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master degree course in AUTOMOTIVE ENGINEERING

Course year 2
Teaching units Unit Simulation and Modelling of Turbulent Flows
To be chosen by the student (lesson)
  • TAF: Optional subjects SSD: ING-IND/06 CFU: 6
Teachers: Andrea CIMARELLI
Exam type written
Evaluation final vote
Teaching language inglese
Contents download pdf download




The course is meant to provide students with the tools to simulate and understand turbulent flows. This includes basic elements of computational fluid dynamics and turbulence modelling. The student learns how to use his/her comprehension in turbulence to select the appropriate simulation techniques for the various applications.

Admission requirements

Vector analysis, differential equations, classical mechanics and fluid dynamics.

Course contents

Numerical fluid dynamics (3 cfu)
1) Brief introduction to the equations of fluid mechanics;
2) Integration and discretization techniques for solving partial differential equations: Euler, Crank-Nicholson, Multi-step, Runge-Kutta methods and finite differences;
3) Convection-diffusion equations. Modified wave-number and Von Neumann stability;
4) Discretization techniques for the Navier-Stokes equations. Projection methods and solutions for the Poisson equation;

Turbulence modelling (3 cfu)
1) Introduction to the physics of turbulence;
2) Direct Numerical Simulation and the need of modelling. The closure problem of turbulence;
3) Reynolds Average Navier-Stokes simulation: eddy viscosity models (k-epsilon, k-omega, k-omega SST) and Reynolds stress models (RSM and ASM);
4) Large Eddy Simulation: eddy viscosity models (Smagorinsky, Germano and one-equation ksgs models), similarity models, mixed models;
5) Mixed approaches: Hybrid RANS/LES simulation and Detached Eddy Simulation;

Teaching methods

During lessons, the subjects are presented by the lecturer, including the explicit proofs of mathematical formulas introduced and the presentation of the methods to solve the problems.

Assessment methods

The exam will be performed at the end of the lesson period. The students will sustain a written exam (3 hours) composed by open questions and followed by an oral discussion of the result of the written exam itself. The final score is given by the result of the written exam possibly refined by the oral discussion and will be communicated at the end of the oral discussion. It is not allowed to consult texts or lecture notes during the exam.

Learning outcomes

1- Knowledge of the numerical techniques for the solution of the Navier-Stokes equations and of their properties of consistency, accuracy and stability;
2- Knowledge of the modelling approaches to turbulence and of their properties in terms of amount and quality of the information provided;


1- "Computational methods for fluid dynamics" - Joel H. Ferziger and Milovan Peric.

1- "Turbulent flows", Stephen B. Pope.
2- "Turbulence", Uriel Frish.

1- ”Turbulence modeling for CFD” – D.C. Wilcox
2- "Large Eddy Simulation for Incompressible Flows", Pierre Sagaut.