Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships

Journal of Bionic Engineering 14 (2017) 770–780

Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships Naga Praveen Babu Mannam1, Parameswaran Krishnankutty1, Harikrishnan Vijayakumaran1, Richards Chizhuthanickel Sunny2 1. Ocean Engineering, Indian Institute of Technology Madras, Chennai 600036, India 2. Texas A&M University, College Station, TX 77840, USA

Abstract The use of biomimetic tandem flapping foils for ships and underwater vehicles is considered as a unique and interesting concept in the area of marine propulsion. The flapping wings can be used as a thrust producing, stabilizer and control devices which has both propulsion and maneuvering applications for marine vehicles. In the present study, the hydrodynamic performance of a pair of flexible flapping foils resembling penguin flippers is studied. A ship model of 3 m in length is fitted with a pair of counter flapping foils at its bottom mid-ship region. Model tests are carried out in a towing tank to estimate the propulsive performance of flapping foils in bollard and self propulsion modes. The same tests are performed in a numerical environment using a Computational Fluid Dynamics (CFD) software. The numerical and experimental results show reasonably good agreement in both bollard pull and self propulsion trials. The numerical studies are carried out on flexible flapping hydrofoil in unsteady conditions using moving unstructured grids. The efficiency and force coefficients of the flexible flapping foils are determined and presented as a function of Strouhal number. Keywords: biomimetic propulsion, flapping foil, penguin locomotion, Strouhal number, tandem arrangement, thrust coefficient Copyright © 2017, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(16)60442-0

Nomenclature α α0 y η P (CX) FX P D R

μ ω Φ ψ ψ0 ρ

Instantaneous angle of attack Maximum angle of attack Instantaneous sway velocity Efficiency Average power coefficient Average thrust coefficient Average thrust Average power Drag force Resultant force Dynamic viscosity of fluid Circular flapping frequency Phase difference between sway and yaw Instantaneous yaw angle Yaw amplitude Fluid density

Corresponding author: Naga Praveen Babu Mannam E-mail: [email protected]

(x, y)

φ A y0 c CX, CY CM f FX , FY L M P p Re s St t

Effective flexible motion coordinates of centerline Conservative scalar quantity Characteristic width of flapping hydrofoil Sway amplitude Chord length of the hydrofoil Force coefficients corresponding to FX, FY Moment coefficient Flapping frequency Forces in x and y directions Lift force Moment due to lift and drag forces Power Pressure Reynolds number Span of the hydrofoil Strouhal number Time

Mannam et al.: Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships

t' T U V y

Non-dimensional time Flapping period Frees stream velocity Control volume Instantaneous sway position

1 Introduction Subsea technology is among most exciting areas of modern engineering that currently experiences revolutionary changes. To flourish subsea applications such as ocean exploration, seafloor mining or submarine communications require intensive use of Autonomous Underwater Vehicles (AUVs) and Autonomous Surface Vehicles (ASVs) having longer endurance and eco-friendly with less disturbances. Marine mammals use less energy for their locomotion and create less noise compared to manmade propulsion systems. This study aims to explore a new design philosophy for autonomous crafts, inspired by propulsion mechanisms of aquatic organisms. A particular focus will be on aquatic birds such as turtles and penguins which are capable of swimming and maneuvering. They are considered to be the best performers among aquatic animal locomotion[1]. The present work will combine experimental and numerical studies of penguin type propulsion and motion mechanisms. This approach will allow identification of hydrodynamic behavior of penguin pectoral fins and its mechanisms in swimming, which benifits in the design and development of hybrid underwater robots which are able to swim and maneuver. Expected design solutions may go well beyond terrestrial applications and appear to be instrumental for development of AUVs for deployments in extraterrestrial seas such as Titan’s Karken Mare[2]. Aquatic birds posses flying ability by using flapping wings to generate sufficient lift to stay afloat in water and produces thrust for propelling in the forward direction. Studies on aquatic animals have shown that flapping foils can be an efficient means of propulsion for marine vehicle[3–19]. Lua et al.[20] studied the aerodynamics of 2D flapping foils in tandem configuration in forward propulsion at Re = 5000 using experimental and numerical techniques. Both tandem foils are subject to symmetrical simple harmonic heave and pitch motions with a Strouhal number of 0.32. The observations show that the foil-wake interaction is favorable to thrust gen-

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eration, which requires the tandem wing has to cross the shear layer shed from the front wing and results in increasing effective angle of attack of the foil. Liu et al.[21] performed numerical and hydrodynamic experiments to analyze the propulsive performance of an ocean wave energy extraction device called a Wave Glide Propulsor (WGP) in regular waves. The propulsor consists of six flapping rigid foils arranged in tandem with asynchronous flapping motion. From the comparison of computational and experimental results, the WGP produces larger thrust force at smaller wave length or at larger wave height. Yuan et al.[22] conducted water tunnel tests, for a pitching-plunging 2D airfoil and a flapping 3D wing. Force and Particle Image Velocimetry (PIV) measurements were carried out to acquire sets of data that are considered acceptable for the validation of corresponding Computational Fluid Dynamics (CFD) simulations. It shows good agreement between the experimental and numerical results. Chae et al.[23] studied the Fluid-Structure Interaction (FSI) response and stability of a flexible foil in dense, turbulent, and incompressible flow using unsteady RANS fluid solver coupled with a 2-DOF solid solver through an efficient and stable numerical algorithm. Politis and Politis[24] discuss about the effect of flapping wings on thrust production and motion control. Schouveiler et al.[25] conducted experimental studies on flapping hydrofoil propulsor, inspired from thunniform swimming mode. The study investigated the effects of variations of the Strouhal number and the maximum angle of attack on the thrust force and on the fin hydro-mechanical efficiency. In the present work, numerical studies are carried out to ascertain the thrust producing mechanism of flapping foils and then experimental studies are performed on a ship model fitted with two flapping foils in tandem at its bottom. The current work attempts to investigate the thrust generation capability and efficiency of the flexible hydrofoil in open water condition and the results of which are compared with experimental ship model in self propulsion mode. The foil flexibility in CFD environment is introduced using an appropriate user defined function. The foils used in the ship model test are also flexible and the flexibility closely matches with the numerically simulated fin.

2 Hydrodynamics of flapping propulsion system Penguins and other marine animals overcome the

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drag by producing thrust using its flapping fins. The fins oscillate in water creating a pressure difference across the surfaces of the fin, and generating an upward force called lift. This principle is the same as the airplane producing lift. The hydrodynamic forces acting on the penguin is shown in Fig. 1a. Aquatic bird wings generate lift and thrust as shown in Fig. 1a and the penguin fin motion is similar to a helicopter rotor which produces a lift force and thrust. Helicopter rotor continuously rotates, where as the penguins flapping wings produce reciprocal motion. The flapping motion is in the form of inclined ellipse or like “8” shape shown in Fig. 1b. Helicopters inclines to fly forward by tilting the nose down, where as penguins tilt the stroke plane of flapping motion. The flapping motion consists of down stroke and upward stroke. At the same time the penguin is moving forward resulting the flow in the backward direction. The upward transverse flow and the backward flow result in a oblique flow with an angle of attack as shown in Fig. 2a. In down stroke flapping motion, shown in Fig. 2a, when the wing moves downwards the transverse flow is in upward direction. It generates lift force “L” perpendicular to the fluid flow and drag force “D” parallel to the wing and the resultant hydrodynamic force R, which is the vector sum of lift and drag forces. The resultant force (R) can also be resolved into components along the penguin body longitudinal axis and normal to it. The

Fig. 1 Penguin motion and force components. (a) Hydrodynamic forces acting; (b) penguins mode wing flapping.

Fig. 2 (a) Flapping wing down stroke; (b) magnitude of resultant force (R) in down stroke and up stroke of flapping wing.

normal component supports the weight of the penguin and the axial component is the forward force (thrust) opposing the body drag. The flapping fin produces a forward thrust in this stroke. In upward stroke, the angle of attack with respect to the fluid motion reduces and it may become zero. The resultant force has smaller magnitude in upstroke and larger magnitude in down stroke. The flapping wing flexes during upstroke. The upstroke is usually a “recovery stroke”. It generates negative thrust during this motion. The flapping wing resultant force magnitude during down stroke and upward stroke is shown in Fig. 2b. The tip of the flapping fin generates most of the thrust. This phenomenon is the same as in the marine screw propeller, where in screw propeller, the tangential velocity of propeller blade increases with radius of the propeller from the axis of rotation. The tangential velocity near the hub region is very low compared to the tip of the propeller blade. The cross section of flapping fin near the body and tip are shown in Fig. 3. At section x (near the body), the flapping speed is less compared to flapping speed at section y (near the fin tip)[26].

3 Experimental study of flapping foil propulsors 3.1 Ship model with flapping foil propulsion system The lift-based penguin type propulsion system installed in a ship model, used in the present study, is shown in Fig. 4. A pair of flexible flapping foils is attached to the ship model bottom. The main dimensions

Fig. 3 Force and velocity diagrams of the fin near the body and near the tip of wing.

Fig. 4 Ship model fitted with flapping foils.

Mannam et al.: Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships

of the ship model are length overall is 2.94 m, breadth 0.66 m, depth 0.25 m and draft 0.18 m. The ship model is operated at a designed speed of 1 m·s−1. Two foils are fitted at the ship model bottom in a tandem orientation. The amplitude and frequency of these flapping foil oscillations are varied using the electro-mechanical systems. 3.2 Flapping foils Two foils are arranged in tandem at the bottom of the ship model. The foils are flexible, made of polyurethane, and are counter-oscillating. The flexibility of flaps enables it to have a variable angle of attack along its span. Flexible flaps are chosen because they are found to be more efficient than rigid ones. The span of the fins is 300 mm with a tapering from root to tip. The dimension of the top cross section has a maximum thickness of 32.6 mm and a chord length of 150 mm. The tip section has a chord length of 120 mm. The distance between two flapping foils, trailing edge of the forward foil to the leading edge of the aft foil, is 100 mm and the gap between the rotating shaft centre to the root of the fin is 50 mm. Fig. 5 shows the dimensions and arrangement of the flapping foils. A right handed cartesian coordinate system, fixed to the moving carriage, is used with the x-axis aligned with the flow and the z-axis pointing upwards and with anti-clockwise rotation taken as positive. The foil “flaps” (rotate) about x axis (rotating shaft).

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formed on the vehicle. Fig. 6 shows the front view of the geometric model of the foils for two different angular positions ± 45° and ± 90°. Considering the flapping motion of the hydrofoil cross section will have only oscillatory sway motion in a cylindrical plane. This is the case when a rigid material is used for the flapping foil. But here flexible material is used which in turn provides each foil cross section a yaw motion along with the sway motion. 3.4 Experimental tests In the experimental study, flapping foil ship model resistance tests, bollard pull tests and self propulsion tests are carried out in the towing tank and the results are discussed below. 3.4.1 Flapping foil propulsion ship – resistance tests The ship model fitted with fins and other fixtures was run in the towing tank to experimentally determine the model resistance at different speeds. The model was built in fiber glass to the scale 1: 15. The towing tank dimensions are 82 m long × 3.2 m wide × 2.5 m (water depth). The model was ballasted to the loaded condition with even keel. Model towing tests were conducted in the speed range covering the design speed. The resistance test results are shown in Fig. 7. The effective

3.3 Motion of flapping foil The two flaps oscillate in opposite directions to cancel out any rotational moment which could be Fig. 6 Flapping foils at ± 45˚ and ± 90˚ orientations.

Fig. 5 Dimensions and arrangement of flapping foil.

Fig. 7 Resistance tests of flapping foil propulsion ship.

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power of the model with fins is 33.15 W at a speed of 1.3 m·s−1. 3.4.2 Bollard pull tests Conventionally, bollard pull tests are done for tugs, where the tug is tied to a bollard (a fixed structure usually land based) using a rope and the tension in the rope is measured by putting the engine throttle to the maximum. In the present study, the ship model is tied to a stationary post and the flaps are oscillated at different frequencies and oscillation amplitudes corresponding to different Strouhal numbers. The thrust developed by the flapping foils is measured at each case using a load cell. Fig. 8 shows the measured thrust variation against flapping frequency (Hz). The maximum thrust was found to be 21.0 N at f = 3 Hz. 3.4.3 Self propulsion tests Self propulsion tests are carried out to determine thrust produced at various speeds. The ship model is fitted with an electric motor and shear beam load cell fitted onboard for the measurement of thrust force. During a self propulsion test, the model speed is matched with the carriage speed to find the speed achieved by the model at different operating conditions of the foils. The foils are operated at different frequencies and amplitudes and the forces are recorded using the data acquisition system. In self propulsion tests, a maximum thrust of 13.18 N was recorded at St = 0.53. The thrust force (N) versus Strouhal number (St) is shown in Fig. 9.

4 Numerical study of flexible flapping hydrofoil 4.1 CFD modeling and solution This study computes the unsteady turbulent

Fig. 8 Thrust in bollard pull condition (experimental study)versus flapping frequency.

flow-fields by solving the Navier-Stokes equations using unstructured triangular grids. The simulations are performed using the commercial software ANSYS Fluent 14.5, based on the control-volume method. The flow field in all runs is assumed to have an associated Re = 40000, and the conservative variables are solved sequentially. The momentum equations are discretized with a second order upwind scheme, while pressure is interpolated at cell faces from calculated cell center values using a second order accurate scheme. An implicit first order scheme is used for temporal discretization and the pressure-velocity coupling of the continuity equation is achieved using the Semi Implicit Method For Pressure Linked Equation (SIMPLE) algorithm. The turbulence model chosen for the current study is the kω-SST turbulence model as it offers significant advantages for non-equilibrium turbulent boundary layer flows. The kω-SST model can be used as a low Re turbulence model, good behavior in adverse pressure gradients and separating flow, when compared to other models like Spalart-Allmaras and k-ε, making it an excellent choice for the current study[23]. All elements in the domain were modeled using unstructured triangular elements as the flapping motion causes large deformations in the mesh. The standard National Advisory Committee for Aeronautics (NACA0012) airfoil section with a chord length c = 0.1 m is used for the hydrofoil modeling. The dynamic mesh technique is employed to model the flapping motion of airfoils by creating User Defined Function (UDF) which could be hooked up to the ANSYS Fluent solver. The hydrofoil is positioned at a distance of 3c from the inlet. The computational domain has a length of 14c and width of 10c. The instantaneous flow velocity in computational domain is dependent motion of the flapping hydrofoil. A no-slip boundary condition is imposed on the hydrofoil surface,

Fig. 9 Thrust in self propulsion mode (experimental study) produced by the flapping foils at different Strouhal numbers.

Mannam et al.: Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships

which takes the tangential velocity of flow at the surface to be zero and perpendicular velocity to be the hydrofoil velocity. The top and bottom boundaries of the domain are defined as no-slip walls. Free stream velocity of 0.4 m·s−1 and zero static pressure conditions are specified at the inflow and outflow boundaries, respectively, with 10325 Pa taken as the reference pressure. The simulations are carried out for flexible flapping foil and rigid flapping foil by varying the parameters α0 and St. The α0 values are taken as 10˚, 15˚ and 20˚, and for each value of α0, St is varied from 0.1 to 0.4 in steps of 0.025. The grid independence study and validation of computational set up were performed for rigid flapping motion case and compared with the experimental data by Schouveiler et al.[25]. 4.2 Kinematic study of flexible flapping hydrofoil The present study needs a numerical simulation to make the hydrofoil undergo a flexible flapping motion. This motion essentially consists of translational sway and rotational yaw motion about a pivot point, with an additional flexibility component. These sway and yaw motions are achieved by: y (t ) = y0 sin(ω t )

ψ (t ) = ψ 0 sin(ω t + φ )

for sway motion, for yaw motion.

(1) (2)

In the above equations, y0 is the sway amplitude, ψ0 is the yaw amplitude, ω is the oscillation frequency (which is taken as same in sway and yaw), φ is the phase difference between sway and yaw. The geometric details and particulars of the flexible flapping hydrofoil used in the current study are shown in Fig. 10. The key idea is to integrate the deflection of the hydrofoil due to its chord wise flexibility and yaws motion into an effective flexible motion, and separately consider the sway motion.

CY =

CM =

FY , 1 ρU 2 cs 2 M , 1 2 2 ρU c s 2

775 (4)

(5)

where U and c are the free stream flow speed and hydrofoil chord length, respectively. s is the span of the hydrofoil and is taken as unity. The thrust force created by flapping hydrofoil is FX(t). The average thrust over n complete flapping cycles over period T is given as: FX =

1 nT

∫

nT

0

FX dt.

(6)

The power required for the flapping hydrofoil, defined as the amount of energy imparted to the hydrofoil to overcome the fluid force, is given as: P(t ) = − Fy (t )

dy dψ − M (t ) . dt dt

(7)

The sign on both the terms in Eq. (7) is negative as vertical force and moment are the reaction forces created by the fluid as the hydrofoil moves through it. The average power over n complete cycles of flapping, over a period T, is given as: P=

P =

1 nT

∫

nT

0

Pd t ,

( P) . 1 3 ρU cS 2

(8) (9)

The hydro-mechanical propulsive efficiency η is defined

4.3 Performance parameters In order to analyze the results obtained from the above simulations, non-dimensional flow variables are used. The forces acting on flapping hydrofoil are resolved in x and y direction as FX(t) and FY(t), respectively, and the moment about the pivot point is denoted as M(t). The corresponding force coefficients are given as: CX =

FX , 1 2 ρU cs 2

(3)

Fig. 10 Definitions of principal parameters of flexible flapping hydrofoil, sway y(t), yaw 2ψ(t).

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as the ratio of the output power FX U to the input power ( P ). From Eqs. (3) and (9), η is given by:

η=

FX U (C X ) = . P P

(10)

A major parameter on the performance of a foil is the maximum nominal angle of attack of flow to the foil. The relative velocity of flow to the hydrofoil results from free stream velocity U, and the sway velocity dy ( y (t ) = ) on the other. If α(t) denotes the instantanedt ous flow angle of attack, referenced at the pivot point, then

tan[α (t ) + ψ (t )] =

y (t ) . U

(11)

Therefore, the instantaneous angle of attack of the foil, i.e., the angle between the effective velocity and the centerline, is given by:

α (t ) = tan −1 (

y (t ) ) − ψ (t ). U

(12)

The maximum values of α(t), denoted as α0, occurs when the phase angle between sway and yaw is π/2, and is given as:

α 0 = tan −1 (

ω y0 U

) −ψ 0 .

(13)

Another parameter of importance in the flapping foil performance is the Strouhal number St, a non- dimensional form of flapping frequency, defined as: St =

fA , U

(14)

where f denotes the frequency of foil oscillation in Hz, ω , and A denotes the oscillation amplitude. i.e. f = 2π Since A is unknown before measurements are made, it is taken as sway amplitude, i.e. A = 2y0. 4.4 Results and discussions Wake characteristics: As the hydrofoil undergoes flapping motion, a jet of water that provides thrust is created downstream of the hydrofoil, which is due to the formation of reverse von Karman vortices. The qualitative assessment of the wake characteristics is done using velocity magnitude con-

tours and absolute vorticity contours. Velocity magnitude contours helps in visualizing the jet flow and the absolute vorticity contours helps in visualizing the vortex shedding pattern. Mathematically, vorticity is defined as the curl of the velocity vector, which for a two-dimensional flow field simplifies down to ∂u ∂u − ωz = . ∂x ∂y The wake characteristics of flexible hydrofoil which undergoes flapping motion with different maximum angle of attack values, viz. 10˚,15˚ and 20˚, at selected Strouhal number values, viz. 0.1, 0.2, 0.3 and 0.4. As the Strouhal number increases from 0.1 to 0.4 are studied here, the flow leaving the trailing edge of the flapping hydrofoil transforms to that like a jet of water, which is the primary reason for thrust production. This is characterized by the formation of alternating reverse von Karman vortices, which increases in strength with increase in Strouhal number up to a certain level and then decreases. Opposed to the von Karman vortex shedding, the upper edge of hydrofoil produce counter clockwise vortices and the lower edge produce clockwise vortices, causing an increase in the flow velocity in the wake region. The number of fully shed vortices visible in the computational domain is proportional to the Strouhal number. For instance, for St = 0.1 the number of fully shed complete vortices is 1, for St = 0.2 it is 2, St = 0.3 it is 3 and St = 0.4 it is 4 (Fig. 11). Also, the vortices formed at higher Strouhal numbers reach stable circular shape faster, when compared to lower Strouhal numbers. The vortices become more prominent as the maximum angle of attack is increased from 10˚ to 20˚ (Fig. 12). At St = 0.1 in 10˚ maximum angle of attack case (Fig. 11a), the visibility of vortex is near zero causing zero effective thrust generation, but as the maximum angle of attack is increased to 15˚ to 20˚, the strength of the vortex increases significantly resulting in gradual increase in thrust production. The absolute vorticity contours for flexible flapping hydrofoil at α0=10˚ and 20˚ for St = 0.1 to 0.4 are shown in Figs. 11 and 12. 4.5 Performance of flexible flapping hydrofoil The performance characteristics of the hydrofoil undergoing flexible flapping and rigid flapping motion is studied in detail. The motion characteristics of the hydrofoil at varying Strouhal numbers (0.1 to 0.4) is

Mannam et al.: Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships t'=0.0

t'=0.5

t'=0.0

t'=0.5

t'=0.0

t'=0.5

t'=0.0

t'=0.5

(a)

(b)

(c)

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this case, as mentioned earlier, the yaw angle amplitude becomes negative, and the average value (CX) is negative (very close to zero). (2) From Fig. 13, the CY profile amplitude increases with St for a given maximum angle of attack. The profile amplitude is also found to be increasing with increase in α0. The CY profile starts showing twin peaks of increasing magnitude from the St values above which the angle of attack profile is no longer similar to sine profile. The effect of twin peaks is less prominent as the α0 is increased to 15˚ and 20˚. The CY profile variation is very smooth in lower angle of attack case (α0 = 10˚) for all St. As the α0 increases, the CY profile corresponding to low St shows sharp fluctuations. This can be attributed to the effect of low ψ0 values. (3) From Fig. 13, the CM profile amplitude increases with increase in St for a given maximum angle of attack. The profile amplitude is also found to increase with α0. The CM profile variation is very smooth in lower maximum angle of attack case (α0 = 10˚) for all St. As the t'=0.0

t'=0.5

(d)

Fig. 11 Absolute vorticity contours of flexible flapping hydrofoil at α0 = 10˚ for (a) St = 0.1; (b) St = 0.2; (c) St = 0.3; (d) St=0.4.

given for each value of maximum angle of attack selected (α0 = 10˚, 15˚, 20˚) and comparison is made with the force and moment coefficients (CX, CY, CM) time series for flexible flapping motion and rigid flapping motion. A study of average thrust coefficient (CX) and efficiency (η) variation with Strouhal number, for the two modes of flapping are also made for each value of maximum angle of attack selected (α0 = 10˚, 15˚, 20˚). Performance characteristics: (1) From Fig. 13, the (CX) profile amplitude increases with increase in St for a given maximum angle of attack, causing an overall increase in (CX) with respect to St. The (CX) profile attains its most positive value at α0 = 10˚, but the most negative value is also attained in the same case, making the average value (CX) over a cycle to go down. Although the positive value of (CX) values are decreasingly lower for α0 = 15˚ and 20˚, the negative values also decrease with increase in α0, causing a systematic increase in (CX) with increase in α0. The exception to this observed at α0 = 20˚ and St = 0.1. For

(a) t'=0.0

t'=0.5

t'=0.0

t'=0.5

t'=0.0

t'=0.5

(b)

(c)

(d)

Fig. 12 Absolute vorticity contours of flexible flapping hydrofoil at α0 = 20˚ for (a) St = 0.1; (b) St = 0.2; (c) St = 0.3; (d) St = 0.4.

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(a)

(b)

(c)

Fig. 13 Variation of (a) CX, (b) CY, (c) CM for St = 0.1 to 0.4 at α0 = 10˚ for flexible foil.

α0 increases, the CM profile corresponding to low St shows sharp fluctuations. This may again attributed to the low ψ0 values.

5 Comparison of experimental and numerical results From the experimental results in self-propulsion mode, the thrust coefficient (CX = 2.75) was measured at St = 0.53. Due to practical limitations of the experi-

mental setup, the ship model flapping foils could be operated only up to St = 0.53. The thrust force was measured using shear beam load cell fitted on the flapping foil rotation axes. From the thrust force, thrust coefficient (CX) is determined by using Eq. (3). Numerical simulations are performed at α0 = 10˚, 15˚, 20˚ for St = 0.1 to 0.8. The maximum thrust coefficient (CX = 2.38) was obtained for α0 = 20˚ at St = 0.55. The thrust coefficient (CX) versus Strouhal number for flexible and

Mannam et al.: Experimental and Numerical Study of Penguin Mode Flapping Foil Propulsion System for Ships

rigid flapping foils are shown in Fig. 14. Furthermore, neglecting fin to fin interactions, it is assumed that the thrust force acting on first and second foils are equal. The total thrust force generated by a pair of tandem foils in experimental self propulsion mode and CFD open water condition can be predicted using numerical and experimental studies. The rigid foil simulations were carried out only up to St = 0.4 and it has been observed that the flexible foils outperform the rigid foils substantially. The efficiency (η) for single foil versus Strouhal number (St) at α0 = 10˚, 15˚, 20˚, for St = 0.1 to 0.8 is shown in Fig. 15. The efficiency (η) profile attains its maximum value near St = 0.25 for all the considered values of α0. The overall average efficiency for a flapping foil decreases with increase in, α0 but the η profile attains its global maximum value for α0 = 15˚, which makes the most efficient combination of parameter as

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α0 = 15˚, St = 0.25. The efficiency of flapping foils have been observed as coming down consistently and gradually after crossing the peak at around St = 0.25 as expected.

6 Conclusion The thrust generation capability and efficiency of flapping hydrofoils at different parametric variations (α0, St, ψ) have been studied experimentally and numerically. Both rigid and flexible foils have been analyzed to evaluate the hydrodynamics performance. The current simulations do not include the hydrodynamic interactive effects between the fins. The current model considered only chord wise flexibility, where as in reality flapping fins undergo chordwise and spanwise deformations. From the studies it is observed that the maximum thrust generation by the flexible flapping hydrofoil occurs at St = 0.4 and α0 = 20˚, where the thrust coefficient (CX) is 1.24 and efficiency (η) is 0.54. But, the most efficient thrust generation occurs at St = 0.225 and α0 = 15˚, where the thrust coefficient (CX) is 0.48 and efficiency (η) is 0.73. The CX and η parameters can be used as a tool for determining suitable flapping parameters to achieve good efficiency for a required thrust coefficient.

Acknowledgment The authors would like to thank Department of Ocean Engineering, Indian Institute of Technology Madras, India for providing financial support for doing this project. Fig. 14 (CX) versus St at α0 = 10˚, 15˚, 20˚ for single flapping foil.

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