2D Navier-Stokes Flow Simulator

The simulator solves the 2D, incompressible Navier-Stokes equations using the vorticity-stream function formulation. This approach eliminates pressure and enforces mass conservation automatically. The governing equation for vorticity ($\omega$) transport is:

Where $\omega = \nabla \times \mathbf{u}$ is the vorticity (a scalar in 2D, measuring local rotation), $\mathbf{u}$ is the velocity vector, $t$ is time, and $Re$ is the Reynolds number. The term $(\mathbf{u}\cdot \nabla)\omega$ represents convection, and $\frac{1}{Re}\nabla^2 \omega$ represents viscous diffusion.

The stream function ($\psi$) is solved from the vorticity to obtain the flow field. It is defined such that velocity components are its derivatives, guaranteeing an incompressible flow:

$$ u = \frac{\partial \psi}{\partial y}, \quad v = -\frac{\partial \psi}{\partial x}$$ Here, $\psi$ is the stream function. Lines of constant $\psi$ are the streamlines you see in the simulation. The velocity components $(u, v)$ are derived from it, ensuring the flow satisfies $\nabla \cdot \mathbf{u} = 0$ (incompressibility).

Chemical Mixing Tanks: The flow inside a stirred tank reactor is very similar to a lid-driven cavity. Engineers use this model to optimize paddle design and placement, ensuring chemicals mix thoroughly and efficiently without dead zones where fluid stagnates.

Microfluidic Devices: In lab-on-a-chip devices, fluids are often manipulated in tiny etched chambers. Understanding the vortex formation at different flow speeds (Reynolds numbers) is crucial for designing efficient cell sorters, mixers, and sensors.

Building Ventilation: Airflow in a room with a ceiling fan or a ventilation inlet creating a shear flow along the ceiling can be modeled as a cavity flow. This helps HVAC engineers design systems that minimize drafts and ensure proper air circulation.

Coating and Painting Processes: When applying a liquid coating (like paint or a protective film) over a surface contained within a raised border, the fluid dynamics resemble a driven cavity. Simulating this helps predict coating thickness uniformity and avoid defects.

When you start using this simulator, there are a few points that are easy to misunderstand. First, you might tend to think "higher Reynolds numbers are closer to real flow," but that's not correct. Increasing Re does make the vortices more complex, but that's a state attempting to mimic "unsteady, near-turbulent behavior." However, this simulation is 2D and calculates under the assumption of "laminar flow." It's fundamentally different from real-world 3D turbulence, so directly using results from, say, Re=10000 for pump design is risky. Remember, it's a tool for observing the "trend" of a phenomenon.

Next, the grid size N setting. With N=32 or 64, calculations are fast, but especially at high Reynolds numbers, the vortex shapes can be coarse, and sometimes the solution can become unstable and oscillate. This happens because numerical diffusion becomes strong and cancels out the actual physical viscous effects. In practice, mesh (grid) resolution is the lifeline of simulation, and a "mesh dependency study"—the essential process of refining the mesh until results no longer depend on its coarseness—is mandatory. With this tool too, you can experience its importance by comparing, for example at Re=1000, how the bottom-left secondary vortex is represented when you change N to 64, 128, and 256.

Finally, understand the implication of the fact that "pressure is not calculated." The vorticity-stream function method is stable and excellent for computation, but it cannot be used directly for phenomena where you "need to know the pressure itself," such as wing lift/drag or pressure loss in pipes. For those cases, you need "primitive variable methods" (like the MAC or SIMPLE methods) that solve for velocity and pressure directly. This tool is your first step for understanding flow "patterns" and "structure."