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 with wave-average ${\bar {\tilde {u_{i}}}}=0[m/s]$

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

The wave-averaged total volume flux is defined as

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle hV_{i}_{=}{\bar {{\int _{z}^{\eta }}{u_{i}dz}}}}

(2)

where

$h$ = wave-averaged water depth $h={\bar {\eta }}-z_{b}$ [m]

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

$u_{i}$ = instantaneous current velocity [m/s]

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

$z_{b}$ = bed elevation with respect to the SWL [m]

For simplicity in the notation, the over bar in subsequent wave-averaged variables is omitted.

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

$hU_{i}=\int _{z}^{\eta }{\bar {u_{i}}}dz$ (2-3)

(3)

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

$Q_{wi}=hU_{wi}={\bar {\int {\tilde {u_{i}}}dz}}$

(4)

where $U_{wi}$ is the depth-averaged wave flux velocity [m/s], and $\eta _{t}$ = wave trough elevation [m]. Therefore the total flux velocity may be written as

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

(5)