[Vect13]

Hi,

I study an unsteady two-dimensional boundary layer problem and I write my own code based on the finite difference method (on Matlab). I have been struggling with it for a while so if someone could find a solution I would really appreciate it.
Roughly I try to solve the boundary layer equations of a jet between two parallel plates. The initial velocity profile along x=0 is parabolic such that u=1-Y^2 for Y in [-1;1] and 0 otherwise. (V=0 along x=0). Since the problem is symmetric, I study it between Y=0 and Y=d, with d>1.

Firstly I wrote the code of the problem without the plates, therefore as Y->+inf a matching condition for u holds, and it worked great: a jet propagates along the x-axis as time t increases, and after the perturbation u is equal to 0 (so e.g., u(x,Y=0,t) is equal to 0 immediately after the perturbation) as the fluid is initially at rest. Let's notice that the pressure gradient dp/dx=0 everywhere.

Then I added the plates. At Y=0, the symmetry boundary conditions hold: du/dY=0 and V=0. However at Y=d, the no-slip condition must be satisfied. So we can set u=0, but since the boundary layer equations are of first order in Y for V, I can't set V=0 at Y=d. So I restored the pressure gradient dp/dx and I used the Newton's method to compute dp/dx such that V=0 at Y=d.
And here is my problem, the profile of dp/dx looks consistent, but
1) u is not equal to 0 anymore after the perturbation. E.g., u(x,Y=0,t) tends to a finite value after the perturbation instead of 0. This value becomes larger as the distance d becomes smaller.
2) Same thing for the pressure gradient dp/dx, it does not tend to 0 after the perturbation.

The fluid being initially at rest, u should becomes 0 after the perturbation. Instead, it goes from 1 to 0.16 for example.

Someone would have a suggestion to overcome this issue? Thank you

[FMDenaro]

I don't understand what you are trying to do...

The problem is steady, not transient. Furthermore, if you start at x=0 with a parabolic velocity profile you are assuming a fully developed boundary layer at the inlet, not a spatially evolving that would require a uniform inlet.