USING STEREO PHOTO MEASUREMENTS TO ANALYZE THE SURFABILITY OF SHIP INDUCED WAVES
S. DE VRIES (1), M.A. DE SCHIPPER (2), J.S.M. VAN THIEL DE VRIES (3), W.S.L. UIJTTEWAAL (4) and M.J.F. STIVE (5) (1) MSc., Delft University of Technology, Stevinweg 1, Delft, 2628 CN, The Netherlands. Sierd.deVries@tudelft.nl (2) MSc., Delft University of Technology, Stevinweg 1, Delft, 2628 CN, The Netherlands. M.A.deSchipper@tudelft.nl (3) MSc., Delft University of Technology, Stevinweg 1, Delft, 2628 CN, The Netherlands. J.S.M.vanThieldeVries@tudelft.nl
(4) PhD. MSc., Delft University of Technology, Stevinweg 1, Delft, 2628 CN, The Netherlands. W.S.J.Uijttewaal@tudelft.nl (5) Professor, PhD., MSc., Delft University of Technology, Stevinweg 1, Delft, 2628 CN, The Netherlands. M.J.F.Stive@tudelft.nl
The objective of this paper is to demonstrate the use of Stereo Photogrammetry (SP) measuring ship-generated waves and to determine under which conditions they are suitable for surfing. A physical experiment is conducted to gain insight in ship wave behavior under specific conditions. Waves are generated by a hull and are measured using SP. It is concluded that the SP technique holds great potential in laboratory use and ship waves can potentially be used for surfing purposes.
Keywords: Stereo Photogrammetry; Wave field measurements; surfing; Ship waves.
1. Introduction
In recent times surfing is gaining popularity. Good waves however are scarce, especially in urban environments where the majority of people live. To satisfy the need for quality surfing waves, attempts have been made to construct surf pools in which waves are artificially generated and forced to break over a bottom topography.
The idea of generating ship waves in a circular pool is patented by Australian surf board designer Greg Webber with the aim of building a surf pool. The pool consists of a circular channel in which waves are generated along the outer ring by a hull. The generated waves propagate through the channel and eventually break on the slope of the inner ring. The breaking waves can be surfed along the entire circle creating, in theory, an endless wave. For an artist impression of the wave pool see figure 1. The pool design requires surfable waves of 2 meters height.
Figure 1. Artist impression of the wave pool concept (illustration by Lew Keilar).
Surfing waves are ideally ‘clean’, i.e. narrow spectrum waves not affected by currents. An important surfability parameter is the peel angle (Henriquez, 2004). The peel angle (α) is the angle between the breaker line and the wave crest and determines the surfer speed, see figure 2 left. A large angle corresponds to a slow breaking wave (for beginners) whereas a small angle produces a fast breaking wave, which is more challenging for advanced surfers. Hutt et. al. (2001) determined that surfable waves should have peel angles ranging between 40° and 60°. In case of the wave pool where a hull is used to generate waves, peel angles are predominantly determined by the wave angle generated by the hull, see figure 2 right. Therefore the focus of this paper is particularly on the wave generation by a hull where ship wave angles and induced currents are of interest. For a description on ship wave heights and numerical modeling of the wave pool reference is made to the work of De Schipper (2007).
Ship waves are qualitatively described as early as 1891 by Lord Kelvin (Thomson, 1891). Ship waves can be divided in a primary and secondary wave system. The secondary wave system is generated by pressure gradients caused by the discontinuities in the hull. These discontinuities are found at the bow and at the stern, both of which emit waves. The emitted waves form transverse and diverging waves. Interference of these transverse and diverging waves lead to the typical wave pattern called the Kelvin wave pattern, see figure 3 right. The angle (θ) between the interference line of the transverse and
diverging waves and the sailing line can be derived to be 19°. Wave angles (φ) of individual waves are generally 55°. Note that these angles are independent of the ship’s speed in deep water. The primary wave system consists of the bow and stern wave and the local water level set down due to increased water velocities beside the hull, see figure 3. In open water the primary wave system is mainly present in the vicinity of the hull whereas in restricted channels the primary wave can dominate over the entire channel.
Figure 2. Left shows the definition of the peel angle. The wave celerity (c) and the peel angle (α) determine the surfer speed (Vsurfer). Right shows that the
peel angle of the wave is highly determined by the wave angle generated by the ship. The color gradient represents the slope on which the wave breaks.
In restricted channels the return flow around the hull can be significant depending on the ship speed (Vs) and blocking percentage of the channel. Using Schijf's theory (Schijf, 1949) the characteristics of the ship wave patterns can be predicted. Sub-, trans- and super-critical regimes are defined depending on the ship's speed and blocking percentage of the channel. When blocking and ship speeds are low, the return flow (Ur) is weak and the regime is sub-critical. When blocking and ship speeds are higher the return flow can reach critical values. The theoretical threshold of the trans-critical regime, where return currents become critical, can be calculated using the vessel speed and blocking percentage. Figure 4 shows the dimensionless limit hull speed as a function of the blocking coefficient as calculated according to Schijf’s theory. In the trans-critical regime, critical return currents account for a dominating primary wave system. In the sub-critical regime the primary wave system is non-existing and the secondary wave system is dominant. The super-critical regime occurs when the ship speed itself becomes critical. In practice the hull would than plane on the water surface. Due to practical limitations concerning the wave pool design the super critical regime is not of interest.
Figure 3, Left shows the primary wave system consisting of bow and stern wave together with the water level set down due to return currents (Ur) around the
hull (Vs represents ship speed). Right shows the secondary wave system with the angle of interference (θ=19°) and wave angle (φ=55°) depicted.
A numerical model to fully predict ship induced wave characteristics is found to be not available for both sub-critical and trans-critical conditions (De Schipper, 2007). To gain insight in the generation of ship waves with respect to surface elevation and flow velocities an experiment is conducted. Stereo Photogrammetry (SP) is applied as a measurement technique in the experiment. This study investigates the possibilities of SP to reconstruct water surface elevations and flow velocities. Traditional measurement devices measure surface elevation and flow velocities at one point only whereas SP shows a high-resolution spatial representation of both. The obtained data can be used to verify future numerical models.
Summarizing, the objectives of this paper are: a) to investigate the surf-ability of ship induced waves and b) to investigate the applicability of SP measuring ship induced wave fields.
Vs
Ur
Sailing line
Wave crest Broken wave
Breaker line α α α Vsurfer c
Figure 4, Dimensionless limit hull speed as a function of blockage.
2. Experiment
The experiment is conducted in the towing tank facility of the Faculty of Mechanical, Maritime and Materials Engineering at Delft University of Technology. The tank is rectangular, 80m long and 2.75m wide and the water depth is 0.3 m. A ship's hull is towed along the sidewall of the tank generating a wave field. Ship speed and blocking percentage of the channel are varied during 24 runs. Schijf’s theory is used to estimate the relevant towing speeds and blocking percentage with respect to the sub- and trans-critical regimes. Blocking percentages where varied changing hull widths and drafts and ranged from 4% to 29.2 %. For a full overview of tests see table 1
Table 1, overview of all runs Run Towing speed
[m/s] Fr,d,hull [-] Regime* Blocking = 12.6% (Vlimit gd = 0.58) 1.1 0.46 0.27 Sub 1.2 0.70 0.41 Sub 1.3 0.82 0.48 Sub 1.4 1.05 0.61 Trans 1.5 1.28 0.75 Trans 1.6 1.65 0.96 Trans Blocking = 8.5% (Vlimit gd = 0.65) 2.1 0.66 0.38 Sub 2.2 0.64 0.37 Sub 2.3 1.04 0.61 Sub 2.4 1.27 0.74 Trans 2.5 1.30 0.76 Trans 2.6 1.52 0.89 Trans 2.7 1.71 1.00 - 2.8 2.17 1.26 Trans Blocking = 4.0% (Vlimit gd = 0.76) 3.1 0.49 0.29 Sub 3.2 0.61 0.36 Sub 3.3 0.83 0.48 Sub 3.4 1.04 0.61 Sub 3.5 1.30 0.76 Sub/Trans 3.6 1.48 0.86 Trans 3.7 1.60 0.93 Trans 3.8 1.74 1.01 Trans Blocking = 29.2% (Vlimit gd = 0.37) 4.1 1.11 0.64 Trans 4.2 1.30 0.75 Trans
Wave fields in the sub-critical and trans-critical regime are generated where surface elevation and flow velocities are of interest. The generated waves are measured using SP. Therefore the wave field is photographed by two (time) synchronized cameras from different positions, see figure 5. The cameras are mounted above the towing tank and photograph the wave field as the hull passes. In the measurement area floats are applied to increase contrast at the water surface. The cameras are triggered at 8Hz. The measurement area covers the entire flume width of 2.75m and about 5 meters in the length direction of the flume. For verification purposes there are also resistance type wave gauges in the flume.
Figure 5. 3D overview of the camera setup. Arrow indicates direction of hull movement
3. Stereo photogrammetry
The images from the two cameras with overlapping fields of view are used to determine a 3D presentation of the wave field in world coordinates. The procedure of stereo reconstruction is to calibrate the cameras, to correlate the two images and to triangulate the 3D world coordinates, see figure 6. The method used for calibration, correlation and triangulation is similar as the method described by Clarke et al. (in prep), a brief description is given in this paper.
Figure 6. 3D point triangulation. The projection x on the image plane of Camera 1 determines a straight line. Given the cameras are calibrated, this straight line can be shown as an epipolarline l′ on the image plane of Camera 2. When on the epipolar line l′, x′ is defined, point X can be reconstructed.
Camera hardware and calibration procedures are adapted from the ARGUS coastal monitoring system (Holman & Stanley 2007). Thorough calibration is important because the accuracy of the triangulation is highly determined by the accuracy of the calibration. The calibration is done for each camera independently in two steps. First the internal camera parameters (e.g. focal length and distortion parameters) are determined by taking images of a reference grid. Second, the external camera parameters (camera location and orientation) are determined measuring the camera positions and 15-16 Ground Control Points (GCPs) relative to a reference frame. Using the camera positions, the GCPs and internal camera parameters the cameras orientations are calculated. With the internal and external camera parameters determined the cameras are calibrated.
The image correlation technique is based on rectified images where the constraint of epipolar geometry is used. The intersection of feature on the image plane (e.g. a pixel) and the camera centre from one camera resembles a (epipolar) line on the other cameras image plane, see figure 6. On this epipolar line the same feature is to be expected. When rectifying the images, images are virtually rotated so that epipolar lines match. Now an automated correlation can be limited to the epipolar and neighboring lines, which saves calculation time and excludes irrelevant (and false) matches.
For triangulation a linear triangulation method described in Hartley and Zisserman (2000) is used. This method estimates the most likely 3D position governed by back projecting correlating pixels of the 2D images.
In addition to the techniques described by Clarke et al., a Particle Tracking Velocimetry (PTV) technique is used to derive surface flow velocities. This technique isolates individual or clustering floats for one camera on the water surface. The image features are then correlated with the consecutive image of that camera in time. A time series of the images from one camera is cross-correlated to determine the 2D spatial movements of the features in the image. Using the movements in the image and the prior knowledge of the known 3D coordinates corresponding to the 2D image coordinates 3D velocity vectors can be determined.
Additional information on the actual camera calibration parameters and procedures can be found in De Vries (2007).
Epipolar line l’
Camera 1 Camera 2
X
With the knowledge of surface elevation, and thus the local water depth, together with the local flow velocities a spatial distribution of the local Froude numbers can be derived according to
[1]
Where Vcurrent is the current velocity relevant to the hull, d is the local water depth and g the gravitational acceleration. The spatial distribution of local Froude numbers can be used to verify the use of Schijf’s theory when predicting sub- and trans-critical regimes.
4. Results
The experimental data processed using the SP technique results in 3D images of the surface elevations and surface flow velocities, figure 7 left and right show such images. Over all processed images, surface waves in the order of 5-20 cm are measured. Triangulation errors are estimated to be 1-2 cm based on the size of the image pixel stamps in world coordinates. Velocities in the order of 0-2 m/s are measured.
Figure 7 left shows the dominant primary wave in the trans-critical regime. The primary wave, i.e. the return flow, the stern wave and the water level set down next to the hull are clearly visible. With respect to the trans-critical regime in all cases no wave train is distinguishable and the measured wave angles relevant to the primary wave range between 30 and 42 degrees. Surface flow velocities appear to be large around the water level set down besides and behind the hull.
Figure 7 right shows an example run of the sub-critical regime. A wave train of limited height is clearly visible behind the hull. With respect to the sub-critical regime, the measured wave angles are between 50 and 60 degrees which are in line with the theoretical 55 degrees. Surface flow velocities do not show a coherent pattern and are relatively small compared to the towing velocity. The maximum velocity measured is 0.33 m/s and is less than half the towing speed for that particular run.
Figure 7. Images of the surface elevation of a wave field generated by a hull. The tank wall is indicated in light gray and the moving hull is depicted in dark gray at the top right corner. The arrow indicates the direction of hull movement. White vectors represent flow velocities. Left image shows the trans critical regime where the primary wave dominates and surface flow velocities are relatively large. Right image shows the sub-critical regime where a wave train is
distinguishable and surface flow velocities are relatively small.
A time series of the images (at a sampling frequency of 8 Hz) is used to generate a time series of the surface elevation on specific positions. These positions are chosen to refer to wave gauge positions and a comparison between SP and wave gauge data is made. The comparison of the two measurement techniques is shown in Figure 8. Although there might need some improvement (on behalf of both techniques) one can argue that the signals correspond well.
Figure 8. Comparison plot between measurements of the stereo photo technique (8Hz) and a traditional wave gauge.
current
V
Fr
gd
=
Surface elevations and surface flow velocities for the different runs are analyzed. The local Froude number distribution is used to identify the sub-critical and trans-critical regimes. It is found that the occurrence of sub- and trans-critical regimes can be successfully determined in accordance with Schijf's theory. To do so, the local Froude number needs to be corrected to take flow velocity relative to the hull into account. Moreover, the measurements show that in the trans-critical regime the primary wave system is dominant and in the sub-critical regime the secondary wave system is dominant, see figure 8.
The wave angles of the primary wave system are too small to serve surfing purposes, furthermore the currents are high and there is no wave train of consecutive waves. This makes the trans-critical regime unsuitable for surfing. The secondary wave system includes suitable wave angles relative to the desired peel angles, also there is a wave train of consecutive waves and currents are small. This makes the secondary wave system most suitable for surfing purposes.
Figure 8, Derived dominating primary and secondary wave fields plotted against schijf ’s theory. The gray and white area account for the sub-critical and trans-critical regime respectively according to Schijf. * markers show data point where a dominating secondary wave system is observed and + markers show
data points where a dominating primary wave system is observed.
5. Conclusions
After evaluating all images for different runs it is concluded that:
1. The Stereo Photogrammetry technique offers great potential to measure surface elevations. Cameras need to be calibrated and contrast needs to be added to the water surface. Measured surface elevations using SP agree well with measured surface elevations of resistant type wave gauges at similar locations.
2. Using a time series of the stereo photos, 3D surface flow velocities can be derived. Using the images, 2D velocities of features on the water surface in the image can be determined. These 2D velocities can be converted to 3D velocities with the a priori knowledge of the 3D reconstruction.
3. The trans-critical regime accounts for a dominant primary wave. This dominant primary wave involves high surface flow velocities and relatively small wave angles. The sub-critical regime shows a dominant secondary wave field with low flow velocities. The 55° wave angle in the sub critical regime is similar to the desired peel angles. Therefore it is concluded that the sub-critical regime is potentially suitable for surfing and the trans-critical regime is not suitable.
References
Clarke, L., Van Thiel de Vries, J.S.M., Holman, R., “High Resolution Morphology from Stereo Video Cameras.”, in preparation
De Schipper, M.A., “On the generation of surfable ship waves in a circular pool, Part I: Physical background & Wave pool design.” Hydraulic Engineering. Delft, TUDelft. MSc: 64, 2007.
De Vries, S. “On the generation of surfable ship waves in a circular pool, Part II: The application of Stereo Photo Technique measuring water surface elevation and surface flow velocities.” Hydraulic Engineering. Delft, TUDelft. MSc: 68, 2007.
Hartley, R. and A. Zisserman. “Multiple view geometry in computer vision.” Cambridge university press, 2000.
Holman, R. A. and J. Stanley., "The history and technical capabilities of Argus." Coastal Engineering 54(6-7): 477-491, 2007.
Hutt, J. A., K. P. Black, et al. “Classification of Surf Breaks in Relation to Surfing Skill.” Journal of Coastal Research(Special Issue): 66-81, 2001.
Schijf, J. B., “Influence of Form and Dimensions of the Cross-Section of the Canal, of the Form, of the Speed and the Propulsion System of Vessels”, XVIIth PIANC, section 1, subject 2, Lisbon, 1949.
Thomson W. (Lord Kelvin), “Stochastic nonlinear shoaling of directional spectra”, Popular lectures and Addresses, Volume III, 79-99, 1891.
