Self-Propulsive Property of Flexible Foil Undergoing Traveling Wavy Motion: A Numerical Investigation

Abstract

The propulsive characteristics of self-propelling 3D flexible foil are numerically studied. Two kinds of dynamic boundary techniques, namely the dynamic mesh technique and overlapping mesh technique, are used to realize the self-propulsion of flexible foil. The effects of aspect ratio (AR), characteristic thickness (d), and section shape on propulsive characteristics are numerically studied. Results demonstrate that the moving velocity increases monotonically with the consistent growth of AR, and a linear relationship is found between them. The peak value of propulsive efficiency can be acquired when AR = 1.0. Moreover, the growth of d shall produce a negative effect on moving velocity. It is suggested that the value of d should be smaller than 0.15 for the sake of acquiring high propulsive efficiency. As for the section shape effect, the foil with a rectangular shape presents the worst propulsive property, while the NACA0015 foil exhibits the best one. Furthermore, the typical vortex structures are also exhibited and analyzed. The conclusions acquired in this study are of great significance for designing a bionic underwater vehicle.

1. Introduction

In the natural selection process of the ‘survival of the fittest’, existing aquatic creatures, especially the predators of advanced aquatic animals in the ocean, have developed a complete set of cruising methods and skills with low energy consumption, high propulsive efficiency and rapid mobility [1,2]. For a long time, researchers have had a peculiar interest in exploring the mystery of these aquatic animals and revealing the inherent movement laws to create the best bionic transportation tool to meet the needs of military and civilian industries [3,4,5].

Fish’s motion can be simplified as oscillating motion [6]. In previous research, the foil was fixed, and the uniform incoming velocity was imposed on the computation domain, assuming that the oscillating foil is in forward motion [7]. The foil does not truly move forward, and the propulsive property cannot be revealed from the mooring simulation. Therefore, the auto-propelled foil also needs to be studied. In the typical work of Wu [8], he studied the impact of motion types on rigid oscillating foil. The corresponding boundary conditions were active heaving and active pitching (AHAP), passive heaving and active pitching (PHAP), and active heaving and passive pitching (AHPP), respectively. Xu [9] numerically investigated the self-propelling performance of 2D NACA0012 foil under numerous traveling wave lengths, and the characteristic Re number was found to change from 500 to 1750. Similarly, the propulsive characteristic of 2D NACA0014 undulatory foil was simulated by Benkherouf [10]. The innovation in the latter work lies in the zero-velocity inlet boundary condition that was introduced. The authors claimed that the pure heaving presents the best propulsive property. The effect of the wall effect on 2D self-propelling foil was investigated by Dai [11], who concluded that the wall produces a surge in the peak value of lift force of oscillating foil, so as to enhance its propulsive property. Similarly, Tang [12] expanded the above 2D research object to the 3D condition and analyzed the effect of the structural property of the foil on its propulsive property. A similar work was conducted by Xu [13], in which two changes stand out. First, the research object was changed from a flexible plate to a rigid oscillating foil. For the second, the viscous effect was ignored in Xu’s work for the sake of utilizing the potential theory method. Apart from the simplified model selected for simulating the self-propulsion of fish, few works have employed the entire fish model [14,15,16]. A typical one is the research of Li [17], who took the tuna fish as the physical model to evaluate the propulsive property by implementing the immersed boundary method. In other research, the devilfish was modeled by Su [18] to analyze the effect of motion frequency and amplitude on its propulsion speed.

Although many works concerning the auto-propelled characteristic of foil have been conducted [19,20,21,22,23,24,25,26], two major deficiencies still exist. First, almost all physical models chosen in the available literature are 2D models, while little attention is paid to the 3D model that more closely resembles the real world. In addition, the aspect ratio (AR) of the 2D model can be considered infinite, while the corresponding value of AR in reality often lies in a small range. Hence, it is particularly necessary to take the 3D effect into account. Second, the research object adopted is limited to NACA series foil. The influence of the geometric parameter is still unknown. In the near future, the flexible traveling foil will be utilized in the propulsion of unmanned underwater vehicles (UUVs) for its good maneuverability and low noise. In order to give practical suggestions for the design of traveling wavy foil-based UUVs, there exists much necessity in investigating the connection between the propulsive properties and shape parameters of flexible foil. In view of this, the current study numerically studied the propulsive characteristics of 3D foil with the variation of aspect ratio (AR), characteristic thickness (d) and section shape. In the following, the research object coupled with the motion equation is presented. In Section 3, the simulation tool is described, followed by the simulation results and discussion. The major conclusions acquired in this study are drawn in Section 4.

2. Research Object

The calculation conditions in the current study are listed in

Table 1. Calculation conditions of the current study.

Section Shape	Aspect Ratio (AR)	Characteristic Thickness (d)
1. Effect of section shape on the propulsive property of 3D flexible foil
NACA0015	AR = 1	d = 0.15 C
Elliptical
Rectangular
2. Effect of aspect ratio on the propulsive property of 3D flexible foil
NACA0015	AR = 0.25~4	d = 0.15 C
3. Effect of thickness on the propulsive property of 3D flexible foil
NACA0015	AR = 1	d = 0.04 C~0.50 C

. As for the shape effect, three distinct shapes are selected, namely NACA015 foil, rectangular shape and elliptical shape. In addition, a comparatively large range of AR, changing from 0.25 to 4.0, is considered to analyze the AR effect. Similarly, a wide range of thickness, lying in the range of [0.04, 0.50], is designed to investigate the thickness effect.

The definition of AR, shown in Figure 1, taking the NACA0015 foil as an example, can be calculated as, AR = B/C. When it comes to the characteristic thickness, it denotes the maximum thickness in the x-axis direction, denoted as d.

As shown in Formulas (1) and (2), the flexible oscillating motion or traveling wave equation is adopted by the above physical model to achieve the forward motion. The parameters λ and f represent the traveling wave length and the traveling frequency.

$$y(x,t)=A(x)\sin ({\displaystyle \frac{2\pi }{\lambda }}x-2\pi ft)$$

(1)

$$A(x)=a_0+a_1·x+a_2·x^2$$

(2)

Figure 2 shows the diagram of the foil’s deformation process in a single period under the various combinations of motion parameters.

3. Introduction to Methodology

3.1. Governing Equations

The 3D incompressible viscous Navier–Stokes equations are selected as the governing equations of the surrounding flow field, and the corresponding equations are expressed in Formula (3), where u is the flow velocity, μ is the kinematic viscous coefficient of the flow, p is the pressure and ρ is density of the fluid.

$$\nabla \cdot u=0, {\displaystyle \frac{\partial u}{\partial t}}+(u\cdot \nabla )u=-{\displaystyle \frac{1}{\rho }}\nabla p+{\displaystyle \frac{\mu }{\rho }}{\nabla }^{2}u$$

(3)

The finite-volume method is used to discretize Formula (3) using ANSYS Fluent software V17.2. The SST k-ω turbulent two-equation model is selected to enclose the governing equation for the sake of comparative high calculation accuracy. The standard wall function is adopted to handle with the flow near the wall. The first-order implicit method is utilized to deal with the time term in the governing equation, the second up-wind is adopted to cope with the convection term, and the central difference scheme is used to discretize the diffusion term. The coupling of pressure and velocity is solved by using the SIMPLE method.

3.2. Computation Domain and Grid

As seen in Figure 3, the cuboid-like domain consists of a computational domain that can be divided into two parts, the overlapping grid domain and the background grid domain. As for the overlapping domain (seen in Figure 3b), the size is designated as (x, y, z) = (3C, 3C, B + 2C). Considering that the AR effect will be analyzed, the value of span length (B) is set as a variable, and so is the length of the overlapping domain along the z-axis direction. The foil is located at the center of the overlapping domain. When it comes to background domain, the relevant size is chosen as (x, y, z) = (38C, 12C, B + 6C), and the overlapping domain is also located in the center.

The boundary conditions setting can be seen in Figure 3a,b. Concerning the computation grid (seen in Figure 3c), the whole-structured grid is imposed on the computation domain to improve the computation accuracy and efficiency. Moreover, the refined grid is designed on the near surface of the foil to precisely capture the vortex structures. The value of y⁺ is set as y⁺ ≈ 1 for the purpose of solving the flow in the region of viscous sub-layer without using any wall functions. In our study, the dynamic grid technique is adopted to capture the moving boundary of the flexible foil. In addition, the grid smoothing technique is applied to achieve slow deformation of the grid. More specifically, the closer the mesh is to the wall, the slower the deformation will be. As a result, the corresponding height of the first layer will remain almost unchanged.

3.3. Solving Procedure

The solving procedure is shown in Figure 4. The UDF of DEFINE_GRID_MOTION defines the foil’s flexible motion, and the dynamic grid technique is applied to realize the corresponding deformation. When the position of the foil is acquired, the corresponding fluid force will be calculated with the help of ANSYS Fluent. Next, the corresponding acceleration will be calculated according to Formula (4), where F is the force acting on the foil, m denotes the mass of the foil, and V denotes the moving velocity.

$${\displaystyle \sum F=m\frac{dV}{dt}}$$

(4)

After specifying the acceleration, the forward velocity can be obtained by using Formula (5), where F_x denotes the force along the forward direction, and uⁿ and uⁿ⁻¹ represent the foil’s forward velocity at this time step and previous time step, respectively.

$$u^n={\displaystyle \frac{F_x}{m}}\Delta t+u^{n-1}$$

(5)

Finally, the UDF of DEFINE_ZONE_MOTION is adopted to acquire the new position of the foil.

The relevant hydrodynamic parameters in the current study are defined in Formulas (6)–(10). Specifically, the non-dimensional drag force coefficient (C_Fx) is expressed in Formula (6), and the input power and the corresponding input power coefficient are defined in Formulas (7) and (8), respectively, where P_in denotes the input power, v_n and p represent the velocity and pressure on the foil surface, and Γ represents the foil surface. The output power is defined as P_out = T_mean·u_Mean, where T_mean is the average magnitude of thrust force. The propulsive efficiency is defined as the ratio of the output power and the input power, namely, η = p_out/p_in.

$$C_{F_x}=F_x / 0.5\cdot \rho {(fA)}^2 BC$$

(6)

$$P_{in}={\displaystyle {\oint }_{\Gamma }{v}_{n}pdl}$$

(7)

$$C_p=P_{in} / 0.5\cdot \rho {(fA)}^2 BC$$

(8)

3.4. Sensitivity Study

The sensitivity tests are designed and shown in this subsection. The corresponding grid numbers are 608,120, 1,696,270, 4,784,860, and 13,468,500. The time steps are chosen as Δt = T/250, Δt = T/500, and Δt = T/1000, where T is the motion period that can be calculated as T = 1/f. The turbulent model is selected as the SST k-ω model, and the corresponding test results are listed in

Table 2. Sensitivity tests of grid number, time steps (ts) and turbulent models.

Serial Number	Grid Number	ts/Period	uMean/(m/s)	η/(%)
No. 1	608,120	500	0.582	60.2
No. 2	1,696,270	500	0.552	56.5
No. 3	4,784,860	500	0.538	52.4
No. 4	13,468,500	500	0.537	52.2
No. 5	4,784,860	250	0.565	57.4
No. 6	4,784,860	1000	0.536	52.2

. The motion parameters are chosen as a₀ = 0.02, a₁ = −0.08, a₂ = 0.16, f = 1, and λ = 1. The physical model is chosen as NACA0015 foil with a magnitude of AR = 1.

As seen in Table 1, the influence of grid number on the fluid dynamics of foil is pretty sensitive. When the grid is relatively coarse (N = 608,120), there exists a tremendous difference in the value of u_Mean and η between different grid numbers. With the continuous rise in grid number (the corresponding conditions are No. 1, 2, 3, and 4), the corresponding difference quickly diminishes. In particular, when the grid number changes from 4,784,860 to 13,468,500, the corresponding difference in u_Mean is merely 0.19% and the relevant difference in η is 0.38%, meaning that a further increase in grid number will not induce an obvious change in the corresponding results. Therefore, the third set of grid (N = 4,784,860) is adopted in the following calculation for the consideration of saving computation resources and memory. When it comes to the time step, the increase in ts also results in a decreasing difference in u_Mean and η (the corresponding conditions are No. 3, 5, and 6). Specifically, when the value of ts changes from Δt = T/500 to Δt = T/1000, the difference in u_Mean and η is merely 0.37% and 0.38%, respectively. The above difference is too small to be considered, hence in the following calculation, the value of ts is set as Δt = T/500.

3.5. Validation Test and Turbulence Model Selection

In this subsection, the validation work will be carried out. Considering that the turbulent model plays a vital role in numerical simulation, the turbulence models will also be selected. Taking the results from Ref. [27] as a validation object, three turbulence models, including the SST k-ω model, SST k–ε model, and Spalart–Allmaras (S–A) model, were selected to carry out the turbulence model sensitivity test. The calculation model in Ref. [27] is 3D foil with an oval section plane, the aspect ratio is set as 4.0, and the ratio between the foil’s density and fluid’s density is 4.0. The comparison results are shown in Figure 5, where Figure 5a exhibits the variation curve of forward velocity with respect to wavy length, and Figure 5b presents the comparison results for input power of the system. As can be seen in Figure 5, the results of the SST k-ω model match well with those of Ref. [27], and the corresponding difference lies in the range of [0.2%, 0.8%]; therefore, the SST k-ω model will be adopted in following calculation.

4. Results and Discussion

4.1. Evolution Process Analysis

In this subsection, the evolution process of the 3D auto-propelled flexible foil is presented and analyzed. Taking the NACA0015 foil with the value of AR = 1.0 as an example, the corresponding variation curves are shown in Figure 6. The relevant parameters are set as a₀ = 0.02, a₁ = −0.08, a₂ = 0.16, f = 1.0, and λ = 1.0.

As for C_Fx and η, seen in Figure 6a, the flexible foil starts to move at the initial time (t = 0), and the corresponding magnitudes of C_Fx and η start to increase considerably. It is noted that the values of C_Fx and η are negative, meaning that the flexible foil starts to move along the negative x-axis. With the further increase in motion time, the variation curves of C_Fx present a trend of periodic fluctuation, and the variation amplitude starts decreasing. The value of u keeps increasing with a comparatively small acceleration. When t/T = 8, the positive value emerges in the variation curves of C_Fx, that is, the drag force starts to act on the foil. The existence of drag force keeps cutting down the acceleration of the foil, and the growth speed of u also presents a consistent decline. When t/T = 11, the evolution curves of C_Fx are nearly symmetric along the x-axis, and the corresponding mean value moves around zero. This means that the flexible foil acquires the condition of balance and the corresponding moving velocity also achieves dynamic balance, i.e., although the moving velocity undergoes periodic change, its mean value remains unchanged.

As for C_p and η, it can be deduced from Figure 6b that the value of C_p is tremendous, implying that much energy will be consumed to activate the forward motion of the foil. With the consistent increase in motion time, the value of C_Fx keeps decreasing, and so does the corresponding value of η. When t/T = 8, the decreasing magnitude of C_p turns into a small value and C_p gradually becomes stable. When t/T = 11, the fluctuation in the value of C_p is pretty small, meaning that C_p changes into a condition of balance. As for the value of η, the changing tendency is opposite to that of C_p. Specifically, the value of η exhibits a trend of gradual decrease, while the foil keeps moving forward and achieves a condition of dynamic balance at t/T =11.0. This conclusion is consistent with the phenomenon of increase in the moving velocity (Figure 6a) and decrease in C_p (Figure 6b).

Considering that there is a close connection between flow field and hydrodynamic performance, it is necessary to analyze these structures. The relevant vortex structures are presented in Figure 7, where the corresponding vortex criterion is λ₂ [28]. The mathematical definition of vortex can be represented as velocity curl, i.e., $\nabla \times v$ (where v denotes the velocity vector of the flow field).

As seen in Figure 7, at t/T = 1, the flexible foil starts to move forward, and the attached vortex structures show up on its surface. The attached vortex structures move backward along the foil surface and finally fall off from it. Specifically, the attached vortex on the foil upper surface moves toward the downside of the center-line, while the vortex on the lower surface moves in the opposite direction. With the consistent progression of motion time (t/T = 2), the shedding vortex in the flow field starts to be stretched and connected with the tip vortex structures, forming a complete vortex ring. With the consistent motion of the foil (t/T = 3), three vortex rings show up in the flow field, and the adjacent two vortex rings present a staggered arrangement. When the foil moves at x = −2.8 (t/T = 6), a series of vortex rings emerge in the wake of the foil. To illustrate the vortex structures more clearly, Figure 8 presents three views of the vortex structures at time t/T = 6.

As seen in Figure 8, the vortex rings present a staggered arrangement and are closely connected with each other. In addition, an angle θ can be observed between the center line of the vortex ring and the horizontal direction, which is defined as oblique (seen in Figure 8a). According to the theory of vortex dynamics, much energy will be released during the formation of the vortex ring, and the existence of an oblique angle will decompose the released energy into the horizontal and vertical directions. The release of energy along the horizontal direction will produce backward momentum. According to the momentum theorem, a resulting forward momentum will be imposed on the foil, pushing it to move forward. Hence, we conjecture that the smaller the oblique angle, the more horizontal energy will be released and the better the propulsive property. On the other hand, it can also be derived from Figure 8b that the span length of the vortex structure approximately equals the span length of the foil (B) when the foil falls off. However, the corresponding span length keeps decreasing, and the vortex structure becomes narrow with the further backward movement of the vortex structure, which can be attributed to the dissipated energy.

4.2. Effect of Aspect Ratio (AR)

The AR value, ranging from 0.25 to 4.0, is selected to analyze the AR effect on propulsive property. The changing curves of fluid dynamics of the flexible foil under various values of AR are shown in Figure 9. To conveniently characterize the propulsive performance of the foil, the absolute value of moving velocity is utilized, and |u_Mean| denotes the absolute mean value of u.

It can be acquired from Figure 9a that the changing curves of |u_Mean| under various values of AR can finally achieve a state of dynamic balance, that is, although the value of moving velocity undergoes consistent change with the variation of motion time, the fluctuating amplitude and period remain unchanged. The difference lies in that the time required for a foil to achieve dynamic balance keeps decreasing with the further increase in AR, while the corresponding fluctuating amplitude keeps expanding. Specifically, when AR is pretty small (AR = 0.25 and 0.5), it will cost at least 16 motion periods to achieve dynamic balance, and the relevant fluctuating amplitude is comparatively small. With the further increase in AR (AR = 1), at least 14 motion periods are consumed. When AR turns into AR = 3 and 4, 10 motion periods are sufficient for the foil to acquire a stable condition.

As for C_Fx, as seen in Figure 9b, the increase in AR leads to a substantial increase in the peak value of C_Fx. According to Formulas (4) and (5), the increasing fluid force acting on the foil will certainly result in a sharp increase in the corresponding value of |u_Mean|, and this conclusion is in good agreement with that acquired from Figure 9a. Concerning moving velocity, as seen in Figure 9c, the magnitude of |u_Mean| increases monotonically with the consistent growth of AR, and a linear relationship is found between the AR and |u_Mean|. This conclusion is of vital significance for the future design of flexible oscillating foil-based bionic underwater vehicles. In addition, it can also be acquired from Figure 9c that when AR is smaller than 1.0, the value of η increases quickly with the increase in AR; however, with the further increase in AR (AR > 1), the magnitude of η presents a tendency of consistent drop, meaning that when AR = 1.0, the highest value of η will be acquired.

In order to give some explanation for the above numerical results, three views of vortex structures under four typical values of AR are shown in Figure 10.

As shown in Figure 10, when the value of AR is comparatively small (AR = 0.25 and 0.5), the phenomenon of breaking down and fracture shows up in the corresponding vortex rings. Moreover, the connection between the adjacent vortex rings is particularly loose, and they start to separate from each other, implying that much energy is consumed. Furthermore, the relevant oblique angle is particularly large, meaning that little energy is released along the horizontal direction. When the value of AR becomes of large magnitude (AR = 1.0 and 2.0), there exists no breaking down and fracture on the corresponding vortex rings. In particular, when AR = 1.0, the corresponding oblique angle reaches the smallest value, meaning that the most energy is released along the horizontal direction; therefore, the peak magnitude of η can be obtained.

4.3. Effect of Thickness

In this section, the value of d ranges from 0.04 C to 0.50 C. The relevant motion parameters are identical with those in Section 4.2. The variation curves of the fluid dynamics of a flexible foil are exhibited in Figure 11.

As for u, seen in Figure 11a, the changing curves under various magnitudes of d tend to be stable, and the time required is almost identical, that is, 14 motion periods will be consumed for the foil to realize stable cruising speed. On the other hand, when the foil becomes thick (corresponding to a large value of d), the changing curve of u starts to move upward and the corresponding absolute value of u starts decreasing (seen in Figure 11c), indicating that the increase in d is unfavorable for the moving velocity of the foil. Concerning C_Fx, Figure 11b reveals that the increase in d leads to the decrease in C_Fx. Hence, the corresponding magnitude of u declines. As for the magnitude of η, it can be concluded from Figure 11c that when the value of d lies in a small range (d/C ≤ 0.15), the increase in d enhances the magnitude of η. However, when d/C > 0.15, the relevant magnitude of η drops sharply, and the thicker the foil is, the more quickly this occurs. That is to say, the peak magnitude of η can be obtained at d/C = 0.15.

Similarly, three views of vortex structures under three typical values of d are presented in Figure 12 to give some explanation for the above results.

It can be intuitively seen from Figure 12 that the increase in the value of d leads to a consistent decrease in the length of the vortex ring along the horizontal direction. This implies that much energy has been consumed and little energy can be utilized to facilitate the forward motion of foil, which may explain why the moving velocity keeps decreasing with the gradual rise in d. On the other hand, from the perspective of oblique angle, when the foil is relatively thick (d/C = 0.3), the corresponding oblique angle is pretty large, resulting in little released energy along the horizontal direction. When the foil becomes comparatively thin (d/C = 0.04 and 0.15), the oblique angle decreases to a small value, facilitating the release of much horizontal energy; hence, a high value of η is obtained.

4.4. Section Shape Effect

In this part, the effect of section shape will be analyzed. Three typical kinds of section shapes are selected, namely, NACA0015 shape, elliptical shape, and rectangular shape. The geometric parameters of the above three foils are set as B = C =1.0, d = 0.15 C. The corresponding motion parameters are set to be the same as in Section 4.2 and Section 4.3. The variation curves of fluid dynamics of the flexible foil under various kinds of section shapes are shown in Figure 13, with Figure 13a depicting the variation curves of moving velocity and Figure 13b illustrating the variation curves of drag force coefficient. The mean values of moving velocity, input power coefficient, and propulsive efficiency are listed in

Table 3. Value of |u_Mean|, C_P-Mean and η under three kinds of section shapes.

	|uMean|	η/(%)	Cp-Mean
Elliptical	0.304	46.1%	−1.15
NACA0015	0.308	52.4%	−1.04
Rectangular	0.242	32.5%	−1.86

.

It can be seen from Figure 13a that only 14 motion periods are required for the NACA0015 foil to acquire dynamic balance. In contrast, at least 20 motion periods are needed for the foils with elliptical shape and rectangular shape to achieve a stable cruising state. From the perspective of moving velocity, the variation curves of the foils with the NACA0015 shape and elliptical shape are close to each other, with merely 1.3% difference between the corresponding mean values, while that of the foil with the rectangular shape changes the least, equaling to 0.242, meaning that the section shape of the foil has a tremendous impact on the propulsive velocity. As for the propulsive efficiency, Table 2 shows that the foil with the NACA0015 shape acquires the highest value of η, followed by the foil with the elliptical shape and then the foil with the rectangular shape. As for the foil with the rectangular shape, since the value of moving velocity is obviously lower than that of the other two foils and the corresponding value of C_p-Mean also reaches the highest value (seen in Table 2), the corresponding value of η is the smallest. As for the foils with the NACA0015 shape and elliptical shape, in spite of the fact that the mean values of moving velocity are close to each other, the value of C_p-Mean for the foil with the elliptical shape is considerably larger than that for the foil with the NACA0015 shape, leading to a smaller value of η.

Figure 14 exhibits the vortex structures of a foil with three kinds of section shapes. It can be clearly seen that the staggered vortex rings all emerge in the wake of the foil with various section shapes, and the lengths of the vortex rings along the horizontal direction are similar to each other. The difference lies in that the obvious breaking down and fracture shows up in the vortex rings of the foil with the rectangular shape, and the vortex rings are no longer complete. In addition, it is more obvious from the top view that the middle part of the vortex ring in the foil with the rectangular shape is almost completely broken and dissipated, indicating a heavy energy loss, hence the corresponding propulsive property is the worst. Moreover, for the foil with an elliptical shape, although no breaking down occurs in the vortex rings along the horizontal direction, a small part of the fracture shows up in the middle part along the span length. Therefore, compared with the foil with the NACA0015 shape, more energy has been dissipated, and the propulsive property is weaker than that of the foil with the NACA0015 shape.

5. Conclusions

In this work, the propulsive characteristics of a 3D foil undergoing flexible oscillating motion are numerically studied with the help of ANSYS Fluent. The effects of geometric parameters of the foil (aspect ratio, thickness along the chord length, section shape) on its propulsive property are systematically analyzed. The main conclusions that can be drawn are as follows:

(1)

The increase in AR value is beneficial for improving the foil’s forward moving velocity, and there exists a linear connection between AR and |u_Mean|. The magnitude of η climbs to the peak value and then decreases when AR keeps increasing, with an optimum value resulting in the highest value of η.

(2)

The growth in thickness will lead to a consistent decrease in |u_Mean|. When the magnitude of d is smaller than 0.15 C, slight changes can be observed in the value of η. However, when the foil becomes thicker (d > 0.15 C), a sharp drop manifests in the value of η.

(3)

The section shape of the foil has a tremendous impact on the corresponding propulsive property. The foil with the NACA0015 shape acquires the best propulsive property, followed by the foil with an elliptical shape, and the foil with rectangular shape obtains the worst propulsive property.

In addition, constrained by the current research conditions, there do exist some limitations in the current study. For one, further validation of the conclusions acquired in the current study has not been conducted, and there exists some difficulty in validating the self-propulsive property through physical experiments. In order to solve this tricky problem, the flexible foil will be applied to the bionic propulsion of a UUV in the very near future. The flexible foil can serve as the ‘propulsive unit’, while the fuselage can serve as the ‘storage unit’. The UUV can move forward with the help of a ‘propulsive unit’. At that time, the conclusions acquired in the current study can be experimentally validated. Another limitation is that the cost of transient numerical simulation is pretty heavy. Taking the case of AR = 1 as an example, it costs roughly 33 h to complete a motion period on a personal computer (Intel core, i7 9700 K, 32 G). In order to accelerate the solving speed, we adopted parallel computing in the super-computing center (96 cores), and it cost about five days to finish a case. Constrained by the computational resources, the mutual interaction between fluid and structure was not considered in the current study. In our future work, the structure solver will be added into our current simulation code to realize the synchronous solution of fluid and solid.