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```vary during a tidal period due to scouring and sedimentation of
the river bed, and additionally, its areal mean value and local actual value may dif-
fer greatly.
The above formulas can only be used when the water depth is much more small-
er than the Ekman depth (cf. VII) , but not too small (e. g. , h> 10m) , and when
there is no wind stress. If h is sufficiently small, nonlinearity of flow has a great in-
fluence on top and bottom turbulent boundary layers. In order to reflect the influence
of h properly, various modified formulas for C and n have been proposed:
Leendertze empirical formula
C = 19. 4 ln(0. 9h) (1. 4 . 30)
Brebbia empirical formula
C = 15 lnh ( I . 4. 31)
Sat0 modified Manning formula
C = ( h - a ) ' l 6 / z , a = 0. 5 - 1. 0 (1. 4 . 32)
Another formula relating roughness to water depth is (in the English system of units)
(1. 4. 33)
Xin Wenjie proposed an inversely proportional relation between n and h , which
When a wind stress z,, is exerted on a free surface, a simple formula is zb= pro,
was utilized in a numerical simulation for the Pearl River estuary in China.
where is a dimensionless constant. Moreover, a formula used in Japan is
(1.4. 34)
3 4
2. Oceanographic approach
Due to the differences in the materials comprising river beds and sea bottoms,
the associated relationships are also different. In oceanography, an empirical formula
in common use is
(1 .4 .35)
where yh=a sea bottom friction coefficient. It is often given a value associated with
C,, in Eq. (1. 4. 6 ) and T in Eq. (1. 4 . 2 6 ) , i.e. , yb=2. 6X10-3 . Another simple
formula is
zh = p u (1 . 4. 36)
where T is the same as in Eq. (1. 4 . 26). In the above-mentioned computation for the
North Sea, took r = 0. 0024. For the deep sea the effect of zb is small, so a simplifi-
When wind occurs over the water surface, a value /3z, should be subtracted from
the right-hand side. The empirical coefficient /I= 0. 35 represents the contribution of
surface turbulent shear stress to that at the sea bottom.
If depth h is very small ( e. g. , h<<3 m ) , to avoid the instability in a numerical
solution, we may use a correction formula for y,, in terms of h , e. g.
z h = PYIu I u I
1
'' = 32(lg148hI2
or try another formula
(1. 4. 37)
(1. 4 . 38)
where d and z =heights from mean sea level down to the sea bottom and up to the
free surface respectively, h=d+z. We usually take H o = H I = 1 m , p= 1.
Besides bottom frictional loss, in 1-D unsteady flow computations, there are
sometimes additional loss terms, due to expansion/contraction of cross-section as well
as local river bends. In 2-D shallow water flow computations, expansion/contraction
losses do not exist when using a rectangular mesh ; however, transversal circulations,
which occur at a river bend but which would disappear after depth-averaging, may
have a significant influence on the local flow field, and it can be considered by modi-
fying the momentum equations.
1'11. BODY FORCES
Body force terms F , , etc. , represent the external forces exerted distributively
on a fluid element per unit mass. Besides gravity, which has been discussed earlier,
two others are often encountered.
1. Geostrophic force
The Coriolis inertial force, stemming from the daily rotation of the earth, gives
rise to clockwise rotational flows in large water-bodies in the northern hemisphere.
Components of the force in the z- and y-directions are
( 1 . 4 . 39) Fx. = f v , FBg =- fu
35
where
f = 2wsinp (1. 4. 40)
f = the Coriolis coefficient; @=angular velocity of the earth in its daily rotation, w
= 7. 29X lop5 l / s ; and p=latitude. The above formulas are also applicable to any
orthogonal coordinate system. When using a non-orthogonal system with axes 4 and
q, it is necessary to project the Cartesian components onto the coordinate axes of that
system.
We often use a dimensionless Rossby number, R o = u / f d , where d is the charac-
teristic water depth, to express the importance of geostrophy in a flow. Ro multiplied
by d / L , where L is the characteristic horizontal length, denotes the ratio of horizon-
tal flow to geostrophic flow. The broader the free surface, the more important must
the geostrophic force be.
Geostrophic motion imposes on the velocity vector field an Ekman spiral struc-
ture, which extends under the action of bottom friction up to a maximum height d , ,
called the Ekman depth. Its value is more or less fixed, in general, about 150m. In
1914 , Theorade proposed that when the wind speed w, > 6m/s, d , = 7. 6 w,/
(n s p ) . When there is no other external force, and the ratio d / d , is smaller
than 0. 3 , the shear stress will be in almost the same direction as velocity, so that it
is permissible to deal with a shallow-water flow by depth-averaging.
2. Tide-raising force
This is Newton’s universal gravitation exerted on a water body and mainly com-
ing from the moon and the sun. The tide-raising force due to the moon is about
0. 056-0. 112 millionth of gravity, while for the sun it is 0. 026-0. 052 millionth.
The force exerted on a unit mass and denoted by F, belongs, like gravity, to the po-
tential force (cf. Section 2. l ) . Specifically, we are able to find a function 17 (x, ,
r 2 , x3) , called the tide-raising potential, whose partial derivatives equal the compo-
nents of F,
P,, = a n / a x , (1. 4. 4 1 )
Taking the center of the earth as a datum, the tide-raising potential of the moon at a
fluid particle on the earth’s surface is
(1. 4. 42)
where po = a universal gravitation constant, po = (6 . 670 & 0. 004 ) X 1 0-3 dyne
cm2/g2; M = t h e mass of the moon; D = the distance between the moon and the
earth; L=the distance between the moon center and the fluid particle. a=distance
between the earth center and the fluid particle; and 0=the angle made by the two
lines connecting the two centers and the fluid particle.
The physical meaning of tide-raising potential is an integral of the infinitesimal
work done by the tide-raising force exerted on a fluid element per unit mass. It can
be expanded into a series, resulting in multi-component tide potentials superposed to-
gether. Detailed tables have been compiled for reference .
Except for vast water bodies like the seas and oceans, the impact of the tide-rais-
ing force can generally be neglected. Tidal action in an estuary is chiefly due to the
variation of water level at the sea interface, and not through a force exerted directly
36
on the river flow. We may use an observed tide hydrograph as boundary condition,
or use the chief regular components obtained by harmonic analysis of astronomic tide.
For example, in dealing with diurnal tides with a period of 12 hours plus 20 minutes,
a common choice makes use of (M2 +&)-tide for determining amplitude (harmonic
constant) , Mz-tide for phase difference, and K2-tide for angle lag. Here, we usually
call M2 lunar tide, S z solar tide, and Kz the semi-diurnal constituent of luni-solay de-
clination.
V I I I . DEPTH-AVERAGED TliRBlILENT (EDDY) VISCOSITY
Turbulent viscosity terms such as v, V 2 u , etc. , represent the momentum ex-
change and energy dissipation resulting from molecular diffusion , turbulent diffu-
sion, vertical variation of horizontal velocity, and nonuniformity of the velocity dis-
tribution over the horizontal plane.
From the physical viewpoint, turbulent viscosity differs from surface and bot-
tom friction terms. For a meandering river, or when the bottom rises and falls great-
ly , a significant transportation of horizontal momentum appears between main flow
and shore wall, convex side and concave side, as well as main channel and flood-
plain, so it is inappropriate to omit the viscosity terms. On the other hand, numerical
experiments show that, if the wall is a non-slip boundary (where flow velocity is ze-
ro) , the calculated```