In
continuum mechanics, 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 field, 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 waves 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 models, 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
postulate
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, 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 parcels – 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
For a pure wave motion (physics), motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of wat ...
,
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 o ...
,
pseudomomentum and
pseudoenergy – and the associated
conservation laws – arise naturally when using the GLM method.
The GLM concept can also be incorporated into
variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
s of fluid flow.
Notes
References
By Andrews & McIntyre
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By others
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* See Chapter 12: "Generalized Lagrangian mean (GLM) formulation", pp. 105–113.
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{{refend
Continuum mechanics
Concepts in physics