u
uα,t + ng∇η + uα ⋅ ∇ α + zα ∇ (uα,t ⋅ ∇h ) + (∇ ⋅ uα,t )∇h
n
(60)
1
( zα + h ) - h2 ∇ (∇ ⋅ uα,t ) + nfl uα + nft uα uα = 0
2
+
2
(1953) recommended the following empirical relationships for the laminar and
turbulent friction factors:
(1 - n)3 ν
fl = αo
(61)
n2 d 2
(1 - n) 1
ft = βo
(62)
n3 d
where ν is the kinematic viscosity of water, d is the characteristic stone size, and
αo and βo are empirical constants that range from 780 to 1,500, and 1.8 to 3.6
respectively.
Simulation of Wave Runup
The runup of waves on shorelines provides an important boundary condition
for predicting wave-induced currents and sediment transport in the surf zone. The
runup limit is also important for determining the minimum crest elevation of
coastal structures to prevent overtopping and/or flooding. A simple runup scheme
has been implemented in BOUSS-2D. Dry computational cells (land points) are
assumed to be porous regions where the phreatic surface elevation and volume-
averaged velocities are calculated simultaneously with the fluid motion in the wet
cells. When the phreatic surface elevation exceeds the elevation of the land point
by a specified threshold, the porous cell is considered flooded and treated as a
wet cell during the next time-step. Alternatively, when the free surface elevation
drops below a specified threshold above the bottom elevation of a wet cell, the
wet cell is assumed to be dry and treated as a porous cell during the next
time-step.
Subgrid Turbulence
BOUSS-2D optionally provides a mechanism to simulate the turbulence and
mixing that occurs in regions with large gradients in the horizontal velocities
such as around the tips of breakwaters. The dissipation term is identical to that
used for wave breaking, i.e., Equation 16. The eddy viscosity is given by
Smagorinsky's (1963) formulation with the turbulent length scale proportional to
the grid size. It can be written as:
25
Chapter 3 Numerical Solution
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