FLOW AROUND A CYLINDER - istiarto

��
V ˆ
h � �� ˆ
��
��
� � 2 � ˆ ��
u 1
��
��
u 2
2
1 2
� ˆ u 1 cos� �� ��
T
�� ��
sin� T ��
��
��
– 2.10 –
The velocity direction, �V, is given by the relations below:
�V � arcsin ˆ ��u
2 � u ˆ 2 cos� ��
T
��
V ˆ ��, if u ˆ 1 � 0
�� h sin� T ��
�V � 180
��
� � arcsin ˆ ��u
2 � u ˆ 2 cos� �� T
��
��
V ˆ
h sin� T ��
�� , if u ˆ 2 � 0
(2.10)
(2.11)
The three relations, Eqs. 2.9, 2.10, and 2.11, are the ones required to describe the threedimensional
instantaneous velocity of the moving target. It is, however, more convenient
to describe the velocity by its components along the Cartesian or cylindrical coordinate
system. The following section discusses the decomposition of the velocity into those
components.
Cartesian and cylindrical velocity components
The Cartesian coordinate axes are defined as x, y, and z for the horizontal, transversal,
and vertical directions, respectively. The cylindrical coordinate axes are defined as r,
���� = � – 180°), and z for the radial, angular, and vertical directions, respectively (see
Fig. 2.4). The origin of the two coordinate systems is defined at the center of the cylinder,
at the original (uneroded) bed level.
The decomposition of the velocity into its components along the Cartesian and cylindrical
coordinate systems depends on the orientation of the ADVP with respect to the
coordinate system. In the measurements, the instrument is positioned along radial planes
around the cylinder. Fig. 2.4 shows a typical placement of the ADVP and the
decomposition of the velocity into its components along the Cartesian and cylindrical
coordinate systems.
By using geometrical relationships, the Cartesian velocity components can be obtained
from the following expressions:
� �
� �
u ˆ � ˆ V h cos � � �R � � V
v ˆ � ˆ V h sin � � �R � � V
w ˆ � w ˆ 1 � w ˆ 2
where the definitions of the angles are given in Fig. 2.4.
(2.12)