Heuristic optimization of submerged hydrofoil using ANFIS–PSO

M.H. Djavareshkian n, A. EsmaeiliFerdowsi University of Mashhad, Mashhad, Iran

a r t i c l e i n f o

Article history:Received 2 March 2013Accepted 27 September 2014

Keywords:OptimizationHydrofoilPSOANFISHeuristicSubmerge

a b s t r a c t

In this research, optimization of shape and operating conditions of a submerged hydrofoil is investigatedby a heuristic optimization approach, been a combination of an adaptive network based fuzzy inferencesystem model (ANFIS) and particle swarm optimization (PSO). The constrained discrete variables such asthe thickness and camber of hydrofoil, angle of attack and submerge distance are clearly defined asdesign variables and the lift to drag ratio is selected as a nonlinear objective function, which is extractedfrom an accurate numerical procedure. The Navier–Stocks equation is numerically solved, and volume offluid (VOF) method has been utilized to simulate two-phase fluid (water and air). The resultsdemonstrate that the resulted body of the hydrofoil in the optimum operating conditions should reacha maximum value of lift to drag ratio.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Hydrofoils play a significant role in the design of hydrofoil-boats, which move on the water surface and applied the hydrofoilto reduce drag force. Hydrofoils are installed under boat’s hull andhave moved underwater, near the free surface of water. Therefore,generated water wave, which is created by boat movement, caninfluence in the hydrofoil performance and the boat revenue,consequently. There is a great volume of published work dealingwith the hydrofoil performances, (De Blasi et al., 2000; Daskovsky,2000; Filippov, 2001; Kouh et al., 2002; Bourgoyne, 2003; Chenand Liu, 2005; Hay and Visonneau, 2005; Carcaterraa et al., 2005;Xie and Vassalos, 2007; Sadathosseini et al., 2008; Ducoin et al.,2009; Münch et al., 2010; Zanette et al., 2010; Kim and Yamato,2005). The ultimate goal is to develop knowledge that helpshydrofoil design and services more effectively and efficiently.As a result, several numerical simulation and optimization methodshave been widely applied in hydrodynamics. Traditional methodsthat need to calculate and analyze for modification of the modelfrequently, had wasted time and had not been in suitable accuracy.On the other hand, the visualization of flow based on CFD has wellestablished, and quite accurate and robust approximate modelsand optimization algorithms have innovated for the last decade,too. Today, several new techniques are widely applied to optimizesome equipment (Hwang et al., 2009; Shafaghat et al., 2008; Schmitzet al., 2004; Hsin et al., 2006; Spogis and Nunhez, 2009). Further-more, deterministic optimization process such as first order gradient

techniques had been applied to reach the maximum value of drag tolift ratio (Tozzi, 2004). Heuristic method such as genetic algorithm isused to optimize hydrofoil in many researches, too (Ouyang et al.,2006; Wang et al., 2012; Yang et al., 2012, 2009; Yang and Shu, 2012;Frunz et al., 2010; Guo et al., 2009; Cocke, 2012).

In the previous studies, neither high accuracy numericalmethods, nor robust optimization algorithms, which can simulta-neously optimize the operating conditions, are used. On the otherwords, almost all of them have focused to find optimum config-urations of hydrofoil with the negligence of the operating condi-tions. Furthermore, design variables in most of them have beencontinuous ones and additionally, these methods would be veryexpensive and time consuming.

In this study, a robust method is designed for optimization ofhydrofoil, moving near the free surface of water. This algorithmcombines ANFIS method and PSO algorithm. Thickness and cam-ber of hydrofoil, angle of attack and submerge distance areconsidered as constrained design variables and value of lift todrag ratio is defined as the objective function, which has anonlinear behavior. The objective function is obtained by numer-ical simulation. On the other hand, the design variables arediscrete ones and ANFIS is estimated the variable performanceand an accurate continuous search space is generated. Finally, PSOalgorithm, which is simple, fast, more efficient, and useful forcomplex and nonlinear problem, is obtained the best configurationof hydrofoil and optimum operating conditions.

2. Governing equations

The basic equations, which describe the conservation of mass,momentum, and scalar quantities can be expressed in the following

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/oceaneng

Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2014.09.0330029-8018/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ98 511 8615100; fax: þ98 511 8436432.E-mail addresses: javareshkian@ferdowsi.um.ac.ir (M.H. Djavareshkian),

aliesmaeli30316@yahoo.com (A. Esmaeili).

Ocean Engineering 92 (2014) 55–63

vector form, which is independent of the coordinate system.

∂ρ∂t

þdiv ρV!� �

¼ Sm ð1Þ

∂ðρV!Þ∂t

þdiv ρV! � V

!� T!� �

¼ S!

v ð2Þ

∂ðρϕÞ∂t

þdiv ρV-ϕ� q

-� �

¼ S-

ϕ ð3Þ

The stress tensor for a Newtonian fluid is:

T-¼ �P I

-ð4Þ

In addition, the Fourier-type law usually gives the scalar fluxvector:

q-¼ Γϕgradϕ ð5Þ

In this study, the k�ε model has been chosen because of beinga public turbulent model. The comparison of experimental andnumerical data shows that the k�ε model is suitable, too.Furthermore, the turbulent intensity is equal to 5% (Yue et al.,2003,, 2005).

∂∂t

ρkð Þþ ∂∂xj

ρujk�Γk∂k∂xj

� �¼ G�ρε ð6Þ

∂∂t

ρεð Þþ ∂∂xj

ρujε�Γε∂ε∂xj

� �¼ C1

ε

kG�C2ρ

ε2

kð7Þ

The turbulent viscosity and diffusivity coefficients are defined as:

μt ¼ Cμρk2

εð8Þ

Γtϕ ¼

μtσtϕ

!ð9Þ

Moreover, the generation term G in Eqs. (6) and (7) is defined as:

G¼ μt∂ui

∂xjþ∂uj

∂xi

� �∂ui

∂xj

� �ð10Þ

The discretization of the above differential equations is carriedout by applying a finite-volume approach. First, the solutiondomain is divided into a finite number of discrete volumes orcells, where all variables are stored at their geometric centers (seee.g., Fig. 1).

The equations are then integrated over all the control volumesby utilizing the Gaussian theorem. The discrete expressions arepresented to refer only one face of the control volume, e, for thesake of brevity. For every variable ϕ (which may also stand for thevelocity components), the result of the integration yields:

∂ν∂t

ρϕð Þnþ1p � ρϕð Þnp

h iþ Ie� Iwþ In� Is ¼ Sϕδν ð11Þ

where I’s are the combined cell-face convection Ic and diffusionID fluxes. The diffusion flux is approximated by central difference.The discretization of the convective flux requires special attentionand it causes to develop the various schemes. A representation ofthe convective flux for cell-face (e) is:

Ice ¼ ρ� V � Að Þeϕe ¼ Feϕe ð12ÞThe value of ϕe is not known and should be estimated from the

values of neighboring grid points by interpolation. The expressionfor the ϕe is determined by Second order Upwind scheme. Thefinal form of the discretized equation from each approximation isgiven as:

Ap � ϕp ¼ ∑m ¼ E;W ;N;S

Am � ϕmþS0ϕ ð13Þ

where A;s are the convection-diffusion coefficients. The term S0ϕ inEq. (13) contains quantities arising from non-orthogonality,numerical dissipation terms and external sources. For the momen-tum equations, it is easy to separate out the pressure-gradientsource from the convection momentum fluxes.

VOF ideas have been used to simulate two-phase fluid (waterand air). The VOF model can model two or more immiscible fluidsby solving a single set of momentum equations and tracking the

Nomenclature

A cell face areaAOA angle of attackBP back propagation algorithmc cord lengthCa camber of hydrofoilCFD computational fluid dynamicCL lift coefficientCD drag coefficient2D two dimensionF mass fluxFr Froude number according cord lengthG generation termg gravity accelerationGA genetic algorithmH water depthh submerge distanceIc convection fluxID diffusion fluxk turbulence model parameterLS least square method_m mass transfer

P pressurePSO particle swarm optimizationq! scalar flux vectorRe Reynolds numberS!

source termt timeT!

stress tensorth thickness of hydrofoilνi normalized ring strengthsVi current position due to new velocity of every particle

V!

velocity vectorVOF volume of fractionX horizontal Cartesian coordinateY vertical Cartesian coordinateθ angle (1)ε turbulence model parameterδν cell volumeρ densityα volume fractionϕ scalar quantityϖi normalized ring strengthsΓ diffusivity coefficientμ dynamic viscosity

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–6356

volume fraction of each of the fluids throughout the domain. Thetracking of the interface between the phases is accomplished bythe solution of the continuity equation for the volume fraction ofone of the phases. For the qth phase, this equation has thefollowing form:

1ρq

∇� αqρq υ!� ¼ Sαqþ ∑

n

p ¼ 1_mpq� _mqp� " #

ð14Þ

where _mpq is the mass transfer from phase q to phase p, and _mqp

is the mass transfer from phase p to phase q. Spq is the source term,so that in this problem is zero. The volume fraction equation willnot be solved for the primary phase. The primary-phase volumefraction will be computed based on the following constraint:

∑n

q ¼ 1αq ¼ 1 ð15Þ

The volume fraction equation may be solved either throughimplicit or explicit discretization. In this research, implicit is used.

3. ANFIS–PSO algorithm

In this study, a new method which has consisted an ANFISmodel (Djavareshkian and Esmaeili, 2013) and PSO algorithm(Kennedy and Eberhart, 1995), is presented. The advantage of themethod is clearly highlighted when the design variables arediscrete type. In fact, the algorithm is anticipated a search spaceaccording to the discrete variables to obtain a continuous surface;therefore, the cost and time consuming are sharply reduced. Thementioned approach can be classified as a branch of surrogatemethods, which applies ANFIS to obtain design (or search) spaceand employs PSO as an optimization algorithm for finding the bestsolution. Therefore, the optimization process can be divided byseveral steps and the flowchart of it is depicted in Fig. 2.

Initially, design variables, the objective function and constraintsare defined according to the problem. In the next step, initialsample selection is chosen by design of experiments method,selection of points in the design space at which the completemodel is evaluated. Subsequently, numerical simulation is alsocarried out based on the initial selection and value of the objectivefunction is resulted and the obtained data are initially applied astraining data. Because of the nonlinear behavior between thedesign variables and objective functions and unknown relation-ship between them, an intelligent system is required to identifythis linkage. Moreover, the search space should be determinate,and it is essential to achieve a high precision of it. As a result,ANFIS is utilized to find the most accurate search space.

Jang (1993) has presented ANFIS. The method has attained itspopularity due to a broad range of useful applications in suchdiverse areas in recent years as optimization of fishing predictions(Nuno et al., 2005), vehicular navigation (Noureldin et al., 2007),identify the turbine speed dynamics (Kishor et al., 2007) and otherstudies that have used this method (Lee and Gardner, 2006; Übeyliand Güler, 2006; Civicioglu, 2007; Qin and Yang, 2007; Daomingand Jie, 2006; Depari et al., 2006; Assaleh, 2007; Huang et al.,2007). All above works manifest that ANFIS is introduced as a gooduniversal approximation, predictor, interpolator and estimator.They demonstrate that each non-linear function of many inputsand outputs can be easily constructed with ANFIS. The searchspace is built using ANFIS prediction as follow function:

f x; yð Þ ¼ ν1xð Þp1þ ν1yð Þq1þ ν1ð Þr1þ ν2xð Þp2þ ν2yð Þq2þ ν2ð Þr2 ð16Þwhere νi is referred to as the normalized ring strengths andpi; qi; ri ði¼ 1;2Þ are the design parameters. It is clear that theANFIS has four layers and two sets of adjustable parameters,namely the premise and consequent parameters. During thelearning process, the premise parameters in the first layer andthe consequent parameters in the fourth layer are tuned until thedesired response of the FIS is achieved. In this paper, the Hybridlearning algorithm, which combines the least square method (LS)and the back propagation (BP) algorithm, is utilized to rapidly trainand adapt the FIS. When the premise parameter values of themembership function are fixed, the output of the ANFIS can bewritten as a linear combination of the consequent parameters(Djavareshkian and Esmaeili, 2013).

Subsequently, the PSO algorithm tries to find the optimumsolution in this design space and distributes its particles thatconstitute a swarm, moving around the search space. They look for

Fig. 1. Finite volume and storage arrangement.

Fig. 2. Flowchart of ANFIS–PSO process.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–63 57

the best solution and each individual flies in the search space witha velocity that is dynamically adjusted according to its own flyingexperience and its companions flying experience (Beni and Jing,1993). This velocity is responsible by the movement imposed tothe particle and its spatial position changes in the search space tofind a better performance thereupon with these movements. Theparticles converge to the optimum position in search space, and allof them almost move to a specified point that it is found as anoutput of the algorithm (Fig. 3).

The final step is to evaluate the quality of design space based onthe output of PSO and an additional numerical simulation. If theobjective function difference between the additional numericalinvestigation and the output satisfies the criteria, the process willbe stopped; otherwise, a new numerical result is added toprevious ones and then all steps will be performed again untilthe difference reaches a given threshold.

4. Result and discussion

In this study, the Froude number (Fr) is equal 1 (according tothe chord of hydrofoil). The setting of numerical simulation hasbeen explained in Table 1.

Grid and domain independence, and comparison the currentresults with the published data have to be examined. The gridstructure in CFD simulation is created by a structured mesh,because of its simplicity and applicability to the current flowconfiguration (i.e., with a near-by free surface). In this process, H-type grid is used, and schematic configuration of this two-dimensional structured grid is illustrated in Fig. 4.

Fig. 5 illustrates the boundary condition, which velocity andpressure are prescribed for the inlet and outlet, respectively. Besides,slip boundary conditions are employed on upper and lower surfacesof the domain and the non-slip condition is applied at the solid walls

for hydrofoil surfaces. To account for the steep variations in turbulentboundary layers near solid walls, wall functions, which define thevelocity profile near the no-slip boundaries, are applied. Further-more, the Yþ value on the upper and lower surfaces of the hydrofoil

Fig. 3. Direct and velocity particles in search space, (a) the initial movement, (b) the advanced movement.

Table 1Setting of numerical simulation.

Flow Turbulent

Solver 2-D Double precisionMomentum equation solver Second order upwindSolver Simpleturbulent model k�ε

Fig. 4. H grid topology and H grid around the hydrofoil.

Fig. 5. Dimension and boundary condition of 2D domain.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–6358

is considered in which the center of cells is located in the sub-layer.Subsequently, the dimension of the domain after doing manydifferent lengths has been selected.

The grid independence is achieved by doing several times. Forexample, the effect of grid size is illustrated in Fig. 6. For othercases, the above process is employed for grid and domainindependences.

To validate the numerical simulation, Table 2 draws an analogybetween the lift coefficients for present numerical results andexperimental data (Kouh et al., 2002), and it can be drawn theconclusion that the numerical results are in a high-quality adapta-tion with experimental data.

As the following, the numerical and experimental pressurecoefficient distributions on the mentioned hydrofoil surfaces arecompared in Fig. 7, which confirms good accuracy of the presentnumerical results.

In this study, the effect of the camber and thickness of thehydrofoil has numerically been investigated in different angle ofattack and several submerge distances. However, the lift and dragcoefficients and lift to drag ratio have been analyzed at threespecial angles of attack 2.51, 51 and 7.51 degrees; moreover, thesubmerge distance falls into three main categories: h/c¼0.5,0.7 and 1. Besides, to attain the best mathematical modeling ofthe submerged hydrofoil performance, the camber and thicknessof hydrofoils should be taken into consideration; therefore, abroad range of them are designated, and both of them are dividedinto 3 segments. The non-dimensional camber is slightly changedfrom 0c to 0.02c and 0.04c; also, the non-dimensional thickness isgently varied from 0.09c to 0.12c and 0.15c. Table 3(a)–(d)represents the lift and drag coefficients and L/D in the variantangles of attack as varying submerge distances (Djavareshkianet al., 2013). Indeed, it demonstrates some significant trends; forexample, the lift and drag coefficients are moved up when the AOAis amplified for all cases.

Also, the lift and drag coefficients are enlarged when h/cbecomes high. Pressure difference between upper and lowersurfaces of hydrofoil is advanced with increscent submerge dis-tance until four time of the chord.

The high L/D provides a net gain in economic efficiency, whichit is one of the principal design parameters of the marine vehicles.In the present research, the aim of the optimization is an approachto maximum L/D. The numerical simulation makes a case forgeneral influences of four parameters (camber, thickness, AOA, h/c)on the objective function (L/D); indeed, to find the optimum shapeand condition, it is needed to go into great details; for example,the searching space should be found by considering all parametersand objective function. Consequently, the ANFIS approach sets outsome powerful rules, which will create the searching space,accurately.

The behavior of L/D in all of the hydrofoils as varying thicknessand AOA which is predicted by ANFIS has been presented in Fig. 8.It can be seen that the L/D is gradually increased by an upturn ofthe thickness in all ranges of AOA. The behaviors can be verifiedphysically, and the results of numerical simulation strenuouslydefend these anticipations. Therefore, the reasons of the behaviorscan be explained by the pressure coefficient distribution aroundthe three hydrofoils in the different thicknesses, NACA0009, 0012

Fig. 6. Effect of grid sizing on pressure distribution on the surface of the NACA4412hydrofoil for an AOA¼51, Fr¼1 and h/c¼1.

Table 2Comparison between numerical and experimental lift coefficients around theNACA4412, AOA¼51, h/c¼1.

Fr CL (present) CL (Kouh et al., 2002) Error%

0.4 1.52 1.227 190.7 1.15 1.347 170.8 0.995 1.094 101 0.758 0.744 1.851.2 0.629 0.62 1.431.4 0.603 0.565 6.3

Fig. 7. Comparison between numerical and experimental pressure coefficientdistribution around the NACA4412, AOA¼51 and h/c¼1.

Table 3The lift and drag coefficients of hydrofoils (a) NACA0009, (b) NACA4409,(c) NACA0015, (d) NACA4415.

h/c 0.5 0.7 1

AOA (1) CL CD CL CD CL CD

(a)2.51 0.1508 0.0458 0.20358 0.04796 0.2686 0.049751 0.29322 0.06048 0.3687 0.06519 0.465 0.069727.51 0.4237 0.08311 0.5327 0.09333 0.65808 0.10208

(b)2.51 0.33941 0.06848 0.4395 0.07295 0.5702 0.0737851 0.4835 0.08677 0.6143 0.09513 0.7795 0.099287.51 0.6301 0.11538 0.7838 0.1285 0.9851 0.13787

(c)2.51 0.1521 0.07411 0.225 0.07921 0.2999 0.0805151 0.2881 0.08666 0.37805 0.09311 0.4862 0.097967.51 0.416 0.10673 0.5358 0.1185 0.6736 0.12637

(d)2.51 0.34108 0.09887 0.4549 0.10609 0.597 0.108751 0.4854 0.11603 0.6099 0.12606 0.788 0.13167.51 0.6202 0.14353 0.782 0.15893 0.99513 0.16893

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–63 59

and 0015, which are depicted in Fig. 9. As it can be seen, thepressure on the upper and lower surfaces of hydrofoil is slightlydeclined when the thickness rises slowly, but the pressure reduc-tion on the upper surface is more than lower ones. Accordingly, anincrease of the lift force is more than the drag one, and the L/D isincreased by rising thickness.

Subsequently, the value of using a robust design will be high-lighted when the variation of L/D is considered according to AOAchanges as it can observe in Fig. 8. In this figure, the performanceof L/D across the growth of AOA is fluctuated, while it is firstlyincreased and then declined gradually. Indeed to confirm thesetrends, the numerical simulation results are also adapted withthem, and the variation of L/D according to various AOA andthickness which are obtained by numerical simulation is illu-strated (Fig. 10). This figure depicts that the L/D reaches a peakconsistently when AOA increases, gradually; then, there is a slightdrop. The figure also illustrates that the thickness of hydrofoil isthe key factor in these trends whereas these behaviors are sharp inlow thickness. The same nonlinear trend can be seen in Fig. 11,variation of L/D according to various AOA and h/c. The figuredemonstrates that the L/D reaches a peak consistently when AOArises; then, there is a small decrease.

As a result, the behavior of L/D is not simplicity and easilypredictable when other parameters (camber, thickness, AOA, h/c)change. It is a clear illustration of the value of using a strong andaccurate optimization process.

Fig. 8. Behavior of L/D as varying thickness (t/c) and AOA for investigated hydrofoils.

Fig. 9. Pressure coefficient distribution around NACA0009, 0012 and 0015 hydro-foils for h/c¼0.7 and AOA¼51.

AOA (deg)

L/D

3 4 5 6 75

6

7

8

NACA 2409NACA 2412NACA 2415

Fig. 10. Variations in L/D as a function of AOA in different thickness and h/c¼1.

AOA(deg)

L/D

3 4 5 6 7

4

5

6

7h/c=0.5h/c=0.7h/c=1

Fig. 11. Variations in L/D as a function of AOA and different h/c.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–6360

ANFIS also anticipates the behavior of L/D as varying camber andh/c and it is presented in Fig. 12. This figure illustrates that L/D has anupward trend when the camber of hydrofoil is increasing, which theperformance is physical, and the ANFIS has predicted it very well.

To verify the accuracy of the prediction, as can be observed in thenumerical results, an increase in the camber is led to drop on thepressure distribution on the upper surface, and rise in the lowersurface of hydrofoil and the lift force grows (Fig. 13(a)). Furthermore,Fig. 13(b) illustrates the velocity profile behind the hydrofoils andconfirms to increase the drag when the camber is picked up. However,outgrowth of the drag force is less than the lift one, and it causes to anupsurge of L/D when the camber increases. Furthermore, the figuremanifests that the L/D is marginally dropped in a very intimate thefree surface of water. The figure broadly supports the view that thebehavior of L/D has a different trend in the unique conditions, and itsets out a strong argument, which a robust optimization approach isneeded.

The optimization algorithm, which has been applied in thisstudy, is a robust stochastic optimization technique based on themovement and intelligence of swarms. The proportion of particlesand their velocity are the key factors in the finding of the optimumconditions. Consequently, the impact of a number of particles andtheir velocity are considered; consequently, the number of particleand its velocity has been chosen 400 and 0.001, respectively.

Pursuant to following sentences, the best shape characteristicsof submerged hydrofoil and operating conditions are obtained byPSO and the optimization results are presented in Table 4. Accord-ing to this result, the best hydrofoil should have 3/2% camber and9/3% thickness; furthermore, the optimum hydrofoil should bealso closed the free surface of water approximately 0.98, and thebest angle of attack is attained just over 3.51. The optimized formis depicted in Fig. 14 according to the optimization results.

Indeed, the optimum hydrofoil with the above condition willproduce the maximum possible L/D which is around 8.5. In orderto verify the anticipation of ANFIS–PSO, the numerical simulation

Fig. 12. Behavior of L/D as varying camber and h/c for investigated hydrofoils.

Fig. 13. Effect of camber on (a) the pressure coefficient distribution and (b) the velocity profile behind the hydrofoil in x/c¼2, and h/c¼0.5, AOA¼51.

Table 4Characterize of optimize hydrofoil.

AOA h/c Camber Thickness L/D

3.51 0.98 0.032 0.093 8.5

Fig. 14. Optimum shape of hydrofoil.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–63 61

of flow around optimum configuration of submerged hydrofoil inthe stated conditions has been performed.

As the following, the lift and drag coefficients and L/D arenumerically obtained 0.664, 0.088 and 7.90, respectively. Finally,the low difference between both L/D lays emphasis on confirmingthe optimization process and proves the high talent of ANFIS–PSO.The estimative L/D of ANFIS–PSO has been compared with CFDsimulation and the error related to these methods is equal to 7.6%.

Error¼ L=DðCFDÞ�L=DðANFIS�PSOÞL=DðCFDÞ

¼ 0:076

Since, the error is reasonable, it can be declared that the ANFIS–PSO method is appropriate for optimization, and the results can berelied on.

In Table 5(a) and (b), a comparison of optimum shape ofhydrofoil and other ones are performed which all of them arenumerically simulated in the optimum conditions. The CL, CDand L/D of the optimum hydrofoil, case 3 (Ca/c¼0.04, th/c¼0.1)and case 4 (Ca/c¼0.04, th/c¼0.12) are compared in Table 5(a).In addition, Table 5(b) illustrates CL, CD and L/D in optimum shape,case 1 (Ca/c¼0, th/c¼0.09) and case 2 (Ca/c¼0.02, th/c¼0.09) foroptimum condition. The comparisons confirm that the L/D of theoptimum hydrofoil is dramatically grown by 5.2% when it iscompared with other hydrofoils having same camber and by 3%in the same thickness. The results prove that the camber ofhydrofoil has more sensitivity on the hydrofoil performance.

Fig. 15 manifests pressure coefficient contribution on the case 1,case 2 and optimum hydrofoil surfaces in optimum conditions.This figure proves that the pressure on the upper surface ofhydrofoil is noticeably reduced and on the lower surface is slightly

increased when the camber is grown. Besides, the drag coefficienthas an upward trend by the camber increasing. Fig. 16 demon-strates that the wake behind the hydrofoil is stronger than that ofin other hydrofoils. Consequently, this phenomenon makes toincrease the drag force. Of course, the increase of the dragcoefficient is less than the lift one and the L/D is gone up whenthe camber is consistently enlarged.

5. Conclusion

The shape optimization and operating conditions of the sub-merged hydrofoil are investigated by a heuristic optimizationapproach. This method is a combination of an adaptive networkbased fuzzy inference system model (ANFIS) and particle swarmoptimization (PSO). Thickness and camber of hydrofoil, angle ofattack and submerge distance are considered as constraineddesign variables and value of lift to drag ratio is defined as anobjective function, which has a nonlinear behavior. The objectivefunction is obtained by numerical simulation. The Navier–Stocksequation is numerically solved, and volume of fluid (VOF) methodhas been utilized to simulate two-phase fluid (water and air). Theresults demonstrate that the obtained configuration of hydrofoil inoptimum operating conditions could reach a maximum value oflift to drag ratio and the hydrofoil with designed section, Ca/c¼0.032, th/c¼0.093, h/c¼0.98 and AOA¼3.51, has the bestperformance in the close free surface of water.

References

Assaleh, K., 2007. Extraction of fetal electrocardiogram using adaptive neuro-fuzzyinference systems. IEEE Trans. Biomed. Eng. 54 (1), 59–68.

Bourgoyne, D., 2003. Flow Over a Hydrofoil with Trailing Edge Vortex Shedding atHigh—Reynolds Number.

Beni, G., Jing, W., 1993. Swarm Intelligence in Cellular Robotic Systems. Robots andBiological Systems: Towards a New Bionics. Springer Berlin Heidelberg, pp.703–712.

Chen, C.K., Liu, H., 2005. A submerged vortex lattice method for calculation of theflow around three-dimensional hydrofoil. J. Ship Mech 9, 2.

Carcaterraa, A., Dessi, D., Mastroddi, F., 2005. Hydrofoil vibration induced by arandom flow: a stochastic perturbation approach. J. Sound Vib. 283, 401–432.

Civicioglu, P., 2007. Using uncorrupted neighborhoods of the pixels for impulsivenoise suppression with ANFIS. IEEE Trans. Image Process. 16 (3), 759–773.

Cocke, W., 2012. Shape Optimization of Hydrofoils. Master of Arts Thesis. Washing-ton University.

Daoming, G., Jie, C., 2006. ANFIS for high-pressure water jet cleaning prediction.Surf. Coat. Technol. 201 (3-4), 1629–1634.

Table 5Comparison of between optimum hydrofoil and other ones (a) thickness effect,(b) camber effect.

Hydrofoil type CL CD L/D

(a)Optimum shape 0.65357 0.08269 7.903858Case 3 0.6628 0.08823 7.512184Case 4 0.64373 0.097545 6.599313

(b)Optimum shape 0.65357 0.08269 7.903858Case 2 0.50237 0.065477 7.672465Case 1 0.3444 0.05611 6.137943

Fig. 15. Pressure coefficient distribution around the case 1, case 2 and optimumshape in optimum condition.

Fig. 16. Effect of hydrofoil camber on the velocity profile behind the hydrofoil in x/c¼2 and optimum condition.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–6362

De Blasi, P., Di Felice, F., Lalli, F., Romano, G., 2000. Experimental study of breakingwave flow field past a submerged hydrofoil by LDV. Int. J. Offshore Polar Eng.10, 1053–5381.

Daskovsky, M., 2000. The hydrofoil in surface proximity, theory and experiment.Ocean Eng. 27 (10), 1129–1159.

Depari, A., Flammini, A., Marioli, D., Taroni, A., 2006. Application of an ANFISalgorithm to sensor data processing. IEEE Trans. Instrum. Meas., 1596–1599.

Djavareshkian, M., Esmaeili, A., 2013. Neuro-fuzzy based approach for estimation ofhydrofoil performance. Ocean Eng. 59, 1–8.

Djavareshkian, M.H., Esmaeili, A., Parsania, A., 2013. Numerical simulation of smarthydrofoil in marine system. Ocean Eng. 73, 16–24.

Ducoin, A., Jacques, A., François, D., François, S.J., 2009. Computational andexperimental investigation of flow over a transient pitching hydrofoil. Eur. J.Mech. B—Fluid 28, 728–743.

Filippov, S., 2001. Flow past a submerged hydrofoil. J. Fluid Dyn 36 (3), 489–496.Frunz, T., Resiga, S.R., Muntean, S., Bernad, S., 2010. Optimization of the hydrofoil

cascade and validation with quasi-analytical solution for hydraulic machinery.IOP Conf. Ser.: Earth Environ. Sci. 12 (1), 75–87.

Guo, W., Songlin, Y., Peng, C., 2009. The analysis of intelligent water jet propulsionsystem optimization on self service gliding hydrofoil craft. In: IEEE SecondInternational Symposium on Computational Intelligence and Design. 250-253.

Hay, A., Visonneau, M., 2005. Computation of free-surface flows with local meshadaptation. Int. J. Numer. Methods Fluids 49 (7), 785–816.

Hwang, I.S., Lee, Y.H., Kim, S.J., 2009. Optimization of cycloidal water turbine andthe performance improvement by individual blade control. Appl. Energy 86,1532–1540.

Hsin, C.Y., Wu, J.L., Chang, S.F., 2006. Design and optimization method for a two-dimensional hydrofoil. J. Hydrodyn. Ser. B 18, 323–329.

Huang, M., Chen, H., Huang, J., 2007. Glaucoma detection using adaptive neuro-fuzzy inference system. Expert Syst. Appl. 32 (2), 458–468.

Jang, J., 1993. ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans.Syst. Man Cybern. 23 (3), 665–685.

Kouh, J.S., Lin, T.J., Chau, S.W., 2002. Performance an analysis of two-dimensionalhydrofoil under free surface. J. Nat. Taiwan Univ, 86.

Kim, S., Yamato, H., 2005. The estimation of wave elevation and wave disturbancecaused by the wave orbital motion of a fully submerged hydrofoil craft. J. Mar.Sci. Technol 10 (1), 22–31.

Kennedy, J., Eberhart, R., 1995. Particle swarm optimization. IEEE Int. Conf. 4,1942–1948.

Kishor, N., Singh, S., Raghuvanshi, A., 2007. Adaptive intelligent hydro turbinespeed identification with water and random load disturbances. Eng. Appl. Artif.Intell. 20 (6), 795–808.

Lee, K., Gardner, P., 2006. Adaptive neuro-fuzzy inference system (ANFIS) digitalpredistorter for RF power amplifier linearization. IEEE Trans. Veh. Technol.55 (1), 43–51.

Münch, C., Ausoni, P., Braun, O., Farhat, M., Avellan, F., 2010. Fluid-structurecoupling for an oscillating hydrofoil. J. Fluids Struct 26 (6), 1018–1033.

Noureldin, A., El-Shafie, A., Reda Taha, M., 2007. Optimizing neuro-fuzzy modulesfor data fusion of vehicular navigation systems using temporal cross-validation.Eng. Appl. Artif. Intell. 20 (1), 49–61.

Nuno, A.I., Arcay, B., Cotos, J.M., Varela, J., 2005. Optimization of fishing predictionsby means of artificial neural networks, ANFIS, functional networks and remotesensing images. Expert Syst. Appl. 29, 356–363.

Ouyang, H., Weber, L., Odgaard, A., 2006. Design optimization of a two-dimensionalhydrofoil by applying a genetic algorithm. J. Eng. Optim 38, 529–540.

Qin, H., Yang, S., 2007. Adaptive neuro-fuzzy inference systems based approach tononlinear noise cancellation for images. Fuzzy Sets Syst. 158 (10), 1036–1063.

Sadathosseini, S., Mousaviraad, S., Firoozabadi, B., Ahmadi, G., 2008. Numericalsimulation of free-surface waves and wave induced separation. Sci. Iran 15 (3),323–331.

Shafaghat, R., Hosseinalipour, S., Nouri, N., Lashgari, I., 2008. Shape optimization oftwo-dimensional cavitators in supercavitating flows using NSGA II algorithm.Appl. Ocean Res. 30, 305–310.

Schmitz, A., Besnard, E., Hefazi, H., 2004. Automated hydrodynamic shape optimi-zation using neural networks. In: SNAME Maritime Technology Conference &Expo.

Spogis, N., Nunhez, J., 2009. Design of high efficiency hydrofoil through the use ofcomputational fluid dynamics and multiobjective optimization. AIChE J. 55 (7),1723–1735.

Tozzi, G.M., 2004. Hydrofoil Shape Optimization by Gradient Methods. DoctoralDissertation. Massachusetts Institute of Technology.

Übeyli, E., Güler, I., 2006. Adaptive neuro-fuzzy inference system to compute quasi-TEM characteristic parameters of micro shield lines with practical cavitysidewall profiles. Neurocomputing 70 (1-3), 296–304.

Wang, C., Feng, F., Chang, X., Guo, C.Y., Liu, Y.H., 2012. The application andcomparison of two different optimization algorithms in the optimizationdesign of hydrofoil. J. Appl. Mech. Mater 178, 2871–2876.

Xie, N., Vassalos, D., 2007. Performance analysis of 3D hydrofoil under free surface.Ocean Eng. 34 (8-9), 1257–1264.

Yang, Z.Y., Gao, T., Pang, Y.J., Ma, Y., 2012. Hydrodynamic integrated optimization oftwo dimensional hydrofoils based on Kriging model. J. Adv. Mater. Res 421,651–656.

Yang, B., Shu, X., 2012. Hydrofoil optimization and experimental validation inhelical vertical axis turbine for power generation from marine current. OceanEng. 42, 35–46.

Yang, S., Xi, W., Ma, L., Zhang, H., Yang, Z., 2009. Optimizing computation ofcontrolling parameters of intelligent propulsion system of a hydrofoil slidingcraft propelled by adjustable pitch screw. In: Fifth International Joint Con-ference on INC, IMS and IDC, IEEE. pp.1815–1818.

Yue, W., Ching-Long, L., Virendra, C.P., 2005. Large eddy simulation of turbulentopen-channel flowwith free surface simulated by level set method. Phys. Fluids17 (2), 1–13.

Yue, W., Ching-Long, L., Virendra, C.P., 2003. Numerical Investigations of TurbulentFree Surface Flow Using Level Set Method and Large Eddy Simulation. DoctoralDissertation. University of Iowa.

Zanette, J., Imbault, D., Tourabi, A., 2010. A design methodology for cross flow waterturbines. Renewable Energy 35, 997–1009.

M.H. Djavareshkian, A. Esmaeili / Ocean Engineering 92 (2014) 55–63 63