| Author | Message | ||
     Suku Kim (skimflexpde) New member Username: skimflexpde Post Number: 1 Registered: 01-2004 |
How are you? One of the Flexpde samples used the following formulation to solve an viscous fluid problem. del2(P) = 2*dens*[dx(U)*dy(V) - dy(U)*dx(V)] + L*(dx(U)+dy(V)) I am wondering if this is a generally accepted method in finite element modeling of fluid and what the formulation should be for 3D case. Is there a better formulation to solve for pressure in incompressible flow system, specifically for Flexpde? I'll appreciate any information. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 100 Registered: 06-2003 |
Fluid flow systems admit of an incredible variety of formulations (See, for example, Gresho and Sani, "Incompressible Flow and the Finite Element Method"). We at PDE Solutions Inc are not fluid dynamicists. We read books and try to implement methods, but I could not tell you what is "a generally accepted method", or if there is such a thing. The strength of FlexPDE is that it is a general purpose system, allowing knowledgeable users to implement whatever system they have need of solving, without being restricted to the particular understanding of a given field by the implementors of the program. The embarrassment with Navier-Stokes equations is that you have an equation for each momemtum component (involving pressure), and a continuity equation, which does not involve pressure. So some kind of intervention is necessary to generate an equation for pressure, while simultaneously enforcing continuity. In the specific case you mention, the first term of the P equation derives from taking the divergence of the momentum equation and combining terms to result in an equation for pressure. This process incorporates the continuity equation, but does not enforce it, so by itself it usually gives very poor results (one could argue that this is because it is a tautology, not an equation with new information). The second term is a penalty to enforce the continuity equation. In practice, it turns out that the first term plays practically no part in generating the solution. The dominant player is the penalty term. So, my preference is to keep the penalty term and throw the rest away, and you will see this in many of our examples. I have constructed various other rationales for the resulting equation, as you will see in notes to "Samples | Steady_State | Fluids | Viscous.pde" and other examples. You can, for example, derive div(grad(P))=L*div(V) from the equations of compressible flow using an "almost incompressible" equation of state. Gresho mentions "Penalty pressure" formulations, in which one assumes P=-L*div(V)and eliminates the pressure equation entirely. I find the resulting systems too harsh for effective computing, and prefer the div(grad(P)) form. I use the div(grad(P))=L*div(V) formulation successfully in 3d in the example "Samples | Steady_state | Fluids | 3D_flowbox.pde". | ||
| Suku Kim (skimflexpde) New member Username: skimflexpde Post Number: 2 Registered: 01-2004 |
Thank you for the answer. I am trying to use Flexpde for air flow simulation (or wind) between building structures (or downtown street). This is a high Reynolds no. fluid simulation. I am first trying with your sample by changing the parameter (density and viscosity) and using the methods of stages and time-dependent. It does not seem to solve the problem yet. Can Flexpde handle such a problem, or has anyone conducted similar job? Thank you. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 101 Registered: 06-2003 |
At high Reynolds numbers, the Navier-Stokes equations become very ill-conditioned, and at some point FlexPDE is no longer able to solve them successfully. The usual approach is, as you have mentioned, to use stages to gradually increase the speed. At very high Reynolds numbers, a different formulation is necessary. One approach is the streamfunction/vorticity model, or perhaps the streamfunction alone (for irrotational flow). Unfortunately, I only know how to do this for two dimensions. (There is one paper I know of that addresses the 3D case, "Three-Dimensional Numerical Study of Boundary-Layer Stability", by J.W. Murdock, AIAA-86-0434, available from the American Institute of Aeronautics and Astronautics.) Perhaps you can find discussions in the literature about approaches to high-Reynolds-Number flow or streamfunction/vorticity models. |