Can pressure Poisson equation (PPE) be used when computing unsteady incompressible flows, were there exist moving boundaries?

As I can see, when derive the PPE, taking divergence of NS-equations, the time dependent term du/dt cancels out as well as the diffusion term. Leaving only the convection terms behind for the source term in the PPE?

As the time dependent term is expelled, where do I then calculate the unsteady pressure?

And why is du/dt term then present in the Neumann boundary condition when solving the PPE?

Grateful for an answer: Svante Hellzén

> As I can see, when derive the PPE, taking divergence of NS-equations, the time dependent term du/dt cancels out as well as the diffusion term. Leaving only the convection terms behind for the source term in the PPE?

du/dt is not the only time dependent term. To say that the "time dependent term du/dt drops out" is inaccurate. The time _derivative_ term cancels out (which by the way under certain conditions may actually be independent of time!). The source term in the PPE is still time dependent.

Now, while the pressure is time-dependent, nothing in the Navier-Stokes equations requires you to evaluate its time derivative - only the spatial gradients. And the PPE essentially intergates these spatial gradients out. The obtained pressure distribution will still be time dependent, because the source term in the PPE is time dependent. So at any timestep, you are going to have different values of the source term - thus, different values of the pressure (in time) for the same spatial position

Adrin Gharakhani

To Adrin and all readers of the forum:

What an insightful reply. If all the posts were of this quality, I'd look forward to reading this forum every day.

Carry on...

Bob

(1). I think, you are right. (2). But I am about to pour a bucket of cold water on you. (3). The question was related to the math, not the fluid dynamics, not even the CFD. (4). With your background (?), you should be able to do more than just " would look forward to reading....". (5). Tell us more about what the school is doing in CFD these days.

Hi,

The answer altogether is not bad, but there is a little inaccuracy. Let's start with the basics: in FVM one can formulate the integral equations not only over a control volume but also over an arbitrary volume - you additionally get the swept volume fluxes and the rate of change of the volume, neither of which cause problems.

> Now, while the pressure is time-dependent, nothing in the > Navier-Stokes equations requires you to evaluate its time > derivative - only the spatial gradients.

This is only correct for incompressible flows, but the rest of the argument holds: if div(U) = 0, the pressure equation IS a Poisson equation on the "new" spatial configuration and there's no d/dt. If the flow is compressible, you get a dp/dt term in the pressure equation but it still causes no trouble.

How can you say that the source term is time dependent when there is no temporal term included in the source term?

Do not have the answer if the PPE can be used for moving boundaries.

I rephrase my question. If there are several measurements of a solenoidal incompressible flow in time, where the velocity component is known and they are contained in a domain with a moving boundary. Will the PPE calculate a pressure that relates each pressure between each timeframe? Or, must I correct each pressure result to relate the pressure between each time intervals?

Svante Hellzén I calculate the PPE by using the Calderon's Extension Theorem,Where pressure gradients from the fluid region is extend to a rectangular computational domain.

John,

You're right, the question was mathematical in nature. But I believe there are plenty of mathematical issues related to fluid mechanics and to CFD to make mathematical questions and answers such as this perfectly appropriate for discussion here. CFD is at least in part a mathematical topic.

As for adding to the discussion, Adrin handled it as well or better than I could, and I had nothing substantive to add over and above what he said, so I didn't. My background is in compressible flows (amongst other things); I've never in fact solved a PPE in my life - so I defer such questions to those with greater experience. Nevertheless, I thought I'd show my appreciation for his adroit reply.

You asked about CFD at Princeton. Things are currently scaled back from the traditional level of activity with the departure of Prof. Jameson and Prof. Orszag, which creates a period of transition for us. Rumors abound about who might be coming to fill those shoes. I don't envy whoever aspires to that task.

Nonetheless we have some CFD programs still underway, some of which is under the direction of Prof. Martinelli who remains. Two major efforts right now are in nonlinear free surface ship hydrodynamics and design (with some spinoff work in submarines and other hydrodynamic applications), and my work, which is modeling relevant to the design of a new concept in hypersonic ground testing which, in principle, might extend the "Mach 8 barrier" to upwards of Mach 12 to maybe Mach 15, by generating stagnation enthalpy not only in the plenum of a blowdown facility, but also in the supersonic regime via radiative means. There is also active research in optimization based on control theory, wherein cost derivatives are generated with a single solution of an adjoint equation, rather than the many solutions required for finite difference based algorithms. Hope that gives you an idea.

Bob

All that you need to know about the PPE, and more, is contained in my recent book with bob sani, INCOMPRESSIBLE FLOW AND THE FINITE ELEMENT METHOD, John wiley 1999, AND , to be released in 2 months as a 2-part updated paperback. Also, don't let the 2nd part of the title scare you off!! There's VERY MUCH material in this 1043 page book that is TOTALLY INDEPENDENT of the numerical method....

One of the MAIN GOALS of this book is to remove the eternal confusion related to the PRESSURE....the PPE, the boundary conditions, the initial conditions, etc etc.

I have worked exclusively with the PPE so let me throw my hat in this ring. First, I think the source terms should have no direct coupling to the time dependent boundary condition. Let me explain.

The source terms are internal and are not boundary derived. The boundary condition should be imposed by a Neumann condition. Example: a horizontal wall being pushed downward (like a flexible artery). In this case the wall-normal inward to the flow is + ey. The normal velocity is V.ey = v. The Neumann boundary condition on this surface (to be implemented for the PPE) reads:

GRAD(P,ey) = -rho * (dv/dt + V . GRAD(v)) + mu *(v,xx + v,yy)

Note that the sign of the normal vector inward will have an effect on this equation.

Normally for PPE solutions with steady conditions at the boundaries, the dv/dt term is zero. Clearly not zero for your application. Also, the convective derivative (V.GRAD(v)) will probably not be zero. I tend to set it to zero for steady solutions (seems to give better results and upwind schemes for convective terms sets this term to zero anyway) Note that the convective term will need to be differenced forward into the domain (at least for non-staggered grids). The upwind values of velocity don't exist. Forward differencing is somewhat destabilizing, having observed this first hand

Finally, the source term (internal source) should be a conventional development, i.e. include a time dependent divergence for mass control (see early MAC paper). Note also that if a Dirichet BC is not imposed somewhere in the domain (example P = 0 at outlet duct), the the source will need to be adjusted to enforce the Neumann condition. If this is not done, the pressure solution will drift (although in most cases depending on the solution method, the proper gradients will be sustained even thought the pressure creaps up or down). If the PPE is tagged to a fixed pressure boundary, this is not an issue.

best regards

Dean