PETER BARKHOLT MULLER AND HENRIK BRUUS PHYSICAL REVIEW E 90, 043016 (2014)
FIG. 2. (Color online) (a) The used triangular mesh with a gradually increasing element size from 0.5 μm at the boundaries to 20 μm
in the bulk. This mesh, chosen as the default mesh, contains 30 246 elements. (b) Rectangular mesh with thin elongated 0.1-μm-by-10-μm
elements at the boundaries and gradually changing to nearly square 10-μm-by-10-μm elements in the bulk. This mesh contains 3308 elements.
(c) The convergence parameter C of Eq. (20) for all first- and second-order fields vs the numerical resolution defined by δs/dbd, where dbd is
the mesh-element size at the boundary. The fields are solved on triangular meshes with different boundary element sizes but all with fixed bulk
element size dbk = 20 μm and growth rate α = 1.3, while the reference solution is calculated for dbd = 0.15 μm, dbk = 2 μm, and α = 1.3.
The vertical dotted line indicates the solution for dbd = 0.5 μm, which is chosen as the default value for the following simulations.
Due to the very different length scales of the channel dimensions
and the boundary layer thickness an inhomogeneous
mesh is necessary, and thus there is a number of ways to refine
the mesh.We used three parameters: maximum mesh-element
size at the boundaries dbd, maximum mesh-element size
in the bulk dbk, and the maximum mesh-element growth
rate α (maximum relative size of neighboring elements).
The convergence of the fields was considered through the
relative convergence parameter C(g) defined in Ref. 20
by
C(g) =
(g − gref )2 dy dz
(gref )2 dy dz
, (20)
where C(g) is the relative convergence of a solution g with
respect to a reference solution gref . Convergence graphs for
all fields as a function of dbd are shown in Fig. 2(c). The mesh
parameters for the reference solution are dbd = 0.15 μm,
dbk = 2 μm, and α = 1.3, whereas other solutions for given
dbd use dbk = 20 μm and α = 1.3. The basis functions for
the first- and second-order velocity and temperature fields are
all fourth order, while for the first- and second-order pressure
they are third order. All fields exhibit good convergence,
and we choose C = 10−3 as our convergence criterion in the
following. The corresponding default triangular mesh has
dbd = 0.5 μm; see Fig. 2(a). In Fig. 2(b), a mesh is shown with
rectangular mesh elements that are nearly square in the bulk of
the channel while very elongated near the walls. This mesh has
been used for testing purposes as it contains approximately ten
times fewer mesh elements compared to the default triangular
mesh, and the resulting fields all show convergence parameters
below C = 10−3 with respect to the triangular reference mesh.
All results have been calculated using the triangular mesh,
but the square mesh provides a huge advantage regarding
calculation speed and memory requirement.
IV. RESULTS
A. Resonance analysis
To determine the acoustic resonance frequency fres corresponding
to the horizontal half-wavelength resonance, we
sweep the actuation frequency around the ideal frequency f0
=
cs/2w, corresponding to the half-wavelength match λ/2 = w,
and we calculate the acoustic energy density Eq. (16), shown
in Fig. 3. The resonance frequency fres is shifted slightly with
respect to the ideal frequency f0 due to the viscous loss in
the boundary layers. This loss also determines the width of
the resonance curve and thus the Q value of the acoustic
cavity.
FIG. 3. (Color online) Graph of the acoustic energy density Eac
Eq. (16) as a function of the frequency f of the oscillating boundary
condition. fres is the resonance frequency at the center of the peak,
while f0 is the ideal frequency corresponding to matching a halfwavelength
with the channel width. The inset shows the magnitude
of the resonant oscillating first-order velocity field va
y1 relative to
the amplitude of the oscillating velocity boundary condition vbc as a
function of the actuation frequency f .
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