# Flow Characteristics of a Fish-Like Body Based on Taguchi Method under Different Strouhal Numbers

Authors: Baoquan Guo; Junjie Xu; Shizhen Zheng; Kai Wang; Jianlong Chang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In the investigations of fish swimming, the relationship between fish swimming and the corresponding characteristics of flow field attracts lots of attention. In this paper, the influence of different Strouhal numbers on the flow field characteristics and mechanical characteristics of fish-like body is investigated. It is found that the strength of the anti-Karman vortex and the thrust on the fish-like body are affected by the Strouhal number. When the Strouhal number is small, the thrust coefficient is small but the propulsion efficiency is high. Similarly, when the Strouhal number is large, the thrust coefficient is large and the propelling efficiency is low. In order to better reveal the swimming status of the flexible body of the fish imitation, the effect of the Strouhal number is analyzed in detail in this paper. Besides, the influence of the three parameters in the Strouhal number formula on the flow field characteristics and thrust coefficient of the fish-like body is further investigated by the Taguchi method. The thrust coefficient is positively correlated with the swing amplitude and frequency, which is found to be negatively correlated with the inflow velocity. Swing amplitude and swing frequency are the main factors affecting the thrust coefficient; there is little effect of the inflow velocity on the thrust coefficient.


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## Body

## 1. Introduction

The problem of how to manipulate the fish-like body and improve the propulsion efficiency of the fish-like body is studied based on the principle of bionics. The manipulation and propulsion methods of fish are developed and imitated by mechanical structure, electronic equipment, and functional materials and are adopted in the underwater navigation. This new type of underwater vehicle can be used in mine warfare, antimine warfare, and military reconnaissance and can also be used as a supporting weapon of submarine in wartime. In peacetime, this new type of underwater vehicle can be used for submarine survey, ocean observation, underwater lifesaving, and so on in complex marine environment, which is of great significance. It is one of the promising research topics to use the bionics principle to research the morphology and motion bionics of typical living fish, in order to reduce energy consumption and improve propulsion efficiency.Compared with the traditional propeller, there are many advantages to using the fish-like body underwater as follows:(a)
At high velocity, the ability of starting, accelerating, and steering of vehicle is improved.(b)
By controlling the tail vortex of the fish-like body, more perfect hydrodynamic properties can be obtained, thus swimming resistance and energy consumption can be reduced.(c)
The noise of the fish-like body is lower than that of the propeller; as a result, it is not easy to be found and recognized by the opposing sonar, which is beneficial to stealth and penetration.(d)
The function of the rudder and paddle in the propeller is replaced by the swinging of the fish-like body, the mechanism is simplified, and the effective volume and load capacity of the underwater vehicle are increased, which is of great practical significance.Many achievements have been made in the investigations of morphology and motion bionics of typical living fish. In terms of theory, the thin body theory was proposed by Lighthill [1], and the two-dimensional aerodynamic wing theory was applied to the research of caudal fin propulsion. The three-dimensional problem of the uniform swimming of the slender fish body was simplified to the two-dimensional unsteady flow problem, and the motion of the crescent tail was investigated. The two-dimensional wave plate theory was put forward innovatively by Wu [2], and the optimal movement mode of fish was analyzed. The mechanism of fish movement was further studied by Cheng et al. [3], a three-dimensional wave plate potential flow model was proposed, and the morphological adaptation of fish caudal fins based on zoological knowledge was discussed.The flow field characteristics of fish-like bodies had been studied by many scholars through numerical simulation, and many useful results had been obtained. The results of numerical simulation method in three-dimensional space were used in the calculation of flexible fish-like body model by Leroyer and Visonneau [4], and the three-dimensional calculation results of coupling RANS equation and Newton’s law of motion were introduced for the first time. Unsteady viscous flow field around a fish-like body was numerically studied by Chen and Doi [5]; the simulation results showed that the fish-like body could improve the propulsion efficiency of swimming under the condition of reliability and effective vorticity control. The swimming of two-dimensional fish body was simulated by Deng et al. [6], the swimming status of fish body under different control parameters was investigated, and the thrust characteristics and flow field characteristics of fish body in water were revealed. The visual modeling of the movement of complex organisms based on CFD was realized by Liu and Kawachi [7–9], and the three-dimensional flow field of tadpole swimming was numerically analyzed. CFD analysis displayed that the special shape of tadpole was matched by tadpole kinematics, and jet propulsion with high propulsion efficiency was generated. The wave motion pattern of anguilli was optimized and analyzed by Kern and Koumoutsakos [10], and quantitative information of three-dimensional fluid body interactions was provided by current results. The influence of the kinematic characteristics of fish swimming in the water channel with antiskid wall on its swimming performance was discussed by Zhang et al. [11] using the three-dimensional simulation method. The results indicated that the thrust generation mechanism was provided by the velocity field and vorticity field around the model, and the influence of kinematics on swimming performance was emphasized.The visualization of the tail structure of the dorsal fin was realized by Drucker and Lauder [12, 13]; using the digital particle velocity measurement technology, the force generated by the movement of the fish body was calculated and the interaction between the flexible long fin and caudal fin in the movement of fish had been investigated. The results indicated that the thrust force could be enhanced by the hydrodynamic effect of the interaction between the dorsal fin and caudal fin; the dorsal fin was used to produce an axial deflection force when a fish was turning. The influence of fish-like body that crowds on swimming efficiency was studied by Li et al. [14]; by changing the spacing of fish-like bodies, four formations (series, square, diamond, and rectangle) were formed. The simulation results showed that when the distance between fish-like bodies was less than 1.25 times of their own length, the tandem array could improve the propulsion efficiency, while when the distance was greater than 1.25 times of their own length, the rectangular array had higher propulsion efficiency than other arrays. The hydrodynamic effect under the interaction of fins was investigated by Liu et al. [15]. In addition, a new perspective on how to promote fish swimming was provided by the interaction between midfin and caudal fin. A new flapping wing propulsion mode was investigated by Karbasian based on the swimming process of fish [16], and the propulsion efficiency of the flapping wings was significantly improved by the model. The δ+-SPH method was applied to fish-like swimming by Sun et al. [17], and the problem of mesh deformation could be effectively avoided by the meshless characteristics of the method, and the ability of the δ+-SPH method to simulate the swimming of fish had been confirmed by the results of numerical studies. The optimal wave propulsion conditions for a single fish and a pair of interacting fish were established by Maertens et al. [18] in 2D and 3D simulations, and the propulsion efficiency of the fish body was greatly improved by the conditions. The greater the flexibility of the tail fin, the more conducive to the fast start of the fish, but it was not conducive to the fast cruise of the fish, which was found by Feng et al. [19] through simulation. In addition, the flexibility of the caudal fin is also beneficial to the directional stability of the fish’s autonomous swimming. A new fish-like swimming model was proposed by Esfahani et al. [20]; the results showed that the best propulsion performance could be obtained when the pitch amplitude of the model was 0.2°∼0.4° and 30°∼40°, respectively. The result was the same as that of fish swimming in nature. The flow characteristics of the fish-like wake were studied by Macias et al. [21]. Through numerical simulation, the results show that when the fish-like body swims, it will generate von Kármán vortex street. At the same time, it is found that the maximum thrust will affect the leading-edge vortex generated by the tail fin. The hydrodynamic characteristics of eel swimming were studied by Borazjani and Sotiropoulos [22]. It was found that compared with the tail fin swing amplitude, the tail fin swing frequency is more likely to affect the thrust. At the same Reynolds number, anguilliform swimming consumes less energy than carangiform swimming.In terms of experiments, the main mechanism of propulsion and transient force generated in the fish-like body and fins swinging in the water, as well as the formation and control of large-scale vortices had been experimentally determined by Triantafyllou et al. [23], and the agility of fish was explained. The complex hydrodynamics and control model in the optimization of flexible fish-like mechanism were investigated. The open water experiment was carried out by using the tail fin of the mechanism, and the basis for establishing the model of the fish-like body was provided, and the feasibility of the fish-like mechanism was demonstrated [24].There are many factors that affect the fish-like body propulsion. In order to get the influence of each factor on the fish-like body propulsion, the Taguchi method is used in this paper. The Taguchi method is a low-cost and high-efficiency quality engineering method, which can accurately predict the results by testing a special combination based on orthogonal array and statistical analysis. The Taguchi method combines engineering experience with statistical principles and uses mathematical methods to comprehensively study the theories and methods of quality management from engineering, technical, and economic perspectives, so as to form a set of unique, effective, universal, and marginal quality robustness design methods. This design method is proposed by Japanese scholar Genichi Taguchi in the 1950s [25]. It is a quality engineering optimization method with low cost and high benefit [26], aiming to improve the quality of manufactured products. The Taguchi method has been widely used in various fields. The purpose of using orthogonal arrays is to reduce research time and obtain the best parameter combination, so as to find the design scheme with the strongest anti-interference ability and the best performance [27]. Moreover, the Taguchi method can also be used to investigate the influence of various factors on the experimental results by means of the analysis of variance. The influence of four factors, namely, caudal fin length-diameter ratio, caudal fin stiffness, oscillation frequency, and spring stiffness, on the swimming velocity of the fish-like body was analyzed by Li et al. [28] through the Taguchi method. Two groups of results were obtained through 25 groups of experiments under orthogonal design. Firstly, the vibration frequency and spring stiffness were the main factors affecting the swimming velocity. Secondly, when the frequency was 12 Hz and the spring stiffness was infinite, the fastest swimming velocity can be achieved.The content of this paper consists of two parts. The first part reveals the influence of the different Strouhal number on the vortex street intensity, the force exerted on the fish body, and the characteristics of the flow field. In the second part, the influence of the parameters of the Strouhal number, i.e., swing amplitude, swing frequency, and inflow velocity on the thrust coefficient and flow field characteristics of the fish body is investigated by the Taguchi method in detail.


## 2. Numerical Method, Model Validation, and Calculation Formula

### 2.1. Numerical Method

In this paper, Fluent is used to carry out the numerical simulation of the swing of fish. By referring to literature, it is found that the model of the swing of the flexible fish-like body (short for fish-like body) can be simplified into a two-dimensional model. In the investigation, the common NACA0012 airfoil is used to simulate the fish profile. The schematic diagram of flow field calculation domain is indicated in Figure1. The leading edge of the airfoil is set as the coordinate origin, the inflow velocity U is the characteristic velocity, the horizontal direction is set as the X-axis direction, and the vertical direction is set as the Y-axis direction, respectively. The chordal length of the airfoil is taken as the characteristic length L = 1.5 m, the radius of the inlet section of the left semicircle is 7 L, and the outflow section is 10 L. The mesh structured of the fish-like body is shown in Figure 2. In addition, in order to better capture the flow changes in the boundary layer of the fish-like body, the mesh around the boundary of the fish-like body is encrypted.Figure 1
Schematic diagram of calculation domain.Figure 2
Structures of the grids.The fluid in the flow field is set as an incompressible fluid, and the fish wall surface is set as no slip condition. The direction of inlet velocity is horizontal to the right, and the upper and lower boundaries are far away from the fish body and will not be affected by the change of the flow field. Therefore, the upper and lower boundaries are set as the velocity inlet, and the direction is horizontal to the right. The outlet is set as a pressure outlet, so that the flow is fully developed.The motion rules of moving mesh are described by user defined function (UDF). By establishing the fish body wave function in UDF, the swing of the fish body can be controlled. According to the current boundary position, time, and velocity increment, the boundary position of the next time can be calculated. By adjusting the grids near the boundary or rebuilding the grids, the grid distortion and negative grids can be avoided, and as a result, the ideal grids motion can be obtained.In the numerical solution, the governing equations are discretized by the finite volume method, and the pressure velocity coupling is carried out by the SIMPLE method. All kinds of spatial differences are set as second-order upwind. The finite volume method is used to solve the steady-state incompressible N–S equation and the continuous equation in the numerical simulation. The N–S equation of two-dimensional steady-state incompressible fluid is(1)∂V⇀∂t+V⇀⋅∇V⇀=−∇P+1Ra∇2V2,∇⋅V⇀=0.The turbulence model used in this paper is the SSTk-ω turbulence model. The basic principle of SST is that the K-ω model is used near the wall, and the k-ε model is adopted at the boundary layer edge and the free shear layer. The influence of Reynolds shear stress transport is introduced by the Bradshaw hypothesis. Thus, the advantages of K-ω, K-ε and JK models are cleverly combined by the SST k-ω turbulence model. In addition to the abovementioned advantages, the transport of turbulent shear stress in the adverse pressure gradient and boundary layer separation can also be well handled, and the more complex flow conditions such as adverse pressure gradient and boundary layer separation can be better predicted by the SST k-ω model. Compared with the standard K-ω model, the SST K-ω model incorporates the cross diffusion from the ω equation, and the turbulent viscosity is also affected by the wave propagation of the turbulent shear stress. The accuracy and reliability of simulation results are improved by these improvements. Therefore, in order to reduce the shear stress and the influence of complex flow conditions, the SST K-ω turbulence model is selected. The two equations turbulence model of SST K-ω is as follows:(2)ρ∂k∂t+ρu∇k=P−ρβ0kω+∇μ+σkμT∇k,ρ∂ω∂t+ρu.∇ω=ργμTP−ρβω2+∇.μ+σωμT∇ω+21−fv1ρσω2ω∇ω.∇k.


### 2.2. Model Validation

In order to verify the accuracy of model adopted in the paper, the lift coefficient of NACA0012 airfoil is taken as reference. The structured grids are used to obtain the lift coefficient of the wing with the change of the angle of attack, when the Reynolds number is 2.21 × 105. At the same time, in order to obtain more reliable simulation results, the grids near the wing are encrypted. The comparison between the lift coefficient obtained in the manuscript and the results of Jha et al. [29] is displayed in Figure 3. It can be seen that the curves are in good agreement, which means that the accuracy of the results is credible.Figure 3
Comparison of lift coefficient variation curve.


### 2.3. Calculation Equation

In order to simulate the swimming process of fish, the transverse wave equation is selected as the condition of the fish body swing, and the equation is applied to the center line of airfoil to realize the fish-like swimming of wing. The motion equation of transverse wave is(3)Fx,t=axncos2π1λx−ft.In the previous formula,an is the swing amplitude, λ is the wavelength, f is the frequency, and t is the time. The λ value is 1.5 m. The origin of the calculated coordinates is the leading edge of the airfoil, and the x and y directions are the flow coordinates and lateral coordinates, respectively. The variation of swing amplitude is affected by the value of n; when n = 0, it is constant amplitude swing, and when n > 0, it is amplitude swing, which is to improve swimming speed and propulsion efficiency. In order to improve swimming velocity and propulsion efficiency, n = 1.1 is selected for discussion in this paper.According to equation (1), loading the transverse wave equation on the center line of the airfoil can guarantee that the thickness of the fish body is constant. The shape change of the fish body within one cycle after loading the transverse wave equation is indicated in Figure 4. As can be seen from the figure, the wave equation can also ensure that the surface of the fish body is always smooth. The simulation accuracy and the accuracy of simulation results are greatly improved.Figure 4
Shape change of fish body swing.The Strouhal number is a similar criterion to characterize the unsteadiness of flow and a physical quantity to indicate the unsteadiness of fluid. The formula of the Strouhal number is as follows:(4)St=fAU,where A is the distance length between the peak and trough of the wave, f is the characteristic frequency, and U is the characteristic velocity.In order to verify the influence of the Strouhal number on the propulsion efficiency of the fish-like body, the propulsion efficiency formula is introduced [30, 31], which is(5)η=121+β−12α2β21−β,where α=λF′L/2πFL, β=U/V, V=λ⋅f.


## 2.1. Numerical Method

In this paper, Fluent is used to carry out the numerical simulation of the swing of fish. By referring to literature, it is found that the model of the swing of the flexible fish-like body (short for fish-like body) can be simplified into a two-dimensional model. In the investigation, the common NACA0012 airfoil is used to simulate the fish profile. The schematic diagram of flow field calculation domain is indicated in Figure1. The leading edge of the airfoil is set as the coordinate origin, the inflow velocity U is the characteristic velocity, the horizontal direction is set as the X-axis direction, and the vertical direction is set as the Y-axis direction, respectively. The chordal length of the airfoil is taken as the characteristic length L = 1.5 m, the radius of the inlet section of the left semicircle is 7 L, and the outflow section is 10 L. The mesh structured of the fish-like body is shown in Figure 2. In addition, in order to better capture the flow changes in the boundary layer of the fish-like body, the mesh around the boundary of the fish-like body is encrypted.Figure 1
Schematic diagram of calculation domain.Figure 2
Structures of the grids.The fluid in the flow field is set as an incompressible fluid, and the fish wall surface is set as no slip condition. The direction of inlet velocity is horizontal to the right, and the upper and lower boundaries are far away from the fish body and will not be affected by the change of the flow field. Therefore, the upper and lower boundaries are set as the velocity inlet, and the direction is horizontal to the right. The outlet is set as a pressure outlet, so that the flow is fully developed.The motion rules of moving mesh are described by user defined function (UDF). By establishing the fish body wave function in UDF, the swing of the fish body can be controlled. According to the current boundary position, time, and velocity increment, the boundary position of the next time can be calculated. By adjusting the grids near the boundary or rebuilding the grids, the grid distortion and negative grids can be avoided, and as a result, the ideal grids motion can be obtained.In the numerical solution, the governing equations are discretized by the finite volume method, and the pressure velocity coupling is carried out by the SIMPLE method. All kinds of spatial differences are set as second-order upwind. The finite volume method is used to solve the steady-state incompressible N–S equation and the continuous equation in the numerical simulation. The N–S equation of two-dimensional steady-state incompressible fluid is(1)∂V⇀∂t+V⇀⋅∇V⇀=−∇P+1Ra∇2V2,∇⋅V⇀=0.The turbulence model used in this paper is the SSTk-ω turbulence model. The basic principle of SST is that the K-ω model is used near the wall, and the k-ε model is adopted at the boundary layer edge and the free shear layer. The influence of Reynolds shear stress transport is introduced by the Bradshaw hypothesis. Thus, the advantages of K-ω, K-ε and JK models are cleverly combined by the SST k-ω turbulence model. In addition to the abovementioned advantages, the transport of turbulent shear stress in the adverse pressure gradient and boundary layer separation can also be well handled, and the more complex flow conditions such as adverse pressure gradient and boundary layer separation can be better predicted by the SST k-ω model. Compared with the standard K-ω model, the SST K-ω model incorporates the cross diffusion from the ω equation, and the turbulent viscosity is also affected by the wave propagation of the turbulent shear stress. The accuracy and reliability of simulation results are improved by these improvements. Therefore, in order to reduce the shear stress and the influence of complex flow conditions, the SST K-ω turbulence model is selected. The two equations turbulence model of SST K-ω is as follows:(2)ρ∂k∂t+ρu∇k=P−ρβ0kω+∇μ+σkμT∇k,ρ∂ω∂t+ρu.∇ω=ργμTP−ρβω2+∇.μ+σωμT∇ω+21−fv1ρσω2ω∇ω.∇k.


## 2.2. Model Validation

In order to verify the accuracy of model adopted in the paper, the lift coefficient of NACA0012 airfoil is taken as reference. The structured grids are used to obtain the lift coefficient of the wing with the change of the angle of attack, when the Reynolds number is 2.21 × 105. At the same time, in order to obtain more reliable simulation results, the grids near the wing are encrypted. The comparison between the lift coefficient obtained in the manuscript and the results of Jha et al. [29] is displayed in Figure 3. It can be seen that the curves are in good agreement, which means that the accuracy of the results is credible.Figure 3
Comparison of lift coefficient variation curve.


## 2.3. Calculation Equation

In order to simulate the swimming process of fish, the transverse wave equation is selected as the condition of the fish body swing, and the equation is applied to the center line of airfoil to realize the fish-like swimming of wing. The motion equation of transverse wave is(3)Fx,t=axncos2π1λx−ft.In the previous formula,an is the swing amplitude, λ is the wavelength, f is the frequency, and t is the time. The λ value is 1.5 m. The origin of the calculated coordinates is the leading edge of the airfoil, and the x and y directions are the flow coordinates and lateral coordinates, respectively. The variation of swing amplitude is affected by the value of n; when n = 0, it is constant amplitude swing, and when n > 0, it is amplitude swing, which is to improve swimming speed and propulsion efficiency. In order to improve swimming velocity and propulsion efficiency, n = 1.1 is selected for discussion in this paper.According to equation (1), loading the transverse wave equation on the center line of the airfoil can guarantee that the thickness of the fish body is constant. The shape change of the fish body within one cycle after loading the transverse wave equation is indicated in Figure 4. As can be seen from the figure, the wave equation can also ensure that the surface of the fish body is always smooth. The simulation accuracy and the accuracy of simulation results are greatly improved.Figure 4
Shape change of fish body swing.The Strouhal number is a similar criterion to characterize the unsteadiness of flow and a physical quantity to indicate the unsteadiness of fluid. The formula of the Strouhal number is as follows:(4)St=fAU,where A is the distance length between the peak and trough of the wave, f is the characteristic frequency, and U is the characteristic velocity.In order to verify the influence of the Strouhal number on the propulsion efficiency of the fish-like body, the propulsion efficiency formula is introduced [30, 31], which is(5)η=121+β−12α2β21−β,where α=λF′L/2πFL, β=U/V, V=λ⋅f.


## 3. Results and Discussion

### 3.1. Flow Field and Thrust Characteristics under Different Strouhal Numbers

In order to verify the influence of the Strouhal number on flow field characteristics and thrust characteristics, the results of the Strouhal number at 0.11, 0.22, and 0.33 are, respectively, investigated. In this paper, different values of the Strouhal number are taken by changing the velocity, as shown in Table1.Table 1
Different Strouhal numbers.
StfAU0.112.30.1022.0930.222.30.1021.0650.332.30.1020.710


#### 3.1.1. Flow Field Characteristics under Different Strouhal Numbers

The vorticity fields at the different Strouhal numbers are displayed in Figure5. It can be seen from the figure that the vorticity intensity reaches the maximum when the Strouhal number is 0.33, followed by 0.22, and the minimum appears when the Strouhal number is 0.11. The vorticity intensity increases with the increase of the Strouhal number. As shown in Figure 5(a), the vortex is in a long and narrow ‘S’ shape. As the vortex dissipates backwards, the shape of the vortex gradually changes into a ‘comma’, which is quite different from the shape of the vortex in Figures 5(b) and 5(c). When St = 0.11, a column of vortices is generated in the wake, while when St = 0.22 and 0.33, two columns of vortices are formed in the wake. This phenomenon indicates that the shape difference and arrangement of vortices are greatly affected by the Strouhal number.Figure 5
Vorticity fields under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)The velocity contours under different Strouhal numbers are indicated in Figure6. It can be seen from Figure 6 that the wake velocity is increased by the swing of the fish body, so that the fluid accelerates backward and the fish body is accelerated forward. Therefore, the fish gets forward thrust by accelerating the wake backward through the swing of its body.Figure 6
Velocity contours under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)By comparing Figures6(a) and 6(b), it can be found that the velocity trajectory generated by the fish body swing is curved, and discontinuous fluid microclusters are generated, which are in pairs in each cycle of the fish body swing. It can be found that the thrust generated by the fish’s swing is not continuous but periodic and discrete. It can be clearly discovered from the three figures in Figure 6 that the velocity distribution on both sides of the fish body is different. Specifically, when the fish body is bent downward, the part with higher velocity on the lower side is in front of the fish, while the upper part of the velocity is higher in the tail of the fish. On the contrary, when the fish body is bent upward, the part with higher velocity on the upper side is in front of the fish, while the lower part of the velocity is higher in the tail of the fish. With the increase of the Strouhal number, the bending amplitude decreases gradually and the velocity value is the smallest at the depression of each turn. It can be concluded that the influence of the Strouhal number on the velocity field in the oscillating wake of fish body is mainly manifested at the difference of velocity magnitude and bending amplitude, but there is no significant influence for the Strouhal number on the overall distribution of the velocity field.


#### 3.1.2. Thrust Characteristics and Propulsion Efficiency under Different Strouhal Numbers

The thrust coefficient curves at different Strouhal numbers are displayed in Figure7. It can be seen from the figures that the period and frequency of the thrust coefficient curves at different Strouhal numbers are the same, but the amplitude and average value of the curves are different. It is obvious that various thrust characteristics are generated by different Strouhal numbers.Figure 7
Thrust coefficient curves under different Strouhal numbers.The maximum and minimum thrust coefficients and the mean values of the thrust coefficients generated by the swing of the fish body at different Strouhal numbers are demonstrated in Table2. The data in Table 2 can be obtained by equation (5). It can be seen that the higher the Strouhal number, the lower the propulsion efficiency. Combined with the data in Table 2, it can be found that when the Strouhal number is 0.11, the thrust coefficient is small and the propulsion efficiency is high. When the Strouhal number is 0.33, the thrust coefficient is large and the propulsion efficiency is low. As the average value increases with the Strouhal number, there is a proportional relationship approximately 3 times larger. Therefore, it is necessary to increase the Strouhal number when traveling in a hurry. On the contrary, when coasting or when the energy consumption needs to be reduced, the Strouhal number should be appropriately reduced. Therefore, it is very important to choose an appropriate Strouhal number.Table 2
Maximum, minimum, and average values of thrust coefficient and propulsion efficiency with different Strouhal numbers.
St = 0.11St = 0.22St = 0.33CT (min)0.00280.00300.0590CT (max)0.07870.26330.7303CT (average)0.03930.13020.3857η0.80330.65430.6029The part of influence of the Strouhal number on the flow field characteristics and thrust characteristics of the fish-like body is investigated in the content mentioned above, and the effect of various factors on the fish-like body cannot be fully demonstrated. In order to better reveal the swimming status of the fish-like body, the three variables in the Strouhal number would be analyzed in the following discussion. In addition, the Taguchi method is used to discuss the degree of influence of various factors on the swimming of the fish-like body, and the influence of various factors on the flow field of the flexible body is further investigated. The results can provide insights for improving the working efficiency of the fish-like body and improve the swimming scheme.


### 3.2. Influence of Different Factors on Flow Field and Thrust Characteristics

#### 3.2.1. Controllable Factors and Levels

On the basis of the previous analysis, the influence of the three variables in the Strouhal number on the flow field and the thrust coefficient is investigated through the Taguchi method. Therefore, the flow velocityU, the swing frequency f, and the swing amplitude A0 are the controllable factors for the investigations, and four levels are set for each factor. According to the Taguchi test and simulation, the wavelength is set as the characteristic length of the fish body, and the controllable factors and levels of the test are shown in Table 3.Table 3
Controllable factors and levels.
LevelControl factorsA0fU10.0420.420.052.50.630.0630.840.073.51According to the three controllable factors including flow velocity, swing frequency, and swing amplitude, as well as four groups of levels, 16 groups of data are obtained through the orthogonal list as shown in Table4. If the full-factor combination scheme is used, 43 = 64 tests are needed, which is 48 more than the 16 combination schemes of the Taguchi method. The number of tests is reduced by 75% and the workload is greatly reduced. Fluent is used to simulate the combined data, and the mean values and thrust coefficient are obtained, as shown in Table 4. It is observed that the smallest thrust coefficient appears in Case 3, while the largest thrust coefficient is in Case 13. Compared with Case 3, the thrust coefficient of Case 13 is increased by 2320%, and the thrust coefficients of Case 13 and Case 14 are much higher than that of other data sets. Therefore, a reasonable selection of three controllable factors is very crucial to the final results.Table 4
Research scheme combination and test results.
CaseControl factorsA0fUCTCase 10.0420.40.1998828Case 20.042.50.60.2894164Case 30.0430.80.1037629Case 40.043.510.6069987Case 50.052.50.80.4547392Case 60.05210.1859725Case 70.053.50.41.1613857Case 80.0530.60.7259077Case 90.06310.9978357Case 100.063.50.81.4445390Case 110.0620.60.4730492Case 120.062.50.40.9219013Case 130.073.50.62.5119235Case 140.0730.42.0029727Case 150.072.510.7893034Case 160.0720.80.5413109


#### 3.2.2. Analysis of Influence Factors

The influence degree of each factor on the thrust coefficient is displayed in Table5, where K represents the average value of the calculated results of each factor at each level, and R is the range, representing the difference between the maximum and minimum value of K. The influence of various factors on the results can be judged according to the value of R. It can be found from Table 5 that the influence on thrust coefficient is Amplitude > Frequency > Velocity. According to the degree of influence on the results, the swing amplitude and swing frequency are the major factors, while the future flow velocity is the minor factor.Table 5
Influence of various factors on thrust coefficient.
A0fUK10.30001520.35005381.0715356K20.63200120.61384011.0000742K30.95933130.95761970.6360880K41.46137761.43121170.6450276R11.16136241.08115790.4265081Through the simulation of 16 times, the average thrust coefficient at different levels of each parameter can be obtained, as shown in Figure8. It can be seen from Figure 8, the thrust coefficient is positively correlated with the swing amplitude and swing frequency, while negatively correlated with the inflow velocity. With the increase of swing amplitude and swing frequency, the growth rate of thrust coefficient is basically unchanged, which is close to the change law of the first order function. With the increase of the inflow velocity, the thrust coefficient decreases slowly, then rapidly, and then slowly. According to the analysis data, the maximum thrust coefficient can be obtained when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s.Figure 8
Thrust coefficient of different parameters at different levels.The oscillating process is a periodic process, and the fish body will be pushed by water in the process of propulsion. In the process of periodic oscillation of the fish body, the thrust coefficient generated also indicates periodic fluctuations, as shown in Figure9. Figures 9(a)–9(c), respectively, display the comparison of four parameters under three conditions of constant swing amplitude, constant swing frequency, and constant inflow velocity. The swing amplitude, swing frequency, and inflow velocity are 0.04 m, 2.5 Hz, and 0.6 m/s, respectively. It can be seen that when the swing amplitude is constant, the mean and amplitude of thrust coefficient increase with the increase of swing frequency and incoming flow. When the oscillating frequency is constant, the mean and amplitude of thrust coefficient increase with the increase of the oscillating frequency but are not affected by the inflow velocity. When the inflow velocity is constant, the mean value and amplitude of thrust coefficient no longer change with the change of a single factor. Moreover, combining with the range size in Table 5, it can be found the influence effect of swing and frequency is similar.Figure 9
Comparison of thrust coefficients under four group of parameters. (a) Constant swing amplitude. (b) Constant oscillation frequency. (c) Constant inflow velocity.
(a)(b)(c)


#### 3.2.3. Flow Field and Thrust Characteristics under the Condition of Optimal Controllable Factor

As discussed previously, when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s, the maximum thrust coefficient can be obtained. In this case, the flow field characteristics and thrust characteristics of the fish-like body are investigated in the following part. The optimized average thrust coefficient is 2.7253642. Compared with Case 13, the thrust coefficient increases due to the decreasing inlet velocity. For example, the inlet velocity changed from 0.6 m/s to 0.4 m/s, and the thrust coefficient increased by 8.5%.The vorticity fields at different moments are indicated in Figure10. It can be found that there are two rows of anti-Karman vortices which are staggered and rotated to the opposite direction in the wake of the fish body. During the process of the fish tail upswing, the attached vortices which rotate counterclockwise are formed on the lower surface of the tail. When the tail swings to the highest point, the attached vortices fall off due to the rapid downward swing. The formation of backward jet between the two vortices is caused by the formation process of anti-Karman vortex street, which makes the fish body receive forward thrust. As shown in Figure 10(a), when the tail swings to the lowest point, the positive vortex ring is shed, and the positive vortex ring is located on the upper row of the two rows of vortices. After the shedding of the positive vortices, as displayed in Figure 10(b), adherent vortices are generated at the tail of the fish body, and the attached vortices fall off to form a negative vortex ring. As shown in Figure 10(c) and 10(d), positive and negative vortices rings appear in pairs and gradually move backward. As the vortex intensity decreases, the vortex rings gradually dissipate. From the analysis from Figures 10(a)–10(e), there are only three pairs of vortex pairs that can be observed within the range of vorticity shown in this figure, and the intensity of previously generated vortices is already too small to be observed.Figure 10
Vorticity fields of fish body in a period of swing. (a)t = 0. (b) t = T/4. (c) t = T/2. (d) t = 3 T/4. (e) t = T.
(a)(b)(c)(d)(e)


## 3.1. Flow Field and Thrust Characteristics under Different Strouhal Numbers

In order to verify the influence of the Strouhal number on flow field characteristics and thrust characteristics, the results of the Strouhal number at 0.11, 0.22, and 0.33 are, respectively, investigated. In this paper, different values of the Strouhal number are taken by changing the velocity, as shown in Table1.Table 1
Different Strouhal numbers.
StfAU0.112.30.1022.0930.222.30.1021.0650.332.30.1020.710


### 3.1.1. Flow Field Characteristics under Different Strouhal Numbers

The vorticity fields at the different Strouhal numbers are displayed in Figure5. It can be seen from the figure that the vorticity intensity reaches the maximum when the Strouhal number is 0.33, followed by 0.22, and the minimum appears when the Strouhal number is 0.11. The vorticity intensity increases with the increase of the Strouhal number. As shown in Figure 5(a), the vortex is in a long and narrow ‘S’ shape. As the vortex dissipates backwards, the shape of the vortex gradually changes into a ‘comma’, which is quite different from the shape of the vortex in Figures 5(b) and 5(c). When St = 0.11, a column of vortices is generated in the wake, while when St = 0.22 and 0.33, two columns of vortices are formed in the wake. This phenomenon indicates that the shape difference and arrangement of vortices are greatly affected by the Strouhal number.Figure 5
Vorticity fields under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)The velocity contours under different Strouhal numbers are indicated in Figure6. It can be seen from Figure 6 that the wake velocity is increased by the swing of the fish body, so that the fluid accelerates backward and the fish body is accelerated forward. Therefore, the fish gets forward thrust by accelerating the wake backward through the swing of its body.Figure 6
Velocity contours under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)By comparing Figures6(a) and 6(b), it can be found that the velocity trajectory generated by the fish body swing is curved, and discontinuous fluid microclusters are generated, which are in pairs in each cycle of the fish body swing. It can be found that the thrust generated by the fish’s swing is not continuous but periodic and discrete. It can be clearly discovered from the three figures in Figure 6 that the velocity distribution on both sides of the fish body is different. Specifically, when the fish body is bent downward, the part with higher velocity on the lower side is in front of the fish, while the upper part of the velocity is higher in the tail of the fish. On the contrary, when the fish body is bent upward, the part with higher velocity on the upper side is in front of the fish, while the lower part of the velocity is higher in the tail of the fish. With the increase of the Strouhal number, the bending amplitude decreases gradually and the velocity value is the smallest at the depression of each turn. It can be concluded that the influence of the Strouhal number on the velocity field in the oscillating wake of fish body is mainly manifested at the difference of velocity magnitude and bending amplitude, but there is no significant influence for the Strouhal number on the overall distribution of the velocity field.


### 3.1.2. Thrust Characteristics and Propulsion Efficiency under Different Strouhal Numbers

The thrust coefficient curves at different Strouhal numbers are displayed in Figure7. It can be seen from the figures that the period and frequency of the thrust coefficient curves at different Strouhal numbers are the same, but the amplitude and average value of the curves are different. It is obvious that various thrust characteristics are generated by different Strouhal numbers.Figure 7
Thrust coefficient curves under different Strouhal numbers.The maximum and minimum thrust coefficients and the mean values of the thrust coefficients generated by the swing of the fish body at different Strouhal numbers are demonstrated in Table2. The data in Table 2 can be obtained by equation (5). It can be seen that the higher the Strouhal number, the lower the propulsion efficiency. Combined with the data in Table 2, it can be found that when the Strouhal number is 0.11, the thrust coefficient is small and the propulsion efficiency is high. When the Strouhal number is 0.33, the thrust coefficient is large and the propulsion efficiency is low. As the average value increases with the Strouhal number, there is a proportional relationship approximately 3 times larger. Therefore, it is necessary to increase the Strouhal number when traveling in a hurry. On the contrary, when coasting or when the energy consumption needs to be reduced, the Strouhal number should be appropriately reduced. Therefore, it is very important to choose an appropriate Strouhal number.Table 2
Maximum, minimum, and average values of thrust coefficient and propulsion efficiency with different Strouhal numbers.
St = 0.11St = 0.22St = 0.33CT (min)0.00280.00300.0590CT (max)0.07870.26330.7303CT (average)0.03930.13020.3857η0.80330.65430.6029The part of influence of the Strouhal number on the flow field characteristics and thrust characteristics of the fish-like body is investigated in the content mentioned above, and the effect of various factors on the fish-like body cannot be fully demonstrated. In order to better reveal the swimming status of the fish-like body, the three variables in the Strouhal number would be analyzed in the following discussion. In addition, the Taguchi method is used to discuss the degree of influence of various factors on the swimming of the fish-like body, and the influence of various factors on the flow field of the flexible body is further investigated. The results can provide insights for improving the working efficiency of the fish-like body and improve the swimming scheme.


## 3.1.1. Flow Field Characteristics under Different Strouhal Numbers

The vorticity fields at the different Strouhal numbers are displayed in Figure5. It can be seen from the figure that the vorticity intensity reaches the maximum when the Strouhal number is 0.33, followed by 0.22, and the minimum appears when the Strouhal number is 0.11. The vorticity intensity increases with the increase of the Strouhal number. As shown in Figure 5(a), the vortex is in a long and narrow ‘S’ shape. As the vortex dissipates backwards, the shape of the vortex gradually changes into a ‘comma’, which is quite different from the shape of the vortex in Figures 5(b) and 5(c). When St = 0.11, a column of vortices is generated in the wake, while when St = 0.22 and 0.33, two columns of vortices are formed in the wake. This phenomenon indicates that the shape difference and arrangement of vortices are greatly affected by the Strouhal number.Figure 5
Vorticity fields under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)The velocity contours under different Strouhal numbers are indicated in Figure6. It can be seen from Figure 6 that the wake velocity is increased by the swing of the fish body, so that the fluid accelerates backward and the fish body is accelerated forward. Therefore, the fish gets forward thrust by accelerating the wake backward through the swing of its body.Figure 6
Velocity contours under different Strouhal numbers. (a)St = 0.11. (b) St = 0.22. (c) St = 0.33.
(a)(b)(c)By comparing Figures6(a) and 6(b), it can be found that the velocity trajectory generated by the fish body swing is curved, and discontinuous fluid microclusters are generated, which are in pairs in each cycle of the fish body swing. It can be found that the thrust generated by the fish’s swing is not continuous but periodic and discrete. It can be clearly discovered from the three figures in Figure 6 that the velocity distribution on both sides of the fish body is different. Specifically, when the fish body is bent downward, the part with higher velocity on the lower side is in front of the fish, while the upper part of the velocity is higher in the tail of the fish. On the contrary, when the fish body is bent upward, the part with higher velocity on the upper side is in front of the fish, while the lower part of the velocity is higher in the tail of the fish. With the increase of the Strouhal number, the bending amplitude decreases gradually and the velocity value is the smallest at the depression of each turn. It can be concluded that the influence of the Strouhal number on the velocity field in the oscillating wake of fish body is mainly manifested at the difference of velocity magnitude and bending amplitude, but there is no significant influence for the Strouhal number on the overall distribution of the velocity field.


## 3.1.2. Thrust Characteristics and Propulsion Efficiency under Different Strouhal Numbers

The thrust coefficient curves at different Strouhal numbers are displayed in Figure7. It can be seen from the figures that the period and frequency of the thrust coefficient curves at different Strouhal numbers are the same, but the amplitude and average value of the curves are different. It is obvious that various thrust characteristics are generated by different Strouhal numbers.Figure 7
Thrust coefficient curves under different Strouhal numbers.The maximum and minimum thrust coefficients and the mean values of the thrust coefficients generated by the swing of the fish body at different Strouhal numbers are demonstrated in Table2. The data in Table 2 can be obtained by equation (5). It can be seen that the higher the Strouhal number, the lower the propulsion efficiency. Combined with the data in Table 2, it can be found that when the Strouhal number is 0.11, the thrust coefficient is small and the propulsion efficiency is high. When the Strouhal number is 0.33, the thrust coefficient is large and the propulsion efficiency is low. As the average value increases with the Strouhal number, there is a proportional relationship approximately 3 times larger. Therefore, it is necessary to increase the Strouhal number when traveling in a hurry. On the contrary, when coasting or when the energy consumption needs to be reduced, the Strouhal number should be appropriately reduced. Therefore, it is very important to choose an appropriate Strouhal number.Table 2
Maximum, minimum, and average values of thrust coefficient and propulsion efficiency with different Strouhal numbers.
St = 0.11St = 0.22St = 0.33CT (min)0.00280.00300.0590CT (max)0.07870.26330.7303CT (average)0.03930.13020.3857η0.80330.65430.6029The part of influence of the Strouhal number on the flow field characteristics and thrust characteristics of the fish-like body is investigated in the content mentioned above, and the effect of various factors on the fish-like body cannot be fully demonstrated. In order to better reveal the swimming status of the fish-like body, the three variables in the Strouhal number would be analyzed in the following discussion. In addition, the Taguchi method is used to discuss the degree of influence of various factors on the swimming of the fish-like body, and the influence of various factors on the flow field of the flexible body is further investigated. The results can provide insights for improving the working efficiency of the fish-like body and improve the swimming scheme.


## 3.2. Influence of Different Factors on Flow Field and Thrust Characteristics

### 3.2.1. Controllable Factors and Levels

On the basis of the previous analysis, the influence of the three variables in the Strouhal number on the flow field and the thrust coefficient is investigated through the Taguchi method. Therefore, the flow velocityU, the swing frequency f, and the swing amplitude A0 are the controllable factors for the investigations, and four levels are set for each factor. According to the Taguchi test and simulation, the wavelength is set as the characteristic length of the fish body, and the controllable factors and levels of the test are shown in Table 3.Table 3
Controllable factors and levels.
LevelControl factorsA0fU10.0420.420.052.50.630.0630.840.073.51According to the three controllable factors including flow velocity, swing frequency, and swing amplitude, as well as four groups of levels, 16 groups of data are obtained through the orthogonal list as shown in Table4. If the full-factor combination scheme is used, 43 = 64 tests are needed, which is 48 more than the 16 combination schemes of the Taguchi method. The number of tests is reduced by 75% and the workload is greatly reduced. Fluent is used to simulate the combined data, and the mean values and thrust coefficient are obtained, as shown in Table 4. It is observed that the smallest thrust coefficient appears in Case 3, while the largest thrust coefficient is in Case 13. Compared with Case 3, the thrust coefficient of Case 13 is increased by 2320%, and the thrust coefficients of Case 13 and Case 14 are much higher than that of other data sets. Therefore, a reasonable selection of three controllable factors is very crucial to the final results.Table 4
Research scheme combination and test results.
CaseControl factorsA0fUCTCase 10.0420.40.1998828Case 20.042.50.60.2894164Case 30.0430.80.1037629Case 40.043.510.6069987Case 50.052.50.80.4547392Case 60.05210.1859725Case 70.053.50.41.1613857Case 80.0530.60.7259077Case 90.06310.9978357Case 100.063.50.81.4445390Case 110.0620.60.4730492Case 120.062.50.40.9219013Case 130.073.50.62.5119235Case 140.0730.42.0029727Case 150.072.510.7893034Case 160.0720.80.5413109


### 3.2.2. Analysis of Influence Factors

The influence degree of each factor on the thrust coefficient is displayed in Table5, where K represents the average value of the calculated results of each factor at each level, and R is the range, representing the difference between the maximum and minimum value of K. The influence of various factors on the results can be judged according to the value of R. It can be found from Table 5 that the influence on thrust coefficient is Amplitude > Frequency > Velocity. According to the degree of influence on the results, the swing amplitude and swing frequency are the major factors, while the future flow velocity is the minor factor.Table 5
Influence of various factors on thrust coefficient.
A0fUK10.30001520.35005381.0715356K20.63200120.61384011.0000742K30.95933130.95761970.6360880K41.46137761.43121170.6450276R11.16136241.08115790.4265081Through the simulation of 16 times, the average thrust coefficient at different levels of each parameter can be obtained, as shown in Figure8. It can be seen from Figure 8, the thrust coefficient is positively correlated with the swing amplitude and swing frequency, while negatively correlated with the inflow velocity. With the increase of swing amplitude and swing frequency, the growth rate of thrust coefficient is basically unchanged, which is close to the change law of the first order function. With the increase of the inflow velocity, the thrust coefficient decreases slowly, then rapidly, and then slowly. According to the analysis data, the maximum thrust coefficient can be obtained when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s.Figure 8
Thrust coefficient of different parameters at different levels.The oscillating process is a periodic process, and the fish body will be pushed by water in the process of propulsion. In the process of periodic oscillation of the fish body, the thrust coefficient generated also indicates periodic fluctuations, as shown in Figure9. Figures 9(a)–9(c), respectively, display the comparison of four parameters under three conditions of constant swing amplitude, constant swing frequency, and constant inflow velocity. The swing amplitude, swing frequency, and inflow velocity are 0.04 m, 2.5 Hz, and 0.6 m/s, respectively. It can be seen that when the swing amplitude is constant, the mean and amplitude of thrust coefficient increase with the increase of swing frequency and incoming flow. When the oscillating frequency is constant, the mean and amplitude of thrust coefficient increase with the increase of the oscillating frequency but are not affected by the inflow velocity. When the inflow velocity is constant, the mean value and amplitude of thrust coefficient no longer change with the change of a single factor. Moreover, combining with the range size in Table 5, it can be found the influence effect of swing and frequency is similar.Figure 9
Comparison of thrust coefficients under four group of parameters. (a) Constant swing amplitude. (b) Constant oscillation frequency. (c) Constant inflow velocity.
(a)(b)(c)


### 3.2.3. Flow Field and Thrust Characteristics under the Condition of Optimal Controllable Factor

As discussed previously, when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s, the maximum thrust coefficient can be obtained. In this case, the flow field characteristics and thrust characteristics of the fish-like body are investigated in the following part. The optimized average thrust coefficient is 2.7253642. Compared with Case 13, the thrust coefficient increases due to the decreasing inlet velocity. For example, the inlet velocity changed from 0.6 m/s to 0.4 m/s, and the thrust coefficient increased by 8.5%.The vorticity fields at different moments are indicated in Figure10. It can be found that there are two rows of anti-Karman vortices which are staggered and rotated to the opposite direction in the wake of the fish body. During the process of the fish tail upswing, the attached vortices which rotate counterclockwise are formed on the lower surface of the tail. When the tail swings to the highest point, the attached vortices fall off due to the rapid downward swing. The formation of backward jet between the two vortices is caused by the formation process of anti-Karman vortex street, which makes the fish body receive forward thrust. As shown in Figure 10(a), when the tail swings to the lowest point, the positive vortex ring is shed, and the positive vortex ring is located on the upper row of the two rows of vortices. After the shedding of the positive vortices, as displayed in Figure 10(b), adherent vortices are generated at the tail of the fish body, and the attached vortices fall off to form a negative vortex ring. As shown in Figure 10(c) and 10(d), positive and negative vortices rings appear in pairs and gradually move backward. As the vortex intensity decreases, the vortex rings gradually dissipate. From the analysis from Figures 10(a)–10(e), there are only three pairs of vortex pairs that can be observed within the range of vorticity shown in this figure, and the intensity of previously generated vortices is already too small to be observed.Figure 10
Vorticity fields of fish body in a period of swing. (a)t = 0. (b) t = T/4. (c) t = T/2. (d) t = 3 T/4. (e) t = T.
(a)(b)(c)(d)(e)


## 3.2.1. Controllable Factors and Levels

On the basis of the previous analysis, the influence of the three variables in the Strouhal number on the flow field and the thrust coefficient is investigated through the Taguchi method. Therefore, the flow velocityU, the swing frequency f, and the swing amplitude A0 are the controllable factors for the investigations, and four levels are set for each factor. According to the Taguchi test and simulation, the wavelength is set as the characteristic length of the fish body, and the controllable factors and levels of the test are shown in Table 3.Table 3
Controllable factors and levels.
LevelControl factorsA0fU10.0420.420.052.50.630.0630.840.073.51According to the three controllable factors including flow velocity, swing frequency, and swing amplitude, as well as four groups of levels, 16 groups of data are obtained through the orthogonal list as shown in Table4. If the full-factor combination scheme is used, 43 = 64 tests are needed, which is 48 more than the 16 combination schemes of the Taguchi method. The number of tests is reduced by 75% and the workload is greatly reduced. Fluent is used to simulate the combined data, and the mean values and thrust coefficient are obtained, as shown in Table 4. It is observed that the smallest thrust coefficient appears in Case 3, while the largest thrust coefficient is in Case 13. Compared with Case 3, the thrust coefficient of Case 13 is increased by 2320%, and the thrust coefficients of Case 13 and Case 14 are much higher than that of other data sets. Therefore, a reasonable selection of three controllable factors is very crucial to the final results.Table 4
Research scheme combination and test results.
CaseControl factorsA0fUCTCase 10.0420.40.1998828Case 20.042.50.60.2894164Case 30.0430.80.1037629Case 40.043.510.6069987Case 50.052.50.80.4547392Case 60.05210.1859725Case 70.053.50.41.1613857Case 80.0530.60.7259077Case 90.06310.9978357Case 100.063.50.81.4445390Case 110.0620.60.4730492Case 120.062.50.40.9219013Case 130.073.50.62.5119235Case 140.0730.42.0029727Case 150.072.510.7893034Case 160.0720.80.5413109


## 3.2.2. Analysis of Influence Factors

The influence degree of each factor on the thrust coefficient is displayed in Table5, where K represents the average value of the calculated results of each factor at each level, and R is the range, representing the difference between the maximum and minimum value of K. The influence of various factors on the results can be judged according to the value of R. It can be found from Table 5 that the influence on thrust coefficient is Amplitude > Frequency > Velocity. According to the degree of influence on the results, the swing amplitude and swing frequency are the major factors, while the future flow velocity is the minor factor.Table 5
Influence of various factors on thrust coefficient.
A0fUK10.30001520.35005381.0715356K20.63200120.61384011.0000742K30.95933130.95761970.6360880K41.46137761.43121170.6450276R11.16136241.08115790.4265081Through the simulation of 16 times, the average thrust coefficient at different levels of each parameter can be obtained, as shown in Figure8. It can be seen from Figure 8, the thrust coefficient is positively correlated with the swing amplitude and swing frequency, while negatively correlated with the inflow velocity. With the increase of swing amplitude and swing frequency, the growth rate of thrust coefficient is basically unchanged, which is close to the change law of the first order function. With the increase of the inflow velocity, the thrust coefficient decreases slowly, then rapidly, and then slowly. According to the analysis data, the maximum thrust coefficient can be obtained when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s.Figure 8
Thrust coefficient of different parameters at different levels.The oscillating process is a periodic process, and the fish body will be pushed by water in the process of propulsion. In the process of periodic oscillation of the fish body, the thrust coefficient generated also indicates periodic fluctuations, as shown in Figure9. Figures 9(a)–9(c), respectively, display the comparison of four parameters under three conditions of constant swing amplitude, constant swing frequency, and constant inflow velocity. The swing amplitude, swing frequency, and inflow velocity are 0.04 m, 2.5 Hz, and 0.6 m/s, respectively. It can be seen that when the swing amplitude is constant, the mean and amplitude of thrust coefficient increase with the increase of swing frequency and incoming flow. When the oscillating frequency is constant, the mean and amplitude of thrust coefficient increase with the increase of the oscillating frequency but are not affected by the inflow velocity. When the inflow velocity is constant, the mean value and amplitude of thrust coefficient no longer change with the change of a single factor. Moreover, combining with the range size in Table 5, it can be found the influence effect of swing and frequency is similar.Figure 9
Comparison of thrust coefficients under four group of parameters. (a) Constant swing amplitude. (b) Constant oscillation frequency. (c) Constant inflow velocity.
(a)(b)(c)


## 3.2.3. Flow Field and Thrust Characteristics under the Condition of Optimal Controllable Factor

As discussed previously, when the swing amplitude is 0.07 m, the swing frequency is 3.5 Hz, and the inflow velocity is 0.4 m/s, the maximum thrust coefficient can be obtained. In this case, the flow field characteristics and thrust characteristics of the fish-like body are investigated in the following part. The optimized average thrust coefficient is 2.7253642. Compared with Case 13, the thrust coefficient increases due to the decreasing inlet velocity. For example, the inlet velocity changed from 0.6 m/s to 0.4 m/s, and the thrust coefficient increased by 8.5%.The vorticity fields at different moments are indicated in Figure10. It can be found that there are two rows of anti-Karman vortices which are staggered and rotated to the opposite direction in the wake of the fish body. During the process of the fish tail upswing, the attached vortices which rotate counterclockwise are formed on the lower surface of the tail. When the tail swings to the highest point, the attached vortices fall off due to the rapid downward swing. The formation of backward jet between the two vortices is caused by the formation process of anti-Karman vortex street, which makes the fish body receive forward thrust. As shown in Figure 10(a), when the tail swings to the lowest point, the positive vortex ring is shed, and the positive vortex ring is located on the upper row of the two rows of vortices. After the shedding of the positive vortices, as displayed in Figure 10(b), adherent vortices are generated at the tail of the fish body, and the attached vortices fall off to form a negative vortex ring. As shown in Figure 10(c) and 10(d), positive and negative vortices rings appear in pairs and gradually move backward. As the vortex intensity decreases, the vortex rings gradually dissipate. From the analysis from Figures 10(a)–10(e), there are only three pairs of vortex pairs that can be observed within the range of vorticity shown in this figure, and the intensity of previously generated vortices is already too small to be observed.Figure 10
Vorticity fields of fish body in a period of swing. (a)t = 0. (b) t = T/4. (c) t = T/2. (d) t = 3 T/4. (e) t = T.
(a)(b)(c)(d)(e)


## 4. Conclusion

In this paper, the influence of the Strouhal number on the flow field and thrust characteristics of the fish-like body is analyzed, the mechanism affecting the thrust and propulsion efficiency is revealed, and the flow field characteristics of the wake of the fish body are intuitively demonstrated. According to the Taguchi method, an orthogonal table L16 is designed, and the coupling effects of swing amplitude, swing frequency, and inflow velocity on thrust performance are investigated. The main conclusions are as follows.(1)
The anti-Karman vortex street generated by the fish oscillation is affected by the Strouhal number, and the intensity of the vortex generated by various Strouhal numbers are different under the same oscillation equation. The fish can swim forward because of the effect of swinging with the fluid nearby, so that the fish body is pushed forward. The thrust on the fish body is discontinuous and periodic, and the thrust is affected by the Strouhal number, but it does not simply increase with the increase of the Strouhal number.(2)
WhenSt = 0.11, the thrust coefficient is small and the propulsion efficiency is high. When the St = 0.33, the thrust coefficient is large, but the propulsion efficiency is low. Therefore, when high running velocity is required, the larger Strouhal number is suitable, while when running velocity is not required and energy saving is needed, the smaller Strouhal number is suitable.(3)
In all the design schemes by the Taguchi method, the optimal group of increasing thrust coefficient and thrust is obtained. That is, when the swing amplitude is 0.07, the swing frequency is 3.5 Hz, and when the inflow velocity is 0.4 m/s, the maximum thrust and thrust coefficient are generated.(4)
Through the investigations, it is found that the thrust coefficient is positively correlated with the swing amplitude and swing frequency, and is negatively correlated with the inflow velocity. The swing amplitude and the swing frequency are the main factors affecting the thrust characteristics of the fish body, while the impact of incoming flow velocity on the results is not as great as the first two.



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*Source: 1592792-2022-11-08.xml*