Course Overview
1 Course Description
This lecture introduces master-level engineering students to the simulation of thermofluids and fluid dynamics using Julia.
The lecture combines:
- physical modeling,
- numerical discretization,
- finite differences,
- finite elements,
- CFD packages,
- machine learning,
- physics-informed neural networks,
- verification and validation.
2 Structure
| Component | Duration | Content |
|---|---|---|
| Introduction | 5 h | Julia, governing equations, discretization, boundary conditions |
| Project 1 | 5 h | Cooling of a hot steel cube |
| Project 2 | 5 h | Cooling of coffee in a mug |
| Project 3 | 5 h | Heating water in an L-shaped pipe |
3 Main Themes
- Transient heat conduction
- Convective boundary conditions
- Internal and external forced convection
- Natural convection
- Finite difference and finite element methods
- CFD workflows in Julia
- Surrogate modeling and parameter identification
- Physics-informed neural networks
4 Governing Equations
4.1 Heat Conduction
\[ \rho c_p \frac{\partial T}{\partial t} = \nabla\cdot(k\nabla T)+\dot{q} \]
For constant \(k\):
\[ \frac{\partial T}{\partial t} = \alpha\nabla^2T + \frac{\dot{q}}{\rho c_p} \]
4.2 Incompressible Flow
\[ \nabla\cdot \mathbf{u}=0 \]
\[ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} \right) = -\nabla p + \mu\nabla^2\mathbf{u} + \rho\mathbf{g} \]
4.3 Energy Equation for Flow
\[ \rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u}\cdot\nabla T \right) = \nabla\cdot(k\nabla T)+\dot{q} \]
5 Dimensionless Numbers
| Number | Definition | Meaning |
|---|---|---|
| Reynolds | \(Re=UL/\nu\) | inertia vs. viscosity |
| Prandtl | \(Pr=\nu/\alpha\) | momentum vs. thermal diffusion |
| Nusselt | \(Nu=hL/k\) | convective heat transfer |
| Biot | \(Bi=hL_c/k_s\) | internal vs. external thermal resistance |
| Fourier | \(Fo=\alpha t/L_c^2\) | dimensionless time |
| Rayleigh | \(Ra=g\beta\Delta T L^3/(\nu\alpha)\) | natural convection strength |