Home-Made Computational Fluid Dynamics (CFD), (Includes CFD code)

     One fine morning, I decided to code the Navier–Stokes equations using the finite-difference method. This post has the results of this adventure 🏞️ (so-far). As is customary with all my CFD work using commercial and home-made CFD codes, this too is inspired by the free lectures of Dr. Lorena Barba.

Internal / External Fluid Dynamics - Lid-Driven Cavity

     I started using Lid-Driven Cavity example. Just because everyone else uses it to validate the code they write. The lid-driven cavity case is giving correct results up to ~Re 1,000 without any turbulence models or wall functions.

     The results shown in Fig. 1 correspond to at Re 1,000. It can be seen that the present code which is just vanilla Navier–Stokes; captures the vortices at both bottom edges well as compared to published data. But why would you want to use a vanilla CFD code which solved only ~Re 1,000?, anyways... Some validation... u-velocity at (0.5, 0.1719) is at -0.3869 m/s as compared to -0.38289 m/s [1]. Meanwhile, v-velocity at (0.2266, 0.5) is 0.3252 as compared to 0.30203. Furthermore, v-velocity at (0.8594, 0.5) is at -0.4128 m/s as compared to -0.44993 m/s. All in all, a good agreement with published data. An animation is uploaded here.  Fig. 1a shows a different variation of lid driven cavity.

Fig. 1, Lid driven cavity post-processing

Fig. 1a, Heated lid driven cavity post-processing

Internal / External Aerodynamics - Backward-Facing Step (BFS)

     The next example is the case of Backward Facing Step (BFS). This case is analogues to flow around a building or a pipe with different diameter or anything with sharp edge along the flow. Let's be honest, flow around a building 🏢 is at much higher Re than the measly ~1,000 this code solves. But for validation, here we are.

 

     The cases are self validating as the re-attachment length of the vortex just after the step is very well documented [4]. Fig. 2 compares the results from the vanilla code with published data. It can be seen that the results are in good agreement.

Fig. 2, Backward Facing Step (BFS) post processing at Re 200

Internal Fluid Dynamics - Aero-Thermal Pipe Flow

     This is the first case I ran as it is self validating. The product of area and velocity should be same at  the outlet as given at the inlet as a boundary condition, which it is! Furthermore, density calculated should be proportional to the temperature calculated, which it is! The results at Re ~3300 are shown in Fig. 3. This case is relevant to bio medical field as Reynolds number in human blood 🩸flow is around same flow regime. Moreover, flow through heated pipes etc. is also relevant.

Fig. 3, Flow through heated pipe / between flat plates

Clot Flow

     Fig. 4 shows an example of flow where a clot appears in a blood 🩸 vessel. Done on this same 🏡made CFD 🌬️ code used throughout this blog. Or in a pipe where oil 🛢️ flows. It can be seen how smooth /less chaotic the flow is without clot (less work for heart ❤️ ). Eat 🦐 🥑 🥙 🥕 🥒 🍲 healthy! Mechanical properties of fluid are taken as follows. 5 ltr/min flowrate. Blood vessel diameter 40 mm. Fluid density at 1,070 Kg/m³ and dynamic viscosity at 4.5 cP.

Fig. 4, Flow inside a blood vessel with and without clot

External Aerodynamics - Flow around a Square Cylinder

     Why not circular cylinder you ask? I am very lazy 😂. It is hard work to draw a circle using code. I am used to CAD. 😇. Again, low Reynolds number are very rare in practical applications but for the sake of completeness, I added this case as well. Fig. 5 shows flow around the cylinder at Re 100. Vorticity is shown in Fig. 5. The results are compared with experimental data. The Strouhal number from the home-made CFD code is at 0.158 as compared to a value of 0.148. The results are within 7%  of published literature [2-3].

Fig. 5, Post-processing

Thank you for reading! If you want to hire me as your most awesome PhD student, please reach out!

Code:

     I present to you the code-able discretized equations for solving fluid flow problems. At first, C++ version is presented followed by the MATLAB version. Equation 1-2 are Poisson equations for pressure. Equation 3 and 4 are x-momentum equations (without source). Equations 5 and 6 are y-momentum equations.

double p_ij = ((pn(i + 1, j) + pn(i - 1, j)) * dy * dy + (pn(i, j + 1) + pn(i, j - 1)) * dx * dx) / (2 * (dx * dx + dy * dy)) - dx * dx * dy * dy / (2 * (dx * dx + dy * dy)) * (rho * (1 / dt * ((u(i + 1, j) - u(i - 1, j)) / (2 * dx) + (v(i, j + 1) - v(i, j - 1)) / (2 * dy)) - ((u(i + 1, j) - u(i - 1, j)) / (2 * dx)) * ((u(i + 1, j) - u(i - 1, j)) / (2 * dx)) - 2 * ((u(i, j + 1) - u(i, j - 1)) / (2 * dy) * (v(i + 1, j) - v(i - 1, j)) / (2 * dx)) - ((v(i, j + 1) - v(i, j - 1)) / (2 * dy)) * ((v(i, j + 1) - v(i, j - 1)) / (2 * dy))));                (1)

p(i, j) = ((pn(i+1, j) + pn(i-1, j)) * dy^2 + (pn(i, j+1) + pn(i, j-1)) * dx^2) ./ (2 * (dx^2 + dy^2)) - dx^2 * dy^2 / (2 * (dx^2 + dy^2)) * (rho * (1/dt * ((u(i+1, j) - u(i-1, j)) / (2 * dx) + (v(i, j+1) - v(i, j-1)) / (2 * dy)) - ((u(i+1, j) - u(i-1, j)) / (2 * dx)).^2 - 2 * ((u(i, j+1) - u(i, j-1)) / (2 * dy) .* (v(i+1, j) - v(i-1, j)) / (2 * dx)) - ((v(i, j+1) - v(i, j-1)) / (2 * dy)).^2));                (2)

double u_ij = un(i, j) - un(i, j) * dt / dx * (un(i, j) - un(i - 1, j)) - vn(i, j) * dt / dy * (un(i, j) - un(i, j - 1)) - dt / (2 * rho * dx) * (p(i + 1, j) - p(i - 1, j)) + nu * (dt / (dx * dx) * (un(i + 1, j) - 2 * un(i, j) + un(i - 1, j)) + dt / (dy * dy) * (un(i, j + 1) - 2 * un(i, j) + un(i, j - 1)));                (3)

u(i, j) = un(i, j) - un(i, j) * dt/dx .* (un(i, j) - un(i-1, j)) - vn(i, j) * dt/dy .* (un(i, j) - un(i, j-1)) - dt / (2 * rho * dx) * (p(i+1, j) - p(i-1, j)) + nu * (dt/dx^2 * (un(i+1, j) - 2 * un(i, j) + un(i-1, j)) + (dt/dy^2 * (un(i, j+1) - 2 * un(i, j) + un(i, j-1))));                (4)

double v_ij = vn(i, j) - un(i, j) * dt / dx * (vn(i, j) - vn(i - 1, j)) - vn(i, j) * dt / dy * (vn(i, j) - vn(i, j - 1)) - dt / (2 * rho * dy) * (p(i, j + 1) - p(i, j - 1)) + nu * (dt / (dx * dx) * (vn(i + 1, j) - 2 * vn(i, j) + vn(i - 1, j)) + dt / (dy * dy) * (vn(i, j + 1) - 2 * vn(i, j) + vn(i, j - 1)));                (5)

v(i, j) = vn(i, j) - un(i, j) * dt/dx .* (vn(i, j) - vn(i-1, j)) - vn(i, j) * dt/dy .* (vn(i, j) - vn(i, j-1)) - dt / (2 * rho * dy) * (p(i, j+1) - p(i, j-1)) + nu * (dt/dx^2 * (vn(i+1, j) - 2 * vn(i, j) + vn(i-1, j)) + (dt/dy^2 * (vn(i, j+1) - 2 * vn(i, j) + vn(i, j-1))));                (6)
    

     Of course constants need to be defined, such as grid spacing in space and time, density, kinematic viscosity. These equations have been validated, as you might have read already! Happy coding!

     If you want to hire me as your PhD student in the research projects related to turbo-machinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.

References

[1] U Ghia, K.N Ghia, C.T Shin, "High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method", Journal of Computational Physics, Volume 48, Issue 3, 1982, Pages 387-411, ISSN 0021-9991, https://doi.org/10.1016/0021-9991(82)90058-4
[2] Khademinejad, Taha & Talebizadeh Sardari, Pouyan & Rahimzadeh, Hassan. (2015). Numerical Study of Unsteady Flow around a Square Cylinder in Compare with Circular Cylinder
[3] Ávila, Ítalo & Santos, Gabriel & Ribeiro Neto, Hélio & Neto, Aristeu. (2019). Physical Mathematical and Computational Modeling of the Two-Dimensional Flow Over a Heated Porous Square Cylinder. 10.26678/ABCM.COBEM2019.COB2019-0854
[4] Irisarri, Diego & Hauke, Guillermo. (2019). Stabilized virtual element methods for the unsteady incompressible Navier–Stokes equations. Calcolo. 56. 10.1007/s10092-019-0332-5.