Total-Internal-Reflection Deflectometry for Measuring Small Deflections of a Fluid Surface
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Noname manuscript No.
(will be inserted by the editor)
Total-Internal-Reflection Deflectometry for Measuring Small
Deflections of a Fluid Surface
Utkarsh Jain · Anaı̈s Gauthier · Devaraj van der Meer
arXiv:2009.00531v2 [physics.flu-dyn] 8 Jun 2021
Received: date / Accepted: date
Abstract We describe a method that uses total inter- 1 Introduction
nal reflection (TIR) at the water-air interface inside a
large, transparent tank filled with water to measure the Measuring instantaneous free surface deformations of
interface’s deflections. Using this configuration, we ob- liquids is of general interest in several practical applica-
tain an optical setup where the liquid surface acts as a tions such as in coating and food industries, in large ap-
deformable mirror. The setup is shown to be extremely plications such as to study ship wakes, and in off-shore
sensitive to very small disturbances of the reflecting wa- engineering [1, 2]. The interest also naturally extends
ter surface, which are detected by means of visualising to more fundamental fluid dynamics and physics prob-
the reflections of a reference pattern. When the water lems such as studying interfacial fluid instabilities [3, 4],
surface is deformed, it reflects a distorted image of the droplet dynamics [5, 6], wave formation and propaga-
reference pattern, similar to a synthetic Schlieren setup. tion on the surface of a fluid [7], and in oceanography
The distortions of the pattern are analysed using a suit- [8, 9].
able image correlation method. The displacement fields The methods to quantitatively measure liquid sur-
thus obtained correlate to the local spatial gradients face behaviour may be broadly divided into two cate-
of the water surface. The gradient fields are integrated gories based on whether they are intrusive or not. Intru-
in a least-squares sense to obtain a full instantaneous sive methods can be used when the extent of intrusion
reconstruction of the water surface. This method is par- is small, and the average flow is not significantly dis-
ticularly useful when a solid object is placed just above turbed. Traditionally, arrays of resistive (or capacitive)
water surface, whose presence makes the liquid surface wave probes have been used to study the variation of
otherwise optically inaccessible. water level in large setups studying waves [9, 10], but
can only be installed in sparse distributions separated
by gaps of (at least) several centimetres. Less intru-
sive methods that rely on flow velocities collected us-
Keywords free surface visualisation · synthetic ing a stereo particle-image-velocimetry setup have also
Schlieren · liquid surface deflectometry been shown to work for large scale systems [11, 12].
Some non-intrusive methods for such measurements,
that only use reflections from the water surface and
a set of multiple cameras for reconstruction have also
Grants or other notes about the article that should go on the been developed [9, 13].
front page should be placed here. General acknowledgments
A non-intrusive method compatible with smaller,
should be placed at the end of the article.
lab scale setups, to resolve deflections of the microm-
U. Jain, A. Gauthier and D. van der Meer eter to millimeter scale of the free surface, is to use
Physics of Fluids Group and Max Planck Center Twente for
Complex Fluid Dynamics, MESA+ Institute and J. M. Burg-
the liquid surface as a refracting or reflecting inter-
ers Centre for Fluid Dynamics, University of Twente, P.O. face. Usually refraction is used, where the water sur-
Box 217, 7500AE Enschede, The Netherlands face acts as the surface of a lens. A reference pattern is
E-mail: u.jain@utwente.nl, d.vandermeer@utwente.nl placed underneath the water bath that is contained in a2 Utkarsh Jain et al.
transparent tank. When the light rays from the pattern between the measured displacement fields and the lo-
emerge through the liquid surface, they are refracted cal spatial gradients of the free surface. Finally we dis-
due to the jump in refractive index. The variation in cuss how this gradient information is integrated in a
heights of the free surface causes further movements least-squares sense to obtain a fully reconstructed liq-
of the refracted image of the reference pattern. These uid surface profile from the imaged snapshot at a given
movements can be recorded using a camera and anal- instant.
ysed to reconstruct the liquid profile. This method is a The main offering of this particular method is that
spin on the well-known Schlieren method, and is known the liquid surface can be visualised when it is not op-
as the free-surface synthetic Schlieren method. It was tically accessible, due to, for instance, the presence of
first proposed by Kurata et al. [14], and since has been an opaque object above the free surface. An example
matured by the works of Moisy et al. [1] and Wilde- of such a situation is when a solid projectile is close to
man [15] to result in a packaged method that is quick slamming onto the liquid surface, and obstructs direct
and inexpensive to arrange. The optics of the problem imaging needed for synthetic Schlieren.
are used to compute the spatial gradients of the liquid As imperfections on a mirror are much easier de-
surface. The gradient fields are then integrated using tected than on a lens, our the method is expected and
a suitable algorithm to obtain a full reconstruction of shown to be inherently more sensitive than classical
the imaged area. Even when a fully quantitative recon- synthetic Schlieren.
struction cannot be obtained, a great deal of qualitative The paper is organised as follows: in section 2, we
information can be learnt, as discussed by Fermigier et introduce the optics which allow the technique to work,
al. [3] and Chang et al. [5, 6]. and details of the setup in which we implemented the
A few other methods use the reflections from the method. The first stage of the technique involves mea-
liquid surface acting this time as a mirror to compute suring the displacements of the reference pattern in the
its spatial profile. Cox & Munk [16] were the first to mirror plane. The methods to quantify these displace-
use the specular reflections of the Sun from the sea sur- ments are discussed in section 3. Next, in section 4, we
face to obtain information about spatial gradients of discuss the relation between these displacements and
the water surface. Direct specular reflections can also the deformation of the water surface from which they
be obtained from suitably placed lamps, a method used originate. In section 5, we discuss some subtleties in-
by Rupnik et al. [17] to reconstruct the liquid profile. volved in performing the inverse gradient operation in
Another category of such methods uses structured light order to finally obtain the final height field, along with
(such as spatially periodic bright bands of light) that an example of the reconstructed surface. In section 6.1
are projected on the free surface. When the surface de- we cover sensitivity, optical limitations and uncertainty
forms, the projections also appear distorted. A cam- estimation. An example of this technique is discussed in
era is used to record the movements of the projected section 7, wherein we show a comparison between the
fringes, whose phase changes are interpreted to recon- measurements and simulations, thereby validating the
struct the height profile of the liquid surface [18, 19]. technique. We end in section 8 with conclusions, the
Such methods have long been used in solid mechanics advantages of this technique, and its limitations when
where extremely small displacements (of the order 10 compared to other methods which may offer a similar
nm) need to be resolved [20–24]. They have come to be range of accuracy in measurements.
known as ‘deflectometry’.
Here we visualise the movements of the water sur-
face by using it as a specularly reflecting surface in 2 Setup requirements
a total-internal-reflection (TIR) configuration. Taking
inspiration from Moisy et al. [1] and Wildeman [15], The setup consists of a water-filled transparent tank
we use a fixed pattern, whose distortions by the mov- with flat walls, a fixed pattern that is allowed to project
ing free surface are interpreted in a synthetic-Schlieren onto the liquid surface of interest, and a camera to im-
sense to obtain displacement fields. Note that contrary age the reflection from the liquid surface. A light source
to Moisy et al. [1] and Wildeman [15], we use the wa- is used to illuminate the fixed pattern as shown in figure
ter surface as a mirror rather than as a lens. From the 1.
point of view of a ray-optics problem, the presence of The light which enters the water tank is refracted
a mirror results in an additional complication as it is towards the interface’s normal, as it enters an optically
the reflecting ‘mirror’ that undergoes deformation, and denser medium. Eventually it reaches the air-water in-
not the apparent object that is behind the mirror. We terface, where depending on the magnitude of angle of
exploit the ray optics in the setup to derive relations the incidence (represented by θ in figure 1), the lightTIR-Deflectometry 3
Apparent object cations. This condition also sets the maximum magni-
behind the 'mirror', O' tude of deformations that can be measured. Indeed, lo-
cal and sharp distortions of the air-water interface pro-
duce large curvatures. Thus, with the condition θ > θc
still holding true, the light rays reflected at the inter-
face can be deflected away from the sensor of the cam-
era. Additionally, even at small deformations, some ray-
TIR crossing may occur, especially where curvature is large,
Original printed making the imaging and interpretation ambiguous.
pattern, O Note that due to arrangement of the optical setup,
the images recorded by an observer at the camera’s lo-
a
er
cation are flattened in the y−direction, i.e., along the
m Water
Ca
direction in which light rays are shown to propagate
in figure 1 (to the reader, the direction in the plane
Light source of the paper). The result is such that a circular object
suspended at the water surface appears elliptical. Thus
Fig. 1 Schematic of TIR setup. A brightly lit, large light
source is used to illuminate the printed pattern O. The image a conversion factor applies to the aspect ratio. This is
from the printed pattern is reflected at the water-air interface found by placing a circular disc at the water surface,
and enters a suitably placed high speed imaging camera. At a and measuring the eccentricity of the ellipse that results
large enough angle of incidence, the interface acts as a mirror from the distortion. There is no such distortion along
due to total internal reflection, and the camera only captures
the mirror image. The light rays illustrate the general optics the x-direction (to the reader, normal to the plane of
of the problem. the paper), and the pattern is reflected as is.
Clearly, also other deformations created by optical
imperfections in the setup (e.g., curved container walls)
rays might either pass into the surrounding optically can be dealt with using standard digital image correla-
rarer medium (here, air) or get specularly reflected as tion techniques performed on the undisturbed image of
if by a mirror. The latter case is what we aim to obtain, the pattern.
known as total internal reflection (TIR). It requires the
angle of incidence at water surface to be greater than
the critical angle θc = arcsin na /nw , where na and nw 3 Quantifying displacement fields
are the refractive indices of air and water respectively.
For TIR to occur at an air-water interface, the angle of An example of the image of a stationary water surface,
incidence needs to be greater than θc ≈ 48.75◦ , which as recorded on camera, is shown in figure 2(a). When a
may require the water bath depth to be of the order of disturbance travels across the water surface, it deforms
the lateral width of the tank. Here we use a tank that the interface such that the reflected image is distorted,
is 50 cm in length and width, and is filled with water as seen in figure 2(b). The disturbances of the water
up to a depth of ∼ 30 cm . surface are recorded with time, and the images are pro-
cessed using an appropriate method to extract displace-
ment vectors from the movements of the pattern. Two
2.1 Operating conditions such methods are discussed.
The method described here can be used to visualise the
motion of air-water interface only if the light passing 3.1 Using cross-correlation
from water to air is fully reflected at the surface, which
is easily obtained with large incident angles. However, Cross-correlation methods are usually deployed on two
TIR cannot be achieved if the air were replaced by a subsequent images from a time series (for instance as
medium optically denser than water, such as glass (n ≈ they are used in particle image velocimetry, PIV), and
1.52) or silicone oil (n ≈ 1.40): the image of the original divide the region of interest into interrogation windows.
pattern (O in figure 1) would always be refracted and In typical PIV measurements, a multi-stage algorithm
never reflected. is used, whereby each image is scanned multiple times,
With the above conditions satisfied, the air-water with successively decreasing size of the interrogation
surface will only act as a mirror if it exists. Any small windows. Cross-correlation techniques, by their very
contamination floating at the surface disrupts the free nature, are best used with images that contain a large
surface, such that the ‘mirror’ disappears at all such lo- number of randomly distributed ‘particles’ (here, dots4 Utkarsh Jain et al.
(a) (b)
(c) (d)
Fig. 2 (a) The reference pattern O is reflected, as is, when the water surface is stationary. (b) Waves passing on the water
surface create disturbances on the reflecting
q ‘mirror’, which results in a distorted image of the reference pattern being reflected
towards the camera. (c) The magnitude u2x + u2y of the displacement vectors (ux , uy ) of bright squares such as shown in
panel (b) are measured using a PIV routine. (d) The magnitude of displacement vectors of the same pattern shown in panel
(b) are measured using Fourier demodulation. See section 3.3 for comparisons between the two methods.
or squares) [25]. Note that although such a random pat- ries are usually compared to a reference image with the
tern may be better suited for use with cross-correlation undisturbed pattern. These methods have been com-
techniques, we here use a pattern with regularly spaced monly used in solid mechanics [21, 24] as they can re-
squares due to demanding illumination requirements. solve extremely small disturbances which are of use in
Any freely available or commercial PIV program may measuring 2D strain fields. Recently these techniques
be used to obtain two-dimensional displacement fields have been introduced in fluid mechanics [15]. The prin-
in the x and y directions. ciple is the following: given a regularly spaced pattern
During the interrogation process, we choose window with a periodicity determined by two orthogonal wave
sizes in keeping with the recommendations made by vectors ks for s = 1, 2, the intensity profile of the undis-
Raffel et al. [25] and Keane & Adrian [26]. However, it turbed pattern, I0 (r) is dominated by the Fourier com-
can be seen in figure 2(c) that the displacement field can ponents corresponding to ks . Here, r is the position
still contain anomalies in some regions. This is due to vector. A disturbed free surface reflects a distorted pat-
how the spatial resolution and displacement resolution tern, such that the reference intensity profile is slightly
are affected by the size of the interrogation window. deformed, and changes to
Most of the noise in the data may be smoothened in
later stages when reconstructing the water surface (see
section 5.2). I(r) = I0 (r − u(r)) , (1)
3.2 Using Fourier Demodulation where u(r) denotes the displacement u of the pattern
at position r. By filtering out only the dominant Fourier
When regularly spaced patterns are used (O in figure 1), modes, I0 (r) transforms into
the images (shown in figure 2) can be processed using
Fourier-demodulation (FD) based methods to extract
displacement fields. In this case, images from a time se- g0 (r) ≈ as exp[iks · r] for s = 1, 2 , (2)TIR-Deflectometry 5
with as constant. Consequently, the deformed pattern 4 Surface movements from projected
I(r) transforms into distortions
g(r) = g0 (r − u(r)) ≈ as exp[iks · (r − u(r))] (3) The last step is to relate the displacement vector ~u(~r) to
for s = 1, 2 , the actual deformation of the liquid surface. To do so,
we consider the ray optics of the setup. As illustrated
i.e., it is phase-modulated by the disturbances u(r) of in figure 3, a source object is placed at position P , from
the pattern. The latter can be extracted by multiplying which a light ray travels towards the ‘mirror’ (here, the
g(r) with the complex conjugate of the filtered reference air-water interface). Although we measure the displace-
pattern g0∗ (r) and determining the phase shift ment fields by tracking the deformation of a fixed pat-
tern (O → O0 in figure 1), the deformations actually
arg(g(r)g0∗ (r)) ≈ −ks · u(r) for s = 1, 2 . (4) take place at the air-water interface. In other words,
it is the mirror that deforms, and makes the apparent-
For each position r this constitutes a pair of linear equa-
object behind it look distorted. The reader is asked to
tions, which can be readily solved for u(r).
refer to figure 3 as a guide. Since the water surface can
An example resulting from this procedure is shown
both move vertically, or just tilt by some angle, we have
in Figure 2(d). Naturally, some restrictions apply. For
here a set of two, generally coupled problems, which we
example, the components in the signal whose wave-
may treat as uncoupled by virtue of the smallness of the
lengths are significantly shorter than the pattern wave-
free surface deformations that we aim to measure: the
length are simply filtered out. The reader can refer to
‘mirror’ may undergo angular deflection (figure 3(a)),
Wildeman [15] for a more detailed discussion on how to
or it may simply move in the vertical direction (figure
select the wave vectors ks of the pattern appropriately.
3(b)).
The first case, where the angular deflection occurs in
isolation, is shown in figure 3(a). A light ray emerging
3.3 Comparisons between the two methods
from P travels towards the ‘mirror’ and gets reflected
to point C, the observer. To the observer at C, this
The main difference between using FD and PIV is that
light ray appears to travel from a point P 0 , the mirror
while the former compares each image on a stack to the
image of P . With the observer still at point C, let the
same reference image (typically the first in the stack) to
mirror tilt by a small angle α. As a result, the point P 0
calculate the displacement, the latter involves compar-
now translates in the horizontal apparent object plane
ing each image to the preceding one in the series. Thus
to point Pa00 . The displacement P 0 Pa00 can be seen by the
when a pattern deforms beyond a certain extent such
observer at C. From figure 3(a), Pa00 is related to, (in
that no amount of (even distorted) periodicity of the ~
this case), the y−component of the height gradient ∇h
pattern can be detected, the FD method will fail to de- 0 00
via tan α = P Pa /2H = ∂h/∂y.
tect a displacement. In such instances auto-correlation
The other case occurs when the water surface only
based PIV will still yield a displacement field, which
undergoes vertical translation, and no angular deflec-
however, will likely contain some inaccuracies.
tion. As shown in figure 3(b), a light ray travelling from
Since PIV divides the total image into multiple win-
P to the mirror, incident at some angle θ, is reflected
dows, the displacements that occur within the outer
to the observer at C. As the mirror is vertically shifted
margins of the image that are half the width of the
by some distance h, the apparent object P 0 moves to
interrogation windows, are not resolved. Additionally,
some other point Pv00 in the apparent-object plane. Us-
the resolution of the displacement field depends on the
ing geometry of the problem as shown in figure 3(b),
overlap between adjacent interrogation windows. Ob-
the displacement P 0 Pv00 as seen by the observer at C
taining a full-pixel resolution between the image and
can be related to h via tan θ. With the same conven-
the displacement field are often computationally very
tion in both cases that displacements to the right of the
expensive. In contrast, FD yields displacement fields at
line segment P 0 P 00 would be considered positive, and
full-pixel resolution as that of the images being pro-
those to the left as negative, we obtain the following
cessed, and no information at the margins of the image
two relations:
is lost.
In both methods, displacements may be measured ~ and
P 0 Pa00 = −2H ∇h, (5)
with sub-pixel resolution, but spatial structures smaller
than the interrogation window (in the case of PIV), or
the wavelength of the pattern (for FD) cannot be easily
resolved. P 0 Pv00 = −2h tan θ, (6)6 Utkarsh Jain et al.
(a) (b)
z
y
x
Fig. 3 Ray diagrams representing the decoupled ‘mirror’ deformation problem, which gives us the relations between dis-
placements recorded in apparent-object plane and the surface deformation h. A light ray coming from P gets reflected and
is seen by an observer at C. However, to the observer at C, the object at P appears to lie at its mirror image P 0 . When the
reflecting ‘mirror’ (in the experiment, the air-water interface, with the undisturbed and disturbed interface denoted by solid
and dashed lines respectively) deforms, to the observer at C, the apparent object moves to a point P 00 . The optics of this
mirror deformation problem is decoupled into two cases: (a) Angular deflection of the mirror and (b) Vertical translation of
the mirror. See main text for details.
which relate the displacements P 0 P 00 of the pattern to 5 Spatial integration of gradient fields
the height variations h. Height of the bath H and an-
gle of incidence θ are obtained from the experimental 5.1 Recasting the integrand using an integrating factor
setup. Denoting the unit vectors in x and y directions
as î and ĵ respectively, the overall displacement at any Note that equation (8) cannot be directly integrated
point, P 0 Pa00 î + P 0 Pa00 ĵ + P 0 Pv00 ĵ is equivalent to the 2- due to the additional dependence on h. Thus we re-
dimensional total displacement field ux î + uy ĵ (where cast the expression using an integrating factor. Equa-
we have used that a tilt of the interface in the x- and tion (10) can be re-written as
y-direction would lead to a shift of the image point in uy ∂h h tan θ
the x- and y-direction respectively, whereas a vertical − = +
2H ∂y H
displacement of the interface causes a shift in the y- ∂ y tan θ/H
direction only). Thus, the above system of equations = e−y tan θ/H e h . (11)
∂y
can be re-written as
Similarly, equation (9) can be re-written using the same
~ − 2h tan θĵ,
~u ≡ ux î + uy ĵ = −2H ∇h (7) integrating factor
which can be rewritten to give the height gradient −ux ∂h
=
2H ∂x
~ = − ~u − h tan θĵ.
∇h (8) = e−y tan θ/H
∂ y tan θ/H
e h . (12)
2H H ∂x
Equation 8 can be separated for the two directions x Equations (11) and (12) can be combined using vector
and y as notation as
~u
~ ey tan θ/H h ,
∂h ux = e−y tan θ/H ∇ (13)
=− and (9) 2H
∂x 2H
or,
y tan θ/H
~ ey tan θ/H h = − e
∂h uy h tan θ ∇ ~u. (14)
=− − . (10) 2H
∂y 2H H
The gradient fields in x and y directions, that are to
The surface h(x, y) is then reconstructed by solving the be integrated over, are expressed in the form shown on
system of equations expressed in equation (8). The nu- the right hand side of equation (14). The result ob-
merical implementation to do so is described in the next tained from surface integration is divided by the factor
section. exp( y tan θ
H ) to obtain the final height field h(x, y).TIR-Deflectometry 7
With equation 14, we have now recast our original
problem in a conservative form
∇f ~
~ = ξ, (15)
where ξ~ is the known vector field, and f is to be deter-
mined. Mathematically such an expression can be di-
~ × ξ~ = ∇
rectly integrated since ∇ ~ × ∇f
~ ≡ 0. However,
since ξ is only approximately known due to unavoid-
able noise in the experiments, some additional care is
needed during the integration.
5.2 Inverse gradient operation Fig. 4 Reconstructed surface profile of water from the dis-
placement field shown in figure 2(d). The arbitrary distur-
bances on the water surface were recorded and measured over
The inverse gradient operation is performed on equa- a small section of the total water surface in the bath, that is
tion (14) to obtain the final result shown above.
y tan θ/H
~ −1 − e
f (x, y) = ey tan θ/H h(x, y) = ∇ ~u + f0 , or, in short, GF ~ Since the number of elements in
~ = ξ.
2H
the variables in the above equations is M N , there are
(16) 2M N knowns (the gradient information) in the system.
where f0 is an integration constant, connected to the However there are only M N unknowns (the compo-
absolute height of the free surface. In the following dis- nents of F) in the above system. Thus, equation (20)
cussion, f0 is set to zero for convenience. One way to is an over-determined matrix system, and cannot be
integrate over the gradient information ξ~ is to start at simply inverted to find F. The inversion is therefore
a reference point (xr , yr ), and integrate along a path performed while minimising a residual cost function of
such that form [1, 27]:
Z x Z y
f (x, y) = ξx (x0 , yr )dx0 + ξy (x, y 0 )dy 0 . (17) kGF ~ 2.
~ − ξk (21)
xr yr
However, using this approach, any noise in the local The least-squares solution thus found has the effect
gradient information may get added over the path of in- of smoothening out local outliers present in the gra-
tegration [1]. Moreover, in a discretised implementation dient fields. An efficient MATLAB implementation was
of this method, it is not clear how the final result would written and made public by D’Errico [28]. More details
be modified if the order of integration along the paths on global least squares reconstruction, and further ad-
in x and y direction were switched. Both drawbacks can vanced methods can be found in the works by Harker
be avoided by using a ‘global’ approach. This is done & O’Leary [27, 29, 30]. We use the implementation by
by building a linear system of equations using a 2nd- D’Errico which is now commonly used in reconstruction
order centred finite difference operator G = (Gx , Gy ) as problems that involve an inverse gradient operation to
the gradient operator. In x and y directions, the matrix be performed on a mesh of spatial gradients [1, 31–33].
system of discretised equations (from equation (14)) has An example of the reconstructed surface profile, based
the form [1] on the typical displacement field shown in figure 2(d),
is shown in figure 4. A more systematic experiment,
Gx F = ξx , and (18) along with comparisons with simulations is discussed
Gy F = ξy . (19) in section 7.
Here F, ξx , and ξy are vectors of length M N corre-
sponding to the M ×N elements defined on the discrete
6 Sensitivity, limitations and error estimation
(x, y)-mesh, and Gx , Gy are sparse M N × M N matri-
ces corresponding to the finite difference operators. The 6.1 Sensitivity
two equations can be written in combined vector nota-
tion as Starting from equation (7) which relates the surface
Gx 0 F ξ profile height h(x, y, t) to the measured displacement
= x , (20) field ~u(x, y, t), one immediately realizes that there are
0 Gy F ξy8 Utkarsh Jain et al.
two manners in which the surface profile height may in- From the above we can immediately conclude that the
fluence the displacement field, namely by a tilting of the deformations that are visible with our method are much
interface (corresponding to the first term on the right smaller than the spatial resolution of the displacement
hand side, ∼ ∇h) ~ or by a vertical shift (the second pattern. For the example of Fig. 4, where H = 30 cm,
term which is proportional to h). We will now address and the typical wavelength of the structure is λ ≈ 2
the sensitivity of the setup, where we will start with as- cm, we find that λ/(4πH) ≈ 0.005. Using a spatial res-
sessing the relative sensitivity of a tilt versus a vertical olution δuy,min = 100 µm, we obtain that the minimal
shift. displacement δhmin,tilt that is discernible through the
Since tilt and shift are usually correlated, we start detection of the tilted interface equals δhmin,tilt = 0.5
by performing a modal decomposition of the surface µm, and that this sensitivity may (at least theoreti-
height profile, where it suffices for our purposes to con- cally) be increased by increasing the distance H be-
centrate on the y-direction only tween camera/pattern to the liquid surface. Similarly,
X 2π we obtain for the sensitivity for a vertical shift that
h(y) = αk sin(ky) with k = . (22)
λ 2 tan θαk ' δuy,min , (28)
k
Rewriting the y-component (10) of equation (7) as
or,
∂h
−2H − 2 tan θh = uy ≡ uy,tilt + uy,shift , (23) αk 1
∂y ' . (29)
δuy,min 2 tan θ
and inserting equation (22) yields, for each of the nodes
separately As expected, the result is independent of the wave-
length and much larger than it is in the case of a tilted
−2Hkαk cos(ky) − 2 tan θαk sin(ky) = uy,tilt + uy,shift . interface. In fact, using the same spatial resolution in
(24) the case of the example of Fig. 4 (θ ≈ 55◦ ) we have
δhmin,shift = 50 µm, i.e., the setup is two orders of mag-
Clearly, the two terms of these equation do not attain nitude less sensitive for a vertical shift than for a tilt.
their maxima in the same points, as a result of the fact Conversely, this means that if two patterns differ by
that sin(ky) is zero where its derivative is maximal and a vertical shift, i.e., h1 (x, y, t) = h2 (x, y, t) + ∆h(t), the
vice versa, but one may easily compute the respective difference between h1 (x, y, t) and h2 (x, y, t) would be
maxima and determine the relative sensitivity as the very difficult to detect, especially if ∆h is of the same
ratio of these order as h1,2 , which would usually be the case in exper-
uy,shift 2 tan θαk tan θ tan θ λ iment. Here, the contribution of ∆h to the signal would
= = = . (25) be typically two orders of magnitude smaller than that
uy,tilt 2Hkαk Hk 2π H
of the surface deformation features. This implies that,
Note that this ratio is independent of the amplitude even in a time series, there may be a shift between the
αk . Since the wavelength of even the largest structures profiles determined at different moments in time that
that are to be observed is usually much smaller as the is extremely hard to detect, if at all. This makes the
distance of the liquid surface and the pattern, i.e., λ method most suitable in the case that there exists a
H, the above ratio is typically much smaller than one, reference point on the interface where no deformation
which implies that the setup is much more sensitive for is expected.
a tilting of the surface than for a vertical shift uy,tilt
uy,shift .
To put this difference in absolute terms, we note 6.2 Limitations
that the detection of the displacement field ~u is bounded
by the sensitivity of the method used to obtain it which The setup has several limitations originating from the
provides a minimum detectable displacement δuy,min fact that it makes use of the liquid surface as a deformed
which is some fraction of the pixel size of the measured mirror, which we will discuss in sequence in this sub-
image. Using uy,tilt > δuy,min , we find that section.
2Hkαk ' δuy,min , (26)
6.2.1 Mirroring condition
yielding
αk λ Total internal reflection will only happen if the angle
' . (27) of incidence φi on the deformed liquid surface is larger
δuy,min 4πHTIR-Deflectometry 9
than the minimal angle φi,min for which total internal first detect the displacement field ~u(x, y, t) in the im-
reflection will take place, i.e., age plane (using PIV or FD) and to secondly compute
h(x, y, t) with the spatial integration method, and are
na
φi > φi,min = arcsin . (30) difficult to assess or control. Here it is crucial to em-
nl ploy a scheme that integrate the displacement field in
Now the angle of incidence is determined by the angle a global least square sense (as discussed in Subsection
θ at which we look at the pattern and the slope of the 5.2), as otherwise especially systematic errors may be
liquid surface in the y-direction ∂h/∂y, namely φi = cumulatively integrated and lead to substantial errors
θ − arctan(∂h/∂y) which limits the slope to in h.
Relevant from the perspective of the current setup is
∂h na how errors in the main parameters H and θ propagate
/ tan θ − arcsin , (31)
∂y nl in the final interface profile h. Based upon the sensitiv-
or, ity results of Subsection 6.1 one may expect that the
influence of errors in H are more significant than those
λ na in θ. More quantitatively, we may use the modal de-
αk / tan θ − arcsin . (32)
2π nl composition (22) in equation (7) to determine how a
variation ∆H in H propagates into a variation ∆αk of
As long as the typical length scale on which the pat-
the amplitude αk of mode k, leading to
tern changes (λ) is sufficiently larger than the ampli-
tude (αk ) we seek to measure, satisfying the above con- ∆αk ∆H 1 ∆H
dition will not be a serious problem, provided θ is not ≈− ≈− ,
αk H 1 + tan θ tan(ky)(λ/(2πH)) H
chosen too close to arcsin(na /nl ).
(34)
6.2.2 Ray crossing where we have used that the wavelength of the observ-
able structures are usually much smaller than H (i.e.,
Two incident, parallel rays will cross before reaching λ/(2πH) 1, such that the second term in the denom-
the camera if the local radius of curvature R of the inator is small everywhere except close to where the
liquid surface is smaller than the distance of the cam- slope of the interface is zero. Similarly, we can write for
era H/ cos θ. Since for small deformations the radius of the propagation of a variation ∆θ in θ that
curvature can be approximated as 1/R ≈ ∂ 2 h/∂y 2 , we
obtain, using modal decomposition (22) ∆αk ∆θ
≈− 2 (35)
2
αk cos θ((2πH/λ) cot(ky) + tan θ)
λ
αk / cos θ. (33) λ θ ∆θ
4π 2 H ≈− , (36)
2πH cos2 θ cot(ky) θ
For the example of Fig. 4 (H = 30 cm, λ ≈ 2 cm,
θ ≈ 55◦ ), this will lead to αk / 19.5 µm. This is quite a where the dominant term (for cot(ky) not too small)
stringent requirement, which can be improved by mov- has been kept in the second approximation. The first
ing the camera closer to the liquid surface, decreasing term is much smaller than one whereas the second is
H. As discussed above, doing so will however lead to a typically of order unity, such that the relative error in
loss of sensitivity. θ is multiplied by a small number. This is good to realize
when setting up the experiment: it is more crucial to
assure that the pattern is positioned such that H can
6.3 Error estimation be considered constant over the region of interest, and
some compromise in the constancy of the value of θ can
The method is prone to some systematic and random be made in order to reach that goal.
errors that in the end will propagate into the measure-
ment result, the deformation of the interface h(x, y, t).
Some of those are quite generic for systems making use 7 Example and validation: Water surface
of high-speed optical image acquisition, and find their deflection due to air cushioning under an
origin in the specifications of the camera (spatial and approaching plate
temporal resolution, motion blur, pixel sensitivity) and
have to be addressed by using a camera that is suitable Validation of the experimental method is difficult due
for the particular problem at hand [34]. Others are re- to the sensitivity of the method. When one tries to use
lated to the quite elaborate image data processing to known or macroscopically observable menisci around10 Utkarsh Jain et al.
(a) (b) (c)
V
D
Air squeezed Air squeezed
out out
Fig. 5 (a) Schematic of an experiment where a flat disc of diameter D is impacted on a stationary water bath at controlled
velocity V . The flow of air due to being squeezed out creates a stagnation point flow under the disc centre, locally pushing
the water surface down. (b) Measured water surface profiles (azimuthally averaged about the disc centre) from the experiment
shown in panel (a) in an experiment at V = 1 m/s are shown at different time instants τ before impact. (c) The amount of
water surface deflection at r = 0 from panel (b) non-dimensionalised using inertial scales D and V , and compared with the
simulation from Peters et al. [35] and [36].
immersed objects, the problem is that the interface dis- tern that is reflected in the water surface. The displace-
turbance close to the object is not observable due to ment of the reference pattern is then quantified using
the large local deflection and curvature. This implies an image correlation method such as PIV or FD.
that one may only observe the far-field exponential de- Secondly, these displacements are interpreted as pro-
cay which is hard to relate to a physical length scale. jections in the two-dimensional image plane, and re-
This leaves the observation of water waves (as has been lated to the instantaneously deforming water surface
done qualitatively in earlier Sections) or the deforma- and its spatial gradients. By decoupling the light paths
tion of the interface due to the impact of an object. when the reflecting surface either undergoes an angular
We will now turn to the latter and, for the purpose of deflection, or a vertical translation, we build a system
validation reproduce some results from [36] in figure 5. of equations that relate the pattern deformation to the
The experimental setup is described in figure 6(a): a local surface deflection. This second step thus involves
flat disc is slammed onto a stationary water bath with recasting the measured displacement fields to a suitable
a controlled velocity. The approaching disc pushes out integrable form, and calculating the final height field.
the ambient air from the gap in between itself and the
A relative drawback of TIR-D arises from the high
water surface. The stagnation pressure set up under
sensitivity it offers: it requires the water surface to be
the disc centre deflects the water surface away. The
very well isolated from external sources of noise. This
(azimuthally averaged) measured profiles are shown in
high degree of isolation from mechanical disturbances
panel (b) at various times before impact (τ ). The mea-
limits the method’s application to well-controlled en-
surements at r = 0 are compared with two-fluid bound-
vironments. Another consequence of the sensitivity is
ary integral simulations described in [35, 37–40]. The
that using menisci of a stationary object for calibration
favourable comparison indicates that the measurement
purposes is difficult, since deflections easily become too
technique is successful at resolving deflections of the or-
large to be measurable.
der of micrometres up to several tenths of millimetres.
For additional information, we refer to [36]. An application of this method was discussed by mea-
suring the water surface deflections due to air-cushioning
under a plate that is about to slam on it. Good com-
8 Conclusions parison of the measurements with boundary integral
simulations validate the technique for measurements
We described a TIR-based method to measure small- up to tens of micrometres. Some more examples of the
scale deformations of a water surface, consisting of two use of this method are described in ref. [41, chapter 6]
steps: First, the movement of the water surface is mea- by measuring micron-scale waves on a water surface,
sured by recording the deformation of a reference pat- and showing successful comparisons with a theoreticalTIR-Deflectometry 11
model, thus showing its effectiveness in resolving pre- 6. C.-T. Chang, J.B. Bostwick, S. Daniel, and P.H.
cise micron scale deformations. Steen. Dynamics of sessile drops. Part 2. Exper-
The method’s greatest merit lies in it using total in- iment. Journal of Fluid Mechanics, 768:442–467,
ternal reflection at the water surface. First, this implies 2015.
that whatever moves above the water surface remains 7. A. Paquier. Generation and growth of wind waves
invisible to the camera. Secondly, it is inherently more over a viscous liquid. PhD thesis, Université Paris-
sensitive than compared to using it as a refracting (lens- Saclay, 2016.
like) surface [1, 15]. However this does not naturally 8. G. Gallego, A. Yezzi, F. Fedele, and A. Bene-
imply a greater precision than these synthetic Schlieren tazzo. A Variational Stereo Method for the Three-
methods - a high degree of precision may be achieved Dimensional Reconstruction of Ocean Waves. IEEE
using either of the methods, depending on the exact transactions on geoscience and remote sensing,
nature and scale of the experiment. The present tech- 49(11):4445–4457, 2011.
nique does, however, make the liquid surface optically 9. A. Benetazzo, F. Fedele, G. Gallego, P.-C. Shih,
accessible in settings where synthetic Schlieren cannot and A. Yezzi. Offshore stereo measurements of
be used. gravity waves. Coastal Engineering, 64:127–138,
2012.
Acknowledgements We would like to thank Ivo Peters for 10. D. Liberzon and L. Shemer. Experimental study of
originally suggesting the idea of using TIR on water in a large the initial stages of wind waves’ spatial evolution.
bath, Francesco Viola and Vatsal Sanjay for helpful discus- Journal of Fluid Mechanics, 681:462–498, 2011.
sions on the inverse gradient operation, and Patricia Vega 11. D.E. Turney, A. Anderer, and S. Banerjee. A
Martı́nez for attempts to validate the method by measuring
the meniscus on an immersed pin. We acknowledge the fund- method for three-dimensional interfacial particle
ing from SLING (project number P14-10.1), which is (partly) image velocimetry (3D-IPIV) of an air–water in-
financed by the Netherlands Organisation for Scientific Re- terface. Measurement Science and Technology,
search (NWO). 20(4):045403, 2009.
12. M. van Meerkerk, C. Poelma, and J. Westerweel.
Scanning stereo-PLIF method for free surface mea-
Conflict of interest surements in large 3D domains. Experiments in
Fluids, 61(1):1–16, 2020.
The authors declare that they have no conflict of inter- 13. J.M. Wanek and C.H. Wu. Automated Trinocular
est. stereo imaging system for three-dimensional sur-
face wave measurements. Ocean Engineering, 33(5-
6):723–747, 2006.
14. J. Kurata, K.T.V. Grattan, H. Uchiyama, and
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