Two methods are investigated to simultaneously obtain both three-dimensional (3D) velocity field and free surface elevations (FSEs) measurements near a surface piercing foil, while limiting the equipment. The combined velocity field and FSE measurements are obtained specifically for the validation of numerical methods requiring simultaneous field data and free surface measurements for a slender body shape. Both methods use stereo particle image velocimetry (SPIV) to measure three component velocities in the flow field and both methods use an off the shelf digital camera with a laser intersection line to measure FSEs. The first method is performed using a vertical laser sheet oriented parallel to the foil chord line. Through repetition of experiments with repositioning of the laser, a statistical representation of the three-dimensional flow field and surface elevations is obtained. The second method orients the vertical laser sheet such that the foil chord line is orthogonal to the laser sheet. A single experiment is performed with this method to measure the three-dimensional three component (3D3C) flow field and free surface, assuming steady flow conditions, such that the time dimension is used to expand the flow field in 3D space. The two methods are compared using dynamic mode decomposition and found to be comparable in the primary mode. Utilizing these methods produces results that are acceptable for use in numerical methods verification, at a fraction of the capital and computing cost associated with two plane or tomographic particle image velocimetry (PIV).

## Introduction

The surface wave field generated from a translating slender body is an important component in the evaluation of the drag characteristics of slender bodies, such as ships or hydrofoils. This aspect of the generated flow field is critically important in the verification of seakeeping prediction programs, which aim to model the physics of dynamic fluid-structure interactions through computer simulation. In the development of computational fluid dynamics methods, canonical problems or analytic solutions are often initially used to provide verification of the numerical approximations, while comparison with experimental data provides validation to the developed methods. Often, the experimental data used for comparison are obtained with the purpose of understanding or measuring a specific physical phenomenon and not for validation of numerical methods. To remedy this, an experiment was designed specifically in support of the validation of numerical methods.

Recent developments of hybrid computational fluid models, which solve the Navier–Stokes equations based on a decomposition of the velocity into an inviscid component and a viscous perturbation, present an interesting challenge for validation since the methods require separate solutions of an inviscid set of equations and a viscous perturbation set of equations, solved in overlapping three-dimensional (3D) domains [1,2]. In particular, velocities within the overlapping viscous and inviscid domains, and quantities at the boundary domains, such as free surface elevations (FSEs), velocities, and pressures at the edges of the viscous domain, must be validated. Hybrid Lattice–Boltzmann method and boundary element method solvers are under development to investigate improvements in computational time for seakeeping problems [3]. This method is implemented using parallelized general purpose graphical processing units to reduce computational time required to solve the methods throughout the domain. Due to the parallelization of local solution methods, such as Lattice–Boltzmann method, it is necessary to validate the method using the velocity field as opposed to validation based on integrated measurements, such as lift or drag.

Although many modern techniques exist for separately obtaining measurements of velocity or free surface elevation in a three-dimensional volume, many of these techniques require large and expensive experimental facilities, a large number of expensive camera systems, or multiple laser systems. The purpose of this paper is to investigate simplified techniques that utilize a standard stereo particle image velocimetry (SPIV) system to obtain three-dimensional measurements of velocity and free surface elevation within a volume defined by the overlapping domain of interest. An emphasis is placed on using a single laser to obtain all measurements, while two techniques are investigated based on the orientation and positioning of the laser plane. A surface piercing foil is used as the example test case since it generates a wave field at the surface, but represents a simple known geometry. The orientation of the laser plane in the present experiments is of note, since the two techniques investigated orient the laser plane parallel and offset from the foil span as well as perpendicular to the foil's direction of motion, respectively. Both of these laser plane orientations are unconventional in studying the flow features around a translating foil, however, they enable the laser plane to intersect the free surface, such that free surface and velocity field measurements may be made simultaneously using the same laser system.

## Background

Stereo particle image velocimetry is a common tool for measuring 3D velocity field components [4,5]. SPIV utilizes two angled cameras and a single laser plane to measure three-component flow field velocities in a two-dimensional (2D) slice. Other techniques using three cameras have been developed to improve accuracy and improve calibration procedures [6,7]. While 3D measurement techniques, such as tomographic particle image velocimetry (PIV), have been shown to improve velocity estimates when studying fluid flows in a 3D area of interrogation, the additional equipment and computations are necessary to make these techniques less widely used for validation experiments [8].

For free surface measurements, traditional methods for obtaining free surface elevations utilize invasive equipment, such as wave probes. In small-scale experiments, this invasive technique can contribute to inaccuracies in the measurement of free surfaces through curvatures induced by surface tension at the penetration point of the probe. At larger scales, the use of wave probes is common practice, since the small scale of the probe can be negligible compared with the relevant forces at the free surface [9]. A second limitation of wave probes is that they only provide a point measurement time history of the free surface at a given spatial location. To obtain a global measurement of a free surface based on wave probes, it is either necessary to have a large grid of probes, which can introduce surface tension problems and interference with the measurement of interest, or individual wave probes must be moved around, requiring statistical assumptions about the problem of interest.

More recent techniques have been developed for determining free surface topography. Fouras developed a topographic free surface imaging technique based on measured light distortions caused by curvature of the free surface compared with a reference image [10]. Images are compared using a PIV algorithm to determine the relative displacements, where the free surface elevations may then be determined based on ray tracing. This technique was modified to incorporate PIV measurements obtained from cameras oriented above the free surface and below the glass bottom of the tank in order to measure free surface topography and velocity fields in the wake of a circular cylinder in a uniform free stream [11]. Differences in PIV measurements from the two cameras are attributed to light refraction at the free surface, such that the free surface may be reconstructed based on the difference between the PIV measurements. This technique is extended by Gomit, where refraction at the free surface is measured using a stereo camera system to reconstruct the free surface, with velocity measured in a two-dimensional slice below the free surface in the wake of a towed ship model [12].

The techniques implemented by Fouras and Gomit are very useful in obtaining free surface and velocity field measurements at the same time, however, extending these techniques to measure velocity fields throughout the fluid volume poses challenges [11,12]. For example, near the free surface, intersection of the laser sheet with the free surface can distort the laser sheet, inhibiting PIV measurements. Additionally, due to the magnification effect of water, varying the location of the laser sheet requires recalibration of the system at new heights. The purpose of this paper is to investigate other methods to simultaneously obtain free surface and velocity measurements in a fluid volume surrounding a surface piercing body, using limiting assumptions about the flow problem of interest. Two methods are investigated: (1) The experiment is conducted multiple times with reorientation of the laser sheet to obtain an average volume measurement of the velocity field with free surface and (2) A single experiment is conducted in a two-dimensional plus time (2D + T) orientation, using the time dependence of the measurement and frame rate of the camera to reconstruct the third spatial dimension of the area of interrogation.

## Experimental Setup

The experiment in this study consists of a surface piercing NACA 0012 foil section that is towed with zero angle of attack through a tank of stationary fluid. The surface piercing body generates a disturbance on the free surface and depth-wise change in flow velocities. Two methods are investigated for simultaneously measuring the three-dimensional three component (3D3C) velocity field and FSE within the area of interrogation, using a standard SPIV system with an additional off-the-shelf digital camera. The first method obtains average field measurements of a two-dimensional slice by repeating the experiment at varying distances between the laser and the foil. The second method reconstructs the three-dimensional field measurements from a single experiment by relating time, by way of frame rate, to the third spatial dimension. Both methods use an area of interrogation three by one-half chord lengths, centered longitudinally on the center of the foil and extending out transversely from the chordline.

The towing tank used in the present experiments had a useful length of 2.6 m, depth of 0.68 m, and width of 0.90 m. The tank had glass viewing walls on all sides to allow an unobstructed view of the test model and an unobstructed path for the laser sheet. The tow tank was fitted with an automated tow carriage which includes an X–Z traverse system, allowing the foil to be precisely positioned across the width of the tank with repeatability less than 0.1 mm. A rotational positioning motor held the foil with a constant angle of attack of 0 deg relative to the forward motion of the carriage. An additional motor and linear traverse was mounted below the tank to carry PIV equipment (cameras and laser) to maintain a constant viewing window with the forward motion of the foil. Figure 1 shows a picture of the experimental setup with the model foil piercing the free surface, PIV cameras visible in the foreground looking through the side wall of the tank, and green laser sheet shining through the bottom of the tank.

An important feature of the towing tank is the ability to conduct highly repeatable experiments through the use of precision servo motors. The movement of both carriages is fully automated using servomotors with 0.1 mm precision, ensuring that experiments are consistently repeated when given sufficient time for flow disturbances to dissipate in the tank. This enables the first experimental method, which utilizes repeated experiments with an SPIV system to determine an average velocity field and free surface elevation field surrounding the surface piercing foil by combining averaged data from a series of experiments.

The foil section used in the experiments was a NACA 0012 foil with a chord length of 70 mm and a submerged span of 3.5 chord lengths. The foil was constructed of aluminum and was anodized black to provide a smooth nonreflective test surface. The hollow foil section was square-tipped, such that the cross section was constant throughout depth and the end of the foil was sealed such that the interior of the foil was dry.

### Experimental Method 1: Average Field Measurements From Longitudinal (Streamwise) Slicing.

For a constant cross section foil, the largest variation in the flow field will normally occur in the x–z plane as defined in Fig. 2, such that if one wants to only measure the 2D velocity field, the laser plane would normally be oriented in the x–z plane, cutting through the cross section of the foil. In the present study, the intersection of the laser plane with the free surface is used to measure the free surface elevation, such that the laser plane must be oriented in the x–y plane to provide an intersection line at the free surface. To carry out this method, the laser is oriented vertically to shine through the bottom of the tank such that the laser plane is oriented parallel to the chord line of the foil. Figures 2 and 3 show schematics of the experimental setup for this method.

In order to measure the 3D3C velocity throughout the area of interrogation, it is necessary for experiments to be run with the laser plane at varying distance from the foil. For each experiment, the foil and the laser plane are translated down the tank in the x direction and 3C velocities are measured along the 2D slice. To complete the 3D area, a series of experiments at different z locations relative to the foil were performed. This method is referred to as longitudinal slicing, since the laser plane is oriented along the longitudinal axis of the tank and the 3D volume is measured by aggregating many slices of 3C velocities.

The SPIV system uses two high-speed cameras (Phantom V10 by Vision Research) to capture the same field of view in order to compute the 3C velocities in the laser plane. The cameras are oriented such that the line of sight of the cameras is near perpendicular to the laser plane, with the center of the camera lenses below the static free surface. The box with a solid outline in Fig. 2 illustrates the region of the camera field of view for velocity field computations. As waves are generated by towing the foil, only concave curvature of the intersection line between the free surface and the laser plane below the mean free surface may be captured by the PIV cameras, since the free surface in the near field of the camera blocks the view of the intersection line for convex curvature above the mean free surface. Due to this limitation, the cameras used for SPIV may not be used to measure free surface elevations and an additional off-the-shelf digital camera (GoPro Hero Black) was used for measuring free surface elevations. The additional digital camera was placed above the free surface and oriented to capture the full curvature of the laser intersection with the free surface. The digital camera field of view is illustrated by the box with a dotted line in Fig. 2. The two camera fields of view overlap slightly, such that a visual validation check may be performed for the computed free surface elevation.

### Experimental Method 2: Two-Dimensional + T Measurements With Transverse (Crossflow) Slicing.

A second method was investigated where 3C velocities were measured at a fixed 2D spatial location, however the velocities in this plane varied in time due to motion of the foil. The method is referred to as 2D + T since the third spatial dimension of the flow is treated as equivalent to the variation in time. This method allows for a reduction in the necessary experimental time by limiting the number of experiments performed to a single experiment. The method is implemented by fixing the cameras and laser plane such that the volume is sliced perpendicular to the direction of foil motion. By assuming that the flow is near steady, a relatively safe assumption for slender bodies, it is possible to treat time as a third spatial dimension. To employ this technique, the laser is aligned vertically and orthogonal to the path of the foil. The foil is then “driven” through the laser plane and the time series of images is collected. Figures 3 and 4 show the relative positions of the foil, laser plane, and cameras used for this method. The distance between each image was computed from the frame rate of the camera and the forward speed of the foil. This method allows for the experimental test matrix for a given Froude number to be reduced to a single experiment.

### Stereo Particle Image Velocimetry System Arrangement, Control, and Calibration.

The SPIV system was a commercially available LaVision SPIV system consisting of two Vision Research Phantom V10 high-speed cameras, Photonix DM30-527PIV pulsed Nd/Yag laser, timing controller, and DaVis 8.1 control and processing software package.

The Phantom V10 high-speed CMOS cameras were capable of a frame rate of 480 Hz with a resolution of 2400 × 1800 pixels. The cameras were mounted in a traditional stereo PIV arrangement; located at the same vertical height, viewing the same field of view with an angular separation of approximately 30 deg. Each camera was equipped with Nikon lens mounts and Scheimpflug adapters. Figure 3 shows a schematic of the camera setup with the field of view.

Two different lenses were used, one for each version of the experiment. For the longitudinal experiments, Nikkor 50 mm f1.2 lenses were used. For the transverse experiments, Nikkor 105 mm f2 lenses were used to obtain an appropriate field of view. In both cases, the lenses were manually focused on the region of interest using a calibration plate to correct for lens distortion, camera angle, and image scaling. A LaVision HighSpeed timing controller was used to control the two high-speed cameras as well as the laser timing. An optical sensor mounted on the PIV carriage was used to trigger the SPIV image acquisition, allowing for precise repeatability in image acquisition for determination of the velocity fields.

The DaVis 8.1 software package by LaVision was used for control of the SPIV system, calibration, and for postprocessing of the velocity fields. A calibration plate (LaVision model 204-15) was carefully positioned with the front face of the plate in line with the laser sheet. Reference images from each camera were taken with the calibration plate in place and the images were calibrated utilizing the DaVis third-order polynomial calibration algorithm.

### Free Surface Camera Arrangement, Control, and Calibration.

Due to the interference of the near field free surface in SPIV images, it was necessary to use a separate camera to obtain free surface elevations from the intersection of the free surface with the laser light sheet. In order to visualize the free surface at a given position, a GoPro Hero4 Black camera was mounted close to the foil, above the free surface, at a 30 deg angle to capture the full wave elevation changes near the foil without obscuring the view of the free surface. In addition, for ease of repetition, the camera was fixed in relation to the laser plane, such that the camera would maintain the same view of the laser plane when the laser plane is moved relative to the position of the foil. Ideally, the laser plane should be as thin as possible to provide better resolution for the free surface measurements, however, in order to perform SPIV simultaneously to free surface elevation measurement, it was necessary to have some thickness to the laser plane. For all experiments, the laser plane was set to a thickness of 2.0 mm. Figure 5 illustrates the setup of the free surface camera in relation to the foil viewing the intersection line of the laser with the free surface.

The small, high-definition camera used for free surface measurements was capable of frame rates up to 240 Hz. For the experiments, the camera was set to record images at 240 Hz with an image size of 1280 × 720 pixels using a narrow view angle with the camera lens settings to limit distortions of the image and improve calibration of the image. Triggering for the camera was performed using the GoPro iPhone application via bluetooth, enabling an automated and repeatable control over image acquisition timed with acquisition of the velocity field.

In order to calibrate the free surface camera, a set of images showing the LaVision 204-15 calibration plate was recorded. These images were used by DaVis to complete a calibration using the DaVis pinhole camera calibration model, which assumes that light ray paths in the image can be traced back to a single pinhole location. This results in calibration of the digital pixels, such that light intensity at the free surface interface may be tracked to indicate elevation changes in the free surface. This calibration procedure was found to yield accurate measurement of the free surface elevation. To demonstrate that the calibration was accurate, the length of the foil in the image was compared to the actual chord length of the foil and overlaid in the independent images from SPIV measurement. Measurement error was indicated to be less than 1% of actual chord length. To avoid changing the location of the camera, a wireless bridge was established using the WiFi adapter of the GoPro. This allowed the images to be downloaded to the processing computer after each experiment without breakdown of the camera system.

The free surface images were synced to the velocity data in two different manners. The longitudinally sliced free surface measurements were synced to the velocity measurements by locating the foil position in the images collected, while the laser plane was on the center line of the foil. Using this information, ProAnalyst tracking lines, described below, were developed relative to this zero position. The same tracking lines were used for each slice to maintain the known position of the foil. In order to reconstruct the free surface from the transversely sliced data, a baseline t = 0 frame was established for both the high-speed camera images and the GoPro images. This was achieved by manually determining the frame where the leading edge of the foil crosses the laser plane. Once the individual frame is known, it is possible to reconstruct the field using the frame rate of the camera and the foil speed to calculate the distance for subsequent frames. The frame spacing distance is then applied to the topography measurements to create the free surface. This method can result in an error in time equal to the time spacing of the individual camera images, which translates to a possible small maximum error in positioning of the free surface relative to the velocity field of 0.4 mm.

### Motion Tracking of Free Surface.

Off the shelf motion tracking software (proanalyst) was used to determine the location of the free surface in images collected by the free surface camera. ProAnalyst can be used to track both one-dimensional (1D) and 2D feature movement within a video. In the present experiments, the intersection line of the laser sheet with the free surface represents a 2D feature, where the elevation of the free surface may be tracked by identifying 1D vertical tracking lines in the image and tracking the maximum intensity of light along the tracking line as a function of time.

The free surface appears in each frame of the video as a line of peak intensity where the laser plane intersects the free surface. This interface is approximately 7–10 pixels thick and directly proportional to the thickness of the laser plane. To determine the free surface location, each tracking line was set to detect the location of highest intensity along the tracking line. Figure 6 shows how the tracking lines are arranged along the high intensity laser plane—free surface intersection. In order to establish a baseline for the still water free surface, a tracking line was placed forward of the foil in the still water region. ProAnalyst calibration was performed using calibrated marks placed on each frame of the video by DaVis. Using the calibration marks, ProAnalyst establishes a relationship between pixel and world coordinates, in this case, 10.4 pixels per mm. The tracking function locates the highest intensity pixel along each tracking line, however, because the free surface is between 7 and 10 pixels in thickness, this leads to variability in the free surface on the order of 0.5 mm.

In addition to the free surface elevation and SPIV velocity data, foil position and velocity were also collected for each experiment. Foil position and velocity were recorded using the encoder feedback from the servo motors driving the carriage. Velocity was also independently measured using a Unimeasure HX-VP1010-200-E1-L7M spring potentiometer.

## Longitudinal Slicing Experiments

### Approach.

A time-resolved 3D velocity field is difficult to measure without a tomographic PIV system or comparably expensive technology. With only an SPIV system available, the present experiments required measurement of three-component velocities at 2D slices in separate experiments, where the average 3D field could be reconstructed from the set of experiments. In order to reconstruct the flow field volume, it was assumed that the flow field near the foil was repeatable and steady on average with minimal flow separation in the wake. A similar technique is used in Metcalf to obtain free surface elevations [9]. In Metcalf, a wave gauge was dynamically positioned in the wave field during the experiment. By averaging free surface heights in time, Metcalf was able to reconstruct the entire surface wave field. This method takes a similar approach; however, separate experiments are conducted in order to move the relative location of the light sheet for measurement of the velocity field and free surface elevation [9].

### Test Matrix.

For a slender, surface piercing body, the underwater and free surface flows are dependent on Froude number (Fr) and Reynolds number (Re). Froude number is given in Eq. (1), where U is the characteristic velocity of the flow, g is the gravitational acceleration, and l is the chord length of the foil. Froude numbers 0.37 and 0.55 were chosen in order to provide a comparison to the Froude numbers used in Ref. [9], although a different NACA section was used. For each Froude number, 15 experiments were repeated with the laser plane oriented at varying distances away from the chord line of the foil, starting at the chordline of the foil, and moving away to a maximum distance of one chord length from the chord line. Slices of the velocity field were obtained at increments of 0.035 chord lengths as shown in Fig. 7. For Fr = 0.37, experiments were performed in order starting at the chord line and moving the light sheet away from the foil with consistently increasing distance away from the foil. For Fr = 0.55, the order of experiments was randomized, such that the distance of the light sheet away from the foil was varied in a random order. This procedure ensured that systematic errors in the positioning system were not present when the results from each experiment were combined to reconstruct the average three-dimensional flow field. A sample of the reconstructed velocity field is illustrated in Fig. 7
$Fr=Ug*l$
(1)

### Velocity Field and Free Surface Calculations and Error Estimate.

Images were corrected and scaled using the calibration data obtained during the experimental setup. The SPIV calculation was performed using a 48 × 48 pixel square interrogation window followed by a 32 × 32 pixel square secondary interrogation panel to determine velocity vectors. Two passes of velocity vector calculations were performed for each interrogation window size and an overlap of 50% over the windows was used to increase the number of computed vectors. The time average and standard deviation were calculated for the vector fields based on the entire series of images. The final result yields a 2D vector field of time-averaged three component velocities at a particular distance from the center line of the foil. Combining all of the slices performed at a specific Froude number, a three-dimensional average velocity field is constructed for the flow field on one side of the foil. This same process is followed for the two Froude numbers tested (Fig. 8).

In Willert, the accuracy of PIV measurements is quantified based on the interrogation window size, the number of seed particles present in the window, and the quality of the computed correlation between images [13]. Based on the PIV algorithm characterization of Willert and Gharib, it is possible to make a conservative estimate of the error present in the PIV measurements [13]. Willert and Gharib use an interrogation window of 32 × 32 pixels to show the error fluctuation in the PIV algorithm as a function of seeding density [13]. Since a final interrogation window of 32 × 32 pixels with an average seeding density of six particles per window is used in the present experiments, a fluctuation of 0.013 pixels in the vertical dimension and 0.026 pixels in the horizontal dimension is estimated. From the calibration, 1.0 mm is equal to 8 pixel, translating to a variability of ±2.08 × 10−4 m/s in the horizontal direction and ±1.04 × 10−4 m/s in the vertical direction.

Free surface measurement data were collected simultaneously with velocity data. The free surface elevation was derived from the previously described procedures. Since the flow and subsequent free surface elevation in the image is assumed to be steady-state, the elevation of the free surface was determined by averaging the height of maximum light intensity determined on each tracking line over all frames of a particular slice. This yields an average free surface profile for a given longitudinal slice. The free surface was reconstructed by plotting each average height relative to the foil position determined from the slice along the center line of the foil. Aggregating all 15 slices, the free surface topography is reconstructed as shown in Fig. 9.

## Transverse Slicing Experiments

By slicing the velocity field parallel to the direction of foil motion, it was necessary to repeat experiments in order to reconstruct the entire three-dimensional velocity field with free surface elevations. If the phenomenon of interest does not have a large unsteady component of the wake or the body is not accelerating, one may assume a steady average flow field, hence reorientation of the light sheet can provide a three-dimensional estimate of flow field velocities assuming that spatial locations in the flow field are steady in time. In the transverse slicing method, the light sheet is oriented perpendicular to the forward motion of the foil and the light sheet remains stationary relative to the foil. The resulting three-dimensional velocity field that is measured varies in time, however for a steady flow, the time variations of the flow field in the light sheet will correspond with spatial variation of the flow field along the length of the foil. This method is referred to as a 2D + T method because two dimensions of the flow field are directly measured, while the third dimension is inferred from the time variation of the flow field in the 2D plane. This technique is common in simulations of complex shapes, such as ship hulls, for limiting the computational intensity of a simulation, but here may be used to significantly decrease the necessary experiments to reconstruct a three-dimensional flow field [14]. In the present experiments, since the light sheet does not need to be moved with separate experiments, the total number of experiments required to reconstruct the flow field is reduced from 15 to 1 experiment.

Specific to the experimental setup of the transverse slicing experiments, the laser system and cameras remained in stationary positions while only the test foil was moving. This had a significant advantage in the quality of vector fields measured, as vibrations of the carriage system do not produce additional variability in the measurement of velocity. The SPIV cameras were placed at the end of the tow tank, positioned to look down the length of the tank. The cameras were oriented at an approximate 20 deg angle away from the path of the foil with the foil 5.0 cm off of the center position between the cameras. This offset and orientation of the cameras was necessary to avoid shadows of the laser system on one side of the foil and to avoid the foil blocking the field of view in the near field. The laser plane was positioned to intersect the path of the foil, perpendicular to the foil direction, at the focal length of the camera lenses. A sample of the reconstructed velocity is found in Fig. 10.

To capture the free surface, the free surface camera was affixed to the side of the tank using a suction cup mount and flexible arm. This allowed the camera to be fixed relative to the laser plane but not interfere with the movement of the foil or track. The camera was positioned to look forward, toward the foil and laser plane intersection. This allowed the collection of free surface elevations on one side of the foil.

One challenge of this method is that the seed particles are at rest to start and the disturbance caused by the foil was very small due to the thin foil section used in the experiments. To overcome this limitation, the frame rate of the SPIV cameras was increased to 1094 Hz to better capture the displacement of the particles. The resolution of the camera was slightly reduced in order to achieve the increased frame rate, however, the effective resolution remained the same as in longitudinal slicing experiments due to the difference in camera lens. Seeding density was similar to that of the longitudinal method. The transverse slice experiments were performed as a comparison with the longitudinal method, hence tests were only performed for a Froude number of 0.37. It is important to note that with the orientation of the system for transverse slicing, the U-component velocities were measured by the stereo calculation, as opposed to the W velocities in the longitudinally sliced data.

### Transverse Method Calculations.

Image sequences were processed in the same manner as the longitudinally sliced experiments using DaVis. It was necessary to reduce the sample rate of the collected images by a factor of three based on the displacement of the particles in each frame. At the full frame rate, the displacement of the particles was not large enough to provide an accurate velocity calculation. A first pass window size of 64 × 64 pixels with 50% overlap was used. The two final passes were made with windows of 48 × 48 pixels and 50% overlap. For this experiment, the average and root-mean-square (RMS) velocities were not calculated because time is used to reconstruct the third spatial dimension for the velocity field assuming a steady flow. To reconstruct the 2D + T field of velocities and free surfaces, each image was considered to be separated by a physical distance dependent on the speed of the foil and frame rate of the cameras, as seen in Eq. (2), where Δd is the change in position of the laser location relative to the foil, U is the foil speed, and n is the camera frame rate
$Δd=Un$
(2)

Calculation of the free surface using the transverse orientation of the laser sheet was accomplished using the same technique previously described for the longitudinal slice method. The entire free surface elevation map was reconstructed using the same transformation in Eq. (2) to translate time to spatial dimensions, assuming an average steady free surface.

## Results

### Longitudinal Slicing.

The component velocities U, V, and W resulting from the longitudinal slicing method for Froude numbers 0.37 and 0.55 are shown in Fig. 8 for a representative depth of y/c = –0.334. The U velocity for all Froude numbers shows a region of increased speed at the location of maximum thickness as expected. V velocities indicate velocities in the downward direction from the leading edge of the foil transitioning to upward velocities at the trailing edge. This is consistent with wave formation on the free surface. W velocities show that the fluid is moving outward from the foil at the leading edge and toward the foil at the trailing edge as expected from the thickness of the foil directing the flow around the foil. The reconstruction of the full average velocity field shows a smooth connection between velocity magnitudes and directions over the entire data space, indicating that the experiments were repeatable and consistent.

Figure 11 shows the variation in velocity with depth for Froude Number 0.37 as an example case. The free surface is located at y/c = 0. Near the free surface, the U-component velocity is highly variable in the presence of the distortion of the free surface. As the depth increases, the U-component establishes a consistent trend that does not significantly vary with depth. This is expected as the chord dimensions of the foil do not vary with depth, hence distortions are only expected near the free surface. The W-component of velocity follows a similar trend as expected. The vertical velocity component, V, is highly variable near the free surface, with decreasing magnitude in the velocity with depth. Again, this is consistent with strong wave formation at the free surface and decay in free surface effects with depth.

Free surface elevations for each Froude number tested are shown in Fig. 9 based on the longitudinal slicing technique. The longitudinal slicing method has a significant deficiency in measuring the free surface very close to the foil section as the intersection of the light sheet with the free surface is difficult to discern, particularly when the light sheet is nearly parallel with the surface of the foil where the sheet intersects with the foil. This results in a distorted free surface and spurious reflections, such that the measurement of free surface in the near vicinity of the foil must be discarded.

The free surface elevations farther from the foil (greater than 0.1 chord length away) show a strong comparison with the similar flow fields obtained by Metcalf et al. [9]. Although the magnitudes of the free surface cannot be directly compared, similarities exist in the wavelength and shape of the free surface in the formation of the Kelvin train.

### Transverse Slicing.

The transverse slicing technique was only performed for a single experiment at Froude number 0.37, so results are only given for a single Froude number. The velocity components obtained through the transverse slicing method are shown in Fig. 12 for Froude number 0.37 at a depth of y/c = −0.334 in order to make a direct comparison with results from the longitudinal slicing method. The transverse method offers greater resolution in measurement of the velocity field due to the increased frame rate and resulting field of view. Each velocity component shows a similar spatial distribution to the velocities observed in Fig. 10; however, some differences exist due to different methods used to obtain the velocity field. One advantage of the transverse slicing method is that with the perpendicular orientation of the light sheet, the intersection of the light sheet with the foil does not produce strange artifacts or increased reflection, such that the velocity field can be measured very close to the foil surface. The high resolution in time results in a high resolution in the available vector fields used to reconstruct the flow field in the x-direction. The velocities shown in Fig. 12 are not averaged in time, since time is used to obtain the third spatial dimension. This results in a higher variability in the velocity field that was not seen in the time averaged fields obtained through the longitudinal slicing method.

Free surface elevations obtained through the transverse slicing method are shown in Fig. 13 (middle). The observed free surface elevations are consistent with those measured using the longitudinal slicing technique (shown in Fig. 13, top) and the shape of the free surface compares well with similar experiments from Metcalf et al. [9]. Due to the increased resolution in the x-direction from the high frame rate and improved ability to measure the free surface near the foil with perpendicular orientation of the laser sheet, the free surface elevations are well defined near the foil. The measurement does not include as much noise as with the longitudinal measurement, as the high frame rate gives good resolution of the free surface elevation. With the NACA 0012 section being thinner than Metcalf, there is significantly less variability in wave elevations due to breaking at the leading edge, resulting in little fluctuation of the measurements in time [9]. The root-mean-square difference (RMSD) was calculated between the two methods resulting in a maximum difference of 7% as seen in Fig. 13, bottom.

## Discussion

The two tested methods for obtaining the velocity field and free surface elevations should yield matching results for both velocity and free surface elevations; however since the transverse slicing method is not time averaged with a single experiment, some discrepancies may exist. Two techniques are used to compare the two experimental sets. First, a qualitative comparison can be made through direct comparison. Figures 9 and 12 show similarities in the velocity distribution near the foil at the depth shown. In both methods, the velocities follow an expected shape for flow over a slender body; however, differences in resolution, differences in the direction for which the stereo calculation of velocity is obtained, and differences in time averaging lead to some differences in the observed velocity fields and free surface elevations. SPIV is least accurate in resolving the velocity of the flow traveling in the normal direction to the laser plane since this direction requires that particles remain in the field of view within the thin light sheet and slight misalignment in calibration can lead to large errors in the estimated velocity component. In the longitudinally sliced data, the W velocities are normal to the laser plane and are expected to have the most error. In addition, these velocities are smallest in magnitude which increases the relative error. In the transversely sliced data, U velocity measurements are normal to the laser plane and are expected to have a larger error. With the larger magnitude velocities in this direction, the frame rate of the camera was increased to properly capture these velocities and the relative error is less due to the larger velocity components in this direction.

Differences existed in the SPIV processing of the velocity fields. The longitudinally sliced data had a final processing window of 32 × 32 square pixels and the transverse sliced data were processed with a final window size of 48 × 48 square pixel size, both with a 50% overlap in window, due to the different resolutions and seeding particle densities in the images. As determined in Ref. [13], window size does not have a direct effect on accuracy of velocity measurements; however, particle density in a particular interrogation window can impact the accuracy of velocity estimates. As the seeding is similar for each data set, the larger window size of the transverse data set allows for more particles to be present in each interrogation window, resulting in less error expected for the SPIV algorithm. To quantitatively compare the two methods, the RMSD was calculated using Eq. (3), where L denotes the longitudinal sliced data, T denotes transverse data, and N is the total number of data points. U, V, and W are the component velocities with L and T indicating the longitudinally and transversely sliced data point, respectively. The RMSD and RMS of time averaged data from the longitudinally sliced data is shown in Table 1. The table divides the comparisons into three separate flow regions: flow forward of the leading edge, flow along the chord length of the foil, and flow aft of the trailing edge. These separate distinctions are made due to the nature of variability in the two data sets. Since the transverse sliced data are not time averaged, one may expect more variability in RMSD in the wake region, where the flow is less steady and time dependence would contribute to spatial variability of the velocity. Similarly, flow forward of the leading edge is expected to be steady, hence one would expect small RMSD between the two methods forward of the leading edge
$RMSD=Σ1N(UT−UL)2+Σ(VT−VL)2+Σ(WT−WL)2N$
(3)

Table 1 gives comparisons of both RMSD and the variance of the time average for the longitudinally sliced data, denoted as RMS in the table. The measure of RMS from the time-averaged longitudinal data gives an estimate of the variability in the computed average, while the RMSD gives a direct comparison of the variability between the two experimental methods. In general, RMSD is fairly low (less than 5%) difference regardless of location and velocity component, except for a few regions of the flow. Near the free surface, there is a larger variability in the computed U and W components of velocity in the region near the foil as seen in Table 1. In the experiments, near the free surface, particles that were slightly buoyant had a tendency to collect near the free surface. This produces a large backscatter in light intensity close to the free surface, which can impact the cross-correlation calculation in determining velocities. This effect was more apparent in the longitudinal sliced data, which also helps to explain the large variability in RMS computed near the free surface. Since the V component velocities were very small, regardless of the method used, the variability in this component between methods was not observed to be large, even near the free surface.

At depth, the variability in measurements is significantly reduced, such that the two methods produce very similar results, with fairly small differences observed between the two methods. This confirms that at depth, the assumption of steady flow in the transverse slicing method is reasonable as it returns velocity components very similar to the average velocities measured by the longitudinal method. To further illustrate the steadiness of the flow field, Fig. 14 shows the RMS variability for each velocity vector component over the entire data space at the representative depth of y/c = −0.334 and for each Froude number tested. This indicates that time dependence to the flow is not strong and a steady flow assumption is valid. The transverse method provides increased resolution of the component velocities, in addition, there is a difference in spatial resolution between the two methods. The longitudinal slices were taken in increments of 0.036 * CL (where CL is the Chord Length) from the foil, whereas the transverse method resolves the entire volume in increments of 0.014 * CL which is limited by frame rate.

### Verification of Free Surface Measurements.

Although the free surface measurements compare well qualitatively with similar experiments from Metcalf, the large angles of the digital camera relative to the intersection line of the laser on the free surface require independent verification of the free surface measurements [9]. Since images were simultaneously obtained through the SPIV system, the SPIV images may be overlaid with the free surface estimations for comparison of the free surface position. In the SPIV images, the free surface position can only be identified for curvature below the free surface, due to blockage from the near field free surface, however, the visible free surface may still be used as verification of the method.

Figure 15 shows an example raw image from SPIV measurements with the independently measured free surface slice for this image overlaid on the image (image is for a longitudinal slice 0.11 chord length from the center line of the foil for Fr = 0.37). The calculated free surface matches well with the illuminated free surface from the light sheet intersection, verifying that the general curvature of the free surface is properly captured with this technique.

To illustrate the interaction of the velocity field with the free surface, Fig. 16 shows a sample image of the free surface overlaid with the velocity field in an x–y slice of the field. The image shows the U-component of velocity at the same location as Fig. 15. One can see from the velocity field that as the free surface height increases, the U-velocity component decreases, while at depth the U-component of the velocity is nearly constant with a slight increase in the vicinity of the foil. The foil leading edge is located at x/c = 0.

## Conclusions

The present experiments demonstrate that using a standard SPIV system with an additional off-the-shelf digital camera, it is possible to simultaneously measure the three-dimensional velocity field and free surface elevations for a steady flow. The present experiments demonstrate this technique on the flow around a NACA 0012 foil section oriented with zero angle of attack. This method requires considerably less expensive equipment than other three-dimensional visualization techniques. For a steady flow, the transverse slicing method is observed to considerably reduce the number of experiments necessary to perform this measurement. While the longitudinal slicing method required a significant number of experiments (15 experiments) to achieve a reasonable resolution of the velocity field, the transverse slicing method achieved a higher spatial resolution using a single experiment. This significantly reduces the cost of performing this type of validation experiment, while providing an improved measurement. Additionally, the transverse slicing method allowed for measurement of the velocity field and free surface very close to the surface of the foil, in contrast with the longitudinal slicing method. While it was not done in the present study, the repetition of experiments using the transverse slicing method could give an ensemble average velocity field to characterize variability between experiments and variability in the flow field that is unavailable from a single experiment.

Based on the results of both methods, the transverse (2D + T) method proves best suited for an economical measurement of free surface elevations and 3D3C velocities surrounding a surface piercing foil. The transverse method allowed the number of experiments to be reduced to one in this case. In more complex experimental cases, such as a foil at an angle of attack or flow around an asymmetrical body, additional experiments would be needed to characterize the full flow field around the body. This could be done with two experiments and with the servo-controlled towing tank used in the present experiments, this could simply be accomplished with a change in the angle of attack of the foil, while the SPIV system remains unchanged, requiring only a single calibration. The reduction of the number of experiments brings a significant savings in time and computing expense. The longitudinal method, although similarly accurate, requires many more individual experiments and corresponding processing time. The longitudinal method is useful for measuring time-averaged data when the body under investigation is symmetrical and the flow is steady with no separation. If the longitudinal method is used on an asymmetrical body, the number of experiments and processing time required to develop the volume is doubled as SPIV must be carried out on both sides of the body, leading to a significant increase in experiment and computational time.

## Funding Data

• Office of Naval Research Global (Grant No. N000141310687).

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