# Computational Fluid Dynamics

Curriculum: HYRIS

Term: 1st year, 2nd Semester

Syllabus

CFU: 6

SSD: CEAR-01/A

Duration and Schedule: available here

Office hours: Wednesday 2-4 PM

OBJECTIVES

At the end of the Course, the student will have acquired a basic knowledge of the numerical methods applied to hydraulic and fluid dynamic analysis, learning how to apply them with awareness, also through the use of dedicated software packages.
The student will also have acquired concepts such as stability, convergence and accuracy of a numerical solution, adequacy of the computational mesh, choice of the most suitable turbulence model, which are fundamental in the application of fluid dynamic calculation methods to characteristic problems of hydraulic
risk prediction, such as those addressed during the second year of the course.
Through the practical lessons, the student will also acquire knowledge of the approaches which are
followed to perform a computational fluid dynamics simulation, through the examination of case studies solved through a CFD software package.

DESCRIPTION

THEORETICAL LESSONS
Convection and diffusion effects. Mathematical character of hyperbolic and parabolic equations. Continuity equation. Divergence-free condition for incompressible flows. Momentum balance. Newtonian and non-Newtonian fluids. Cauchy equations of continuum mechanics. Viscous stress in a Newtonian fluid. Navier-Stokes equations. Mathematical properties of the equations of fluid dynamics.
From continuum to discrete: discretization in space and time. Finite difference (FD) method: backward, forward and central differencing. Order of accuracy. Structured vs. unstructured computational meshes.
Time integration of the linear convection equation. Instability of the explicit Euler method. 1st order upwind scheme: Courant condition, CFL number and numerical diffusion. Lax-Friedrichs scheme: artificial viscosity, odd-even decoupling. Implicit Euler method. Dirichlet and Neumann Boundary conditions.
Numerical examples of the solution of the linear convection equation. Solution of the diffusion equation. Stability conditions on the numerical Peclet number.
Extension to the non-llinear case: explicit schemes for the Burgers equation, Predictor-Corrector methods, linearization Time integration schemes. Runge-Kutta multi-stage schemes.
Finite Volume method. FV cell-centered, node-centered and cell-vertex schemes. Flux evaluation in Finite Volume scheme: central differencing, upwind and linear upwind. Gauss method for the solution of linear systems.
Solution of tridiagonal systems: Thomas algorithm. Iterative methods for linear systems: Jacobi and Gauss-Seidel methods
Solution of the incompressible Navier-Stokes equations: Ladhizenskaya theorem, Projection method.
Free-surface flows: Volume-of-Fluid (VoF) method. Aribtrary Lagrangian Eulerian (ALE) methods. Introduction to turbulence. Direct Numerical Simulation (DNS) and Kolmogorov length scale. Reynolds decomposition. Reynolds Averaged Navier-Stokes (RANS) equations. Reynolds stress tensor. Boussinesq hypothesis. Prantl mixing-length. One-equation models: transport equation for the turbulent kinetic energy. Two-equation models: the k-ε model.

PRACTICAL LESSONS
Introduction to CFD software packages. Computational optimisation: “environmental”; and “industrial” models. Flexible meshing for environmental applications. Common features of “industrial”; CFD packages.
Solution of the Reynolds Averaged Navier-Stokes (RANS) equations. LES and DNS of turbulent flows.
Solution of continuity and momentum conservation equations. Coupled and segregated solution in steady and unsteady conditions. Under-relaxation, Algebraic Multi-Grid and Pre-conditioning methods.
Case studies: transonic flow around a blunt body in a wind tunnel; 2D vortex shedding downstream of a cylinder; thermal sewer discharges in a dock basin.

REQUIREMENTS

Basic knowledge in Fluid Mechanics: Eulerian and Lagrangian description, hydrostatic pressure distribution, continuity and momentum equations, Navier-stokes equations
Basic knowledge in Numerical Analysis: numerical approximation and rounding error, solution of linear systems, iterative root-finding methods

REFERENCES

J.H. Ferziger, M. Peric. Computational methods for fluid dynamics. Springer.

ASSESSMENT

Written exam consisting of 4 open questions, 2 on the topics of the theoretical lessons and 2 on the topics of the practical lessons. Each question is assigned a maximum rating of 8 points, modulated on the level of coherence of the answer to the question asked, on the completeness of the general arguments and on the correctness of the theoretical-mathematical demonstrations reported.

Instructor: Stefano Sibilla: official webpage and CV

Institution: DICAR (UNIPV)

E-mail: stefano.sibilla@unipv.it

Voice: +39 0382 985320

Fax:    +39 0382 985589

Skype: stefano.s.sibilla

Bio: My main research interests are: Lagrangian and Eulerian numerical methods in environmental fluid mechanics.Development of algorithms and methods for Smoothed Particle Hydrodynamics (SPH).  Application of SPH to unsteady open-channel flows and sediment transport. Analysis of log transport in flood flows by coupled Discrete Element (DE) and Finite Volumes method. Application of Finite Volume methods to complex convection-diffusion problems in water bodies.