52 CHAPTER 6. STREAMING STUDIES
Figure 6.1: Analytical solution for the steady streaming velocity eld
v2
in parallel-plate channels.
(a)
v2
(arrows) and its magnitude color plot from 0 (black) to maximum vstr (white) in the vertical
y-z-cross section of a parallel-plate channel with =2 = w and aspect ratio = h=w = 1:2. The normalized
spatial coordinates are ~y = y=(w
2 ) and ~z = z=( h
2 ). (b) The same as (a), but for = 0:2. (c) The same
as (b) but for a standing full wave = w. (d) Line plot of the horizontal streaming velocity along the
horizontal centerline z = 0,
vy2(y; 0)
, divided by the maximum streaming velocity vstr, for =2 = w and
aspect ratios = 0:2, 0.5, 0.8, and 1.2. (e) Line plot of the maximum
vy2(y; 0)
max of the center-axis
streaming velocity, divided by the maximum streaming velocity vstr, as function of aspect ratio, and for
the resonances n=2 = w, with n = 1, 2, and 3, respectively. Figure adapted from ref. 11 Appendix B.
towards the center, resulting in a lower streaming magnitude in the center plane. Figure
6.1(e) shows the analytical prediction of the maximum streaming velocity in the horizontal
center plane z = 0 as a function of the aspect ratio = h
w of the channel cross section, with
curves for the rst-, second-, and third-order resonances, n = 1; 2; 3. The graphs show how
the streaming velocity in the center plane is signicantly smaller for larger aspect ratios
and higher-order resonances. The eect of the no-slip condition on the sidewalls is not
included in Fig. 6.1, and for a visual interpretation of this eect the reader is referred to
Ref. 11 Appendix B.