Adding my final project

[cbell14]

No description provided.

[labarba]

Good effort, but there are some fundamental misconceptions that sent you in the wrong direction. The comments below follow a linear reading of the notebook.

"Due to the Laplace operators"—>just delete this preamble to the statement. The system is 3D for physical reasons, and the Laplace operator can be written in any dimension.

"Cavity Flow will be set up as pressure driven flow"—> I wonder what your rationale is for this? Pressure-driven flows are, for example, flow in a channel or flow in a pipe. If you add a source term F to the u-momentum equation, you are, in fact, adding a horizontal pressure gradient everywhere, one that will drive the whole flow field in that direction. Cavity flow is, rather, driven by the boundary conditions: you have one side of the domain with an imposed u-velocity. But that only applies on the boundary, not the entire domain! Anyway, it looks like this statement did not carry through to the rest of your work, because I see no source term F in the momentum equation.

At some point, you changed from lower-case to upper-case P for the pressure.

"As you probably noticed in the Burger's Equation above…"—>That's the Navier-Stokes equation, not Burgers' ... and note that it is Mr. Burgers, so Burgers' equation (not Burger's).

"which when combined with an equation such as PV=nRT"—>unfinished or malformed sentence.

"To form the Poisson Equation a source term is added to the equation, thus providing our driving pressure for our pressure driven flow." —> There is a conceptual error here. The pressure Poisson equation does have a source term on the right, but that does not make this a pressure-driven flow.

Boundary & Initial Conditions
... u(x,y,t) & v(x,y,t)=0 everywhere else —> I think you mean to say on the boundaries
and thus should restrict this statement to specific values of x and y.

"image courtesy of ..." —> Generally, this means you are using the image with permission from the copyright holder. In your case, you're probably resorting to fair use (as this is an educational purpose), and you should add a "Credit" instead (perhaps as an image caption).

The code in this notebook follows closely the one in step 11 of CFD Python. I noticed that your pressure-Poisson iterative solver works with a while-loop and a convergence criterion, rather than a fixed number of iterations like our simpler solver in step 11. This is an improvement (although you should also have a maximum number of iterations to exit if the solver is not converging).

In your function Burger2D(), it looks like you added an update of the pressure field following the Helmholtz equation, right before solving the pressure Poisson equation. I don't think you can just solve these two equations sequentially within each time iteration to somehow get an acoustic version of cavity flow!! You're changing the physics of the Navier-Stokes equation by an update on the pressure with a different equation. Remember, the role of the pressure-Poisson equation is to ensure that continuity is satisfied in the incompressible-flow situation. You are altering the pressure field here, so I suspect that your code does not satisfy continuity, which could be why it blows up for longer simulation times.

A collocated grid does not make your code blow up: it should work just fine, only with some checker-board pattern on the pressure field.

After your cavity flow calculation, you present some post-processing with plt.psd and plt.specgram but with no explanation. I'm not sure what you are trying to do there.

"there must be a negative pressure gradient in the flow pulling the flow down into the cavity" —> all the discussion in this paragraph is confusing. And certainly an immersed boundary method is not required to solve in an extended domain in the shape of a T.

By the way, there is no point in trying to compare different results without a matching Reynolds number! (Physically, they should not match.)

I'm afraid you reach incorrect conclusions from applying a flawed simulation approach in the first place. I don't think you can just plug in a pressure update using a Helmholtz equation into a Navier-Stokes solver. You need to think deeply about what it means to have acoustic waves in an incompressible-flow situation.

Typos
to advanced a project—advance
we will set are—>we will set our
descritze—>discretize
discritization—>discretization
The flow will be model —> modeled
we will set are —> our
some advanced was to look —> ways to look
spacial —>spatial
we will be using will have because—> ??
python defines by the y variable first unlike our intuition—> what??
staggerred—>staggered
immerssed—>immersed
check are Navier-Stokes results—>our
studys—>studies
lessor—>lesser ("to a lesser effect" is badly written anyway)
curtesy—>courtesy (but we should have a credit line instead)
differeces—>differences
repsonse—>response
Move forward—>Moving forward
to narrowly—> too narrowly