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2016 | OriginalPaper | Buchkapitel

4. Depth-Averaged Modelling Equations for Single-Phase Material Flows

verfasst von : Ioana Luca, Yih-Chin Tai, Chih-Yu Kuo

Erschienen in: Shallow Geophysical Mass Flows down Arbitrary Topography

Verlag: Springer International Publishing

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Abstract

In this chapter we deduce depth-averaged model equations for thin mass flows down arbitrary topographies, be they stationary or active, when the flowing material is assumed a single-phase (or one-component) continuum body.

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Vorheriges Kapitel Differential Operators and Balance Laws in the Topography-Fitted Coordinates

Nächstes Kapitel Closure Relations for the Depth-Averaged Modelling Equations

This assumption is only made to simplify the jump across ${\mathcal {S}}_b$ of the mass balance equation, see the forthcoming (4.9).

To such a topographic bed we will henceforth refer as being active, according to the convention made at the beginning of Sect. 2.2.

Notation $[\![f ]\!]$ stands for the jump of f at the moment t across a given surface. That is, for the case that the surface is ${\mathcal S}_{b}$, separating $\varOmega $ from $\varOmega _{b}$, at each time t the function f is assumed continuous on $\varOmega \cup {\mathcal S}_{b}$ and $\varOmega _{b}\cup {\mathcal S}_{b}$, but may be discontinuous on ${\mathcal S}_{b}$; the difference between the limits of f on ${\mathcal S}_{b}$ taken from both parts $\varOmega $, $\varOmega _{b}$,

$$[\![f ]\!]_{Q} \equiv \lim _{\tiny {\begin{array}{l}P\rightarrow Q\\ P\in \varOmega \end{array}}}f - \lim _{\tiny {\begin{array}{l}P\rightarrow Q\\ P\in \varOmega _b\end{array}}}f\,,\quad \; Q\in {\mathcal S}_{b}\,, $$

is the jump of f across ${\mathcal S}_{b}$ at the point Q.

If the topographic bed is deformable and no erosion/deposition processes occur, we simply set $\mathcal{U}=0$ in (4.9), i.e., $[\![\rho {\varvec{v}}]\!]\cdot \varvec{n}_b=0$.

Suppose that other scales than those which we have adopted here are used for the physical quantities entering the 3D modelling equations. That is, assume the physical quantity $Q_\mathrm{dim}$ to be non-dimensionalized as $Q_\mathrm{dim}=\beta \tilde{Q}$, while in this book $Q_\mathrm{dim}=\alpha {Q_\mathrm{dimless}}$, $\alpha \not =\beta $. Then, our final modelling equations, expressed in terms of quantities like ${Q_\mathrm{dimless}}$, can be immediately written in terms of quantities like $\tilde{Q}$ by the identification ${Q_\mathrm{dimless}}=\frac{\beta }{\alpha }\tilde{Q}.$

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Titel: Depth-Averaged Modelling Equations for Single-Phase Material Flows
verfasst von: Ioana Luca
Yih-Chin Tai
Chih-Yu Kuo
Verlag: Springer International Publishing
Buch: Shallow Geophysical Mass Flows down Arbitrary Topography
Print ISBN: 978-3-319-02626-8

Electronic ISBN: 978-3-319-02627-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-02627-5_4

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