CMS-Flow Hydrodnamics: Variable Definitions

The instantaneous current velocity u_i is split into:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_i = \bar{u_i} + \tilde{u_i} + u_i^'}

in which

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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

$hV_{i}$_ = ${\bar {{\int _{z}^{\eta }}{u_{i}dz}}}$

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]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle hU_i = \int^\eta_{z} \bar{u_i}dz} (2-3)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_i} is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_{wi} = hU_{wi} = \bar{\int \tilde{u_i} dz}}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{wi}} is the depth-averaged wave flux velocity [m/s], and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \eta_t} = wave trough elevation [m]. Therefore the total flux velocity may be written as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_i = U_i + U_{wi}}