Estimation of River Banks Influence on Tachoida Shape at the Meridian - Experimental Methods in Hydraulic Research - page 216

z
z 0

K ð z Þ¼ K 0 1 þ (2)

u x ð 0 Þ¼ 0 (3)

where H is the depth of the water, K ( z ) is the eddy viscosity coefficient of water,
K 0 is the eddy viscosity coefficient at the bottom, z, x is the basic set of coordinates
( x directed horizontally, z vertically upward), z 0 is the linear measure of bed
roughness,

t b is the shear
stress at the bottom level. These simplifications were discussed in the previous
papers (Meyer 2009a , b ). Under such assumptions, the solution has the following
form:

t zx ð z Þ is the component of turbulent shear stress, and

U H

u k ð 1 þ U Þ ln 1 þ z

H

u x ð z Þ¼ (4)

U

and

" #

Z

H

2 ln 1

u k ð þ U Þ ð þ U Þ UU ð þ Þ

1
H

1 1 1

2

u 0 ¼ u x ð z Þ d z ¼ (5)

U 2

0

where

p

gHI

H
z 0 ;

U ¼ u ¼ (6)

and similarly as in (Prandtl 1956 ):

K 0

z 0 ¼ (7)

k u

where I is the slope of the free water surface, g is the acceleration due to gravity, k is
the von Karmann constant, and u * is the shear velocity. It can be noticed that for
x

1 (such cases often occur in practice):

h i

u k

ln z

z
H

u x ð z Þ¼ ln U þ

H

(8)

and

u k ln U

e 3 = 2

u 0 ¼

(9)