FLOW AROUND A CYLINDER - istiarto

� fe fD,1 �
cs – 2.8 –
� � � �
�u ˆ 1 sin� D,1 � w ˆ 1�1�
cos�D,1 ��
(2.3)
By analogy, similar relations as Eqs. 2.1 to 2.3 exist for the Doppler frequency recorded
� �
by the transducer T1 T3 (see Fig. 2.2d). The analogy of Eq. 2.1 for the transducers T1 T3
reads:
� fe fD �
c
�� s
��
��
ˆ
��
�
V 1
� �� �
e 1 � V ˆ �� ��
3 � e 3��
��
and by using geometrical relationships (see Fig. 2.2d), it can be shown that:
�
V ˆ 1
� �� �
e 1 � V ˆ
1
� � ˆ
ˆ
��
V 3 � ��
e 3 � ˆ �
V 3 � ˆ
�
u 1 sin �D,1
�
w 1
� � �
� ˆ cos�D,1
w 1
Thus, one can rewrite Eq. 2.4 as:
� fe fD,1 �
cs (2.4)
(2.5)
� � � �
�u ˆ 1 sin� D,1 � w ˆ 1�1�
cos�D,1 ��
(2.6)
� �
If the systems of T1 T3 and T3T1 are symmetrical about the T3-axis and the measuring
volume of the two systems is the same, one may write:
� � � � � �
�D,1 � �D,1 � �D,1 , ˆ � ˆ � u ˆ 1 , and ˆ � ˆ � ˆ
u 1
u 1
w 1
Therefore, the instantaneous velocity component u ˆ 1 and w ˆ 1 can be extracted from Eqs.
�
system as follows:
�
2.3 and 2.6, obtained from the T1 T3T1 � � � � fD,1�
u ˆ 1 � cs fD,1 2 fe sin�D,1 w 1
� � � � fD,1�
and w ˆ 1 � cs fD,1 2 fe 1� cos� D,1
w 1
� � (2.7)
The above relations give the two-dimensional velocity component, ˆ V 1�u ˆ 1, ˆ �, of the
three-dimensional velocity of the target, ˆ
i
V ˆ
�
on the tristatic plane formed by the T1 T3T1 w 1
i
�u , v ˆ , w ˆ �; this is the projection of ˆ V �u ˆ , v ˆ , w ˆ �
�
transducers. The other two-dimensional
velocity component, ˆ V 2�u ˆ 2, w ˆ 2�,
can be worked out by analogy to Eq. 2.7 for the tristatic
�
. This yields the following expressions:
�
plane T2 T3T2 � � � � fD,2�
u ˆ 2 � cs fD,2 2 fe sin�D,2 � � � � fD,2�
and w ˆ 2 � cs fD,2 2 fe 1� cos� D,2
� � (2.8)