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### Transcript of Integral Solutions to the Poisson Equation gtryggva/CFD-Course/2011-Lecture-30.pdfآ  the Poisson...

• Computational Fluid Dynamics

Vortex Methods!

Grétar Tryggvason! Spring 2011!

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

Flow over a body!

Irrotational outer flow!

Boundary layer: the flow is viscous and rotational!

Rotational wake!

Computational Fluid Dynamics

Integral Solutions to the Poisson

Equation !

Computational Fluid Dynamics

∇ 2φ = σ

∇2φ = 1

r 2 ∂ ∂r

r 2 ∂φ ∂r

= σδ r( )

1 r 2

d dr

r 2 dφ dr

= 0 ⇒ d r 2 dφ dr

⎛ ⎝⎜

⎞ ⎠⎟ = 0 ⇒

dφ dr

= C r 2

⇒φ = − C r 2

To evaluate the constant we integrate the equation over a small sphere !

To find a solution to the Poisson equation!

We start by considering a point source at the origin.!

Except at the origin the RHS is zero so we can integrate!

∇2φ

V∫ dv = ∇φ ⋅n dsS∫ = ∂φ ∂n

ds S∫ = 4πr

2 ∂φ ∂r

= 4πr 2 −C r 2

⎛ ⎝⎜

⎞ ⎠⎟ = −4πC = σ

φ = −σ

4πrThe solution is therefore:!

Computational Fluid Dynamics

∇2φ = σ To solve the equation with a distributed source we integrate!

The solution in a two-dimensional unbounded domain is!

φ(x) = 1 2π

σ (x')ln rda' A∫

The solution in a three-dimensional unbounded domain!

φ(x) = −1 4π

σ (x') r

dv ' A∫

r = x − x'

Vortex Methods! Computational Fluid Dynamics

u x( ) = −∂ψ ∂y ,∂ψ ∂x

⎝ ⎜

⎠ ⎟ =

−1 2π

ω(x') −∂ ln r ∂y

,∂ ln r ∂x

⎝ ⎜

⎠ ⎟ da'A∫�

∇2ψ = −ω

∂ ln r ∂y

= ∂ ∂y ln (x − x ')2 + (y − y ')2( ) = (y − y ')(x − x')2 + (y − y')2

u x( ) = −12π

ω (x') −( y − y '),(x − x ')( ) r 2

da ' A∫ =

1 2π

k × r r 2

ω (x') da ' A∫

ψ(x) = −1 2π

ω(x')ln rda' A∫Solution!

Velocity for 2D flow!

since!

Vortex Methods!

• Computational Fluid Dynamics

Why Vortex Methods?!

Computational Fluid Dynamics

u =∇φ + ∇ ×Ψ

Helmholtz decomposition:! Any vector field can be written as a sum of !

Take the curl!

∇ ⋅u = ∇⋅∇φ = ∇ 2φ = 0

∇ × u = ∇ × ∇ ×Ψ( ) = ω

Take divergence!

By a Gauge transform this can be written as!

∇ 2Ψ = −ω

Vorticity!

Computational Fluid Dynamics

∂ω ∂t

+ u∇ ⋅ω = ω ⋅∇( )u + ν∇ 2ω

For incompressible flow with constant density and viscosity, taking the curl of the momentum equation yields:!

or:!

Dω Dt

= ω ⋅∇( )u + ν∇ 2ω

Helmholtzʼs theorem:! Inviscid Irrotational flow remains irrotational!

Vorticity! Computational Fluid Dynamics

Ψ = (0,0,ψ)In two-dimensions:!

or:!

ω = ∂v ∂x

− ∂u ∂y

ω = (0,0,ω)

Dω Dt

=ν∇ 2ω

∇ 2ψ = −ω

Dω Dt

= 0Zero viscosity: ! The vorticity of a fluid particle does not change!!

Vorticity!

Computational Fluid Dynamics

For inviscid unbounded flows, the motion everywhere, is completely determined by the vorticity. Thus, the evolution of the flow can be predicted by following the motion of the vorticity containing fluid ONLY.!

Dω Dt

= ω ⋅∇( )u

Dω Dt

= 0 u x( ) = 12π

k × r r 2

ω (x') da ' A∫

u x( ) = 14π

ω × r r3

dv ' V∫ 3D!

2D!

Computational Fluid Dynamics

Point Vortex Methods for 2D

Flows!

• Computational Fluid Dynamics

∇2ψ = −ω

Euler Equation for two-dimensional inviscid incompressible flow!

dω dt

= 0

u = ∇ × (ψk)

u x( ) = 1 2π

K A∫ x,x'( )ω x'( )dA'

where K is the appropriate velocity kernal!

The vorticity of each material particle is constant!

Vortex Methods! Computational Fluid Dynamics

K x,x'( ) = k × x − x'( ) x − x' 2

= (−(y − y '),(x − x')) r2

For unbounded domain! �

u x( ) = 1 2π

K A∫ x,x'( )ω x'( )dA'

The velocity depends only on the vorticity distribution!

Vortex Methods!

Computational Fluid Dynamics

u x( ) = 1 2π

K A∫ x,x'( )ω x'( )dA'

= 1 2π

K x,x'( )ω x'( )dA' A j ∫

j=1

N

= 1 2π

K x,x'( ) ω x'( )dA' A j ∫

j=1

N

= 1 2π

K x,x'( )Γ j j=1

N

x

x'j

x'j �

x'j

δ

Vortex Methods! Computational Fluid Dynamics

dxi dt

= 1 2π

K xi ,x j( )Γ j j=1

N

d dt (xi,yi) =

1 2π

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

2 + (yi − y j ) 2

j=1 j≠ i

N

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

Vortex Methods!

Computational Fluid Dynamics

d dt (xi,yi) =

1 2π

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! Computational Fluid Dynamics

Use point vortices to approximate a smooth distribution of vorticity !

Vortex Methods!

• Computational Fluid Dynamics

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!

Vortex Methods! Computational Fluid Dynamics

The N2 Problem!

Computational Fluid Dynamics

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 Methods! Computational Fluid Dynamics

Vortex-In-Cell!

Advect point vortices, but solve !

∇2ψ = −ω to find the velocities!

Vortex Methods!

Computational Fluid Dynamics

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 !

Vortex Methods! Computational Fluid Dynamics

Multipole expansion!

dx j dt

− i dy j dt

= q* = 1 2π i

Γ l z j − zll=1

j≠ l

N

For 2D flow we can rewrite the governing equations using complex numbers:!

z = x + iy

• Computational Fluid Dynamics

z j − z l = z j − z0 + z0 − z l = z j0 + z0l

q j

* = 1

2π i Γ l

z jo + zoll=1

N

1 z jo + zol

= 1

z jo −

zol z jo

2 + zol

2

z jo 3 −

zol 3

z jo 4 + .......

q j * =

1 2π i

Γ l l=1

N

∑ { 1z jo −

zol z jo

2 + zol

2

z jo 3 −

zol 3

z jo 4 + .......}

= 1

2π i { 1

z jo Γ l

l=1

N

∑ − 1z jo2 Γ l

l=1

N

∑ zol + 1

z jo 3 Γ l

l=1

N

∑ zol2 − 1

z jo 4 Γ l

l=1

N

∑ zol3 − .......}

Multipole expansion!

q j

* = 1

2π i Γ l

z j − zll=1

N

Computational Fluid Dynamics

q j

* = 1

2π i { 1

z jo Γ l

l=1

N

∑ − 1z jo2 Γ l

l=1

N

∑ zol + 1

z jo 3 Γ l

l=1

N

∑ zol2 − 1

z jo 4 Γ l

l=1

N

∑ zol3 − .......}

Multipole expansion!

The first term is the point vortex approximation but higher accuracy can be obtained using more terms. !

Computational Fluid Dynamics

Vortex Methods for 3D Flows!

Computational Fluid Dynamics

u(x) = −Γ

4π r

(r 2 + δ 2 )3/2C∫ × ∂ x ∂α

u j =

−Γ 4π

x j − x l+1/2 (| x j − x l+1/2 |

2 +δ 2 )3/2l=1

N

∑ × (x l+1 − x l )

3D vortex methods! Filament method!

For a thin filam