I see numerous refernces to "Parabolized Navier Stokes" (PNS) methods for CFD, but they all assume that I know what this phrase means. Can someone enlighten me?

(1). From the steady-state Navier-Stokes equations, one can simplify the equations by eliminating some specific terms. (2). If you eliminate the second order viscous terms and keep only the convection and the pressure gradient terms, you have the so-called inviscid equation (Euler equation). (3). If you eliminate only the stream-wise second order viscous terms (say in the x-direction along the surface , down-stream direction) you have the so-called "parabolized Navier-Stokes equations". (4). In this simplified equation, just like the old boundary layer eqaution, one can obtain the solution by "marching" in the stream-wise direction (say x-direction) from some known initial "location or station". This is possible because the stream-wise momentum equation (say x-momentum eq.) has only first-order terms in stream-wise direction. (with the second order terms being eliminated already) This one-pass integration is fast. (5). The name "parabolized" means that the resultant equations is "parabolic" equations. This is a type of equation which satisfies certain conditions based on the coefficients of the terms in the equation. It also will accept one real solution in the streamwise direction. (6). The original steady state Navier-Stokes equations is "elliptic", and the Euler's equation is "hyperbolic". A better place to start is the partial differential equation chapters in the advanced engineering math text books. Or you can check out the CFD books. The type of the equation determines the nature of the equation and its solution, and how the solutions can be obtained. (7). The simplified equation can propagate the information only in the down-stream direction, thus it is not suitable for general flow situation especially flows with separation.

Thanks! This may be what I need. I am already using a somewhat similar (but much cruder) scheme, but it is too limited in its capabilites.

I want to find the steady-state or time-averaged leakage rate of a compressible or incompressable, viscous fluid through a 'crack' which is the line-of-closest-approach betweeen two surfaces that are cylindrical (so the lamilar flow would be 2D) but not necessarily circular in cross-section. One surface might be more like a prism, but my present algorithm can't handle this case. Also, my algorithm fails on the down-stream side.

(1). Flow through a gap between two curved walls (one has polygon shape and the other has cylindrical shape) requires the solution of Navier-Stokes equations. (2). In 2-D (cut), it is not very complicated. For incompressible flow, you can run several cases with specified inlet velocity (thus mass flow rate is given) , compute the flow field, and then compute the pressure field to give you the pressure difference between the inlet and the outlet. (3). For the compressible flows, you will have to use transient formulation and specify the inlet total pressure and the exit static pressure as boundary conditions. (4). Hope this is what you are looking for?

(1). I have received your e-mail about your attempt to solve the flow through a crack using the parabolized equation. (2). It is hard to know the exact reason why the solution failed further downstream. The only way to check it out is to run some simpler cases first, and then gradually change the geometry to simulate the real one. (3). In the process, you may have to approximate the geometry to avoid large area variation. Marching through the separation point may be great, it is still not the job of the parabolized equation. In that area, you can put in a smoother wall shape in the calculation, instead of the real one. This may not be ideal for accurate solution, but at least, you will have some converged solution to work with. (4). I must say that, for ideal flow cases, the flow will follow the wall and you will not have flow separation and loss. For viscous flows, there will be wall shear stresses and boundary layer displacement effect. But if the crack or the gap is small, it is likely that flow will be viscous everywhere and it is hard to use the classical inviscid + viscous boundary layer approach. (5). Anyway, try to run some cases with relaxed wall geometry (or smoothed wall shapes), keeping the gap size the same, and see whether you can gradually gain control of you program. So, back to the simpler test cases first. Hope this will help. (6). In case, everything fail, you can consider solving the full Navier-Stokes equations. In 2-D, I would suggest the stream function-vorticity formulation for the incompressible flows or sub-sonic flows. You should have no problem in finding such formulation and examples in CFD books.