An optical sensor for detecting the dynamic contact location of a gas-liquid interface along the length of a body is described. The sensor is developed in the context of applications to supercavitating bodies requiring measurement of the dynamic cavity contact location; however, the sensing method is extendable to other applications as well. The optical principle of total internal reflection is exploited to detect changes in refractive index of the medium contacting the body at discrete locations along its length. The derived theoretical operation of the sensor predicts a signal attenuation of 18 dB when a sensed location changes from air-contacting to water-contacting. Theory also shows that spatial resolution (d) scales linearly with sensor length (Ls) and a resolution of 0.01Ls can be achieved. A prototype sensor is constructed from simple components and response characteristics are quantified for different ambient light conditions as well as partial wetting states. Three methods of sensor calibration are described and a signal processing framework is developed that allows for robust detection of the gas-liquid contact location. In a tank draining experiment, the prototype sensor resolves the water level with accuracy limited only by the spatial resolution, which is constrained by the experimental setup. A more representative experiment is performed in which the prototype sensor accurately measures the dynamic contact location of a gas cavity on a water tunnel wall.

Sensing the dynamic contact location of a gas-liquid interface on a body has several applications, particularly related to seaborne vessels. For example, one may desire to locate the contact of a free-surface on a ship hull, measure the submergence of a floating platform or detect the contact location of a gas ventilated supercavity on a cavity-riding vehicle. The latter application motivates the research reported in this paper. Supercavitation refers to the encapsulation of an object traveling in liquid by a gas bubble; this may be accomplished by either vaporizing the liquid (natural supercavitation) or injecting gas around the object (ventilated supercavitation).1,2 Supercavitating vehicles have been the subject of intense research due to the potential for drag reduction that enables increased speed.3 Knowledge of the planing forces generated on partially wetted afterbody surfaces could lead to more robust control of such vehicles.4–9 Measurement of the dynamic gas-liquid-body contact location, which determines the planing area, is thus important for vehicle control. This application demands a sensor that can be embedded in or mounted on a surface and that can measure the dynamic gas-liquid contact location in a quasi-continuous manner along the length of the body. Although the sensor developed in this paper can be used for many fluids, we focus on water and air as the working liquid and gas, respectively. Practical constraints arising from application to supercaviting bodies include: (1) the number of hull pass-throughs should be minimized (e.g., mounting holes, wires, cables, etc., passing through a vehicle hull) and (2) the sensor should be robust to electrical noise and to changes in ambient pressure, temperature, and salinity.

One measurement approach uses an array of wall-mounted pressure sensors, as have been used for turbulence measurements, to determine whether liquid or a gas contacts the body.10 Each sensor, however, would require a separate hull penetration. Electrical capacitance sensors have been used to distinguish phases in various multi-phase flow applications, but are sensitive to changes in temperature and salinity.11–13 

The problem of resolving the gas-liquid contact location is similar to a liquid level measurement, a problem that has received much attention in the literature. For the present application, the most relevant liquid level sensors are optical devices that exploit the refractive index difference between gas and liquid. Optical sensors are attractive due to their electrical noise robustness and high sensitivity.14 Vazquez and Montero15 provided a recent review of optical liquid level sensors and several instruments are discussed here. Discrete (or point) measurement sensors, such as those reported in Refs. 16–18 transmit light along a fiber optic cable to a transducing tip immersed in the gas or liquid medium. The amount of light reflected depends on the refractive index of the medium and is typically sensed by a photodiode. Discrete measurement sensors have been used to measure liquid level in a quasi-continuous manner by arranging the sensors in an array.17,19 Using arrayed discrete transducers is impractical for the supercavity detection problem due to the additional hardware and necessity for many hull penetrations. Romo-Medrano and Khotiaintsev19 demonstrated a self-contained quasi-continuous sensor consisting of an array of discrete optical transducing elements and a mutliplexing scheme to reduce the number of light sources and detecting elements required. The sensor had good spatial resolution (5 mm) and high sensitivity to whether the contacting medium was air or water (∼28 dB power attenuation).

In another approach to liquid level sensing, an optical fiber with modified core or cladding is used as the transducing element. Either the wavelength or intensity of light transmitted is sensitive to the refractive index of the surrounding medium. Multi-mode fibers14 or fiber optics with gratings20–22 are typically used to induce wavelength shifts, and possess the ability to sense continuously and excellent sensitivity when the liquid refractive index is close to that of the fiber and cladding. However, the total length tends to be small (typically <100 mm) and the sensitivity is reduced when the refractive index of the liquid deviates from that of the fiber and/or cladding.21 

Still other optical sensors relate liquid level to light attenuation in an optical fiber. Pérez-Ocón et al.24 and Chandani and Jaeger23 each demonstrated single-source, single-receiver fiber optic sensors with spatial resolution on the order of 1 mm. The measurement range of fiber optic sensing methods can be extended by making only discrete sections of the fiber active sensing areas to produce a quasi-continuous sensor. Betta et al.25 developed a robust sensor using a multiplexed array of six fibers with several discrete sensing regions. Lomer et al.26 constructed a quasi-continuous sensor with 2.2 mm resolution and 39 mm range using a plastic multi-mode fiber wound around a cylinder.

The sensor described in this paper is not fiber optic based but does exploit the optical phenomenon of total internal reflection (TIR) to sense refractive index changes in a body-contacting medium. This transduction method has been used in sensors for detecting rain on a vehicle windshield.27,28 The sensor uses a solid transparent guide in which collimated light rays propagate, as shown in Figure 1. The incidence angle of the rays with respect to the guide surface normal vector is greater than the critical angle for the guide-gas interface. Therefore, the light undergoes total internal reflection and continues propagating within the guide as long as gas contacts the exterior. If at a certain point along the guide the contacting medium changes to a material with larger refractive index (e.g., liquid), then only a fraction of the light will reflect within the guide. The reflected light intensity is measured at several discrete locations along the guide to synthesize a quasi-continuous gas-liquid contact location sensor. There are no discrete external transducing elements as the guide itself acts as the transducer, which reduces the complexity of the instrument and mitigates the need for hull penetrations for individual elements. A single light source is required and an array of photodiodes sense the internally reflected light along the guide length. Because a photodiode is used at each sensing location, the sensitivity to changes in the contacting medium is very large; theory predicts a change from air to water will cause an 18 dB signal attenuation. The large sensitivity comes at the cost of using multiple photodiodes. The spatial resolution scales linearly with the overall length of the sensor. The sensor concept is amenable to surface mounting on a body hull or, if the application allows, to direct integration into the hull. This paper describes the theory of operation and presents the characteristics and calibration of a prototype sensor. A transient gas cavity water tunnel experiment demonstrates the accuracy of the prototype sensor in resolving a dynamic gas-liquid-body contact location.

FIG. 1.

Implementation of the optical sensor on a supercavitating body. The sensor makes use of the principle of total internal reflection to detect where the gas-liquid interface contacts the body.

FIG. 1.

Implementation of the optical sensor on a supercavitating body. The sensor makes use of the principle of total internal reflection to detect where the gas-liquid interface contacts the body.

Close modal

Figure 1 shows schematically an implementation of the optical sensor on a supercavitating body. Guide interface B directly contacts the surrounding medium, which may be either gas or liquid. Refractive indices of the gas and liquid are denoted

$n_\text{g}$
ng and
$n_\text{L}$
nL
, respectively, and are both smaller than
$n_\text{s}$
ns
. Collimated light rays directed from a source into the guide make an angle
$\theta _\text{i}$
θi
with the surface normal vector of interface B. When light is incident upon the interface between two different media, the angles of the incident and transmitted light with respect to the surface normal vector are governed by Snell's law,29
$n_\text{i} \sin \theta _\text{i} = n_\text{t} \sin \theta _\text{t}$
nisinθi=ntsinθt
, where subscripts i and t denote incident and transmitted media, respectively. If
$n_\text{i} > n_\text{t}$
ni>nt
, then the light will undergo total internal reflection when
$\theta _\text{i}$
θi
is greater than or equal to the critical angle, defined as29 

\begin{equation}\theta _\text{c} = \sin ^{-1} \left( \frac{n_\text{t}}{n_\text{i}} \right).\end{equation}
θc=sin1ntni.
(1)

The critical angles for the guide-gas and guide-liquid interfaces are thus

$\theta _{\text{c}_{\text{s-g}}} =\sin ^{-1} ( n_\text{g} / n_\text{s})$
θcs-g=sin1(ng/ns) and
$\theta _{\text{c}_{\text{s-L}}} = \sin ^{-1} ( n_\text{L} / n_\text{s} )$
θcs-L=sin1(nL/ns)
; assuming
$n_\text{L} > n_\text{g}$
nL>ng
we see that
$\theta _{\text{c}_{\text{s-L}}} > \theta _{\text{c}_{\text{s-g}}}$
θcs-L>θcs-g
. Therefore, by setting the incident light angle in the range
$\theta _{\text{c}_{\text{s-g}}} \le \theta _\text{i} < \theta _{\text{c}_{\text{s-L}}}$
θcs-gθi<θcs-L
, total internal reflection will be achieved when gas contacts the guide, preventing any light from leaving the guide as it propagates down the length of the sensor. However, when the contacting medium is liquid, a portion of light will be transmitted from the guide into the liquid producing a measurable change in the amount of light reflected from interface B. In this paper, we use photodiodes placed at interface A to sense this change in reflected light. Figure 2 shows one possible sensor package mounted external to the body hull. The collimated source (e.g., laser diode) directs light into the guide via a right angle prism. The photodiodes are shown directly contacting interface A, a configuration we demonstrate in Sec. III. This setup would require two hull penetrations: one for the light source power and one for the photodiode signals.

FIG. 2.

Schematic of an externally mounted implementation of the optical sensor with photodiodes used to sense the light reflected within the guide.

FIG. 2.

Schematic of an externally mounted implementation of the optical sensor with photodiodes used to sense the light reflected within the guide.

Close modal

In describing the theory of operation, we consider the sensor model shown in Figure 2. The photodiodes are responsible for measuring a portion of the light reflected from interface B. The amount of light measured by photodiode Sj indicates the medium contacting interface B at location xj. Locations at which gas contacts the sensor are said to be in a dry (D) state, the first location in contact with liquid is in the W1 state, the second liquid-contacting location is in the W2 state and so on (see Figure 2).

Reflectance, R, is the ratio of light power reflected from an interface,

$P_\text{r}$
Pr⁠, to power incident on the interface,
$P_\text{i}$
Pi
.29 Reflectance is defined by two components, R and R, referring to the light that is polarized perpendicular and parallel to the plane of incidence, respectively. For simplicity, we assume the plane of polarization is perpendicular to the plane of incidence and the reflectance is then given as29 

\begin{equation}R = R_{\perp } = \left( \frac{n_\text{i} \cos \theta _\text{i} - n_\text{t} \cos \theta _\text{t}}{n_\text{i} \cos \theta _\text{i} + n_\text{t} \cos \theta _\text{t}} \right)^2.\end{equation}
R=R=nicosθintcosθtnicosθi+ntcosθt2.
(2)

Figure 3 plots the reflectance at interface B for a solid-gas (Rs-g) and solid-liquid interface (Rs-L) as a function of incident angle for representative refractive indices (

$n_\text{s}$
ns = 1.49,
$n_\text{L}$
nL
= 1.333, and
$n_\text{g}$
ng
= 1). The vertical dotted lines denote the critical angle associated with each interface. The difference in reflectance, ΔR = Rs-gRs-L gives the change in light power reflected from interface B when the sensor goes from a dry to a wet state at a given location, which is the signal of interest. For small
$\theta _\text{i}$
θi
, ΔR is small as most of the light power is transmitted through the interface. However, for
$\theta _\text{i} = \theta _{\text{c}_{\text{s-g}}}$
θi=θcs-g
, ΔR is maximal as all light is reflected when gas contacts interface B. When
$\theta _\text{i} = \theta _{\text{c}_{\text{s-L}}}$
θi=θcs-L
, ΔR = 0 because total internal reflection occurs for both gas and liquid contacting the sensor. In order to maximize the sensitivity of the sensor, the incidence angle should be chosen as close as possible to
$\theta _{\text{c}_{\text{s-g}}}$
θcs-g
so as to maximize ΔR. However, a relatively large design space exists for the incident angle for this representative case.

FIG. 3.

Reflectance at interface B as a function of angle of incidence for representative solid, gas, and liquid refractive indices.

FIG. 3.

Reflectance at interface B as a function of angle of incidence for representative solid, gas, and liquid refractive indices.

Close modal

The theoretical power sensing requirements of the photodiodes are now derived. The initial power in the light beam after entering the guide is defined as Ps. The power in the beam incident on photodiode

$\text{S}_\text{N}$
SN is
$P_{\text{i}_\text{N}}$
PiN
and the power in the beam exiting
$\text{S}_\text{N}$
SN
(i.e., traveling toward location
$x_\text{N+1}$
xN+1
) is
$P_{\text{e}_\text{N}}$
PeN
. The ratio of exiting to incident power is defined as
$\chi _\text{N} \equiv P_{\text{e}_\text{N}}/P_{i_\text{N}}$
χNPeN/PiN
; nominally, the power difference is entirely absorbed by photodiode
$\text{S}_\text{N}$
SN
. The power incident on
$\text{S}_\text{N}$
SN
can be expressed as

\begin{equation}P_{\text{i}_\text{N}} = P_s e^{-\alpha L N} R_\text{N} \prod _{j=1}^{N-1} \chi _j R_j,\end{equation}
PiN=PseαLNRNj=1N1χjRj,
(3)

where Rj is the reflectance at location xj and the exponential accounts for power attenuation within the guide material. The optical path length between detecting elements is L and α is the attenuation coefficient of the material, typically on the order of 5 × 10−5 mm−1 for acrylics.30 Assuming no other losses, the difference between incident and exiting power gives the power absorbed by

$\text{S}_\text{N}$
SN⁠,

\begin{equation}P_\text{N} = P_{\text{i}_\text{N}} - P_{\text{e}_\text{N}} = P_s e^{-\alpha L N} R_\text{N} \left( 1 - \chi _\text{N} \right) \prod _{j=1}^{N-1} \chi _j R_j.\end{equation}
PN=PiNPeN=PseαLNRN1χNj=1N1χjRj.
(4)

With

$\text{S}_\text{N}$
SN in the W1 state, Rj = Rs-g = 1 for j = 1 to N−1. Therefore, the ratio of power absorbed by
$\text{S}_\text{N}$
SN
in the D versus W1 states is
$P_{\text{D}_\text{N}}/P_{\text{W1}_\text{N}} = 1/R_{\text{s-L}}$
PDN/PW1N=1/Rs-L
. This power ratio defines the signal attenuation caused by a change from dry to wet state,
$A_s = 10 \log _{10} (P_{\text{D}_\text{N}}/P_{\text{W1}_\text{N}})$
As=10log10(PDN/PW1N)
. Using the refractive index values defined in Sec. II A and
$\theta _\text{i} = 45^{\circ }$
θi=45
, the theoretical attenuation is As = 18 dB. For context with respect to other quasi-continuous sensors, Romo-Medrano and Khotiaintsev19 reported an attenuation of ∼28 dB and Lomer et al.26 reported an attenuation magnitude of 0.17 dB, each with air and water as the working gas and liquid, respectively.

The theoretical power resolution requirement for the Nth photodiode is given as

\begin{eqnarray}\Delta P_\text{N} & = & P_{\text{D}_\text{N}}-P_{\text{W1}_\text{N}}\nonumber \\& = & P_s e^{-\alpha L N} \left( 1 - R_{\text{s-L}} \right) \left( 1 - \chi _\text{N} \right) \prod _{j=1}^{N-1} \chi _j.\end{eqnarray}
ΔPN=PDNPW1N=PseαLN1Rs-L1χNj=1N1χj.
(5)

Figure 4(a) plots ΔP normalized by the max power absorbed by the first photodiode,

$P_{\text{D}_\text{1}}$
PD1⁠, as a function of photodiode number, N, for different values of χ with α = 0. The horizontal blue line denotes
$\Delta P/P_{\text{D}_\text{1}} = 0.01$
ΔP/PD1=0.01
, which is a conservative estimate of the power resolution of the photodiodes (i.e., 1% of max power to be sensed). The plot demonstrates that the number of photodiodes that can be used for a given power resolution increases dramatically with increasing χ. Coupled with the geometry of the sensor, the number of photodiodes determines the spatial resolution of the sensor for a given length.

FIG. 4.

(a) Difference in absorbed power between D and W1 states normalized by the max power absorbed by the first photodiode,

$P_{\text{D}_\text{1}}$
PD1⁠, as a function of photodiode number, N, for different values of χ. (b) Theoretical spatial resolution as a function of sensor length with
$\theta _\text{i} = 45^{\circ }$
θi=45
; data points from previous studies are also plotted.

FIG. 4.

(a) Difference in absorbed power between D and W1 states normalized by the max power absorbed by the first photodiode,

$P_{\text{D}_\text{1}}$
PD1⁠, as a function of photodiode number, N, for different values of χ. (b) Theoretical spatial resolution as a function of sensor length with
$\theta _\text{i} = 45^{\circ }$
θi=45
; data points from previous studies are also plotted.

Close modal

As Figure 2 illustrates, the sensor length, Ls, spatial resolution, d, guide thickness, h, and incident light angle,

$\theta _\text{i}$
θi⁠, are related by

\begin{equation}d = \frac{L_s}{\text{N}} = 2h \tan \theta _\text{i}.\end{equation}
d=LsN=2htanθi.
(6)

The optical path length between photodiodes is then given as

\begin{equation}L = \frac{d}{\sin \theta _\text{i} }.\end{equation}
L=dsinθi.
(7)

For a given sensor length and incident light angle, Eqs. (5)–(7) can be solved iteratively to find the achievable spatial resolution with the power constraint

$\Delta P/P_{\text{D}_\text{1}} \ge 0.01$
ΔP/PD10.01⁠; the guide thickness is allowed to take on whatever value is required to supply the spatial resolution. Figure 4(b) plots the theoretical spatial resolution as a function of sensor length for different values of χ with
$\theta _\text{i} = 45^{\circ }$
θi=45
and α = 5 × 10−5 mm−1. The spatial resolution scales linearly with sensor length, which is attributable to the minimal attenuation in the material. The slope provided by each χ value is denoted in the figure. Also plotted are data points from several studies previously reported in the literature to demonstrate that the design space available for the present sensor affords spatial resolution that is commensurate with other technologies.

In practice, the upper and lower bounds on spatial resolution will be constrained by the guide thickness. The upper bound on thickness will be driven by packaging considerations, while the lower bound will be constrained by the light beam geometry. If the light source shown in Figure 2 consists of a laser with circular beam of radius r, then the minimum allowable guide thickness is given by

\begin{equation}h_\text{min} = \frac{r}{\sin \theta _\text{i} }.\end{equation}
hmin=rsinθi.
(8)

Therefore, the smallest achievable spatial resolution is

\begin{equation}d_\text{min} = \frac{2r}{\cos \theta _\text{i} }.\end{equation}
dmin=2rcosθi.
(9)

In the practical realization of the sensor constructed for laboratory testing, photodiodes in housings are placed near the outside surface of an acrylic wall, which serves as the guide. Figure 5(a) shows the photodiodes incorporated into a liquid level experiment that is described in Sec. III B. Because interface A is exposed to air, the light nominally experiences total internal reflection at the interface. However, as depicted in the close-up view of the photodiode, a fraction of the light scatters out of the acrylic guide due to imperfections at interface A. This scattered light is sensed by the photodiode and the amount of light sensed indicates the medium contacting interface B at the corresponding location. Figure 5(b) shows a more sophisticated design concept for transmitting a fraction of the beam through interface A to the photodiode. In this design, a partially reflective coating is placed on interface A and a right-angle prism is attached to the interface using an optically transparent adhesive. This design would likely be more robust, provide better control over the amount of light kept within the guide (i.e., better control over χ) and increase the light power reaching the photodiode, which may be necessary for field applications. However, for the laboratory experiments discussed here, the method of light sensing shown in Figure 5(a) is adequate to demonstrate and characterize the sensor operation.

FIG. 5.

(a) Photodiodes incorporated into a liquid level experiment. The photodiodes sense light scattered out of the guide at interface A to determine the medium contacting interface B at the corresponding location. (b) A more sophisticated design for transmitting a fraction of the beam to the photodiode.

FIG. 5.

(a) Photodiodes incorporated into a liquid level experiment. The photodiodes sense light scattered out of the guide at interface A to determine the medium contacting interface B at the corresponding location. (b) A more sophisticated design for transmitting a fraction of the beam to the photodiode.

Close modal

The sensor prototype uses silicon photodiodes with an active sensor area of 5.1 mm2 operated in photovoltaic mode. The purpose of the sensor calibration is to establish the voltage response of each photodiode in the D and W1 states. Three approaches to calibration are proposed. The first option is to measure the photodiode voltage responses with each integrated into the final sensor package. As the responses with each diode in the first wetted position (W1 state) are required, the entire sensor must be moved relative to a gas-liquid interface, or vice versa.

Alternatively, with the photodiodes integrated into the final sensor package and the entire sensor in a dry state, the power to voltage relationship can be established by varying the input power from the light source. In photovoltaic mode, the photodiode voltage output is a logarithmic function of the incident power and the normalized voltage response is expressed as

\begin{equation}V^{*} = K_\text{1} \ln P^{*} + K_\text{0},\end{equation}
V*=K1lnP*+K0,
(10)

where P* = P/P0 and V* = V/V0; V0 and P0 are the photodiode voltage and power corresponding to a chosen reference state, respectively. The constants

$K_\text{1}$
K1 and
$K_\text{0}$
K0
are computed for each photodiode through calibration of Eq. (10). It is convenient to have the reference state coincide with the actual dry state of the sensor such that P0 = PD; the reference voltage for each photodiode becomes the dry state voltage, V0 = VD, which can be easily measured in situ. Normalizing the power and voltage in this way allows for calibration of Eq. (10) without explicit knowledge of the power reaching each photodiode. Assuming all losses and reflections are independent of light power amplitude, the ratio P/P0 is equal to the ratio of a given source light power to the reference state source light power. Therefore, only measurements of source light power and the corresponding photodiode voltages are required for calibration. Equation (10) can be used to predict the voltage response of each photodiode in the W1 state, VW1, by letting
$P^* = P_{\text{W1}}^* = R_{\text{s-L}} P_{\text{D}}^*$
P*=PW1*=Rs-LPD*
.

In the final calibration option, photodiodes are exposed to known light power to generate a calibration of the form of Eq. (10) prior to integrating them into the sensor package. As before, the values of VW1 would be predicted from Eq. (10) with knowledge of V0 and

$P_{\text{D}}^*$
PD*⁠.

The prototype sensor shown in Figure 5(a) is calibrated using the first and second methods. For the second method, the light power is varied by placing beamsplitters at the exit of the laser aperture. The laser is a Thorlabs HNL150L HeNe 15 mW continuous wave laser. The power in the beam exiting the beamsplitters is

$P_\text{m} = b P_\text{L}$
Pm=bPL⁠, where
$P_\text{L}$
PL
is the laser power and b is the fraction of power passed in the in-line direction;
$P_\text{m}$
Pm
is measured using a Coherent®Fieldmaster laser power meter. We choose the reference power to coincide with the D state power, P0 = PD, which occurs when b = 1. The normalized power reaching a given photodiode is then P* = P/P0 = b, which is measured for each different input power level during the calibration. Figure 6 shows the calibration curve for each photodiode demonstrating that the logarithmic model of Eq. (10) is appropriate.

FIG. 6.

Normalized voltage response of photodiodes as a function of normalized incident power. Logarithmic fits are shown for each detector.

FIG. 6.

Normalized voltage response of photodiodes as a function of normalized incident power. Logarithmic fits are shown for each detector.

Close modal

All three calibration methods assume a stable light source and a constant relationship between voltage and power. The stability of the light source could be monitored in situ using a photodiode that always remains dry to compensate for changes in source power. The most likely factor that would alter the relationship between voltage and power is the presence of ambient light. The influence of ambient light on photodiode response is examined using the experimental setup in Figure 5(a) with no beamsplitters placed at the exit of the laser. The amount of ambient light is varied by turning on fluorescent lights in succession. At each ambient lighting condition, the voltage of each photodiode is recorded in the D state with the laser on (V) and with the laser off (Va). Figure 7(a) plots V against Va for each diode; the voltage response with the laser on is mostly constant until a threshold level of ambient light is reached. The difference in voltages across the photodiodes is caused by alignment rather than light attenuation. The black symbols in Figure 7(b) show the data in Figure 7(a) normalized by V0 for each detector, which is measured with the laser on and ambient light at a minimum. This normalization collapses all data and shows that V is constant and equal to V0 until the power from the ambient light is sufficiently large to generate a voltage of Va ≈ 0.8V0. For Va/V0 > ≈1.2, V is approximately equal to Va indicating that the power from ambient light dominates the signal. The robustness to ambient light can be enhanced by either increasing the power reaching the photodiodes from the laser (and thus increasing V0) or by reducing the affect of ambient light by, for example, using optical filters on the photodiodes. The specific relationship between Va and ambient light power incident on the photodiode will depend on the spectral response of the detector and spectral content of the ambient light. However, the voltage response trend demonstrated in Figure 7(b) should be consistent.

FIG. 7.

(a) Voltage response of each photodiode with the laser on (V) as a function of voltage caused by ambient light (Va). (b) Normalized voltage response in the D (black symbols) and W1 (gray/red symbols) states as a function of normalized voltage caused by ambient light.

FIG. 7.

(a) Voltage response of each photodiode with the laser on (V) as a function of voltage caused by ambient light (Va). (b) Normalized voltage response in the D (black symbols) and W1 (gray/red symbols) states as a function of normalized voltage caused by ambient light.

Close modal

Finally, the normalized voltage response for each photodiode when in the first wetted position (W1 state) is plotted with light gray symbols in Figure 7(b). Similar to the response in the D state, the normalized voltages in the W1 state are flat until a threshold voltage. Figure 7(b) represents the nominal result of the first calibration procedure, wherein the voltage response is characterized over a range of ambient light conditions. The red line denotes the predicted value of

$V^*_{\text{W1}}$
VW1* for photodiode 4 computed using the second calibration method while the red symbols are the measured values. The prediction is fairly accurate up to the threshold voltage at which
$V^*_{\text{W1}}$
VW1*
begins to rise linearly (results are similar for other sensors). Therefore, calibration methods 2 and 3 are valid if the ambient light remains below a threshold value that should be determined during calibration.

Once calibrated, a processing strategy for determining the state (D or W1) of each photodiode is required. A simple thresholding approach is adopted with the threshold voltage defined as

\begin{equation}V_{\text{t}} = \frac{V_{\text{D}} + V_{\text{W1}}}{2} + a \frac{\Delta V}{2},\end{equation}
Vt=VD+VW12+aΔV2,
(11)

where ΔV = VDVW1 and a is a weight factor used to bias the threshold in the direction of the less uncertain state voltage. Additionally, if the voltage response as a function of Va has been calibrated, then the ambient light can be compensated for by pulsing the light source to actively sample Va and adjusting VD and VW1 accordingly. The D and W1 calibration data shown in Figure 7(b) are fit with the piece-wise functions

\begin{equation}V^* =\left\lbrace \begin{array}{@{}l@{\quad }l@{}}C &\text{if $V^* \le C$}\\V_{\text{a}}^* &\text{if $V^* > C$,} \end{array}\right.\end{equation}
V*=CifV*CVa*ifV*>C,
(12)

where C is a constant determined separately for the D and W1 data.

The weight factor is constructed to be a function of the uncertainty on the D and W1 state voltages, uD and uW1, respectively. Calibration method 1 enables calculation of an empirical weight factor of the form

\begin{equation}a = a_0 \left( 2 \frac{u_{\text{W1}}}{u_{\text{D}} + u_{\text{W1}}} - 1 \right),\end{equation}
a=a02uW1uD+uW11,
(13)

which varies linearly between −a0 and a0 as a function of the relative uncertainty on VW1. The uncertainty on each voltage is defined as

\begin{eqnarray}u_{\text{D}} & = & \pm \sqrt{\sigma _{\text{D}}^2 + \epsilon _{\text{D}}^2} \nonumber \\[-7pt]\\[-7pt]u_{\text{W1}} & = & \pm \sqrt{\sigma _{\text{W1}}^2 + \epsilon _{\text{W1}}^2},\nonumber\end{eqnarray}
uD=±σD2+εD2uW1=±σW12+εW12,
(14)

where σ is the standard deviation of the voltage signal and ε is the standard error of the fits (Eq. (12)) to

$V_{\text{D}}^*$
VD* and
$V_\text{W1}^*$
VW1*
. Both σ and ε are computed over a range of ambient light conditions via the first calibration procedure. For the present study, σD/V0 was found to be constant for all
$V_{\text{a}}^*$
Va*
, while σW1/V0 increased linearly with the normalized standard deviation of voltage caused by ambient light, σa/V0. Given estimates of the uncertainties on each voltage, we define a signal-to-noise ratio as

\begin{equation}SNR = 20 \log _{10} \left( \frac{\Delta V}{\sqrt{u_{\text{D}}^2 + u_{\text{W1}}^2}} \right),\end{equation}
SNR=20log10ΔVuD2+uW12,
(15)

which Figure 8 plots over the range of

$V_{\text{a}}^*$
Va*⁠. The voltage difference ΔV is computed from the empirical fits to
$V_{\text{D}}^*$
VD*
and
$V_\text{W1}^*$
VW1*
. The prototype sensor demonstrates good SNR (>10 dB) up to
$V_{\text{a}}^* \approx 0.8$
Va*0.8
.

FIG. 8.

Signal-to-noise ratio (SNR) over the range of

$V_{\text{a}}^*$
Va* for the prototype sensor.

FIG. 8.

Signal-to-noise ratio (SNR) over the range of

$V_{\text{a}}^*$
Va* for the prototype sensor.

Close modal

Finally, once the state of each photodiode is determined from the thresholding procedure, the location of the gas-liquid interface can be estimated. If

$\text{S}_\text{N}$
SN is the last photodiode in the dry state then the gas-liquid interface is located between xN and xN + 1. This assumes that the dry portion is continuously dry, but in Sec. III C the influence of partial wetting on sensor response is explored and methods for compensating for partial wetting are considered.

The signal processing method was validated with a series of measurements performed in the setup shown in Figure 5(a) in which the liquid level was varied over 350 mm for each test. Ambient light was varied statically and dynamically across several tests causing typical values of ambient voltage in the range

$V_{\text{a}}^* \approx$
Va* 0.3–0.9. Liquid level measurements using the sensor were compared to image based measurements from a Nikon D200 digital single-lens reflex (SLR) camera. Across all tests, the maximum value of rms error in liquid level location was erms/u0 = 0.59, where u0 = d/2 is the zeroth-order uncertainty of the sensor. The maximum error magnitude over all tests was emax/u0 = 1.14, which occurred for the test with the highest level of static ambient light. Thus, the sensor has been shown to resolve a slowly moving liquid level with accuracy comparable to the spatial resolution. Section IV describes validation of the sensor with a more representative experiment.

Because the applications relate to measurement in multi-phase flows, investigation into the response of the sensor in the presence of liquid droplets that may remain on the otherwise dry guide surface is warranted. To test this, water droplets were placed on a horizontal acrylic guide at the sensing location of a given photodiode. Figure 9 shows the normalized voltage response of this photodiode for four different tests in which the size of the droplet was monotonically increased. The abscissa plots droplet diameter normalized by the diameter of the laser spot at the intersection of the laser with the top of the guide. Droplet diameter is measured as the diameter of the droplet-guide contact region. The centers of the droplets were never displaced from the center of the laser spot by more than one droplet radius. The voltage response decreases rapidly until

$D_\text{drop}/D_\text{laser} \sim 2.5$
Ddrop/Dlaser2.5⁠, at which point the response asymptotes to the W1 state. The significance of
$D_\text{drop}/D_\text{laser} \sim 2.5$
Ddrop/Dlaser2.5
is simply that the probability of the droplet completely covering the laser spot approaches unity at this size. Variations in the response for
$D_\text{drop}/D_\text{laser} < 2.5$
Ddrop/Dlaser<2.5
are likely due to variations in the location of the droplet relative to the laser spot, which governs the portion of light that fails to reach the photodiode. The schematics inset in Figure 9 show representative laser paths caused by the droplets corresponding to the color coded data points. For some larger droplets (e.g., green data point), the light is internally reflected within the droplet and re-enters the guide with a path and angle offset. Depending on the offsets, this may cause a voltage response that falsely indicates a dry state. The practical ramification of partial wetting by droplets is that the voltage responses will be modulated between the nominal D and W1 values depending on the amount of overlap between individual droplets and laser spots and the size of the droplets. A droplet causing a W1 voltage reading on an otherwise dry sensor is considered a false reading.

FIG. 9.

(a) Photodiode response as a function of normalized droplet diameter. The inset figures are color coded to two data points and schematically show the path and angle offset of the ray caused by the droplet.

FIG. 9.

(a) Photodiode response as a function of normalized droplet diameter. The inset figures are color coded to two data points and schematically show the path and angle offset of the ray caused by the droplet.

Close modal

The frequency of false readings caused by droplets could be reduced via hardware. For example, multiple sensors could be used in parallel with spatial offsets to reduce the probability of droplets covering laser spots. However, software can also compensate for droplets by altering the response associated with D and W1 states. If

${\rm S}_\text{N}$
SN registers as being in the W1 state and the remainder of the sensor is truly wetted, then the power absorbed by each ensuing photodiode is
$P_{\text{N+}k} = R_{\text{s-L}}^{(k+1)} P_{\text{D}_{\text{N+}k}}$
PN+k=Rs-L(k+1)PDN+k
for k = 1 to M–N (the sensor has M total photodiodes). However, if there is a droplet at
$x_\text{N}$
xN
and
${\rm S}_\text{N}$
SN
registers in the W1 state, the power absorbed by photodiodes N+k to M will be
$P_{\text{N+}k} = R_{\text{s-L}} P_{\text{D}_{\text{N+}k}} = P_{\text{W1}_{\text{N+}k}}$
PN+k=Rs-LPDN+k=PW1N+k
. Therefore, one can examine the response of photodiodes downstream of the first wetted location to determine if the sensor is partially or fully wetted. If it is determined that a droplet resides at
$x_\text{N}$
xN
then the remaining length of the sensor can be interrogated with the new D and W1 state power redefined by multiplying Eq. (5) by Rs-L. The redefined D and W1 voltages could be updated from the calibration curves, if using calibration method 2 or 3. If using calibration method 1, then the voltage response of photodiodes in several wetted locations should be measured (e.g., W1, W2, W3, ...). The caveat of this compensation is that only downstream photodiodes capable of measuring the reduced light power provide useful information.

A liquid layer of finite thickness covering the sensor surface may also introduce a light path offset that causes false readings. Figure 10 shows the normalized response of two photodiodes as a function of thickness of the liquid layer, z, normalized by the distance between photodiodes, d. As shown in the schematics in Figure 10, the light escapes the guide, is internally reflected at the liquid-gas interface, and re-enters the guide at the original propagation angle with a path offset. When the path offset is an integer multiple of d, a subset of the photodiodes will falsely indicate dry states. This effect causes the voltage spike in photodiode 2 for liquid layer thickness z/d = 0.3. For the supercavity contact location application, for which the body is immersed in an effectively infinite liquid, errors due to a finite liquid layer are not a likely occurrence.

FIG. 10.

Response as a function of normalized water layer thickness, z/d. False dry readings occur when z causes a laser path offset that is an integer multiple of d.

FIG. 10.

Response as a function of normalized water layer thickness, z/d. False dry readings occur when z causes a laser path offset that is an integer multiple of d.

Close modal

In order to validate the operation of the sensor in an environment representative of the application to supercavitating bodies, the contact location of a ventilated gas cavity in a water tunnel is measured using the experiment shown in Figure 11(a). The water tunnel cross section is 305 mm × 305 mm in the test region and has optical access on all sides via acrylic windows. The acrylic window on the top of the tunnel served as the guide for the optical sensor (h = 25.4 mm) and the laser and photodiodes were arranged as shown in Figure 11(a) yielding a spatial resolution d = 57 mm. The large spatial resolution is attributable to the wall thickness and not to a limitation of the sensor operation. With the water tunnel flowing, nitrogen gas injected into the low-pressure wake of a cavitator with a blunt afterbody mounted to the top acrylic window (sensor guide) formed a ventilated cavity. Images captured with a Nikon D200 digital SLR camera synchronized to the sensor data acquisition system provided ground truth measurements of the gas-liquid-solid interface location. A diffused strobe supplied backlight illumination for the images; a sample image is shown in Figure 11(b). The mounting locations of the 8 photodiodes are evident in the image and the laser spot locations are marked with red “x” symbols. By varying the tunnel velocity and gas flow rate over several tests, the appearance of the cavity contact ranged from a sharp gas-liquid interface to a very bubbly two-phase region at the location of contact with the sensor guide. The width of this bubbly two-phase region, w, is marked in Figure 11(b). In defining the interface contact location from images, the guide surface is considered dry only when in contact with the continuous gas cavity.

FIG. 11.

(a) Ventilated gas cavity water tunnel experimental setup and (b) sample image captured with the SLR camera depicting the interface contact location and width of the bubbly two-phase region, w.

FIG. 11.

(a) Ventilated gas cavity water tunnel experimental setup and (b) sample image captured with the SLR camera depicting the interface contact location and width of the bubbly two-phase region, w.

Close modal

A full sensor calibration was not performed for the water tunnel experiment, but the mean voltage response for all photodiodes was measured in situ for the D, W1, and W2 states. Based on previous observations that the uncertainty is lower on the dry voltage, the weight factor in Eq. (11) was set to a = 0.25.

Gas-liquid contact location measurements and error are plotted for four tests in the left hand column of Figure 12. The vertical dashed lines in each plot denote the time of the image shown in the right hand column. In the first test (top row in Figure 12), for which the gas-liquid interface is sharp, the sensor very accurately tracks the contact location. The rms and maximum normalized error are erms/u0 = 0.76 and emax/u0 = 1.67, respectively. For the tests depicted in the second and third rows of Figure 12, the cavity contact location contains more bubbles and the rms and maximum errors increase in both cases. An increase in the uncertainty in measuring the contact location from images contributes to the increased error, but reflections from the bubbly contact region represent the dominant error source. The test in the third row of Figure 12 demonstrates this very clearly. At location x4 preceding the fourth photodiode, a two-phase bubbly flow contacts the guide. The true state of this location likely oscillates between wet and dry causing light to periodically escape the guide. However, as pointed out in the image, the escaping light is scattered by the bubbles. This scattered light is sensed by S4 and produces a response large enough to register a D state. Therefore, the sensor tends to track the leading edge of the two-phase cavity contact region. This error is examined more closely by measuring w from images, which is somewhat subjectively defined as the width of the two-phase region that advects at the speed of the interface contact line (i.e., does not include bubbles that advect downstream faster than the contact line). The dependence of the mean and rms contact location measurement error on the rms width of the two-phase region is shown in Figure 13 (both axes are normalized by u0). When the rms width of the two-phase region is less than or equal to the zeroth-order uncertainty, then the error is attributable to the sensor spatial resolution. For values of

$w_\text{rms}$
wrms greater than u0, the error increases with the increasing width of the two-phase region.

FIG. 12.

Results from four tests of the gas cavity water tunnel experiment. Plots in the left hand column show the gas-liquid contact location tracked via the optical sensor and from images. Snapshots in the right hand column are representative images from the corresponding tests.

FIG. 12.

Results from four tests of the gas cavity water tunnel experiment. Plots in the left hand column show the gas-liquid contact location tracked via the optical sensor and from images. Snapshots in the right hand column are representative images from the corresponding tests.

Close modal
FIG. 13.

Dependence of mean and rms contact location measurement error on the rms width of the two-phase region.

FIG. 13.

Dependence of mean and rms contact location measurement error on the rms width of the two-phase region.

Close modal

Finally, for the test shown in the bottom row of Figure 12, a water droplet remained on the surface at location x7 after the cavity contact location passed. Applying the standard processing routine yields the contact location denoted by the black markers. The sensed location jumps as the two-phase contact region passes x7 and x8; however, after the cavity has grown past x8, the contact location is falsely measured as a lower value because the droplet causes S7 to register in the W1 state. However, application of the software droplet compensation logic described in Sec. III C results in accurate contact location measurements during the entire growth of the cavity (green markers). The error plotted in this figure is computed for the droplet compensated data.

An optical sensor for measuring the contact location of a gas-liquid interface in a quasi-continuous manner along the length of a body was developed and demonstrated. The sensor makes use of simple components—for example, an acrylic guide and photodiodes—to sense changes in light intensity from a collimated source caused by different media contacting the guide surface. An application of particular interest involves detection of the contact location of a gas supercavity on a body. The proposed sensor presents a viable solution for this application as it requires minimal body hull penetrations and—being optical—is robust to electrical noise and to changes in ambient pressure, temperature, and salinity. Also, the spatial resolution scales linearly with length making the sensor flexible for many applications. Sensor power requirements, sensitivity, and spatial resolution were theoretically derived and shown to be comparable to other state-of-the-art optical liquid level sensors.

A prototype sensor was constructed with an acrylic tank wall acting as the guide. Three calibration methods were presented and used to either directly measure or predict photodiode voltage in dry and various wet states. Because the photodiode voltage response was flat up to a critical ambient light value, a simple thresholding argument could be used with good signal-to-noise ratio to discriminate between dry and wet states. When liquid droplets remain on an otherwise dry surface, a particular sensor location will register as wet if a droplet covers the laser spot and Ddrop is on the order of Dlaser. As the droplet grows larger and larger, it behaves more like a thin water layer in which the light internally reflects and re-enters the guide with a path offset. Offsets that are integer multiples of the spatial resolution will cause false dry readings. A software compensation method for false wet readings caused by droplets was proposed and demonstrated in a water tunnel experiment.

An experiment with a tank being slowly drained and filled validated sensor performance in a liquid level measurement mode. The accuracy was shown to be bounded by the spatial resolution. A more representative experiment for supercavitating bodies was performed in a water tunnel with a ventilated gas cavity. For sharply defined gas-liquid interfaces, the contact location measurement accuracy was again bounded by the spatial resolution. In cases where the contact location was characterized by a bubble-laden two-phase flow, the accuracy of the sensor scaled with the size of the two-phase region. The sensor provides a relatively simple, robust, and accurate instrument for gas-liquid contact location measurements.

The authors gratefully acknowledge funding from the Office of Naval Research under task number N0001413WX20545 monitored by program officer Dr. Ronald Joslin (ONR Code 331). We also thank Albert Fredette for data acquisition and experimentation support and Dr. Robert Kuklinski for support and guidance throughout this work.

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