Integral Solutions to the Poisson Equationgtryggva/CFD-Course/2013-Lecture-29.pdf · Equation !...

Computational Fluid Dynamics

Vortex Methods!

Grétar Tryggvason!Spring 2013!

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!


Integral Solutions to the Poisson

Equation !


∇2φ = σ

∇2φ =

1r 2

∂∂r

r 2 ∂φ∂r

= σδ r( )

1r 2

ddr

r 2 dφdr

= 0 ⇒ d r 2 dφdr

⎛⎝⎜

⎞⎠⎟= 0 ⇒

dφdr

=Cr 2 ⇒φ = −

Cr

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

dsS∫ = 4πr 2 ∂φ

∂r= 4πr 2 −C

r 2

⎛⎝⎜

⎞⎠⎟= −4πC = σ

φ =

−σ4πrThe solution is therefore:!


�

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

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!

φ(x) =−14π

σ (x')r

dv 'A∫

�

r = x − x'

Vortex Methods!Computational Fluid Dynamics

�

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( ) = −1

2πω (x') −( y − y '),(x − x ')( )

r 2 da 'A∫ =

12π

k × rr 2 ω (x') da '

A∫

�

ψ(x) = −12π

ω(x')ln rda'A∫Solution!

Velocity for 2D flow!

since!

Vortex Methods!


Why Vortex Methods?!


�

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!


�

∂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:!

�

DωDt

= ω ⋅∇( )u + ν∇ 2ω

Helmholtz’s 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)

�

DωDt

=ν∇ 2ω

�

∇ 2ψ = −ω

DωDt

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

Vorticity!


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.!

DωDt

= ω ⋅∇( )u

DωDt

= 0 u x( ) = 1

2πk × r

r 2 ω (x') da 'A∫

u x( ) = 1

4πω × r

r3 dv 'V∫ 3D!

2D!


Point Vortex Methods for 2D

Flows!


�

∇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) = 1

2πΓ 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!


�

ddt(xi,yi) = 1


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 !


Use point vortices to approximate a smooth distribution of vorticity !

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!


The N2 Problem!


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 !


Multipole expansion!

dx j

dt− i

dy j

dt= q* =

12π i

Γ l

z j − zll=1j≠ l

N

∑

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

z = x + iy


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

q j

* =1

2π iΓ l

z jo + zoll=1

N

∑

1z jo + zol

=1

z jo

−zol

z jo2 +

zol2

z jo3 −

zol3

z jo4 + .......

q j* =

12π i

Γ ll=1

N

∑ { 1z jo

−zol

z jo2 +

zol2

z jo3 −

zol3

z jo4 + .......}

=1

2π i{ 1

z jo

Γ ll=1

N

∑ −1

z jo2 Γ l

l=1

N

∑ zol +1

z jo3 Γ l

l=1

N

∑ zol2 −

1z jo

4 Γ ll=1

N

∑ zol3 − .......}


q j

* =1

2π iΓ l

z j − zll=1

N

∑


q j

* =1

2π i{ 1

z jo

Γ ll=1

N

∑ −1

z jo2 Γ l

l=1

N

∑ zol +1

z jo3 Γ l

l=1

N

∑ zol2 −

1z jo

4 Γ ll=1

N

∑ zol3 − .......}


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


Vortex Methods for 3D Flows!


u(x) =

−Γ4π

r(r 2 + δ 2 )3/2C∫ ×

∂ x∂α

dα

u j =

−Γ4π

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 filament:!

Discretization gives:!


dx i

dt=−14π

| x i − x j |2 +(5 / 2)2

(| x i − x j |2 +2 )5/2j=1

N

∑ (x i − x j ) × Ω j

dΩi

dt=

14π

{j=1

N

∑| x i − x j |2 +(5 / 2)2

(| x i − x j |2 +2 )5/2 Ωi × Ω j

+ 3| x i − x j |2 +(5 / 2)2

(| x i − x j |2 +2 )5/2 (Ωi ⋅ ((x i − x j ) × Ω j ))(x i − x j )

+1054

ReVoliΩ j − Vol jΩi

| x i − x j |2 +2 )9/2}

Ω j =ω j ⋅Vol j

3D vortex methods!

Vorton method!


A “vorton” simulation of the collision and reconnection of two vortex rings!


For more information:!

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


Vortex Sheets!


Often the vorticity is concentrated in a thin vortex sheet. Its strength is given by the integral of the vorticity across the sheet:!

γ = ω dn =∫ u1 − u2( ) ⋅ s

Its velocity is given by!

u x( ) = 1

2πk × r

r 2 ω (x') da 'A∫ =

12π

P k × rr 2 γ (s') ds '

S∫

Where P stands for the “principal value” of the integral.!


u x( ) = 1

2πk × r

r 2 + δ2 γ (s') ds '

S∫

To eliminate the need to consider a principal value integral and to stabilize short waves, the integral is usually regularized by adding a small numerical parameter!

The sheet is usually discretized by replacing it with a row of point vortices !

�

ddt(xi,yi) = 1


2 + (yi − y j )2 + δ 2j=1

N

∑


Computation of the rollup of a vortex sheet by a regularized point vortex method. Left: evolution in time. Top: effect of reducing delta.!


Formation of streamwise structures during the late stages of a Kelvin-Helmholtz instability!


“Contour Dynamics”!


For 2D patches of of uniform vorticity it is possible to rewrite the integral over the vorticity as an integral over the contour defining the patch. This leads to “contour dynamics” ! Example: merging of

co-rotating patches!


Vortex Methods for Stratified Flows!



DγDt

+ γ ∂U∂s

⋅ s = 2A DUDt

⋅ s + 18∂γ 2

∂s⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+ 2Agj ⋅ s − 2

ρ1 + ρ2

∂ p1 − p2( )∂s

γ = u1 − u2( ) ⋅ s

U =

12

u1 + u2( )

ρi

∂ui

∂t+ ui ⋅∇ui

⎛

⎝⎜⎞

⎠⎟= −∇p − ρigj

Dui

Dt=∂ui

∂t+ U ⋅∇ui

A =

ρ2 − ρ1

ρ2 + ρ1

Define the vortex sheet strength, the average velocity and the acceleration following an interface point:!

By subtracting the tangential component of the inviscid Euler equations on either side of the interface!

We derive an equation for the evolution of the vortex sheet strength!


DγDt

+ γ ∂U∂s

⋅ s = 2A DUDt

⋅ s + 18∂γ 2

∂s⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪+ 2Agj ⋅ s − 2

ρ1 + ρ2

∂ p1 − p2( )∂s

The evolution of the vortex sheet strength is found by!

Once the vortex sheet strength is known, the velocity is found by integrating over the sheet!

u x( ) = 1

2πk × r

r 2 + δ2 γ (s') ds '

S∫

The integration is sometimes replaced by the Vortex in Cell method!


The Rayleigh Taylor Instability!


The Rayleigh Taylor Instability for different density ratios!


Free surface deformation due to a Kelvin-Helmholtz instability modeled by a vortex sheet (left) and a uniform vortex layer (right).!


High Reynolds number density current!

Example!

Vortex Methods!


Simulation of a 3D vortex ring colliding with an initially flat interface!


Methods for potential flows were important in the early development of CFD for aerospace applications.!

Panel methods, based on the integral solution of the Laplace equation, resulted in a reasonably efficient method to find potential flows over complex bodies. These methods have, however, for the most part lost their importance and will not be treated here.!

Integral Solutions to the Poisson Equationgtryggva/CFD-Course/2013-Lecture-29.pdf · Equation !...

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