other cfd methodsgtryggva/cfd-course2010/2010...computational fluid dynamics i! where! and at each...

Computational Fluid Dynamics I

Other!CFD Methods!

Grétar Tryggvason!Spring 2010!

http://users.wpi.edu/~gretar/me612.html!Computational Fluid Dynamics I

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!Vortex Methods and other particle methods!

Here we will briefly examine those!


Spectral !

Methods!


Consider the initial-boundary value problem for the one dimensional heat equation!

By standard technique (Fourier sine transform) the solution is!

�

∂f∂t

= ∂2 f

∂x 20 < x < π, t ≥ 0

f (0,t) = f (π,t) = 0f (x,0) = f (x)

�

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

∞

∑an (t) = fne

−n 2t

fn =2π

f (x)sinnx dx0

π∫


Assuming that the time integration is exact, the error by using only N terms is!

�

f − fN ≈ e−N 2t for all time!

This is faster than any power of N. Recall that the error for finite difference/finite volume methods is!

�

O(hα ) ≈ 1N α

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:!

�

dandt

= −n2an; an (0) = fn (n =1,2,…N)


For nonlinear equations the use of spectral methods becomes much more involved!

�

∂f∂t

+ f ∂f∂x

= ν ∂2 f∂x 2

Expand the solution in a Fourier series!

�

f (x,t) = ak (t)e−i 2π

Lkx

|k|


�

Ni(1) = x − xi−1

Δxi; Ni−1

(1) = xi − xΔxi

�

Ni(2) = xi+1 − x

Δxi+1; Ni+1

(2) = x − xiΔxi+1

For element 2:!

For element 1:!

�

i

�

i +1

�

i −1�

Ni(1)

�

Ni−1(1)

�

Ni(2)

�

Ni+1(2)

Linear basis functions!

element 1! element 2!


�

∂Ni(1)

∂x= 1Δxi

; ∂Ni−1(1)

∂x= −1Δxi

�

Ni(1) = x − xi−1

Δxi; Ni−1

(1) = xi − xΔxi

�

Ni(2) = xi+1 − x

Δxi+1; Ni+1

(2) = x − xiΔxi+1

For element 2:!

For element 1:!

�

∂Ni(2)

∂x= −1Δxi+1

; ∂Ni+1(2)

∂x= 1Δxi+1


Example: Elliptic Problem!

�

∂∂x

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = q

�

R = ∂∂x

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ − q

�

W (x)R(x)dxΩ∫ = 0

�

W ∂∂x

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ dx

Ω∫ = qWdxΩ∫

The residual is:!

The weighted residual is:!

Or:!

W(x): Weighting function!


Integrate by parts!

�

W ∂∂x

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ dx

Ω∫ = qWdxΩ∫

�

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ∂W∂x

dxΩ∫ +

∂∂xW k ∂f

∂x⎛ ⎝ ⎜

⎞ ⎠ ⎟ dx

Ω∫ = qWdxΩ∫

�

k ∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ∂W∂x

dxΩ∫ + W k

∂f∂x

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥ 2− W k ∂f

∂x⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥ 1

= qWdxΩ∫

Evaluating the second term:!

Cancel for adjacent elements!


Galerkin Method!Take weighting function equal to the interpolation function!

�

f x( ) = f I NI x( )I∑

k ∂ f∂x

⎛⎝⎜

⎞⎠⎟∂W∂x

dxΩ∫ = qW dxΩ∫

For node i:!

f jj= i−1

i+1

∑ k ∂N j∂xi−1i+1

∫∂Ni∂x

dx − qNi (x)i−1

i+1

∫ dx = 0

�

∂Ni(1)

∂x= 1Δxi

;

∂Ni−1(1)

∂x= −1Δxi

�

∂Ni(2)

∂x= −1Δxi+1

∂Ni+1(2)

∂x= 1Δxi+1


f jj= i−1

i+1

∑ k ∂N j∂xi−1i+1

∫∂Ni∂x

dx − qNi (x)i−1

i+1

∫ dx = 0

ki+1/2fi+1 − fiΔx2

− ki−1/2fi − fi−1Δx2

−qi−1 + 4qi + qi+1

6= 0

ki+1/2 =1Δx

ki

i+1

∫ dx

integrating!

where! Notice that a finite difference method would give the same result except with qi on the RHS! �

∂Ni(1)

∂x= 1Δxi

;

∂Ni−1(1)

∂x= −1Δxi

�

∂Ni(2)

∂x= −1Δxi+1

∂Ni+1(2)

∂x= 1Δxi+1


Finite Element approximation of an unsteady problem:!

�

∂f∂t

+U ∂f∂x

= 0

�

f x( ) = f I NI x( )I∑�

∂f∂tW dx

Ω∫ + U ∂f∂xW dxΩ∫

= 0

Weak formulation!

Introduce the finite element representation!


�

f x( ) = f I NI x( )I∑

�

∂f∂tW dx

Ω∫ + U ∂f∂xW dxΩ∫

= 0

Introduce the finite element representation!

�

∂f I∂tI

∑ NINJ dxΩ∫ +U NI dNJdx dxΩ∫

= 0

or!

�

∂f I∂t

MIJ +U KIJI∑ = 0

I∑

�

MIJ = NINJ dxΩ∫ ; KIJ = NI dNJdx dxΩ∫

;

“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!


Lattice Boltzmann Methods!


Lattice Boltzmann methods use “particles” on hexagonal grids where the particles move according to discrete rules. It can be shown that the macroscopic motion of the particles resembles the Navier-Stokes equations. It is now believed that these methods can yield results comparable in accuracy to other numerical methods!

Lattice Boltzmann methods !


where!

and at each node!

�

e1�

e2

�

e5

�

e6

�

e3

�

e7

�

e4

�

e8

Update the velocity distributions at each node by:!

∂ fγ∂t

+ eγ ⋅∇fv =1ετ

fγ(eq) − fγ( ), γ = 0,1,,M

ρ = fγγ =1

M

∑ ρu = fγγ =1

M

∑ eγ

fγ(eq) = wγ ρ 1+ 3 eγ ⋅u( ) + 92 eγ ⋅u( )

2−32u ⋅u( )2⎡

⎣⎢⎤⎦⎥

Lattice Boltzmann methods !Computational Fluid Dynamics I

�

Δt = Δx = Δy = ε

�

α = Δt eγ Δx ; β = Δx ε

�

fγ x,t + Δt( ) − fγ x,t( ) = −α fγ x,t( ) − fγ x −Δteγ ,t( )( )− βτ

fγ x −Δteγ ,t( ) − fγ(eq ) x −Δteγ ,t( )( )where!

taking! And replacing!

�

fγ x + Δteγ ,t + Δt( ) − fγ x,t( ) = 1τ fγ(eq ) x,t( ) − fγ x,t( )( )

�

∂fγ∂t

+ eγ ⋅ ∇fv =1ετ

fγ(eq ) − fγ( ), γ = 0,1,,M

Approximate the time derivative by a first order forward in time, the advection term by upwind and the collision by downwind, we get!

�

Δx = Δt eγ

�

x + Δteγ yields!by!

�

x



�

Δt = Δx = Δy = ε

�

α = Δt eγ Δx ; β = Δx ε

�

fγ x,t + Δt( ) − fγ x,t( ) = −α fγ x,t( ) − fγ x −Δteγ ,t( )( ) − βτ fγ x −Δteγ ,t( ) − fγ(eq ) x −Δteγ ,t( )( )

Set!

taking!

Replace!

�


�

Δx = Δt eγ

�

x + Δteγby!

�

x

�

fγ x + Δteγ ,t + Δt( ) − fγ x + Δteγ ,t( ) = −Δt eγΔx

fγ x + Δteγ ,t( ) − fγ x,t( )( ) − Δt eγετ fγ x,t( ) − fγ(eq ) x,t( )( )

And crossing out the common term yields!

Proof!Lattice Boltzmann methods !


eγ =ΔxΔt

cos π2

⎛⎝⎜

⎞⎠⎟γ −1( ),sin π

2⎛⎝⎜

⎞⎠⎟γ −1( )⎛

⎝⎜⎞⎠⎟, γ = 1,3,5,7

eγ = 2ΔxΔt

cos π2

⎛⎝⎜

⎞⎠⎟γ −1( ) + π

4,sin π

2⎛⎝⎜

⎞⎠⎟γ −1( ) + π

4⎛⎝⎜

⎞⎠⎟, γ = 2,4,6,8

�

e0 = 0

w0 = 0wγ = 4 9 γ = 1,3,5,7wγ = 1 36 v = 2,4,6,8 �

e1�

e2

�

e5

�

e6

�

e3

�

e7

�

e4

�

e8

The weights must be taken to be!

The displacement vectors are!



�

e1�

e2

�

e5

�

e6

�

e3

�

e7

�

e4

�

e8

�


�

ρ = fγγ =1

M

∑

�

ρu = fγγ =1

M

∑ eγ�

fγ(eq ) = wγ ρ 1+ 3 eγ ⋅u( ) + 92 eγ ⋅u( )

2− 32u ⋅u( )2⎡ ⎣ ⎢

⎤ ⎦ ⎥

where!

and at each node!

Update the velocity distributions at each node by:!Summary:!

Lattice Boltzmann methods !Computational Fluid Dynamics I


Advantages!

Very easy to program–parallelization easy!

Fast–in part because incompressibility is not enforced!

Disadvantages!

Difficult to implement new physics—must find a discrete equation that corresponds to the new physics!

The flow is not completely incompressible!



By now it is understood that LBM are give results comparable to second order finite difference/volume methods!

LBM has been extended to more complex flow, including multiphase flows with sharp interfaces!

The figure shows a comparison of the unsteady rise of a buoyant bubble computed by LBM (left) and front tracking finite 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) 109–116.!


Particle Methods!


Particle methods cover a large range of physical situations, some overlapping with more conventional methods. Particle methods include:!

!Point vortex/vortex blob methods!

!Particle-in-Cell codes!!Smoothed particle hydrodynamics!!Dissipative particle methods!!Molecular dynamics!!and many more!

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


Vortex Methods!


�

∇2φ = σSolution to!

The solution in a two-dimensional unbounded domain is!

�

φ(x) = 12π

σ (x')ln rda'A∫

The solution in a three-dimensional unbounded domain is!

�

φ(x) = −14π

σ (x')r

dv 'A∫

�

r = x − x'

Vortex Methods!Computational Fluid Dynamics I

�

u x( ) = −∂ψ∂y,∂ψ∂x

⎛

⎝ ⎜

⎞

⎠ ⎟ =

−12π

ω(x') −∂ ln r∂y

,∂ ln r∂x

⎛

⎝ ⎜

⎞

⎠ ⎟ da'A∫�

∇2ψ = −ω

�

∂ ln r∂y

= ∂∂yln (x − x ')2 + (y − y ')2( ) = (y − y ')(x − x')2 + (y − y')2�

u x( ) = −12π

ω(x') −(y − y '),(x − x ')( )r2

da'A∫

�

ψ(x) = −12π

ω(x')ln rda'A∫Solution!

Velocity!

since!

Vortex Methods!


�

∇2ψ = −ω

Euler Equation for two-dimensional inviscid incompressible flow!

�

dωdt

= 0

�

u = ∇ × (ψk)

�

u x( ) = 12π

KA∫ x,x'( )ω x'( )dA'

where K is the appropriate velocity kernal!

The vorticity of each material particle is constant!


�

K x,x'( ) = k × x − x'( )x − x' 2

= (−(y − y '),(x − x'))r2

For unbounded domain!�

u x( ) = 12π

KA∫ x,x'( )ω x'( )dA'

The velocity depends only on the vorticity distribution!

Vortex Methods!


�

u x( ) = 12π

KA∫ x,x'( )ω x'( )dA'

= 12π

K x,x'( )ω x'( )dA'A j∫

j=1

N

∑

= 12π

K x,x'( ) ω x'( )dA'A j∫

j=1

N

∑

= 12π

K x,x'( )Γ jj=1

N

∑�

x

�

x'j

�

x'j�

x'j

�

δ


�

dxidt

= 12π

K xi ,x j( )Γ jj=1

N

∑

�

ddt(xi,yi) =

12π

Γ j (−(yi − y j ),(xi − x j ))(xi − x j )

2 + (yi − y j )2

j=1j≠ i

N

∑

Point vortices in an unbounded domain. The motion of each point is determined by !

Vortex Methods!


Use point vortices to approximate a smooth distribution of vorticity !


�

ddt(xi,yi) =

12π

Γ j (−(yi − y j ),(xi − x j ))(xi − x j )

2 + (yi − y j )2 + δ 2j=1

N

∑

Generally, the point vorticies are too singular to make a practical numerical method and the point vortices must be regularized. The simplest way is to find the velocity by !

Numerical parameter !

Vortex Methods!


Small viscosity can be added to vortex methods either by “random walk” or localized averaging!

In three-dimension it is necessary to account also for stretching and tilting of vortex lines, but the basic methodology still works!


High Reynolds number density current!

Example!

Vortex Methods!


The N2 problem:!To find the velocity of each vortex, it is necessary to sum over all the other vortices. This leads to large computational times when the number of vortices, N, is large!

Remedies:!Grid based Vortex-In-Cell methods!Fast summation methods!


Vortex-In-Cell!

Advect point vortices, but solve !

�

∇2ψ = −ωto find the velocities!

Vortex Methods!


Fast summation method!

A vortex far away from a group of vortices “sees” the group as a single large vortex!

By grouping the vortices together in an intelligent way, it is possible to reduce the operation count significantly !


Related Methods!

!Panel and boundary integral !! !method for flow over solid bodies!!Boundary Integral Methods for free !! !surface flows!!Contour dynamics methods for “patches”!! !of constant vorticity!

Vortex Methods!


For more information:!

Vortex Methods: Theory and Applications by Georges-Henri Cottet & Petros D. Koumoutsakos !


Other Particle Methods!

Smooth Particle Hydrodynamics: the fluid mass is lumped into smoothed blobs that are moved using Newtonʼs second law directly, without an underlying mesh!

Dissipative Particles Dynamics: particles represents molecules or fluid blobs but while the method has similarities with SPH, it is generally 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 CFD technique but can be used to simulate small (really small) fluid regions!


There is no doubt that new solution strategies will continue to be developed. However, incremental advances of current approaches are likely to be the main vehicle for future advances in the computations of complex flows. The finite volume approach currently at the core of CFD has, in particular, proven to be exceedingly robust and versatile.!

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