CMS-Flow:Hydro Eqs

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Continuity and Momentum Equations

Hydrodynamics

Governing Equations

Phillips (1977), Mei (1983), and Svendsen (2006) provide a detailed deri-vation of the depth-integrated and wave-averaged hydrodynamic equa-tions. Here, only variable definitions are provided and derivations may be obtained from the preceding references. The instantaneous current velocity ui is split into:

in which

= current (wave-averaged) velocity [m/s]

= wave (oscillatory) velocity with wave-average

= turbulent fluctuation with ensemble average = 0 and wave average = 0 [m/s]

The wave-averaged total volume flux is defined as

=

where

= wave-averaged water depth [m]

= total mean mass flux velocity or simply total flux velocity for short [m/s]

= instantaneous current velocity [m/s]

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

= 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

(2-3)

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

where is the depth-averaged wave flux velocity [m/s], and = wave trough elevation [m]. Therefore the total flux velocity may be written as

On the basis of the above definitions, and assuming depth-uniform cur-rents, the general depth-integrated and wave-averaged continuity and momentum equations may be written as (Phillips 1977; Mei 1983; Svendsen 2006)

(2-6)

Failed to parse (syntax error): {\displaystyle + \frac {\partial}{\partial x_j} {\left(v_{t}h \frac {\partial V_i} {\partial x_j} \right)} - \frac{1}{\rho} \frac{\partial}{\partial x_j} {\left(S_{ij} + R_{ij} - \rho hU_{wi}U_{wj}\right)} + \frac{\tau_{si}{\rho} /math>'' (2-7) Assuming depth-uniform currents in the presence of oscillatory waves, the general depth- and phase-averaged continuity and momentum equations may be written as (Phillips 1977, Mei 1983, and Svendsen 2006) {{Equation|1=<math>\frac{\partial h}{\partial t}+\frac{\partial (h{{V}_{j}})}{\partial {{x}_{j}}}=0} |2=1}}

where is the total water depth , is the water surface elevation, is the still water depth, is the wave-averaged, depth-integrated mass flux velocity defined as , where is the phase- and depth-averaged current velocity, and is the depth-averaged wave velocity (Phillips 1977)

  (2)

where is the wave energy, is the surface roller energy, is the water density, and is the wave celerity, and is the wave unit vector where is the wave direction.

The momentum equation can be written as

  (3)

where is the gravitational constant, is the Coriolis parameter, is the atmospheric pressure, is a reference water density, is the turbulent eddy viscosity, is the surface wind stress, and is the combined wave-current mean bed shear stress. is the permutation parameter equal to 1 for = 1,2, -1 for = 2,1; and 0 for .

The equations above are similar to those derived by Svendsen 2006, ex-cept for the inclusion of the water source/sink term in the continuity equation, and the atmospheric pressure and surface roller terms in the momentum equation. It is also noted that the horizontal mixing term is formulated differently as a function of the total flux velocity similar to the Generalized Lagrangian Mean (GLM) approach (Andrews and McIntyre, 1978; Walstra et al, 2000). This approach is arguably more physically meaningful and also simplifies the descritization.

Bottom Stress

The mean (wave averaged) bed shear stress is calculated as

  (4)

where is the nonlinear wave enhancement factor, is a bed slope friction coefficient, is the bottom friction coefficient, and is the Eulerian current magnitude. For additional information on the bottom friction wee Bottom and Wall Friction

Wave and Roller Stresses

The wave radiation stresses are calculated using linear wave theory

  (6)

where is frequency, is the direction, =1 for , =0 for , and . For more information on the CMS-Wave model see CMS-Wave.

The roller stresses are given by are calculated as

  (7)

where is the roller energy. For more information on the surface roller see Surface Roller.

Numerical Methods

General Transport Equation: Discretization

All of the governing equations may be written in general form

 

(8)

where is a general scalar, is time, is the total water depth, is the depth averaged current velocity, is the diffusion coefficient for , is the gradient operator, and includes all other terms. Note that in the case of the continuity and momentum equations is equal to 1 and respectively.

Temporal Term

The temporal term of the momentum equations is discretized using a first order implicit Euler scheme

 

(9)

where is the cell area, and is the hydrodynamic time step.

Advection Term

The advection scheme obtained using the divergence theorem as where is the outward unit normal on cell face f, is the cell face length and is the total water depth linearly interpolated to the cell face. Here the overbar indicates a cell face interpolation operator described in the following section. For Cartesian grids the cell face unit vector is always aligned with one of the Cartesian coordinates which simplifies the calculation. Defining the cell face normal velocity as the above equation simplifies to

 

(10)

where is the outward unit normal on cell face f, is the cell face length and is the total water depth linearly interpolated to the cell face. Here the overbar indicates a cell face interpolation operator described in the following section. For Cartesian grids the cell face unit vector is always aligned with one of the Cartesian coordinates which simplifies the calculation. Defining the cell face normal velocity as the above equation simplifies to

 

(11)

where , , with being the basis vector. is equal to 1 for West and South faces and equal to -1 for North and East cell faces. Lastly, is the advective value of on cell face f, and is calculated using either the Hybrid, Exponential, HLPA (Zhu 1991) schemes. The cell face velocities are calculated using the momentum interpolation method of Rhie and Chow (1983) described in the subsequent section. The advection value is calculated as , where the superscripts and indicate low and high order approximations and the superscripts and indicate either explicit and implicit treatment. The explicit term is solved directly while the implicit term is implemented through a deferred correction in which the terms are approximated using the values from the previous iteration step.

Cell-face interpolation operator

The general formula for estimating the cell-face value of is given by

 

(12)

where is a linear interpolation factor given by and is the gradient operator in the direction parallel to face f. By definition . Note that for neighboring cells without any refinement and are zero and thus the above equation is consistent with non-refined cell faces.

Diffusion term

The diffusion term is discretized in general form using the divergence theorem

 

(13)

The discritization of the cell-face gradient is described in the next section. On a Cartesian grid the above expression may be further simplified as

 

(14)

where is gradient in the direction perpendicular to the cell face and .

Hydrodynamic Solver

Wetting and drying

In the numerical simulation of the surface water flows with sloped beaches, sand bars and islands, the water edges change with time, with part of the nodes being possibly wet or dry. In the present model, a threshold flow depth (a small value such as 0.02 m in field cases) is used to judge drying and wetting. If the flow depth on a node is larger than the threshold value, this node is considered to be wet, and if the flow depth is lower than the threshold value, this node is dry. Because a fully implicit solver is used in the present model, all the wet and dry nodes participate in the solution. Dry nodes are assigned a zero velocity. On the water edges between the dry and wet nodes, the wall-function approach is applied.

References

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Variable Index

Symbol Description Units
Time sec
Total water depth m
Still water depth m
Water surface elevation with respect to the still water elevation m
Current velocity in the jth direction m/sec
Sum of Precipitation and evaporation per unit area m/sec
Gravitational constant m/secsup2/sup
Water density kg/msup3/sup
Atmospheric pressure Pa
Turbulent eddy viscosity msup2/sup/sec



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