Simulation of Thermofluids and Fluid Dynamics using Julia

CFD, heat transfer, FEM, machine learning, and PINNs for master-level engineering students

Simulation of Thermofluids and Fluid Dynamics using Julia

A master-level engineering lecture combining classical thermofluid modeling, numerical simulation, finite differences, finite elements, CFD libraries, machine learning, and physics-informed neural networks.

1 Course at a Glance

1.1 5 h Introduction

Core Julia syntax, thermofluid governing equations, heat equation, convection, boundary conditions, numerical stability, and visualization.

Go to overview

1.2 3 Projects × 5 h

Application-oriented simulation projects involving cooling, heating, internal flow, natural convection, and boundary layers.

View projects

1.3 Julia Ecosystem

Use plain Julia, DifferentialEquations.jl, Gridap.jl, WaterLily.jl, Lux.jl, and NeuralPDE.jl where appropriate.

Software setup

1.4 Deliverables

Students submit code, plots, model assumptions, verification checks, parameter studies, and engineering interpretation.

Assessment

2 Learning Objectives

After this lecture, students should be able to:

  1. Formulate thermofluid problems using conservation laws.
  2. Identify suitable models for conduction, convection, advection-diffusion, and incompressible flow.
  3. Implement finite-difference, finite-volume, and finite-element models in Julia.
  4. Use selected Julia packages for thermofluid simulation.
  5. Build simple ML surrogates and PINN models for thermal and flow problems.
  6. Verify numerical results using analytical solutions, correlations, grid studies, and energy balances.
  7. Interpret simulation results from an engineering perspective.

3 Projects

3.1 Project 1

3.1.1 Cooling of a Hot Steel Cube

Transient conduction FEM PINN

A steel cube of side length 70 mm is initially at 1000 °C and cooled by air and a water-ice mixture.

Open project

3.2 Project 2

3.2.1 Cooling of Coffee in a Mug

Lumped model Parameter fitting ODEs

A mug of hot coffee cools through convection, radiation, evaporation, and conduction through the mug wall.

Open project

3.3 Project 3

3.3.1 Heating Water in an L-Shaped Pipe

Internal flow Nusselt correlations Surrogate

Water flows through an L-shaped pipe with external wall temperature of 100 °C.

Open project

4 Methods Covered

4.1 Finite Differences

Explicit and implicit discretization of heat and advection-diffusion equations.

Finite difference notes

4.2 Finite Elements

Weak forms, Robin boundary conditions, transient heat conduction, and FEM implementation.

FEM notes

4.3 CFD Libraries

Overview of helpful Julia libraries such as WaterLily.jl, Gridap.jl, Trixi.jl, and Oceananigans.jl.

CFD libraries

4.4 ML and PINNs

Surrogate models, parameter identification, neural ODEs, and physics-informed neural networks.

ML and PINNs

6 Quick Start

Install Julia and the course packages:

using Pkg

Pkg.add([
    "Plots",
    "CairoMakie",
    "DifferentialEquations",
    "OrdinaryDiffEq",
    "SparseArrays",
    "LinearAlgebra",
    "Statistics",
    "Gridap",
    "Lux",
    "Optimization",
    "OptimizationOptimisers",
    "NeuralPDE"
])