Nomenclature
1. Environments
U: Forward speed
λW: Wavelength of the vortex wake
λ: Wavelength of a water wave
K: Wave number of a water wave
v: Wave frequency of a water wave
ωe: Encounter frequency
k: Reduced frequency
ρ: Fluid density
2. Foil Geometry and motion, velocity
l: Length of chord
η: Mean camber line
vm: Complex amplitude of the oscillatory velocity
vg: Vertical velocity of a wave field
v0: Vertical velocity on y = 0
vmβ: Relative vertical velocity of the deflection to vm0
W: Complex velocity amplitude of the wave field
q: Heave (plunging) motion
α: Pitch motion
β: Deflection of an aileron
a: Axis point of the pitch
p: Starting point of aileron
θq,θα,θβ,θW: Phase of heave, pitch, deflection of aileron, vertical wave velocity
3. Forces and mathematical functions
Lm , Lg: Lift force of a flat plate
Qm , Qg: Vortex strength at the leading edge of flat plate
H0, H1: Zeroth and first-order Hankel functions of the second kind
S: Extended Sears function
J0, J1: Zeroth and first-order Bessel functions of the first kind
μ: Argument of Bessel function
Lma: Additional lift force induced from an aileron
Qma: Additional vortex strength at the leading edge due to the aileron deflection
I1,I2,I3: Integral functions used in thrust
C: Theodorsen function
Ymn: Transformed integral functions used in thrust
TLma, TLβma,TLβga, TSmg, TSma: Additional Thrust induced from an aileron
Hαβ, Hqβ, Hββ, HWβ: QTFs between aileron and other parameters
CTa: Thrust coefficient induced from an aileron
i: Unit imaginary number
1. Introduction
Many researchers in various fields have analyzed a flapping foil (Wu, et al., 2020), mainly to determine the mechanism of thrust and the maneuvering of fish, cetaceans, or hovering of birds, and insects (e.g., Lighthill, 1975; Triantafyllou et al., 2000). Recently, several kinds of machines that mimic this mechanism in nature have been applied in many fields (e.g., Liu, 2002; Shyy et al., 2010).
Compared with the interest in a flapping foil, analysis of a propagating foil in unsteady flow fields, i.e., gust problem, has received less attention. Sears (1941) considered the effect of an unsteady wave field on the lift. They analyzed the lifting force using the so-called Sears function. Greenberg (1947) considered lift and thrust on an airfoil in an unsteady wave field. In gust problems, the lift of a foil or change in the angle of attack, stability by gust, and related flow control have been studied. Gursul and Ho (1992) examined a stationary airfoil in an unsteady water channel. They investigated the stream line of fluid near the foil and confirmed that the lift coefficient could be increased in an unsteady stream. The gust effect has also been applied to various situations. For example, Wang and Zhao (2016) examined the characteristics of the unsteady aerodynamic force of an air-foil under a freestream. Williams and King (2018) discussed a gust alleviation system for efficient flow control.
Researchers in the field of naval architecture and ocean engineering have considered the interactions between floating or submerged bodies and unsteady waves. In several topics, research on a foil attached to a ship in waves has attracted considerable attention. An anti-roll fin using the lifting force of a foil is one of the representatives and has been used widely for a long time. An anti-pitch fin located near the bow or stern has been evaluated to reduce the pitch motion of the ship (e.g., Pournaras, 1958; Becker and Duffy, 1959). Some reports have shown that it could reduce the pitch motion and the resistance of the ship, as reviewed in Naito and Isshiki (2005). An attached foil on a ship in waves has been used to produce a thrust directly because the energy from waves could be transferred to the ship and foil. Wu (1972) analyzed a flapping foil in waves, and Grue et al. (1986) extended Wu’s theory to consider the free-surface effect. Isshiki (1982a, 1982b), Isshiki and Murakami (1983, 1984), Naito et al. (2001), and Nagahama et al. (1986) studied theoretically and experimentally the generation of thrust by a wing attached to a ship. They reported that a wing attached to a ship in waves could be used for motion control, reducing resistance, and producing a positive thrust.
Choi (2012) derived an analytic solution on the thrust of a flapping flat plate in unsteady waves using linear potential-flow theory and transformed it into a quadratic transfer function (QTF) of heave (plunging), pitch, and unsteady wave velocity. A parametric study has been conducted for these physical quantities at different positions of the pitch axis. In addition, a sailing ship in waves with an attached foil near the bow has been considered (Choi, 2013). The added resistance could be reduced, and a thrust could occur by the hydrofoil, as studied by previous researchers. Politis and Tsarsitalidis (2014) and Tsarsitalidis and Politis (2015) adopted the boundary element method (BEM). They obtained the thrust at various Strouhal numbers for a wing and showed that it could be used as an energy-saving device in a ship. Bockmann and Steen (2013, 2016) showed experimentally and numerically that a wing attached to the bow of a ship could reduce the pitch motion and provide sufficient thrust.
Most of the above research examined a flat plate, which has no camber effect in the analysis of force. High-lift foil generally has a camber line or aileron near the trailing edge. These are generally chosen to obtain a large force in the same chord length. For a foil with an aileron, Theodorsen (1935) derived a closed-form solution in the potential-flow theory and analyzed flutter. Garrick (1937) extended the method of Theordosen and studied the thrust generation of a foil with an aileron. Many researchers have attempted to obtain both force and aeroelastic responses related to ailerons for various environmental conditions (e.g., Narkiewicz et al., 1995; Leishman, 1994; Alighanbari, 2002).
This study examined the thrust of a propagating foil with an aileron in an unsteady wave field. As an initial study, this study adopted two-dimensional linear potential-flow theory, which has been used to analyze the thrust on a flat plate in Choi (2012). The study was extended to analyze the additional thrust obtained by the deflection of an aileron. The analytic solution was derived and transformed into quadratic transfer functions (QTFs) of heave (plunging), pitch, unsteady wave velocity, and aileron deflection. The effects of these physical parameters on the additional thrust were investigated.
2. Mathematical Formulation
2.1 Problem Definition
This study adopted the same definition used by Choi (2012), which is briefly explained in this chapter. This study introduces a wing that propagates in a positive X-direction with time-harmonic unsteady oscillation, as shown in Fig. 1. The parameter of the unsteady motion of a wing described as l, λW, and U are the length of a chord, the wavelength of vortex wake, and the forward speed of a wing. In addition, it was assumed that a wing propagates in a vertically oscillating wave flow field, as shown in Fig. 2. The unsteady waves propagate in a negative X-direction with time-harmonic oscillation. The region of wave flow is infinite. Thus, the free-surface effect was not considered. The parameters of the wave-flow field are s λ, K, v, which are wavelength, wave number, and wave frequency of unsteady flow field. vg and W are the vertical velocity and complex velocity amplitude of an unsteady wave field.
In the body-fixed coordinate, the unsteady flow comes to a negative x-direction with forward speed U. Thus, the unsteady motion of a wing has encounter frequency. The velocity of the unsteady flow field is expressed in this form:
Under these conditions, this study used linear potential-flow theory for which the amplitude of time-harmonic oscillation and thickness of a wing are small. In addition, the amplitude of unsteady flow was much smaller than forward speed U. The linear lifting problem with the perturbation scheme could be addressed provided that the fluid is inviscid and the flow field is irrotational. Under this assumption, the flow at each side of the wing could be approximated by the value, y = 0, and the wing could be regarded as the mean camber line. Thus, the body boundary condition is expressed in the following form:
where v0 is vertical velocity on y = 0. Here, the Kutta condition was applied to the trailing edge. Eq. (4) could be divided by two conditions using the linear superposition scheme. The first was an oscillating wing in calm water (motion problem), and the other was negative unsteady flow with a fixed wing (gust problem). They are expressed in the following form (Choi, 2012):
where vm is the complex amplitude of the oscillatory velocity.
2.2 Lift and Moment of a Foil
After the vortex distribution on a foil was made, the lift and moment were analyzed by considering the vortex on the foil. They were calculated by integrating the pressure of two sides of a foil (Newman, 1977). First, the lift of the oscillating foil is expressed in the following form (Theodorsen, 1935):
where ρ is the fluid density. C(k) is the Theodorsen function; k is reduced frequency. The reduced frequency is defined in the following form:
According to Theodorsen (1935), lift and moment are composed of circulatory and non-circulatory forces. The Theodorsen function is related to circulatory force as follows:
where H0 and H1 are the Hankel functions of the second kind. F(k) and G(k) are real and imaginary parts of the Theodorsen function, respectively. The function C(k) shows the memory effect of the wake. When C(k) = 1, the wake effect is not contained, so the circulatory force becomes quasi-static.
The vortex strength at the leading edge should be calculated to obtain the leading edge suction force, which will be used to calculate the thrust. The asymptotic solution of vortex strength is as follows (Choi, 2012):
The lift and vortex strength for the fixed wing with an oscillatory flow field could be defined by substituting the flow velocity (−Weikx) instead of the oscillating velocity of a wing into Eq. (8) and Eq. (11) as follows:
where J0 and J1 are Bessel functions of the first kind. S is an extended Sears function. They are written as follows:
Provided that the argument in function m is much smaller than 1, the chord length is smaller than the wavelength of unsteady flow. In that case, the extended Sears function has the following form:
If m becomes infinite, the wavelength of an unsteady flow field is very small. In that case, the extended Sears function goes to zero, and the force does not occur, as expressed in Eq. (12). Using the linear superposition method, the original problem can be expressed by summing the above two problems. They are summarized in the following form:
2.3 Thrust of a Wing
This section addresses the formulation of thrust in unsteady linear potential theory. The thrust becomes a second-order force in linear theory and is expressed by the quadratic product of linear quantities after taking the time mean value into account. This article only describes the additional thrust induced from an aileron. Choi (2012) confirmed the mathematical formulation of the thrust of a flat plate.
The additional thrust induced by the deflection of an aileron can be expressed as
Lma and Qma are the additional force and vortex strength by the aileron deflection, respectively. α is the unsteady time-harmonic rotational motion (pitch) of a foil, and β is the deflection of an aileron, which are defined as
The thrust force is obtained using three types of integrations for calculating the force of a foil as follows:
where z = l/2 for a flat plate and z = pl for an aileron. p means the starting point of deflection. Using the above definition, each thrust induced from the deflection of an aileron could be summarized in this form.
Coupling between the pitch and additional lift from an aileron (Fig. 3)
Coupling between the deflection of an aileron and additional lift from an aileron (Fig. 4)
Coupling between the deflection of an aileron and the additional lift induced by gust on the aileron (Fig. 5)
Leading edge suction induced from deflection of an aileron and other motions (Fig. 6)
Leading edge suction induced from gust effect and deflection of an aileron (Fig. 7)
where both c.c. and asterisk (*) mean complex conjugate.
2.4 Vertical Velocity Profile and Integration for the Calculation of Thrust
The mean camber line and vertical velocity profile should be given to obtain the thrust from the equation. The integration results were calculated using the vertical velocity profile in each case.
2.4.1 Mean camber line
Obtaining the vertical velocity profile requires first considering the mean camber line of a wing (Fig. 8).
If a wing has an aileron, it has additional deflection, which has an arbitrary angle β. In this case, the mean camber line of the plate can be expressed as follows.
where p is the axis of the deflection of an aileron with a range of from −0.5 to 0.5, as shown in Fig. 9.
2.4.2 Vertical velocity profile
The vertical velocity profile in flapping motion could be obtained considering the mean camber line of a wing as follows:
where vmβ is the relative velocity of deflection to vm0. ωβ is the angular frequency of deflection. In this study, it was assumed that the same was true of the encounter frequency.
2.4.3 Integration I1, I2, I3
The thrust of a plate with an aileron was obtained considering the partial integration obtained by deflection and flapping motion of an aileron for the deflection region, as described by Theodorsen (1935). Each of them is summarized as follows:
2.5 QTFs of Thrust
Thrust is composed of the quadratic product of several linear quantities. Thus, they are expressed using the QTFs of these physical variables, which are summarized as follows. The thrust of a flat plate is composed of the quadratic product of physical quantities, such as heave (q), pitch (α), vertical velocity of the wave field (W), and deflection of an aileron (β) if a wing has an aileron. The additional thrust obtained by deflection was analyzed by re-expressing it by the QTFs of a non-dimensional amplitude of each physical quantity such as |α0|, |β0|,
| q 0 l | | W U |
After some algebra, each component of QTFs by deflection of an aileron is summarized as follows:
(1) Coupling between the deflection of aileron and pitch
(2) Coupling between the deflection of an aileron and heave
(3) Aileron deflection only
(4) Coupling between the deflection of an aileron and the vertical velocity of a wave
3. Parametric Study on QTFs
This chapter discusses details of the QTFs of thrust obtained in the previous chapter. The introduced thrust coefficients under specific conditions are then introduced, and the maximum values are deduced under the given conditions.
3.1 Lifting Force
The lifting force was compared with the result obtained by the formula reported by Theodorsen (1935) to confirm the integrals in Eqs. (38)–(48). The lifting force of a plate with an aileron was calculated using the value in Table 1 and Eq. (13) in Theodorsen (1935). The results were compared with the force result in Eq. (8). Fig. 11 presents the time series of additional lifting force obtained from aileron deflection. Two different positions of deflection were selected, and the starting points of deflection were p = −0.25 (c = 0.5 in Theodorsen) and p = 0.0, respectively. The reduced frequency was set arbitrarily to 1.0, and the amplitude of deflection was 0.1. Fig. 11 shows the time series of the lifting force with both methods. The force at each deflection point showed good agreement.
3.2 Thrust QTF of a Flat Plate with Aileron
The QTFs of additional thrust obtained by the deflection of an aileron were analyzed. Using Eqs. (47)–(59), the QTFs were calculated for different centers of pitch motion (α), the starting point of deflection (pl). In addition, the QTFs with a phase difference between the deflection (β) and other heave motions (plunging, q), pitch motions (α), and vertical velocity of wave (W) of an aileron were mainly considered. The discussion on each QTF was as follows:
3.2.1 Thrust induced from pitch and deflection (Hαβ)
Fig. 12 shows the QTF of additional thrust by pitch (α) and deflection of an aileron (β). The starting point of deflection changed from −0.375 to 0.0 in 0.125 intervals, and three locations of the pitch axis were selected (−1/4l, 0, 1/4l). The amplitude of the thrust QTF generally decreases as the starting position of deflection approaches the trailing edge. This appears natural because QTF is composed mainly of a lifting force and rotational motion. The additional lifting force by deflection decreases as the deflection region decreases. Therefore, the additional thrust is also reduced. Next, the position of the pitch axis does not have a substantial effect at a low reduced frequency (k). In the high reduced frequency (k) region, however, the positive values decrease, and negative values approach a positive value. Finally, the phase difference between the pitch (α) and deflection of an aileron (β) has a substantial influence on the trend of the QTF value at each pitch axis location. For example, the left figures show that a large phase difference generally gives a positive value. The QTF value becomes a negative value as the phase difference decreases. On the other hand, the right figures show that the QTF value depends on a reduced frequency (k) and phase difference.
3.2.2 Thrust induced from heave and deflection (Hqβ)
Fig. 13 shows the thrust QTF of the heave (plunging, q) and deflection (β) of the aileron. The amplitude of thrust QTF decreased as the deflection region decreased, similar to the previous thrust QTF for pitch (α) and deflection (β). The largest value was obtained when the phase difference between the heave and deflection of the aileron became 0.5 π and it decreased as the phase difference was far from 0.5 π. The QTF generally shows a larger amplitude at a high reduced frequency (k) region than at a low reduced frequency (k) region.
3.2.3 Thrust induced from wave field and deflection (HWβ)
Fig. 14 shows the QTFs of the additional thrust induced by the vertical velocity of the wave field (W) and deflection (β) of the aileron. According to the phase difference between the two variables, the QTF generally shows an opposite trend. A small phase difference has a positive value at a low reduced frequency (k) region, and it decreases as the reduced frequency (k) increases. On the other hand, the large phase difference has a negative value at a low k region, and it increases as k increases. Similar to other QTFs of the thrust component, the amplitude becomes small as the starting position as the aileron approaches the trailing edge.
3.2.4 Thrust induced from deflections (Hbb)
Fig. 15 shows the thrust QTFs of the additional thrust induced only by the deflection (β) of an aileron. In this figure, each line means the QTF value at each starting position of deflection. The QTF generally has a negative value, but it moves in the positive direction as the reduced frequency(k) increases. The position of the starting point of deflection has relatively little effect on the QTF.
4. Conclusions
The QTFs of additional thrust for propagating foil with an aileron in unsteady wave fields were derived using two-dimensional potential-flow theory. The derived formulations for the thrust are transformed into QTFs, which are composed of deflection and other motions such as heave, pitch, and vertical velocity of the wave field. A parametric study on each QTF was conducted, and the effects of the phase difference between deflection and other motions were considered. The following results were obtained:
(1) Thrust induced from the pitch and deflection of the aileron had a large value when the phase difference was large, except in the high-frequency region.
(2) The thrust induced from the heave and deflection of the aileron had the largest value when the phase difference was 0.5 π and the value decreased as the phase difference became far from 0.5π.
(3) The thrust induced from the wave vertical velocity and deflection of the aileron showed a different trend depending on the phase difference. The thrust increased with frequency when the phase difference was large and vice versa.
(4) The thrust induced from the deflection of the aileron only generally had a negative value.
This study used the basic assumptions based on linear potential theory, which has some limitations. In particular, the viscous effect on the leading edge suction force was large, and detailed research on the leading edge vortex (LEV) when the angle of attack is relatively large is needed. Designers can use this material as reference data for comparison with the result of a more sophisticated scheme.
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