WSP09211

• File Number: 8220.101.10
• Description: Colorado River-Water Projects-Glen Canyon Dam/Lake Powell-Glen Canyon Adaptive Management-
• Basin: Colorado Mainstem
• Water Division: 5
• Date: 1/1/2005
• Author: Wiele-Torizzo
• Title: Modeling of Sand Deposition in Archaeologically Significant Reaches of the Colorado River in Grand Canyon
• Water Supply Pro - Doc Type: Report/Study

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00625

Modelling of sand deposition in Colorado River 361

where x is the direction normal to the upstream boundary, y is the direction normal
to x, u is the vertically averaged velocity in the x-direction, v is the vertically averaged
velocity in the y-direction, h is the flow depth, '1 is the bed surface elevation, S is the
average reach slope, E is the eddy viscosity, g is gravity, p is density of water, and T.
and T, are the shear stresses in the x- and y-directions, respectively. Equations
(14.1HI4.3) are in Cartesian coordinates. For applications to reaches with signifi.
cant curvature, [he equations were modified with the metric of Smith and McLean
(\ 984) for calculations based on an orthogonal curvilinear coordinate system with a
variable radius of curvature.
A friction coefficient, Cr, is used to relate the resolved shear stress, T, to the
resolved velocity,

T = pcrlUlU

(14.4)

where V = (.l + Vl)112 is the magnitude of the resolved velocity. The x and y
components are determined from the relations:

T. = PCfUV

(14.5)

and

Ty = pcrvU

(t4.6)

The friction coefficient is defined by

( )2
It.
Cf -
- Int- I

(14.7)

where K is von Karman's constant and Zo is the roughness parameter. Equation
(\4.7) is derived by vertically averaging the logarithmic velocity profile (Keulegan,
1938). The value of to at each node depends in part on the thickness of the sand
cover. A value for to was initially computed based on bathymetric measurements
with Zo = O.lb.., where b.. is the 84th percentile of the deviations of the local
bathymetric measurement from a straight line drawn between two adjacent nodes.
The coefficient O. 1 is typically used to relate to to a distribution of gravel sizes. Where
the local sand thickness exceeds the Zo computed from bathymetric records, then the
local to was computed based on estimated bedform dimensions, as will be discussed
in more detail later.
The eddy viscosity, E, is defined by

E(Z) = u.ltz(l - z/h)

(14.8)

where u. is the shear velocity (Tlp)'/l, and z is the distance above the bed. Equation
(14.8) is venicaUy averaged for use in equations (14.1) and (\4.2):

U.t<h
E=-
6

(14.9)

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