Hydrodynamics

Fluids are all around us at all times and on all scales. Understanding fluid mechanics means understanding the mechanics of our surroundings, particularly air and water. As such, hydrodynamics is part of a physicist's basic knowledge.

Hydrodynamics is an introduction to the mechanics of incompressible perfect fluids (Euler) and viscous Newtonian fluids (Navier-Stokes). Classical flows are presented, as well as the concepts of boundary layer, instability, and turbulence. The emphasis is placed more on physical ideas than on advanced mathematical or numerical solution methods.

Master the basic concepts and flows of fluid dynamics in order to be able to tackle more advanced courses, either in numerical solution of hydrodynamic equations, complex fluids in M2, or magnetohydrodynamics (UE Fluid Dynamics in Astrophysics and Cosmology in M2) and applications in the fields of biophysics, colloid physics, or coastal engineering, to name but a few examples.

Newtonian Dynamics 1&2 in L

Recommended prerequisites:

The courses Hydrodynamics, Elasticity and Hydrodynamics, and Mathematics for Physics (L) are useful but not mandatory. In general, a solid background in physics is necessary to take this course.

CCI 100 %

Hydrodynamic description. Eulerian and Lagrangian perspectives. Equation of conservation of mass, incompressibility. Ideal fluid: Euler's equation, momentum balance and stress tensor of the ideal fluid, boundary conditions, hydrostatics, Bernoulli's theorem and applications, conservation of vorticity, potential flow, examples of ideal flows.

Viscous stress tensor, viscosity, Newtonian fluid, boundary layer. Navier-Stokes equation and boundary conditions. Euler ↔ Navier-Stokes transition, role of the Reynolds number, examples and orders of magnitude. Simple classic examples of steady and unsteady viscous flows: plane and cylindrical Couette flow, flow in a pipe, Poiseuille's formula. Viscous fluid set in motion by a tangential stress and relationship with diffusion. Highly viscous fluid, flow and Stokes' formula.

Laminar boundary layer, Prandtl theory, boundary layer shape. Boundary layer separation and its role in turbulence. Concepts related to turbulent boundary layers.

Instability: the principle of calculating stability conditions. Empirical conditions and Reynolds number, unstable flows. Examples: Von Karman alley, Couette instability in rotating cylinders, Kelvin-Helmholtz instability, flow instability in a pipe (Reynolds) and between two parallel planes, Rayleigh-Taylor instability, etc. The path to turbulence.

Turbulence: statistical description of turbulent fluid, correlations, Reynolds tensor, role and interpretation. Example of application: the logarithmic velocity profile near an obstacle. Kolmogorov cascade and energy dissipation in a fluid.

Gravity waves: boundary conditions at the interface between two fluids (with and without surface tension). Linear boundary conditions, waves in deep water and finite depth, dispersion relation, interpretation and role of phase velocity and group velocity. Stokes waves (periodic nonlinear solution). Applications.

The course is illustrated with numerous applications from physics and everyday life, as well as environmental applications.