5
boundary conditions for the first-order velocity become
top: vy1 = 0, vz1 = 0, (17a)
bottom: @zvy1 = 0, vz1 = 0, (17b)
left-right: vy1 = vbc sin(!t), vz1 = 0. (17c)
The boundary conditions on the second-order velocity
are set by the zero-mass-flux condition n · v = 0 on
all boundaries, as well as zero parallel velocity component
on the top, right and left wall boundaries, and zero
orthogonal derivative of the parallel component of the
mass flux on the bottom symmetry boundary. The explicit
boundary conditions for the second-order velocity
become
top: vy2 = 0, vz2 = 0, (18a)
bottom: @z
��
0vy2 + 1vy1
= 0, vz2 = 0, (18b)
left-right: 0vy2 + 1vy1 = 0, vz2 = 0. (18c)
C. Spatial resolution
The physical fields are discretized using fourth-order
basis functions for v1 and v2 and third-order basis functions
for p1 and p2. The domain shown in Fig. 1(a) is
covered by basis functions localized in each element of
the spatial mesh shown in Fig. 1(c). Since the streaming
flow is solved in the time domain, the computational
time quickly becomes very long compared to the computational
time of solving the usual steady streaming flow.
Thus we have optimized the use of precious few mesh elements
to obtain the best accuracy of the solution. We use
an inhomogeneous mesh of rectangular elements ranging
in size from 0.16 μm at the boundaries to 24 μm in the
bulk of the domain. The convergence of the solution g
with respect to a reference solution gref was considered
through the relative convergence parameter C(g) defined
in Ref. 11 by
C(g) =
vuuuuut
Z ��
g − gref
2
dy dz
Z ��
gref
2
dy dz
. (19)
In Ref. 11, C(g) was required to be below 0.001 for
the solution to have converged.
The
solution for the
steady time-averaged velocity
vfd
, calculated with
2 (∞)
the mesh shown in Fig. 1(c) and 1(d), has C = 0.006 with
respect to the solution calculated with the fine triangular
reference mesh in Ref. 11, which is acceptable for the
present study.
D. Temporal resolution
The required temporal resolution for time-marching
schemes is normally determined by the Courant–
Friedrichs–Lewy (CFL) condition 29, also referred to
as just the Courant number
CFL =
cs t
r ≤ CFLmax, (20)
where t is the temporal discretization and r is the
spatial discretization. This means that the length over
which a disturbance travels within a time step t should
be some fraction of the mesh element size, ultimately
ensuring that disturbances do not travel through a mesh
element in one time step. A more accurate interpretation
of the CFL-condition is that it ensures that the error on
the approximation of the time-derivative is smaller than
the error on the approximation of the spatial-derivatives.
Consequently, the value of CFLmax depends on the specific
solver and on the order of the basis functions. For
fourth-order basis functions and the generalized alpha
solver, Ref. 29 reports a value of CFL4th
max = 0.05, which
is an empirical result for a specific model. Due to the inhomogeneity
of the mesh, two values for the upper limit
for the temporal resolution can be calculated based on
Eq. (20); t = 8 × 10−10 ns ≈ t0/600 for the bulk mesh
size of 24 μm and t = 5 × 10−12 ns ≈ t0/95000 for the
boundary mesh size of 160 nm.
To determine a reasonable trade-off between numerical
accuracy and computational time, we study the convergence
of the transient solution towards the steady
solution for different values of the temporal resolution
t0/t. The acoustic energy Eac(t) is shown in Fig. 2(a)
for different values of t and normalized by the steady
time-averaged energy
Efd
ac(∞)
of the frequency-domain
calculation, and it is thus expected to converge to the
unity for long times. In Fig. 2(b), Eac(1000t0)/
Efd
ac(∞)
is plotted versus the temporal resolution t0/t, which
shows how the accuracy of the time-domain solution increases
as the temporal resolution is increased. In all
subsequent simulations we have chosen a time step of
t = t0/256, the circled point in Fig. 2(b), for which the
time-domain energy converge to 99.4% of the energy of
the steady calculation. The chosen value for the time step
is larger than the upper estimate t0/600 of the necessary
t based on the CFL-condition. This might be because
our spatial domain is smaller than the wavelength, and
consequently a finer spatial resolution is needed, compared
to what is usually expected to spatially resolve a
wave.
We have noted that the fastest convergence is obtained
when actuating the system at its (numerically determined)
resonance frequency fres. When shifting the actuation
frequency half the resonance width 1
2f away
from fres, the energy Eac(t) for t = t0/
256 converged
to only 95% of the steady value
(calculated in
Efd
ac(∞)
the frequency domain), thus necessitating smaller time
steps to obtain reasonable convergence.
The computations where performed on a desktop PC
with Intel Xeon CPU X5690 3.47 GHz 2 processors, 64-
bit Windows 7, and 128 GB RAM. The computations
took approximately one hour for each time interval of
width 100t0 with t = t0/256, and the computational