6
FIG. 2. (Color online) Numerical convergence and temporal
resolution. (a) Graphs of the build-up of acoustic energy
Eac(t) in the time-domain simulations calculated with different
fixed time steps t. The energy of the time-domain simulations
is normalized with respect to the energy
Efd
ac(∞)
of the steady solution in the frequency domain, and should
thus converge towards unity. In all simulations the actuation
frequency equals the resonance frequency discussed in
Section IVA. (b) Acoustic energy Eac(1000 t0) at t = 1000 t0,
normalized by
Efd
ac(∞)
, and plotted versus the temporal resolution
t0/t of the oscillation. The inset is a semilog plot of
the relative deviation of Eac(1000 t0) from
Efd
ac(∞)
. The circled
point in each graph indicates the time step t = t0/256
used in all subsequent simulations.
time was not limited by RAM, as only less than 2 GB
RAM was allocated by Comsol for the calculations.
IV. ONSET OF ACOUSTIC STREAMING
In this section the fluid is initially quiescent. Then, at
time t = 0, the oscillatory velocity actuation is turned on,
such that within the first oscillation period its amplitude
increases smoothly from zero to its maximum value vbc,
which it maintains for the rest of the simulation. We
study the resulting build-up of the acoustic resonance
and the acoustic streaming flow.
A. Resonance and build-up of acoustic energy
To determine the resonance frequency, the steady
acoustic energy
Efd
ac(∞)
Eq. (13b) was calculated for
a range of frequencies based on the frequency-domain
equations (9)-(10). In Fig. 3 the numerical results (cir-
FIG. 3. (Color online) Resonance curve and build-up of
acoustic
energy. (a) The numerical acoustic energy density
Eac(∞)
fd
/V (circles) for different frequencies of the boundary
actuation and a Gaussian fit (full line) to the numerical
data. fres is the fitted resonance frequency at the center of the
peak, while fideal is the frequency corresponding to matching
a half-wavelength with the channel width. The inset shows
the numerical build-up of the acoustic energy (full line) for
actuation at the resonance frequency, ! = 2fres, along with
the analytical prediction Eq. (23) (dashed line) for a single
harmonic oscillator with the same resonance frequency and
quality factor Q = fres/f.
cles) are shown together with a Gaussian fit (full line),
while the inset exhibits the fitted resonance frequency
fres, the full width f at half maximum, and the quality
factor Q = fres/f.
The build-up of the acoustic energy in the cavity is
well captured by a simple analytical model of a single
sinusoidally-driven damped harmonic oscillator with
time-dependent position x(t),
d2x
dt2 + 2��!0
dx
dt
+ !2
0x =
1
m
F0 sin(!t). (21)
Here, �� is the non-dimensional loss factor, !0 is the resonance
frequency of the oscillator, 1
mF0 is the amplitude
of the driving force divided by the oscillator mass, and
! is the frequency of the forcing. The loss factor is related
to the quality factor by �� = 1/(2Q), and in the
underdamped case �� < 1, the solution becomes
sin(!t + )
x(t) = A
−
! e−��!0t
!0√1 − ��2
sin
p
1 − ��2 !0t +
. (22)
The amplitude A and the phase shift between the forcing
and the response are known functions of F0
m , !0, !,
and ��, which are not relevant for the present study. From
Eq. (22) we obtain the velocity dx/dt, leading to the total
energy E of the oscillator,
2m!2
E = 1
0x2 + 1
2m
dx
dt
2
. (23)