# CMS-Flow Hydrodnamics: Variable Definitions

The instantaneous current velocity ui is split into:

 ${\displaystyle u_{i}={\bar {u_{i}}}+{\tilde {u_{i}}}+u_{i}^{'}}$ (1)

in which

${\displaystyle {\bar {u_{i}}}}$ = current (wave-averaged) velocity [m/s]
${\displaystyle {\tilde {u_{i}}}}$ = wave (oscillatory) velocity [m/s]with wave-average ${\displaystyle {\bar {u_{i}}}=0}$ below the wave trough
${\displaystyle u_{i}^{'}}$ = turbulent fluctuation [m/s] with ensemble average ${\displaystyle \langle u_{i}^{'}\rangle }$ = 0 and wave average ${\displaystyle {\bar {u_{i}^{'}}}}$ = 0

The wave-averaged total volume flux is defined as

 ${\displaystyle hV_{i}={\overline {{\int _{z_{b}}^{\eta }}{u_{i}dz}}}}$ (2)

where

${\displaystyle h}$ = wave-averaged water depth ${\displaystyle h={\bar {\eta }}-z_{b}}$ (Figure 1) [m]
${\displaystyle V_{i}}$ = total mean mass flux velocity or simply total flux velocity [m/s]
${\displaystyle \eta }$ = instantaneous water level with respect to the Still Water Level (SWL) [m]
${\displaystyle {\bar {\eta }}}$ = wave-averaged water surface elevation with respect to the SWL (Figure 2-1) [m]
${\displaystyle z_{b}}$ = bed elevation with respect to the SWL (Figure 1) [m]

Figure 1. Vertical conventions used for the bed and mean water surface elevation.

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

 ${\displaystyle hU_{i}=\int _{z_{b}}^{\bar {\eta }}{\bar {u_{i}}}dz}$ (3)

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

 ${\displaystyle Q_{wi}=hU_{wi}={\overline {\int _{\eta _{t}}^{\eta }{\tilde {u_{i}}}dz}}}$ (4)

where

${\displaystyle U_{wi}}$ = depth-averaged wave flux velocity [m/s]
${\displaystyle \eta _{t}}$ = wave trough elevation [m]

Therefore the total flux velocity may be written as

 ${\displaystyle V_{i}=U_{i}+U_{wi}}$ (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.