In this post, details will be shared about a simplified version of the Immersed 🤿 Boundary Method for fluid dynamics 💨 simulations which yours truly has developed in abundant spare time ⏲, referred to as S-IBM 🤓. The purpose of this code is for teaching 🎓 purposes only. The mesh is Cartesian and the method used has no measures in place to resolve the stairstep 𓊍 mesh. The syntax share in this post is for Python 🐍, but this code can be translated to other languages such as C or MATLAB, easily.
The process starts with creating a polygon, a circular cylinder ⭕ for this post. A polygon is created using parametric equations for the circle. NOTE: Before writing these lines, several parameters such as domain size, mesh size, number of points in each coordinate direction etc. have been already defined and calculated. For more information, read here.
theta = np.linspace(0, 2 * np.pi, 100) # create equally spaced angles from 0 to 2 * pi
Path from matplotlib is used to create a polygon approximation of a circle or an airfoil by using the x and y co-ordinates of the circle (x1, y1) or any shape. Next, coordinate pairs of the grid points are created using vstack and ravel commands. A check is performed to verify if a grid point falls inside the polygon. custom_curve is a boolean array. The values inside this array are True only for the points inside the polygon. Transpose is used to keep the curve orientation consistent with the Navier-Stokes equations.
Another boolean mask is created for boundary points using the boundary_mask statement. Points that are inside the polygon (circle) are marked using using the interior statement.
boundary_mask = np.zeros_like(curve, dtype=bool)
interior = curve[1:-1, 1:-1]
Next a check is performed to mark each interior point's immediate neighbor. right means point to the right of each interior point, bottom means point to the bottom of each interior point etc. boundary statement is used to identify the boundary points (fluid-solid interface) according to the following logic. The point itself is inside the circle (interior = True) and at least one neighbor is outside. boundary_mask[1:-1, 1:-1] is used to fill the boolean with boundary. Consider a 5×5 a grid (zoomed near the polygon boundary). X marks solid points (inside circle), O marks fluid points, as shown in Fig. 1. For example, if interior = True, right = True, left = True, top = True and bottom = True → boundary = True & ~(True) = False i.e. an interior point. And for example, interior = True, right = False, left = True, top = True, bottom = True → boundary = True & ~(False) = True, i.e. a point on the right boundary. For points outside the polygon, interior = False → boundary = False (regardless of neighbors).
Fig. 1, A grid example
right = curve[2:, 1:-1]
left = curve[:-2, 1:-1]
top = curve[1:-1, 2:]
bottom = curve[1:-1, :-2]
boundary = interior & ~(right & left & top & bottom)
boundary_mask[1:-1, 1:-1] = boundary
custom_curve_boundary = boundary_mask
In the next step, boundary_indices obtains indices of all boundary points. right_neighbors, left_neighbors etc. calculates indices of neighboring points in all directions. valid_right, valid_left etc. checks if the neighbor is within domain bounds (not at edge) and is outside the circle (fluid point). This allows proper application of no-slip boundary conditions. If for example, valid_right is True, pressure is taken from the right neighbor to be applied to the corresponding boundary index. A visualization of the points is shown in Fig. 2.
Fig. 2, Depiction of grid points in various regimes
While plotting Fig. 2, several points are skipped. This is done to make plot clearer. This code can be easily implemented in other languages such as MATLAB (inpolygon), C (ray-casting) etc. At Re 200, this code predicts a Drag coefficient of 1.395 and a Lift coefficient of 0.053. These values are well within rage of the values reported by [1]. In this code, there is no interpolation of scheme or smoothing employed. The lift and drag force coefficients for one shedding cycle are shown in Fig. 3. Within Fig. 4, u and v velocities (top left and right), pressure and streamlines (bottom left and right) are shown.
Fig. 3, red = Cd and black = Cl
Fig. 4, Post processing from the simulations.
References
[1] Braza M, Chassaing P, Minh HH. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. Journal of Fluid Mechanics. 1986;165:79-130. doi:10.1017/S0022112086003014
Thank you for reading! If you want to hire me as a post-doc researcher in the fields of thermo-fluids and / or fracture mechanics, do reach out!
This is probably the final step in the series. Unless, one fine morning yours truly randomly decides to write about moving and morphing airfoils 😑 i.e. a true virtual wind tunnel 🤓.
This whole series is a continuation of a series created by Dr. Barbra 👩🏫 to teach about coding 🖳 of the Navier-Stokes 🌬 equations. The code made available this blog is an independent continuation of her open-source work. Yours truly has coded the scaled version of the equations with a modified pressure Poisson equation. The original series ends with the example of the channel flow 🔚. So far, yours truly has added flow around obstacles both curved and non-curved; flow around moving obstacles and species transport to the series. The Fluid-Structure Interaction (FSI) version of the code 💻 is available here.
This post is about flow around shape changing 🪱 obstacles i.e. morphing wings. The development of these codes shared here, including the one mentioned in this post, started in the year 2020 and continued till the fall of 2023. All this effort was to create a digital CV so that yours truly can get himself accepted in to a funded PhD program. That happened in fall 2023 😌.
The frequency of oscillations, the equations for amplitude and lateral oscillations 〰️ are all taken from the published literature about swimming fish 🐟. As a starting point, the NACA 0012 airfoil is used. A polygon is then created. Each grid index is then tested for presence inside or outside the polygon. An index is marked as the boundary of the polygon if one index has anyone neighboring index outside the polygon. The neighboring boundary indices are used to apply Neumann boundary conditions for pressure. The boundary indices are used to apply no-slip Dirichlet boundary conditions on the polygon boundary. More details can be read here.
Because the camber of airfoil changes at each time-step, approximately 400 points are required on the airfoil boundary to properly capture the time-varying camber of the airfoil i.e. the fish 🐋. The output from the code is shown in Fig. 1. Within Fig. 1, top row has u and v components of velocity. Meanwhile, the bottom row shows the pressure and streamlines. The animation is shown in Fig. 2.
If you plan to use these codes in your scholarly work, do cite this blog as:
Fahad Butt (2025). 18th Step of the 12 Steps to Navier-Stokes (https://fluiddynamicscomputer.blogspot.com/2025/05/18th-step-of-12-steps-to-navier-stokes.html), Blogger. Retrieved Month Date, Year
Fig. 1, The result from code
Fig. 2, Animation
Code
#Copyright <2025> <FAHAD BUTT>
#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
y1 = foil_center_y + np.concatenate([y_top, y_bottom[::-1]]) # foil y
foil_path = Path(np.column_stack((x1, y1))) # foil region (polygon)
points = np.vstack((X.ravel(), Y.ravel())).T # check if inside region
custom_curve = foil_path.contains_points(points).reshape(X.shape) # test all grid points for inclusion in airfoil, contains_points returns True for each point inside the airfoil (polygon)
curve = custom_curve.T # transpose to match grid orientation
right = curve[2:, 1:-1] # boolean arrays indicating if neighbors of each interior point are inside airfoil (polygon)
left = curve[:-2, 1:-1]
top = curve[1:-1, 2:]
bottom = curve[1:-1, :-2]
boundary = interior & ~(right & left & top & bottom) # mark if point is inside foil and at least one neighbor is outside foil (mark interface between solid and fluid)
boundary_mask[1:-1, 1:-1] = boundary # fill the empty boundary
custom_curve_boundary = boundary_mask.T # transpose back
boundary_indices = np.where(custom_curve_boundary.T) # mark boundary indices
right_neighbors = (boundary_indices[0] + 1, boundary_indices[1]) # right index (neighboring grid points in each direction)
left_neighbors = (boundary_indices[0] - 1, boundary_indices[1]) # left index
top_neighbors = (boundary_indices[0], boundary_indices[1] + 1) # top index
bottom_neighbors = (boundary_indices[0], boundary_indices[1] - 1) # bottom index
# identify valid neighbor indices, ensure the neighbor is not inside airfoil (polygon) i.e. is inside fluid)
FOANSS, i.e. Fadoobaba's Open Advanced Navier-Stokes 🌬 Solver is now capable to simulate flow with moving objects 🤓. In this example, the objects are square and circular cylinders. These blog posts could have been a master's thesis or a peer-reviewed publication but yours truly decided to are write here for educational purposes! The
The simulated case is called the case of flow around an obstacle. For validation, read. This code and blog post is not endorsed or approved by Dr. Barba, I just continue the open-source work of her. The 13th, 14th, 15th and 16th Steps are available for reading along with code. The resulting motion of the square cylinder is shown in Fig. 1. The Fig. 1 is colored using the horizontal velocity. The Fig. 2 shows the example of moving circle. Within Fig. 2, the results of u and v velocities are shown on top while the pressure and streamlines are shown at the bottom. The code for stationery curved / arbitrary boundaries is made available here. A customary remainder that you are reading a blog, not a Q1 journal 😃.
If you plan to use these codes in your scholarly work, do cite this blog as:
Fahad Butt (2025). 17th Step of the 12 Steps to Navier-Stokes (https://fluiddynamicscomputer.blogspot.com/2025/04/17th-step-of-12-steps-to-navier-stokes.html), Blogger. Retrieved Month Date, Year
Fig. 1, The animation
Code
#%%
#Copyright <2025> <FAHAD BUTT>
#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
plt.contourf(X, Y, u1.T, 128, cmap = 'jet') # plot contours
# plt.colorbar()
plt.gca().set_aspect('equal', adjustable='box')
plt.xticks([0, L])
plt.yticks([0, D])
plt.xlabel('x [m]')
plt.ylabel('y [m]')
plt.show()
plt.figure(dpi = 200)
plt.contourf(X, Y, v1.T, 128, cmap = 'jet') # plot contours
# plt.colorbar()
plt.gca().set_aspect('equal', adjustable='box')
plt.xticks([0, L])
plt.yticks([0, D])
plt.xlabel('x [m]')
plt.ylabel('y [m]')
plt.show()
plt.figure(dpi = 200)
plt.contourf(X, Y, p1.T, 128, cmap = 'jet') # plot contours
# plt.colorbar()
plt.gca().set_aspect('equal', adjustable='box')
plt.xticks([0, L])
plt.yticks([0, D])
plt.xlabel('x [m]')
plt.ylabel('y [m]')
plt.show()
plt.figure(dpi = 200)
plt.streamplot(X, Y, u1.T, v1.T, color = 'black', cmap = 'jet', density = 2, linewidth = 0.5, arrowstyle='->', arrowsize = 1) # plot streamlines
plt.gca().set_aspect('equal', adjustable='box')
plt.xticks([0, L])
plt.yticks([0, D])
plt.xlabel('x [m]')
plt.ylabel('y [m]')
plt.show()
Code (Curved / Slanted Boundaries)
#Copyright <2025> <FAHAD BUTT>
#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
This post is about the simulation of flow 🍃 around and through various objects 🪈 and obstacles ⭕. The simulated cases include flow through a partially blocked pipe, shown in Fig. 1. Flow around flat plates arranged in the shape of the letter "T", shown in Fig. 2. Flow around a wedge and flow around a triangle 🔺, which are shown in Fig. 3 and 4 respectively.
To create obstacles in Python 🐍, the Path statement is used. This is similar to the inpolygon statement in MATLAB 🧮. For example, the following code 🖳 is used to create an ellipse ⬭.
custom_curve_boundary = np.zeros_like(custom_curve, dtype=bool) # find boundary of shape
for i in range(1, Nx-1):
for j in range(1, Ny-1):
if custom_curve.T[i, j]: # inside shape
if (not custom_curve.T[i+1, j] or not custom_curve.T[i-1, j] or
not custom_curve.T[i, j+1] or not custom_curve.T[i, j-1]):
custom_curve_boundary.T[i, j] = True # mark as boundary
boundary_indices = np.where(custom_curve_boundary.T) # mark boundary
right_neighbors = (boundary_indices[0] + 1, boundary_indices[1]) # right index
left_neighbors = (boundary_indices[0] - 1, boundary_indices[1]) # left index
top_neighbors = (boundary_indices[0], boundary_indices[1] + 1) # top index
bottom_neighbors = (boundary_indices[0], boundary_indices[1] - 1) # bottom index
valid_right = ~custom_curve.T[right_neighbors] # check if points are on shape boundary
valid_left = ~custom_curve.T[left_neighbors]
valid_top = ~custom_curve.T[top_neighbors]
valid_bottom = ~custom_curve.T[bottom_neighbors]
The arrays x1 and y1 are created using the equation for ellipse with the required parameters. The ellipse region is marked using the Path statement. Indices inside the ellipse are marked using a Boolean mask. Nested loops are used to mark the ellipse boundary using a curve. The right, left, top and bottom neighbor points are identified to be used for the application of Neumann boundary conditions for pressure (no-slip). Within the time loop, "where" statement is used to identify and apply the Neumann boundary conditions on the ellipse wall. The same method applied on any shape, for example aero-foils, circles, nozzles etc. The complete code to reproduce Fig. 1 is made available 😇.
It should be noted that, that the mesh is still stairstep. It doesn't matter how small the mesh resolution 😲. This code is developed for educational and research purposes only as there is not much application to stairstep mesh in real world❗
Code
#Copyright <2025> <FAHAD BUTT>
#Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
#The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
#THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
