HYDRODYNAMICS OF A FLAPPING FOIL IN THE WAKE OF A D-SECTION CYLINDER*

SHAO Xue-ming, PAN Ding-yi

State Key Laboratory of Fluid Power Transmission and Control, Department of Mechanics, Zhejiang University, Hangzhou 310027, China, E-mail: mecsxm@zju.edu.cn

HYDRODYNAMICS OF A FLAPPING FOIL IN THE WAKE OF A D-SECTION CYLINDER*

SHAO Xue-ming, PAN Ding-yi

State Key Laboratory of Fluid Power Transmission and Control, Department of Mechanics, Zhejiang University, Hangzhou 310027, China, E-mail: mecsxm@zju.edu.cn

The water environment of swimming fish in nature is always complex which includes various vortices and fluctuations. In order to study the interaction between the fish and its surrounding complex flow, the physical model with a D-section cylinder placed at the front of a flapping foil is employed. The D-section cylinder is used to produce vortices to contact with the foil as well as the vortices shed from the foil. According to the experimental work of Gopalkrishnan et al., there are three interaction modes between vortices shed from the cylinder and the flapping foil, which are expanding wake, destructive interaction and constructive interaction. Here in this article, three of those typical cases are picked up to reproduce the vortices interaction modes with the modified immersed boundary methods and their hydrodynamic performances are studied further. Results show that, for expanding wake mode and destructive interaction mode, the incoming vortices contact with the foil strongly, inducing relative low pressure domains at the leading-edge of the foil and enlarging the thrust of foils. For constructive mode, the foil slalom between the shed vortices from the D-section cylinder do not contact with them obviously and the foil’s thrust is only enlarged a little.

flapping foil, fish propulsion, vortices interaction, thrust enlargement

Introduction

Fish swimming in nature has many advantages such as high speed, high efficient, high maneuvering and low noise producing. Unlike traditional propeller based ship propulsion, fishes oscillate their body or fins to obtain thrust and swim forward and their outstanding propulsion performances are far beyond those of current manmade ships or underwater vehicles. Hence, studies on fishes’ distinctive swimming abilities and their hydrodynamic mechanism may provide new ideas on ship design and illuminate the future of marine industry and human life.

A great amount of endeavor has been devoted to fish swimming studies in the past decades. Among these, there are two simple models to represent fish swimming, the travelling wave plate and the flappingfoil (wing). Here in this article, the flapping foil model will be employed for study. The flapping foil (wing) undergoing a combined motion of heaving and pitching can be considered as a simple model of the caudal fin motion in the thunniform propulsion and has been well studied in the last two decades. In the 1980s, the wake structures of a plunging and pitching airfoil was captured in laboratories and the reversed Kàrmàn vortex street was observed when the foil owed thrust. An approbatory explanation of the flapping foil thrust is that an unsteady jet is formed behind the flapping foil and generates thrust acting on the foil. After that, the research group of Prof. Triantafyllou at Massachusetts Institute of Technology (MIT) conducted a series of experiments on the flapping foil. They first investigated the relationship between the flapping Strouhal (St) number and the foil propulsion performance. Flapping foil usually reach its maximum propulsion efficiency when its Strouhal number ranges from 0.25 to 0.35[1]and these results coincide with the fish swimming behavior in nature. They also systematically studied the propulsion performance of the twodimensional flapping foil which includes the thrust owing mechanism[2], effects of each flapping parameter[3], effects of angles of attack[4]and performanceof asymmetrical flapping[5]. Besides, there has also been a lot of numerical work in the last decade. Wang[6]first employed a single heaving elliptical foil to study the relationship between its forward motion and the heaving motion. Lewin and Haj-Hariri[7]presented the formations of leading-edge vortices under differentStnumbers and their influence on the foil thrust and propulsion efficiency using the similar model of Wang[6]. Guglielmini and Blondeaux[8]extended the model of Wang[6]to the fully coupled heaving and pitching foil and two reference coordinates were used to describe this coupled motion. Results showed that the additional pitching motion makes the thrust and efficiency of the flapping foil become larger. Lu et al.[9]elucidated three modes of the flapping foil leading-edge vortices formation and pointed out that different modes may result in the reverse Kàrmàn vortex street or normal Kàrmàn vortex street in the foil wake. Recently, Gao and Lu[10]investigated the ground effects on flapping foil propulsion. Hu[11]employed a new flapping model to represent the pectoral fins stroke. Shao et al.[12]simulated the flow fields of three-dimensional flapping wings and discussed the effects of aspect ratios on the wing propulsion.

All the work mentioned above placed the flapping foil (wing) into uniform incoming flow fields or quiescent flow fields. However, as a matter of fact, the real environment of swimming fish is always complex with some vortices or fluctuations around the fish bodies. In early years of the 21th century, scientists first observed the behavior of real fish swimming in complex flow in laboratory. Liao et al.[13-15]placed a live fish behind a D-section cylinder. When the incoming fluid flowed over the cylinder, vortices shed in the wake of the cylinder and the fish were located in the resulting vortex street. They observed that fish slalomed between vortices rather than through them. Compared to the swimming fish in free stream, its tail-beat frequency decreased and the body wavelength increased. They also measured that the activity of fish’s muscle decreased which indicated that fish might extract energy from the incoming vortices. Shao et al.[16]first simulated the flow fields similar with the above experiment and concluded that both the low pressure domain at the fish head and the reverse flow around the fish surface resulted from the incoming vortices make the owed thrust of fish enlarged. Besides these, a previous experimental work also intrigued the authors. Gopalkrishnan et al.[17]placed an oscillating D-section cylinder and a flapping foil with tandem arrangement in the incoming flow. The original motivation of this work is to make the vorticity active control by using the flapping foil. Three interaction modes between the vortices shed from the cylinder and the foil were concluded in their work. Obviously, the experimental setup also has some similarity with the real fish experimental work by Liao et al.[13]. Nevertheless, as a simply fish model, the flapping foil in Gopalkrishnan et al.’s experiments is still lack of a full investigation on its hydrodynamic performance and the interaction mechanism also need to be further studied. Thus, the authors try to fill this gap in this article numerically, in which the same model as that in Gopalkrishnan et al.’s experiments is employed. In the following sections, the physical model and the numerical methods will be first outlined and the analysis and discussion of our numerical results will be presented in Sections 3 and 4. The concluding remarks are given in the final section.

Fig.1 Physical model

1. Physical models

In this article, the two-dimensional flow fields of an oscillating D-section cylinder (D-cylinder) and a flapping foil with tandem arrangement are numerically studied. As is shown in Fig.1, the fluid flows over the D-cylinder and the flapping foil consequently, and the velocity of the incoming flow isU0, chosen as the characteristic velocity. The diameter of the cylinder equalsd, the foil chord length isC, and the distance between the cylinder and the foil isS. Select the foil chord lengthCas the characteristic length, the diameter is set asd=0.5C. The Reynolds number can be defined asRe=U0C/ν. Hereνrepresents the kinematic viscosity of the fluid. In this article, the Reynolds number is fixed asRe=1100 which is identical with the number taken in Gopalkrishnan et al.’s experiments[17].

The oscillating motion of the D-cylinder and the foil flapping can be described as follows:

wherehc(t) andhf(t) represent the heave motion of the cylinder and the foil respectively, andθf(t)represents the pitch motion of the flapping foil,Ac,Afandθ0are the corresponding amplitudes. All these three sinusoidal functions have the same frequencyf. Meanwhile,ψrepresents the phase difference between the heave motions of the cylinder and the foil, andΦrepresents the phase difference between the heave motion and the pitch motion of the foil. The whole computational fluid domain is 28Clong and 16Cwide. For the boundary conditions, a uniform constant velocity,U0, is specified at the domain entrance, as well as the top and the bottom boundaries and a no-reflect boundary condition is used on the outlet.

Table 1 Parameters list of the studied cases

According to the experiments of Gopalkrishnan et al.[17], three typical cases are picked up for study in this article marked with DF1, DF2 and DF3 (D-cylinder and Foil (DF)). The parameters selections are listed as follows in Table1. For comparison, two additional Cases SF1 and SF2 (Single Foil (SF)) for a single flapping foil with the same parameters are also considered. Thus, two groups are created and cases in each group have the same parameter selections, Group I: Cases DF1, DF2 and SF1, Group II: Cases DF3 and SF2.

The flow fields around the cylinder and foil are numerical simulated with the modified immersed boundary methods[18,19]. For the moving body immersed in flow fields, a virtual force can be added into the governing Navier-Stokes equations to represents the solid boundary. Thus, only the Cartesian grid was used with an additional interpolating procedure between the flow grids and the solid boundary. The efficiency of pre-process could be improved. The Navier-Stokes equations with the virtual force are shown as

HereVandPrepresent the velocity and pressure of the flow field respectively.Fis the virtual force.

The “discrete forcing approach”[19]is adopted here and the virtual force can be calculated as,

Table 2 Validation of the time step and grid independences

Fig.2 Vortex structures of the flow fields outside the D-section cylinder and flapping foil with tandem arrangement in a single period, mode1: Af/C=0.25, S/C=2.125 and θ0=30o

2. Analysis of the flow structures

In the experimental work of Gopalkrishnan et al.[17], a series of cases were studied by varying the physical parameters which include the heaving amplitudes of the D-cylinder and flapping foil and the distance between the cylinder and the foil. According to the interactions between vortices shed from the cylinder and the foil, in their work, three typical modes were concluded, which are expanding wake, destructive interaction and constructive interaction. Here in this article, three of those typical cases are picked up and numerically studied intending to reproduce the vortices interaction modes similar to the experimental work with our numerical approach. In addition, the hydrodynamic performances of the foils in different modes will be investigated to obtain a full understanding of flow structures and interaction modes. In this section, the flow fields of different cases will be analyzed first and the investigation of the foil hydrodynamic performances will be demonstrated in the next section.

2.1 Mode 1: Expanding wake

The Case DF1 which results in the expanding wake mode is analyzed first in this subsection. The parameters are chosen as Af/C=0.25, S/C= 2.125 and θ0=30o. The flow structures of the numerical results and the experiments’ hand-drawn figures are shown in Fig.2.

For each case, four subfigures corresponding to four different time levels (tU0/C=0.25T, tU0/ C=0.5T, tU0/C=0.75T and tU0/C=T, where T denotes the non-dimensional period) in a single period are presented. As is shown in Fig. 2, the fluid flows over the D-section cylinder and the flapping foil respectively, the vortices shed from the cylinder are marked alphabetically (such as A, B, C, …) and the vortices shed from the foil are marked with the numbers (such as 1, 2, 3, …).

Fig.3 Vortex structures of the flow fields outside the D-section cylinder and flapping foil with tandem arrangement in a single period, Mode 2:Af/C=0.25,S/C=1.8 andθ0=30o

Fig.4 Vortex structures of the flow fields outside the D-section cylinder and flapping foil with tandem arrangement in a single period, Mode 3: Af/C=0.333, S/C=2.625 and θ0=30o

2.2Mode 2: Destructive interactionmes shorter and the other parameters are still kept the same. Since the distance becomes shorter, the shed vortices from the cylinder would arrive at the foil’s leading-edge earlier. As is shown in Fig.3, the cylinder vortex contacts with the foil. It is destroyed in the numerical results and the flow field becomes disordered. After that, the cylinder vortex moves to the wake of the foil and contacts with the shed vortex from the foil with the opposite sign. Through a closer inspection, at the instance tU0/C=0.25T, the cylinder vortex “C” arrives at the trailing-edge of the foil and pairs up with the foil vortex “3”. When tU0/C= 0.5T and tU0/C=0.75T, the vortex “D” interacts with vortex “4” and vortex “E” contacts with vortex “5” at tU0/C=T. Unlike the above case, the shed vortices from cylinder and vortices with opposite sign contact with each other strongly and destructively in current numerical results, the intensity of the newly formed vortices become weak and their formation at the foil wake becomes disordered. 2.3 Mode 3: Constructive interaction

The flow structures of the third mode, constructive interaction, are shown in Fig.4. The parameters are fixed as Af/C=0.333, S/C=2.625 and θ0=30o. Unlike previous two cases, the heave amplitudes of the cylinder and foil are larger in this case. Hence, the resulting Kàrmàn vortex street in the wake of the D-section cylinder becomes wider and it is possible for the flapping foil to slalom between the cylinder shed vortices.

As is shown in Fig.4, the vortex formation around the flapping foil is regular and the foil does not contact with the cylinder vortices completely as a result of the wider vortex street. The flapping foil does slalom between the cylinder shed vortices. In the wake of the flapping foil, large vortices with opposite sign are arranged along the centre line of the flow domain. Obviously, these large vortices are constructed by the shed vortices from the cylinder and the foil respectively.When tU0/C=0.25T,thecylindervortex“B” has arrived at the foil’s trailing-edge and merges with the foil vortex “2” and the resulted vortex “2(B)”with the same sign of vortex “B” is located at the center line. At the same instance, the flapping foil is at the bottom of its heave stroke, and the incoming cylinder vortex “C” is upward, thus, there is no interaction happening between the cylinder vortex and the foil. WhentU0/C=0.5T, the cylinder vortex “C” merges with the foil vortex “3” in the wake of the foil. Moreover, during the construction procedure, the cylinder vortices dominate the foil vortices and the newly vortices have the same sign as the cylinder vortices.

3. Hydrodynamic performance of the flapping foil

In this section, the hydrodynamic performance of the flapping foil in the vortex street will be studied based on the numerical results. In the previous work of Gopalkrishnan et al.[17], the force measurements of the foil were also conducted in their experiments. However, unlike their flow-visualization experiments mentioned above, the Reynolds number for those force measurements experiments were set to 20 000. For such a high Reynolds number, the flow structures around the flapping foil may be different from that observed in the visualization experiments. Hence, the Reynolds number discrepancy between these two parts cannot create a correspondence between the flow structures and the force acting on the foil those both obtained in Gopalkrishnan et al.’s work. To eliminate this discrepancy, we extract the hydrodynamic data from our previous numerical results with the flow Reynolds number equaling 1 100.

Table 3 Hydrodynamic coefficients of the studied cases

Table 3 gives the mean thrust coefficients of the flapping foil in different cases. Here,represents the total thrust coefficient,the thrust coefficient caused by pressure andthe thrust coefficient caused by viscosity. For Cases DF1, DF2 and SF1, they have the same amplitudes, and they are assembled together as Group I for comparison. Cases DF3 and SF2 are classified into Group II. Especially Cases SF1 and SF2 represent the cases that single foil flap in uniform incoming flow without a cylinder located up front.

Fig.5 Evolutions of the instantaneous thrust coefficients in Cases DF1, DF2 and SF1 within certain time interval,tU0/C=30-40

As is shown in Table 3, for Group I cases, the mean thrusts of foils in Cases DF1 and DF2 are much larger than that in Case SF1. Therefore, in this group cases, the vortex streets play positive roles in the foil thrust enhancement, and for details, both the thrusts caused by pressure and viscous are enlarged by the vortex street. However, for Group II cases, the thrust of foil located in the vortex street (Case DF3) is only enlarged a little compared with that in Case SF2. Hence, the flow structures with different interaction modes mentioned in last section may also influence the thrust owing process of the foil. In order to get a clear insight, we exact the developments of instantaneous thrust coefficients within certain time interval and the results are shown in Figs.5 and 7 for the two groups respectively. As is shown in Fig.5, for Group I, the trends of thrust developments of all the three cases are similar, and the thrust coefficients oscillate periodically with approximately similar periods. However, the coefficient in Case SF1 is relatively larger. Figure 6 gives the instantaneous pressure and vortical contours around the foils at certain time corresponding with the black squares marked in Fig.5. For the DF cases, the foils are located in the vortex streets, and contact with the incoming vortices at their leadingedge. Usually, it is a relatively low pressure domain where the vortex is located. Hence, the vortices “E” in both Fig.6(a) and Fig.6(b) make the local pressure around the foils’ leading-edges lower in thrusts. Especially for Case DF2, the foil is closer to the D-cylinder and the pressure deducing caused by the incoming vortex “E” is larger and then its thrust is the largest in Group I.

For Case DF3, the comparison with the SF Case SF2 is shown in Fig.7. The evolutions of the thrust coefficients of these two cases are close. In this comparison group, for the DF Case DF3, the flapping foil thrust is not enlarged very much. To explain this, we make a review on the shedding vortex interaction mode of Case DF3. As is shown in Fig.4, the intera-ction mode corresponding to Case DF3 is the constructive interaction. For this parameter selection, the resulting shedding vortex street in the wake of the D-cylinder is wide, and the flapping foil can slalom in the vortex street without contacting with the incoming vortices. Therefore, the thrust enhancement process is absent in this interaction mode.

Fig.6 Pressure contours and vorticity contours of the flow fields around the flapping foils in different cases

Fig.7 Evolutions of the instantaneous thrust coefficients in Cases DF3 and SF2 within certain time interval, tU0/C=30-40

4. Conclusions

The flow fields within oscillating D-section cylinder and flapping foil have been numerically studied by using the modified immersed boundary method. As a simple model of swimming fish, flapping foil has been well studied in last two decades. On the other hand, the water environment of swimming fish in nature is always complex with various vortices and fluctuations. The D-section cylinder placed at the front of the foil is used to produce vortices to contact with the foil as well as the foil shed vortices.

Based on the experimental work of Gopalkrishnan et al.[17], according to the interactions between vortices shed from the cylinder and the foil, three typical modes can be concluded which are expanding wake, destructive interaction and constructive interaction. Here in this article, three of those typical Cases DF1, DF2 and DF3 are picked up and numerically studied to reproduce the vortex interaction modes similar to those in the experimental work with our numerical approach.

In addition, the hydrodynamic performances of the flapping foils in those cases have also been investigated. For the expanding wake mode and destructive interaction mode, the incoming vortices contact with the foil strongly and induced relatively low pressure domains at the leading edge of the foil to enlarge the foil thrust compared to the single flapping foil Case SF1. For the constructive mode, the foil slalom between the shed vortices from the D-section cylinder without contacting with them strongly. Compared to single flapping foil case with same parameter selection (SF2), when foil flapping behind the D-cylinder with the constructive mode, its thrust is only enlarged a little.

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June 13, 2011, Revised July 8, 2011)

* Project supported by the National Natural Science Foundation of China (Grant No.10872181), the National Key Basic Research Program of China (973 Program, Grant No. 2009CB724303) and the Fundamental Research Funds for the Central Universities (Grant No. 2010QNA4015).

Biography: SHAO Xue-ming (1972-), Male, Ph. D., Professor

2011,23(4):422-430

10.1016/S1001-6058(10)60132-1