I’ve always been fascinated by the Navier-Stokes equations. They’re like the “laws of physics” for fluids, telling us how fluids like water or air move under various forces. It’s amazing how these equations, which come from Newton’s second law and account for things like pressure and viscosity, can predict the complex behavior of fluids.
One problem that’s always intrigued me is the cavity flow problem. It’s a classic scenario in fluid dynamics where you have a square box filled with fluid. The top of the box (the lid) slides to one side, but the rest of the box stays still. This sets up a circulating flow inside the box, which is fascinating to watch and study.
The Python code I’ve been looking at solves this problem using the Navier-Stokes equations. It starts by setting up a grid for the box and defining the properties of the fluid. Then it uses a function called build_up_b to calculate the right-hand side of an equation related to pressure. This equation, called the pressure Poisson equation, comes from the continuity equation and the Navier-Stokes equations.
Next, the code uses a function called pressure_poisson to solve the pressure Poisson equation. This is done iteratively, which means the function keeps refining its solution until it’s good enough. This ensures that the amount of fluid going into any point in the box equals the amount of fluid coming out, which is a fundamental principle in fluid dynamics.
Then the code uses a function called cavity_flow to calculate the velocity of the fluid at each point in the box for each moment in time. It does this by updating the velocity based on the Navier-Stokes equations and the pressure calculated earlier.
Finally, the code plots the pressure and velocity of the fluid at the final moment in time. This gives a detailed picture of the fluid’s motion inside the box, which is really cool to see!
I find it amazing how we can use mathematics and programming to solve such complex problems and gain insights into the behavior of fluids. It’s a testament to the power of computational fluid dynamics!