Numerical Analysis of Unsteady Viscous Flow through a Weis-Fogh-Type Water Turbine

International Conference on Emerging Trends in Computer and Image Processing (ICETCIP'014) Dec. 15-16, 014 Pattaya (Thailand) Numerical Analysis of Unsteady Viscous Flow through a Weis-Fogh-Type Water Turbine K. D. Ro, S. Y. Han, H. G. Ju, and J. G. Kim Abstract In this study, a rotating-type water turbine model that applied the principle of the Weis-Fogh mechanism was proposed, and its hydrodynamic characteristics were calculated by advanced vortex method. Primary condition was set at wing chord C=1, rotation radius of wing shaft r=1.5c, and rotating angle velocity ω=1.0. The unsteady flow field and pressure field around the wing for two revolutions were calculated by changing uniform flow from U=1.0~3.0 and the maximum opening angle of the wing at α=0, 30, and 36 to each calculation parameter, which are important design factors. The summary of results is as follows. The average thrust increased as the rotating angle velocity increased. The maximum efficiency for one wing of the water turbine was 45.3% at the maximum opening angle of the wing α=36 and velocity ratio U/rω=.0. The flow field of the water turbine is very complex because the wing rotates and moves unsteadily in the channel. However, using the advanced vortex method, accurate calculation was possible. Keywords Computational Fluid Dynamics, Advanced Vortex Method, Unsteady Flow, Water Turbine. T I. INTRODUCTION HE Weis-Fogh mechanism[1,], which was modeled on the hovering flight of a small bee about 1mm in size, called Encarsia formosa, is gaining attention[3-6] among several scientists who study hydrodynamics because of its unique, efficient lift-generation mechanism. Recently, engineering applications[7-14] of the mechanism have also been actively attempted. The engineering applications to date are as follows. Furber and Ffowcs Williams[7] reported that by applying the principle of this mechanism to an axial compressor, they improved the efficiency of the compressor. Tsutahara et al.[8] applied a two-dimensional model of the mechanism to a ship s propeller and showed that it works very effectively as a propulsion mechanism. In addition, Tsutahara et al. constructed a pump[9] and a fan[10] that applied this mechanism, and through characteristics tests, they showed that K. D. Ro is professor in the Department of Mechanical System Engineering, Gyeongsang National University, Gyeongnam, 650-160 Republic of Korea (corresponding author to provide phone: +8-55-77-9103; fax: +8-55-77-9109; e-mail: rokid@gnu.ac.kr). S. Y. Han, is undergraduate student in the Department of Mechanical System Korea (e-mail: 1115111@naver.com). H. G. Ju, is undergraduate student in the Department of Mechanical System Korea (e-mail: jhg8911@naverl.com) J. G. Kim, is graduate student in the Department of Mechanical System Korea (e-mail: pmsco@ymail.com) the mechanism is feasible as a pump or a fan. Recently, Ro et al. conducted a model ship sailing test[11] and a performance improvement study[1] using a spring on various Weis-Fogh type propulsion models to realize the practical application of the mechanism. Also, Ro proposed a reciprocating water turbine model using this mechanism, and through numerical calculation, he showed that this mechanism works effectively as a water turbine[13]. Ro studied power coefficients and efficiency acting on the wing to various experimental parameters, and gained result of maximum efficiency of 6% at the opening angle of 36 and velocity ratio(u/v) of.0[14]. Because most of the currently used generator for water turbine are rotating types and the water turbine mentioned above is a reciprocating one, mechanical loss is unavoidable during the conversion of rotating motion to reciprocating motion. Therefore, in this study, a rotating Weis-Fogh type water turbine model, which has a similar motion to the reciprocating type, is proposed, and the hydrodynamic characteristics of the water turbine model will be studied using advanced vortex method[15, 16] to examine the possibility of practical application. Meanwhile, vortex method[17], which is a kind of a boundary integral equation, is a method that expresses vorticity distribution that is present in the fluid as discrete vortices and traces it in Lagrangical way to analyze the flow field. This calculation method is simpler to modelize the flow compared to other calculation methods, the physical meaning is easily understandable, and it does not need grid generation. Therefore it is very useful in analyzing unsteady flow field such as this water turbine where the wing rotates in the water channel. II. CALCULATION METHOD A. Reciprocating and rotating type water turbine models Figure 1 shows (a)the reciprocating type and (b)the rotating type models of Weis-Fogh type water turbine. The wing movement in the reciprocating model (a) is as follows. By only setting the opening angle α at the beginning of each stroke, because of uniform flow U, a lift is generated on the wing, which makes point p to move translationally in the y direction at the velocity of V. More specifically, the wing rotates and opens from the lower wall, with point p as the center pivot point (opening stage), then moves translationally, maintaining the opening angle α (translational stage), and finally rotates and closes at the upper wall, with point p as the 4

International Conference on Emerging Trends in Computer and Image Processing (ICETCIP'014) Dec. 15-16, 014 Pattaya (Thailand) pivot point (closing stage). Then, the wing repeats the motion: it Fig. 1 Models of Weis-Fogh type water turbine rotates and opens from the upper wall, moves translationally, and finally rotates and closes at the lower wall. Also in the rotating type water turbine model (b), if opening angle α is set at the beginning of each stroke, by uniform flow U, a lift is generated on the wing, which makes point p, which corresponds to the wing shaft, to rotate counter-clockwise at the angle velocity of ω, maintaining rotation radius r at point q, the center of the water channel. At this point, the trailing edge of the wing is touching the lower wall, and it rotates and opens with point p as its pivot point (opening stage). Then, maintaining a certain opening angle α, it moves upward (translating stage) and closes with the leading edge touching the upper wall with point p as the pivot point (closing stage). The wing repeats the motion: it rotates and opens from the upper wall with the wing shaft as the pivot point, moves downward, and finally rotates and closes at the lower wall. B. Calculation of the flow fields by the advanced vortex method The calculation of the flow fields - the velocity and the pressure fields - by the vortex method is based on vorticity transport equation derived from the rotation of Navier-Stokes equation and pressure-poisson equation derived from divergence. dω = ( ω grad) u+ ν ω dt (1) p = ρ div( u grad u) () Here, u and ω each represents velocity vector and vorticity vector that is defined as ω= rot u. Advanced vortex method is calculated by applying Biot-Savart law[15] for the velocity, and integral equation formulated by Uhlman[18] for the pressure. Meanwhile, in the vortex method, the distribution of vorticity in the flow field is expressed by introducing discrete vortex element into the flow field. In calculating this flow field, since the same method was used when calculating the flow field of reciprocating type Weis-Fogh propulsion mechanism[16] for boundary condition, the introduction of nascent vortex elements, and calculation method of velocity field and pressure field, the specific method will not be elaborated. C. Calculation of fluid force and definition of characteristic coefficients The flow force F acting on the wing of the water turbine is calculated by integrating the normal component of the pressure p and the tangential component of shear stress τ w to the wing surface as follows: F = ifu + jfv = {( p n ) + τ w t } ds0 (3) S0 Here, F u and F v each represent the components of force I n the U and V directions, which corresponds to x and y directions respectively, and S 0 represents the integral path according to the wing surface. Meanwhile, force coefficients C u and C v in the U and V directions, which represent the hydrodynamic characteristics of the water turbine, are non-dimensionalized by uniform flow U and calculated as follows: Fu Cu = (4) 1 ρ U S Fv Cv = (5) 1 ρ U S Here, ρ represents the density of fluid, and S represents the area of the wing below the water surface. Also, the efficiency of the water turbine η is the ratio of the net output generated from the wing to the input and is calculated as follows: Mω η = 100 (6) 1 3 ρ AU Here, A represents the cross-sectional area of the water channel. Also, M represents the torque, and the numerator of Equation (6), that is, the output, can be calculated as Mω r = F ( ) ω = F v( xp xq) Fu ( yp yq) ω. III. RESULTS AND DISCUSSIONS First, the conditions for calculations were similar to those in the reciprocal type calculations[13,14] done previously. For 5

International Conference on Emerging Trends in Computer and Image Processing (ICETCIP'014) Dec. 15-16, 014 Pattaya (Thailand) primary conditions, the wing chord was set at C = 1, the rotation radius of the wing shaft was set at r = 1.5C, the rotating angular velocity was set at ω = 1, and the distance from the trailing edge to the wing shaft was set at r = 0.5C. Meanwhile, the validity of the calculation method such as comparing calculation and experimental results has been sufficiently discussed in the reciprocating type calculations[19], therefore it will not be elaborated. Figure shows the changes in the wing positions according to rotating angle φ and the time variation of C u and C v during two revolutions. p C u and C v both start from positive values, but as the rotating angle increases, both values decrease sharply then increase; then, as the wing gets close to the lower wall, the values have a tendency to decrease sharply then increase. Figure 3 and Figure 4 show pressure distributions around the wing surface with the same conditions as Fig., when the wing is at the center of the upstream and downstream water channels. Fig. 3 The pressure distributions around the wing at the center of the water channel(θ=180, U=3.0, α=36 ) Fig. Positions of a wing and time variation of Cu and Cv with rotating angle of rotor for two revolutions(α=36, U=3.0) Here, rotating angle φ means the rotating angle from the origin of the wing shaft, that is, φ = 3 π / θ. The movement of the wing in Fig. 3(i) shows that the wing opens from the lower wall, pivoting on the wing shaft, then rotates while maintaining a certain opening angle a, and finally closes at the upper wall. Then, the wing repeats the motion: it rotates and opens from the upper wall, moves translationally, and finally rotates and closes at the lower wall. In Fig. (i), (a), (e), (d), and (g) are the center points of rotating angle during opening and closing stages. (b), (c), (f), and (g) are points 1/3 and /3 of translational stage. Each position (a) (h) in Fig. (ii) corresponds to each position (a) (h) in Fig. 3(i). Also, in Fig. 3(ii), the dotted line C u represents force coefficient in U direction, and the solid line C v represents force coefficient in V direction. When C u and C v curves are compared, overall, both curves oscillate in the same direction according to rotating angle φ, and in the same rotating angle, C v is bigger than C u. In the first 1/ rotation((a)~(d)), at the beginning of the opening stage, C u and C v both have negative values, but as the rotating angle increases, both values also increase, then as the wing closes, the values decrease very sharply. In the next 1/ rotation((e)~(h)), at the beginning of the opening stage, In Fig. 3 and Fig. 4, arrows facing toward the surface of the wing represent positive(+) pressure, and the arrows facing outward represent negative(-) pressure. First, in Fig. 3, to the uniform flow, positive pressure is mostly acting on the pressure face, and negative pressure is acting on the back face. When associated with the opening angle of the wing, this tendency demonstrates that upward lift is acting on the wing. Next, in Fig. 4, to the uniform flow, positive pressure is mostly acting on the pressure face, and negative pressure is acting on the back face. When associated with the opening angle of the wing, this tendency demonstrates that downward lift is acting on the wing. Also, Fig. 4 The pressure distributions around the wing at the center of the water channel(θ=0, U=3.0, α=36 ) 6

International Conference on Emerging Trends in Computer and Image Processing (ICETCIP'014) Dec. 15-16, 014 Pattaya (Thailand) considering the size and the directions on the surfaces of the two wings, C u and C v values both have + values as shown in Fig. (ii), but each value is bigger when the wing is in the upstream than the downstream. When the considering the values with the rotating radius r in Fig. 1, there is clockwise rotation torque acting on both wings. Figure 5 and Figure 6 show vortex distributions at each position and equi-vorticity contours as the wing rotates once, with the same conditions as Fig.. In Fig. 5 and Fig. 6, (a)~(h) each corresponds with the wing position (a)~(h) in Fig. (i). In the vortex distribution in Fig. 5, the red dots represent clockwise vortex, and the blue dots represent counter-clockwise vortex. Therefore, we can see that under the lower water channel, clockwise vortex is generated, and under the upper water channel, counter-clockwise vortex is generated. When (b), (c), and (d) of Fig. 5 and Fig. 6 are examined consecutively and simultaneously, shedding vortices that is generated from the leading and trailing edges of the wing are joined together as time goes on to slantly connect with the vortices on the lower wall. Also, when (f), (g), and (h) are consecutively examined, in (f), big shedding vortices can be seen around leading and trailing edges of the wing opposite of the direction of the wing movement, but in (g), the shedding vortex around the leadingedge is separated from the wing and joins the shedding vortex around the trailing edge to flow to the wake stream. Especially, when comparing the vortex patterns of (b) and (f) in Fig. 5 and Fig. 6, (f) has bigger shedding vortex than (b). This is due to the fact that in (b), the wing moves in the opposite Fig. 6 Equi-vorticity contour around one wing with positions ((a) (h) on the figure correspond to (a) (h) on Fig., U=3.0, α=36 ) direction to the uniform flow, whereas in (g), it moves in the same direction as the uniform flow which makes the relative opening angle bigger in (f) than in (b), as shown in Fig. (ii). Table 1 shows the average value of C u, C v and η according to the changes of uniform flow during two revolutions of the wing In the table, ( U / rω ) is the ratio of uniform flow to the circumferential speed. As shown in the table, as uniform flow increases, C u and C v both increase. However, average efficiency η increases as uniform flow increases, showing maximum value of 45.3% at U =.5, then decreases at U = 3.0. The velocity ratio( U / rω ) at the point of maximum average efficiency is.0. AVERAGE VALUE C u, v TABLE 1 C AND η WITH UNIFORM FLOW(Α=36 ) Fig. 5 Vortex distributions around one wing with positions ((a) (h) on the figure correspond to (a) (h) on Fig., U=3.0, α=36 ) In the previous study of the flow field calculation of the reciprocal Weis-Fogh type water turbine[14], the maximum value of average efficiency was also at the opening angle α = 36 and the velocity ratio U / V =.0. Therefore we can see that the efficiency of the water turbine depends heavily on the opening angle and the velocity ratio. Also, as shown in Table 1, in various velocity ratio, the average efficiencies are very high, more than 38.0%, and the efficiency can be 7

International Conference on Emerging Trends in Computer and Image Processing (ICETCIP'014) Dec. 15-16, 014 Pattaya (Thailand) increased by increasing the number of wings, which brings expectation for practical usage. IV. CONCLUSION In this study, a rotating-type water turbine model that applied the principle of the Weis-Fogh mechanism was proposed, and its hydrodynamic characteristics were calculated by advanced vortex method. Primary condition was set at wing chord C = 1, rotation radius of wing shaft r = 1.5C, and rotating angular velocity ω = 1. As calculating coefficients, uniform flow was set from U = 1.0 to 3.0, and the maximum opening angles were set at α = 0, 30, and 36. To each calculating parameter, unsteady flow fields and pressure fields around the wing for revolutions were calculated. The summary of results is as follows. 1) C u and C v acting on the wing oscillated greatly in the same direction with the change in rotating angle. ) During the translational stage, shedding vortices were formed in the leading and trailing edges of the wing, and the size was bigger where the uniform flow was smaller. 3) Average values of C u and C v increased as the uniform flow and the opening angle increased. 4) The maximum efficiency for one wing of the water turbine was 45.4% at the opening angle α = 36 and velocity ratio U / rω =.0. [10] Tsutahara. M. and Kimura. T, Study of a Fan Using the Weis-Fogh Mechanism (An Experimental Fan and Its Characteristics), Transactions of the JSME, Vol.60, No.571, 1994, pp.910-915. [11] K.D. Ro, J.Y. Seok, Sailing Characteristics of a Model Ship of Weis-Fogh Type, Trans KSME, Vol.34, 010, pp. 45-5. [1] K.D. Ro, Performance Improvement of Weis-Fogh Type Ship s Propulsion Mechanism Using a Wing Restrained by an Elastic Spring, J Fluids Eng, Vol.13, 010, pp. 041101-1~041101-6. [13] Ki-Deok Ro, "Numerical Calculation of Unsteady Flow Fields: Feasibility of Applying the Weis-Fogh Mechanism to Water Turbines," Journal of Fluids Engineering, Vol. 135, No.10, 01, pp. 101103-1-101103-6. [14] Ki-Deok Ro, "Calculation of Hydrodynamic Characteristics of Weis-Fogh Type Water Turbine Using the Advanced Vortex Method," Trans. of the KSME(B), Vol.38, No.3, 014, pp. 03-10. [15] K. Kamemoto, On Attractive Features of the Vortex Methods, Computational Fluid Dynamics Review 1995 Wily & Sons, 1995, pp. 334-353. [16] K.D. Ro, B.S. Zhu, H.K. Kang, Numerical Analysis of Unsteady Viscous Flow Through a Weis-Fogh Type Ship Propulsion Mechanism Using the Advanced Vortex Method, J Fluids Eng, Vol.18, 006, pp. 481-487. [17] Leonard, A., Vortex methods for flow simulations, Journal of Computational Physics, Vol. 37, 1980, pp.89-335. [18] J.S. Uhlman, An Integral Equation Formulation of the Equation of Motion of an Incompressible Fluid, Naval Undersea Warfare Center T R., 199, pp. 10-086. ACKNOWLEDGMENTS This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning(No. 01-0555) REFERENCES [1] Weis-Fogh, T, Quick Estimates of Flight Fitness in Hovering Animals, Including Novel Mechanism for Lift Production, Journal of Experimental Biology, Vol.59, 1973, pp. 169-30. [] Lighthill, M. J, On the Weis-Fogh Mechanism of Lift Generation, Journal of Fluid Mechanics, Vol.60, Part 1, 1973, pp. 1-17. [3] Maxworthy, T., "Experiments on the Weis-Fogh Mechanism of Lift Generation by Insects in Hovering Flight. Part 1. Dynamics of the 'Fling'," Journal of Fluid Mechanics, Vol. 93, 1979, pp. 47-63. [4] Ro, K. D. and Tsutahara. M., "Numerical Analysis of Unsteady Flow in the Weis-Fogh Mechanism by the 3D Discrete Vortex Method with GRAPE3A," Journal of Fluids Engineering, Vol. 119, 1997, pp. 96-10. [5] Maxworthy, T., "The formation and maintenance of a leading edge vortex during the forward motion of animal wing," Journal of Fluid Mechanics, Vol. 587, 007, pp. 471-475. [6] Kolomenskiy, D., Moffatt, H.K., Farge, M. and Schneider, K., The Lighthill-Weis-Fogh clapfling-sweep mechanism revisited, Journal of Fluid Mechanics, Vol. 676, 011, pp.573-606. [7] Furber, S. B. and Ffowcs Williams. J. E, Is the Weis-Fogh Principle Exploitable in Turbomachinary?, Journal of Fluid Mechanics, Vol.94, Part 3, 1979, pp. 519-540. [8] Tsutahara, M. and Kimura. T, An Application of the Weis-Fogh Mechanism to Ship Propulsion, Transactions of the ASME Journal of Fluids Engineering, Vol.109, 1987, pp. 107-113. [9] Tsutahara, M. and Kimura. T, A Pilot Pump Using the Weis-Fogh Mechanism and Its Characteristics, Trans JSME, Vol.54, 1988, pp. 393-397. 8