Hydrodynamics

Fluids are all around us at all times and on every scale. Understanding fluid mechanics means understanding the mechanics of our surroundings—air and water, in particular. As such, hydrodynamics is part of a physicist’s basic knowledge.

The course “Hydrodynamics” provides an introduction to the mechanics of ideal incompressible fluids (Euler) and Newtonian viscous fluids (Navier-Stokes). Classical flow patterns are presented, along with the concepts of the boundary layer, instability, and turbulence. The focus is on physical concepts rather than on advanced mathematical or numerical solution methods.

Master the basic concepts and flow patterns of fluid dynamics in order to be able to tackle more advanced courses, such as numerical solution of hydrodynamic equations, complex fluids in the M2 program, or magnetohydrodynamics (Course Unit: Fluid Dynamics in Astrophysics and Cosmology in the M2 program) and their applications in the fields of biophysics, colloid physics, or coastal engineering, to name just a few examples.

Newtonian Dynamics 1 & 2 in L

Recommended prerequisites:

The L-level courses “Hydrodynamics,” “Elasticity and Hydrodynamics,” and “Mathematics for Physics” are helpful but not required. Generally speaking, a solid foundational 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 its 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 classical examples of steady and unsteady viscous flows: planar and cylindrical Couette flow, flow in a pipe, Poiseuille’s formula. Viscous fluid set in motion by a tangential stress and its relationship to diffusion. Highly viscous fluid, flow, and Stokes’ formula.

Laminar boundary layer, Prandtl's theory, boundary layer shape. Boundary layer separation and its role in turbulence. Basics of the turbulent boundary layer.

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

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

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

The course includes numerous examples of physics in everyday life and environmental contexts.