CMS-Flow Hydrodnamics: Variable Definitions

The instantaneous current velocity u_i is split into:

$u_i=\overline{u_i}+\widetilde{u_i}+u_i^{\text{'}}$

(1)

in which

$\overline{u_i}$ = current (wave-averaged) velocity [m/s]

$\widetilde{u_i}$ = wave (oscillatory) velocity [m/s]with wave-average $\overline{\widetilde{u_i}}=0$ below the wave trough

$u_i^{\text{'}}$ = turbulent fluctuation [m/s] with ensemble average $⟨u_i^{\text{'}}⟩$ = 0 and wave average $\overline{u_i^{\text{'}}}$ = 0

The wave-averaged total volume flux is defined as

$h{V}_i=\overline{\underset{z_b}{\overset{\eta }{\int }}{u}_{i}dz}$

(2)

where

$h$ = wave-averaged water depth $h=\overline{\eta }-z_b$ (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]

$\overline{\eta }$ = 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

$h{U}_i=\underset{z_b}{\overset{\overline{\eta }}{\int }}\overline{u_i}dz$

(3)

where $U_i$ is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by

$Q_{wi}=h{U}_{wi}=\overline{\underset{{\eta }_t}{\overset{\eta }{\int }}\stackrel{~}{u_i}dz}$

(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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