Writing Code for CFD (Computational Fluid Dynamics)

Introduction

• Purpose: Understanding how to write a CFD code from scratch
• Use Case: Trying new schemes or methods not available in software like ANSYS
• Example Problem: Lid-driven cavity problem

Problem Description

• Domain: Square domain with stationary left, right, and bottom walls (u, v = 0)
• Moving Wall: Top wall moving in X direction (u = U, v = 0)
• Comparison: Importance of benchmarking results for correctness

Discretization and Mesh Creation

• Mesh Type: Simple Cartesian structured mesh
• Domain Division: Divide square into smaller squares
• Interior Nodes: Points inside the square (green), boundary nodes (red)
• Example Mesh Size: 7x7 for representation

Avoiding Checkerboard Oscillations

• Issue: Checkerboard oscillations in direct equation solving
• Solution: Staggering the grids
• Staggering Method:
  • Displace u-velocities in Y direction
  • Displace v-velocities in X direction

Grid Details

• u-velocity Grid: 7x8 mesh (one additional row)
• v-velocity Grid: 8x7 mesh (one additional column)
• Pressure Grid: 8x8 mesh
• Complex Grid: Combining all variables (u, v, pressure)

Control Volumes

• Purpose: Define control volumes for solving the equations
• X-Momentum Equation:
  • Control Volume Definition
  • Equations for u-velocity
  • Terms:
    • Time derivative (∂u/∂t)
    • Convective terms (u²/∂x, uv/∂y)
    • Pressure gradient (∂p/∂x)
    • Viscous terms (∆²u/∂x², ∆²u/∂y²)
• Y-Momentum Equation: Similar to X-momentum, but in terms of v-velocity

Continuity Equation

• Original Form: Divergence of velocity = 0
• Artificial Compressibility: Adding a term to simplify and directly obtain pressure
• Discretization:
  • Time derivative of pressure
  • Spatial derivatives of velocities (∂u/∂x, ∂v/∂y)

Boundary Conditions

• Importance: Defining conditions on domain boundaries
• u-velocity:
  • Assign zero values at stationary walls
  • Adjust values near the boundary using additional layers (to ensure average velocities match physical conditions)
• v-velocity: Similar to u-velocity adjustments
• Pressure: Ensure no flow across boundaries by equalizing interior and exterior pressure points

Summary

• Learning Outcome: Understanding grid creation, equation discretization, and boundary condition implementation in CFD coding
• Next Steps: Follow-up video with actual code implementation
• Conclusion: Preparation for hands-on coding and application of discussed concepts