Hi,

I am working with the incompressible Navier-Stokes equations (NSE) in a confined Cartesian geometry and I am trying to use the potential representation of the velocity v= curl((Psi1)*i)+ curl((Psi2)*j), where Psi1 and Psi2 are scalar potentials and i,j are unit vectors in X and Y directions. I apply the curl operator to the NSE so as to eliminate the pressure gradient and thus obtain two PDEs of fourth order for the potentials Psi1 and Psi2. For these two equations I need some more boundary conditions which does not follow trivially from the NSE.

Do you have some experience in solving this problem or some useful references?

Sincerely, Ulian Spassov

I have a question for you:

You are basically defining the velocity as the curl of a vector potential; i.e.

curl(psi(i,j)) = curl [ (Psi1)*i + (Psi2)*j ]

However, unless I'm missing something here, the above definition is only part of the story! ANY vector can be decompesed _uniquely_ (by the Helmholtz decomposition rule) into the curl of a vector potential and the gradient of a scalar potential:

velocity = curl(vector psi) + grad(scalar phi)

So, your decomposition is not unique to begin with!

Your approach is basically the same as what is known as the vorticity-velocity formulation (the curl of NSE gives you an equation in terms of vorticity)

The issue of BC in such a formulation is quite complicated and beyond the scope of this discussion. Suffice it to say that you really don't have to solve a 4th order PDE!

For a review of (and good references for) BC's related to the above formulation please read:

A. Gharakhani, "A Review of Grid-Free Methods for the Simulation of 3-D Incompressible Flows in Bounded Domains," SAND97-2256, SANDIA Contractor Report, September 1997

Or (not as many references to BC, but gives you the idea about vector decomposition, etc.)

A. Gharakhani and A. F. Ghoniem, "Three Dimensional Vortex Simulation of Time Dependent Incompressible Internal Viscous Flows," Journal of Computational Physics, Vol. 134, No. 1, pp. 75-95, 1997

Adrin Gharakhani

(1). The vorticity equation, the vorticity-velocity formulation, and the vorticity-stream function formulation are fairly standard. These can be found in most text books. (2). Are you inventing equations? Or are you looking for solutions?

Hi, Ulian.

This, obviously, is the same difficulty which is usually encountered when vorticity is used as an unknown: one needs additional boundary condition for it, because by taking curl of NSE the order of the system was increased. This is a well-known problem. Not too diffcult, really.

I do not remember a suitable reference but it is Saturday and I am alone far from home (-:, so I have 15 min to spare: see how it should be done.

I went to www.excite.co.uk (would be www.excite.com in USA. Excite search engine may be not such impressive as Altavista or Yahoo, but if you are after discovering a complicated information it works much better: it really tries to find what you need and not all that vaguely resembles what you need).

I typed in

+vorticity +velocity +variables +flow +navier +numerical +Spalding

chose Entire Web as a target for the search and pressed Search button. (Spalding is a man who, as I remember, wrote something on b.c. for vorticity in one of his books).

I got a list of sites, and in one of them, namely

http://www.postech.ac.kr/center/AFERC/bradshaw/refs255

I found, using the find button of the brouser and looking for 'vorticity', the following

..27.2,25.5: 3D stream function/vorticity approach* CARTER, J.E.; DAVIS, R.L.; EDWARDS, D.E.; HAFEZ, M.M.* Three-dimensional separated viscous flow analyses* Tenth Int. Conf. Num. Methods in Fluid Dyn. (Lec. Notes in Phys., vol. 264) (Zhuang, F.G. and Zhu, Y.L. eds, Springer), p. 147* 1986. `

..25.5: vector potential / vorticity formulation - not as neat as Koh's* GATSKI, T.B.; GROSCH, C.E.; ROSE, M.E.* The numerical solution of the Navier-Stokes equations for 3-dimensional, unsteady, incompressible flows by compact schemes* J. Comp. Phys. 82, 298* 1989. `

Looks like this is quite close to what you are doing. So, hope this helps and good luck.

Sergei.

P.S. And there are more sites found by Excite to explore.

Hi, Adrin.

>However, unless I'm missing something here, the above definition is only part of the story! ANY vector can be decompesed _uniquely_ (by the Helmholtz decomposition rule)

I am missing something here, too. Helmholtz decomposition is not unique. If you add any potential field to your 'vector psi', your velocity will not change. Ulian is doing something else, he is using a two-streamfunction approach. Helmholtz and two-streamfunction approaches are related, but different. In the former you express the velocity using 4 scalar functions, in the latter by using only 2.

However, you did noticed something that I forgot. Two-streamfunction decomposition of a velocity field is not unique, and this can require intorducing additional boundary conditions which do not follow from the original NSE formulation. It is similar in nature to the non-uniqueness of the Helmholtz decomposition. If we add to Psi1 and Psi2 dW/dx and dW/dy correspondingly, where W=f(x,y)+g(z) with arbitrary f(x,y) and g(z), the velocity will not change.

To Ulian:

Therefore, Ulian, you can use any additional b.c. you like as far as they are eliminating exactly this amount of non-uniqueness, in addition to those conditions derived from NSE similar to vorticity-velocity formulation case. I would propose to choose these additional conditions taking into account numerical stability of the scheme you are using and the desired orientation of the streamline surfaces, say, far upstream: it might be convenient, for example, to have them coinciding with x-y and y-z planes. A hint: consider Psi1 and Psi2 for (u,v,w)=(1,0,0).

Sergei

>I am missing something here, too. Helmholtz decomposition is not unique. If you add any potential field to your 'vector psi', your velocity will not change. Ulian is doing something else, he is using a two-streamfunction approach.

Hi Sergei,

You are of course right about (non)uniqueness of the Helmholtz decomposition. Poor verbage on my part. What I meant to imply is that the uniqueness of the velocity will be preserved by the addition of grad(scalar potential) which basically lumps all the non-uniqueness from the curl(psi) into grad(phi), which when combined with incompressibility constraint and the normal flux BC gives the recipe for finding the correct velocity. (actually this "free" term is quite a powerful capability in terms of coming up with different types of recipes for obtaining the velocity).

In contrast, the curl(psi) operation on its own misses a term that could be used to account for the correct velocity - this is independent of whether psi _itself_ is unique or not because we are dealing with a differential operator.

As for the two streamfunction operation, I'm wondering what the advantage of it would be compared to a vorticity-velocity formulation. It seems that we end up with more unknowns and higher order PDE's ...

Adrin Gharakhani

Hi, Adrin.

Sure, we agree on everything, really.

>As for the two streamfunction operation, I'm wondering what the advantage of it would be compared to a vorticity-velocity formulation. It seems that we end up with more unknowns and higher order PDE's ...

Ask Ulian

I was once attracted by the possibility to obtain streamfunctions directly, with a view to further visualisation. But I did not try it, really.

Sergei