FLOW AROUND A CYLINDER - istiarto

3.2.4 Vertical velocity (downward flow)
– 3.15 –
The vertical velocity component manifests itself as a downward flow, notably in the
upstream planes close to the cylinder. The downward flow along the cylinder face is
driven by the pressure gradient in the vertical (z) and angular (�) directions. It is one of
the important signatures of the flow around the cylinder; together with the reversed flow,
they are usually referred to as the horseshoe vortex.
It has been previously shown (see Fig. 3.1) that the downward velocity has its maximum
value near the cylinder (at the cylinder face). The variation of this downward velocity
along the cylinder face (at r = 10 [cm]) observed in the upstream planes (� ≤ 90°), is
shown Fig. 3.6a. The data are normalized by the maximum value, wm , attained in the
corresponding plane. Note that the data from the additional measurements in the planes
� = 15°, 30°, 60°, and 75° (see Chapter 2) have been included in the figure. It is
interesting to see that the maximum values in all planes fall at 40% of the local depth
measured from the bed, �z � zbed� h � 0.4. Their profile, notably below the maximum
value, falls in a narrow band, showing a near similarity.
A different result was reported by Dey et al. (1995), who found that the maximum
downward flow was at the original bed level, z ≈ 0, irrespective of the plane �.�
In the case of flat channel bed (see Graf and Yulistiyanto, 1998), a somewhat different
result was reported. In the plane � = 0° the downward velocity is increasing with depth,
in the plane � = 45° it is almost uniformly distributed over the depth, and in the plane
� = 90° it is diminishing with depth. The downward flow peak, in the plane � = 0, is
found at z/h = 0.15, which is much closer to the bed than what is observed in the present
case with scoured channel bed. This is due partly to the lower depth (less friction loss by
the cylinder) and, to a lesser extent, to the smaller separation zone. Consequently, the
horseshoe vortex in the flat channel bed case is squeezed to the bottom corner of the
cylinder.
A similar observation was reported by Ahmed and Rajaratnam (1998). The maximum
downward flow is attained near or at the bed in the absence of scour hole, and below the
original bed level in the presence of a scour hole.
Fig. 3.6b depicts the distribution of the maximum downward velocity in the radial
direction for � ≤ 90°. Plotted in the figure are the maximum values of downward velocity
attained at any particular distance, denoted by w ˜ m , which are normalized by the largest
of these values, wm . What is interesting here is that the profiles fall in a narrow band,
notably for r Dp �1.8 where a similarity can be observed. Regression through all data
points yields an exponential relation:
w ˜ m wm � 2.9683 exp �1.6736 r Dp � � ��
(3.1)