2011-lecture-29

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Computational Fluid Dynamics

Other

CFD Methods

Grtar Tryggvason!Spring 2011!

http://www.nd.edu/~gtryggva/CFD-Course/!


Finite Volume Methods are used in the vast majority

of CFD codes. However, there are other ways to

compute the motion of fluids. Some of the more

common alternatives include:!

Spectral Methods!

Finite Element Methods!

Lattice Boltzmann Methods!

Smoothed Particle Hydrodynamics!

Vortex Methods and other particle methods!

Here we will briefly examine those!


Spectral

Methods


Consider the initial-boundary value problem for the onedimensional heat equation!

By standard

technique

(Fourier sinetransform) the

solution is!

f

t=

2f

x2

0 < x < , t 0

f(0,t) = f(,t) = 0

f(x,0) = f(x)

f(x, t) = an(t)sinnxn=1

an(t) = fnen 2t

fn =2

f(x)sinnx dx0


Assuming that the time integration is exact, the error by

using only N terms is!

f fN eN2t for all time!

This is faster than any power of N. Recall that the error

for finite difference/finite volume methods is!

O(h)

1

N

Spectral methods are said to have spectral or infinite

order accuracy!

This property is not retained if the solution has

discontinuities!


A spectral approximation to the heat equation is now

simply!

f(x,t) = an(t)sinnxn=1

N

Sum to N only!!

Substituting into the heat equation we get:!

dan

dt= n

2an; an (0) = fn (n =1,2,N)

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For nonlinear equations the use of spectral methods

becomes much more involved!

f

t+ f

f

x=

2f

x2

Expand the solution in a Fourier series!

f(x,t) = ak(t)ei

2

Lkx

|k|< K

dal

dt+

i

L3/ 2

l alm am =

2l

L

|m|< K

|l -m|< K

2

al

We get the following equation for the coefficients:!


Computing the nonlinear term can becomeexpensive, particularly for three-dimensional

systems. By using the Fast Fourier Transform,the computation can be made faster by taking

the inverse transform and do the multiplication

in real space. Then Fourier transform theproduct back into spectral space. The

multiplication can, however, lead to aliasingproblems where the resolved wave numbers arecontaminated by unresolved ones.!


The fast Fourier transform!

fj fle

i2

Nl j

l=1

N

Can be evaluated in 2N log2N operations!

inverse!

fl fjei

2

Nlj

l=1

N

The Cooley-Tukey algorithm!


Although spectral methods can have very highaccuracy, they can be difficult to apply to nonlinear

problems, problems with complex boundaryconditions, and complicated geometries.!

In addition to Fourier series, several other basisfunction can be used for other geometries and to

control the resolution near boundaries!

Spectral elements methods, where a complexdomain is broken into large blocks and each block

is treated spectrally can be used treat somewhat

complex boundaries !


Finite

ElementMethods


f x( ) = fINI x( )I

f

x= fI

NI

xI

Finite element methods are similar to spectral methods

in that we expand the solution in terms of a known basisfunction. Unlike spectral methods, where the basisfunctions are defined globally over the whole

computational domain, in the finite element method thebasis functions are defined locally on each element.!

Element 1! Element 2!

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Ni

(1)=

x xi1

xi

; Ni1

(1)=

xi x

xi

Ni

(2)=

xi+1 x

xi+1

; Ni+1

(2)=

x xi

xi+1

For element 2:!

For element 1:!

i

i +1

i1

Ni

(1)

Ni1

(1)

Ni

(2)

Ni+1

(2)

Linear basis functions!

element 1! element 2!


Ni

(1)

x=

1

xi

;N

i1

(1)

x=

1

xi

Ni

(1)=

x x i1

xi

; Ni1

(1)=

xi x

xi

Ni

(2)=

xi+1 x

xi+1

; Ni+1

(2)=

x xi

xi+1

For element 2:!

For element 1:!

Ni

(2)

x=

1

xi+1

;N

i+1

(2)

x=

1

xi+1


Example: Elliptic Problem!

xkf

x

= q

R =

xkf

x

q

W(x)R(x)dx

= 0

W x

kfx

dx

= qWdx

The residual is:!

The weighted residual is:!

Or:!

W(x): Weighting function!


Integrate by parts!

W

xkf

x

dx

= qWdx

kf

x

W

xdx

+

xW k

f

x

dx

= q Wdx

kf

x

W

xdx

+ W kf

x

2

W kf

x

1

= q Wdx

Evaluating the second term:!

Cancel for adjacent elements!


Galerkin Method!

Take weighting function equal to theinterpolation function!

f x( ) = fINI x( )I

kf

x

W

x

dx

= q W dx

For node i:!

fjj=i1

i+1

kNj

xi1

i+1

Ni

xdx qNi (x)

i1

i+1

dx = 0

Ni

(1)

x

=

1

x i

;

Ni1

(1)

x=

1

xi

Ni

(2)

x=

1

xi+1

Ni+1

(2)

x=

1

xi+1


fjj=i1

i+1

kNj

xi1

i+1

Ni

xdx qNi (x)

i1

i+1

dx = 0

ki+1/2fi+1 fi

x2

ki1/2fi fi1

x2

qi1 + 4qi + qi+1

6= 0

ki+1/2

=1

xk

i

i+1

dx

integrating!

where! Notice that a finite

difference method

would give the sameresult except with q

i

on the RHS!

Ni

(1)

x

=

1

x i

;

Ni1

(1)

x=

1

xi

Ni

(2)

x=

1

xi+1

Ni+1

(2)

x=

1

xi+1

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Finite Element approximation of anunsteady problem:!

f

t+U

f

x= 0

f x( ) = fINI x( )I

f

tW dx

+ Uf

xW dx

= 0

Weak formulation!

Introduce the finite element representation!


f x( )=

fINI x( )I

f

tW dx+ U

f

x W dx= 0

Introduce the finite element representation!

fI

tI

NINJ dx

+ U NIdNJ

dxdx

= 0

or!

fI

tM

IJ+ U K

IJ

I

= 0I

MIJ

= NIN

Jdx

; KIJ = NIdN

J

dxdx

;

Mass matrix! Stiffness matrix!


The mass matrix makes it complex to use

simple explicit time integrators. This can

be overcome by lumping the matrix!


In two-dimensions triangular (or tedrahedron)elements are frequently used!


While finite elements dominate solid

mechanics, their role in computational fluiddynamics has been secondary to finitevolume methods. Indeed, most

methodological progress has been firstdeveloped in the context of finite volumes

and then implemented in the finite element

framework. Some areas, such assimulations of viscoelastic flows have,

however, relied heavily on finite elementsfor historical reasons.!


Lattice Boltzmann

Methods

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Computational Fluid DynamicsLattice Boltzmann methods !

Advantages!

Very easy to programparallelization easy!

Fastin part because incompressibility is not enforced

Disadvantages!

Difficult to implement new physicsmust find a

discrete equation that corresponds to the newphysics!

The flow is not completely incompressible!

Computational Fluid DynamicsLattice Boltzmann methods !

By now it is understood that

LBM are give resultscomparable to second orderfinite difference/volumemethods!

LBM has been extended tomore complex flow, includingmultiphase flows with sharpinterfaces!

The figure shows a comparisonof the unsteady rise of abuoyant bubble computed byLBM (left) and front trackingfinite volume method (right).!

From: K. Sankaranarayanan, I.G. Kevrekidis, S.

Sundaresan , J. Lu, G. Tryggvason. A comparative

study of lattice Boltzmann and front-tracking finite-difference methods for bubble simulations. Int. J.

Multiphase Flow 29 (2003) 109116.!


Smooth ParticleHydrodynamics


Smooth Particle Hydrodynamics: the fluid mass is

lumped into smoothed blobs that are moved using

Newtons second law directly, without an underlying mesh!

Smooth Particle Hydrodynamics!

x

1 x

2 x

3


In SPH the fluid is modeled as a collection of smooth blobs

or particles !

A(r) = A(r ') | r r ' |( )dr '

We start by the relationship!

And then approximate the deltafunction by a smoother kernal!

A(r) = A(r ')W(| r r ' |,h) dr '

W(r,h) dr =1 & W(r,h) h0 r( )

Where:!

W(r,h) =1

3/2hexp

r

2

h2

We could select thekernal to be a

Gaussian, althoughusually we desire acompact support!



i= (r

i) = m

k

k

k

k

W(| ri r

k|,h) = m

k

k

W(ri rk,h),

A(r) = A(r ')W(| r r ' |,h) dr '

= A(r ')(r ')

(r ')W(| r r ' |,h) dr '

= mk

k A

k

k

W(| r rk

|,h)


m

k (r ') dr '

Approximate the integral by a sum over

the smoothed particles:!

For density we have!

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Other

ParticleMethods

Computational Fluid DynamicsParticle methods cover a large range of physical situations, someoverlapping with more conventional methods. Particle methodsinclude:!

!Point vortex/vortex blob methods!

!Particle-in-Cell codes!

!Direct Simulation Monte Carlo (rarified flow)!

!Dissipative particle methods !

!Molecular dynamics!

!and many more!

In many cases a grid is used also, either for efficiency or toaccount for some of the physics!

Not really forcontinuum flows!


Dissipative Particles Dynamics: particles

represents molecules or fluid blobs but while

the method has similarities with SPH, it isgenerally used to model mesoscale behavior

of complex fluids and more liberties are taken

in modeling the interactions of the particles!

Molecular Dynamics: Not really a CFDtechnique but can be used to simulate small(really small) fluid regions!

2011-lecture-29

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