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.
- Introduction to turbulence models: Reynolds experience. Evolution equation for the average velocity and the average kinetic energy of the fluctuating motion. Spectral density of kinetic energy and dissipation. Kolmogorov analysis. Direct numerical simulations (DNS). Application to the flow between two planes: linear law in the viscous sublayer and logarithmic law. Turbulence models with  0,1,2 equations.

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