NUMERICAL STUDY OF THERMOVISCOUS EFFECTS IN . . . PHYSICAL REVIEW E 90, 043016 (2014)
FIG. 4. (Color online) (a) Time-averaged second-order fluid velocity field v2 (vectors) and its magnitude color plot ranging from 0 mm/s
(black) to 0.13 mm/s (white) in the vertical channel cross section calculated at T0
= 25 ◦C and fres
= 1.966 992 3 MHz. (b) The horizontal
velocity component vy2 plotted along y = w/4 indicated by the magenta line in (a). The velocity field has been calculated for the five actuation
frequencies shown in the inset resonance curve, and normalized to the analytical Rayleigh streaming magnitude vR
str
= (3/8)(va
y1)2/cs, which is
calculated based on the corresponding first-order solutions. The symbols are plotted in selected points illustrating the five numerical solutions
(black lines) that coincide. (c) Normalized streaming magnitude vstr/vR
str (symbols) vs equilibrium temperature T0, calculated for different
channel heights h. The full curve shows the analytical single-wall result by Ref. 15. (d) Normalized streaming magnitude vstr/vR
str (symbols)
vs channel height h, calculated for different equilibrium temperatures T0. The full curves show the analytical single-wall result by Ref. 15,
while the dashed lines show the results of a one-dimensional analytical model with a Poiseuille backflow; see Eq. (21).
B. Time-averaged second-order velocity
The time-averaged second-order velocity field v2 is shown
in Fig. 4(a), calculated for the default 380-μm-by-160-μm
rectangular geometry, at T0
= 25◦C, and at the resonance
frequency fres
= 1.966 992 3 MHz. It exhibits the well-known
pattern of four flow rolls each λ/4 wide. To investigate the
magnitude of the streaming velocity, Fig. 4(b) shows the
velocity along a line perpendicular to the bottom wall at
y = w/4. The streaming velocity field has been calculated
for five different frequencies shown in the inset resonance
curve. The streaming velocities have been normalized to the
classical result by Lord Rayleigh for the magnitude of the
acoustic streaming vR
str
= (3/8)(va
y1)2/cs , where va
y1 is taken
from the corresponding first-order solutions. The five numerical
solutions (black lines) coincide completely, showing
that the rescaled second-order velocity field is the same for
off-resonance actuation frequencies. This is important for
our further analysis, as we do not need to determine the
exact resonance frequency as it changes due to variations in
temperature T0 and channel height h. The magnitude of the
streaming velocity vstr is determined by the maximum value
of vy2 along the line y = w/4 as indicated in Fig. 4(b).
In Fig. 4(c), we show the normalized magnitude of the
streaming velocity vstr/vR
str versus the equilibrium temperature
T0. The streaming velocity has been calculated for different
channel heights indicated by different colors and symbols. The
full line is the analytical single-wall solution by Rednikov and
Sadhal 15 for a standing acoustic wave parallel to a single
planar wall. For all channel heights, the streaming velocity
shows an almost linear dependence with positive slope on
the equilibrium temperature. The numerical results for the tall
channel h = 500 μmagree well with the analytical single-wall
prediction, while for more shallow channels the streaming
velocity is significantly lower. At 25 ◦C, the streaming velocity
is 19% larger than the classical Rayleigh result, while for 50 ◦C
this deviation has increased to 39%.
To elaborate on the dependence of the streaming velocity
on the height of the channel, we plot in Fig. 4(d) the normalized
streaming velocity versus the channel height for three
equilibrium temperatures. The numerical results are shown by
symbols, while the analytical single-wall predictions for each
temperature are shown by full lines. The numerical results for
the rectangular channel deviate from the analytical single-wall
prediction as the channel height is decreased. To qualitatively
explain this deviation, we make a simple one-dimensional
analytical model along the z dimension of the rectangular
channel in which we impose a boundary-driven flow. The first
part of the model is a plug flowwith an exponential dependence
close to the wall, vplug(z) = v0
{1 − exp−(z + h/2)/δs} for
−h/2 < z < 0. This approximates the z dependence of the
streaming velocity field inside the viscous boundary layer,
where v0 corresponds to the analytical single-wall solution
15. As the water is pushed toward the sidewall, a pressure
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