After the end of the course the students shoud expect the following outcomes: - familiarity with methods of computational fluid-dynamics and ability to carry out mathematical models for the analysis of the main unsteady events occurring in thermal and hydraulic machine components; - capability of applying the theoretical concepts illustrated during the lessons to the solution of physical problems; - skills in designing numerical models for both thermal and fluid dynamics characterization of the thermal and hydraulic machines. Ability to select the correct numerical technique depending on accuracy requirements and available experimental data. - skills in the evaluation of the numerical tool performance and in the critical review of the achieved results. In general, the main objective of the course is to tranfer expertise on computational fluid dynamics. The knowledge of computational fluid dynamics becomes effective only when the student is able to apply the theoretical concepts to the solution of problems of engineeering relevance.

Fundamentals of thermodynamics: first and second laws of thermodynamics, polytropic evolutions, concepts of work and heat, reversible and irreversible evolutions. Fundamentals of fluid mechanics: conservation of mass and momentum principles, Euler's equations, Navier-Stokes' equations, concepts of incompressibility, wall friction, Bernoulli's equation. Fundamentals of Thermal and Hydraulic machines: nozzles and diffusers, flows in compressors and gas turbines, flows in hydraulic machines.

Analytical models for characterization of energy systems and components. - Mathematical classification of PDEs; hyperbolic, parabolic and elliptic equations of physical interest; theory of the well-posed engineering problem; boundary conditions; system of equations. - Model equations in fluid-dynamics: linear advection and linear viscous equations, inviscid, viscous and thermal models, nonlinear equations . - Euler’s equations for one-dimensional flows; wall friction; calculus of the wave propagation speed; method of characteristics for the solution of the Euler equations. General formulation of the conservative equations in the presence of shocks, theory of the boundary conditions for wave propagation problems. - Shock velocity and Rankine-Hugoniot jump conditions. Methods for the numerical computation of flows - Finite difference methods for unsteady flows: explicit and implicit difference formulas, upwind e centered schemes; numerical accuracy of methods; the concepts of consistency, stability and convergence; Lax’s theorem for convergence of numerical solutions and general formulation of Von Neumann’s method to evaluate the stability of numerical schemes; spectral analysis of numerical errors. - Finite volume method and conservative differentiation: numerical fluxes in convection and diffusion equations; Lax-Friedrichs and Lax-Wendroff schemes; - Flux vector splitting (upwind conservative discretization). - Riemann’s problem and high-resolution numerical schemes to reduce numerical oscillations in the presence of flow discontinuities (such as aerodynamic shocks or cavitation collapse); Godunov methods. Flux-Difference vs. Flux-Vector Splitting methods. High resolution techniques. - Applications of the developed concepts to the numerical simulation of: unsteady processes in induction and exhaust systems of internal combustion engines and pressure waves and shock propagation in 1D nozzles. Numerical modelling of turbulent flows: the Reynolds-Averaged Navier-Stokes (RANS) equations. Mathematical closure of turbulence models: one- and two-equation models. Applied lectures and Laboratory work programme. Numerical and graphic exercises are carried out and proposed, involving: evaluation of numerical stability, spectral error analysis, effects of conservativeness in the simulation of shock waves, method of characteristics in bidimensional domains, unsteady evolutions in nozzles, diffusers and pipes. Numerical simulations are presented by the teacher to show the expect outcomes from the laboratory work of the students. In their Laboratory Work, the students are requested to model 1D or 2D processes by using homemade and/or commercial tools.

The subject consists of 48 hours of lectures and 12 hours of applied lectures: - the 48 hours of lessons consist of: 10 hours on classification of partial differertial equations, 10 hours on model equations, Euler's equations and general conservation law theory, 4 hours on shock and expansion wave theory, 20 hours on methods for numerical computation in flows and 4 hours on turbulence modelling; - the 12 hours of applied lectures consist of: 2 hours on Von Neumann's stability analysis, 3 hours on performance of schemes, 3 hours on method of characteristics, 4 hours on unsteady nozzles, diffusers and pipes with wall frition and heat transfer. For each applied lecture, the students are expected to prepare a report in which they present the outcomes of the assigned activities . These consist in developing numerical codes and in using such codes to produce the required graphs.

Notes, diagrams and charts are available to students at the end of the lecture. Video-recored lessons and Powerpoint slides in pdf format through the subject web page. For further reference and reading students may consult the followings: - R.J. Leveque, “Finite Volume Methods for Hyperbolic Problems”, Cambridge University Press, N.Y., 2002. - J.C. Tannehill, D.A. Anderson, , R.H. Pletcher, “Computational Fluid Mechanics and Heat Transfer”, second edition, McGraw-Hill, N.Y., 1997 - C. Hirsch, “Numerical Computation of Internal and External Flows” Vol. 1: Fundamentals of Numerical Discretization, John Wiley & Sons, 2007 - E. F. Toro “Riemann Solvers and Numerical Methods for Fluid Dynamics”, Springer-Verlag, Berlin, 1997. - C. B Laney "Computational GasDynamics" Cambridge University Press, 1998.

Exam: Compulsory oral exam; Group essay;

The structure of the exam is coherent with the objectives of the course (cf. Expected Learning Outcomes). Students are evaluated with an oral examination based on the assessment of the home work assigned during the applied lectures. The subjects of the 4 reports are: 1) Von Neumann's stability analysis of schemes; 2) analysis of performance of numerical schemes with smooth and Riemann initial data; 3) method of characteristics in two-dimensional domains ; 4) unsteady evolution in nozzles, diffusers and pipes. Students can work in team to realize the numerical codes that are illustrated during each applied lecture. The oral exam starts from an analysis of the final report concerning the applied lectures and each student should be able to answer the questions showing knowledge of the various methods and techniques illustrated during the lessons. In particular, it is expected that the candidate is able to justify any choices made in the presented laboratory work on the basis of the concepts acquired during the theoretical lessons. The oral exam lasts approximately 30 minutes. The final score is the sum of the score assigned to the laboratory work ( 2/3 of the final score) and the score assigned to the oral discussion (1/3 of the final score). The maximum mark is 30/30 cum Laude.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.

Exam: Compulsory oral exam; Group essay;

The structure of the exam is coherent with the objectives of the subject (cf. Expected Learning Outcomes). Students are evaluated with an oral examination based on the assessment of the home work assigned during the applied lectures. The subjects of the 4 reports are: 1) Von Neumann's stability analysis of schemes; 2) analysis of performance of numerical schemes with smooth and Riemann initial data; 3) method of characteristics in two-dimensional domains ; 4) unsteady evolution in nozzles, diffusers and pipes. Students can work in team to realize the numerical codes that are illustrated during each applied lecture. The oral exam starts from an analysis of the final report concerning the applied lectures and each student should be able to answer the questions showing knowledge of the various methods and techniques illustrated during the lessons. In particular, it is expected that the candidate is able to justify any choices made in the presented laboratory work on the basis of the concepts acquired during the theoretical lessons. The oral exam lasts approximately 30 minutes. The final score is the sum of the score assigned to the laboratory work ( 2/3 of the final score) and the score assigned to the oral discussion (1/3 of the final score). The maximum mark is 30/30 cum Laude.

Exam: Compulsory oral exam; Group essay;

The structure of the exam is coherent with the objectives of the subject (cf. Expected Learning Outcomes). Students are evaluated with an oral examination based on the assessment of the home work assigned during the applied lectures. The subjects of the 4 reports are: 1) Von Neumann's stability analysis of schemes; 2) analysis of performance of numerical schemes with smooth and Riemann initial data; 3) method of characteristics in two-dimensional domains ; 4) unsteady evolution in nozzles, diffusers and pipes. Students can work in team to realize the numerical codes that are illustrated during each applied lecture. The oral exam starts from an analysis of the final report concerning the applied lectures and each student should be able to answer the questions showing knowledge of the various methods and techniques illustrated during the lessons. In particular, it is expected that the candidate is able to justify any choices made in the presented laboratory work on the basis of the concepts acquired during the theoretical lessons. The oral exam lasts approximately 30 minutes. The final score is the sum of the score assigned to the laboratory work ( 2/3 of the final score) and the score assigned to the oral discussion (1/3 of the final score). The maximum mark is 30/30 cum Laude.