Experiments in Fluids 30 2001) 381±390 Ó Springer-Verlag 2001 Simultaneous free-surface deformation and near-surface velocity measurements D. Dabiri, M. Gharib 381 Abstract A newly developed non-intrusive approach has surface with the near-surface turbulence is by no means been devised for studying near-surface ¯ows where it is simple. The near-surface turbulence causes deformations important to be able to construct correlations between of the free surface, thereby storing energy in this defor- small-sloped free-surface deformations and near-surface mation. When the deformation relaxes, energy is then velocities. This method combines digital particle image released back into the ¯uid, thereby affecting the near- velocimetry DPIV)and the re¯ective mode of the free- surface turbulence. Should the deformations be large, it is surface gradient detector FSGD)technique into a single possible for the deformations to release their energy in the measurement system, providing us with an approach to be form of radiating waves that travel away, thereby affecting able to characterize correlations between elevation and the near-surface ¯ow elsewhere. To understand these kinematic properties, such as velocity and vorticity, which phenomena, it is important to be able measure both the is essential in understanding near surface turbulence. near-surface velocity, and the free-surface deformation Furthermore, as the free-surface elevation is directly pro- simultaneously, at least within a two-dimensional domain, portional to the pressure for low Froude number ¯ows, in order to obtain insights into the physics of such this method will allow for the measurement of pressure interactions. near the free surface. This will also be useful in calculating While velocity measurements have enjoyed dramatic the pressure-velocity term in the turbulent kinetic energy progress through the development of DPIV techniques, it equation for near-surface ¯ows. The approach is explained is only recently that free-surface measurements have been and demonstrated by measuring these correlations for a developed to the degree that would allow for both spatial vertical shear layer intersecting a free surface. and temporal measurements. Cox 1958)®rst developed a single point, single slope component detector, capable of 1 measuring water slopes through time. More recently, Introduction Zhang and Cox 1994)and Zhang 1995),using a dis- Free-surface ¯ow phenomena are a subject of great interest cretized color palette, developed a FSGD based on a re- as their effects can be seen both in nature and in matters of fractive color encoding scheme, capable of measuring two practical importance. Wave breaking, both spilling and slope components for a given area at a single point in time plunging, is responsible for large air, and therefore oxy- for short wind wave studies. For more accurate results, gen, entrainment into the oceans, and is therefore a major JaÈhne 1997)and Balschbach et al. 1998)incorporated a contributor to the existence of life within the oceans. From continuous color palette with the refractive FSGD tech- a more practical view, ship wakes have been known to nique. Such gradient and elevation schemes are a subset of persist for over several hundred kilometers, thus making the more general shape-from-shading techniques as them detectable to different remote sensing techniques. In explained by Klette et al. 1998). order to understand these phenomena, it is important to Initially, these methods, which relied on image acqui- understand not only the ¯uid mechanics beneath the free sition, were applied to single photographic images. How- surface, but also the free-surface deformation, and their ever, with the advent of technology, it became possible to interaction with each other. The interaction of the free apply these methods to data acquired through video technology, and thereby attain time-evolving results. For Received: 2 August 1999/Accepted: 23 July 2000 example, a real-time acquisition version of the refractive color-encoding method using a discretized color palette was developed for the study of time-evolving free-surface D. Dabiri, M. Gharib Graduate Aeronautics Laboratories deformations Zhang et al. 1996; Dabiri et al. 1997). In a California Institute of Technology, Pasadena, CA 91125, USA clever approach, Hering et al. 1996)and Wierzimok et al. e-mail: [email protected] 1996)obtained both free-surface elevation results and particle paths from sparsely seeded ¯ows, using a com- Correspondence to: D. Dabiri bined digital particle tracking velocimetry method DPTV) and a refractive color-encoding FSGD technique. Since The authors would like to thank Patrice Maheo for allowing us to waves were studied, the refractive color-encoding scheme use Fig. 6 from his thesis, which was used to describe the same facility used for both our projects. The authors gratefully was used, as it was necessary to resolve large slopes. acknowledge the support of the Of®ce of Naval Research under Interestingly, the work of Dommermuth et al. 1994) the research grant number N00014-97-1-0303. provides a method whereby the near-surface pressure ®eld can be obtained through the free-surface deformation rections. When a lens is placed at a distance of one focal ®eld. In their study, it is shown that by looking at low length away from the color palette, all of the rays of each Froude number ¯ows, it is possible to express the equa- color will become parallel; yet will be oriented in different tions of motion as directions with respect to other colored light rays. This setup will then create a system of parallel color beams, oUr oUr oPr 1 o2Ur oUr i Ur i À i af Ur i 0 ; which is used to illuminate the free surface. Figure 2 shows ~ j 2 i ot oxk ox~i Re oxj ox~i how the different free-surface slopes are color-coded. When the free surface is illuminated with the parallel color o2Pr oUr oUr j i ~ beams, there is only one free surface slope that will re¯ect 2 À for z 0 ; oxj ox~i ox~j a particular color towards an observer located far away. In this way, a one-to-one correspondence is achieved between 382 1 Wr 0 and gr ÀP Pr for ~z 0 color and free-surface slopes. Fr2 a where the superscript r refers to components that con- 2.2 tribute to the free surface roughness, g is the free-surface Color palette elevation, P is the pressure normalized by the density, U is For a re¯ective system, the angle of the re¯ected beams the velocity, the subscript a refers to atmospheric condi- from the free surface is equal to the incident angle onto the tions, a is a source coef®cient, and W is the vertical ve- free surface. Therefore, as shown in Fig. 3, the largest angle locity component and f is a linear operator. Most that can be measured, amax, is a function of the diameter of interesting is the last of these equations that shows that the lens, D, its focal length, f , and the diameter of the color palette, Ds: with the neglect of the atmospheric pressure, the surface elevation is hydrostatically balanced by the vortically-in- D D 1 a tanÀ1 s tanÀ1 tanÀ1 : duced pressure. This insight provides us with a way to max 2f 2f 2F# measure near-surface pressures through the measure of free-surface elevations. 1 To better understand free-surface ¯ows, as a ®rst step, When the size of the palette is equal to the size of the lens, we propose to examine the interaction of the free surface the maximum measurable angle can be determined by the with the near-surface turbulence without the added com- F# of the lens. The maximum measurable area denoted by plexity of wave radiation due to steep slopes. However, for L depends on the size of the lens, D, the angle between small free-surface deformations, the refractive FSGD will max the optical axis and the free surface normal, b , the normal not provide enough resolution to properly map out the 0 distance from the center of the lens to the free surface, H, deformation ®eld. It is therefore the intent of this paper to and the largest possible angle between the light rays and present a combined technique that incorporates a real- the free surface normal, b see Fig. 3), and is given by time digital free-surface re¯ective gradient detector max the relation: FSGD)system with DPIV, to provide simultaneous, time- evolving ¯ow and surface deformation ®elds for free- D L D cos b À H À sin b tan b surface ¯ows with small deformation slopes. As will be max 0 2 0 max shown below, this is important in terms of identifying D correlations between surface elevation and various kine- À H sin b tan b À 2b : 2 matic properties, as well as being able to calculate the near 2 0 max 0 surface pressure spectra. A sample color palette used is shown in Fig. 4. Given this color palette, the dimension of each of the color elements 2 can be denoted by de. Therefore, the smallest angle change Free surface gradient detector that can be measured is 2.1 da tanÀ1 de=f ; 3 Operating principle and is de®ned as the slope sensitivity. The viewing angle of The operating principle behind the FSGD is to color code the camera is the different slopes of the free surface with different colors À1 in order to establish a one-to-one correspondence between w tan Dc=L ; 4 color and slope Dabiri et al. 1997; Zhang et al. 1996; Zhang and Cox 1994). There are basically two modes that where Dc is the diameter of the camera lens, and L is the can be employed.