Download Wormald Coupland On measuring 3D flow within inkjet droplet

On measuring 3D flow within inkjet droplet streams
S.A. Wormald* and J.M. Coupland
Wolfson School of Mechanical and Manufacturing Engineering, Loughborough
University, Loughborough, United Kingdom
Abstract
With a view to measuring the internal and external flow fields in an inkjet droplet stream,
this paper discusses the problem of imaging through droplet surfaces and more generally
interfaces in two-phase flows. First the propagation of optical fields through interfaces
between media of different refractive index is discussed with reference to scalar
diffraction theory. Some approximations suitable for droplet imaging are then discussed
and the use of a-priori information is then explained. The imaging technique is applied to
holograms recorded using a digital holographic microscope and is illustrated in the
synthesis of a 3D image that is reconstructed through a cylindrical telecommunication
fibre. Subsequently we demonstrate for the first time, high resolution imaging throughout
an inkjet droplet. The practical implementation and improvement of these imaging
methods for the measurement of two-phase flows is then discussed.
Keywords: digital holographic microscope, two-phase flow analysis, droplet surface
reconstruction.
1. Introduction
Central to the design of products such as inkjet printers [1], fuel injectors [2] and
asthma inhalers [3], two-phase flows represent one of the most challenging areas of
modern fluid dynamics. Flows depend strongly on the interactions between phases
and are greatly influenced by parameters such as viscosity and surface tension [1].
Nevertheless, certain two-phase flows such as inkjet droplet streams can be made to
be some of the most steady and predictable of any flow phenomena [4].
Experimental measurements of two-phase flows have been limited. The shape
of fuel injector plumes has been measured using light sheet methods [5] and the
droplet size and velocity distributions has been estimated using phase Doppler
*
Corresponding author. Email: [email protected]
anemometry [6]. Two-dimensional (2D) measurements of the internal velocity field
have been attempted in relatively large (4mm) droplets using particle image
velocimetry (PIV) [7]. At smaller scales the 3D flow within a droplet in a channel was
recently measured using confocal microscopy [8]. In this case the droplet was in
contact with the channel walls and imaging through a planar interface was possible.
In essence, the large refractive index change between different phases causes
refraction and reflection at fluid interfaces and this can be attributed to the failure of
commercial optical diagnostics. For the most part optical metrology has been
developed for use in homogenous fluids or for operation through planar windows but
in some cases can be modified for use through windows of known form. For example,
corrective optics have been used in PIV measurements made through the glass
cylinder of an optical engine [9] and phase conjugation and ray tracing techniques
have been used in similar measurements using holographic PIV (HPIV) [10]. This
work shows that if the optical distortion or aberration is known then it can be
compensated; however, a clear requirement for a non-intrusive measurement
technique which compensates for the irregular boundaries in two phase flows remains.
Optical tomography is one candidate technology. In essence, optical
tomography synthesises an image from measurements of the field scattered under a
set of different illumination conditions [11]. At present, however, two problems
prohibit its use in two phase flow mapping. First, a significant time is required to
record the scattered data sets and this precludes tomographic studies of transient
phenomena. Second, the strong scattering caused by large changes in the refractive
index require sophisticated non-linear inversion methods to compute an image [12].
Although a-priori knowledge can be used to expedite the inversion process [13], it is
unlikely that the computational resources to compute 3D images of actual size
droplets will be available in the near future.
Holography, provides a single record of the scattered field and a holographic
reconstruction can be thought of as a simple tomographic image [14]. Since a
hologram is typically a recording of the optical field at a surface, the reconstruction
can only be considered as a 3D image if it is interpreted with a-priori knowledge. For
example, the identification and location of particles in holographic PIV relies on the
fact that the particles are known to be sparsely distributed, point-like objects. For the
case of a seeded two-phase flow we also have considerable a-priori knowledge. In
addition to knowledge of the seeding particle properties, we often know the refractive
indices of the various phases. For the case of droplet imaging, the droplets themselves
are frequently well dispersed and we are usually dealing with phases bounded by a
finite number of closed surfaces. In addition to knowledge of the incident wave-front
this information is sufficient to make detailed 3D measurements of sparse flows.
In the following we consider the problem of imaging within inkjet droplets
with radii of curvature of around 50-200 m. The scattered field is recorded using a
digital coherent microscope. First we consider the general problem of reconstructing
fields in two phase media and discuss some of the simplifications that can be made to
reduce the problem. We demonstrate the technique first by imaging through a
telecommunications fibre which has particles and debris on its outer surface. It is
shown that the strong lensing effect of the fibre can be efficiently compensated by
application of a-priori knowledge. The more general problem of imaging through
seeded inkjet droplets is then considered.
2. Theory
Let us consider the propagation of light through a water droplet in ambient air as
shown in Figure 1. According to scalar diffraction theory, the fields, E w , and Ea ,
within the droplet and that in surrounding air respectively, obey the Helmholtz
equations,

2
 k w2 E w (r )  0

(1)

2
 k a2 E a (r )  0

(2)
where k w  2 w and k a  2 a are wave-numbers in water and air at
wavelengths,  w and  a . In addition to these relations the external and internal fields
are coupled via the boundary conditions. In this case both the complex amplitude of
the field and its normal derivative must be continuous at the boundary, rs , such that,
E a (rs )  E w (rs )  0 and

E a (rs )  E w (rs )   0
n
In order to calculate the fields for a particular illumination condition, the field
outside the droplet is written as the sum of the light scattered by the object, E s (r) ,
(3)
and the reference field, Er (r) , that is defined by the illumination in the absence of
the object, such that,
E a (r)  E r (r)  E s (r)
(4)
In principle, for a given illumination condition, equations 1-4 can be solved
using the boundary element method to find E s (rs ) and E w (rs ) [15]. It is then
possible to calculate the fields within the regions using the Fresnel-Kirchhoff
diffraction formula [16].
A more direct solution, that we have found to be quite useful for calculating
the field scattered by smooth objects, is based on the assumption that the internal and
scattered fields can be written as surface integrals such that,
E w (r)   S  (r  )

exp jk w r  r 
r  r

E s (r)   S  (r  )


exp jka r  r 
r  r
 ds
(5)
 ds
(6)
where   and   are surfaces just outside of and just inside the boundary as
shown in Figure 2, and S  and S  are source strengths per unit area. In this way the
internal field, E w (r ) , must be a solution to equation 1 since it is produced by sources
that are external to the region of interest and similarly, E s (r) , must be a solution to
equation 2.
In a numerical environment it is relatively simple to solve these equations by
replacing the integral with a discrete set of sources uniformly spaced on internal and
external surfaces. It is found that both the spacing of the internal and external surfaces
and the sources should be approximately equal and substantially less than one
wavelength. In general, all of the sources contribute to the boundary fields and
consequently the solution includes the effects of multiple scattering within the droplet.
For the purposes of the work described here, however, we simplify the droplet
scattering problem further by assuming, first, that the internal field can be adequately
modelled by integrating equation 5 over a finite portion of the droplet surface and
second, that the field at the droplet surface is dominated by local sources. In this case,
  , is the front region of the droplet surface (closest to the recording device) and we
assume,
Sw (r ) 
1
E m (r )
j w
7.
With these assumptions we have neglected the effects of multiple scattering
both within the droplet and any external refraction and/or reflection that causes light
to re-enter the droplet. We will return to the validity of our assumptions later; for the
moment, however, we note that in practice the droplet surface is usually fairly smooth
and although present, these effects are of secondary importance.
The reconstruction process is illustrated for the case of imaging through a
multimode telecommunications fibre that was sparsely coated with 3.7 m PMMA
micro-spheres by dipping in a diluted aqueous suspension of these particles. The fibre
was placed under the coherent microscope shown in Figure 3 with its axis
approximately perpendicular to the z-direction defined by the optical axis of the
instrument. This microscope illuminates the sample from behind with a plane wave
from a diode pumped Nd:YLF laser of wavelength  a = 523 nm. The microscope
utilises an objective lens of NA = 0.55 to provide diffraction limited resolution with a
cut-off frequency (amplitude) of 1000 cycles/mm when imaging in air.
Reconstruction of the 3D scattered field using this microscope has been discussed
elsewhere and the reader is referred to reference [17] for details. It is noted, however,
that with the comparatively large NA and high resolution imaging offered by this
microscope, the Fresnel approximation (that is equivalent to the paraxial
approximation [16]) cannot be justified.
3.1 Images of a telecommunications fibre
First a holographic reconstruction is performed with the assumption that the particles
are suspended in air; Figure 4 shows the absolute value of the reconstructed
holographic image in 3 distinct x-y planes. Figure 4a) shows the image in a plane
which is tangential to the front surface (closest to the objective) of the fibre. Here only
a small portion of the surface is illuminated since the plane wave illumination is
focused by the rear surface of the fibre. In the illuminated region, images of particles
on the front surface of the fibre can be seen. Figure 4b) shows a focal plane 70 m
further from the objective and approximately half way through the fibre. In this figure
the edge of the fibre is brought into focus and particles on the edge can clearly be
seen. The diffracted images of the particles that are on the front surface can also be
identified. Figure 4c) shows a focal plane 70 m still further from the objective that
approximately corresponds to a plane that is tangential to the rear surface of the fibre.
No images can be identified in this plane due to the image aberrations introduced by
the front of the fibre.
Figure 5 shows an x-z section through the reconstruction. It can be seen that as
the illumination passes through it is brought to focus slightly in front of the fibre due
to the curvature of both the front and rear surfaces. It is noted that figure 5 does not
resemble the fibre section and shows that although holography is closely related to
tomography a single hologram does not produce an accurate 3D image [14]. However,
there is significant a-priori knowledge with which to improve the image. In this case,
it is known that the fibre is a cylinder we may assume that it has a refractive index of
approximately n=1.5. From the best focus centre-plane, the edge of the fibre was
identified and the fibre diameter was measured (140 m). This is sufficient
information to estimate the fibre surface. Accordingly regularly spaced points were
defined on a cylindrical surface, +, in the reconstruction space and the complex
valued field interpolated at these points. The field approximately 70 m into the fibre
(i.e. the central plane) was then calculated using equations 7 and 5. The complex field
propagating throughout the fibre volume was then calculated in the usual way.
Figure 6 shows the absolute value of the reconstructed holographic image in 3
distinct x-y planes corresponding approximately to those in Figure 4. Figure 6a)
shows the image in a plane which is tangential to the front surface (closest to the
objective) of the fibre. Here, the same particles that were identified in Figure 4a) can
be seen. Figure 6b) shows the central plane approximately half way through the fibre.
Here there are no particles in focus but the diffracted images of the particles that are
on the front and rear surfaces surface can be identified. Figure 6c) shows the x-y plane
that is tangential to the rear surface of the fibre and the particles on the rear surface
are clearly brought into focus. The characteristic spot at the centre of these particles
(an image of the source) and comparison with Figure 4a) shows that the
reconstruction is good quality and is approximately diffraction limited.
In the following section the same basic approach is used to produce high
fidelity images in a two-phase flow.
3.2. Images of an ink jet
To investigate the suitability of this method to the study of droplet break-up the same
basic technique was applied to the imaging of a pulsed ink jet. The ink jet was created
using filtered de-ionised water that was driven through a 100 m sapphire nozzle at a
pressure of 1.3 Bar. The jet was seeded with 3.7 m PMMA micro-spheres at a
concentration of approximately 1% by volume. To form a sequence of droplets the jet
was controlled by a The Lee Company INKX0514300AA solenoid valve which
supplied a square pulse stream at 5Hz with 0.1% duty cycle. With strobed
illumination a regular stream of droplets was observed with an exit velocity of
approximately 5 m/s.
The ink jet was placed under the coherent microscope with the nozzle axis
approximately perpendicular to the optical axis (z-axis) of the instrument. The
instrument was focused approximately at the position of the nozzle axis. Figure 7.
shows the reconstructed hologram of a droplet close to the nozzle exit approximately
1ms after the solenoid valve opened. At this stage the droplet is in the form of an
extruded filament of fluid. Out-of-focus and aberrated images of particles within the
fluid stream can be seen. In order to estimate the position of the droplet surface, the
plane where edge of the droplet was in best focus was found. In this case, the droplet
has a similar geometry to the telecommunications fibre described previously and was
assumed to be cylindrical. The diameter of the droplet and the direction of
propagation was estimated from the best focus reconstruction. It was found that the
diameter was approximately 92 m. The field on surface,   , corresponding to the
front surface of the droplet was then interpolated according to equation 7. and the
internal field at the jet axis, calculated using equation 5. The complex field throughout
the droplet was then calculated.
Figure 8 shows the field in three planes separated by 45 m. Figure 8a) is
tangential to the front surface and particles located in this plane are in good focus and
are relatively free from aberration. Moving through the planes in Figure 8b), particles
move into and out of focus as expected. In Figure 8c) closest to the rear of the fibre
the best focus images show signs of astigmatism in regions A and B. Here the particle
images are elliptical in shape and exchange major and minor axes as they pass
through focus. It is clear that the effect of the front surface, that is similar to a
cylindrical lens, has not been completely compensated and further measures, that we
will discuss later, are required.
Figure 9 shows a holographic reconstruction of a droplet further from the
nozzle exit where a droplet is beginning to form. The reconstruction assumes
homogenous air and is focussed on the axial plane. In this case, the droplet is far from
cylindrical and for this reason represents a more challenging reconstruction. It is
reasonable, however, to assume that the droplet has axial symmetry, at least in the
first instance. The surface now is formed by estimating the edge of the droplet and
calculating the surface of revolution about an estimate of its central axis. The
intersection of a portion of this surface subtending an angle of ±25° with the focused
image has been superposed in Figure 10 for illustrative purposes. As before the field
on surface,   , corresponding to the front surface of the droplet was then interpolated
according to equation 7. and the internal field at the jet axis, calculated using equation
5.
Figure 11 shows the reconstruction of the internal field of the droplet. Figure
11(a) is once again tangential to the front surface, like the reconstruction of the
filament in Figure 8(a), the particles focus to a well defined spot in this plane. Figures
11(b) and (c) show planes close to the central axis of the droplet. Most of the particles
in focus in these planes are suffering from astigmatism, which we attribute to an
incorrect model of the droplet surface. Figure 11(d) shows a plane close to where we
estimate the back plane of the droplet. Here particles can still be identified, but they
are more severely aberrated.
4. Discussion
Our preliminary results are based on certain assumptions and a-priori information and
these deserve further discussion. For the general problem of holographic imaging
through seeded droplets we typically know the following;



Refractive index of the droplet
Size and optical properties of the seeding
Illumination conditions
In the procedure outlined in this paper we have exploited some, but not all of
this knowledge, and have used this with additional assumptions in order to construct a
3D image. The most critical step in the process is estimating the droplet surface.
Inherent in this process was the assumption that the illumination was unperturbed by
other droplets and the outline of the droplet provided a good estimate of the plan form
of the droplet. With the refractive index and the assumption that the droplet was
axisymmetric, this is sufficient information to estimate the droplet surface and image
within it.
It is evident from Figure 8 that the image reconstruction is not always perfect
and, in this case signs of astigmatism are clear. In this case it is likely that the axis of
the jet is not perpendicular to the optical axis (z-axis) of the microscope but rather
slopes such that it is closer to the objective at the top of the image. This information is
not apparent from the plan form of the jet shown (assumed to be the axial plane)
shown in Figure 7. It is noted, however, that in this preliminary work, knowledge of
the illumination wave-front has not been used. For the case of an axisymmetric
droplet the paraxial focus of the illumination after propagation through the droplet can
be used to give an unambiguous estimate of the droplet surface. If the drop is not
quite axisymmetric, as is the case of Figure 10, then the relative power of the front
and back surfaces will affect this estimate. It is therefore necessary to make better use
of the other a-priori information.
In our current work we have made use of the fact that we know what the
particle images should look like to check the quality of our reconstruction. In
principle it is possible to modify the surface shape to optimise the image quality
throughout the field. In practice this could be done by defining a low order
polynomial perturbation to the surface shape and calculating the polynomial
coefficients required to optimise the particle images identified at a number of control
points. In essence this would modify the reconstructed wave-fronts in a manner
similar to the wave-shaping devices used in adaptive optics [18], albeit in a virtual
environment. If the front surface of the fibre was found in this way then a similar
polynomial perturbation to the back surface could in principle be used to minimise the
error between a measurement and a calculated estimate of wave-front corresponding
the transmitted illumination. In this way, we believe it is possible to deduce the shape
of the front and rear surfaces of an irregular droplet.
We also note that our method calculates the internal field from a measurement
of the field transmitted by the front surface without taking full account of the
influence of the remaining parts of the droplet surface. Mathematically, we have
assumed that the surface integral in equation 5 is dominated by the front portion of the
surface. In effect we have neglected the contributions due to multiple reflections
within the droplet and typically these will increase the level of background noise. In
certain cases front and back reflections may give rise to more obvious effects as the
dominant illumination beam propagates through the interfaces. We believe that the
central “line feature” in Figure 6(c) is an example of this effect. For the case of a
water interface at normal incidence the amplitude reflection coefficient is
approximately 14% and since two reflections (front and back) are required in this
instance, the amplitude is expected to be reduced to around 2% of its original value.
Nevertheless the focusing effect of the surfaces will concentrate the light and make
the effect more noticeable. In principle a more complex reconstruction using a greater
portion of the droplet surface and optimisation based on the appearance of this feature
could be used to improve the estimate of droplet shape.
Finally, we should comment on the computational time required to produce
droplet images. All of the algorithms used in this work were written in MATLAB
programming language and implemented on a PC with 32GB memory and dual quad
core 2.6GHz Xeon processors. The holograms were recorded at a resolution of
4008x2672 pixels and demodulated to give the complex image of the field in the
object plane at a resolution of 882x588 pixels with 0.5 m spacing. This sampling is
optimal for the 0.55 NA of the objective lens at  = 523 nm. To produce the droplet
image, by far the most costly computation was the surface integral defined in equation
5 which was performed by direct numerical integration. This operation required
approximately 6 hours for the largest droplet (Figure 10). In contrast reconstruction of
a 3D image to find the surface field and a similar step to reconstruct the internal field,
made use of the FFT algorithm and each of these steps required almost 2 minutes. It is
clear that any of the optimisation methods that we have suggested will require
considerably faster computation and we are currently testing 3D FFT based
algorithms for this purpose.
5. Conclusions
In this paper we have shown for the first time how it is possible to reconstruct high
resolution holographic images through irregular interfaces with large refractive index
steps such as those found in two-phase flows. We have discussed scalar diffraction
theory in systems of this kind and proposed some useful approximations that can be
applied to the problem of imaging through droplets. The critical step in this process is
estimating the droplet surface. Our preliminary work has considered near
axisymmetric droplets and in this case the surface form was estimated from the plan
form. Using knowledge of the refractive index, the internal field was calculated. It is
clear from these results that multiple holograms or multiple exposure holograms could
be used to measure changes in the droplet shape. By seeding both the droplet and the
ambient air, detailed 3D maps of both the internal and external velocity distribution
should be possible taking advantage of complex amplitude correlation techniques
[19].
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
Er
Es

2
2

 k a2 E a  0

 k 2w E w  0
Figure 1. Scattering from a droplet suspended in air
Es
Es
Es
Er
Ew
-
+
Es
Es
Figure 2. Geometry of boundary surfaces
50/50 Beam
Splitter Cube
Mirror
Object
Microscope
Objective
Image array

Mirror
Spatial
Filter
Figure 3. Holographic microscope
Mirror
(a)
(b)
(c)
Figure 4. Holographic reconstruction of external field a) front plane, b) axial plane
and c) far plane.
Figure 5. Holographic reconstruction of external field x-z section.
(a)
(b)
(c)
Figure 6. Holographic reconstruction of the internal field a) front plane, b) axial plane
and c) far plane.
Figure 7. Holographic reconstruction of external field in axial plane
(a)
(b)
(c)
Figure 8. Holographic reconstruction of the internal field a) front plane, b) axial plane
and c) far plane.
Figure 9. Holographic reconstruction of external field in axial plane
Figure 10: Holographic reconstruction of external field in axial plane with field at
estimated droplet surface superimposed
(a)
(b)
(c)
(d)
Figure 11. Holographic reconstruction of the internal field a) front plane, b) and c)
internal planes and d) far plane.