I'm trying to solve the Plane-Poiseulle Flow throuh my own code. The problem is I cannot figure out how to put the boundary conditions at the exit. Mathematically fully developed flow only occurs at x = infinity. Numerically one only has a finite channel length and therefore the first derivative of the u-velocity with respect to x is not zero along the exit cross-section no matter how large the channel length. So what one must do?

Saad

One alternative would be to put static pressure = 0 (relative to atmospheric pressure). That way, you would not have to worry about channel length. In fact this condition can take care of the situation where even reverse flow can happen at the outlet boundary.

Your assumption is not quite correct. Even 'mathematically', the entrance length is not infinite. Boundary layers start growing from the inlet of the channel, on both walls. The boundary layer thickness increases in axial direction until both layers merge in the center of the channel. That happens at a finite (!) axial distance from the inlet. The entrance length / channel height ratio is a function of the Reynolds number, and can be determined analytically for laminar flow. Knowing the entrance length in your case, you can create a grid with appropriate length and use any standard outlet condition.

Boundary Layer theory is an approximation to the full NS equations and hence yields a finite entrance length. It is a parabolic equation in the axial coordinate which means that the downstream conditions cannot affect the upstream conditions of the flow (only one boundary condition can be satisfied in the axial direction) and I think this is the reason why one gets a finite entrance length. This is not the case in the NS equation (two boundary conditions are needed) in which information about the flow is communicated in both directions. Further the assumption that after a finite length all the derivatives of the u-velocity with respect to the axial coordinate become zero is not mathematically consistent. This is I think analogous to assuming a curve (defined by a single funcional equation) in the x-y plane, which after a certain value of x converts into a straight line. Now as far as I know no such function exists. Neither any differential equation exists which defines such a function.

Fully-developed flow in a uniform channel means that the flow is translationally invariant, that the flow at a point is identical to the flow at a point downstream. This translates into the statement about vanishing axial derivatives.

Now, about my computational experience (2D) on the subject. I use the finite element method with divergence-free elements. Divergence-free means that the functions are also the curl of a stream function element. It also means that any matrix element involving the integral of the product of a divergence-free test (weight) function and the gradient of a potential (the pressure) vanishes (they are orthogonal by the Helmholtz decomposition). Thus, the pressure gradient immediately disappears from the flow equation. The flow is controlled by the boundary condition on the stream function. The pressure can later be recovered from the flow, and the pressure gradient related to the magnitude of the flow or the b.c. on the stream function. The elements are of the Hermite type and involve stream function and velocity degrees-of-freedom.

Now, application to channel flow. If I set up a mesh and apply stream function and no-slip b.c. and no boundary conditions on the inflow or outflow, the computational solution converges to fully-developed flow in a single nonlinear iteration, regardless of the size of the mesh (even if it is one element long). Apparently this "no outflow boundary condition" has been seen by others (but I have seen no reference to "no inflow b.c.s), and there is at least one paper (by D. Griffiths) on this, though the mathematical analysis is in one dimension. If I specify a uniform flow inlet b.c. with a long-enough mesh I get the usual developing flow.

This "no outflow condition" also works for the backwards-facing step/ step diffuser when the mesh is cut off in the middle of the recirculation bubble. (Works in the sense that the stream lines are visually indistinguishable from those generated with a longer mesh which resolves the full bubble.) In fact, it is so successful that I use no other outflow b.c.

The trouble with the boundary layer assumptions is more apparent at the inlet, not so much in the downstream region. However, for such simple flows (in the laminar case), analytical solutions of the boundary layer equations are often at least as accurate (if not more accurate) as numerical solutions of the N-S equations. By accurate I refer to comparison with experimental data, which provides the only true evaluation of accuracy, if short of an exact solution to the equations. I see no reason to assume that the steady-state (laminar) fully developed flow inside an infinite straight duct is not parabolic by nature. The full N-S equations include a whole lot of possibilities, not all of which may really take effect in a given flow. For example, for air flow there is no such thing as true incompressibility. Air is a compressible fluid. When we speak of incompressible flow we simply mean flow in which the dependence of density on pressure is negligible, because it does not come into play. Any problem you tackle, analytically or numerically, includes a set of assumptions, that do not exactly reflect reality but are still appropriate to model it. Sometimes the assumptions are already applied to obtain an approximate set of analytical equations you would like to solve. Even the full N-S equations imply the assumptions of local thermal equilibrium in a continuum, not to mention Newtonian stress relations (in most cases). Often, additional assumptions are applied to then solve those approximate equations.

Keep in mind that you are doing numerics. You are introducing errors, due to discretization, incomplete convergence, etc. Whatever boundary conditions you apply, they will also be evaluated numerically, not exactly. Depending on what your focus is, you will probably not need to look for a boundary condition that is more accurate than your numerical scheme, as long as it is consistent with the approximate numerical solution you are searching, and with the approximate equations you are solving. If axial gradients downstream of a finite entrance length are not even detectable in experiments, I would not bother trying to accommodate for such gradients in a numerical scheme. All you have to do is assume a length at which axial gradients become numerically unresolvable. The entrance length obtained by B-L methods or experiments may serve as an estimate for the minimum required length.

If you are looking for fully developed flow, you may as well apply the conditions that are given in fully developed flow, as it is defined. You didn't specify what you are trying to solve, but in case you are interested in the entrance flow problem and you really insist on applying boundary conditions at infinity, you may take a look at this paper:

http://www3.interscience.wiley.com/c...73124/ABSTRACT

In this case, your downstream condition may be less critical than the inflow condition, about which you also have to make certain assumptions. Uniform flow is not realistic, because in the inlet region the elliptic nature of the equations really comes into play, so what do you do about that? Model the region upstream of the duct entrance and around the duct to infinity? If you are only looking at the fully developed region inside the duct you don't have to worry about all this.