Basic introduction to compressible gasdynamics. Includes some fundamental thermodynamics, thermal and caloric equations of state, derivation of Euler’s equations by control volume approach. Also, includes the theory of steady flows in ducts with area changes, adiabatic frictional flows, duct flows with heat transfer, normal and oblique shock waves, Prandtl-Meyer expansion wave, moving shock and rarefaction waves, shock tubes, and wind tunnels. The lectures are supplemented by problem sets. Reference book: Anderson, J.D., Modern Compressible Flow with Historical Perspective.

Moment closure methods are considered for kinetic equations governing a range of complex transport phenomena, including non-equilibrium, rarefied, gaseous flow phenomena as described by the Boltzmann kinetic equation, multi-phase poly-disperse spray behaviour described by the Williams-Boltzmann equation, and radiative heat transfer in non-gray, participating media as described by the radiative transfer equation. Moment closure methods offer a means of significant complexity reduction when constructing approximate solutions to such high-dimensional equations, providing a good compromise between computational efficiency and accuracy for many practical engineering applications. Closure techniques based on classical Grad-type methods, maximum-entropy considerations, as well as quadrature formulations will all be considered.

Overview of numerical modelling techniques for the prediction of turbulent flows with emphasis on the capabilities and limitations of engineering approaches commonly used in computational fluid dynamics (CFD) for the simulation of turbulence. Topics include: Introduction to turbulent flows; definition of turbulence; features of turbulent flows; requirements for and history of turbulence modeling. Conservation equations for turbulent flows; Reynolds and Favre averaging; velocity correlations, Reynolds-averaged Navier-Stokes equations (RANS); Reynolds stress equations; effects of compressibility. Algebraic Models; eddy viscosity and mixing length hypothesis; Cebeci-Smith and Baldwin-Lomax models. Scalar-field evolution models; turbulence energy equation; one- and two-equation models; wall functions; low-Reynolds-number effects. Second-order closure models; full Reynolds-stress and algebraic Reynolds stress models. Large-Eddy Simulation (LES) techniques. Direct Numerical Simulation (DNS) Methods.

Introduction to upwind finite-volume methods widely used in computational fluids dynamics (CFD) for thehe solution of high-speed inviscid and viscous compressible flows. Topics include: Brief review of conservation equations for compressible flows; Euler equations; Navier-Stokes equations; one- and two-dimensional forms; model equations. Mathematical properties of the Euler equations; primitive and conserved solution variables; eigensystem analysis; compatibility conditions; characteristic variables, Rankine-Hugoniot conditions and Riemann invariants; Riemann problem and exact solution. Godunov's method; hyperbolic flux evaluation and numerical flux functions; solution monotonicity; Godunov's theorem. Approximate Riemann solvers; Roe's method. Higher-order Godunov-type schemes; semi-discrete form; solution reconstruction including least-squares and Green-Gauss methods; slope limiting. Extension to multi-dimensional flows. Elliptic flux evaluation for viscous flows; diamond-path and average-gradient stencils; discrete-maximum principle. High-order methods; essentially non-oscillatory (ENO) schemes.