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```Fluid Flow:
Application of Numerical Methods
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Section 5 – Fluid Flow
Objectives
Module 2: Numerical Methods
Page 2

Understand the application of numerical methods.


Investigate discretization of equations.

Compare different numerical methods.

Understand the process of numerical analysis.

Become familiar with the use of CFD software, such as Autodesk
Simulation Multiphysics.
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Section 5 – Fluid Flow
Understanding Numerical Methods
Module 2: Numerical Methods
Page 3

Numerical Methods are used when an approximate analysis can be
deemed accurate enough.

Due to the nonlinear and complex nature of flow equations, exact
solutions are possible for only a handful of cases.

Even when using numerical methods, simplifications have to be

Three discretization schemes used in numerical methods are:
Finite Element Method
(FEM)
Finite Difference Method
(FDM)
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Finite Volume Method
(FVM)
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Section 5 – Fluid Flow
Domain Discretization: Part I
Module 2: Numerical Methods
Page 4

In Numerical Fluid flow analysis, a continuous domain is replaced by
a discrete domain using a grid.

In a continuous domain, a result (e.g., velocity) can be found at any
point in the domain.
Continuous Domain

In a discrete domain, results are calculated only at the grid points
(nodes) or at the centers of control volumes (CVs) defined by those
grid points.


Discrete Domain
Values for other positions are extrapolated from grid point results.
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Section 5 – Fluid Flow
Domain Discretization: Part II
Module 2: Numerical Methods
Page 5

Similarly, when solving fluid flow in a CFD software application, the
flow domain must be discretized into a number of nodes.

These elements can be quadrilateral or triangular.
Cell
Triangular Mesh
Nodes
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Section 5 – Fluid Flow
Discretization of Equations- Techniques
Module 2: Numerical Methods
Page 6

Numerical discretization techniques used in commercially popular
applications are:
Finite Element Method (FEM) – popular in structural mechanics
 Finite Volume Method (FVM) – popular in CFD
 Finite Difference Method (FDM) – popular in optimization and flow studies
requiring less accuracy
 The differences lie in how the equations are discretized, or converted into
discrete form over a number of points.

FEM is mainly popular for
structural analysis (left)
but can also be applied for
CFD analysis (right)
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Discretization of Equations
Taylor Series: Part I
Section 5 – Fluid Flow
Module 2: Numerical Methods
Page 7

Once the domain is discretized, the equation also needs to be
discretized, or converted into discrete form over a number of points.

The Taylor Series is given below:
df
d 2 f (x)
d n f (x) n
f ( x  x)  f ( x)  (x)  2
 ......... n
dx
dx 2
dx
n!
from this series:
df { f ( x  x)  f ( x)}

dx
x
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Section 5 – Fluid Flow
Discretization of Equations
Taylor Series: Part II

Module 2: Numerical Methods
Page 8
Replace the continuum with discrete points:
x  ....,xi 1 , xi , xi 1 ,....
u  ....,ui 1 , ui , ui 1 ,....

Approximate derivatives:

Central

Backward

Forward
ui 1  ui 1
 u 
  
2  x
 x  i
ui  ui 1
 u 
  
x
 x  i
ui 1  ui
 u 
  
x
 x  i
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Section 5 – Fluid Flow
Discretization of Equations, Taylor Series III
Module 2: Numerical Methods
Page 9

Example of partial differential
equation (PDE) with spatial and
temporal derivatives:

For Space discretization


Index “i” is used with Backward
differencing scheme.
For Time discretization
Index “n” is used with Forward
differencing scheme.
 Notice that the PDE has been reduced
to an algebraic equation.

u
u
 a
0
t
x
ui  ui 1
 u 
   a 
x
 t i
n 1
i
u
u u
 a 
t
x
a  x n
n 1
n
n
ui  ui 
(ui  ui 1 )
t
u
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n
i
n
i
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n
i 1
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Section 5 – Fluid Flow
Discretization of Equations
Finite Difference Method: Part I

Module 2: Numerical Methods
Page 10
A system of flow governed by the following equation:
 u 
   ui  0
 x i
x
1

2
4
Can be discretized using Taylor series algebraic equations:
 u1  (1  x) u2  0
 u2  (1  x) u3  0
 u3  (1  x) u4  0
i  2
i  3
i  4
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Discretization of Equations
Finite Difference Method: Part II

Section 5 – Fluid Flow
Module 2: Numerical Methods
Page 11
If there is a boundary condition (B.C) of u1=0, then:
0
0
0   u1  1
1


 1 1  x



0
0  u2  0


0
 1 1  x
0  u 3  0 

   
0
 1 1  x  u4  0
0




This matrix can be solved using a direct or iterative matrix method.
More nodes = more equations to solve.
The higher the accuracy of a Taylor Series, the more terms in the
equation.
A computer can greatly help to solve the complex system of
equations resulting from a large, finely meshed domain.
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Section 5 – Fluid Flow
FDM vs “FEM” and “FVM”
Module 2: Numerical Methods
Page 12

FDM is an easy to implement, easy to understand and easy to
program scheme.

FDM does not show good results for unstructured meshes.

Compared to FEM and FVM, FDM is very a crude scheme.

In-house CFD codes based on FDM do exist, but most commercial
software for CFD are based on either FEM or FVM.

In the next slide, differences between FEM and FVM are explored.
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Section 5 – Fluid Flow
Comparison between FEM and FVM for CFD
Module 2: Numerical Methods
Page 13
FEM
Solves both structural mechanics and
flow/thermal problems
Equations are discretized over a number
of points
More stable compared to FVM
Requires high amount of memory, limits
solution of large flow domains
Is capable of solving cases involving
Fluid–Solid Interaction (FSI)
Solves non-Newtonian fluid flow (e.g.,
plastic flow in molds) much better than
FVM
Discretizes conservative form of
equations
FVM
Is used only for flow/thermal problems
Governing equations are solved over
discrete control volumes (CV)
Less stable, convergence can sometimes
require manipulation
Requires less memory, a mesh with up to
5 million CVs can be solved on a PC
Schemes for FVM based FSI have been
devised, but are difficult to implement
Can solve non-Newtonian fluids, but not
as effective as FEM
Recasts and discretizes integral form of
equations
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Section 5 – Fluid Flow
Process of Numerical Analysis
Module 2: Numerical Methods
Page 14

To solve a problem numerically, the following steps are required:
(First simplify geometry if possible)
 Establishing problem boundaries and flow assumptions
(e.g., inlet/outlet, walls, density constant)
 Discretization of the domain
 Generation of equations for each nodal point (by using FDM, FEA)
 Solving those equations (using direct or iterative matrix scheme)
Actual Geometry
Simplified Geometry
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Geometry
simplification
often
involves elimination of unnecessary
curves and details that may have
negligible or no influence on the
flow. This helps mesh creation or
domain discretization by reducing
complexity.
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Section 5 – Fluid Flow
Flow Process
Module 2: Numerical Methods
Page 15
FVM
Numerical
Analysis
Simplification
FDM
Initial / Boundary
Conditions
FEM
Discretization
Solving
Convergence
Results
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Section 5 – Fluid Flow
Questions for establishing workflow
Module 2: Numerical Methods
Page 16

The first step of the analysis process is to formulate the flow problem by
seeking answers to the following questions:
 What is the objective of the analysis?
 What is the easiest way to obtain that objective?
 What geometry should be included?
 What are the freestream and/or operating conditions?
 What dimensionality
•
of the spatial model is required?
(1D, 2D, axisymmetric, 3D)
 What should the flow domain look like?
 What is the nature of the viscous flow? (inviscid,
laminar, turbulent)
 How should the fluid be modelled? (compressible or incompressible)
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Section 5 – Fluid Flow
Using CFD software
Module 2: Numerical Methods
Page 17

A CFD software application breaks down the analysis using the
following steps:
Pre-Processing
Solving (number crunching)
Post-Processing
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Section 5 – Fluid Flow
Video: Application of Numerical Methods
Module 2: Numerical Methods
Page 18

The video for this module on application of numerical methods
covers:
 Domain discretization
 Discretization of equations
 The concept of numerical analysis
 How computers have helped
 Types of discretization and their applications
Measuring the
circumference
of a circle
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Section 5 – Fluid Flow
Summary
Module 2: Numerical Methods
Page 19

Navier–Stokes is a complex equation and can be highly nonlinear for
many flow cases.

There are relatively few cases where an exact solution to this
equation can be found, and they involve a great amount of
assumptions and simplification.

We replace these equations with small linear equations which are
applicable at very small intervals.

This is called domain discretization and discretization of equations.

The result is a large number of simultaneous equations.
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Section 5 – Fluid Flow
Summary
Module 2: Numerical Methods
Page 20

To solve these equations, computers are used.

Because of advancements in computer technology, large flow
domains can now be solved.

FEA, FDM and FVM are different types of discretizing schemes that
have found applications in different areas.

For instance, FVM is widely popular for CFD.

FEA is used largely in structural analyses and also in complex CFD
problems