Some spatially integrated time-averaged identities for fluids

Li Chun-yan; Fang Guo-hong

doi:10.1007/BF02842825

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Some spatially integrated time-averaged identities for fluids

Updated：2023-05-16

- Some spatially integrated time-averaged identities for fluids
- Journal of Oceanology and Limnology Vol. 17, Issue 4, Pages: 315-330(1999)
- Affiliations：
  
  1. CCPO
  2. Institute of Oceanology Chinese Academy of Sciences
- Author bio：
- Funds：
- DOI：10.1007/BF02842825
  CLC：
- Published：1999
- Accepted：
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Chun-yan, L., Guo-hong, F. Some spatially integrated time-averaged identities for fluids., Chin. J. Ocean. Limnol. 17, 315–330 (1999). https://doi.org/10.1007/BF02842825
DOI：

Chun-yan, L., Guo-hong, F. Some spatially integrated time-averaged identities for fluids., Chin. J. Ocean. Limnol. 17, 315–330 (1999). https://doi.org/10.1007/BF02842825 DOI：

摘要

Abstract

This paper deals with the derivation of a series of seemingly simple but ignored relations

and presents several identities for fluids never obtained before (relations of integrals over different sets of independent variables and connected by an invertible mapping). These identities are based on mass conservation and a mathematical transform with no restriction on dynamics. They are

however

crucial to some fundamental concepts and the interpretation of results from dynamics. The identity for momentum is of most importance and

when averaged over time

yields a relationship between the spatially integrated and time-averaged Lagrangian momentum and the spatially integrated and time-averaged Eulerian momentum. For a constant density fluid

the averaged identity reduces to a relation between the integrated mean displacement of the particles and the integrated mean Eulerian velocity. For an exactly oscillatory flow in an Eulerian description this identity yields a zero integrated mean displacement of particles. In the case of a progressive surface gravity wave

which is periodic but not exactly oscillatory in an Eulerian description such that there are no particles above the trough during part of the period

the mean momentum identity ensures that the integrated mean Eulerian momentum is equal to the integrated mean momentum of the particles. Therefore

there is essentially no superiority as to which description gives a better estimate of the total momentum or total transport at the same order of approximation. The widely used relation

that the Lagrangian velocity equals the Eulerian velocity plus the Stokes velocity

is not based on a 1 to 1 invertible mapping and is therefore ambiguous. This relation is not valid where there is a discontinuity

particularly above the wave trough. It can therefore give an incorrect result on the Lagrangian velocity. The Generalized Lagrangian Mean (GLM) theory uses a mapping between the mean positions of particles and the particles themselves and apparently avoids the non-invertibility. However

this mapping is actually not invertible in general because different particles may have the same mean position.

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references

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