Physics & Simulation Modeling

This section addresses a fundamental question: How do I understand the physics itself before writing CFD / simulation code? How are mathematics, equations, models, and engineering scenarios connected? Where can approximations be made, and where must we be rigorous?

1. Mathematical Foundations

Not re-teaching a mathematics textbook, but listing the essential mathematical foundations needed for CFD / simulation. Detailed derivations will be in technical notes; this serves as an outline.

• 1.1 Scalars, Vectors, Tensors & Notation How physical quantities are expressed in tensor form, unified notation conventions (Einstein summation, index positions).
• 1.2 Multivariable Calculus Gradient, divergence, curl operators in Cartesian and curvilinear coordinates; relationships between volume, surface, and line integrals (Gauss, Stokes theorems).
• 1.3 Linear Algebra & Spectral Perspective Linear operators, eigenvalues, spectral radius, condition numbers—foundations for stability analysis and Jacobian linearization.
• 1.4 ODEs and Stability Modal analysis of simple ODE systems, linear system stability, intuitive understanding of timestep constraints.
• 1.5 PDE Classification & Boundary Conditions Elliptic, parabolic, hyperbolic equations; common boundary conditions (Dirichlet, Neumann, Robin) and their physical meanings.

2. Governing Equations for Fluids

• 2.1 Conservation Principles & Control Volumes Deriving mass, momentum, energy conservation from fixed volume principles; Eulerian vs Lagrangian descriptions.
• 2.2 Mass / Momentum / Energy Equations Integral and differential forms, transitioning via Gauss theorem; approximations and assumptions at each step.
• 2.3 Compressible / Incompressible & Equation of State Ideal gas, general equations of state, Mach number, low-Mach approximations, when incompressibility is valid.
• 2.4 Rotating Frames, Body Forces & Source Terms How centrifugal, Coriolis, gravity, and volumetric heat sources enter governing equations—essential for rotating machinery.

3. Models & Approximations

In engineering simulation, governing equations are just the starting point. The key is: which models to choose for different conditions, understanding what approximations help and what they sacrifice.

• 3.1 Turbulence Models
  • RANS: k–ε / k–ω / SST—applicable scenarios, pros/cons, experience in compressors/turbines.
  • LES / DES: grid scales, timesteps, boundary layer treatment, computational requirements.
  • Engineering choices: when to use RANS, when LES is justified, where LES still requires engineering judgment.
• 3.2 Multiphase Models
  • VOF / Level-Set: free surface problems (fuel sloshing, interface oscillations).
  • Euler–Euler: high volume fraction particles, slurry, multiphase mixtures.
  • Euler–Lagrange / DEM: particle resolution, collisions, wall erosion.
• 3.3 Heat Transfer & Combustion
  • Convection, conduction, radiation models; common simplifications (e.g., ignoring radiation).
  • Combustion: fully mixed / premixed, mixture fraction-based or global reaction models.
  • Engineering simplifications: temperature-only analysis in some compressor/turbine regions.

4. Engineering Scenarios

• 4.1 Fan & Compressor Internal Flows Blade rows, gaps, secondary flows, tip leakage, rotor-stator interactions.
• 4.2 Combustors Recirculation zones, flame stabilization, mixing and dilution; why experiments remain primary, simulation for trends.
• 4.3 Turbines & Cooling Strong 3D effects, complex blade cooling, film cooling holes; where commercial software struggles, custom methods needed.

5. My Practice

Current work focuses on full-flowpath engine modeling:

• Fan/compressor aerodynamics and structural requirements; balancing efficiency, strength, and computational cost in model selection.
• Combustor: which parts can use rough models for trends, which require experiments.
• Turbine cooling and film holes: fine scales beyond commercial software, addressed through custom models.

Specific modeling trade-offs, comparisons, and lessons will be organized in projects and technical notes.