Fluid Flow Simulator Pro (MATLAB)

An interactive, lightweight 2D Potential Flow and Pressure Field solver built purely in native MATLAB OOP.

Overview

Modeling fluid mechanics via the full Navier-Stokes equations from scratch often results in numerical explosion due to non-linear convective acceleration terms. This application solves the fundamental Pressure-Poisson Equation ($\nabla^2 P = b$), which acts as the mandatory "Pressure Correction" step in advanced computational fluid dynamics (CFD) algorithms like SIMPLE and MAC.

By calculating the steady-state pressure field across a user-defined grid, the simulation recovers the 2D velocity vector field via momentum gradient descent.

Mathematical Formulation

1. The Governing Equation

We solve the 2D Poisson equation for pressure $P(x,y)$:

$$\frac{\partial^2 P}{\partial x^2} + \frac{\partial^2 P}{\partial y^2} = b(x,y)$$

Where the synthetic source term driving the flow is defined as: $$b(x,y) = -2\pi^2 \sin(\pi x) \sin(\pi y) $$

2. Discretization (The 5-Point Stencil)

Using second-order central finite differences over step sizes $\Delta x$ and $\Delta y$, the continuous Laplacian is mapped to a system of linear algebraic equations ($A \cdot P = b$):

$$P_{i,j} \left( \frac{-2}{\Delta x^2} + \frac{-2}{\Delta y^2} \right) + \frac{P_{i+1,j} + P_{i-1,j}}{\Delta x^2} + \frac{P_{i,j+1} + P_{i,j-1}}{\Delta y^2} = b_{i,j}$$

3. Velocity Recovery

Velocity vectors $\vec{V} = (u, v)$ are recovered from the spatial pressure gradient scaled by the stepping parameter $dt$:

$$u = -dt \frac{\partial P}{\partial x}, \quad v = -dt \frac{\partial P}{\partial y}$$

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Features

• Dual Numerical Engines: Switch instantly between a Direct Solver (LU Decomposition) and an Iterative Solver (Gauss-Seidel) to observe trade-offs in memory vs. convergence speed.
• Asymptotic Sparse Assembly: The coefficient matrix $A$ is pre-allocated as a MATLAB sparse() data type. This allows the app to solve thousands of simultaneous equations instantly without triggering MATLAB memory limits.
• Strictly Bound Dirichlet Walls: Outer boundary nodes are locked to $P = 0$, guaranteeing a well-posed matrix that cannot become singular.
• Zero Dependencies: Written entirely in Vanilla MATLAB. Requires no extra Toolboxes (e.g., no PDE Toolbox or CFD Toolbox needed).

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Installation & Usage

1. Clone this repository or download FluidFlowSimulatorApp.m.

2. Open MATLAB and navigate to the directory containing the file.

3. In the MATLAB Command Window, instantiate the application:

   FluidFlowSimulatorApp
   

Contributors

Name	GitHub
Prakhar Gupta	@PrakharG8651
Arun Chauhan	@Arun-Chauhan-24
Prince Meena	@xenoz27
Aman Agrawal	@Amanag185
Aadi Chhajed	@Aadi-Chhajed
Rajinder Kumar	@raman9728152450
Abhishek Parth	@Abhishek-Parth
Ankit Kumar Saini	@ankitkumar971