continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...

, the generalized Lagrangian mean (GLM) is a formalism – developed by – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the

flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psych ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, but appointed to fixed Eulerian coordinates.

Background

In general, it is difficult to decompose a combined wave–mean motion into a mean and a wave part, especially for flows bounded by a wavy surface: e.g. in the presence of

surface gravity wave In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...

s or near another undulating bounding surface (like atmospheric flow over mountainous or hilly terrain). However, this splitting of the motion in a wave and mean part is often demanded in

mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...

s, when the main interest is in the mean motion – slowly varying at scales much larger than those of the individual undulations. From a series of postulates, arrive at the (GLM) formalism to split the flow: into a generalised Lagrangian mean flow and an oscillatory-flow part. The GLM method does not suffer from the strong drawback of the Lagrangian specification of the flow field – following individual

fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...

s – that Lagrangian positions which are initially close gradually drift far apart. In the Lagrangian frame of reference, it therefore becomes often difficult to attribute Lagrangian-mean values to some location in space. The specification of mean properties for the oscillatory part of the flow, like: Stokes drift,

wave action In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of t ...

, pseudomomentum and pseudoenergy – and the associated

conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...

s – arise naturally when using the GLM method. The GLM concept can also be incorporated into variational principles of fluid flow.

Notes

References

By Andrews & McIntyre

* * * *

By others

* * See Chapter 12: "Generalized Lagrangian mean (GLM) formulation", pp. 105–113. * * {{refend Continuum mechanics Concepts in physics