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## Introduction to Computational Fluid Dynamics Lecture 5: Discretization, Finite Volume Methods

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Title: Introduction to Computational Fluid Dynamics Lecture 5: Discretization, Finite Volume Methods

1
Introduction to Computational Fluid
DynamicsLecture 5 Discretization, Finite Volume
Methods
2
Transport Equations
• Mass conservation
• The integral form of mass conservation equation
is
• where ? is the density in domain O , v the
velocity of the fluid and n the unit normal to
the boundary, S.

3
Transport Equations
• Momentum Conservation
• T Stress tensor, n normal to the boundary
• b body force (gravity, centrifugal, Coriolis,
Lorentz etc..)

4
Transport Equations
• Energy transport
• T temperature, k thermal conductivity, c
specific heat at constant pressure, Q heat flux
• (Species transport is similar no specific heat
term)

5
Finite Volume Methods
• See class slides for finite volume methods

6
Discretization

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7
Overview
• Why discretization?
• Discretization Methods
• Dealing with Convection and Diffusion
• Discretization Errors

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8
• The Navier-Stokes equations equations governing
the motion of fluid, in this instance, around a
vehicle, are highly non-linear, second order
partial differential equations (PDEs)
• Exact solutions only exist for a small class of
simple flows, e.g., laminar flow past a flat
plate
• A numerical solution of a PDE or system of PDEs
consists of a set of numbers from which the
distribution of the variable f can be obtained
from the set
• The variable f is determined at a finite number
of locations known as grid points or cells. This
number can be large or small

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9
What is discretization?
• Discretization is the method of approximating the
differential equations by a system of algebraic
equations for the variables at some set of
discrete locations in space and time
• The discrete locations are grid/mesh points or
cells
• The continuous information from the exact
solution of PDEs is replaced with discrete
values

Pipe discretized into cells
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10
Discretizing the domain
• Transforming the physical model into a form in
which the equations governing the flow physics
can be solved can be referred to as discretizing
the domain

Illustration of the cells
Discretized domain
Continuous domain
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11
Solving the PDEs
• The are a number of methods for the solution of
the governing PDEs on the discretized domain
• The most important discretization methods are
• Finite Difference Method (FDM)
• Finite Volume Method (FVM)
• Finite Element Method (FEM)

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12
Finite Difference Method - Introduction
• Oldest method for the numerical solution of PDEs
• Procedure
• Start with the conservation equation in
differential form
• Solution domain is covered by grid
• Approximate the differential equation at each
grid point by approximating the partial
derivatives from the nodal values of the function
giving one algebraic equation per grid point
• Solve the resulting algebraic equations for the
whole grid. At each grid point you solve for the
unknown variable value and the value of its
neighboring grid points

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13
Finite Difference Method - Concept
• The finite difference method is based on the
Taylor series expansion about a point, x

Subtracting the two eqns above gives
Adding the two eqns above gives
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14
Finite Difference Method - Application
• Consider the steady 1-dimensional
convection/diffusion equation
• From the Taylor series expansion, get

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15
Finite Difference Method - Algebraic form of PDE
• Substitute the discrete forms of the
differentials to get

Algebraic form of PDE
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16
Finite Difference Method - Summary
• Discretized the one-dimensional
convection/diffusion equation
• The derivatives were determined from a Taylor
series expansion
• Advantages of FDM simple and effective on
structured grids
• Disadvantages of FDM conservation is not
enforced unless with special treatment,
restricted to simple geometries

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17
Finite Volume Method - Introduction
• Using Finite Volume Method, the solution domain
is subdivided into a finite number of small
control volumes by a grid
• The grid defines to boundaries of the control
volumes while the computational node lies at the
center of the control volume
• The advantage of FVM is that the integral
conservation is satisfied exactly over the
control volume

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18
Finite Volume Method - Typical Control Volume
• The net flux through the control volume boundary
is the sum of integrals over the four control
volume faces (six in 3D). The control volumes do
not overlap
• The value of the integrand is not available at
the control volume faces and is determined by
interpolation

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19
Finite Volume Method - Application
• Consider the one-dimensional convection/diffusion
equation
• The finite volume method (FVM) uses the integral
form of the conservation equations over the
control volume
• Integrating the above equation in the x-direction
across faces e and w of the control volume and
leaving out the source term gives
• The values of f at the faces e and w are needed

N
n
E
P
W
e
w
s
S
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20
Finite Volume Method - Interpolation
• Using a piecewise-linear interpolation between
control volume centers gives

• linear interpolation between nodes
• face is midway between nodes
• equivalent to Central Difference Scheme (CDS)

where
• Under assumption of continuity, discrete form of
PDE from FVM is identical to FDM

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21
Finite Volume Method - Exact Solution
• Exact solution with boundary conditions
• f fo at x 0, f fL at x L
• Peclet number, Pe, is the ratio of the strengths
of convection and diffusion
• When the Peclet number is high (positive or
negative), the profile is highly non-linear
• The Central-Difference Scheme relies on a linear
interpolation which will fail to capture the
gradient changes in the variable f

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22
Finite Volume Method - Interpolation
• The piecewise-linear or CDS interpolation may
give rise to numerical errors (oscillatory or
checkerboard solutions). CDS was used only as an
example of discretization and is inappropriate
for most convection/diffusion flows
• A large number of interpolation techniques in
FLUENT software that are improvements on the CDS.
Some of these, in increasing level of accuracy,
are
• First-Order Upwind Scheme
• Power Law Scheme
• Second-Order Upwind Scheme
• Higher Order
• Blended Second-Order Upwind/Central Difference
• Quadratic Upwind Interpolation (QUICK)

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23
Sources of Numerical Errors - FDM FVM
• Discretization Errors from inexact interpolation
of nonlinear profile (FVM)
• Truncation Errors due to exclusion of Higher
Order Terms (FDM)
• Domain discretization not well resolved to
capture flow physics
• Artificial or False Diffusion due to
interpolation method and grid

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24
False Diffusion (1)
• False diffusion is numerically introduced
diffusion and arises in convection dominant
flows, i.e., high Pe number flows
• Consider the problem below
• If there is no false diffusion, the temperature
along the diagonal will be 100ºC
• False diffusion will occur due to the oblique
flow direction and non-zero gradient of
temperature in the direction normal to the flow
• Grid refinement coupled with a higher-order
interpolation scheme will minimize the false
diffusion

Diffusion set to zero k0
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25
False Diffusion (2)
Second-order Upwind
First-order Upwind
8 x 8
64 x 64
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26
Finite Volume Method - Summary
• The FVM uses the integral conservation equation
applied to control volumes which subdivide the
solution domain, and to the entire solution
domain
• The variable values at the faces of the control
volume are determined by interpolation. False
diffusion can arise depending on the choice of
interpolation scheme
• The grid must be refined to reduce smearing of
the solution as shown in the last example
• Advantages of FVM Integral conservation is
exactly satisfied, Not limited to grid type
(structured or unstructured, cartesian or
body-fitted)

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27
Finite Element Method - Introduction
• Using Finite Element Method, the solution domain
is subdivided into a finite number of small
elements by a grid
• The grid defines to boundaries of the elements
and location of nodes for higher-order
elements, there can be mid-side nodes also
• FEM uses multi-dimensional shape functions which
afford geometric flexibility and limit false
diffusion

Mid-side node
Finite element
Computational node
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28
Finite Element Method - Typical Element
• Within each element, the velocity and pressure
fields are approximated by
• where ui, vi, pi are the nodal point
unknowns and fi and yi are interpolation
functions
• Quadratic approximation for velocity, linear
approximation for pressure required to avoid
spurious pressure modes

nodes with u, v, p
nodes with u, v
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29
Finite Element Method - Interpolation
• The solution on an element is represented as

fi
where N are the basis functions. We choose
basis functions that are 1 at one node of the
element and 0 at all other the nodes.
fi1
node
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30
Finite Element Method - Application
• Recall the one-dimensional convection/diffusion
equation
• Most often, the finite element method (FEM) uses
the Method of Weighted Residuals to discretize
the equation
• Multiply governing equation by weight function Wi
and integrate over the element
• How do we choose the Wi? For Galerkin FEM,
replace Wi by Ni, the shape or basis functions

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31
Finite Element Method - Weak form
• Use integration by parts to obtain the weak
formulation involves first derivatives rather
than second derivatives
• We can now substitute the interpolation function
for f
• and evaluate the required integrals to
produce the discrete equation

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32
Finite Element Method - Wiggles (1)
• Wiggles occur in FEM when linear weighting
functions are used
• Typical cures are
• Petrov-Galerkin method where the weighting
function is different from the shape function.
The weighting function is asymmetric, being
skewed in the direction of the upwind element

Wi
u
Ni
i
i1
i-1
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33
Finite Element Method - Wiggles (2)
• Artificial tensor viscosity
• the viscosity in the streamline direction is
augmented
• this is equivalent to skewing the weighting of
the convection terms towards the upstream
direction

flow
shapes of Galerkin and Streamline Upwind
weighting functions for nodal point C
D
U
C
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34
Finite Element Method - Summary
• FEM solves the weak form of the governing
equations
• weak form requires continuity of lower order
operators only
• very similar to using the divergence theorem in
FVM
• The technique is conservative in a weighted sense
• The weight functions an easily be made
multi-dimensional
• this limits false diffusion

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35
Summary
• The concept of discretization was introduced.
The three main methods of discretization, FDM,
FVM, and FEM were detailed.
• The one-dimensional convection/diffusion was used
to illustrate the different methods of
discretizing the partial differential equations
into algebraic equations.
• The process of discretization was shown to
introduce errors (though minimal in some cases)
through truncation of higher order terms (FDM),
approximation of the integrals (FVM).
• The correct interpolation of the convective
fluxes across the cell faces minimizes errors in
both discretization methods.

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36
Designing Grids for CFD

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37
Outline
• Why is a grid needed?
• Element types
• Grid types
• Grid design guidelines
• Geometry
• Grid import

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38
Why is a grid needed?
• The grid
• designates the cells or elements on which the
flow is solved
• is a discrete representation of the geometry of
the problem
• has cells grouped into boundary zones where
b.c.s are applied
• The grid has a significant impact on
• rate of convergence (or even lack of convergence)
• solution accuracy
• CPU time required

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39
Element Types
• Many different cell/element and grid types are
available choice depends on the problem and the
solver capabilities
• Cell or element types
• 2D
• 3D

triangle (tri)
prism with quadrilateral base (hexahedron or
hex)
tetrahedron(tet)
prism with triangular base (wedge)
pyramid
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40
Grid Types (1)
• Single-block, structured grid
• i,j,k indexing to locate neighboring cells
• grid lines must pass all through domain
• Obviously cant be used for very complicated
geometries

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41
Grid Types (2)
• Multi-block, structured grid
• uses i,j,k indexing with each block of mesh
• grid can be made up of (somewhat)
arbitrarily-connected blocks
• More flexible than single block, but still limited

cross-section
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42
Grid Types (3)
• Unstructured grid
• tri or tet cells arranged in arbitrary fashion
• no grid index, no constraints on cell layout
• There is some memory/CPU overhead for
unstructured referencing

CFD stands for Cow Fluid Dynamics!
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43
Grid Types (4)
• Hybrid grid
• use the most appropriate cell type in any
combination
• triangles and quadrilaterals in 2D
• tetrahedra, prisms and pyramids in 3D
• can be non-conformal grids lines dont need to
match at block boundaries

prism layer efficiently resolves boundary layer
tetrahedral volume mesh is generated automatically
triangular surface mesh on car body is quick and
easy to create
non-conformal interface
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44
Grid Design Guidelines Quality (1)
• Quality cells/elements are not highly skewed
• Two methods for determining skewness
• 1. Based on the equilateral volume
• Skewness
• Applies only to triangles and tetrahedra

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45
Grid Design Guidelines Quality (1)
• 2. Based on the deviation from a normalized
equilateral angle
• Skewness (for a quad)
• Applies to all cell and face shapes
• High skewness values ? inaccurate solutions
slow convergence
• Keep maximum skewness of volume mesh lt 0.95
• Possible classification based on skewness

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46
Grid Design Guidelines Resolution
• Pertinent flow features should be adequately
resolved
• Cell aspect ratio (width/height) should be near 1
where flow is multi-dimensional
• Quad/hex cells can be stretched where flow is
fully-developed and essentially one-dimensional

better
flow
OK!
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47
Grid Design Guidelines Smoothness
• Change in cell/element size should be gradual
(smooth)
• Ideally, the maximum change in grid spacing
should be 20

smooth change in cell size
sudden change in cell size AVOID!

Dxi
Dxi1
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48
Grid Design Guidelines Total Cell Count
• More cells can give higher accuracy downside is
increased memory and CPU time
• To keep cell count down
• use a non-uniform grid to cluster cells only
where theyre needed
• use solution adaption to further refine only
selected areas

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49
Geometry
• The starting point for all problems is a
geometry
• The geometry describes the shape of the problem
to be analyzed
• Can consist of volumes, faces (surfaces), edges
(curves) and vertices (points).

Geometry can be very simple...
or more complex
geometry for a cube
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50
Geometry Creation
• A good preprocessor provides tools for creating
and modifying geometry.
• Geometry can also be imported from other CAD
programs.
• Various file types exist
• IGES
• ACIS
• STL
• STEP
• DXF
• various proprietary (Universal files, etc.)

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51
• Q how do you ensure adequate grid resolution,
when you dont necessarily know the flow
features?
• A solution-based grid adaption!
• The grid can be refined or coarsened by the
solver based on the developing flow
• solution values
• along a boundary
• inside a certain region

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52
Grid Import
• Grids can also be created by many CAD programs
(I-DEAS, Patran, ANSYS, etc.).
• These can be imported into the solver.
• Be sure to check grid quality!
• A grid acceptable for stress analysis may not be
good enough for CFD.
• Repair/improve if necessary.

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53
Summary
• Design and construction of a quality grid is
crucial to the success of the CFD analysis.
• Appropriate choice of grid type depends on
• geometric complexity
• flow field
• cell/element types supported by solver
• Hybrid meshing offers the greatest flexibility.