CMS-Flow Hydrodnamics: Variable Definitions: Difference between revisions
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:<math>\bar{u_i}</math> = current (wave-averaged) velocity [m/s] | :<math>\bar{u_i}</math> = current (wave-averaged) velocity [m/s] | ||
:<math>\tilde{u_i}</math> = wave (oscillatory) velocity with wave-average <math>\bar{\tilde{u_i}} = 0 | :<math>\tilde{u_i}</math> = wave (oscillatory) velocity [m/s]with wave-average <math>\bar{\tilde{u_i}} = 0</math> below the wave trough | ||
:<math>u_i^'</math> = turbulent fluctuation with ensemble average <math>\langle u_i^' \rangle</math> = 0 and wave average <math>\bar{u_i^'}</math> = 0 [m/s] | :<math>u_i^'</math> = turbulent fluctuation with ensemble average <math>\langle u_i^' \rangle</math> = 0 and wave average <math>\bar{u_i^'}</math> = 0 [m/s] | ||
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where | where | ||
:<math>h</math> = wave-averaged water depth <math>h=\bar{\eta} - z_b </math> [m] | :<math>h</math> = wave-averaged water depth <math>h=\bar{\eta} - z_b </math> (Figure 2-1) [m] | ||
:<math>V_i</math> = total mean mass flux velocity or simply total flux velocity for short [m/s] | :<math>V_i</math> = total mean mass flux velocity or simply total flux velocity for short [m/s] | ||
:<math> | :<math>\eta</math> = instantaneous water level with respect to the Still Water Level (SWL) [m] | ||
:<math>\eta</math> = | :<math>\bar{\eta}</math> = wave-averaged water surface elevation with respect to the SWL (Figure 2-1) [m] | ||
:<math>z_b</math> = bed elevation with respect to the SWL [m] | :<math>z_b</math> = 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 | 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 | ||
{{Equation| | {{Equation| | ||
<math>hU_i = \int^\bar{\eta}_{ | <math>hU_i = \int^\bar{\eta}_{z_b} \bar{u_i}dz</math>|3}} | ||
|3}} | |||
where <math>U_i</math> is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by | where <math>U_i</math> is the depth-averaged current velocity. Similarly, the wave volume flux is defined as by | ||
Revision as of 17:21, 11 August 2014
The instantaneous current velocity ui is split into:
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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^'} |
(1) |
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]
- 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 \tilde{u_i}} = wave (oscillatory) velocity [m/s]with wave-average 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{\tilde{u_i}} = 0} below the wave trough
- 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^'} = turbulent fluctuation with ensemble average 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 \langle u_i^' \rangle} = 0 and wave average 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^'}} = 0 [m/s]
The wave-averaged total 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 h V_i = \overline{{\int_z^\eta} {u_i dz }}} |
(2) |
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 h} = wave-averaged water depth 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 h=\bar{\eta} - z_b } (Figure 2-1) [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 V_i} = total mean mass flux velocity or simply total flux velocity for short [m/s]
- 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} = 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 \bar{\eta}} = wave-averaged water surface elevation with respect to the SWL (Figure 2-1) [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 (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
|
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^\bar{\eta}_{z_b} \bar{u_i}dz} |
(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} = \overline{\int_{\eta_t}^\eta \tilde{u_i} dz}} |
(4) |
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}} = depth-averaged wave flux velocity [m/s]
- 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}} |
(5) |