CMS-Flow Hydrodnamics: Variable Definitions

The instantaneous current velocity u_i is split into:

$u_{i}={\bar {u_{i}}}+{\tilde {u_{i}}}+u_{i}^{'}$

(1)

in which

${\bar {u_{i}}}$ = current (wave-averaged) velocity [m/s]

${\tilde {u_{i}}}$ = wave (oscillatory) velocity [m/s]with wave-average ${\bar {\tilde {u_{i}}}}=0$ below the wave trough

$u_{i}^{'}$ = turbulent fluctuation [m/s] with ensemble average $\langle u_{i}^{'}\rangle$ = 0 and wave average ${\bar {u_{i}^{'}}}$ = 0

The wave-averaged total volume flux is defined as

$hV_{i}={\overline {{\int _{z_{b}}^{\eta }}{u_{i}dz}}}$

(2)

where

$h$ = wave-averaged water depth $h={\bar {\eta }}-z_{b}$ (Figure 2-1) [m]

$V_{i}$ = total mean mass flux velocity or simply total flux velocity [m/s]

$\eta$ = instantaneous water level with respect to the Still Water Level (SWL) [m]

${\bar {\eta }}$ = wave-averaged water surface elevation with respect to the SWL (Figure 2-1) [m]

$z_{b}$ = bed elevation with respect to the SWL (Figure 2-1) [m]

The total flux velocity is also referred to as the mean transport velocity (Phillips 1977) and mass transport velocity (Mei 1983). The current volume flux is defined as

$hU_{i}=\int _{z_{b}}^{\bar {\eta }}{\bar {u_{i}}}dz$

(3)

where $U_{i}$ is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by

$Q_{wi}=hU_{wi}={\overline {\int _{\eta _{t}}^{\eta }{\tilde {u_{i}}}dz}}$

(4)

where

$U_{wi}$ = depth-averaged wave flux velocity [m/s]

$\eta _{t}$ = wave trough elevation [m]

Therefore the total flux velocity may be written as

$V_{i}=U_{i}+U_{wi}$

(5)

References

• Mei, C. 1989. The applied dynamics of ocean surface waves. New York: John Wiley.

• Phillips, O. M. 1977. The dynamics of the upper ocean. (2nd Edition). Cambridge, England: Cambridge University Press.

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