圆柱体绕流的数值研究3

Aerodynamics Drag coefficient Axial jet experiences more drag than the feather model. Unlike the synthetic model, the feather shuttlecock is associated with a swirling flow towards the end of the skirt. The effect of the twist angle of the feathers on the drag as well as the flow has also been studied. port cocks due to such badm nd at thers made of nylon/ t of the synthetic etic shuttlecocks to differences in erred choice for professional badminton. the shuttlecock, in flight, is at a zero angle of attack, aligned axially with the direction of flow. Despite the immense Contents lists available at ScienceDirect Journal of Fluids and Structures Journal of Fluids and Structures 41 (2013) 89–98 0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.01.009 n Corresponding author. Tel.: þ 91 512 259 7906; fax: þ91 512 259 7561. E-mail address: smittal@iitk.ac.in (S. Mittal). Both types of shuttlecocks share a similar weight distribution. The cork is much heavier than the skirt. Therefore, all shuttlecocks tend to align themselves cork-first to the direction of motion. The most stable and dominant configuration of expensive. They wear out or break within a short duration of play. Synthetic shuttlecocks share sim features such as the cork and a conical shaped skirt. However, the skirt of the synthetic shuttlecock is plastic. In order to replicate the aerodynamic effect of overlap of feathers in a feather shuttlecock, the skir shuttlecock is associated with a non-uniform distribution of the porosity of the net. Although the synth are relatively cheaper and more durable than the feather shuttlecocks, the difference in their design leads their aerodynamic characteristics and their trajectory. As of today, feather shuttlecocks are the pref imparts certain aerodynamic properties that are exploited by skilled players. However, feather shuttlecocks are brittle and ilar geometrical 1. Introduction Badminton is a popular racquet s this game use a shuttlecock. Shuttle time, they decelerate much faster affects its trajectory. It should be in skilled players. The modern game of information on the game may be fou made from sixteen overlapping fea & 2013 Elsevier Ltd. All rights reserved. and, unlike many other racquet sports that are played with a ball, the players in can achieve speeds larger than the projectiles used in other sports. At the same the larger drag that they experience. The drag on a shuttlecock significantly a range so that the shuttlecock remains within the allowed limits of the court for inton is played with two variants of the shuttlecock: feather and synthetic. More http://en.wikipedia.org/wiki/Badminton. The conventional shuttlecock is usually arranged in a conical form. The asymmetric shape of the feather shuttlecock Aerodynamics of badminton shuttlecocks Aekaansh Verma, Ajinkya Desai, Sanjay Mittal n Department of Aerospace Engineering, Indian Institute of Technology Kanpur, UP 208 016, India a r t i c l e i n f o Article history: Received 10 May 2012 Received in revised form 11 January 2013 Accepted 20 January 2013 Available online 13 March 2013 Keywords: Shuttlecock a b s t r a c t A computational study is carried out to understand the aerodynamics of shuttlecocks used in the sport of badminton. The speed of the shuttlecock considered is in the range of 25–50 m/s. The relative contribution of various parts of the shuttlecock to the overall drag is studied. It is found that the feathers, and the net in the case of a synthetic shuttlecock, contribute the maximum. The gaps, in the lower section of the skirt, play a major role in entraining the surrounding fluid and causing a difference between the pressure inside and outside the skirt. This pressure difference leads to drag. This is confirmed via computations for a shuttlecock with no gaps. The synthetic shuttle journal homepage: www.elsevier.com/locate/jfs popularity of the game and its rich historical background, the studies related to the aerodynamic properties of shuttlecocks have been few, in contrast to other bluff bodies (Asai and Kamemoto, 2011; Behara and Mittal, 2011). Cooke (1996) carried out an experimental study to qualitatively understand the aerodynamics of a feather shuttlecock. The wake of the shuttle is associated with a strong axial jet surrounded by an annular stagnant zone. The axial jet entrains the air surrounding the shuttlecock. It was concluded that the gaps in the skirt lead to an increase in the drag of the shuttlecock. Alam et al. (2009) conducted experiments for feather and synthetic shuttlecocks of varying diameter, mass and length. For a shuttlecock moving with a speed of 60 km/h the drag coefficient was found between 0.50 and 0.70 for five varieties of shuttlecock made of synthetic rubber. It was also concluded that, compared to the feather shuttlecock, the synthetic shuttlecock undergoes higher deformation at high speeds. This streamlining effect leads to lesser aerodynamic drag. Kitta et al. (2011) conducted experiments on a feather shuttlecock. They presented the variation of drag coefficient with Reynolds number for a certain model of the feather shuttlecock. The effect of gap as well as that of the rotation was investigated. They found 2. Problem set-up A. Verma et al. / Journal of Fluids and Structures 41 (2013) 89–9890 2.1. Geometric models of shuttlecock Three models of shuttlecock have been used for the computational study: synthetic, feather and a gapless model. The models are shown in Fig. 1. The synthetic model is a close replica of a popular model of the Mavis 350 model from Yonex. It should be pointed out that the details for this model have been compiled from simple measurements from an off-the-shelf piece of the shuttlecock available in open market. No sophisticated instrumentation was used to make the measurements. In that sense, the present computer model is an approximate replica of the Mavis 350. The model for the feather shuttlecock is based on the details provided by Kitta et al. (2011). Both models are assumed to have non-porous walls. The feather model consists of sixteen trimmed feathers glued to the cork. Similarly, the synthetic shuttle comprises of sixteen panels. For both the models, the cork consists of a hemisphere and a short cylindrical section. Each panel, in the feather and synthetic shuttlecock, subtends an angle of 22.51 at the centre of hemisphere associated with the cork. The diameter, D, of the circle circumscribing the end of the skirt is 65 mm for both the models. The length of the shuttlecock, L, is the distance between the nose of the cork to the centre of the largest circle circumscribing the skirt. It is 80 mm for the synthetic and 85 mm for the feather shuttlecock. The third model is for the gapless shuttlecock. All its dimensions are same as that for the feather shuttlecock. However, its skirt is a frustum of a cone and has no gaps. This model is expected to provide information on the role of gaps in the flow past a shuttlecock. Fig. 1. Description of the various models of the badminton shuttlecock used in the present study: (a) synthetic, (b) feather and (c) gapless models. that the drag coefficient for the shuttlecock without gap is significantly smaller than that for the ordinary shuttlecock. However, the effect of rotation is marginal. The shuttlecock with spin experiences a marginally larger drag. Chan and Rossmann (2012) studied the flight performance of four models of shuttlecock: two each of feather and synthetic kind. The shuttlecock follows a variety of trajectories depending on the type of shot played. Four configurations were selected to represent different kinds of shots. The speeds chosen for these are: 7.7 m/s for netshots, 10 m/s for serve shots and 47 m/s for smashes and high clear shots. For a serve shot, the shuttlecock generally follows a parabolic trajectory till the highest point and then dips suddenly. Typically, the plastic shuttlecocks fly faster under smash shots. This is because of the larger deformation of the skirt, causing a reduction in drag, of synthetic shuttlecocks. One of the aims of the present study is to understand, in detail, the features of the flow past a shuttlecock. Both, the feather and synthetic models were investigated. An attempt is made to bring out the difference between the flow for the two models. To study the effect of gaps in the skirt, computations are carried out for a shuttlecock without any gaps. The role of different parts of the shuttlecock in contributing to the overall drag is studied. The break-up of the drag as pressure and viscous contributions is also studied. The pressure distribution for different models of the shuttlecock is studied. The effect of refinement of the mesh as well as utilizing an alternate model for turbulence is presented. Wherever possible, comparison with the results from earlier studies is presented. For a feather shuttlecock, the effect of the angle of twist of the feather, on the drag and flow, is presented. The effort is entirely computational. The flow speeds considered are 25–50 m/s. 2.2. Solution method The computations are carried out in a frame of reference attached to the shuttlecock. The flow is assumed to be incompressible. The steady solution to the Reynolds Averaged Navier Stokes (RANS) equations is sought. A turbulence model is utilized for closure. Most computations have been carried out with the v22f model. A few studies, to bring out the sensitivity of the computations to the turbulence model, have been carried out with the realizable k–E turbulence model. The rk E is a two-equation model, while the v22f model utilizes four equations for modeling of turbulence. A finite volume method capable of close to second-order discretization, in primitive variables formulation, is utilized. The SIMPLE algorithm is used for pressure–velocity coupling. The Reynolds number is defined as Re¼ rU1D=m, where U1 is the free-stream speed of the flow relative to the shuttlecock, D is the maximum diameter of the skirt of the shuttlecock, r is the density of the fluid and m is its coefficient of viscosity. The coefficient of drag is defined as CD ¼ Fd=0:5rU21A. Here, Fd is the drag on the shuttlecock and A is its reference area defined as A¼ pD2=4. All the test cases in the present study assume a rigidly held shuttlecock at zero angle of attack, without spin or yawing motion. The shuttlecock, including its skirt, is assumed to be rigid, i.e., it does not deform under the action of fluid forces. Experiments from earlier studies have shown that at large speeds, the skirt of the synthetic shuttlecock can undergo large deformations. This will be investigated in a later study. 2.3. Boundary conditions Free-stream condition is assigned for the velocity at the upstream boundary. The gauge pressure as well as the shear stress is prescribed to be zero at the downstream boundary. No-slip condition is imposed on the velocity on the walls of the shuttlecock. On the lateral outer boundary, the component of velocity normal to the surface as well as the shear stress vector in the tangential direction is assigned a zero value. A. Verma et al. / Journal of Fluids and Structures 41 (2013) 89–98 91 Fig. 2. A typical mesh for the feather shuttlecock. 3. The mesh and convergence study The shuttlecock resides in a cylindrical outer domain. The upstream and downstream boundaries are located at x¼�143 mm (x=L¼�1:682) and x¼577 mm (x/L¼6.788), respectively, from the nose of the shuttlecock. The diameter of the outer domain is 310 mm. Fig. 2 shows a typical mesh used for the computation for the feather shuttlecock. This mesh consists of 3.2 million unstructured, tetrahedral elements and 0.62 million nodes. We refer to this mesh as M1feather. A cross-sectional view of the mesh is also shown. The mesh is very fine close to the surface of the shuttlecock and it coarsens, gradually as one moves away from the shuttlecock. To facilitate the generation of the mesh, the entire computational domain is divided into three parts: the region in a frustum shaped inner most domain, an intermediate region and the coarse outer region between two cylindrical surfaces. The adequacy of the spatial resolution of the descritization is tested via computations on a finer mesh. A second mesh, M2feather is generated for the feather shuttlecock. It consists of 4.8 million elements and 0.85 million nodes. Computations are carried for Re¼ 2:22� 105. The drag coefficient computed with mesh M1feather is 0.479 and it is 0.480 with mesh M2feather. These values are in very good agreement. Compared to the fine mesh, the error in CD with mesh M1feather is 2%, approximately. A similar study for mesh convergence is carried out for the synthetic shuttlecock. Two meshes are generated: M1synthetic with 3.2 million elements and 0.59 million nodes and M2synthetic with 5.6 million elements and 0.98 million nodes. The computed drag coefficient with mesh M1synthetic is 0.632 and 0.668 with M2synthetic. Again, these values are in very good agreement. This confirms the adequacy of mesh M1 in resolving the flow. All the computations presented in this work are, therefore, carried out with mesh M1. 4. Results Flow past the synthetic and feather shuttlecock is investigated for 1:11� 105rRer2:22� 105. This corresponds to a shuttle speed of 25–50 m/s. A qualitative comparison with the description provided by Cooke (1996), as well as a comparison of the variation of coefficient of drag with Reynolds number, for the synthetic and feather shuttlecock models with the experimental studies conducted by Alam et al. (2009) and Kitta et al. (2011) is presented. Detailed flow analysis is carried out for Re¼ 2:22� 105 corresponding to a shuttle speed of 50 m/s. The effect of the twist of feathers on the aerodynamic performance is also investigated. A. Verma et al. / Journal of Fluids and Structures 41 (2013) 89–9892 4.1. Drag coefficient Fig. 3 shows the variation of the coefficient of drag with Reynolds number. Results for both synthetic and feather shuttlecock models, along with the data from earlier studies of Alam et al. (2009) and Kitta et al. (2011) are presented. The coefficient of drag for synthetic and feather shuttlecock is found to be 0.658 and 0.491, respectively at Re¼ 1:11� 105 and 0.631 and 0.479 at Re¼ 2:22� 105. The drag coefficient decreases with Reynolds number. The decrease is more rapid for low inlet speeds. At higher speeds, the CD is virtually independent of Re. The results from the present computations are in good agreement with those from earlier studies. Interestingly, the agreement between the results for synthetic shuttle is better than that for the feather shuttle. For all the cases studied, the (rigid) synthetic shuttlecock experiences more drag than its feather counterpart. This is also consistent with the results from earlier studies. Fig. 3 also shows the CD for the gapless shuttlecock for Re¼ 2:22� 105. Of all the three shuttlecock models, it is associated with the least drag. This shows that the gaps in the skirt lead to increased drag. Also shown in Fig. 3 are the values of CD for the synthetic and feather shuttlecocks with the realizable k�E turbulence model for Re¼ 2:22� 105. The values are slightly different than the ones obtained with the v2�f turbulence model. The difference between the results for the synthetic shuttlecock is 6% approximately, while it is much smaller for the feather shuttlecock. The relative aerodynamic performance of the feather and synthetic shuttlecocks is about the same with the different turbulence models. For both the models of shuttlecock the results from the v2�f model are closer to those from experiments. Therefore, the v2�f model has been used for obtaining most of the results in this study. The shuttlecock is a bluff body. However, the relative contribution of the viscous and pressure forces to the drag has not been investigated in earlier studies. Our analyses for the flow at Re¼ 2:22� 105 shows that the viscous drag accounts for only 4.60% and 1.98% of the total drag for the feather and synthetic shuttlecocks, respectively. For the gapless shuttlecock, Fig. 3. Variation of drag coefficient with Reynolds number for various models of shuttlecock. The values from earlier studies are also shown. Table 1 Re¼ 2:22� 105 flow past a synthetic shuttlecock: the pressure (CDP) and viscous (CDv) contribution to the drag coefficient, from various regions of the shuttlecock, as a percentage of the total drag. Region CDP (%) CDv ð%Þ (1) 10.754 0.366 (2) �0.473 0.011 (3) 8.205 0.034 (4) 34.936 0.146 (5) 39.564 1.015 (6) 5.031 0.404 Fig. 4. Pressure distribution (Cp) on the x�z plane for the (a) synthetic, (b) feather and (c) gapless shuttlecock for the Re¼ 2:22� 105 flow. Fig. 5. Variation of the coefficient of pressure (Cp) inside and outside the skirt along a typical stalk for the synthetic, feather and gapless shuttlecock for the Re¼ 2:22� 105 flow. A. Verma et al. / Journal of Fluids and Structures 41 (2013) 89–98 93 the viscous forces account for 5.86% of the total drag. Also, it is of interest to know the contribution of the various parts of the shuttlecock to its total aerodynamic drag. To this extent, a detailed analysis has been carried out for the synthetic A. Verma et al. / Journal of Fluids and Structures 41 (2013) 89–9894 shuttlecock at Re¼ 2:22� 105. The shuttlecock is divided into six regions as shown in Table 1. The cork forms the region 1. The two rings form regions 2 and 3. The net on the skirt has two parts. These are assigned as regions 4 and 5. All the rhombic stalks are clubbed in region 6. The contribution of the viscous and pressure contributions, for the various regions, as a percentage of the total drag is listed in this table. Most of the drag (� 75:66%) is from the net of the shuttlecock. Interestingly, the cork contributes only 11.12% of the total drag. While the rings and stalks are an integral part of the structure of the shuttlecock, their contribution to drag is relatively small. The shape and the density of the net as well as the shape of the second ring (region 3) are important design parameters in terms of generating the drag and the required moment for autorotation of the shuttlecock. In this study, the spin of the shuttlecock has not been accounted for. Fig. 6. Variation of the difference in coefficient of pressure (DCp) outside and inside the skirt along a typical stalk for the synthetic, feather and gapless shuttlecock for the Re¼ 2:22� 105 flow. 4.2. Coefficient of pressure Fig. 4 shows the distribution of the pressure coefficient on the x�z plane for the fully developed flow at Re¼ 2:22� 105 for the three models of the shuttlecock. In all the cases, close to stagnation pressure is observed in the region surrounding the nose of the cork. The pressure inside and outside the shuttlecock, near the skirt show interesting differences for the three models. To bring this out more clearly, Fig. 6 shows the variation of the coefficient of pressure, along a stalk, inside and outside the skirt. In general, the pressure inside the skirt of the shuttlecock is lower than that on the outside. This indicates the tendency of the shuttlecock skirt to compress as it moves through the flight. The difference between the inner and outer pressure contributes to the drag. Larger is the pressure difference (DCp), larger is the drag. We note that in the gap region there is only one value of the pressure. Therefore, these regions do not contribute to drag directly. It is only the solid/gapless surface of the skirt that directly contribute towards pressure drag. Fig. 5 shows that owing to the presence of gaps in the net on the skirt of the synthetic shuttlecock, there are alternating regions of low and high Cp. For the feather shuttlecock the pressure on the outer and inner surface of the stock are relatively comparable before t