# Numerical Method

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 June 15, 2005, 10:04 Numerical Method #1 Andrew Hayes Guest   Posts: n/a I have been working on a simple numerical program to calculate the velocity profile between two plates. I assume the initial Pressure, the exit Pressure=0, plate length,x, and distance between plates,h. I can then calculate an inlet velocity. If I solve the implicit finite difference model in steady state form I get a typical parabolic velocity profile. And, when I average the output velocities I get a velocity very close to the inlet velocity - some error due to small grid size. But, when I move to the time dependent model I cannot get the velocities to average to the inlet velocity any more. I do, however, get the velocity profile shapes (square in the beginning to parabolic). When I evaluate the dP/dx term in the NV equation I simply calculate the deltaP for the whole pipe and then multiply it by the distance at which the velocity profile is being evaluated [(P/x)*dx], then I divide that by the distance, dx. That worked with the steady state model, but doesn't work with the time dependent model. Any suggestions would be very helpful. thanks

 June 15, 2005, 14:09 Re: Numerical Method #2 Andrew Hayes Guest   Posts: n/a NV should be NS..navier-stokes..sorry

 June 16, 2005, 14:52 Re: Numerical Method #3 Mani Guest   Posts: n/a Provide some details on how you made your model time dependent. The flow itself is still supposed to be steady, since you use steady boundary conditions, correct?

 June 17, 2005, 08:30 Re: Numerical Method #4 Andrew Hayes Guest   Posts: n/a it is developing flow between 2 plates, 1-d, no-slip boundary condition at the wall. I know the inlet Pressure, P, the length of the plates, x, and the distance between them, h. So, if I were to define a time-step and look at the velocity profile at the first time step it should be square in form, which I get. Then, as the time increases it becomes more parabolic. I get that too. But, I am under the impression that if I average the velocities at all the grid points at a certain time step I should come up with my Vin, which I don't get. my equation: (standard Navier-Stokes equation) (dUx/dt) = -(1/rho)(dP/dx) + nu(d^2Ux/dy^2)

 June 17, 2005, 09:03 Re: Numerical Method #5 Jim_Park Guest   Posts: n/a The advection terms are negligible at steady state, as is the second of the momentum equations. I'd think that, during a startup transient, a lot of things are happening in the entrance region of the flow regime that are not 1-d. In that case, the terms missing from your equation could screw things up during the startup. Put another way, you'll get that Uinlet = avg(Ux) when dUx/dt = 0 everywhere. Perhaps not before. If you haven't done so, run your simulation out to convergence and see if that solves your problem.

 June 17, 2005, 09:17 Re: Numerical Method #6 Andrew Hayes Guest   Posts: n/a I have run it at steady state (dUx/dt = 0) and have gotten the parabolic profile, and when I sum the velocities at each grid point I get the Vin. That was my only question; about the velocities at each grid point at each time step averaging to the Vin. The profiles look as the should, but the velocities do not average. thanks

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