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Module 1. Fluid Mechanics and High-Performance Computing

1.1. Computational Fluid Dynamics

1.1.1. The Origin of the Turbulence
1.1.2. The Need for Modeling
1.1.3. CFD Work Process

1.2. The Equations of Fluid Mechanics

1.2.1. The Continuity Equation
1.2.2. The Navier-Stokes Equation
1.2.3. The Energy Equation
1.2.4. The Reynolds Averaged Equations

1.3. The Problem of Closing Equations

1.3.1. The Bousinesq Hypothesis
1.3.2. Turbulent Viscosity in a Spray
1.3.3. CFD Modeling

1.4. Dimensionless Numbers and Dynamic Similarity

1.4.1. Dimensionless Numbers in Fluid Mechanics
1.4.2. The Principle of Dynamic Similarity
1.4.3. Practical Example: Wind Tunnel Modeling

1.5. Turbulence Modeling

1.5.1. Direct Numerical Simulations
1.5.2. Large Eddy Simulation
1.5.3. RANS Methods
1.5.4. Other Methods

1.6. Experimental Techniques

1.6.1. PIV
1.6.2. Hot Wire
1.6.3. Wind and Water Tunnels

1.7. Supercomputing Environments

1.7.1. Supercomputing. Ide Future
1.7.2. Supercomputer Operation
1.7.3. Tools for Use

1.8. Software in Parallel Architectures

1.8.1. Distributed Environments: MPI (Message Passing Interface)
1.8.2. Shared Memory: GPU
1.8.3. Data Engraving: HDF5

1.9. Grid Computing

1.9.1. Description of Computer Farms
1.9.2. Parametric Problems
1.9.3. Queuing Systems in Grid Computing

1.10. GPU, the Future of CFD

1.10.1. GPU Environments
1.10.2. GPU Programming
1.10.3. Practical Example: Artificial Intelligence in Fluids using GPUs

Module 2. Advanced Mathematics for CFD

2.1. Fundamentals of Mathematics

2.1.1. Gradients, Divergences and Rotations. Total Derivative
2.1.2. Ordinary Differential Equations
2.1.3. Partial Derivative Equations

2.2. Statistics

2.2.1. Averages and Moments
2.2.2. Probability Density Functions
2.2.3. Correlation and Energy Spectra

2.3. Strong and Weak Solutions of a Differential Equation

2.3.1. Function Bases. Strong and Weak Solutions
2.3.2. The Finite Volume Method. The Heat Equation
2.3.3. The finite volume method. Navier-Stokes

2.4. Taylor's Theorem and Discretization in Time and Space

2.4.1. Finite Differences in 1 Dimension. Error Order
2.4.2. Finite Differences in 2 Dimensions
2.4.3. From Continuous Equations to Algebraic Equations

2.5. Algebraic Problem Solving, LU method

2.5.1. Algebraic Problem Solving Methods
2.5.2. The LU Method on Full Matrices
2.5.3. The LU Method in Sparse Matrices

2.6. Algebraic Problem Solving, Iterative Methods I

2.6.1. Iterative methods. Waste
2.6.2. Jacobi's Method
2.6.3. Generalization of Jacobi's Method

2.7. Algebraic Problem Solving, Iterative Methods II

2.7.1. Multi-grid Methods: V-cycle: Interpolation
2.7.2. Multi-grid Methods: V-cycle: Extrapolation
2.7.3. Multi-grid Methods: W-cycle
2.7.4. Error Estimation

2.8. Eigenvalues and Eigenvectors

2.8.1. The Algebraic Problem
2.8.2. Application to the Heat Equation
2.8.3. Stability of Differential Equations

2.9. Non-linear Evolution Equations

2.9.1. Heat Equation: Explicit Methods
2.9.2. Heat Equation: Implicit Methods
2.9.3. Heat Equation: Runge-Kutta Methods

2.10. Stationary Non-linear Equations

2.10.1. The Newton-Raphson Method
2.10.2. 1D Applications
2.10.3. 2D Applications

Module 3. CFD in Research and Modeling Environments

3.1. Research in Computational Fluid Dynamics (CFD)

3.1.1. Challenges in Turbulence
3.1.2. Advances in RANS
3.1.3. Artificial Intelligence

3.2. Finite Differences

3.2.1. Presentation and Application to a 1D Problem. Taylor's Theorem
3.2.2. 2D Applications
3.2.3. Boundary Conditions

3.3. Compact Finite Differences

3.3.1. Objective. SK Lele's Article
3.3.2. Obtaining Coefficients
3.3.3. Application to a 1D Problem

3.4. The Fourier Transform

3.4.1. The Fourier Transform. From Fourier to the Present Day
3.4.2. The FFTW Package
3.4.3. Cosine Transform: Tchebycheff

3.5. Spectral Methods

3.5.1. Application to a Fluid Problem
3.5.2. Pseudo-spectral Methods: Fourier + CFD
3.5.3. Placement Methods

3.6. Advanced Time Discretization Methods

3.6.1. The Adams-Bamsford Method
3.6.2. The Crack-Nicholson Method
3.6.3. Runge-Kutta

3.7. Structures in Turbulence

3.7.1. The Vortex
3.7.2. The Life Cycle of a Turbulent Structure
3.7.3. Visualization Techniques

3.8. The Characteristics Method

3.8.1. Compressible Fluids
3.8.2. Application: A Breaking Wave
3.8.3. Application: Burguers Equation

3.9. CFD and Supercomputing

3.9.1. The Memory Problem and the Evolution of Computers
3.9.2. Parallelization Techniques
3.9.3. Domain Decomposition

3.10. Open Problems in Turbulence

3.10.1. Modeling and the Von-Karma Constant
3.10.2. Aerodynamics: Boundary Layers
3.10.3. Noise in CFD Problems

Module 4. CFD in Application Environments: Finite Volume Methods

4.1. Finite Volume Methods

4.1.1. Definitions in FVM
4.1.2. Historical Background
4.1.3. FVM in Structures

4.2. Source Terms

4.2.1. External Volumetric Forces

4.2.1.1. Gravity, Centrifugal Force

4.2.2. Volumetric (Mass) and Pressure Source Term (Evaporation, Cavitation, Chemical)
4.2.3. Scalar Source Term

4.2.3.1. Temperature, Species

4.3. Applications of Boundary Conditions

4.3.1. Inputs and Outputs
4.3.2. Symmetry Condition
4.3.3. Wall Condition

4.3.3.1. Tax Values
4.3.3.2. Values to be Solved by Parallel Calculation
4.3.3.3. Wall Models

4.4. Boundary Conditions

4.4.1. Known Boundary Conditions: Dirichlet

4.4.1.1. Scalars
4.4.1.2. Vector

4.4.2. Boundary Conditions with Known Derivative: Neumann

4.4.2.1. Zero Gradient
4.4.2.2. Finite Gradient

4.4.3. Cyclic Boundary Conditions: Born-von Karman
4.4.4. Other Boundary Conditions: Robin

4.5. Temporary Integration

4.5.1. Explicit and Implicit Euler
4.5.2. Lax-Wendroff Time Step and Variants (Richtmyer and MacCormack)
4.5.3. Runge-Kutta Multi-Stage Time Step

4.6. Upwind Schematics

4.6.1. Riemman's Problem
4.6.2. Main Upwind Schemes: MUSCL, Van Leer, Roe, AUSM
4.6.3. Design of an Upwind Spatial Scheme

4.7. High-Order Schemes

4.7.1. High-Order Discontinuous Galerkin
4.7.2. ENO and WENO
4.7.3. High-Order Schemes. Advantages and Disadvantages

4.8. Pressure-Velocity Convergence Loop

4.8.1. PISO
4.8.2. SIMPLE, SIMPLER and SIMPLEC
4.8.3. PIMPLE
4.8.4. Transient Loops

4.9. Moving Contours

4.9.1. Overlocking Techniques
4.9.2. Mapping: Mobile Reference System
4.9.3. Immersed Boundary Method
4.9.4. Overlapping Meshes

4.10. Errors and Uncertainties in CFD Modeling

4.10.1. Precision and Accuracy
4.10.2. Numerical Errors
4.10.3. Input and Physical Model Uncertainties

Module 5. Advanced Methods for CFD

5.1. Finite Element Method (FEM)

5.1.1. Domain Discretization. Finite Elements
5.1.2. Form Functions. Reconstruction of the Continuous Field
5.1.3. Assembly of the Coefficient Matrix and Boundary Conditions
5.1.4. Solving Systems of Equations

5.2. FEM: Case Studies Development of a FEM Simulator

5.2.1. Form Functions
5.2.2. Assembling the Coefficient Matrix and Applying Boundary Conditions
5.2.3. Solving Systems of Equations
5.2.4. Post-Process

5.3. Smoothed Particle Hydrodynamics (SPH)

5.3.1. Fluid Field Mapping from Particle Values
5.3.2. Evaluation of Derivatives and Particle Interaction
5.3.3. The Smoothing Function. The Kernel
5.3.4. Boundary Conditions

5.4. SPH: Development of a Simulator Based on SPH

5.4.1. The kernel
5.4.2. Storage and Sorting of Particles in Voxels
5.4.3. Development of Boundary Conditions
5.4.4. Post-Process

5.5. Direct Simulation Monte Carlo (DSMC)

5.5.1. Kinetic-Molecular Theory
5.5.2. Statistical Mechanics
5.5.3. Molecular Equilibrium

5.6. DSMC: Study Methodology

5.6.1. Applicability of the DSMC Method
5.6.2. Modeling
5.6.3. Considerations for the Applicability of the Method

5.7. DSMC: Applications

5.7.1. Example in 0-D: Thermal Relaxation
5.7.2. Example in 1-D: Normal Shock Wave
5.7.3. Example in 2-D: Supersonic Cylinder
5.7.4. Example in 3-D: Supersonic Corner
5.7.5. Complex Example: Space Shuttle

5.8. Lattice-Boltzmann Method (LBM)

5.8.1. Boltzmann Equation and Equilibrium Distribution
5.8.2. From Boltzmann to Navier-Stokes. Chapman-Enskog Expansion
5.8.3. From Probabilistic Distribution to Physical Magnitude
5.8.4. Conversion of Units. From Physical Quantities to Lattice Quantities

5.9. LBM: Numerical Approximation

5.9.1. The LBM Algorithm. Transfer Step and Collision Step
5.9.2. Collision Operators and Momentum Normalization
5.9.3. Boundary Conditions

5.10. LBM: Case Study

5.10.1. Development of a Simulator Based on LBM
5.10.2. Experimentation with Various Collision Operators
5.10.3. Experimentation with Various Turbulence Models

Module 6. Turbulence Modeling in Fluids

6.1. Turbulence. Key Features

6.1.1. Dissipation and Diffusivity
6.1.2. Characteristic Scales. Orders of Magnitude
6.1.3. Reynolds Numbers

6.2. Definitions of Turbulence. From Reynolds to the Present Day

6.2.1. The Reynolds Problem. The Boundary Layer
6.2.2. Meteorology, Richardson and Smagorinsky
6.2.3. The Problem of Chaos

6.3. The Energy Cascade

6.3.1. Smaller Scales of Turbulence
6.3.2. Kolmogorov's Hypothesis
6.3.3. The Cascade Exponent

6.4. The Closure Problem Revisited

6.4.1. 10 Unknowns and 4 Equations
6.4.2. The Turbulent Kinetic Energy Equation
6.4.3. The Turbulence Cycle

6.5. Turbulent Viscosity

6.5.1. Historical Background and Parallels
6.5.2. Initiation Problem: Jets
6.5.3. Turbulent Viscosity in CFD Problems

6.6. RANS Methods

6.6.1. The Turbulent Viscosity Hypothesis
6.6.2. The RANS Equations
6.6.3. RANS Methods. Examples of Use

6.7. The Evolution of LES

6.7.1. Historical Background
6.7.2. Spectral Filters
6.7.3. Spatial Filters. The Problem in the Wall

6.8. Wall Turbulence I

6.8.1. Characteristic Scales
6.8.2. The Momentum Equations
6.8.3. The Regions of a Turbulent Wall Flow

6.9. Wall Turbulence II

6.9.1. Boundary Layers
6.9.2. Dimensionless Numbers of a Boundary Layer
6.9.3. The Blasius Solution

6.10. The Energy Equation

6.10.1. Passive Scalars
6.10.2. Active Scalars. The Bousinesq Approach
6.10.3. Fanno and Rayleigh Flows

Module 7. Compressible Fluids

7.1. Compressible Fluids

7.1.1. Compressible and Incompressible Fluids. Differences
7.1.2. Equation of State
7.1.3. Differential Equations of Compressible Fluids

7.2. Practical Examples of the Compressible Regime

7.2.1. Shock Waves
7.2.2. Prandtl-Meyer Expansion
7.2.3. Nozzles

7.3. Riemann's Problem

7.3.1. Riemann's Problem
7.3.2. Solution of the Riemann Problem by Characteristics
7.3.3. Non-linear Systems: Shock Waves Rankine-Hugoniot Condition
7.3.4. Non-linear Systems: Waves and Expansion Fans. Entropy Condition
7.3.5. Riemannian Invariants

7.4. Euler Equations

7.4.1. Invariants of the Euler Equations
7.4.2. Conservative vs. Primitive Variables
7.4.3. Solution Strategies

7.5. Solutions to the Riemann Problem

7.5.1. Exact Solution
7.5.2. Conservative Numerical Methods
7.5.3. Godunov's Method
7.5.4. Flux Vector Splitting

7.6. Approximate Riemann Solvers

7.6.1. HLLC
7.6.2. Roe
7.6.3. AUSM

7.7. Higher Order Methods

7.7.1. Problems of Higher Order Methods
7.7.2. Limiters and TVD Methods
7.7.3. Practical Examples

7.8. Additional Aspects of the Riemann Problem

7.8.1. Non-Homogeneous Equations
7.8.2. Splitting Dimensional
7.8.3. Applications to the Navier-Stokes Equations

7.9. Regions with High Gradients and Discontinuities

7.9.1. Importance of Meshing
7.9.2. Automatic Mesh Refinement (AMR)
7.9.3. Shock Fitting Methods

7.10. Compressible Flow Applications

7.10.1. Sod Problem
7.10.2. Supersonic Wedge
7.10.3. Convergent-Divergent Nozzle

Module 8. Multiphase Flow

8.1. Flow Regimes

8.1.1. Continuous Phases
8.1.2. Discrete Phase
8.1.3. Discrete Phase Populations

8.2. Continuous Phases

8.2.1. Properties of the Liquid-Gas Interface
8.2.2. Each Phase a Domain
8.2.2.1. Phase Resolution Independently
8.2.3. Coupled Solution
8.2.3.1. Fluid Fraction as a Descriptive Phase Scalar
8.2.4. Reconstruction of the Gas-Liquid Interface

8.3. Marine Simulation

8.3.1. Wave Regimes. Wave Height vs. Depth
8.3.2. Input Boundary Condition. Wave Simulation
8.3.3. Non-Reflective Output Boundary Condition. Numerical Beach
8.3.4. Lateral Boundary Conditions. Lateral Wind and Drift

8.4. Surface Tension

8.4.1. Physical Phenomenon of the Surface Tension
8.4.2. Modeling
8.4.3. Interaction with surfaces. Angle of Wetting

8.5. Phase Shift

8.5.1. Source and Sink Terms Associated with Phase Change
8.5.2. Evaporation Models
8.5.3. Condensation and Precipitation Models. Nucleation of Droplets
8.5.4. Cavitation

8.6. Discrete Phase: Particles, Droplets and Bubbles

8.6.1. Resistance Strength
8.6.2. The Buoyancy Force
8.6.3. Inertia
8.6.4. Brownian Motion and Turbulence Effects
8.6.5. Other Forces

8.7. Interaction with the Surrounding Fluid

8.7.1. Generation from Continuous Phase
8.7.2. Aerodynamic Drag
8.7.3. Interaction with Other Entities, Coalescence and Rupture
8.7.4. Boundary Conditions

8.8. Statistical Description of Particle Populations. Packages

8.8.1. Transportation of Stocks
8.8.2. Stock Boundary Conditions
8.8.3. Stock Interactions
8.8.4. Extending the Discrete Phase to Populations

8.9. Water Film

8.9.1. Water Sheet Hypothesis
8.9.2. Equations and Modeling
8.9.3. Source Term from Particles

8.10. Example of an Application with OpenFOAM

8.10.1. Description of an Industrial Problem
8.10.2. Setup and Simulation
8.10.3. Visualization and Interpretation of Results

Module 9. Advanced CFD Models

9.1. Multiphysics

9.1.1. Multiphysics Simulations
9.1.2. System Types
9.1.3. Application Examples

9.2. Unidirectional Cosimulation

9.2.1. Unidirectional Cosimulation. Advanced Aspects
9.2.2. Information Exchange Schemes
9.2.3. Applications

9.3. Bidirectional Cosimulation

9.3.1. Bidirectional Cosimulation. Advanced Aspects
9.3.2. Information Exchange Schemes
9.3.3. Applications

9.4. Convection Heat Transfer

9.4.1. Heat Transfer by Convection. Advanced Aspects
9.4.2. Convective Heat Transfer Equations
9.4.3. Methods for Solving Convection Problems

9.5. Conduction Heat Transfer

9.5.1. Conduction Heat Transfer. Advanced Aspects
9.5.2. Conductive Heat Transfer Equations
9.5.3. Methods of Solving Driving Problems

9.6. Radiative Heat Transfer

9.6.1. Radiative Heat Transfer. Advanced Aspects
9.6.2. Radiation Heat Transfer Equations
9.6.3. Radiation Troubleshooting Methods

9.7. Solid-Fluid-Heat Coupling

9.7.1. Solid-fluid-heat coupling
9.7.2. Solid-Fluid Thermal Coupling
9.7.3. CFD and FEM

9.8. Aeroacoustics

9.8.1. Computational Aeroacoustics
9.8.2. Acoustic Analogies
9.8.3. Resolution Methods

9.9. Advection-Diffusion Problems

9.9.1. Diffusion-Advection Problems
9.9.2. Scalar Fields
9.9.3. Particle Methods

9.10. Coupling Models with Reactive Flow

9.10.1. Coupling Models with Reactive Flow. Applications
9.10.2. System of Differential Equations. Solving the Chemical Reaction
9.10.3. CHEMKINs
9.10.4. Combustion: Flame, Spark, Wobee
9.10.5. Reactive Flows in a Non-Stationary Regime: Quasi-Stationary System Hypothesis
9.10.6. Reactive Flows in Turbulent Flows
9.10.7. Catalysts

Module 10. Post-Processing, Validation and Application in CFD

10.1. Post-processing in CFD I

10.1.1. Post-processing on Plane and Surfaces

10.1.1.1. Post-processing on the Plane
10.1.1.2. Post-processing on Surfaces

10.2. Post-processing in CFD II

10.2.1. Volumetric Post-processing

10.2.1.1. Volumetric Post-processing I
10.2.1.2. Volumetric Post-processing II

10.3. Free CFD Post-processing Software

10.3.1. Free Post-processing Software
10.3.2. Paraview
10.3.3. Paraview Usage Example

10.4. Convergence of Simulations

10.4.1. Convergence
10.4.2. Mesh Convergence
10.4.3. Numerical Convergence

10.5. Classification of Methods

10.5.1. Applications
10.5.2. Types of Fluid
10.5.3. Scales
10.5.4. Calculation Machines

10.6. Model Validation

10.6.1. Need for Validation
10.6.2. Simulation vs Experiment
10.6.3. Validation Examples

10.7. Simulation Methods. Advantages and Disadvantages

10.7.1. RANS
10.7.2. LES, DES, DNS
10.7.3. Other Methods
10.7.4. Advantages and Disadvantages

10.8. Examples of Methods and Applications

10.8.1. Case of a Body Subjected to Aerodynamic Forces
10.8.2. Thermal Case
10.8.3. Multiphase Case

10.9. Good Simulation Practices

10.9.1. Importance of Good Practices
10.9.2. Good Practices
10.9.3. Simulation errors

10.10. Free and Commercial Software

10.10.1. FVM Software
10.10.2. Software for Other Methods
10.10.3. Advantages and Disadvantages
10.10.4. CFD Simulation Futures

The interactive summaries for each module will allow you to consolidate your understanding of the application of the finite volume method in a more dynamic way”