This course is part of the Computational Science and Engineering track in the Master Program in Mathematical Engineering. It aims to provide students with the fundamentals of mathematical models and numerical methods for partial differential equations that describe the motion of fluids in both laminar and turbulent regimes. This is a very dynamic area of study full of new contributions. Therefore the course, in addition to the basic aspects, aims to encourage the study of the latest methodological approaches. Finally, since the course is oriented to the use of computer for the simulation of physical and engineering processes, an important part of the course will be devoted to the programming lab activities.

This course is also offered in a reduced 8 CFU version 052030 - COMPUTATIONAL FLUID DYNAMICS [C.I.]. The present sheet defines the objectives, programs and learning outcomes for both courses.

The student will be able to:

- analyze a physical problem involving fluid dynamics phenomena and identify the mathematical model which better describes it

- derive the incompressible Navier-Stokes equations in conservative and differential forms

- derive the weak formulation of Stokes and Navier-Stokes equations for different choices of boundary conditions

- derive and understand the Galerkin formulation and be able to prove the main stability and convergence results

- formulate suitable preconditioning strategies for the solution of saddle point problems

- formulate and analyze iterative schemes for treating the nonlinearity in Navier-Stokes equations

- understand and formulate stabilized methods for inf-sup non compatible discretizations as well as convection dominated problems

-  formulate and analyze time advancing schemes for the solution of time dependent Navier-Stokes equations

- select among the different discretization approaches the one that guarantees the best compromize in terms of accuracy and computational efficiency for a given fluid dynamics problem

- formulate different numerical strategies for the solution of free surface problems

- formulate the statistical quantities that are useful to describe the main features of a turbulent flow

- understand and derive the Kolmogorov theory for statistically homogeneous, stationary and incompressible turbulence

- understand the limitations of Kolmogorov theory and derive some models proposed in literature to overcome such limitations

- (*) characterize the main transport and diffusion properties of a passive scalar in a turbulent flow.

- (*) understand the features of two-dimensional turbulence and the main differences with respect to three-dimensional turbulence

- (*) characterize and describe the laminar and turbulent boundary layers

- understand and formulate the main families of turbulent models: Reynolds Averaged Navier-Stokes (RANS) and Large Eddy Simulation (LES)

- understand the properties and limits of the different turbulent models

- implement a block preconditioner for saddle point problems

- implement stabilized finite element solvers

- implement different time-advancing schemes (including semi-Lagrangian and projection methods) for the time dependent Navier-Stokes equations

- implement simple turbulence models in finite element solvers.

(*) In the reduced 8 CFU version of the course the points marked with an asterisk will not be included.

- Navier-Stokes equations in primitive variables.

- Weak formulation of Stokes and Navier-Stokes equations with different choices of boundary conditions.

- Galerkin/finite elements discretization methods.

- Compatibility between finite element spaces for velocity and pressure.

- Preconditioners for saddle point problems.

- Brief discussion of other discretization techniques: finite differences, finite volumes, spectral methods.

- Steady state Navier-Stokes equations: fixed point and Newton algorithms.

- Unsteady Navier-Stokes equations: time marching schemes; treatment of the convective term; projection methods.

- Numerical approaches for free surface problems.

- Brief dicussion of numerical schemes for compressible flows.

- Statistical description of turbulence. Kolmogorov's theory and subsequent developments; intermittency.

- (*) Transport and diffusion of a passive scalar in a turbulent flow.

- (*) Two-dimensional turbulence.

- (*) Laminar boundary layer; Blasius and Falkner-Skan equations. Turbulent boundary layer; wall law.

- Turbulence models for Reynolds Averaged Navier Stokes (RANS) equation; turbulent viscosity models, mixing length,

two equation models: k-epsilon model, k-omega model.

- Large Eddy Simulation (LES) turbulence models; filtering technique, Smagorinsky turbulent viscosity model,

similarity scale models, dynamic models, dynamic mixed models, anisotropic models.

(*) In the reduced 8 CFU version of the course the points marked with an asterisk will not be included.

Students are required to:

- know basic linear algebra

- know differential calculus for two- and three-dimensional domains for scalar and vector quantities

- know the main properties of partial differential equations

- have basic notions of statistics

- know the main properties of numerical approximation of elliptic and parabolic differential equations

The student will attend a written test including theory and programming exercises, followed by an oral examination. A minimum mark of 18/30 in the written exam evaluation is required for admission to the oral part.

The student proves to know the main definitions and concepts inherent in weak formulation of Stokes and Navier-Stokes equations with different boundary conditions.

He must be able to critically discuss the governing equations for incompressible flows, the related approximated techniques for their numerical solution, the different theoretical descriptions of turbulent phenomena (*), and the modelling approaches for the numerical simulation of turbulent flows.

He must be able to propose the most adequate analytic, modelling and numerical approach that guarantees the best compromize in terms of accuracy and computational efficiency for a given fluid dynamics problem (*).

In particular during the oral part the student should be able to present complex arguments in a rigorous, clear and concise way.

(*) In the reduced 8 CFU version of the course, the knowledge of transport and diffusion of a passive scalar in a turbulent flow, of two-dimensional turbulence, and of boundary layer theory are not included.