FLOW AROUND A CYLINDER - istiarto

– 3.29 –
3.4 Bed shear-stresses along the plane of symmetry
Determination of the bed shear-stress, � o , from the experimental data is a rather difficult
task. While a direct measurement technique would be most desirable, here we must
content ourselves with the estimate values based on available data from the velocity and
shear-stress distributions (see Fig. 3.1 and Fig. 3.9). Three alternative methods were used:
1. Based on the velocities, u and w, measured closest to the bed, z ≈ 4 [mm], a velocity
parallel to the bed —at a distance of � n — was calculated, V t ; the bed shear-stress
was calculated as:
���Vt
��
�o,1 � � �t ���n
�� � � � ��Vt
��
t ���n
��
(3.5)
where � t �1.3 �10 �5 [m 2 s 2 ] was taken as the measured eddy-viscosity in the
approach flow.
2. Based on the measured shear-stress distribution (see Fig. 3.9), a (rather subjective)
extrapolation towards the bed was used to obtain the bed shear-stress, or:
�o,2 � �� u ��w
��
� � bed cos� (3.6)
3. Based on a relation for the velocity distribution (see Graf and Altinakar, 1998, pp.
73-74) the bed shear-stress was evaluated by:
�o,3 � ��u �,3�
2
� � U g
C 2
�� ��
�� ��
�� ��
2
� ��0.07 U�
2 (3.7)
where U is the local depth-averaged flow velocity and C is Chezy coefficient taken as
C = 44 [m 1 2 s] (uniform sand bed with d 50 = 2.1 [mm]).
A numerical simulation of the flow around the cylinder has also been performed (see
Chapter 5), from which the bed shear-stresses can also be obtained; this is denoted as
� o,4 . Its value was obtained with a similar equation as Eq. 3.5, in which the k-�
turbulence model-equation and the logarithmic law-of-the-wall were used to get the eddy
viscosity, � t , and the velocity gradient, �V t �n, (see Chapter 4, Eq. 4.82). For clarity, the
expression used to compute � o,4 is rewritten below:
3 4 1 2
�o,4 � �c � k
�V t
� � (4.82)
�
ln E �n where c � = 0.09 is a k-� model constant, k turbulent kinetic-energy, � = 0.4 Karman
constant, E roughness coefficient of the bed, � n
� � u� � n � dimensionless normal
distance from the bed, and � the kinematic viscosity of water. The sign convention of
� o,4 is the same as that of the velocity.