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PPT – Introduction to Computational Fluid Dynamics Lecture 5: Discretization, Finite Volume Methods PowerPoint presentation | free to download - id: 3cfa7e-NjJhM

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Introduction to Computational Fluid

DynamicsLecture 5 Discretization, Finite Volume

Methods

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.

Transport Equations

- Momentum Conservation
- T Stress tensor, n normal to the boundary
- b body force (gravity, centrifugal, Coriolis,

Lorentz etc..)

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)

Finite Volume Methods

- See class slides for finite volume methods

Discretization

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Overview

- The Task
- Why discretization?
- Discretization Methods
- Dealing with Convection and Diffusion
- Discretization Errors

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The Task

- 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

Courtesy Fluent, Inc.

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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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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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)

Courtesy Fluent, Inc.

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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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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Finite Difference Method - Application

- Consider the steady 1-dimensional

convection/diffusion equation - From the Taylor series expansion, get

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Finite Difference Method - Algebraic form of PDE

- Substitute the discrete forms of the

differentials to get

Algebraic form of PDE

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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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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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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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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

Courtesy Fluent, Inc.

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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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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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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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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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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False Diffusion (2)

Second-order Upwind

First-order Upwind

8 x 8

64 x 64

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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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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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Finite Element Method - Typical Element

- 9-noded quadrilateral
- 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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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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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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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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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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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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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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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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Designing Grids for CFD

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Outline

- Why is a grid needed?
- Element types
- Grid types
- Grid design guidelines
- Geometry
- Solution adaption
- Grid import

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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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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

2D prism (quadrilateral or quad)

triangle (tri)

prism with quadrilateral base (hexahedron or

hex)

tetrahedron(tet)

prism with triangular base (wedge)

pyramid

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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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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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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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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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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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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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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

inadequate

better

flow

OK!

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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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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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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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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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Solution Adaption

- 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
- gradients
- along a boundary
- inside a certain region

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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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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.
- Take advantage of solution adaption.

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