Visualization of three-dimensional vortex dynamics and fluid transport in translating plates, using defocusing DPIV

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1 Visualization of three-dimensional vortex dynamics and fluid transport in translating plates, using defocusing DPIV Daegyoum Kim 1, Morteza Gharib 2 1: Division of Engineering and Applied Science, California Institute of Technology, Pasadena, USA, daegyoum@caltech.edu 2: Division of Engineering and Applied Science, California Institute of Technology, Pasadena, USA, mgharib@caltech.edu Abstract Three-dimensional flow fields around translating plates were mapped by using defocusing digital particle image velocimetry. As a first experiment, vortex structures generated by impulsively translating low aspect-ratio plates with a 90 angle of attack were studied. Rigid and flexible thin plastic plates were used to find the effect of flexibility on vortex formation. The tip vortex motion is one of the obvious differences between the flat-rigid and flexible plates. While the tip vortex moves upward in the flat-rigid plate case, it stays near the tip in the flexible plate case. In order to map the deformation of the flexible plate, three-dimensional positions of the plate edges were obtained through the defocusing technique. As a second experiment, the dynamics of the vortex near a corner region was compared among three different corner angles for impulsively translating plates. For a large corner angle, the forward movement of the vortex tends to be uniform along the plate edges. However, for a small corner angle, the vortex close to the corner moves forward following the plate while the vortex away from the corner retards its forward movement. The three-dimensional trajectories of fluids near the corner were also tracked in order to visualize fluid mixing by the corner vortex. The amount of fluid entrained by the corner is related to the motion of the corner vortex. 1. Introduction Vortex formation by objects starting from rest has been studied extensively in the past. The rollup process of starting vortex sheets has been used to explain lift generation by starting airfoils (Prandtl and Tietjens 1934). The starting vortex formation of various two-dimensional models has also been studied (e.g. Pullin 1978, Pullin and Perry 1980, Saffman 1995). While twodimensional or axisymmetrical starting vortices have been studied intensively, experimental studies of three-dimensional starting vortices are relatively sparse. In particular, the vortex formation in the translating plate of low aspect-ratio with a high angle of attack has been rarely studied experimentally. Here, we study vortex formation of impulsively translating thin plates with a 90 angle of attack by using Defocusing Digital Particle Image Velocimetry system (DDPIV) also known as V3V system (Willert and Gharib 1992, Pereira and Gharib 2002, Lai et al. 2008). This study is composed of two parts. In the first part, the effect of flexibility on vortex formation near the tip of the plate is investigated by comparing vortex structures among rigid plate and flexible plate cases. In the second part, in order to understand the effect of the corner angle on the dynamics of the vortex created near a corner, several cases with different corner angles are examined. The trajectories of fluid particles near the corner were also tracked to visualize the process of fluid entrainment into the corner vortex

2 2. Experimental Setup Fig. 1 shows the experimental setup. A water tank of size mm 3 was used for the experiment. For the first experiment (part 1) to study flexibility effects on vortex formation, three polycarbonate plates were used to represent flat-rigid, flexible, and curved-rigid plates. The thickness of the rigid plates is 1.52 mm, and that of the flexible plate is 0.25 mm. Even though the plates with 1.52 mm thickness are not completely rigid, the deformation during translation is negligible. In the middle of the tank, 150 mm of the plate with width 40 mm was immersed vertically. Part of the plate above a free surface was attached to the traverse. A traverse with a lead-screw (Velmex Inc.) was used to translate the plate. The traverse accelerated for 0.25 sec at the start and moved with a constant velocity (U) of 50 mm/sec. The angle of attack was 90. The Reynolds number based on the constant traverse velocity and plate width was To make the curved shape for the curved-rigid plate, a hole near a tip edge was penetrated and connected to the traverse with a thread while keeping the curved shape. We tried to make the curved-rigid plate bend at the same degree as the flexible plate at its maximum deformation. However, there were some deviations in curvatures because it was difficult to match the shapes exactly. Fig. 1 Experimental setup (left) and the mechanical model from the camera view (right). The arrow indicates the moving direction of a traverse. For the second experiment (part 2) to study corner-angle effects on vortex formation, an acrylic plate with thickness 1.46 mm was immersed vertically so that a corner region is included in the DDPIV camera probe volume. Three plates with different corner angles (60, 90 and 120 ) were used. In the 90 angle case, the area immersed in water has a height 160 mm and width of 200 mm. The position of the corner vertex in the mapping volume is the same for three different corner-angle cases. The vertical edge on the other side of the plate was closely aligned with the tank wall to avoid water leakage through the gap. Even though we used the plates with finite length, ideally there should be no characteristic length so that the vortex formation processes are dependent only on corner shapes. Plates accelerated for 1 sec at the start, and translated with a constant velocity (U) of 20 mm/sec

3 Fig. 2 Positions of the plate model and the camera probe volume inside the tank for 60 cornerangle (left), 90 corner-angle (center) and 120 corner-angle (right) cases of the part 2 experiment. In this figure, the plate moves into the page, and the camera is placed on the right side of the tank. Thick continuous lines indicate the plate model. Thick dashed lines are the volume in which the flow field is mapped. For DDPIV setup, the distance between the water tank and the camera which was placed in front of the water tank was adjusted to position the camera probe volume in the middle of the tank. The tank was seeded with silver-coated glass spheres of mean diameter 100 µm (Conduct-o-fil, Potters Industries Inc.). An Nd:YAG laser (Gemini PIV, New Wave Research Inc.) was placed to the left side of the camera and optical lenses were used to make a laser cone which covered the camera probe volume. A stepper motor control computer sent trigger pulses to synchronize the DDPIV camera, the laser, and the motor start. Image pairs were captured at a rate of 5 pairs/sec. The time gap between two laser pulses to take a pair of images was sec for the part 1 experiment and sec for the part 2 experiment. The images taken from the camera were processed with the DDPIV software based on Pereira and Gharib (2002) and Pereira et al. (2006). First, three-dimensional coordinates of the particles inside the tank were found. Then, from this information, velocity vectors of particles were calculated using a relaxation method of three-dimensional particle tracking. The flow field with mm 3 volume was cropped during these processes. The number of cubic grids with size mm 3 in the mapped domain was quite large compared to the number of randomly-spaced velocity vectors obtained from one case. To increase the density of randomlyspaced velocity vectors in a fluid domain, the experiment was repeated 20 times for the part 1 experiment and 25 times for the part 2 experiment under the same conditions with an interval of 90 sec. For each time step, the randomly-spaced velocity vectors obtained from 20 (or 25) cases were collected and interpolated into cubic grids to produce a velocity field. After removing outlier vectors and applying a smoothing operator to velocity vectors, the vorticity field was obtained by a central difference scheme. For smooth rendering of three-dimensional iso-surfaces of vorticity magnitude, vorticity data were also smoothed. When the plate moved outside of the camera probe volume, the initial position of the plate was moved back 80 mm for the part 1 experiment 120 mm for the part 2 experiment so that the plate showed up in the camera volume later. Then, another set of the experiment was conducted for the later stage of the plate motion. The total fluid volume mapped by two sets of the experiment was mm 3 for the part 1 experiment mm 3 for the part 2 experiment

4 The flexible plate of the part 1 experiment deforms severely when it translates impulsively. In addition, it is necessary to map the curved shape of the curved-rigid plate. To map the deformation of the plate, seeding particles were attached along the edge of the plate by using tape. The plate was immersed into clean water. Particles illuminated by the laser were captured by the DDPIV camera with the same conditions as the flow mapping experiment mentioned above. Images were processed to get the coordinates of the particles. From this coordinate information, the deformed plate could be rendered. To track fluid particle trajectories for the part 2 cases, the Newman software v3.1 was used (Toit 2010). At t = 0, fluid particles are placed in a fluid volume of mm 3. If the origin of the coordinate system is at the initial position of the plate vertex (Fig. 2), the particles are distributed from x = -60 mm, y = -30 mm, z = -30 mm to x = 180 mm, y = 30 mm, z = 30 mm. The distance between particles is 3 mm in all three directions. Time, vorticity were non-dimensionalized with the plate width l (40 mm for the part 1 and 200 mm for the part 2 experiment) and the constant velocity U; t = t d U / l, ω = ω d l / U, where a subscript d means a dimensional variable. 3. Results and Discussion 3.1 Flexiblity effect on vortex formation Fig. 3 depicts plate shapes for the three cases considered in the part 1 experiment: flat-rigid, flexible and curved-rigid plate cases. For the flexible plate case, the deformation is shown at several times. Figs. 4 to 6 show the vortex formation process of the translating plates. Isosurfaces of vorticity magnitude were used to represent vortex structures. Throughout the part 1 experiment, the vortices generated along the vertical long edges of the plate are termed leadingedge vortices (LEV) and the vortex along the horizontal short edge is termed a tip vortex (TV). The plates shown in these figures are all immersed in water. The top area of the plate is beyond the camera probe volume and, thus, its flow field is absent

5 Fig. 3 Shapes of the plates immersed in water during traverse movement. The plates are rearranged to have the same base position in order to compare x-directional deviation of the tip from a vertical line. A horizontal continuous line at the bottom is the distance which the traverse travels for t = 1. First, we explain the flat-rigid plate case (Fig. 4). At the onset of the plate movement, a vortex sheet begins to roll up along all edges. After t = 2, LEV starts to get away from the plate in the mid-section of each side. Whereas LEV near the tip remains attached to the plate, LEV in the upper part detaches from the plate and slants into the xz-plane while moving outward in the z- direction. Near t = 4, TV moves upward continuously by the tip flow. The upward motion of TV, which is concurrent with the tilting of LEV, continues during the full observation time of the experiment. This continuous upward motion of TV accentuates the upward propagation of LEV tilting. Fig. 4 Vortex structures of the flat-rigid plate case at t = 2, 4 and 6. Iso-surfaces of ω = 2.4 are used. In the flexible plate case (Fig. 5), the forward motion of the lower part of the plate is delayed due to the flexibility of the plate, and the vortex system develops first in the upper part. As the lower part of the plate starts to move forward, LEV and TV begin to develop in the lower part as well. After t = 3, the curved shape of the plate is relaxed toward an equilibrium state (Fig. 3). The vortex morphology follows the shape of the plate without exhibiting any distinct deformation until t = 4. However, after this time, the vortex deforms into a horseshoe shape in the lower region of the plate. During this deformation process of the lower LEV part, the vortex core in the upper part continues to elongate in the x-direction. Note that, for the flat-rigid plate case, LEV in the upper part of the plate moves away from the plate early. However, in the flexible plate case, LEV in the upper part does not move outward from the plate edge and ends up having an elongated vortex core. When LEV in the lower part tilts outward to make a horseshoe-shaped vortex, LEVs in the upper part show a tendency to move inward. However, LEV's inward movement is limited due to the imposed symmetric condition at z = 0. Instead, the LEV core in the upper part continues to elongate in the x-direction

6 Fig. 5 Vortex structures of the flexible plate case at t = 2, 4 and 6. Iso-surfaces of ω = 2.4 are used. For the curved-rigid plate case (Fig. 6), the vortex in the lower part of the plate grows faster than that of the flexible plate case. The formation of the vortex into a horseshoe shape in the lower region is more distinct than that of the flexible plate case. The separation of the lower LEV from the plate edge is more distinct than that of the flexible plate case. The vortex system in the lower part of the plate becomes nearly a circular vortex and the vortex core of the lower part becomes corrugated similar to the flexible plate case. This deformation is concurrent with the severe elongation of the LEV core in the upper part of the plate. Fig. 6 Vortex structures of the curved-rigid plate case at t = 2, 4 and 6. Iso-surfaces of ω = 2.4 are used. 3.2 Morpho-dynamics of corner vortices Figs. 7 to 9 show the vortex formation process near the corner of the plate with a different corner angle (60, 90 and 120 ). At the start, the vortex sheet rolls up along the plate edge. Therefore, the shape of the vortex is similar to the shape of the plate edge. However, as the plate moves farther, the vortex core begins to separate non-uniformly from the edge and finally lose its initial shape. Here, the term vortex separation is used to indicate that a vortex core does not follow the plate edge and retards its forward motion. Note that its definition is subjective. We use this term mainly for the purpose of comparing vortex positions relative to the plate edge

7 Fig. 7 Vortex formation process near the corner of the 60 corner-angle plate at t = 0.3, 0.6 and 0.9. (a) is from back view and (b) is from side view. Iso-surfaces of ω = 17 are used. Fig. 8 Vortex formation process near the corner of the 90 corner-angle plate at t = 0.3, 0.6 and 0.9. (a) is from back view and (b) is from side view. Iso-surfaces of ω = 17 are used

8 Fig. 9 Vortex formation process near the corner of the 120 corner-angle plate at t = 0.3, 0.6 and 0.9. (a) is from back view and (b) is from side view. Iso-surfaces of ω = 17 are used. As can be seen in (a) of Figs. 7 to 9, the corner angle affects the morphology of the corner vortex. As the corner angle is small (60 ), the vortex close to the corner follows the forward motion of the plate without noticeable separation from the edge. Meanwhile, the vortex far from the corner separates from the corner early. This trend is weakened as the corner angle gets larger. For the 120 case, the corner vortex tends to separate from the corner earlier than that of the lower angle cases. Its x-directional position is quite uniform along the edge. If there is a thin vortex tube in a flow field without a moving object, the vortex tube can change its position by its self-induction. As the curvature of the thin vortex tube gets bigger, the vortex tube has the larger self-induced velocity component in the direction bi-normal to the curved vortex tube (Batchelor 1967). This theoretical finding may be used in explaining the dependence of the vortex forward motion on the corner angle. As the corner angle gets smaller, the curvature of the corner vortex becomes larger, and the larger displacement of the vortex is anticipated in the direction bi-normal to the plate edge (x-direction in our model). It is well known that, in the absence of a moving object, the velocity field in the fluid domain can be obtained from the vorticity induction equation. This approach makes it easier to explain selfinduction of a vortex or mutual interaction with other vortices. However, in the case of a vortex near a moving body in an infinite field, the potential flow effect caused by a moving body should be taken into consideration as well because the velocity field must be constructed from both vorticity distribution and potential flow effect by the plate motion (Batchelor 1967). Thus, selfinduction of the vortex is not enough to explain the morpho-dynamics of the vortex generated by the plate. To fully understand the vortex motion behind the plate, the effect of the plate motion should be considered. While the plate translates, fluids close to the back of the plate are under strong potential flow effect enough to follow the plate motion. As the distance between a fluid - 8 -

9 particle and the plate becomes larger, the induction effect by the plate becomes smaller. For this reason, it is difficult for the vortex, which is far from the plate, to catch up with the plate motion. As well as the forward motion of the corner vortex, the inward motion of the corner vortex is also different among the three corner-angle cases studied here ((b) of Figs. 7 to 9). As the plate starts to translate, a low pressure region is generated behind the plate and an inward flow (the flow from the outside toward the backside of the plate) is induced behind the plate. Due to the low pressure zone behind the plate, the corner vortex moves inward gradually. Since the corner vortex of the 90 case is closer to the plate in the x-direction, it is influenced by pressure suction more than that of the 120 case, and its inward motion is more distinct. In the 60 corner-angle case, two edges are in a short distance from each other. Vorticity near the corner, which is transported by the inward flow, is eventually entrained by vortices away from the corner in both edges. For this reason, the corner vortex of the 60 case does not develop as well as that of the other two cases. 3.3 Fluid transport near corners Fig. 10 shows the fluid transport process near the corner. Four different colors were used to distinguish particles along the x-axis. Blue particles are positioned behind the initial position of the plate while particles of other colors are initially in front of the plate. Once fluid particles move out of the measured flow field, they are not tracked anymore even though they can return to the fluid volume later. Particles initially behind the plate are dragged by the translating plate, and they are eventually entrained by the vortex. Particles initially in front of the plate turn around the edge and are also entrained by the vortex. For the plate with a larger corner angle, more fluids behind and in front of the corner are displaced by the plate motion. Thus, in the larger corner-angle case, more fluids are involved in the swirling and mixing of fluid particles behind the corner

10 Fig. 10 Fluid transport near the corner for three different corner-angle cases at t = 0, 0.3, 0.6 and 0.9. The first row is for 60, the second row is for 90, and the third row is for 120. Thin black lines are the outer boundary of the measured flow field. The motion of the corner vortex is influenced by the amount of fluid entrained by the vortex. The vortex and fluid trajectories are compared in Fig. 11 for the 90 and 180 (actually straight edge) corner angle cases. In the 180 case, the plate area was extended in the y-direction from the plate of the 90 case, and the bottom edge of the plate was closely aligned with the tank floor. Compared to the 90 case, the entrainment process of fluid particles is more active in the 180 case; more fluids behind and in front of the plate are involved in this process. As the plate continues to translate, the volume of fluid swirling around the vortex becomes bigger, and the vortex core is eventually separated from the plate. The swirling fluid also expands the gap between the plate edge and the vortex core. Thus, the vortex core is pushed away from the plate edge. For example, at t = 6 sec of the 180 case in Fig. 11, most of the blue fluid particles, which were originally behind the plate, swirl around the vortex center. Meanwhile, at t = 6 sec of the 90 case, only a small number of blue particles are around the corner vortex. It implies that the corner vortex can still follow the forward motion of the plate without being pushed away from the plate by entrained fluids. Fig. 11 Comparison of fluid entrainment into the vortex between 90 (upper row) and

11 (lower row) cases at t = 0, 0.3, 0.6, 0.9. All subfigures are in top view. Thin black edge-lines are the outer boundary of the measured flow field. Blue surfaces are iso-surfaces of vorticity magnitude ( ω = 17). 4. Concluding Remarks The three-dimensional flow condition near the tip of the low aspect-ratio plate and the corner with various angles reveals interesting features of vortex dynamics. Here, we showed that the deformation of the plate tip changes the dynamics of the vortex near the tip. In addition, we demonstrated that the dynamics of the growing corner vortex was strongly dependent on the shape of the corner. By using the defocusing technique, we mapped the deformation of the flexible plate. It helps us better understand the fluid-deformable plate interaction and the consequent vortex formation process. We also visualized the fluid transport process near the corner. Fluids initially behind the plate are swirled and mixed with fluids initially in front of the plate. The amount of fluid entrained into the corner vortex is closely related to the motion of the corner vortex. References Batchelor GK (1967) An introduction to fluid dynamics. Cambridge University Press, Cambridge Lai W, Pan G, Menon R, Troolin D, Graff EC, Gharib M, Pereira F (2008) Volumetric threecomponent velocimetry: a new tool for 3D flow measurement. 14th Int Symp on Applications of laser Techniques to Fluid Mechanics, Lisbon, Portugal Pereira F, Gharib M (2002) Defocusing digital particle image velocimetry and the threedimensional characterization of two-phase flows. Meas Sci Technol 13: Pereira F, Stuer H, Graff EC, Gharib M (2006) Two-frame 3D particle tracking. Meas Sci Technol 17: Prandtl L, Tietjens OG (1934) Applied hydro- and aeromechanics. McGraw-Hill, New York and London Pullin DI (1978) Large-scale structure of unsteady self-similar rolled-up vortex sheets. J Fluid Mech 88: Pullin DI, Perry AE (1980) Some flow visualization experiments on the starting vortex. J Fluid Mech 97: Saffman PG (1995) Vortex dynamics. Cambridge University Press, Cambridge Toit PD (2010) Transport and separatrices in time-dependent flows. California Institute of Technology Willert C, Gharib M (1992) Three-dimensional particle imaging with a single camera. Exps Fluids 12: