Numerical Simulation on Vortex Shedding from a Hydrofoil in Steady Flow

Journal of Marine Science and Engineering

Article Numerical Simulation on Vortex Shedding from a Hydrofoil in Steady Flow

Jian Hu, Zibin Wang, Wang Zhao, Shili Sun, Cong Sun and Chunyu Guo * College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] (J.H.); [email protected] (Z.W.); [email protected] (W.Z.); [email protected] (S.S.); [email protected] (C.S.) * Correspondence: [email protected] or [email protected]

Received: 19 February 2020; Accepted: 3 March 2020; Published: 12 March 2020

Abstract: This paper presents a numerical modeling procedure for the idealization of vortex shedding effects in the wake flow field of a NACA0009 hydrofoil. During the simulation, the lift and drag acting on the hydrofoil were monitored, and the vortex-shedding frequency of the hydrofoil was analyzed. The effects of inflow velocity, trailing-edge thickness, angle of attack, and maximum hydrofoil thickness on vortex shedding were investigated. The results indicate that an increase in the inflow velocity led to an increase in the vortex-shedding frequency and a negligible change in the Strouhal number. Furthermore, as the thickness of the trailing edge increased, the vortex-shedding frequency decreased gradually, whereas the Strouhal number first increased and then decreased. Vortex shedding and lift curve oscillations ceased altogether after the angle of attack of the hydrofoil increased beyond a certain threshold. When the maximum hydrofoil thickness was increased while keeping the thickness and chord length of the trailing edge constant, the vortex-shedding frequency decreased.

Keywords: vortex shedding; wake; ﬂow-induced vibrations

1. Introduction Vortex shedding from the trailing edge of a foil is a significant phenomenon in various engineering disciplines. Flows separated by a foil will recombine at the upper and lower surfaces of the trailing edge of the foil, forming periodic eddies in the wake field and inducing mechanical vibrations. This phenomenon was first observed in aircraft airfoils; because airfoils typically exhibit very high wake-vortex shedding frequencies, they are at a considerable risk of fatigue damage. The damage caused by vortex shedding is particularly severe if the vortex-shedding frequency approaches a natural frequency of the airfoil. A similar phenomenon is known to occur in propellers and is particularly concerning in marine propellers because an intense degree of vortex shedding leads to periodic fatigue stresses in the blades of the propeller that shorten their service life. Because vortex shedding can lead to a series of failures in engineered structures, it is important to study the causes of this phenomenon and elucidate the behavior of vortex shedding in a variety of operating conditions. The mechanical vibrations induced by vortex shedding have been widely investigated. Vincenc Strouhal discovered that the audible tone of a wire “singing” in the wind is proportional to the quotient of the wind speed by the wire thickness (i.e., tone = proportionality coefficient wind speed/wire × thickness) and that each cross-section has a certain proportionality coefficient in the noncritical range of Reynolds numbers. This is known as the Strouhal number, and it is an important parameter to evaluate the vortex-shedding frequency. Theodore von Kármán linked this phenomenon to the stable and staggered vortices that form behind a cylindrical object, which was later termed the Kármán vortex street effect. In addition, he proposed the theory of vortex street stability [1]. Perry et al. [2]

J. Mar. Sci. Eng. 2020, 8, 195; doi:10.3390/jmse8030195 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 195 2 of 20 obtained images of the vortex-shedding process from the wake field of a circular cylinder by tracing the motion of aluminum particles in long-exposure photography. Griffin [3] and Williamson et al. [4] investigated the alternating shedding of vortices from a cylinder and discovered that this process is caused by interactions between vortices on the upper and lower surfaces of the cylinder; once a vortex on a given surface reaches a sufficient strength, it draws the opposing shear layer across the near wake, causing the vortices to shed alternatingly from the upper and lower surfaces. Gerrard [5] observed vortex shedding from circular and bluff cylinders with different cross-sectional shapes and deduced that the vortex generation is affected by the end conditions of the cylinders. Gerich [6] proved that the shape of the ends of a cylinder affect the mechanism of vortex shedding from these ends. In addition, Achenbach and Heinecke [7] conducted wind tunnel experiments and discovered that the wake of a cylinder became more regular as the roughness of the cylinder’s surface increased. Furthermore, the Strouhal number of a vortex shedding from a rough cylinder was estimated to be much smaller than that of a vortex shedding from a smooth cylinder. Through the various studies conducted on the vortex-shedding phenomenon, it was observed that intense resonance can occur if the vortex-shedding frequency of the Kármán vortex street is equal to one of the eigenfrequencies of a given object. This discovery shifted the focus of the study of vortex shedding from simple cylindrical objects to objects of practical interest, such as hydrofoils. A series of analytical studies were also conducted on the basis of experimental data. Ausoni [8] experimentally investigated the effects of cavitation on vortex shedding from symmetrical hydrofoils with a blunted trailing edge and analyzed the wake structure, vortex-shedding frequency, and hydrofoil resonance frequency associated with each stage of cavitation development. He discovered that the vortex-shedding frequency can “lock-in” to an eigenfrequency of a hydrofoil. Lotfy and Rockwell [9] investigated vortex shedding in the near-wake of an oscillating trailing edge. Prasad and Williamson [10] analyzed the vortex dynamics of hydrofoil wakes based on the instability of the shear layer separating the water from a bluff body. Dwayne et al. [11] investigated vortex shedding at a high Reynolds number from hydrofoils with different terminating bevel angles on their trailing edges and found that the vortex-shedding intensity is higher in blunter and thicker trailing edges. Zobeiri et al. [12] conducted an experimental study on the effects of the trailing-edge shape on vortex shedding and flow-induced vibrations and found that oblique trailing edges significantly reduce flow-induced vibrations. This was attributed to the collisions between upper and lower vortices in an oblique trailing edge, which leads to vorticity redistribution. Numerical modeling has been widely employed for the study of vortex shedding. Many previous numerical studies focused on vortex shedding from circular objects. For example, Williamson [13] numerically simulated the wake flows of a cylindrical object. William [14] used complex analysis to formulate an analytical expression for the shedding frequency of Kármán vortex streets from the trailing edge of an airfoil. However, the shedding frequencies predicted by this expression were typically much lower than the experimentally observed ones. Huerre and Monkewitz [15] and Oertel et al. [16] analyzed the effects of flow instabilities on vortex-shedding behaviors of blunt hydrofoils analytically. Potential flow theory is often used to study the wake-vortex shedding of swinging or flapping objects. However, current studies on vortex shedding focus on the formation of Kármán vortex streets behind static objects, which is essentially a boundary layer dissociation problem; these problems are very difficult to model with the potential flow theory. Another significant feature of this problem is that the trailing edge of the hydrofoil is often very thin, which leads to very large local Reynolds numbers. This results in unique physical phenomena that can only be modeled by precise numerical simulations. Lee et al. [17] studied the vortex shedding of hydrofoils, numerically considering a variety of trailing-edge shapes, and compared their findings with the experimental findings of Ausoni and Zobeiri. In addition, they investigated the effects of periodically varying free-stream flows on vortex shedding. Many studies using numerical simulations of or experimental tests on vortex shedding have been conducted, and the subsequent hydrodynamic force fluctuations of hydrofoils have been analyzed. J. Mar. Sci. Eng. 2020, 8, 195 3 of 20

The main purpose of the abovementioned studies was to evaluate the amplitude and frequency of force fluctuation, which have significant effects on marine structures. Ausoni [8] experimentally studied the vortex-shedding frequency of hydrofoils with perpendicularly truncated trailing edges and found that the increase in frequency was almost linear with the inflow velocity. Dwayne et al. [11] studied the vortex-shedding frequency of hydrofoils with obliquely truncated trailing edges and found that thicker hydrofoils or blunter trailing edges correspond to higher vortex-shedding frequencies. Another experimental study was conducted by Zobeiri [12], wherein it was found that the obliquely truncated trailing edges can significantly reduce the vortex-shedding strength, and subsequently, the amplitude of the force fluctuation. Analytical studies of vortex shedding were performed by Blake [14] and Oertel [16] on hydrofoils and cylinders, respectively. However, the force of analytical results is much less than that by experiments. The vortex shedding of hydrofoils is strongly related to the formation of viscous boundary layers. From this perspective, methods based on the potential flow assumptions can hardly predict vortex shedding precisely. Instead, computational fluid dynamics (CFD) simulations can effectively represent vortex shedding. Although Reynolds-averaged Navier–Stokes simulations [8] or large Eddy simulations (LES) [17] cannot fully simulate turbulent flows, their numerical results generally agree well with experimental results and can reflect the physical characteristics of vortex shedding. The above discussion shows that vortex shedding from static hydrofoils is a problem of practical significance and academic interest as it plays a significant role in flow-induced vibrations and noise and fatigue damage. However, research on this problem is still insufficient, and further elucidations are required. In this study, we conducted CFD simulations of two-dimensional flows around a NACA0009 hydrofoil in a variety of operating conditions to observe vortex shedding at the trailing edge of a hydrofoil. We investigated the lift curve and trailing-edge vortex shedding of the hydrofoil with different inflow velocities, angles of attack, truncation points, and maximum hydrofoil thicknesses to reveal their correlations with the vortex-shedding frequency. The findings of this study will provide general rules for the investigation of flow-induced vibrations in objects such as hydrofoils and marine propellers.

2. Materials and Methods

2.1. Mathematical Modeling In this study, the finite volume method, which satisfies the conservations of mass and momentum, was used to solve the integral equations. The vertex-centered method was adopted to form the control elements; the second-order upwind scheme was used for the discrete differentials; the semi-implicit method for pressure linked equations scheme was used for the flow solution. It firstly assumes a velocity distribution u∗, v∗, Φ∗(u, v, Φ) in the flow field and then calculates the coefficients and constants in the discrete-form momentum equations at the first iteration. Then, it guesses the pressure p∗; it solves the discrete momentum equation from the pressure field and the pressure correction equation from the velocity field. After correcting the calculated pressure and velocity, all other variables of the control equation are solved. Finally, the results are verified and iterated until convergence. Numerical simulations were performed using the Spalart–Allmaras (S-A) turbulence model, which is suitable for high-precision boundary-layer computations, e.g., lift computations. Spalart and Allmaras [18] argued that energy and information flow from large scales to smaller scales in free shear flows and represented the turbulent eddy viscosity coefficient by two terms: a production term and a diffusion term. They constructed a one-equation turbulence model by utilizing the experience gained from other turbulence models and by amending the other turbulent flow-field models. In the S-A equation, the definition of Reynold stress is: u u = 2vtS , S ∂U /∂x + ∂U /∂x /2 (1) − i j ij ij ≡ i j j i J. Mar. Sci. Eng. 2020, 8, 195 4 of 20

where, the turbulent eddy viscosity vt can be expressed as:

χ3 v˜ vt = v˜ f , f = , χ , (2) v1 v1 3 + 3 ≡ v χ cv1 where v is the kinematic molecular viscosity and v˜ is the working variable. From the above equations, we obtain the transport equation: Dv˜ 1 h 2i cb1 v˜ 2 = cb [1 ft2]S˜v˜ + ((v + v˜) v˜) + cb ( v˜) cw1 fw ft2 + ft1∆U , (3) Dt 1 − σ ∇ ∇ 2 ∇ − − κ2 d where S(~) is the magnitude of vorticity; b is the distance to the nearest wall; ∆U is the velocity difference between a point in the fluid and a point on the wall; ft1, ft2, and fw are functional equations, and the rest of the symbols represent constants. The specific definitions of these symbols are as follows:

v˜ χ S˜ S + f , f = 1 (4) 2 2 v2 v2 ≡ κ d − 1 + χ fv1   w2 h i  t 2 2 2  ft1 = ct1 gt exp ct2 d + g d , gt min(0.1, ∆U/wt∆x), (5) − ∆U2 t t  ≡

2 ft = ct exp c χ , (6) 2 3 − t4  1/6 1 + c6  w3  6 v˜ fw = g  , g = r + cw2 r r , r , (7)  6 + 6  − ≡ ˜ 2 2 g cw3 Sκ d The S-A turbulence model is more computationally eﬃcient than two-equation turbulence models, and it is also very stable and accurate. The S-A model accurately simulates the region of the wake where strain rates are dominated by vorticity, and it predicts the turbulent eddy viscosity.

2.2. Test Case and Computational Mesh In this study, a two-dimensional ﬂow method was used to study vortex shedding. The model used in this study was a NACA0009 hydrofoil with a chord length c of 100 mm and a trailing-edge thickness b of 3.22 mm, which is identical to the model used by Ausoni et al. (2006). Figure1 shows a diagram of the hydrofoil model with the chord length (c), trailing-edge thickness (b), and angle of attack (α). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 20

Figure 1. Diagram of the hydrofoil model. Figure 1. Diagram of the hydrofoil model.

To avoid the effects of the computational boundaries illustrated in Figure2 on the flow simulation, the domain was set to relatively large dimensions (13c 6c). Typically, the outlet boundary has a × greater effect than the inlet boundary owing to the downstream vortex shedding. Thus, the hydrofoil

Figure 2. Computational domain.

Figure 3. Computational domain mesh.

The grid around the hydrofoil is shown in Figure 3. Two mesh types were used in the computational domain, namely trimmed and prism layer meshes. The trimmed mesh was the primary type of mesh used in the fluid region. This part of the mesh consisted of quadrilateral elements parallel to the inflow; this configuration reduces truncation errors in the calculation. Prism layer meshes were used to generate the boundary layers. A 3c × 2c refined region was set up around the hydrofoil to refine the flows around it, and the mesh size in this region was 5% of that of the unrefined region. The computational domain contained a total of 2,495,989 mesh elements.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 20

J. Mar. Sci. Eng. 2020, 8, 195 5 of 20 was further from the outlet than from the inlet. Neglecting the mutual effect between the inlet boundary J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 5 of 20 and the hydrofoil, the inlet was set as the velocity inlet. To achieve better conservation and more stable simulations, the outlet was set as the pressure outlet rather than the velocity outlet. To further avoid the effects of the boundaries, the upper and lower boundaries were set as symmetrical and slipping, different from the nonslipping surfaces of the hydrofoil. Figure 1. Diagram of the hydrofoil model.

Figure 1. Diagram of the hydrofoil model. Figure 2. Computational domain.

The grid around the hydrofoil is shown in Figure3. Two mesh types were used in the computational domain, namely trimmed and prism layer meshes. The trimmed mesh was the primary type of mesh used in the fluid region. This part of the mesh consisted of quadrilateral elements parallel to the inflow; this configuration reduces truncation errors in the calculation. Prism layer meshes were used to generate the boundary layers. A 3c 2c refined region was set up around the hydrofoil to refine × the flows around it, and the mesh size in this region was 5% of that of the unrefined region. The computational domain contained a total of 2,495,989 mesh elements. Figure 2. Computational domain.

Figure 3. Computational domain mesh.

FigureThe grid4 shows around the boundary the hydrofoil layer gridis shown of the hydrofoil.in Figure In 3. theseTwo computations, mesh types were physical used variables in the tendcomputational to vary most domain significantly, namely around trimmed boundaries. and prism Therefore, layer a meshes. refined boundary The trimmed layer mesh mesh was was used the inprimary the calculations. type of mesh The used thickness in the of fluid the first region. mesh This layer part was of determined the mesh consisted by the Y+ ofvalue. quadrilateral Because theelements Reynolds parallel number to the in inflow the simulation; this configuration was quite largereduces (Re truncation= 1,194,257.2), errors the in fluid the calculation. flow became Prism fully turbulentlayer meshes at positions were used close to generate to the surface the boundary of the hydrofoil. layers. A Therefore, 3c × 2c refined the Y+ regionvalue was setset toup 1, around which yieldedthe hydrofoil a first to mesh refine layer the thickness flows around of 1.97 it, and10 the6 m. mesh The growthsize in this rate region of the prismwas 5% layer of that mesh of wasthe × − setunrefined to 1.2, andregion. the The boundary computational layer mesh domain was configured contained a with total a of total 2,495 of,989 20 layers mesh toelements. ensure a smooth transition into the outer mesh. The total thickness of the boundary layer mesh was 3.67 10 4 m. × −

J. Mar. Sci. Eng. 2020, 8, 195 6 of 20

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 6 of 20

Figure 4. Boundary layer mesh of the hydrofoil’s trailing edge. Figure 4. Boundary layer mesh of the hydrofoil’s trailing edge.

2.3. ConvergenceFigure 4 shows Analysis the boundary layer grid of the hydrofoil. In these computations, physical variables tend to vary most significantly around boundaries. Therefore, a refined boundary layer 2.3.1.mesh Meshwas used Convergence in the calculations. The thickness of the first mesh layer was determined by the Y+ value.Using Because the meshingthe Reynolds strategy number reported in the by Ausonisimulation et al. was [8], aquite refined large mesh (Re was= 1,194, configured257.2), the to assess fluid theflow mesh became convergence. fully turbulent at positions close to the surface of the hydrofoil. Therefore, the Y+ value was setThe to medium 1, which mesh yielded had a a first total mesh of 2,495,989 layer thickness elements, of and 1.97 the × 10 thickness−6 m. The of growth the boundary rate of the layer prism was 3.67layer mesh10 4 m.was The setmesh to 1.2, size and was the further boundary increased layer mesh and reduced was configured to verify thewith mesh a total convergence, of 20 layers and to × − theensure resulting a smooth mesh transition had medium into the and outer fine mesh. grids withThe total 1,562,259 thickness and of 4,027,882 the boundary elements, layer respectively. mesh was When3.67 × 10 the−4 gridm. scale was changed between coarse, medium, and fine while keeping the number of boundary layers constant, the extension rates of the layers changed to 1.4, 1.3, and 1.2, respectively. 2.3. ConvergenceThen, the time Analysis step was set to 1 10 6 s. Table1 lists the drag coe fficient C , maximum lift × − d coefficient Cl (max), and Strouhal number (St) obtained with different grid scales. 2.3.1. Mesh Convergence Using the meshing strategyTable reported 1. Mesh by convergence Ausoni et al. analysis [8], a refined results. mesh was configured to assess the mesh convergence. Mesh Cd Cl (max) St The medium mesh had a total of 2,495,989 elements, and the thickness of the boundary layer Coarse 0.0270 0.06018 0.190 was 3.67 × 10−4 m. The mesh size was further increased and reduced to verify the mesh convergence, Medium 0.0258 0.05605 0.192 and the resulting mesh hadFine medium 0.0253 and fine grids 0.05521 with 1,562, 0.197259 and 4,027,882 elements, respectively. When the grid scale was changed between coarse, medium, and fine while keeping the number of boundary layers constant, the extension rates of the layers changed to 1.4, 1.3, and 1.2, From Table1, it can be observed that the calculation results with the three grid conditions are respectively. similar. The mesh size was further reduced−6 under the medium mesh condition, and the resulting Cd, Then, the time step was set to 1 × 10 s. Table 1 lists the drag coefficient 퐶푑 , maximum lift Cl, and St changed by 1.9%, 1.5%, and 2.6%, respectively. The degree of change in data is small, and coefficient 퐶푙 (max), and Strouhal number (푆푡) obtained with different grid scales. thus, it can be considered grid-independent. Table 1. Mesh convergence analysis results. 2.3.2. Time-Step Convergence 푪 푪 푺 We searched for theMesh time step that was풅 required to풍 (max) converge the simulation풕 when the mesh Coarse 0.0270 0.06018 0.190 4 contained 2,495,989 elements. The time step of the simulation was gradually decreased from 2 10− 6 Medium 0.0258 0.05605 0.192 × to 1 10 s. The C , C (max), and St that were obtained with each time step value (Dt) are presented × − d l in Table2. Fine 0.0253 0.05521 0.197

From Table 1, it can be observed that the calculation results with the three grid conditions are

similar. The mesh size was further reduced under the medium mesh condition, and the resulting 퐶푑, 퐶푙, and 푆푡 changed by 1.9%, 1.5%, and 2.6%, respectively. The degree of change in data is small, and thus, it can be considered grid-independent.

2.3.2. Time-Step Convergence

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Table 2. Time step convergence.

Dt (s) Cd Cl (max) St 2 10 4 0.0145 0.000299 0.112 × − 1 10 4 0.0145 8.98 10 5 0.150 × − × − 5 10 5 0.0157 0.01663 0.0171 × − 6 10 6 0.0239 0.05293 0.188 × − 2 10 6 0.0252 0.05599 0.197 × − 1 10 6 0.0258 0.05605 0.197 × −

From Table2, it can be observed that the results become increasingly similar as the time step shortens. In particular, the results obtained at Dt = 2 10 6 and 1 10 6 s are very similar, indicating × − × − that the results obtained with Dt = 1 10 6 s have converged. × − 2.3.3. Turbulence Model Selection The mesh and time-step convergence analyses show that a converged result can be obtained from the simulation with a total mesh number of 2,495,989 and a time step of 1 10 6 s. Therefore, these × − two parameters were used to verify the convergence of the S-A, shear stress transport k-omega, and K-epsilon turbulence models. The comparison results are shown in Table3.

Table 3. Comparison of Cl and St obtained by diﬀerent turbulence models.

Turbulence Model Cl (max) St S-A 0.05605 0.192 Shear stress transport k-omega 0.06199 0.220 K-epsilon 0.05602 0.186

Ausoni [8] conducted a comparative test between a normal hydrofoil and a hydrofoil with rough stripes at the leading edge. In the latter case, a strong turbulent boundary layer was developed. With an inflow velocity of 12 m/s, the normal and rough-stripe hydrofoils had St of 0.245 and 0.174, respectively, indicating that the vortex-shedding frequency is highly related to the occurrence of turbulence. However, there is insufficient understanding of the turbulence around the hydrofoil. The calculation results obtained with the three models are all within the experimental value range and have no significant differences. The S-A model could accurately calculate the near-wall flow of the hydrofoil, and thus, was used for the subsequent simulations.

3. Results

3.1. Simulation Results Based on the results of the mesh and time-step convergence analyses, a base mesh size of 0.005 m and time step of 1 10 6 s were used as the computational parameters of the simulations. The × − inflow velocity Ure f was set to 12 m/s. The lift and drag of the hydrofoil were then obtained from these simulations. A dimensionless time quantity, t = TU/c, was defined as the abscissa to plot the relevant curves. Figure5 shows the changes in the lift curve from the beginning of the simulation until a steady state was reached. Two distinct stages can be identified from these plots—at the beginning of the simulation, the flow field is in an unsteady state, and the amplitude of the lift curve oscillations increases; after some time, the amplitude of the lift curve oscillations stops changing and the flow field becomes stable. Figure6 shows the Fourier transform plot of the convergence-time history curve. From this figure, the frequency (735.3 Hz) and amplitude (0.0689) of the lift curve oscillations under the given set of operating conditions were obtained. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 20

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 8 of 20 increases; after some time, the amplitude of the lift curve oscillations stops changing and the flow field becomes stable. Figure 6 shows the Fourier transform plot of the convergence-time history increases; after some time, the amplitude of the lift curve oscillations stops changing and the flow curve. From this figure, the frequency (735.3 Hz) and amplitude (0.0689) of the lift curve oscillations field becomes stable. Figure 6 shows the Fourier transform plot of the convergence-time history under the given set of operating conditions were obtained. curve. From this figure, the frequency (735.3 Hz) and amplitude (0.0689) of the lift curve oscillations J. Mar. Sci. Eng. 2020, 8, 195 8 of 20 under the given set of operating conditions were obtained. Fully developed 0.10

Fully developed 0.10 0.05

0.05

C 0.00

-0.05 Unsteady

-0.05 Unsteady -0.10 0 1 2 3 4 5 -0.10 t 0 1 2 3 4 5 t Figure 5. Convergence-time history curve of 퐶푙. Figure 5. Convergence-time history curve of Cl. Figure 5. Convergence-time history curve of 퐶푙. 0.08

0.07 (735.30,0.0689) 0.08 0.06 0.07 (735.30,0.0689) 0.05 0.06 0.04 0.05 0.03 Amplitude 0.04 0.02 0.03 Amplitude 0.01 0.02 0.00 0.01 0 2,000 4,000 6,000 8,000 10,000 0.00 f [Hz] 0 2,000 4,000 6,000 8,000 10,000

Figure 6. Characteristics of the liftf curve[Hz] in the frequency domain. Figure 6. Characteristics of the lift curve in the frequency domain. Figure7 illustrates the changes in the drag curve of the hydrofoil over time. Because the flow field Figure 6. Characteristics of the lift curve in the frequency domain. was unsteadyFigure 7 inillustrates the early the stages changes of the in simulation, the drag curve large of changes the hydrofoil were initially over time. observed Because in the the drag flow coefieldfficient. was unsteady Once the in flow the field early became stages of stable, the simulation, the drag coe largefficient changes still exhibited were initially periodic observed oscillation, in the Figure 7 illustrates the changes in the drag curve of the hydrofoil over time. Because the flow similardrag coefficie to the liftnt. curve.Once the The flow steady-state field became oscillation stable, frequency the drag of coefficient the drag still curve exhibit (1460.92ed periodic Hz) is field was unsteady in the early stages of the simulation, large changes were initially observed in the approximatelyoscillation, similar twice that to theof the lift lift curve curve,. indicatingThe steady that-state a pair oscillation of vortex frequency shedding form of the one drag oscillation curve drag coefficient. Once the flow field became stable, the drag coefficient still exhibited periodic of(1 the460.92 liftHz while) is approximately a single vortex twice shedding that fromof the the lift upper curve, or indicating lower surface that a forms pair of one vortex oscillation shedding of J.oscillation, Mar. Sci. Eng. similar 2020, 8, x FOR to the PEER lift REVIEW curve . The steady-state oscillation frequency of the drag curve9 of 20 theform drag. one oscillation of the lift while a single vortex shedding from the upper or lower surface forms (1460.92Hz) is approximately twice that of the lift curve, indicating that a pair of vortex shedding one oscillation of the drag. form one oscillation of the lift while a single vortex shedding from the upper or lower surface forms 0.32 one oscillation of the drag.

0.28

d 0.24 Fully developed

Unsteady 0.20

0.16

0 1 2 3 4 5 t

Figure 7. Drag curve of the hydrofoil under convergent conditions. Figure 7. Drag curve of the hydrofoil under convergent conditions. Here, we deﬁne the pressure at some arbitrary point as p, which can be nondimensionalized as 1Here,p p we define the pressure at some arbitrary point as p, which can be nondimensionalized as C = − ∞ , where p = 0. p 2 U2 ρ re f ∞ 1 pp  CpFigure 8 shows2 that the pressure distribution on the upper and lower surfaces of the hydrofoil 2 Uref are similar at t = 2.68;, where because p∞ = 0. at this time, the diﬀerence in pressures between the upper and lower Figure 8 shows that the pressure distribution on the upper and lower surfaces of the hydrofoil are similar at t = 2.68; because at this time, the difference in pressures between the upper and lower surfaces is small, the corresponding lift coefficient is also small (0.016). Figure 8b,c show the surface pressure distributions of the hydrofoil when the lift coefficient was at its minimum (t = 2.88) and maximum (t = 3.12), respectively. In these figures, it may be observed that a significant pressure difference arises after x/c = 0.75, which induces a lift.

1.0 1

Upper 0.5 Upper 0

p p Cut

C C 0.0 Cut Lower -1 -0.5 Lower

-1.0 -2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 x/c x/c

(a) t = 2.68 (b) t = 2.88

Lower

p 0

C Cut

Upper -1

0.00 0.25 0.50 0.75 1.00 x/c

Figure 8. Distribution of the pressure on the surfaces of the hydrofoil over time.

Two monitoring points (Point 1 and Point 2) were set up at the upper and lower turning points of the trailing edge of the hydrofoil. The velocity and pressure at the monitoring points are denoted

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 9 of 20

0.32

0.28

d 0.24 Fully developed

Unsteady 0.20

0.16

0 1 2 3 4 5 t

Figure 7. Drag curve of the hydrofoil under convergent conditions.

Here, we define the pressure at some arbitrary point as p, which can be nondimensionalized as

1 pp  Cp  2 2 Uref , where p∞ = 0.

J. Mar. Sci.Figure Eng. 20208 shows, 8, 195 that the pressure distribution on the upper and lower surfaces of the hydrofoil9 of 20 are similar at t = 2.68; because at this time, the difference in pressures between the upper and lower surfaces is small, the corresponding lift coefficient is also small (0.016). Figure 8b,c show the surface surfacespressure is distributions small, the corresponding of the hydrofoil lift coe whenfficient the islift also coefficient small (0.016). was at Figure its minimum8b,c show ( thet = 2.88) surface and pressuremaximum distributions (t = 3.12), respectively.of the hydrofoil In these when figures the lift, itcoe mayfficient be observed was at its that minimum a significant (t = 2.88) pressure and maximumdifference (arisest = 3.12), after respectively. x/c = 0.75, which In these induces figures, a lift. it may be observed that a significant pressure difference arises after x/c = 0.75, which induces a lift.

1.0 1

Upper 0.5 Upper 0

p p Cut

C C 0.0 Cut Lower -1 -0.5 Lower

-1.0 -2 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 x/c x/c

(a) t = 2.68 (b) t = 2.88

Lower

p 0

C Cut

Upper -1

0.00 0.25 0.50 0.75 1.00 x/c

FigureFigure 8. 8.Distribution Distribution of of the the pressure pressure on on the the surfaces surfaces of of the the hydrofoil hydrofoil over over time. time.

TwoTwo monitoring monitoring points points (Point (Point 1 1 and and Point Point 2) 2) were were set set up up at at the the upper upper and and lower lower turning turning points points ofof the the trailing trailing edge edge of of the the hydrofoil. hydrofoil. The The velocity velocity and and pressure pressure at at the the monitoring monitoring points points are are denoted denoted as U and P, respectively, and we further define a dimensionless variable u = U/Ure f to simplify the velocity term. The dimensionless velocity–time and velocity–pressure curves obtained at these points are presented in Figure9. In the figure, the velocity and pressure curves at both the monitoring points exhibit similar oscillatory amplitudes and frequencies but in opposing phases. This indicates that the vortices on the upper and lower surfaces of the trailing edge are symmetrical and that they are shed alternatingly. At each monitoring point, the velocity and pressure curves have identical periodicities but opposing amplitudes, i.e., the maximum of one curve always corresponds to the minimum of the other curve. Figure 10 presents a vorticity map of the shed vortices. As can be seen from the figure, vortices are shed alternatingly from the upper and lower surfaces of the trailing edge. The vorticities at the core of vortices 1, 2, 3, 4, 5, and 6 are 26,000, 20,000, 16,000, 13,500, 12,000, and 11,000, respectively. Therefore, the vorticities of the shed vortices dissipate as they move away from the trailing edge, and the vorticity dissipation is most pronounced when the vortices first begin to dissociate from the surface of the hydrofoil. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 10 of 20 as U and P, respectively, and we further define a dimensionless variable 푢 = 푈/푈푟푒푓 to simplify the velocity term. The dimensionless velocity–time and velocity–pressure curves obtained at these points are presented in Figure 9. In the figure, the velocity and pressure curves at both the monitoring points exhibit similar oscillatory amplitudes and frequencies but in opposing phases. This indicates that the vortices on the upper and lower surfaces of the trailing edge are symmetrical and that they are shed alternatingly. At each monitoring point, the velocity and pressure curves have identical periodicities but opposing amplitudes, i.e., the maximum of one curve always corresponds to the minimum of the J. Mar. Sci. Eng. 2020, 8, 195 10 of 20 other curve.

1.00 Point1 Point2

0.75

u 0.50

0.25

0.00 0 1 2 3 4 t

(a) (b)

Point1 Point2 0.4 1.00 0.4

Cp 0.2 0.75 0.2

0.0 u

p C 0.50 0.0

-0.2 0.25 -0.2 -0.4 u

0 1 2 3 4 0.00 -0.4 0 1 2 3 4 t t

FigureFigure 9. VelocityVelocity and pressure variations at the trailing edge of the hydrofoil. ( (aa)) Positions Positions of of the the monitoringmonitoring points. (b)) VelocityVelocity variations variations at at the the two two monitoring monitoring points. points. (c) ( Pressurec) Pressure changes changes at the at twothe two monitoring points. (d) Relationship between velocity and pressure at Point 1. J. Mar. monitoringSci. Eng. 2020 points., 8, x FOR (d PEER) Relationship REVIEW between velocity and pressure at Point 1. 11 of 20

Figure 10 presents a vorticity map of the shed vortices. As can be seen from the figure, vortices are shed alternatingly from the upper and lower surfaces of the trailing edge. The vorticities at the core of vortices 1, 2, 3, 4, 5, and 6 are 26,000, 20,000, 16,000, 13,500, 12,000, and 11,000, respectively. Therefore, the vorticities of the shed vortices dissipate as they move away from the trailing edge, and the vorticity dissipation is most pronounced when the vortices first begin to dissociate from the surface of the hydrofoil.

Figure 10. Vortex shedding from the trailing edge (t = 3.12). Figure 10. Vortex shedding from the trailing edge (t = 3.12).

BecauseBecause vortex vortex shedding shedding often often leads leads to to vibrations vibrations in in various various structures structures and and objects objects in in practical practical applications,applications, different diﬀerent operating operating conditions conditions were set set up up to to investigate investigate the the factors factors that that influence inﬂuence vortex vortex sheddingshedding and and to to elucidate elucidate the the vortex vortex-shedding-shedding behaviors behaviors..

3.2. Frequency of Vortex Shedding at Different Velocities

The value of 푈푟푒푓 was varied to examine the effects of the inflow velocity on vortex shedding. −6 The value of 푈푟푒푓 was changed by altering the velocity at the inlet. A time step of 1 × 10 s was used, and the thickness of the boundary layer was recalculated considering Y+ = 1 and 푈푟푒푓 that was used in the simulation. The wake-vortex shedding frequencies that were obtained with different inflow velocities are presented in Table 4, where f1 is the frequency that was obtained in this simulation, f2 and f3 are the experimental values obtained by Ausoni et al. [8] in their smooth and rough leading- edge experiments, respectively, and f4 is the result that they obtained through a CFD simulation.

Table 4. Vortex-shedding frequency at different inflow velocities; f1: frequency obtained in this study;

f2 and f3: frequencies obtained from smooth and rough leading edge experiments, respectively [8]; f4: frequency obtained through a CFD simulation in this study.

푼풓풆풇 f1 (Hz) f2 (Hz) f3 (Hz) f4 (Hz) 푺 pre. (m/s) 풕 10 601.78 795.24 532.04 599.51 0.194 11 672.02 912.15 590.42 0.197 12 735.30 912.15 649.31 0.197 13 780.69 912.15 717.00 0.193 14 861.62 999.18 775.09 0.198 15 902.63 1096.76 833.92 0.194 16 963.06 1174.99 883.14 957.87 0.194 17 1024.57 1223.34 921.82 0.194 18 1118.94 1291.91 989.51 0.200 19 1146.51 1378.94 1058.08 0.194 20 1207.48 1418.49 1116.10 1212.35 0.194 21 1268.45 1184.66 0.194

Figure 11 compares the vortex-shedding frequencies that were obtained in the present simulation with those obtained by Ausoni et al. (2006). As can be seen from the figure, the CFD calculation results do not differ significantly from each other, but they do deviate from the experimental results to some extent. The simulated results are similar to those obtained in the rough leading-edge experiment both qualitatively and quantitatively, and the results of the latter may be approximated by a linear function of U (f = 60.97U − 11.925). Ausoni et al. argued that the differences between the numerical and experimental values were a result of the differences in flow field around the hydrofoil in the numerical simulation and experiments. Because a turbulence model was used in

J. Mar. Sci. Eng. 2020, 8, 195 11 of 20

3.2. Frequency of Vortex Shedding at Diﬀerent Velocities

The value of Ure f was varied to examine the effects of the inflow velocity on vortex shedding. The value of U was changed by altering the velocity at the inlet. A time step of 1 10 6 s was used, and re f × − the thickness of the boundary layer was recalculated considering Y+ = 1 and Ure f that was used in the simulation. The wake-vortex shedding frequencies that were obtained with different inflow velocities are presented in Table4, where f 1 is the frequency that was obtained in this simulation, f 2 and f 3 are the experimental values obtained by Ausoni et al. [8] in their smooth and rough leading-edge experiments, respectively, and f 4 is the result that they obtained through a CFD simulation.

Table 4. Vortex-shedding frequency at diﬀerent inﬂow velocities; f 1: frequency obtained in this study; f 2 and f 3: frequencies obtained from smooth and rough leading edge experiments, respectively [8]; f 4: frequency obtained through a CFD simulation in this study.

Uref (m/s) f 1 (Hz) f 2 (Hz) f 3 (Hz) f 4 (Hz) St pre. 10 601.78 795.24 532.04 599.51 0.194 11 672.02 912.15 590.42 0.197 12 735.30 912.15 649.31 0.197 13 780.69 912.15 717.00 0.193 14 861.62 999.18 775.09 0.198 15 902.63 1096.76 833.92 0.194 16 963.06 1174.99 883.14 957.87 0.194 17 1024.57 1223.34 921.82 0.194 18 1118.94 1291.91 989.51 0.200 19 1146.51 1378.94 1058.08 0.194 20 1207.48 1418.49 1116.10 1212.35 0.194 21 1268.45 1184.66 0.194

Figure 11 compares the vortex-shedding frequencies that were obtained in the present simulation with those obtained by Ausoni et al. (2006). As can be seen from the figure, the CFD calculation results do not differ significantly from each other, but they do deviate from the experimental results to some extent. The simulated results are similar to those obtained in the rough leading-edge experiment both qualitatively and quantitatively, and the results of the latter may be approximated by a linear function of U (f = 60.97U 11.925). Ausoni et al. argued that the differences between the numerical and J. Mar. Sci. Eng. 2020,− 8, x FOR PEER REVIEW 12 of 20 experimental values were a result of the differences in flow field around the hydrofoil in the numerical simulationthe numerical and experiments.simulations, the Because flows around a turbulence the hydrofoil model waswereused assumed in the to numericalbe fully turbulent. simulations, In the the flowsmodel around experiments, the hydrofoil there werewas a assumed distinct turning to be fully point turbulent., where the In fluid the model flows experiments,changed from there laminar was a distinctto turbulent. turning point, where the fluid flows changed from laminar to turbulent.

Smooth (Ausoni(2006)) 2000 Rough leading edge (Ausoni(2006)) Present CFD 1500

1000

[Hz]

f 500

0 0.0 5.0x105 1.0x106 1.5x106 2.0x106 2.5x106 3.0x106 3.5x106 Re

FigureFigure 11. 11. OscillationOscillation frequency with with different diﬀerent inflow inﬂow velocities. velocities.

Figure 12 presents the values of 푆푡 as a function of inflow velocity. These values in the “smooth” experiment generally fluctuated around 0.24, and the changes in these values with respect to inflow

velocity were significant. The values of 푆푡 that were obtained in the “rough leading edge” experiment are similar to those obtained in our CFD simulation (0.19 versus 0.18), and the changes

in 푆푡 with respect to inflow velocity were insignificant in both cases.

0.50 Smooth(Ausoni(2006)) Present CFD Rough leading edge(Ausoni(2006))

t 0.25

0.00 10 15 20

Uref [m/s]

Figure 12. Strouhal number at different inflow velocities.

3.3. Effects of the Trailing-Edge Thickness A sharp trailing edge is usually undesirable in marine propellers and hydrofoils. To avoid resonances in marine propellers, an appropriate trailing-edge thickness must be selected to shift the vortex-shedding frequency away from the natural frequencies of the propeller. Therefore, numerical studies on the relationship between the vortex-shedding frequency and trailing-edge thickness are highly important. The NACA0009 hydrofoil was truncated at different positions to investigate the

effects of the trailing-edge thickness (b) on vortex shedding. The inflow velocity 푈푟푒푓 was set to 12 m/s, and the thickness of the boundary-layer mesh was set according to the chord length of the hydrofoil. Each set of vortex shedding data in Table 5 corresponds to a hydrofoil shape with a different truncation.

Table 5. Vortex-shedding frequency corresponding to each trailing-edge thickness.

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 12 of 20

the numerical simulations, the flows around the hydrofoil were assumed to be fully turbulent. In the model experiments, there was a distinct turning point, where the fluid flows changed from laminar to turbulent.

Smooth (Ausoni(2006)) 2000 Rough leading edge (Ausoni(2006)) Present CFD 1500

1000

[Hz]

f 500

0 0.0 5.0x105 1.0x106 1.5x106 2.0x106 2.5x106 3.0x106 3.5x106 Re J. Mar. Sci. Eng. 2020, 8, 195 12 of 20 Figure 11. Oscillation frequency with different inflow velocities.

FigureFigure 12 12 presents presents the the values values ofofS 푆t 푡as as a a function function of of inflow inflow velocity. velocity These. These values values in in the the “smooth”“smooth” experimentexperiment generally generally fluctuated fluctuated around around 0.24, 0.24, and and the the changes changes in in these these values values with with respect respect to to inflow inflow velocityvelocity were were significant. significant. The The values values of St that of 푆 were푡 that obtained were obtained in the “rough in the leading “rough edge” leading experiment edge” areexperiment similar to are those similar obtained to those in ourobta CFDined simulationin our CFD (0.19 simulation versus (0.19 0.18), versus and the 0.18), changes and the in Schangest with respectin 푆푡 with to inflow respect velocity to inflow were velocity insignificant were insignificant in both cases. in both cases.

0.50 Smooth(Ausoni(2006)) Present CFD Rough leading edge(Ausoni(2006))

t 0.25

0.00 10 15 20

Uref [m/s]

FigureFigure 12. 12.Strouhal Strouhal number number at at di differentfferent inflow inflow velocities. velocities. 3.3. Effects of the Trailing-Edge Thickness 3.3. Effects of the Trailing-Edge Thickness A sharp trailing edge is usually undesirable in marine propellers and hydrofoils. To avoid A sharp trailing edge is usually undesirable in marine propellers and hydrofoils. To avoid resonances in marine propellers, an appropriate trailing-edge thickness must be selected to shift the resonances in marine propellers, an appropriate trailing-edge thickness must be selected to shift the vortex-shedding frequency away from the natural frequencies of the propeller. Therefore, numerical vortex-shedding frequency away from the natural frequencies of the propeller. Therefore, numerical studies on the relationship between the vortex-shedding frequency and trailing-edge thickness are studies on the relationship between the vortex-shedding frequency and trailing-edge thickness are highly important. The NACA0009 hydrofoil was truncated at different positions to investigate the highly important. The NACA0009 hydrofoil was truncated at different positions to investigate the effects of the trailing-edge thickness (b) on vortex shedding. The inflow velocity Ure f was set to 12 m/s, effects of the trailing-edge thickness (b) on vortex shedding. The inflow velocity 푈 was set to 12 and the thickness of the boundary-layer mesh was set according to the chord length of푟푒푓 the hydrofoil. Eachm/s, set and of vortexthe thickness shedding of data the boundaryin Table5 corresponds-layer mesh to was a hydrofoil set according shape with to the a di chordfferent length truncation. of the hydrofoil. Each set of vortex shedding data in Table 5 corresponds to a hydrofoil shape with a different truncation.Table 5. Vortex-shedding frequency corresponding to each trailing-edge thickness.

Table 5. VortexC-(mm)shedding frequencyB (mm) correspondingF (Hz) to each trailingSt -edge thickness. 29.66 9.00 239.92 0.180 65.45 6.14 464.25 0.237 79.39 4.04 666.22 0.224 88.00 2.69 845.18 0.190 93.30 1.54 1216.55 0.157

Figure 13 illustrates the lift curves that were obtained with different hydrofoil truncations. It can be inferred from the figure that the lift curves always undergo a period of change before stabilization. The lift curve has a rather irregular shape when b = 9 mm, as indicated by the different maximum and minimum values in each cycle, even in the steady state. The changes in lift curve when b = 1.54 mm are different from those observed with the other trailing-edge thicknesses; in addition to the initial growth in oscillation amplitude, the oscillations of the b = 1.54 mm lift curve also decrease in amplitude before they eventually stabilize. The trailing-edge thickness gradually decreases as the truncation approaches the rear end of the hydrofoil. As this occurs, the lift-curve oscillations gradually decrease in amplitude but increase in frequency. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 13 of 20

C (mm) B (mm) F (Hz) 푺풕 29.66 9.00 239.92 0.180 65.45 6.14 464.25 0.237 79.39 4.04 666.22 0.224 88.00 2.69 845.18 0.190 93.30 1.54 1216.55 0.157

Figure 13 illustrates the lift curves that were obtained with different hydrofoil truncations. It can be inferred from the figure that the lift curves always undergo a period of change before stabilization. The lift curve has a rather irregular shape when b = 9 mm, as indicated by the different maximum and minimum values in each cycle, even in the steady state. The changes in lift curve when b = 1.54 mm are different from those observed with the other trailing-edge thicknesses; in addition to the initial growth in oscillation amplitude, the oscillations of the b = 1.54 mm lift curve also decrease in amplitude before they eventually stabilize. The trailing-edge thickness gradually decreases as the truncation approaches the rear end of the hydrofoil. As this occurs, the lift-curve oscillations J. Mar.gradually Sci. Eng. decrease2020, 8, 195 in amplitude but increase in frequency. 13 of 20

1.5 0.2

1.0 0.1 0.5

C 0.0 0.0

-0.5 -0.1 -1.0

-1.5 -0.2 0 5 10 15 0 1 2 3 4 5 6 7 t t

(a) c = 29.66 mm, b = 9.00 mm (b) c = 65.45 mm, b = 6.14 mm

0.15 0.08

0.10 0.04 0.05

C 0.00

-0.05 -0.04

-0.10 -0.08 -0.15 0 1 2 3 4 0 1 2 3 4 t t

0.03

0.02

0.01

C 0.00

-0.01

-0.02

-0.03 0 1 2 3 4 t

(e) c = 93.30 mm, b = 1.54 mm

Figure 13. Variation of the lift coeﬃcient Cl with respect to the trailing-edge thickness, b.

Figures 14–16 indicate that the oscillation frequency of the lift curve (f ) decreases with increase in b. Figure 16 shows that St ﬁrst increases and then decreases with increase in b; the value of St peaks at b = 6.138 mm. Figure 14 indicates that the steady-state lift curve is rather irregular when b is large, as indicated by the multiple frequency modes in the lift curve (the higher-order modes are signiﬁcantly stronger when b is large). Figure 17 illustrates the changes in vortex shedding that occurred as the truncation shifted rearward (thus increasing chord length and decreasing b). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 20 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 14 of 20

Figure 13. Variation of the lift coefficient 퐶푙 with respect to the trailing-edge thickness, b. Figure 13. Variation of the lift coefficient 퐶푙 with respect to the trailing-edge thickness, b. Figures 14–16 indicate that the oscillation frequency of the lift curve (f) decreases with increase Figures 14–16 indicate that the oscillation frequency of the lift curve (f) decreases with increase in b. Figure 16 shows that 푆푡 first increases and then decreases with increase in b; the value of 푆푡 in b. peFigureaks at 16 b =shows 6.138 mm.that Figure푆푡 first 14increases indicates and that then the decreasessteady-state with lift increasecurve is ratherin b; the irregular value ofwhen 푆푡 b is peakslarge, at b =as 6.138 indicated mm. Figure by the 14 multiple indicates frequency that the steady modes-state in the lift liftcurve curve is rather (the higher irregular-order when modes b is are large,significantly as indicated stronger by the multiple when b isfrequency large). Figure modes 17 in illustratesthe lift curve the (the changes higher in - vortexorder modes shedding are that significantly stronger when b is large). Figure 17 illustrates the changes in vortex shedding that J.occurred Mar. Sci. Eng.as the2020 truncation, 8, 195 shifted rearward (thus increasing chord length and decreasing b). 14 of 20 occurred as the truncation shifted rearward (thus increasing chord length and decreasing b).

b/c Ratio b/c Ratio 1.5 0.303 1.5 0.303 0.094 0.094 0.051 1.0 0.051 0.031 1.0 0.031 0.017 0.017 0.5 0.5

Amplitude

Amplitude 0.0

0.0 2200 5 2000 2200 1800 5 2000 1600 4 1800 1400 4 1600 1200 3 1400 f [Hz]1000 1200 800 3 f [Hz]1000 600 2 800 400 2 600 200 1 400 0 200 0 1 -200 -200 Figure 14. Vortex shedding characteristics of each model in the frequency domain. FigureFigure 14. Vortex 14. Vortex shedding shedding characteristics characteristics of each of eachmodel model in the in fr theequency frequency domain domain..

1500 1500

1000 1000

[Hz]

f 500 500

0 0 0 3 6 9 0 3 b6 [mm] 9 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 20 b [mm] FigureFigure 15.15. ChangeChange in ff with various values of b. Figure 15. Change in f with various values of b.

0.24

0.21

0.18

0.15 0 3 6 9 b[mm]

Figure 16. Change in St with various values of b. Figure 16. Change in 푆푡 with various values of b.

(a) c = 29.66 mm, b = 9.00 mm (b) c = 65.45 mm, b = 6.14 mm

Figure 17. Vortex shedding with different trailing-edge thicknesses.

3.4. Effects of Angle of Attack In most scenarios, the hydrofoil has a certain angle of attack, α, that influences the drag and lift of the hydrofoil. In this section, the value of α of the hydrofoil was varied to examine its effects on

wake vortex shedding. The parameters of the hydrofoil model were c = 88 mm, b = 2.69 mm, and 푈푟푒푓 = 12 m/s. The mesh parameters are the same as those in the previous simulations.

Table 6. Effects of α on wake vortex shedding.

Α (°) 푪풍 F (Hz) 푺풕 0 0 845.18 0.190 1 0.086 865.05 0.194 2 0.230 856.60 0.192 3 0.343 855.43 0.192 6 0.618 812.98 0.183 9 1.042 757.99 0.170 12 1.304

As shown in Table 6, the shedding of wake vortices directly depends on α. At small values of α (Figure 18), the vortex-shedding behavior is similar to that at α = 0°. As α increases to a certain value (Figure 19), the boundary layer at the frontal edge of the hydrofoil begins to dissociate from the

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 15 of 20

0.24

0.21

0.18

0.15 0 3 6 9 b[mm] J. Mar. Sci. Eng. 2020, 8, 195 15 of 20

Figure 16. Change in 푆푡 with various values of b.

(a) c = 29.66 mm, b = 9.00 mm (b) c = 65.45 mm, b = 6.14 mm

FigureFigure 17. 17. VortexVortex shedding shedding with with different diﬀerent trailing trailing-edge-edge thicknesses thicknesses..

3.4.3.4. E Effectsffects of of Angle Angle of of Attack Attack InIn most most scenarios, thethe hydrofoilhydrofoil hashas aa certain certain angle angle of of attack, attack,α ,α that, that influences influences the the drag drag and and lift lift of ofthe the hydrofoil. hydrofoil. In thisIn this section, section, the the value value of α ofof α the of hydrofoilthe hydrofoil was was varied varied to examine to examine its eff itsects effects on wake on vortex shedding. The parameters of the hydrofoil model were c = 88 mm, b = 2.69 mm, and U = 12 wake vortex shedding. The parameters of the hydrofoil model were c = 88 mm, b = 2.69 mm, andre f 푈푟푒푓 =m 12/s. m/s. The The mesh mesh parameters parameters are theare samethe same as those as those in the in previousthe previous simulations. simulations. As shown in Table6, the shedding of wake vortices directly depends on α. At small values of α (Figure 18), the vortex-sheddingTable behavior 6. Effects is of similar α on wake to that vortex at α shedding.= 0◦. As α increases to a certain value (Figure 19), the boundary layer at the frontal edge of the hydrofoil begins to dissociate from the surface, Α (°) 푪 F (Hz) 푺 causing the boundary layer near the trailing풍 edge to become thinner. In this scenario,풕 the vortices shed from the suction0 surface are much0 weaker than those845.18 shed from the pressure0.190 surface. Therefore, the vortex street in the1 wake is dominated0.086 by vortices from865.05 the pressure surface.0.194 The vortices from the suction surface move2 along the vortices0.230 from the pressure856.60 surface and rapidly0.192 dissipate. When α increases further (Figure3 20), the boundary0.343 layer dissociates855.43 from the surface0.192 of the hydrofoil at its frontal edge, and the6 K ármán vortex street0.618 is no longer observed812.98 in the wake0.183 field. 9 1.042 757.99 0.170 12 Table 6. E1.304ffects of α on wake vortex shedding.

As shown in Table 6, theA ( ◦shedding) ofC wakel vorticesF (Hz) directly dependsSt on α. At small values of α (Figure 18), the vortex-shedding0 behavior 0is similar to 845.18that at α = 0°. As 0.190 α increases to a certain value (Figure 19), the boundary layer1 at the frontal 0.086 edge of 865.05 the hydrofoil 0.194 begins to dissociate from the 2 0.230 856.60 0.192

3 0.343 855.43 0.192 6 0.618 812.98 0.183 9 1.042 757.99 0.170 12 1.304

By comparing Figure 21a,b and Figure 22, it is found that there is always a stagnation point on the pressure side at which Cp = 1.0 and that the stagnation point moves away from the leading edge as the angle of attack increases. As shown in Figure 21a, at α = 3, the lift on the hydrofoil is relatively small. At the trailing edge, the pressure on the pressure side is lower than that on the suction side. As shown in Figure 21b, when the angle of attack increases to 6◦, the flow becomes stronger, and the J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 20 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 20 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 20 surface, causing the boundary layer near the trailing edge to become thinner. In this scenario, the surface,J. Mar. Sci. causing Eng. 2020 the, 8, 195 boundary layer near the trailing edge to become thinner. In this scenario,16 ofthe 20 vorticessurface, shedcausing from the the boundary suction surfacelayer near are muchthe trailing weaker edge than to those become shed thinner. from the In pressurethis scenario, surface the. vorticesTherefore, shed the from vortex the streetsuction in surface the wake are ismuch dominated weaker bythan vortices those shed from from the pressure the pressure surface. surface The. Therefore,vortices shed the from vortex the street suction in surface the wake are is much dominated weaker by than vortices those fromshed thefrom pressure the pressure surface. surface The. vorticespressure from difference the suction between surface the pressure move along and suction the vortices sides increases, from the whichpressure eff ectively surface increasesand rapidly the vorticesTherefore, from the the vortex suction street surface in the move wake along is dominated the vortices by vortices from the from pressure the pressure surface and surface. rapidly The dissipate.vorticeslift on the from When hydrofoil. the α suctionincreases surface further move (Figure along 20), thethe vorticesboundary from layer the dissociates pressure surfacefrom the and surface rapidly of dissipate.the hydrofoil When at its α frontalincreases edge, further and the(Figure Kármán 20), vortexthe boundary street is layerno longer dissociates observed from in thethe wake surface field. of thedissipate. hydrofoil When at its α frontal increases edge, further and the (Figure Kármán 20), vortex the boundary street is nolayer longer dissociates observed from in the the wake surface field. of the hydrofoil at its frontal edge, and the Kármán vortex street is no longer observed in the wake field.

Figure 18. Vortex shedding when α = 1°. FigureFigure 18. 18. VortexVortex shedding shedding when when α = 1°1◦. . Figure 18. Vortex shedding when α = 1°.

Figure 19. Vortex shedding when α = 9°. Figure 19. Vortex shedding when α = 9°. Figure 19. Vortex shedding when αα == 9°9◦..

J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEWFigure 20. Vortex shedding when α = 12°. 17 of 20 Figure 20. Vortex shedding when α = 12°. Figure 20. Vortex shedding when α = 12°. By comparing Figures 21a,bFigure and 20.22,Vortex it is found shedding that when thereα =is12 always◦. a stagnation point on the By comparing Figures 21a,b and 22, it is found that there is always a stagnation point on the pressure side at which Cp = 1.0 and that the stagnation point moves away from the leading edge as By comparing1.5 Figures 21a,b and 22, it is found that there2 is always a stagnation point on the pressurethe angle side of attack at which increases. Cp = 1.0 As and shown that inthe Figure stagnation 21a, atpoint α = 3,moves the lift away on thefrom hydrofoil the leading is relatively edge as pressure side at which Cp = 1.0 and that the stagnation point moves away from the leading edge as thesmall. angle At1.0 theof attack trailing increases. edge, the As pressure shown inon Figure the pressure 21a, at sideα = 3,is thelower lift thanon the that hydrofoil on the suction is relatively side. small.the angle At theof attacktrailing increases. edge, the As pressure shown onin Figurethe pressure 21a, at side α = is3, lowerthe lift than on the that hydrofoil on the suction is relatively side. As shown0.5 in Figure 21b, when the angle of attack increases to 6°, the flow becomes stronger, and the Assmall. shown At thein Figure trailing 21b, edge, when the the pressure angle ofon attack the pressure increases side 0to 6°,is lowerthe flow than becomes that on stronger, the suction and side. the pressure p difference between the pressure and suction sides increases, which effectively increases the 0.0 As shownC in Figure 21b, when the angle of attack increases to 6°, the flow becomes stronger, and the pressure difference between the pressure and suction sidesp increases, which effectively increases the liftpressure on the difference hydrofoil. between the pressure and suction sidesC increases, which effectively increases the lift on the-0.5 hydrofoil. lift on the hydrofoil. -2 -1.0

-1.5 0.00 0.25 0.50 0.75 1.00 -4 x/c 0.00 0.25 0.50 0.75 1.00 x/c

(a) α = 3° (b) α = 6°

FigureFigure 21. 21.Pressure Pressure distributiondistribution onon thethe surfacessurfaces of of the the hydrofoil. hydrofoil.

Figure 22 shows the pressure distribution on the surfaces of the hydrofoil with a stable lift coefficient. As shown in Figure 23, where α is large, the lift curve of the hydrofoil stabilizes after a short period of growth. It can be inferred from the figure that the pressure on the suction and pressure surfaces of the trailing edge gradually converge to the same value. Then, because these pressures are the same, the lift can no longer change.

2.5

0.0

-2.5

C -5.0

-7.5

-10.0

-12.5 0.00 0.25 0.50 0.75 1.00 x/c

Figure 22. Surface pressure distribution of the hydrofoil when α = 12°.

2.00

1.75

1.50

C 1.25

1.00

0.75

0.50 0 5 10 15 t

Figure 23. Lift curve of the hydrofoil when α = 12°.

3.5. Effects of Hydrofoil Thickness Based on the NACA00 series of hydrofoil models, we designed a set of hydrofoil models with a trailing-edge thickness and chord length of b = 2.69 mm and c = 88 mm, respectively. These models

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1.5 2

J. Mar. Sci.1.0 Eng. 2020, 8, x FOR PEER REVIEW 17 of 20

0.5 0

p 0.0

1.5 C 2 -0.5 1.0 -2 -1.0 0.5 0 -1.5 p 0.00 0.25 0.50 0.75 1.00

C 0.0

p -4

x/c C 0.00 0.25 0.50 0.75 1.00 -0.5 x/c -2 -1.0 (a) α = 3° (b) α = 6°

-1.5 Figure 21. Pressure distribution on the surfaces of the hydrofoil. 0.00 0.25 0.50 0.75 1.00 -4 x/c 0.00 0.25 0.50 0.75 1.00 x/c Figure 22 shows the pressure distribution on the surfaces of the hydrofoil with a stable lift coefficient. As shown(a )in α =Figure 3° 23, where α is large, the lift curve of (theb) αhydrofoil = 6° stabilizes after a short period of growth. It can be inferred from the figure that the pressure on the suction and pressure surfaces of the trailingFigure edge 21. gradually Pressure convergedistribution to on the the same surfaces value. of theThen, hydrofoil. because these pressures are J.the Mar. same, Sci. Eng. the2020 lift, 8 ,can 195 no longer change. 17 of 20 Figure 22 shows the pressure distribution on the surfaces of the hydrofoil with a stable lift coefficient. As shown in Figure 23, where α is large, the lift curve of the hydrofoil stabilizes after a short period of growth. It can be inferred2.5 from the figure that the pressure on the suction and pressure surfaces of the trailing edge gradually0.0 converge to the same value. Then, because these pressures are the same, the lift can no longer change. -2.5

C -5.0 2.5 -7.5 0.0 -10.0 -2.5 -12.5

p 0.00 0.25 0.50 0.75 1.00

C -5.0 x/c -7.5 Figure 22. Surface pressure distribution of the hydrofoil when α = 12◦. Figure 22. Surface-10.0 pressure distribution of the hydrofoil when α = 12°.

Figure 22 shows the pressure-12.5 distribution on the surfaces of the hydrofoil with a stable lift coeﬃcient. As shown in Figure 23,0.00 where α 0.25is large, 0.50 the lift 0.75 curve 1.00 of the hydrofoil stabilizes after a 2.00 x/c short period of growth. It can be inferred from the ﬁgure that the pressure on the suction and pressure 1.75 surfaces of the trailingFigure edge 22. gradually Surface pressure converge distribu to thetion same of the value. hydrofoil Then, when because α = 12°. these pressures are the same, the lift can no longer change.1.50

C 1.25 2.00 1.00 1.75 0.75 1.50 0.50 l 0 5 10 15 C 1.25 t

1.00 Figure 23. Lift curve of the hydrofoil when α = 12°. 0.75

0.50 3.5. Effects of Hydrofoil Thickness 051015 t Based on the NACA00 series of hydrofoil models, we designed a set of hydrofoil models with a Figure 23. Lift curve of the hydrofoil when α = 12 . trailing-edge thickness andFigure chord 23. length Lift curve of b of= 2.69the hydrofoil mm and when c = 88 α =mm, 12°.◦ respectively. These models 3.5. Effects of Hydrofoil Thickness 3.5. Effects of Hydrofoil Thickness Based on the NACA00 series of hydrofoil models, we designed a set of hydrofoil models with a trailing-edgeBased on thickness the NACA00 and chord series lengthof hydrofoil of b = models,2.69 mm we and designedc = 88 mm, a set respectively. of hydrofoil Thesemodels models with a trailing-edge thickness and chord length of b = 2.69 mm and c = 88 mm, respectively. These models were placed in a Ure f = 12 m/s flow field to calculate their flow and drag coefficients. The mesh parameters used in this simulation are the same as those used in the previous simulations. Vortex shedding data for different hydrofoils are shown in Table7.

Table 7. Results obtained after changing the thickness of the hydrofoil.

Hydrofoil Type f (Hz) Cl (max) St 0009 845.18 0.0561 0.190 0012 794.18 0.0497 0.178 0014 743.03 0.0353 0.167 0016 697.51 0.0287 0.157 0018 654.81 0.0215 0.147

The test yielded a time-averaged Cl value of the symmetrical hydrofoil of 0 at α = 0◦. Therefore, Cd is used to compare the experimental and simulated values. Owing to the lack of test results on hydrofoils with truncated trailing edges, we only compared the results of the original hydrofoil test with the simulation results, as shown in Table8. As shown in the table, the S-A model predicted the drag fairly precisely. J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 20 J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 18 of 20 were placed in a 푈푟푒푓 = 12 m/s flow field to calculate their flow and drag coefficients. The mesh parameterswere placed used in a in푈 푟푒푓this = simulation 12 m/s flow are field the sameto calculate as those their used flow in theand previous drag coefficients. simulations. The Vortex mesh sheddingparameters data used for in d ifferentthis simulation hydrofoils are are the shown same inas Tablethose 7.used in the previous simulations. Vortex shedding data for different hydrofoils are shown in Table 7. The test yielded a time-averaged 퐶푙 value of the symmetrical hydrofoil of 0 at α = 0°. Therefore, The test yielded a time-averaged 퐶푙 value of the symmetrical hydrofoil of 0 at α = 0°. Therefore, 퐶푑 is used to compare the experimental and simulated values. Owing to the lack of test results on hydrofoil퐶푑 is useds witto compareh truncated the trailing experimental edges, andwe only simulated compared values. the Owingresults ofto thethe originallack of testhydrofoil results test on withhydrofoil the simulations with truncated results trailing, as shown edge ins ,Table we only 8. As compared shown in the the results table, theof the S-A original model hydrofoilpredicted testthe dragwith fairlythe simulation precisely. results , as shown in Table 8. As shown in the table, the S-A model predicted the drag fairly precisely. Table 7. Results obtained after changing the thickness of the hydrofoil. Table 7. Results obtained after changing the thickness of the hydrofoil. Hydrofoil f (Hz) 푪 (max) 푺 HydrofoilType 풍 풕 f (Hz) 푪풍 (max) 푺풕 Type0009 845.18 0.0561 0.190 00120009 794.18845.18 0.04970.0561 0.1780.190 00140012 743.03794.18 0.03530.0497 0.1670.178 00160014 697.51743.03 0.02870.0353 0.1570.167 00180016 654.81697.51 0.02150.0287 0.1470.157 J. Mar. Sci. Eng. 2020, 8,0018 195 654.81 0.0215 0.147 18 of 20 Table 8. Results obtained after changing the thickness of the hydrofoil (Re = 3.4 × 10−6). Table 8. Results obtained after changing the thickness of the hydrofoil (Re = 3.4 × 10−66). Table 8. Results obtained after changing the thickness of the hydrofoil (Re = 3.4 10− ). Hydrofoil × Hydrofoil 푪풅 (exp.) [19] 푪풅 (pre.) HydrofoilType Type 푪 C(exp.)d (exp.) [19] [19 ] 푪 (pre.)Cd (pre.) Type 풅 풅 NACA0009 0.00654 0.00654 0.00649 0.00649 NACA0012NACA0009 0.006570.00654 0.00657 0.006520.00649 0.00652 NACA0018NACA0012 0.007250.00657 0.00725 0.007200.00652 0.00720 NACA0018 0.00725 0.00720 ByBy observing observing the the vortex vortex-shedding-shedding states states in in Figures Figures 2424 andand 25,25, it it is is clear clear that that vorticity vorticity of of the the shed shed By observing the vortex-shedding states in Figures 24 and 25, it is clear that vorticity of the shed vorticesvortices induced induced by by thicker thicker hydrofoils hydrofoils are are significantly significantly lower. lower. Table 6 Table indicates 6 indicates that the that vortex-shedding the vortex- vortices induced by thicker hydrofoils are significantly lower. Table 6 indicates that the vortex- sheddingfrequency frequency decreases decreases with increase with in increase hydrofoil in hydrofoil thickness. thickness. Furthermore, Furthermore, the effects the of vortexeffects sheddingof vortex shedding frequency decreases with increase in hydrofoil thickness. Furthermore, the effects of vortex sheddingon the lift on also the decrease lift also with decrease increase with in hydrofoilincrease in thickness. hydrofoil From thickness. Figure From26, it canFigure be observed26, it can that be shedding on the lift also decrease with increase in hydrofoil thickness. From Figure 26, it can be observedincreases that in the increases hydrofoil in thickness the hydrofoil lead tothickness larger position-dependent lead to larger position changes-dependent in the surface changes pressure; in the observed that increases in the hydrofoil thickness lead to larger position-dependent changes in the surfacethe position pressure; that producesthe position a pressure that produces differential a pressure also becomes differential closer also to becomes the trailing closer edge. to the trailing edge.surface pressure; the position that produces a pressure differential also becomes closer to the trailing edge.

Figure 24. Vortex shedding of the 0012 model. FigureFigure 24. 24. VortexVortex shedding shedding of the 0012 model model..

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Figure 25. Vortex shedding of the 0018 model. Figure 25. Vortex shedding of the 0018 model. Figure 25. Vortex shedding of the 0018 model.

1.0 1.0

0.5 0.5

p C 0.0 0.0 C

-0.5 -0.5

-1.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 x/c x/c

(a) (b)

FigureFigure 26. 26.Surface Surface pressure pressure distribution distribution on di ﬀonerent different hydrofoil hydrofoil models. (amodels) Surface. ( pressurea) Surface distribution pressure ofdistribution Model 0012 of ( CModel= 0.0282). 0012 (퐶 (b푙 )= Surface 0.0282). pressure (b) Surface distribution pressure distribution of Model 0018 of Model (C = 00180.0232). (퐶푙 = −0.0232). l l − 4. Conclusions We simulated vortex shedding in the wake flow of an NACA0009 hydrofoil using the S-A turbulence model. The main purpose of the simulation was to investigate the factors that affect the vortex-shedding frequency, including the inflow velocity, angle of attack, and trailing-edge thickness. Firstly, it was found that the increase in vortex-shedding frequency was almost linear with the inflow velocity. However, the nondimensional Strouhal number remained almost constant at 0.19. Secondly, the oscillation of vortex shedding could be reduced by increasing the angle of attack. When the angle of attack was sufficiently large, the oscillation was negligible. Thirdly, increases in the trailing-edge breadth led to increases in the vortex-shedding frequency. Importantly, there was a specific breadth corresponding to the maximum Strouhal number. Finally, it was found that a larger maximum thickness of the hydrofoil led to lower frequency and more condense vorticity of vortex shedding.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

References

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4. Conclusions We simulated vortex shedding in the wake flow of an NACA0009 hydrofoil using the S-A turbulence model. The main purpose of the simulation was to investigate the factors that affect the vortex-shedding frequency, including the inflow velocity, angle of attack, and trailing-edge thickness. Firstly, it was found that the increase in vortex-shedding frequency was almost linear with the inflow velocity. However, the nondimensional Strouhal number remained almost constant at 0.19. Secondly, the oscillation of vortex shedding could be reduced by increasing the angle of attack. When the angle of attack was sufficiently large, the oscillation was negligible. Thirdly, increases in the trailing-edge breadth led to increases in the vortex-shedding frequency. Importantly, there was a specific breadth corresponding to the maximum Strouhal number. Finally, it was found that a larger maximum thickness of the hydrofoil led to lower frequency and more condense vorticity of vortex shedding.

Author Contributions: Conceptualization, J.H. and C.G.; Formal analysis, S.S. and C.S.; Investigation, Z.W. and W.Z.; Methodology, J.H. and C.G.; Software, Z.W. and W.Z.; Writing—original draft, Z.W.; Writing—review & editing, J.H. and Z.W. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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