COMPUTATIONAL FLUID DYNAMICS

The course provides an overview of some methods used in the numerical solution of the systems of equations that describe the motion of fluids, both compressible and incompressible. Some general concepts (for example, those related to hyperbolic systems of conservation laws and the corresponding numerical methods) can be applied in a much broader context. PLEASE NOTE: Information for students with disabilities and / or SLD To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs. It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.

Credit Value: 6

Scientific field: MATH/05-A: Numerical analysis
Taught classes: 150 hours in total: 108 hours of individual study, 47 hours of lectures
Term / Semester: 2°

First part: 3 CFU - 21 hours of theory (prof. A. Coco)

Second part: 2 CFU - 14 hours of theory (prof. S. Boscarino)

Third part: 1 CFU - 12 hours of laboratory activities (prof. E. Macca)

Total: 35 hours of theory and 12 hours of laboratory activities

The course consists of lectures and practical sessions, during which some of the methods covered in class will be implemented on the computer.

PLEASE NOTE: Information for students with disabilities and / or SLD To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview so to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs. It is also possible to contact the referent of CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) of the Department.

The course aims to be as self-contained as possible, so that it can be followed successfully even by those without a strong background in numerical methods or without prior knowledge of gas dynamics and fluid dynamics.

However, it is strongly recommended to have a basic understanding of numerical methods and techniques for solving differential equations. Such skills can be acquired by attending the courses in Numerical Calculus (second year of the Bachelor’s degree in Mathematics) and Numerical Analysis (first year of the Master’s degree in Mathematics).

Fluid Dynamics Equations

Derivation of the compressible and incompressible Euler and Navier–Stokes equations.

Numerical methods for incompressible Fluid Dynamics

Derivation of the incompressible Euler and Navier–Stokes equations. Finite difference methods for the Euler and Navier–Stokes equations in primitive variables. Chorin’s projection method and the Poisson solver. MAC (Marker-and-Cell) type discretization. Gauge method. Monolithic method. Ghost-point methods for problems in domains with obstacles.

Numerical methods for Compressible Fluid Dynamics

Review of hyperbolic systems for conservation and balance laws. Finite volume methods. Three-point methods: upwind methods, the Lax–Friedrichs method, and the Lax–Wendroff method. Godunov’s method and its properties. The numerical flux function. Construction of high-order methods. High-order essentially non-oscillatory (ENO) reconstructions. Weighted ENO (WENO) reconstructions. Conservative finite difference methods. Time integration with Strongly Stability Preserving (SSP) Runge–Kutta methods. Treatment of source terms. Implicit–Explicit (IMEX) Runge–Kutta methods for time integration.

Shallow Water Equations
Introduction to the Saint-Venant model for shallow water. Finite volume and finite difference methods for the Saint-Venant equations in one and two spatial dimensions. Well-balanced numerical methods.

Practical Exercises. 
The course includes exercises that demonstrate the implementation of the main methods covered during the lectures. In particular, some methods for solving the incompressible Navier–Stokes equations will be implemented and compared.

Note.
Should the course be delivered in blended or remote mode, the necessary adjustments may be introduced compared to what has been previously stated, in order to adhere to the program described in the syllabus.

Contribution of the course to the objectives of the 2030 Agenda for Sustainable Development

Goal No. 4: QUALITY EDUCATION
Ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.

Goal No. 5: GENDER EQUALITY
Achieve gender equality and empower all women and girls (enhancing strength, self-esteem, and awareness).

Goal No. 8: DECENT WORK AND ECONOMIC GROWTH
Promote sustained, inclusive, and sustainable economic growth, full and productive employment, and decent work for all.

The following are some texts covering topics in CFD that may be used during the course:

1. John D. Anderson Jr., Computational Fluid Dynamics: The Basics with Applications, McGraw Series in Mechanical Engineering, 1995.

A classic in CFD. Written by a professor of aerospace engineering. Very application-oriented. Not particularly sophisticated from a mathematical perspective. Somewhat dated.

2. Dimitris Drikakis, William Rider, High-Resolution Methods for Incompressible and Low-Speed Flows, Springer, 2005.

Fairly up-to-date, provides a simple description of the mathematical formulation of gas dynamics equations.

3. Joel H. Ferziger, Milovan Peric, Computational Methods for Fluid Dynamics, Springer, 2002.

Primarily numerically oriented, very detailed on schemes, but rather lacking in modeling and mathematical aspects.

4. Randall Le Veque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2004.

Specialized in finite volume methods for systems of hyperbolic conservation laws.

5. Randall Le Veque, Numerical Methods for Conservation Laws, Lecture Notes in Mathematics, ETH Zürich, Birkhäuser, Second Edition, 1999.

Excellent for providing a mathematical treatment of systems of conservation laws and some of the recent shock-capturing numerical methods for various conservation law systems.

6. Roger Peyret, Thomas D. Taylor, Computational Methods for Fluid Flows, Springer-Verlag, 1983.

A concise text, focusing mainly on incompressible fluid dynamics. Very advanced when it was first published, but now somewhat dated.

7. Pieter Wesseling, Principles of Computational Fluid Dynamics, Springer Series in Computational Mathematics, 1991.

A good introductory text. Contains much more material than can be covered in the course.

8. G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, 1974.

An excellent text on mathematical models describing wave phenomena.

9. Boscarino, Sebastiano, Lorenzo Pareschi, and Giovanni Russo. Implicit-explicit methods for evolutionary partial differential equations. Society for Industrial and Applied Mathematics, SIAM 2024.
Excellent textbook for IMEX methods.

Criteria for assigning marks: both for the project and for the oral exam, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and that they are able to carry out at least the simplest of the assigned exercises.

The following criteria will normally be followed to assign the grade:

Not approved: the student has not acquired the basic concepts and is not able to carry out the exercises.

18-23: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest, they are able to solve simple exercises.

24-27: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good, they solve the exercises with few errors.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; they solve the exercises completely and without mistakes.

Information for students with disabilities and / or SLD
Students with disabilities and/or SLD must contact the teacher, the CInAP representative of the DMI (Prof. Daniele) and CInAP well in advance of the exam date to communicate that they intend to take the exam using the appropriate compensatory measures (which will be indicated by CInAP).

PLEASE NOTE: The learning assessment can also be carried out electronically, should the conditions require it.

Derivation of the Navier–Stokes equations. Comparison between cell-centered and vertex-centered discretizations: advantages and disadvantages with respect to boundary conditions. Accuracy of the projection method. Boundary conditions for the auxiliary function in the Gauge method. Accuracy of ghost-point methods. Efficiency of monolithic methods.