33933 - Computational Thermo-Fluid Dynamics M

Course contents

- Conservation equation for mass, energy and momentum. Conservative and convective forms. Constitutive equations.
- Simple iterative methods for linear systems: Jacobi, Gauss-Seidel, SOR.
- Cauchy problem and characterization of second-order partial differential equations: elliptic PDEs (Laplace and Poisson equations), hyperbolic PDEs (wave propagation) and parabolic PDEs (heat diffusion).
- Parabolic PDEs: general properties. One-dimensional non-stationary  heat conduction equation.  Explicit and implicit discretization(Crank-Nicholson). Stability conditions. Boundary conditions: fixed temperature or heat flux known. Ghost points.
- Elliptic PDEs: Dirichlet and Neumann problems. Laplace and Poisson equations. Discretization of the heat conduction equation with internal generation in a rectangular domain. Symmetry boundary conditions.
- Quasi-linear hyperbolic PDEs of first-order: characteristic curves and their reduction to a system of ODEs. Numerical integration along the characteristic curve. Propagation of discontinuities in the first-order equations: discontinuities of the initial data or of the derivative. Explicit discretization on Cartesian grids. Lax-Wendroff method, CFL condition for numerical stability. Comparison of various schemes (centered, upwind and Godunov) for the Kelvin-Helmholtz instability.
- Continuity equation: its discretization for incompressible fluids with finite volumes and finite differences.
- Navier-Stokes equations. Different time discretization schemes: first-order and second-order forward scheme,  the leapfrog scheme. Spatial discretization on two-dimensional Cartesian staggered grids: the viscous and convective terms. Centered scheme, first-order upwind, QUICK.
- Poisson's equation for the pressure field.

Readings/Bibliography

- Instructor notes
- S.V. Patankar, Numerical heat transfer and fluid flow, McGraw-Hill Inc.,US (1980)
- S.B. Pope, Turbulent flows, Cambridge University Press (2000)
- G. Tryggvason, R. Scardovelli, S. Zaleski, Direct numerical simulations of gas-liquid multiphase flows, Cambridge University Press (2011)

Teaching methods

The lectures are integrated by exercises with the computer

Assessment methods

Oral exam which include the discussion of a short paper

Teaching tools

Projector, PC, computer labs

Office hours

See the website of Ruben Scardovelli

See the website of Beatrice Pulvirenti