Simulating Pipe Flow on a Staggered Grid in Python | with Inflow & Outflow 

Thank you so much for this amazing feedback ❤️ Before I created this video, I also struggled for a long time with a staggered grid implementation. No-one really practically walked me through how to navigate the "indexing-mess". So, I am very happy that this video is of great help! Thanks again for this kind feedback. :)

Thanks :). You're welcome!

Thanks a lot 😊

Thanks :). And welcome to the channel!

Thanks for the kind words, appreciate it a lot 😊 I'm sure this is possible, similar to the outflow to the right, you could have a patch of the top boundary to be of a similar type. One then just has to ensure that the global mass inflow and mass outflow match (to ensure global incompressibility). One thing I am worried about though, is that depending on the size of the leakage (I think the smaller, the worse), you could already run into the turbulent flow regime for which this simulation is no longer accurate. However, I don't have any experience with leakage simulations, so take these words with a grain of salt. I hope they are still helpful 😊.

Thanks :)

You're very welcome. 😊 Yes, that's one way to fix the additional degree of freedom when solving pressure poisson problems.

Great suggestion! This will definitely come in the future. Computational Fluid Dynamics, as presented so far, is just one of many nice simulation topics. I will note your suggestion down. Do you maybe have a concrete example, you think is particularly interesting?

@@MachineLearningSimulation Thank you for your quick response. Right now I'm trying to develop a 2d thermoelasticity solver , to simulate arc welding. Instead of FE , I'm experimenting with FV and FD schemes. It's quite an interesting take , since most elasticity solvers are FE based. Plus the elasticity equations are not as well covered (online at least) , so I think a lot of people will be interested. All the best !

@@jimpal5119 That's a great insight. I will note that particular application down. In particular, I would like to also have some FEM tutorials, including the challenges in assembles the sparse global matrix. You are right, elasticity equations are usually way less covered than NS equations for fluid flow.

Hi, thanks for the great question and for the kind feedback. 😊 Unfortunately, I don't have experience with modeling blood vessels. My first guess would be to decide how the fluid and the wall interact. If it is only the wall deforming, one could apply the movement as a respective Dirichlet boundary condition on the flow field. However, since the domain changes, it might require remeshing (and as such unstructured meshes) if no immersed boundary method is used. Either way, I guess that this might be beyond this simple method I presented in the video.

Hi, many thanks for the kind words. I'm happy that you're enjoying the videos. What exactly do you mean by introduction comments? Are you referring to the thumbnail, the section before the intro, the doc string in python or sth else?

@@MachineLearningSimulation for the doc string

Oh, yes probably you mean how to create the charts and the mathematical characters. For the latter I use a website called "Unicode it", you could just Google "latex to Unicode". The charts are created by the "unicodecharts" library in Julia. I then just copy / paste them into the doc string?

@@michalislefkiou5505 Can you please tell me where this scripts is written , I mean where ( terminal or something like that ???? I am using spyder , but it is not showing any plot even though I have used matplotlib for the same code and the code is running without producing the plots . please help

In the plotting section of the final code (github.com/Ceyron/machine-learning-and-simulation/blob/26de0997676a49bfcaab305076ae9795ef1d4420/english/simulation_scripts/pipe_flow_with_inlet_and_outlet_python.py#L552 ), the staggered grid (with the ghost cells) is transformed into a vertex centered collocated grid (without any ghost ceels). This includes the value on the boundary (that should adhere to the prescribed BCs). Ultimately, this grid is plotted. Hope that answers the question 😊. Let me know if it is still unclear.

Hi, thanks for the comment and the kind feedback :). Ghost cells are cells that lie outside the domain, often used to enforce the boundary condition. We need them for the staggered grid, since some degrees of freedom do not end up lying directly on the boundary (but half a cell-length adjacent to it). Take for instance the y-velocities on the left boundary of the domain. In order to enforce a Dirichlet BC, we had to prescribe the value of the quantity on the boundary. We can do so by setting the average of the two to the desired. In a sense, this is just a linear interpolation between the value inside the domain closest to the boundary and the one in the ghost cell. Let's assume we had a wall (=no-slip) BC, where the value is homogeneous (=0). Then, we would get (v_out + v_in)/2 = 0. If you rearrange, you see that we enforce a homogeneous Dirichlet BC by v_out = - v_in Hope that helped :)

@@MachineLearningSimulation Thank you for the quick feedback. Your answer makes perfect sense I am just not used to this method yet. I am trying to implement the staggered grid method to the lid-driven cavity problem because when I make the mesh finer I get a RunTimeWarning.

@@manostsistrakis8241 You're welcome :). The RunTimeWarning seems more like you are encountering NaNs (Not a Number) in your computation, which often happens if your simulation turns unstable. Probably, a staggered grid will not help you there. Your problem might be in an unstable choice of the delta-t.

Hi, thanks so much for the kind feedback :). I am happy if the videos are of help. Be aware that the solution strategies presented here are more proof-of-concept type approaches, they lack certain properties that you would for correct CFD simulations. Unfortunately, there are no equally comprehensible resources (yet) that go beyond toy examples. Some general literature recommendations are "Computational methods for fluid dynamics " by Ferziger and Peric as well as "The finite volume method in computational fluid dynamics" by Moukalled which comes with examples in MATLAB and OpenFOAM.

Hi, thanks for the comment, the kind word and the suggestion 😊 Can you elaborate a bit on this picture? How is it different from the one that is already in the docstring in the beginning of the file? If you already have a good visualization in an image format, feel free to open a pull request on GitHub 😊, then everyone could benefit.

@@MachineLearningSimulation Hi, Thanks for the kind reply. I meant if you can add numbers beside rows and columns on the grid. During the video, you explain it by voice , I mean adding on every column if you could its number .In line code 290 , where is the column ( -2) . The video is very well prepared and as I am beginer, I could understand the method but my confusion is where is row number (-1) , where is column (-2) . That is why I asked , if you could put the numbers on columns and rows. You did by voice in some parts but it would be very helpful if you upload a picture with the structured grid with ghost cells but with numbering each row and each column and I do not know how to use GitHub. But I hope I could elaborate what I mean.

Got you 👍 At the moment, my time is a bit limited to create such an image. If I remember correctly, then there are elaborate schematics in Versteeg's book.

I haven't used it (yet). It's probably to be preferred over the legacy FEniCs.

Did you use the vanilla version from GitHub or did you do any modifications of the file? :)

That problem has been resolved, but sir , I am not getting the steady state, before reaching to the steady state the flow is getting vanished . Can you pl tell me the possible mistakes for it?? I have used the same boundary condition And one more thing , can you please reply to my mail, I guess I would require you going forward, if possible kindly reply me in my mail. I have already sent you a mail.

I can try ;)

@@MachineLearningSimulation Hey, Thanks for reply. I have written the code, but it doesn't seems to run as expected. Is it possible if you can check it once?

Unfortunately, I can't check full code bases, that exceeds my available time 😅 If you can reduce it to a minimal working example and post it here, I will give a look.

@@MachineLearningSimulation Hey, thanks for the reply. Here is the code pertaining to energy equation only - - - # Initial condition temperature_prev = np.ones_like(pressure_prev) temperature_next = np.ones_like(temperature_prev) temperature_centered = np.zeros((N_POINTS_Y, n_points_x)) # For visualization for iter in tqdm(range(N_TIME_STEPS)): temperature_diffusion = ( ( + temperature_prev[1:-1, 2: ] + temperature_prev[2: , 1:-1] + temperature_prev[1:-1, :-2] + temperature_prev[ :-2, 1:-1] - 4 * temperature_prev[1:-1, 1:-1] ) / ( cell_length**2 ) ) temperature_advection_x = ( ( velocity_x_prev[ :-2, :-1] + velocity_x_prev[ :-2, 1: ] + velocity_x_prev[2: , :-1] + velocity_x_prev[2: , 1: ] ) / 4 * ( temperature_prev[1:-1, 2: ] - temperature_prev[1:-1, :-2] ) / ( 2 * cell_length ) ) temperature_advection_y = ( ( velocity_y_prev[ :-1, :-2] + velocity_y_prev[ :-1, 2: ] + velocity_y_prev[1: , :-2] + velocity_y_prev[1: , 2: ] ) / 4 * ( temperature_prev[ :-2, 1:-1] - temperature_prev[2: , 1:-1] ) / ( 2 * cell_length ) ) temperature_next[1:-1, 1:-1] = ( temperature_prev[1:-1, 1:-1] + TIME_STEP_LENGTH * ( temperature_diffusion - ( temperature_advection_x + temperature_advection_y ) ) ) # Advance in time temperature_prev = temperature_next # Visualization if iter % PLOT_EVERY == 0: for j in range(0, np.size(N_POINTS_Y) - 1): for i in range(0, np.size(n_points_x) - 1): temperature_centered[i][j] = ( temperature_next[i][j] + temperature_next[i+1][j] + temperature_next[i][j+1] + temperature_next[i+1][j+1] ) / 4 plt.contourf( coordinates_x, coordinates_y, temperature_centered, levels=20, )

@@rahulagarwal2555 Can you also send the error message that is printed when you run the code?

Hi, thanks for the comment :). This update process arises, once you discretize the momentum equation with an Explicit Euler time stepping. This introduces a u at the next point in time and one at the current point in time. Then, you rearrange for the u at the next point in time. You can do this, because for this video all other terms in the equation (diffusion & convection) are treated explicitly. In simple terms, the NS equations are: du/dt + convection = - pressure grad + diffusion Then you discretize (u_next - u_prev)/dt + convection_prev = - pressure_grad_prev + diffusion_prev Then you rearrange for u_next u_next = u_prev + dt * (- pressure_grad_prev + diffusion_prev - convection_prev) Hope that helped :)

@@MachineLearningSimulation Thank you so much it clear now.

@@aliabdullah2474 Beautiful, you're very welcome!