# CMS-Flow Hydrodnamics: Variable Definitions

The instantaneous current velocity ui 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 $\bar{{u_i}} = 0$ 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

 $h V_i = \overline{{\int_{z_b}^\eta} {u_i dz }}$ (2)

where

$h$ = wave-averaged water depth $h=\bar{\eta} - z_b$ (Figure 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 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

 $hU_i = \int^\bar{\eta}_{z_b} \bar{u_i}dz$ (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.