Fluid Flow Simulation

This project implements a real-time 2D fluid flow simulation using a well-known method developed by Jos Stam. The simulation and visualization are written in C++, with SDL2 for graphical output.

Table of Contents

• Overview
• Background
• Methods
• Implementation
• Usage
• Results
• References

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Overview

The simulation showcases how an incompressible fluid, governed by the Navier-Stokes equations, develops instability at the interface of regions with differing velocities. The dye-based visualization allows direct observation of vortex formation and mixing.

Background

Kelvin–Helmholtz instability occurs when there is velocity shear in a single continuous fluid or between two fluids. Small disturbances at the interface grow, leading to turbulence and vortices. Real-time simulation of such flows is computationally intensive, but Stam's method provides a stable, efficient approach suitable for visualization.

Methods

The simulation is based on the discretized incompressible Navier-Stokes equations:

• Projection Method: The velocity field is projected onto a divergence-free field to enforce incompressibility.
• Periodic Boundary Conditions: The fluid grid wraps in both x and y directions (torus topology).
• Simulation Steps:
  1. Force Addition: External forces perturb the fluid.
  2. Advection: Fluid properties are advected using the velocity field.
  3. Diffusion: Viscosity is applied using Fast Fourier Transform (FFT) due to periodic boundaries.
  4. Projection: The velocity field is made divergence-free in Fourier space.

A dye field is evolved alongside the velocity field to visualize the flow.

Implementation

• Language: C++
• Graphics: SDL2 library for real-time rendering.
• Main code: Located in src/fluid_flow.cpp.
• Build system: Use the Makefile in the src/ directory.
• Visualization: The dye concentration is mapped to pixel colors; users can interactively inject dye with the left mouse button.

Arrays Allocation

• Four N x (N+2) arrays are used for velocity components (with extra columns for FFT).
• Force fields are stored separately and can be toggled during the simulation.

Notable Features

• Interactive dye injection (left mouse button).
• Real-time rendering of vortex evolution.
• Simple mechanism for applying and removing forces.

Usage

Prerequisites

• C++ compiler
• SDL2 development libraries

Building

1. Navigate to the src/ directory:

   cd src

2. Run make to build the executable:

   make

Running

• After building, run the simulation from the run/ directory:

  ./fluid_flow

• The GUI window will display the evolving fluid flow.
• Use the left mouse button to add a circular blob of dye for visual study.

Results

The simulation produces evolving patterns that visualize the KH instability. Below are sample outputs at various simulation steps:

20 steps	100 steps	300 steps	500 steps

Another scenario applies a leftward force to a central region:

20 steps	100 steps	300 steps	500 steps

• Increasing viscosity causes the fluid to stabilize quickly.
• Decreasing viscosity enhances instability and vortex formation.

References

• Jos Stam. 2001. "A Simple Fluid Solver Based on the FFT." Journal of Graphics Tools, 6(2), 43–52. doi
• Jos Stam. 1999. "Stable fluids." SIGGRAPH '99. ACM Press/Addison-Wesley Publishing Co., USA, 121–128.
• SDL2 Library GitHub
• Kelvin–Helmholtz instability (Wikipedia)

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For further details, see the full report.pdf.