Transcript
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Computational Fluid Dynamics
Fluent Modeling CourseFirst: An Introduction to CFD
Lecturer: Ehsan.A.Saadati
Sharif Uniersity of !echnology
"#$ %roup&!ehran: Second Edition Fall '(()
ehsan.saadati*gmail.com
###.petrodanesh.ir
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Contents
+. Mathematical Modeling
+. %oerningE,uations'. !he %eneral Scalar !ransport E,uation
'. -umerical Methods
. Mesh !erminology and !ypes
. Discretiation Methods
.
Solution of Discretied E,uations. Accuracy/ Consistency/ Sta0ility and Conergence
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Contents
1. Fluid Flo#+. Discretiation of the Momentum E,uation
'. Discretiation of the Continuity E,uation
1. !he SIM2LE Algorithm
3. !he SIM2LE4 Algorithm
5. !he SIM2LEC Algorithm
6. Direct s. Iteratie Methods
7. Multigrid Methods
3. Solution 4esiduls
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Mathematical Modeling
%oerningE,uations
!he Momentum E,uation 8 Considered in 9&Direction
!he Energy E,uation
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Mathematical Modeling
The differential momentum equation for a Newtonian fluid with constant
density and viscosity
+
+
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=
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2
2
2
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2
2
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( ) ( ) ( )0=
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Convection Piezometric pressure gradient Viscous termsLocal acceleration
Continuity Equation ( ) ( ) ( )
0=
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u
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Mathematical Modeling
%oerningE,uations
!he Species E,uation
The equation for the conservation of mass for a chemical specie i may "e written in
terms of its mass fraction# Yi# where Yi is defined as the mass of species i per mass ofmi$ture! %f &lic'(s law is assumed valid# the governing conservation equation is
Giis the diffusion coefficient for Yiin the mi$ture and Riis the rate of
formation of Yithrough chemical reactions!
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Mathematical Classification of
2artial Differential E,uations
Elliptic2artial Differential E,uations
Let us consider steady heat conduction in a *ne+
,imensional sla"# as shown in &igure! The governing equation
and "oundary conditions are given "y
This simple pro"lem illustrates important properties of elliptic
P,Es! These are-
.+The temperature at any pointx in the domain is influencedby the temperatures on both boundaries.
/+%n the a"sence of source terms# T(x)is "ounded "y the
temperatures on the "oundaries! %t cannot "e either higher or
lower than the "oundary temperatures!
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Mathematical Classification of
2artial Differential E,uations
;yper0olic2artial Differential E,uations
consider the one+dimensional flow of a fluid in a channel# as shown in &igure! The velocity of the
fluid# 1# is a constant2 also 13! &or t35# the fluid upstream of the channel entrance is held at
temperature T! The properties 6 and Cp are constant and '5! The governing equations and "oundary
conditions are given "y-
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Mathematical Classification of
2artial Differential E,uationsThe solution to this pro"lem is-
The solution is essentially a step in T, traveling in the positive x direction with a velocity U, as
shown in igure!
8e should note the following a"out the solution-
.! The upstream boundary condition (x"9 affects the solution in the domain! Conditions
downstream of the domain do not affect the solution in the domain!
/! The inlet "oundary condition propagates with a finite speed# U.
:! The inlet "oundary condition is not felt at pointx until t" x#U
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Mathematical Classification of
2artial Differential E,uations
!emperature
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Mathematical Classification of
2artial Differential E,uations
2ara0olic2artial Differential E,uations
Consider unsteady conduction in the sla" in &igure! %f k,
and Cpare constant, $nergy Equation may "e written in terms
of the temperature Tas!
%t is clear from this pro"lem that the varia"le tbehaves
very differently from the variablex. The variation in tadmits
only one%way influences, whereas the variablexadmits two%
way influences. tis sometimes referred to as the marching or
parabolic direction.
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Mathematical Classification of
2artial Differential E,uations
Some More Specification a0out 2ara0olicE,uations Are:
.+ The "oundary temperature T& influences the temperature T(x,t) at every point in
the domain# =ust as with elliptic P,E>s!
/+ *nly initial conditions are re'uired (i.e., conditions at t"9! No final conditions are
required# for e$ample conditions at tinfinite! 8e do not need to 'now the future to solve
this pro"lem?
4+ The initial conditions only affect future temperatures, not past temperatures.
+ The initial conditions influence the temperature at every point in the domain for all
future times! The amount of influence decreases with time, and may affect different
spatial points to different degrees!)+ @ steady state is reached for t infinite ,Aere# the solution "ecomes
independent of Ti(x,&).%t also recovers its elliptic spatial "ehavior!
0+ The temperature is "ounded "y its initial and "oundary conditions in the a"sence
of source terms!
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Mathematical Classification of
2artial Differential E,uations
=ehaior of the Scalar !ransport E,uation:
The general scalar transport equation we derived earlier Bmentioned "elow9 has much
in common with the partial differential equations we have seen here!
The elliptic diffusion equation is recovered if we assume steady state and there is noflow!
The same pro"lem solved for unsteady state e$hi"its para"olic "ehavior!
The convection side of the scalar transport equation e$hi"its hyper"olic "ehavior!
%n most engineering situations# the equation e$hi"its mi$ed "ehavior# with the
diffusion terms tending to "ring in elliptic influences# and the unsteady and convection
terms "ringing in para"olic or hyper"olic influences!
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Mesh !erminology and !ypes
The physical domain is discretized "y meshing or gridding it! The fundamental unit of
the mesh is the cell Bsometimes called the element9! @ssociated with each cell is the
cell centroid. cell is surrounded "y faces, which meet at nodes or vertices. n three
dimensions, the face is a surface surrounded "y edges. %n two dimensions# faces
and edges are the same!
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Mesh !erminology and !ypes
-ode&=ased and Cell&=ased Schemes
Numerical methods which store their primary un'nowns at the node or verte$
locations are called node%based or vertex%based schemes. Those which store them
at the cell centroid# or associate them with the cell# are called cell%based schemes.
inite element methods are typically node+"ased schemes# and many finite volume
methods are cell+"ased! &or structured and "loc'+structured meshes composed of
quadrilaterals or he$ahedra# the num"er of cells is appro$imately equal to thenum"er of nodes# and the spatial resolution of "oth storage schemes is similar for
the same mesh! &or other cell shapes# there may "e quite a "ig difference in the
num"er of nodes and cells in the mesh! &or triangles# for e$ample# there are twice as
many cells as nodes# on average! This fact must "e ta'en into account in deciding
whether a given mesh provides adequate resolution for a given pro"lem! &rom the
point of view of developing numerical methods# "oth schemes have advantages and
disadvantages# and the choice will depend!
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Mesh !erminology and !ypes
4egular and =ody&fitted Meshes
%n many cases# our interest lies in analyzing domains which are regular in shape-
rectangles# cu"es# cylinders# spheres! These shapes can "e meshed "y regular
grids# as shown in &igure! The grid lines are orthogonal to each other# and conform
to the "oundaries of the domain! These meshes are also sometimes called
orthogonal meshes!
a-egular and
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Mesh !erminology and !ypes
&or many practical pro"lems# however# the domains of interest are irregularly shaped
and regular meshes may not suffice! @n e$ample is shown ! Aere# grid lines are not
necessarily orthogonal to each other# and curve to conform to the irregular geometry!
%f regular grids are used in these geometries# stair stepping occurs at domain
"oundaries# as shown! 8hen the physics at the "oundary are important indetermining the solution# e!g!# in flows dominated "y wall shear# such an
appro$imation of the "oundary may not "e accepta"le!
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Mesh !erminology and !ypes
Structured/ =loc> Structured/ and Unstructured Meshes
The meshes shown in &igure aare e$amples of structured meshes. *ere, every
interior verte$ in the domain is connected to the same num"er of neigh"or vertices!
&igure 0shows a "loc'+structured mesh! Aere# the mesh is divided into "loc's# and
the mesh within each "loc' is structured! Aowever# the arrangement of the "loc's
themselves is not necessarily structured!
&igure a
&igure "
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Mesh !erminology and !ypes
Structured %rid Created 0y an Elliptic %enerator for a -ACA Airfoil
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Mesh !erminology and !ypes
Structured/ =loc> Structured/ and Unstructured Meshes
&igure c shows an unstructured mesh! Aere# each verte$ is connected to an ar"itrary
num"er of neigh"or vertices! 1nstructured meshes impose fewer topological
restrictions on the user# and as a result# ma'e it easier to mesh very comple$
geometries!
&igure c
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Mesh !erminology and !ypes
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Mesh !erminology and !ypes
Conformal and -on&Conformal Meshes
@n e$ample of a non+conformal mesh is shown in &igure d! Aere# the vertices of a
cell or element may fall on the faces of neigh"oring cells or elements! %n contrast# the
meshes in &igures a# 0and care conformal meshes!
&igure d
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Mesh !erminology and !ypes
Cell
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Mesh !erminology and !ypes
Cell Shapes Deshes may "e constructed using a variety of cell shapes! The most widely used are
quadrilaterals and he$ahedra! Dethods for generating good+quality structured
meshes for quadrilaterals and he$ahedra have e$isted for some time now! Though
mesh structure imposes restrictions# structured quadrilaterals and he$ahedra are
well+suited for flows with a dominant direction# such as "oundary+layer flows! Dore
recently# as computational fluid dynamics is "ecoming more widely used foranalyzing industrial flows# unstructured meshes are "ecoming necessary to handle
comple$ geometries! Aere# triangles and tetrahedral are increasingly "eing used#
and mesh generation techniques for their generation are rapidly reaching maturity!
@nother recent trend is the use of hy"rid meshes! &or e$ample# prisms are used in
"oundary layers# transitioning to tetrahedral in the free+stream!
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@re
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@re
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Discretiation Methods
Finite Difference Methods The finite difference method is "ased on the Taylor series e$pansion
a"out a point#x
( )xxuu
x
x
ux
x
uuu
i
ii
ii
+
+
+
+=
+
+
asdefinediswhere
H.O.T.2
1
2
2
2
1
( )xxuu
x
x
ux
x
uuu
i
ii
ii
+
+
=
asdefinediswhere
H.O.T.2
1
2
2
2
1
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Discretiation Methods
Finite Difference Methods
&inite difference methods appro$imate the derivatives in the governing differential
equation using truncated Taylor series e$pansions!
y including the diffusion coefficient and dropping terms of *BDI/9
&inite difference methods do not e$plicitly e$ploit the conservation principle in deriving
discrete equations! Though they yield discrete equations that loo' similar to other
methods for simple cases# they are not guaranteed to do so in more complicated cases#
for e$ample on unstructured meshes!
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Discretiation Methods
Finite Element Methods
8e consider again the one+dimensional diffusion equation! There are different 'inds of finite
element methods! Let us loo' at a popular variant# the Faler'infinite element method! Let F "e
an appro$imation to F!
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Discretiation Methods
Thus# we require
The weight functions +i are typically local in that they are non%ero over element i, "ut are
zero everywhere else in the domain! &urther# we assume a shape function for F# i!e!#
assume how Fvaries "etween nodes! Typically this variation is also local! &or e$ample we
may assume that Fassumes a piece+wise linear profile "etween points . and / and
"etween points / and : in &igure /!.! The Faler'in finite element method requires that
the weight and shape functions "e the same! Performing the integration in Equation /!..
results in a set of alge"raic equations in the nodal values of F which may "e solved "y a
variety of methods! 8e should note here that "ecause the Faler'in finite element methodonly requires the residual to "e zero in some weighted sense# it does not enforce the
conservation principle in its original form! 8e now turn to a method which employs
conservation as a tool for developing discrete equations!
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Discretiation Methods
Finite
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Discretiation Methods Finite
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Discretiation Methods Finite
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Discretiation Methods Finite
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Solution of Discretiation E,uations
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Solution of Discretiation E,uations?hat are pro0lems of direct solution method@
@ solution for fis guaranteed if A&+ can "e found! Aowever# the operation count for the inversion of an N$N matri$ is of *BN/9! Consequently# inversion is almost never employed in practical pro"lems!
Dore efficient methods for linear systems are availa"le! &or the discretization methods of interest here# A is sparse# and for structured meshes it is "anded!
,irect methods are not widely used in computational fluid dynamics "ecause of large computational and storage requirements! Dost industrial C&, pro"lems today involve hundreds of thousands of cells# with +. un'nowns per cell even for simple pro"lems! Thus
the matri$ A is usually very large# and most direct methods "ecome impractical for these large pro"lems! &urthermore# the matri$ A is usually nonlinear# so that the direct method must "e em"edded within an iterative loop to update nonlinearities in A. Thus# the direct
method is applied over and over again# ma'ing it all the more time+consuming!
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Solution of Discretiation E,uations
Iteratie Methods %terative methods are the most widely used solution methods in computational fluid dynamics! These methods employ a guess+and+correctphilosophy which progressively improves the guessed solution "y repeated application
of the discrete equations!
Let us consider an e$tremely simple iterative method# the Fauss+
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Solution of Discretiation E,uations The neigh"or values# f$and f+ are re'uired for the update of f2. These are assumed 'nown at prevailing values! Thus# points which have already "een visited will have recently updated values of f and those that have not will have old values!
:!
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Accuracy @ccuracy refers to the correctness of a numerical solution when compared to an e$act solution! %t is more useful to tal' of the truncation error of a discretiation method. The truncation error associated with the diffusion term using the finite difference method is 6((DI9 /9 as shown "y Equation! This simply says that if
diffusion term is represented by the right hand side# the terms that are neglected are of 6((DI9 /9!
Thus# if we refine the mesh# we e$pect the truncation error to decrease as ((DI9 /9! %f we dou"le the $+direction mesh# we e$pect the truncation error to decrease "y a factor of four! The truncation error of a discretization scheme is the largest truncation error of each of the individual terms in the equation "eingdiscretized! The order of a discretiation method is n if its truncation error is 6((DI9 n9!
Dethods of very high order may yield inaccurate results on a given mesh! Aowever# we are guaranteed that the error will decrease more rapidly with mesh refinement than with a discretization method of lower order!
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Conergence
8e distinguish "etween two popular usages of the term convergence! 8e may say that an iterative method has converged to a solution# or that we have o"tained convergence using a particular method! y this we mean that our iterative method has successfullyo"tained a solution to our discrete alge"raic equation set ! 8e may also spea' of convergence to mesh independence! y this# we mean the process of mesh refinement# and its use in o"taining solutions that are essentially invariant with further refinement!
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Discretiation of the Momentum E,uation
Consider the two+dimensional rectangular domain shown in "elow &igure! Let us
assume for the moment that the velocity vector < and the pressure p are stored at
the cell centroids! Let us also assume steady state! The momentum equations in the
x and y directions may be written as!
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Discretiation of the Momentum E,uation
8e see that Equations )!. and )!/ have the same form as the general scalar transport equation#
and as such we 'now how to discretize most of the terms in the equation! Each momentumequation contains a pressure gradient term# which we have written separately# as well as a
source term B4u or 4v) which contains the body force term# as well as remnants of the stress
tensor term! Let us consider the pressure gradient term! %n deriving discrete equations# we
integrate the governing equations over the cell volume! This results in the integration of the
pressure gradient over the control volume! @pplying the gradient theorem# we get
@ssuming that the pressure at the face centroid represents the mean value on the face# we write
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Discretiation of the Momentum E,uation
Completing the discretization# the discrete u+ and v+momentum equations may "e
written as
Fiven a pressure field# we thus 'now how to discretize the momentum equations!
Aowever# the pressure field must "e computed# and the e$tra equation we need for its
computation is the continuity equation! Let us e$amine its discretization ne$t!
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Discretiation of the Continuity E,uation
&or steady flow# the continuity equation# which ta'es the form-
%ntegrating over the cell 2 and applying the divergence theorem, we get
@ssuming that the 6
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Determining Density and 2ressure
&or practical C&, pro"lems# sequential iterative solution procedures are frequently adopted
"ecause of low storage requirements and reasona"le convergence rate! Aowever there is a difficultyassociated with the sequential solution of the continuity and momentum equations for incompressi"le
flows! %n order to solve a set of discrete equations iteratively# it is necessary to associate the discrete
set with a particular varia"le! &or e$ample# we use the discrete energy equation to solve for the
temperature!
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Determining Density and 2ressure
Density =ased for Compressi0le Flo#s
2ressure =ased for Incompressi0le flo#s
There are a num"er of methods in the literatures which use the density as a primary
varia"le rather than pressure! This practice is especially popular in the compressi"le flow
community! &or compressi"le flows# pressure and density are related through an equation of
state! %t is possi"le to find the density using the continuity equation# and to deduce the pressure
from it for use in the momentum equations!
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Determining Density and 2ressure
Direct Solution is "ut of 4ich
Se,uential Iteratie Method ?ill =e Used
%t is important to realize that the necessity for pressure+ and density+"ased
schemes is directly tied to our decision to solve our governing equations sequentially
and iteratively! %t is this choice that forces us to associate each governing differential
equation with a solution varia"le! %f we were to use a direct method# and solve for thediscrete velocities and pressure using a domain+wide matri$ of size :1x:1, (three
variables % u#v#p + over N cells9# no such association is necessary! Aowever# even with
the power of today>s computers# direct solutions of this type are still out of reach for most
pro"lems of practical interest! *ther methods# including local direct solutions at each
cell# coupled to an iterative sweep# have "een proposed "ut are not pursued here!
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!he SIM2LE Algorithm
The SIMPLEBSemi+Implicit Method for 2ressure Lin'ed Equations9 algorithm and
its variants are a set of pressure+"ased methods widely used in the incompressi"le flow
community! The primary idea "ehind
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!he SIM2LE Algorithm
%f we su"tract equation of momentum which using initial pressures from the momentum
equation uses real pressure valuesBsatisfies continuity equation9 we can drive an
equation for relation "etween pressure correction and velocities correction values!
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!he SIM2LE Algorithm
8e now ma'e an important simplification! 8e appro$imate resultant Equations as
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!he SIM2LE Algorithm
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!he SIM2LE Algorithm
earranging the last equation we ta'e "elow one for pressure correction which can
"e solved iteratively
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!he SIM2LE4 Algorithm
The SIMPLERalgorithm has "een shown to perform "etter than SIMPLE! This is primarily "ecause the
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!he SIM2LEC Algorithm
The
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Direct s. Iteratie Methods
Linear solution methods can "roadly "e classified into two categories# direct or iterative! ,irectmethods# such as Fauss elimination# L1 decomposition etc!# typically do not ta'e advantage of matri$
sparsity and involve a fi$ed num"er of operations to o"tain the final solution which is determined to
machine accuracy! They also do not ta'e advantage of any initial guess of the solution! Fiven the
characteristics of the linear systems outlined a"ove# it is easy to see why they are rarely used in C&,
applications! %terative methods on the other hand# can easily "e formulated to ta'e advantage of the
matri$ sparsity!
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Multigrid
8e saw "efore that the reason for slow convergence of Gauss-Seidelmethod is
that it is only effective at removing high frequency errors! 8e also o"served that low
frequency modes appear more oscillatory on coarser grids and then the Fauss+
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Multigrid
8e 'now that in general the accuracy of the solution depends on the discretization2therefore we would require that our final solution "e determined only "y the finest
gridthat we are employing! This means that the coarse grids can only provide us
with corrections or guesses to the fine grid solution and as the fine grid residuals
approach zero
*ne strategy for involving coarse levels might "e to solve the original differential
equation on a coarse grid! *nce we have a converged solution on this coarse grid#
we could interpolate it to a finer grid! *f course# the interpolated solution will not in
general satisfy the discrete equations at the fine level "ut it would pro"a"ly "e a
"etter appro$imation than an ar"itrary initial guess! 8e can repeat the process
recursively on even finer grids till we reach the desired grid!
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Multigrid
Coarse %rid Correction
@ more useful strategy# 'nown as coarse grid correction is based on the error e'uationecall that the error esatisfiesthesamesetof equations as our solutionif we replace the source vector "y the residual-
we can already see that the strategy outlined a"ove has the desired properties! &irst of all# note that if the fine
grid solution is e$act# the residual will "e zero and thus the solution of the coarse level equation will also "e zero!
Thus we are guaranteed that the final solution is only determined "y the finest level discretization! %n addition#
since we start with a zero initial guess for the coarse level error# we will achieve convergence right away and not
waste any time on further coarse level iterations! @nother useful characteristic of this approach is that we use
coarse level to only estimate fine level errors! Thus any appro$imations we ma'e in the coarse level pro"lem only
effect the convergence rate and not the final finest grid solution!
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Multigrid
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Solution 4esiduals
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Solution 4esiduals
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Solution 4esiduals
@t the end of each solver iteration# the residual sum for each of the conserved varia"les is computed and stored! *n a
computer with infinite precision# these residuals will go to zero as the solution converges! *n an actual computer# theresiduals decay to some small value Bround+off9 and then stop changing Blevel out9! &or single+precision computations
Bthe default for wor'stations and most computers9# residuals can drop as many as si$ orders of magnitude "efore hitting
round+off! ,ou"le+precision residuals can drop up to twelve orders of magnitude!
Definition of 4esiduals for the 2ressure&=ased Soler
@fter discretization# the conservation equation for a general varia"le at a cell P can "e written as
Aere aP is the center coefficient# an" are the influence coefficients for the neigh"oring cells# and " is the contri"ution of
the constant part of the source term
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Solution 4esiduals
This is referred to as the Mun+scaled residual! %t may "e written as
This Mscaled residual is defined as
The scaled residual is a more appropriate indicator of convergence for most pro"lems
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Solution 4esiduals
The pressure+"ased solver(s scaled residual for the continuity equation is defined as
The denominator is the largest a"solute value of the continuity residual in the first five iterations!
%t is sometimes useful to determine how much a residual has decreased during calculations as an
additional measure of convergence! &or this purpose# &L1ENT allows you to normalize the residual
Beither scaled or un+scaled9 "y dividing "y the ma$imum residual value after D iterations# where D is set
"y you in the esidual Donitors panel in the Iterationsfield under -ormaliation!
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4eferences
Purdue 1niversity+ C&, oo'let
&luent )!: ,ocumentation
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!he End
y- Ehsan
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