Momentum Conservation

When the horizontal length scales are much larger than the vertical length scale, volume conservation implies that the vertical velocity is small. The Navier-Stokes vertical momentum equation can be used to justify that pressure is nearly hydrostatic. In the absence of baroclinic pressure gradients (variable density), strong wind forcing and non-hydrostatic pressure, a vertically-averaged version of the momentum equation is adequate. Vertical velocity and vertical derivative terms can be safely neglected (in both mass and momentum equations). The shallow water equations are obtained:

1)

$\begin{array}{l}\displaystyle \displaystyle \frac{\partial \boldsymbol{V}}{\partial t} + (\boldsymbol{V} \cdot \nabla ) \boldsymbol{V} + f_c \boldsymbol{k} \times \boldsymbol{V} = -g \nabla z_s + \frac{1}{h} \nabla \cdot (\boldsymbol{\nu}_t h \nabla \boldsymbol{V}) - \frac{1}{\rho h} \nabla \cdot \left( h \boldsymbol{D} \right) - \frac{\boldsymbol{\tau}_b}{\rho R} + \frac{\boldsymbol{\tau}_s + \boldsymbol{\tau}_w}{\rho h} - \frac{1}{\rho}\nabla p_a - \left(a + b | \boldsymbol{V} | \right) \boldsymbol{V}\end{array}$

where

$\begin{array}{l}\boldsymbol{V}\end{array}$ : velocity vector [L/T]
$\begin{array}{l}g\end{array}$ : Gravitational acceleration [L/T²]
$\begin{array}{l}z_s\end{array}$ : Water surface elevation [L]
$\begin{array}{l}\nabla\end{array}$ : gradient operator [1/L]
$\begin{array}{l}\boldsymbol{\nu}_t\end{array}$ : horizontal eddy viscosity coefficient tensor [L²/T]
$\begin{array}{l}{\boldsymbol{\tau}_b\end{array}$ : bottom shear stresses vector s [M/L/T²]
$\begin{array}{l}R\end{array}$ : Hydraulic radius [L]
$\begin{array}{l}\boldsymbol{D}\end{array}$ : dispersion stress tensor [M/L/T²]
$\begin{array}{l}{\boldsymbol{\tau}_s\end{array}$ : surface wind stress vector [M/L/T²]
$\begin{array}{l}{\boldsymbol{\tau}_w\end{array}$ : wave radiation stress gradient vector [M/L/T²]
$\begin{array}{l}h\end{array}$ : water depth [L]
$\begin{array}{l}\boldsymbol{k}\end{array}$ : is the unit vector in the vertical direction [-]
$\begin{array}{l}f_c\end{array}$ : Coriolis parameter [1/T]
$\begin{array}{l}p_a\end{array}$ : atmospheric pressure [M/L/T²]
$\begin{array}{l}a\end{array}$ : linear (viscous) drag coefficient [1/T]
$\begin{array}{l}b\end{array}$ : quadratic (inertial) drag coefficient [1/L]

The left-hand side of the equation contains the acceleration terms. The right-hand side represents the internal or external forces acting on the fluid. The left- and right-hand side term are typically organized in such a way as to be in accordance with Newton's second law, from which the momentum equations are ultimately derived. From left to right the terms are the unsteady acceleration, convective acceleration, Coriolis term, barotropic pressure term, momentum diffusion, bottom friction, wind and wave stresses, and flow drag.

In the case where the porosity < 1, the velocity vector represents the pore velocity which is also known as the intrinsic or simply flow velocity. The average fluid velocity over the medium (including both the solid and fluid material) is given by the Dupuit–Forchheimer relationship $\begin{array}{l}\boldsymbol{v} = \phi \boldsymbol{V}\end{array}$ . The velocity $\begin{array}{l}\boldsymbol{v}\end{array}$ is known as the filtration velocity, seepage velocity, superficial velocity, Darcy velocity, macroscopic velocity, and volumetric flux velocity. The interstitial velocity is also known as the intrinsic, pore velocity, or simply flow velocity.

It is noted that the notation for the Coriolis term is not strictly correct due to the inconsistent length of vectors. However, this notation is used for shorthand notation and simplicity. Every term of the momentum equation has a clear physical counterpart. A dimensional analysis shows that when the water depth is very small the bottom friction term dominates the equation. As a consequence, the momentum equation for dry cells takes the limit form V = 0. As before, dry cells are computationally treated as a special case, but the result is continuous and physically consistent during the process of wetting or drying.

Since the conservation of momentum is directionally invariant, the momentum equation may be in any direction. In HEC-RAS, momentum is computed normal to each face.

2)

$\begin{array}{l}\displaystyle \displaystyle \frac{\partial u_N}{\partial t} + (\boldsymbol{V} \cdot \nabla ) u_N - f_c u_T = -g \frac{\partial z_s}{\partial N} + \frac{1}{h} \nabla \cdot (\boldsymbol{\nu}_t h \nabla u_N) - \left[ \frac{1}{\rho h} \nabla \cdot \left( h \boldsymbol{D} \right) \right] \cdot \boldsymbol{n} - \frac{\tau_{b,N}}{\rho R} + \frac{\tau_{s,N}}{\rho h} - \frac{1}{\rho} \frac{\partial p_a}{\partial N}- \left(a + b | \boldsymbol{V} | \right) u_N\end{array}$

Acceleration

The Eulerian acceleration terms on the left, can be condensed into a Lagrangian derivative acceleration term taken along the path moving with the velocity term:

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{D \boldsymbol{V}}{Dt} = \frac{\partial \boldsymbol{V}}{\partial t} + (\boldsymbol{V} \cdot \nabla )\boldsymbol{V}\end{array}$

Other names usually given to this term are substantial, material and total derivative. The use of the Lagrangian derivative will become evident in subsequent sections when it will be seen that its discretization reduces Courant number constraints and yields a more robust solution method.

Bottom Friction

The bottom shear stress is given by

4)	$\begin{array}{l}\displaystyle \displaystyle \boldsymbol{\tau_b} = \rho C_D \| \boldsymbol{V} \| \boldsymbol{V}\end{array}$

where $\begin{array}{l}\rho\end{array}$ is the water density and $\begin{array}{l}C_D\end{array}$ is the drag coefficient computed using the Manning’s roughness coefficient as

5)	$\begin{array}{l}\displaystyle \displaystyle C_D = \frac{n^2 g}{R^{1/3}}\end{array}$

where

$\begin{array}{l}n\end{array}$ : Manning's roughness coefficient [T/L^1/3]
$\begin{array}{l}R\end{array}$ : hydraulic radius [L]
$\begin{array}{l}g\end{array}$ : gravitational acceleration [L/T²]

The drag coefficient, $\begin{array}{l}C_D\end{array}$ , is related to the nonlinear friction coefficient, $\begin{array}{l}c_f\end{array}$ , by

6)	$\begin{array}{l}\displaystyle \displaystyle c_f = \frac{C_D}{R} \| \boldsymbol{V} \| = \frac{n^2 g}{R^{4/3}} \| \boldsymbol{V} \|\end{array}$

The shear velocity is given by:

7)	$\begin{array}{l}\displaystyle \displaystyle u_* = \sqrt{\tau_b / \rho }\end{array}$

Coriolis Effect

The last term of the momentum equation relates to the Coriolis Effect. It accounts for the fact that the frame of reference of the equation is attached to the Earth, which is rotating around its axis. The vertical component of the Coriolis term is disregarded in agreement with the shallow water assumptions. The apparent horizontal force felt by any object in the rotating frame is proportional to the Coriolis parameter given by:

8)	$\begin{array}{l}\displaystyle \displaystyle f_c = 2 \omega \text{sin} \varphi\end{array}$

where $\begin{array}{l}\omega\end{array}$ = 0.00007292115855306587 1/s is the sidereal angular velocity of the Earth and $\begin{array}{l}\varphi\end{array}$ is the latitude.

Flow Drag

The linear and quadratic flow drag coefficients are designed to be generic and flexible so that they can be used to represent different processes such as flow drag through porous media or flow drag from larger flow obstructions such as vegetation, buildings, etc.