Free surface measurement by stereo-refractionltces.dem.ist.utl.pt/lxlaser/lxlaser2012/online_cd/... · Abstract An optical method for the measurement of the surface topography, the

$Page 1: Free surface measurement by stereo-refractionltces.dem.ist.utl.pt/lxlaser/lxlaser2012/online_cd/... · Abstract An optical method for the measurement of the surface topography, the$
16th Int Symp on Applications of Laser Techniques to Fluid Mechanics Lisbon, Portugal, 09-12 July, 2012

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Free surface measurement by stereo-refraction

Guillaume Gomit1,*, Damien Calluaud1, Ludovic Chatellier1, Laurent David1

1: Institut Pprime, CNRS, Université de Poitiers, ENSMA, Chasseneuil, France

* Correspondent author: [email protected] Abstract An optical method for the measurement of the surface topography, the height, the normal, and the velocity in free surface flows is presented. This method of surface reconstruction is based on the analysis of the apparent displacement between a reference and refracted images of a laser sheet through an interface view by two viewpoints. This technique is applied in a towing tank to measure the flow in the wake of a ship model. The results allow relating the shape of the surface with the 2D vector fields under the surface in the wake of a ship model. The instantaneous measurement of free surface allows reconstructing the global wave field pattern behind the ship. These results, obtained with a non-intrusive technique and with a high spatial resolution, allow an accurate analysis of the flow behind the ship. Tests were conducted for different Froude numbers in a range between 0.24 and 0.52. The results highlight the effect of the Froude number on established waves system. 1. Introduction The measurement of free surface has multiple applications in the field of research in fluid mechanics. With the aim of analysing free surface flows, several techniques of surface measurements by optical means have been developed over the past three decades. These optical measurement methods have the advantages of being non-intrusive and have the ability to measure on an area and not at a punctual location. The method of free surface measurement describes in this paper is based on the refraction of light. Several techniques using the refraction of light have been developed by different research teams. Zhang and Cox [1994] developed a method for the measurement of the two components of the surface gradient using a source screen coloured under the surface and a large lens. The analysis of coloured images recorded with a camera located above the surface allows to determine the surface gradient. From the surface gradient, the elevation of the surface can be calculated by integration. This refraction slope detector has been used to measure statistical properties of short wind waves. More recently, Fouras et al. [2008], Ng et al. [2010] or Moisy et al. [2009] measured local slope of the interface by comparing reference and refracted images of a random pattern. Moisy et al. [2009] analyse the apparent displacement between a refracted image and a reference image obtained when the surface is flat using a digital image correlation algorithm. In this study, the reference pattern is a random dot pattern located under the transparent tank used for the experiments. With this method, they measured wavy surface and circular wave surface generated by a water drop impact. With a similar principle of measuring the apparent displacement between reference and refracted image, Fouras et al. [2008] and Ng et al.[2010] proposed a technique capable of simultaneous measurement on both free surface topography and the velocity vector field of free surface flows. They used the refraction visualization associated with PIV measurements. A laser sheet replaces the random dot pattern and a camera below the water surface provides the reference image. The gradient surface is estimated by a PIV analysis between the refracted and reference image. The authors applied this method to the study of the wake of the cylinder at low Reynolds number. These two approaches use Snell's law of refraction and hypothesis on the camera geometry to compute the gradient of the surface from the apparent displacement of the pattern trough the interface. In these techniques, an


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integration step is then necessary to estimate the height of the surface. Morris and Kutulakos [2005] proposed a method, called Dynamic Refraction Stereo, based on light refraction and on multi-view system to reconstruct the 3D position and normal of points on an unknown refractive surface. The authors showed that two viewpoints of the apparent displacement of a known pattern are sufficient to solve this problem. They reconstruct a moving surface from the apparent displacement of a checkerboard placed below the interface. This method, based on two viewpoints avoids the integration step. This is an advantage, in particular in the case of a sudden change in the average level of water, which can be problematic for the integration. This paper presented a new method for free surface measurement derived on the principle of the refraction of light proposed by Morris [2005] and applied on the apparent displacement of a laser sheet. This technique allows to reconstruct the position and the normal of the surface and the velocity field below the interface from particles images velocimetry algorithm. This method is used to measure simultaneously both the surface and velocity in the free-surface flow in the wake of a ship model in towing tank. 2. Light refraction in stereoscopic configuration The method proposed here is based on the apparent displacement on a reference pattern through a deformed free surface. This deformation due to light refraction follows the Snell-Descartes law and can be written in vector form as follows:

𝜂!"# 𝒊×𝒏 = 𝜂!"#$"% 𝒓×𝒏 (1)

where ηair and ηliquid are the refractive index of air and liquid respectively, i the normalized incident ray of light in air and r the normalized refractive ray in liquid and n the normal at the surface. These three vectors are coplanar. Using the Snell-Descartes law, the normal vector can be written as follows:

𝒏 = (𝛽𝒊 − 𝒓)/ 𝛽𝒊 − 𝒓 (2)

where β is equal to ηair / ηliquid . Assume an incidence ray from the centre of a camera, C, to a point P(x, y ,z), the image seen by the camera is the point on the reference pattern located in X(x', y', zref), Figure 1.a. Knowing points C, P and X, the incident and refracted vector can be computed:

𝒊 = !!!!!!

, 𝒓 = !!!!!!

(3)

Given an incident and a refractive vector, the normal vector to the hypothetical surface, n, can be computed with equation (2). However, for an incident vector i and a given point X on the reference pattern, there are an infinity of couples P(x, y, z) and n as shown on Figure 1.b. To close the problem, a second point of view is required. This second point of view (C2), Figure 1.c, provides a second incident vector i2 passing by the point P(x, y, z) supposed on the surface. The apparent displacement view by C2, the point X2, allows to estimate the refractive ray r2. From incident and refracted rays, the normal vector n2 can be computed with equation 2. The second camera allows to check whether the point P is on the surface or not. If the point P is located on the surface, the two normal vectors n1 and n2 are equal. A criteria of equality between


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the normal vector is used to verify if the hypothetical point P is on the surface. Here we use the criteria proposed in Morris [2004] and defined as follows:

𝐸! = cos−1(𝒏𝟏 ∙ 𝒏𝟐) (4)

When the two vectors, n1 and n2 , become collinear, En goes to zero.

a) b) c)

Figure 1 Light refraction in a bi-dimensional case: a) Refraction of a incident ray. b) Multiple solution given a incident ray i and the refracted point X. c) Stereo refraction configuration

3. Experimental requirements and reconstruction algorithm This method is based on the knowledge of a reference pattern located below the surface and its deformed image recorded by two points of view above the free surface. In this study, the reference pattern is created with a laser sheet that illuminates seeded particles in the flow. One camera below the tank provides the reference image and two cameras above the surface provide the deformed state of the laser sheet through the interface. The apparent displacement between this reference image and the deformed images is computed with a PIV algorithm. The reconstruction of the surface requires the calibration of the three cameras in the same physical volume of reference. Indeed, to compute the optical rays from the apparent displacement of the pattern, the cameras above the surface and the camera below the tank must be calibrated in the same coordinate system. The camera below the tank is calibrated bi-dimensionally in the plane of the laser sheet, and the two cameras above the free surface are calibrated tri-dimensionally in the same reference. This calibration is performed by translating a calibration target perpendicularly to its plan to create a set of three dimensionally distributed markers [Tsai 1987]. The reconstruction algorithm for solving the surface can be summarized as follows: 1: Given a position (u1, v1) in pixel on camera 1, an incident vector i1 is computed using the calibration relationship. 2: Assuming a point at a height z on the ray define by i1, the point P(x,y,z) can be calculated 3: From the apparent displacement in (u1, v1) obtain by image correlation, the point X is determinate, and the vector r1 can be estimated. 4: With equation 3, the vector n1 is calculated.


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5: Given the point P(x,y,z), the steps (3,4) are applied to the second camera located above the surface and the vector n2 is calculated. 6: The optimization criteria 𝐸! define by equation 4 is estimated for the assuming z. 7: A process of optimization is used to minimize 𝐸! as a function of z on the ray i1

This process is performed on each pixel located on a Cartesian grid on the image of the camera 1. The apparent displacement is computed with classical a PIV algorithm. However, if we apply the algorithm directly to the original images, the correlation can be difficult because of large displacements and deformations due to refraction cause by the angle of view. Figure 2.a shows a result of apparent displacement computed on the original images. The value of displacement in pixel is large, more than 60 pixels, and introduce error in the vector map. An important component of the apparent displacement is due to the angle of view of the camera. This effect can be removed by applying a transformation on the images of the camera located below the tank. This transformation is performed calculating the apparent displacement when the surface is at rest, i.e. 𝑧 = 𝑧!"# and 𝒏 = 0,0,1 . Assuming z and n known, the relationship between a pixel of the camera located above the surface and a pixel of the camera below the surface can be easily computed from the equation of refraction. Each pixel of the camera below the tank can then be projected onto the image reference of the two cameras above the interface assuming the surface at rest. This transformation is equivalent to keep only the displacement due to the deformation of the interface by subtracting the displacement due to the angle of view. Figure 2.b shows the result of correlation after the transformation of images. In this case, the displacement computed by correlation is smaller, of the order of five pixels, and there are less spurious vectors. In the reconstruction algorithm, the vector fields determined with correlation on transformed images is added to the apparent displacement estimated when the surface is at rest to compute the total apparent displacement.

a) b) Figure 2 Apparent displacement on the images from correlation with and without transformation: a)

Correlation from original images. b) Correlation with projected images.

4. Experimental set-up The ship wake measurements were conducted in the towing tank of the Institut Pprime, Poitiers, France. This towing tank is 25 meters long, 1.5 meters wide and 1.2 meters deep, equipped with a drive carriage (maximum towing speed of 2.5 m.s-1). The towing tank is equipped by windows, placed on the sides and on the bottom. Those windows allow us to use optical measurement systems


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like cameras or laser. The ship model used is L=1.2 m long and H=0.193 m wide. To generate the reference pattern, the water is seeded with PIV particles with a diameter of 56 µm and a density of 1.02 g/cm3. A laser sheet is generated from a Nd-Yag laser. The laser sheet is parallel to the free surface at rest and located 28 mm below the surface. The stereo-refraction system comprises of three cameras JAI (1600*1200 pixels). Two cameras are located above the surface in stereoscopic configuration with an angle of approximately 20° with the normal to the initial free surface. The reference camera, located below the tank, records images of the laser plane across a window located on the bottom of the tank. The Hiris software synchronizes the three cameras. The acquisition frequency is 10 Hz, in double frame mode with a Δt depending on the ship speed. The system is triggered from the ship motion. The common field of view is 400mm*400mm in the laser plane and located at the centre of the towing tank. The two cameras above the surface are calibrated in 3D by translating a plane calibration target in air. The reference camera is calibrated in the water in the plane of the laser sheet. The target is translated with a micrometer displacement table with an accuracy of the order of 1/1000 mm. To ensure a calibration of the three cameras in the same coordinate system, the calibration target used is a double-sided target. Two Cartesian coordinate systems are used to represent the results. The first coordinate system, (0*/X*,Y*,Z*), is related to the measurement area. The origin corresponds to the centre of the measurement area located at the centre of the tank. The X*, Y*, Z* axes are directed, respectively, in the opposite direction of ship motion, transversely and upward. All collected data are triggered from the ship motion. This synchronization procedure allows us to reconstruct the wave field around the model ship in a coordinate system, (0ship/Xship,Yship,Zship), related to the ship model. The Xship, Yship, Zship axes are directed with the same orientation as the system (0*/X*,Y*,Z*) and the origin corresponds to the centre of the ship model. The relationship between the two coordinate systems can be written as follows:

𝑋!!!" = 𝑋∗ + 𝑈! . 𝑡 − 𝑋!𝑌!!!" = 𝑌∗

𝑍!!!" = 𝑍∗ (5)

where Ub is the velocity of the boat, t is the acquisition time and X0 the position of the boat at the initial acquisition time in the coordinate system (0*/X*,Y*,Z*). Figure 3 presents a schematic of the experimental set-up.

Figure 3 Experimental set-up


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Tests are conducted for various Froude numbers based on the water depth:

𝐹! =!!!!

(6)

where Ub is the ship velocity, and h the water depth. The tests were performed with a constant water depth equal to h=0.45 meters. In this study, the Froude number depends only on the speed of the ship model. Three Froude numbers were tested equal to 0.36, 0.48 and 0.52 5. Results The method developed here allows to measure simultaneously both the topography of the free surface and the instantaneous 2D velocity in the laser plane in the wake of the ship. For each acquisition time, we can reconstruct the free surface and the 2D velocity field 28 mm under the interface. Figure 4 provides an example of instantaneous free surface height and velocity under the interface for a Froude number of 0.48. The acquisition time presented here corresponds to a time just after the passage of the ship model. The free surface representation reveals two wave crests on each side of the measurement area. These two waves correspond to the diverging waves generated by the stern of the ship. The 2D velocity field above the free surface can be related with the wave patterns. This velocity field can be decomposed into two zones. A first zone, near the sailing line of the model ship, along the line Y* = 0, where the velocity in the flow are important. In this zone, the fluid is lead by the ship. The second zone, on each side of the central line, corresponds to a flow below regular waves. The velocity profile reveals the same angle as waves on the free surface representation.

Figure 4 Instantaneous measurement: a) Free surface. b) Velocity norm.

From all the fields of instantaneous measured height and considering the wave field as stationary relative to the ship, the wave fields behind of the ship can be reconstructed in the system (0ship/Xship,Yship,Zship) related to the model ship using equation (5). Figure 5 shows the wave pattern around the ship model for three Froude numbers. The colour map corresponds to the measurement


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performed by the method presented in this paper. The grayscale elevation field was obtained with another technique of free surface measurement based on stereoscopic method (Chatellier et al. [2010]) and allows a global view of the wave fields around the ship model. The results obtained with the technique of refraction are consistent with those obtained by stereoscopy. The wave pattern consists of two systems of waves generated at the bow and the stern of the ship. On each side of the ship, these two systems are reflected by the sidewalls and interact behind the ship. This interaction is revealed by a point located on the sailing line of the ship where the wave heights are significant. This representation allows to highlight the effect of the Froude number, therefore the effect of speed ship, on the wave fields. The heights of the waves and the wavelength increase with the Froude number.

Figure 5 Wave pattern behind the ship for different Froude number: a) Fr=0.36 b) Fr=0.48 c)

Fr=0.54


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6. Conclusion The measurement technique presented in this paper allows the measurement of the instantaneous surface topography and of the 2D velocity field under the interface. This method based on the analysis of the apparent displacement between a reference and refracted images of a laser sheet through an interface allows the study of complex flow. The application in a towing tank provides access at the topography of the free surface and the instantaneous 2D velocity under the surface in the wake of a ship model. From the instantaneous height fields measured during the ship passage, the field of waves behind the boat can be reconstructed. These data, obtained with a non-intrusive technique with a high spatial resolution, allow the analysis of the evolution of wave fields in a confined channel according to the Froude number and can also provide a benchmark database for numerical computations. Acknowledgements The grant of the PhD thesis of G. Gomit is supported by the Direction Générale de l'Armement. Bibliography Chatellier L, Jarny J, Gibouin F, David L (2010) Stereoscopic measurement of free surface flows. 14th International Conference on Experimental Mechanics, Poitiers, France Fouras A, LoJacono D, Sheard G.J, Hourigan K (2008), Measurement of instantaneous velocity and surface topography in the wake of a cylinder at low Reynolds number. Journal of Fluids and Structures, 24:1271–1277. Moisy F, Rabaud M, Salsac K (2009) A synthetic Schlieren method for the measurement of the topography of a liquid interface. Experiments in Fluids 46:1021–1036 Morris N.J.W (2004) Image-based water surface reconstruction with refractive stereo. Master’s thesis, Department of Computer Science, University of Toronto Morris N. J. W and Kutulakos K. N (2005) Dynamic Refraction Stereo. Proc. 10th Int. Conf. Computer Vision. 1573–1580 Ng I, Kumar V, Sheard G. J, Hourigan K, Fouras A (2011) Experimental study of simultaneous measurement of velocity and surface topography: in the wake of a circular cylinder at low Reynolds number. Experiments in Fluids , 50: 587-595 Tsai R, (1987) A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses. Robotics and Automation, 3: 323 - 344 Zhang X, Cox C. S (1994) Measuring the two-dimensional structure of a wavy water surface optically: A surface gradient detector. Experiments in Fluids, 17:225-237

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Free surface measurement by stereo-refractionltces.dem.ist.utl.pt/lxlaser/lxlaser2012/online_cd/... · Abstract An optical method for the measurement of the surface topography, the