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Cap1_Weiyan_Shallow_Water_Hydrodynamic

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rate of k ) , or two other similar equa- 
tions (e. g. , in terms of k and mixed length l ) . One of the most noticeable multi-e- 
quation models is a 3-equation model taking stress z, , k and E as unknowns, and an- 
other one is expressed in terms of turbulent shear stress and transport flux. 
Boussinesq' s approximation is the most classical technique for the establishment 
of an empirical relationship between the Reynolds stress and time-averaged velocity 
(a 0-equation model). By introducing a turbulent viscosity coefficient the Reynolds 
stress is written as 
(1. 3. 7) 
fiL and v1 are called dynamic and kinetic turbulent viscosity (or eddy viscosity, 
turbulent momentum exchange, turbulent diffusivity ) coefficients, cooresponding to 
/L and v respectively. As stated above, they do not originate from the viscosity prop- 
erty of the fluid but from the vortices produced by turbulence, and they depend on 
the characteristics of the flow field such as the velocity gradient. Turbulent and 
molecular viscosity are also quite different from each other in the scale and strength 
of fluid motion, as p<<pl, generally with a ratio about lo-*. Then the for- 
mula of turbulent stress Eq. (1. 3. 6 ) , (1. 3. 6a) become 
20 
(1. 3. 8 ) 
(1. 3. 8a) 
In a 2-D shallow-water flow computation, the simplest 0-equation model has 
been used in most cases, and even the Reynolds stress term is replaced by some em- 
pirical hydraulic resistance formula (cf. Section 1. 4 ) . When the turbulent structure 
varies gradually along a flow, a 2-D flow computation based on the Reynolds equa- 
tions and some turbulence model has been more or less successful. For example, at- 
tempts have been made to use a depth-averaged k-& model for calculating the distribu- 
tion of depth-averaged turbulent viscosity. It seems that at present the k-e model is 
the most promising one for calculating flow fields with small-scale circulation, espe- 
cially when there exists a separation between the fluid and the solid wall, or there are 
wakes behind an object. 
N . TURBULENT EDDY VISCOSI!lT 
In general, turbulent viscosity, v, , is an order-2 asymmetric tensor composed of 
nine components, whose values are subject to large variations with the flow behavior 
(near-wall turbulent flow or free turbulent flow-including jet, wake, etc. ) and ve- 
locity distribution, depending on the mechanism of turbulent energy production. In a 
flow field, the larger the shear deformation due to velocity gradient, the greater the 
value of v,. It is also affected by wind stress exerted on the water surface. Thus, vis- 
cosity should be a function of space and time, and it also depends on the space-time 
scale of flow (it increases with the scale of flow). In numerical solutions, however, 
because vortices are reproduced by using.a mesh, which imposes a limitation on their 
size, vt is often assumed to be a constant for the sake of simplification. In this case, 
the solution may be close to that for a viscous laminar flow on account of the similari- 
ty between the two expressions for z, so sometimes it is called a quasi-laminar flow. 
The value of v, may be chosen empirically according to the state of flow or through 
laboratory experiment, but an estimate based on observations can only be applied to 
dynamically similar flows. Of course, this choice would be influenced by personal 
judgement, leading to a range as large as several orders of magnitude and yielding 
quite different solutions unavoidably. Substituting Eq. ( 1 . 3. 8) with k= 0 into Eq. 
(1. 3. 5 ) , and assuming that all the components of turbulent viscosity are identical, 
we have 
(1. 3. 9 ) 
When turbulent viscosity varies in a flow field, the order-2 viscosity terms in 
Eq. ( 1. 3. 9 ) , should be written in the form of - ( v 2) . For brevity of notation, 
we shall omit below all the time-averaging marks over the symbols of various flow 
variables. 
a a ; 
axJ ax, 
21 
Seeing that in shallow-water flows, vertical velocity and acceleration are much 
smaller in magnitude than horizontal ones, we distinguish horizontal viscosity V~ from 
vertical viscosity v,, only, so that 
1 a; 
=- - - + ipBi + v,&, v, = va( i = 1 , 2 ) or v,"(Z = 3) (1. 3. 10) 
at P ai 
In applications their values adopted are in a range wide enough, reaching 1 - 1 O3 and 
1 OP4 - 1 0-'m2/s, respectively. The commonly-used range of values of vth is 5- 100 
m2/s, while that of vtV in a middle layer of seawater is 10-2-10-4, and at the sea 
surface 5 X 1 O-'- 5 X lo-', corresponding to wind speed of 5-22 m/s. In a flow 
of geophysical scale, the orders of magnitude for vth and vl. are 1 O2 and 1 0-2, respec- 
tively. 
The cause of such a great difference can be interpreted as follows. Horizontal 
momentum exchange is affected chiefly by the vortices generated by the geometry of 
the boundary of a water body, while vertical vortices leading to vertical exchange 
have three sources, i. e. , underwater topography, wind over the water surface, and 
turbulence due to a vertical gradient of horizontal velocity. As shallow-water is char- 
acterized by its relatively large horizontal length scale, the energy and sizes of hori- 
zontal vortices are also much larger than vertical ones, but with much lower wave 
frequencies. The former influences mainly the distribution of the velocity, and has 
only a slight effect on water level, while frequency is the chief mechanism of dissipa- 
tion. 
Besides the simplest assumption of constancy of v w , there are several empirical 
formulas in use. 
( 1 ) Neglect the variation of v , ~ with depth, and assume that v, is a bilinear func- 
tion of depth and depth-averaged velocity, e. g. 
(1. 3.11) 
where C=an empirical coefficient. 
( 2 ) Assume that va is a linear function of the horizontal gradient of horizontal 
velocity. 
(3) Based on the Prandtl-Karman mixing-length theory, vw is estimated by 
vlh = Ch +/m 
(1. 3. 12) 
where the Karman constant K-0. 4 , and y is a height above bottom. 
Empirical formulas for vrV are exemplified in the following. 
( 1) In the studies of European offshore waters in recent years, ignore the varia- 
tion of vth with depth, and assume that it is approximately proportional to the square 
of the depth-averaged velocity 
k 
V," = - (UZ + u2) 
U 
(1. 3. 13) 
where the dimensionless constant k= 2 X 1 O-5. Another constant u, the frequency of 
longwave (similar to the Coriolis coefficient of geostrophic motion), may be taken as 
u = lo-' 1/s. If it is required to consider the variation of vtu with depth, the above 
value is applicable to the middle layer of the seawater. At the sea surface, we need to 
22 
consider the effects of wind speed and water waves, whose height and period are re- 
lated to wind speed w,(m/s) and wind run L (km). So vt,, can be estimated by an 
empirical formula, e. g. , vt,, = (0. 1695 X 1 0-4 )Lo . wi, 6. At the sea bottom, v,,, de- 
pends on velocity and characteristic roughness length. From the above a vertical pro- 
file of vt,, can be established. 
(2 ) Let v,,, be a function of the square of the vertical gradient of local horizontal 
velocity, e. g. 
(1 . 3. 1 4 ) 
where I= mixing length showing the turbulent scale, determined empirically. 
( 3 ) For a large-scale shallow-water flow, vt,, can simply be taken as a linear or 
parabolic function of depth. In a computation for the Aegean Sea, the following for- 
mula was used 
( 1 . 3. 15) 
where u , =frictional velocity a t water surface, u* = m p ; t,” =shear stress a t 
water surface; 3, = calibration coefficient of the same order as 1 , h = 0 ( 1 ) ; z = 
height above mean sea level; d=height between sea bottom and mean sea level. If 
the fluid density varies in the vertical direction, the right-hand side of the above e- 
quation should be multiplied by a certain empirical function of the Richardson number 
(cf. Section 2. 2). 
A deeper study should utilize some turbulence model, to get the spatial distribu- 
tion