Fluid Dynamics Simulator Lecture Notes

Overview

• Objective: Build a fluid simulator to study various fluid dynamical phenomena (lift, drag, vortex shedding, etc.).
• Importance of building our own simulator:
  • Hands-on understanding of fundamentals.
  • Personal belief in the concepts and equations.
  • Opportunity to manipulate physics and observe phenomena.

Foundations of Fluid Simulation

• Start at the subatomic level to understand molecular interactions.
• Simplification techniques:
  • Average out molecules by treating the fluid as a continuous field.
  • Spatial and temporal discretization.
  • Proper boundary treatment.
• Goal: Understand the mathematical foundation of Computational Fluid Dynamics (CFD).

Microscopic Perspective of Fluid Dynamics

• Importance of aligning perspectives on how fluids work:
  • Discrete colliding particles vs. continuous flowing medium.
  • Aim: Reduce overwhelming amounts of data to focus on relevant information.
• Key insights:
  • Too many molecules to represent individually.
  • Statistical information is often more relevant than individual molecule behavior.

Quantum Mechanics and Its Implications

• Quantum mechanics focuses on measuring small particles (e.g., electrons).
• Measurement issues:
  • Interaction changes the state of the particle being measured (e.g., photon-electron interaction).
  • Heisenberg uncertainty principle limits precision in measuring location and momentum.
• Probabilistic approach:
  • Wave function describes the state of particles and their probabilities.
  • Joint probabilities for multi-particle systems complicate calculations.

Transition to Molecular Dynamics

• Limitations of quantum simulations for fluid dynamics:
  • Computationally expensive due to high-dimensional wave functions.
  • Need for a surrogate model for efficiency.
• Shift to classical mechanics:
  • Model atoms as classical particles with distinct trajectories.
  • Use inter-atomic potentials to represent forces and interactions.

Inter-Atomic Interaction and Potentials

• Nucleus-electron interaction explained by Coulomb potential:
  • Electrons react quickly compared to nucleus motion.
  • Electrons effectively seen as responding instantaneously to nuclei positions.
• Potential energy surface derived from wave functions:
  • Energy levels determine electron behavior around nuclei.
• Common potentials used:
  • Lennard-Jones potential for weak Van der Waals forces.

Kinetic Theory of Gases

• Combines atomic particles to model molecular behavior:
  • Molecule represented as point mass with effective collision radius.
  • Assumes instantaneous collisions for computational efficiency.
• Focus on global fluid behavior:
  • Shift from individual particle interactions to statistical behavior of the fluid.

Recap and Key Takeaways

• Progression from quantum mechanics to classical mechanics leads to:
  • Understanding fluid dynamics as particle interactions.
  • Building up to the kinetic theory of gases.
• Emphasis on reducing information complexity:
  • Break down complex problems into manageable parts and identify underlying principles.
• Next steps: Explore macroscopic perspectives of fluid dynamics.