dt i is the running index
for position
Using second order central difference
to discretize the space variation
n
  2T  T n
 2T n
 T n
   i 1 i i 1  ( Dx 2 )
 x 2  D 2
 i x
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Finite Difference Method for unsteady state heat conduction [3]

The time dependent heat Discretised equation using forward difference in time
conduction equation
Tin 1  Tin n n n
Ti 1  2Ti  Ti 1
T  2T 

t x 2 Dt Dx 2

Transient term Spatial term

discretization discretization

n
Ti  Tin 1 n n n
Ti 1  2Ti  Ti 1

Dt Dx 2
Discretised equation using backward difference in time
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 Dr Edmondo Minisci EngAn3-CFD

Finite Difference Method for unsteady state heat conduction [4]

 the discretised equation using forward difference in time can be rearranged as :

Dt
Ti n 1
 Ti  
n

Dx 2
T n
i 1  2Ti
n
 T n
i 1 

time

n 1 the value of the temperature at Tin 1

t
depends only on temperatures known
from the previous time steps.
n
t
 in this case the numerical scheme is said,

t n 1
Explicit Scheme

xi 1 xi xi 1
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 Dr Edmondo Minisci EngAn3-CFD

Finite Difference Method for unsteady state heat conduction [5]

 the discretised equation using backward difference in time can be rearranged as

Tin  
Dt
Dx 2
T n
i 1  2Ti
n
 T n
i 1   Ti
n 1

time

t n 1 there are more than one temperature

at the highest time step t n

tn  in this case the numerical scheme is said,

Implicit Scheme

t n 1
xi 1 xi xi 1
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How good are numerical solutions?

Consistency, stability and convergence

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 Dr Edmondo Minisci EngAn3-CFD

How good are numerical solutions? [1]

 numerical solutions are approximate solutions as errors arise from each part
of the process.

 these errors are classified in three categories

1- Modeling errors: difference between actual solution and the exact solution
of the mathematical mode

2- Discretisation errors: difference between the exact solution of the system

of algebraic equations generated by the numerical
scheme and the exact analytical solution of the PDE problem
3- Round-off errors : difference between exact solution of algebraic
equations and the solution by iterative model
(i.e., solution given by the computer)

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Consistency: analysis of truncation error [1] - RECALL

 a Finite Difference is identify by replacing every term by the corresponding

Taylor series expansions in the equation

an example, consider the Finite Difference given by:

Ti 1  Ti 1
2Dx
Terms in this finite difference can be written using
Taylor series expansion by:
Dx 2 Dx3
Ti 1  T ( xi  Dx)  T ( xi )  DxT ' ( x)  T ' ' ( x)  T ' ' ' ( x)  ...
2! 3!
Dx 2 Dx3 '''
Ti 1  T ( xi  Dx)  T ( xi )  DxT ' ( x)  T ' ' ( x)  T ( x)  ...
2! 3!
then substituting we have :
2
2DxT ' ( x )  Dx3T ' ' ' ( x )  ...
Ti 1  Ti 1 3!

2Dx 2Dx
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Consistency: analysis of truncation error [2] - RECALL

which is written as

Ti 1  Ti 1 1
 T ' ( x)  Dx 2T ' ' ' ( x)  ...  T ' ( x)  ( Dx 2 )
2Dx 3!

T
 we conclude that this finite difference is a second order approximation to ( x)
x

Ti 1  Ti 1 1
 T ' ( x)  Dx 2T ' ' ' ( x)  ...  ( Dx 2 )
2Dx 3!

The truncation errors

Consistency means the discretization should become

the exact solution as the grid spacing Dx tends to zero.

Rule : Grid spacing should be chosen to minimize discretization error

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Stability analysis of numerical Schemes: basics

T  2T

Consider the heat conduction equation t x 2

Tin 1  Tin
with an associated difference equation Tin1  2Tin  Tin1

Dt Dx 2

Din 1  Din
If D denotes the exact solution of the Din1  2 Din  Din1
finite difference equation then we have 
Dt Dx 2

The solution SC computed by

the computer is not the exact SC  D   Round-off error
solution:

Because the computer is

programmed to solve the difference SC in 1  SC in SC in1  2 SC in  SC in1
equation, the evolution equation 
for SC is also
Dt Dx 2
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Stability analysis of numerical Schemes: basics

Combining equations for D and Sc it follows

 in 1   in  in1  2 in   in1

Dt Dx 2

So the iteration error evolves in time following the same numerical scheme

Stable numerical scheme is one in which the error does not increase as we
progress from time step n to step n+1

 in 1
1
i n

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Stability analysis of numerical Schemes: basics

oscillations

A decreasing residual indicating Iteration errors cause oscillation at

convergence of numerical solution early stage of the solution before
smoothing out

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