Helmholtz Decomposition Algorithm

DrFigus · February 17, 2015, 14:12

Hi everyone,

I'm currently validating an in-house CFD code against some cases of 3D compressible homogeneous isotropic turbulence decay. At the moment I need to perform an Helmholtz decomposition on the turbulent velocity vectors in order to compute the solenoidal (divergence-free) and dilatational (curl-free) components and to study them separately.

I found hundreds of publications explaining the interest of this decomposition and its theoretical basis, but nothing on how to perform it practically. So, I was wondering if anyone could suggest a reference with the description of its implementation or, even better, an open-source code with the algorithm itself (no matter what programming language).

Thanks a lot!
Figus.

FMDenaro · February 17, 2015, 15:20

Quote:

Originally Posted by DrFigus

Hi everyone,

I'm currently validating an in-house CFD code against some cases of 3D compressible homogeneous isotropic turbulence decay. At the moment I need to perform an Helmholtz decomposition on the turbulent velocity vectors in order to compute the solenoidal (divergence-free) and dilatational (curl-free) components and to study them separately.

I found hundreds of publications explaining the interest of this decomposition and its theoretical basis, but nothing on how to perform it practically. So, I was wondering if anyone could suggest a reference with the description of its implementation or, even better, an open-source code with the algorithm itself (no matter what programming language).

Thanks a lot!
Figus.

Assume a known vector field a decompose according to the decomposition

a = b + Grad f

and bc.s

n.a = n.b + df/dn

If you take the divergence of the decomposition you get a Poisson equation to be solved with Neumann bc.s. That provides the field f and then b.

If you want details about the decomposition I wrote this paper some years ago

http://onlinelibrary.wiley.com/doi/1...d.598/abstract

mprinkey · February 17, 2015, 15:21

I think you would use projection, like fractional step pressure methods. If you vector field is

and you want to decompose into the scalar and vector potentials, $\phi$ and

. Just start with the Helmholtz decomposition form:

$V = \nabla\phi + \nabla\times A$

Then take the divergence:

$\nabla\cdot V = \nabla^2\phi$

That gives a Poisson system for $\phi$ . Solve it as you would any other.

EDIT: Beat me by one minute! 8)

DrFigus · February 18, 2015, 03:48

Thanks a lot to both of you!

February 17, 2015, 14:12	Helmholtz Decomposition Algorithm	#1
DrFigus New Member John Ross Join Date: Jun 2010 Posts: 2 Rep Power: 0	Hi everyone, I'm currently validating an in-house CFD code against some cases of 3D compressible homogeneous isotropic turbulence decay. At the moment I need to perform an Helmholtz decomposition on the turbulent velocity vectors in order to compute the solenoidal (divergence-free) and dilatational (curl-free) components and to study them separately. I found hundreds of publications explaining the interest of this decomposition and its theoretical basis, but nothing on how to perform it practically. So, I was wondering if anyone could suggest a reference with the description of its implementation or, even better, an open-source code with the algorithm itself (no matter what programming language). Thanks a lot! Figus.

February 17, 2015, 15:21		#3
mprinkey Senior Member Michael Prinkey Join Date: Mar 2009 Location: Pittsburgh PA Posts: 363 Rep Power: 25	I think you would use projection, like fractional step pressure methods. If you vector field is and you want to decompose into the scalar and vector potentials, $\phi$ and . Just start with the Helmholtz decomposition form: $V = \nabla\phi + \nabla\times A$ Then take the divergence: $\nabla\cdot V = \nabla^2\phi$ That gives a Poisson system for $\phi$ . Solve it as you would any other. EDIT: Beat me by one minute! 8) DrFigus, FMDenaro and TurbJet like this.

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February 18, 2015, 03:48		#4
DrFigus New Member John Ross Join Date: Jun 2010 Posts: 2 Rep Power: 0	Thanks a lot to both of you!