Lecture

Symmetries of Navier-Stokes Equations

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In course

ME-467: Turbulence

This course provides an introduction to the physical phenomenon of turbulence, its probabilistic description and modeling approaches including RANS and LES. Students are equipped with the basic knowledge to tackle complex flow problems in science and engin ...

Description

This lecture covers the symmetries of the Navier-Stokes equations, including the preservation of symmetry in dynamics, spontaneous symmetry breaking, and the periodic boundary conditions. It also discusses the known symmetries, such as translations, Galilean transformations, rotations, and scaling. The lecture emphasizes the importance of symmetry restoration at high Reynolds numbers and the invariance of periodic boundary conditions. Various proofs and examples are provided to illustrate the symmetries and their implications.

Instructor

Tobias Schneider

Tobias Schneider is an assistant professor in the School of Engineering at EPFL, the Swiss Federal Institute of Technology Lausanne. He received his doctoral degree in theoretical physics in 2007 from the University of Marburg in Germany working on the transition to turbulence in pipe flow. He then joined Harvard University as a postdoctoral fellow. In 2012 Tobias Schneider returned to Europe to establish an independent Max-Planck research group at the Max-Planck Institute for Dynamics and Self-Organization in Goettingen. Since 2014, he is working at EPFL, where he teaches fluid mechanics and heads the 'Emergent Complexity in Physical Systems' laboratory. Tobias Schneider's research is focused on nonlinear mechanics with specific emphasis on spatial turbulent-laminar patterns in fluid flows transitioning to turbulence. His lab combines dynamical systems and pattern-formation theory with large-scale computer simulations. Together with his team, Schneider develops computational tools and continuation methods for studying the bifurcation structure of nonlinear differential equations such as those describing the flow of a fluid. These tools are published as open-source software at channelflow.ch. Publications: Google Scholar

Official source

Related concepts (32)

Galilean transformation

In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group.

Galilean invariance

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary.

Lorentz transformation

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.

Principle of relativity

In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell equations have the same form in all inertial frames of reference. In the framework of general relativity the Maxwell equations or the Einstein field equations have the same form in arbitrary frames of reference.

History of Lorentz transformations

The history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product . In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.

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