MIN Shao-song,PENG Fei,WANG Zhan-zhi,ZHANG Tao

(1.Department of Naval Architecture,Naval University of Engineering,Wuhan 430033,China;2.The 91872th Unit of PLA,Beijing 102442,China)

Method of Estimating the Effect of Marine Fouling on Frictional Resistance of Ships

MIN Shao-song1,PENG Fei1,WANG Zhan-zhi1,ZHANG Tao2

(1.Department of Naval Architecture,Naval University of Engineering,Wuhan 430033,China;2.The 91872th Unit of PLA,Beijing 102442,China)

Marine fouling has a severe adverse effect on the hydrodynamics of a ship’s hull.In this paper,the marine fouling was approached as a type of hull roughness,and the use of the integral method in estimating its effect on the frictional resistance of ships was studied.Through the calculation of the friction factor of a smooth surface,the accuracy of the integral method was verified and confirmed.The integral method was then adopted to calculate the friction factor of a U.S.Navy FFG-7 class ship under three different calcareous fouling conditions,which gave a result consistent with that from other researchers,indicating the feasibility and accuracy of the integral method in estimating the effect of marine fouling on hull resistance.The result using the integral method for a FFG-7 ship was compared with result from the 1957-ITTC equation and its roughness allowance ΔCf.The result shows that the ΔCfequation proposed by ITTC is not applicable to characterize the impact of fouling on hull resistance.

marine fouling;frictional resistance;hull roughness;boundary layer integral method

0 Introduction

Marine fouling has long been a more problematic issue compared with corrosion.The robust vitality of fouling organism has made fouling problem a formidable barrier for human to conquer the ocean[1].Marine fouling organisms comprise a wide range of organisms,including all animals,plants,and microorganisms that are attached to the surfaces of immersed objects.Fouling results in an uneven hull surface,largely increasing the roughness of the hull and,hence,the frictional resistance.According to monitoring data of ship resistance from different countries,the total resistance of a ship could increase by up to 60%when severely fouled by calcareous organisms[2].For a newly built ship,a roughness allowance,ΔCf,is usually used to calculate the impact of surface roughness on frictional resistance.Further research shows thatΔCfincludes not only the increase of resistance from surface roughness,but also the difference from various friction equations,scale functions,and the efficiency,wake,and thrust-deduction factor of propellers[3].

Since fouling has a very negative impact on hull resistance,many researchers have been studying the effect of fouling and its estimation method.Townsin[4]summarized the previous studies,suggested a better method of understanding and estimating the effect of fouling,and determined that the effect of calcareous fouling(such as that caused by barnacles)is easier to estimate than bacterial or botanical fouling.Schultz[5]compared the results from studying fouling by barnacles with different sizes and coverages,and found that the size of the barnacle is the dominant factor determining the resistance.Based on the resistance data on the experimental scale and on the similarity law of boundary layers,Schultz[6]estimated a ship’s resistance and power loss under many different fouling conditions.Schultz[7]analyzed the economic loss of fouling on U.S.Navy DDG-51 class ships,and showed that current fouling conditions could result in a ～＄560 million annual economic loss.However,the reliability of Schultz’s estimation method needs further verification.

This work investigated a method of estimating a ship’s frictional resistance under fouling conditions based on the integral method of boundary layers.Taking a U.S.Navy ‘Perry’class corvette(aka a ‘FFG-7’ ship)as the object of study,the ship’s resistance was estimated and analyzed under three different calcareous fouling conditions.

1 Estimation method

In addition to conventional physical roughness,fouling can also be considered as hull roughness.In this work,fouling is treated as a type of hull roughness for further study.Previous workers[8]estimated and analyzed the development of a turbulent boundary layer on a rough surface based on the integral method,and the same method is adopted in this work to estimate a ship’s resistance under fouling conditions.

1.1 Characterization of fouling effect

The main effect of roughness on the flow near a surface is to change its average velocity distribution[9].Therefore,the decrease of average velocity distribution can be used as a parameter to characterize the effect of roughness on surface flow,as a roughness function ΔU+.According to Coles’equation,which describes the average velocity distribution of flow near a smooth surface,and combining it with the roughness function ΔU+,the velocity distribution in a turbulent boundary layer near a rough surface,U+,can be described as

where y is the horizontal axis in the boundary layer with the plate surface as the origin,uτis the friction velocity,δ is the boundary-layer thickness,B0is the logarithmic intercept of the smooth surface,κ is Kármán’s constant(usually taken as 0.41),υ is the kinetic viscosity of fluid,and Π is wake parameter that is usually a function of x and dependent on the pressure gradient in the flow direction and on the surface roughness,is a function referred to the ‘law of the wake’.

The roughness function ΔU+is dependent on the substantial characteristics of the rough surface and the Reynolds number[10].The format of the roughness function is usually considered to have a logarithmic relationship with respect to the roughness factor,described as

where h′is the roughness factor and B and C are constants that depend on particular conditions of roughness.There are many methods of characterizing the roughness factor h′,such as the height,interval,and shape features of the roughness element.

Therefore,the effect of roughness is represented in the equation of velocity distribution within a turbulent boundary layer,directly and indirectly,as the roughness function ΔU+and wake parameter Π,respectively.

1.2 Numerical method of calculating the turbulent boundary layer of the fouling surface

(1)Characteristic parameter relationship of the fouling surface

When y=δ,U=Ue,U is the average velocity at position y,Ueis the free velocity,the velocity of the rough surface,Eq.(1),is converted to

Letting Eq.(3)subtract Eq.(1),anduτ/Ue,in which Cfis the local friction coefficient,the boundary layer can be derived into another form:

The relationship between displacement of boundary-layer thickness δ*and momentum thickness θ can be further derived into

(2)Momentum integral equation

The two-dimensional momentum integral equation at zero-pressure gradients is

For the equilibrium boundary layer,the wake parameter Π is independent of x,i.e.,dΠ/dx=0.By substituting the characteristic parameter equation of the boundary layer,Eq.(5),into Eq.(6),an ordinary differential equation can be derived as

in which

(3)Rough surface equation

A rough surface equation is derived according to the velocity distribution within the boundary layer from Eqs.(1)and(2):

By substituting the characteristic parameter equation of the boundary layer,Eq.(5),into Eq.(9)and performing a derivation to x,an ordinary differential equation can be derived as

in which

(4)Friction coefficient of fouling surface

Combining Eqs.(7)and(10),linear equations can be obtained with dδ/dx and dw/dx as variables.Ordinary differential equations with respect to δ and w can be solved using the Crammer principle.Given a known roughness function ΔU+and velocity distribution at initial conditions,the distribution of δ and w at the rough surface in the flow direction can be calculated by the numerical integral method,resulting in the distribution of the local friction coefficient Cfialong the flow direction.The local frictional resistance fican be further calculated.Integration of fialong the flow direction gives the frictional resistance F and friction coefficient CF.

where Usiis the potential velocity at position xiand V the velocity.

2 Verification of the method

To verify the accuracy of the integral method,the frictional resistance of a smooth surface is calculated before estimating the frictional resistance of a fouled hull.The result is compared with the 1957-ITTC equation,as shown in Fig.1.For a smooth surface,the roughness function ΔU+equals 0.The boundary-layer thickness δ0at the beginning position of the integral(x0)is estimated based on the power-law empirical equation of the smooth surface.As seen in Fig.1,when the Reynolds number ReL(ReL=UL/v,in which U is velocity and L is length)is greater than 106,the result from the integral method matches that from the 1957-ITTC equation.Therefore,the integral method proposed in this work has high accuracy for estimating resistance for regular ships.

Fig.1 Comparison of frictional resistance coefficient on smooth plates between the integral method and 1957-ITTC formula

3 Cases

According to the evaluation method set forth in Naval Ships’Technical Manuals(NSTMs)[11],hull fouling is classified into 10 fouling ratings(FRs),in which FR10-FR30 are membrane or grass fouling,or soft fouling;FR40-FR100 are hard fouling and composite fouling.Since the dominant form in this latter category is calcareous species such as barnacles and tubeworms,fouling in this category is collectively called calcareous fouling,which is the object of study in this work.Schultz et al[6]measured the roughness factor of surfaces subject to different calcareous fouling conditions,as shown in Tab.1,Rtm5( )0 is maximum peak to trough height over a 50 mm sampling length,seen in Ref.

[12].

Tab.1 Roughness factor under different calcareous fouling conditions

Since the barnacle is the dominant species in the biotic community germane to this study,it was chosen as our object of research.The usual shape of a barnacle is conical,and the diameter of its opening is smaller than that of the substrate(the intersection angle between wall and substrate is less than 90°)[13],here,the geometrical setting for a barnacle is a tangent value of intersection angle between wall and substrate of 2.5 and a diameter of the substrate being 3 times the diameter of the opening.According to the definition of the roughness factor h′by the Ship Performance Group at the University of Newcastle upon Tyne(United Kingdom)[14],the roughness factor h′can be calculated as

where Rqis root mean square roughness height(μm),Sais mean slope of the surface profile.

Fig.2 The shapes of barnacles(Left:the barnacles attached to hull surfaces;Right:modeling of barnacles)

Since we cannot obtain the roughness function ΔU+of a barnacle,the constants B and C in the ΔU+equation have to be estimated.Considering that the conical shape of a barnacle is similar to a pyramid,and assuming that a barnacle is firmly attached to the hull,the adhesion shape of a barnacle can be regarded to have the same roughness as that of the experimental surface described in Ref.

[15].The constants B and C in the ΔU+equation can be assigned values of 0.32 and-2.78,respectively.ΔU+is thus derived as

Schultz studied the resistance of a ‘Perry’class corvette under different fouling conditions[6].Taking a FFG-7 class ship as the object of our study,the friction coefficient is estimated under three calcareous fouling conditons using the integral method and roughness function.The friction coefficient Cfis calculated under three calcareous fouling conditions using the integral method.The total coefficient CTis then calculated based on the composition of resistance of a FFG-7 ship.The coefficient increase of Cfand CTcaused by fouling is then calculated and compared with the result from Schultz’s research,as shown in Tab.2.

As seen in the table,at 15 kns,moderate calcareous fouling on a FFG-7 ship increases CTby 53.7%;at 30 kns,the value is 35.9%for CT.Schultz’s results show that moderate calcareous fouling on a FFG-7 ship increases CTby 52%at 15 kns and by 36%at 30 knots[16].Calculational results in this work match Schultz’s results well.Therefore,the integral method proposed in this work shows good feasibility and accuracy in estimating the effect of fouling on hull resistance.

Tab.2 Comparison of results obtained from integral method and from Schultz’s research

Fig.3 compared the aforementioned result of the friction coefficient Cfof a FFG-7 ship with the result of 1957-ITTC equation,along with 1957-ITTC result added the roughness allowance ΔCfsuggested by ITTC.As shown in Fig.3,the roughness allowance suggested by ITTC is much larger than that calculated by the integral method,and it is independent of Reynolds number(the line of 1957-ITTC equation is parallel to the line of 1957-ITTC result added the roughness allowance ΔCfsuggested by ITTC).Therefore,the roughness allowance suggested by ITTC is not applicable to the evaluation of the effect of fouling on hull resistance.

4 Conclusions

In this work,marine fouling is treated as a category of ship roughness worthy of study.Based on the integral method of turbulent boundary layers on a rough surface,the effect of fouling on hull resistance is studied and estimated.For a smooth surface,the friction coefficient calculated using the integral method exhibits good consistency with that calculated using the 1957-ITTC equation for ships of regular size.The integral method is then used to study the hull resistance of a FFG-7 ship under three calcareous fouling conditions caused by barnacles,and the results are in good agreement with those of other research groups.This demonstrates that the integral method has good feasibility and accuracy for estimating the effect of fouling on hull resistance.The integral method result has been compared to that obtained using the 1957-ITTC equation as well as the roughness allowance,ΔCf,equation,showing that the ΔCfvalue obtained by using the ITTC equation is much larger than that obtained by using the integral method.Therefore,the roughness allowance suggested by the ITTC equation is not applicable for evaluating the effect of fouling on hull resistance.

Acknowledgements

We thank LetPub(www.letpub.com)for its linguistic assistance during the preparation of this manuscript.

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海洋污損對船體摩擦阻力影響的預(yù)測方法

閔少松1，彭 飛1，王展智1，張 濤2

（1.海軍工程大學(xué) 艦船工程系，武漢430033；2.中國人民解放軍91872部隊，北京102442）

海洋污損對船體阻力具有嚴(yán)重的不利影響。文章將海洋污損歸類到船體粗糙度的范疇內(nèi)，基于邊界層積分法研究了污損對船體摩擦阻力影響的預(yù)測方法。通過對光滑平板摩擦阻力系數(shù)的計算驗證了積分法的準(zhǔn)確性。然后采用積分法對美軍FFG-7艦在3種鈣質(zhì)污損狀況下的摩擦阻力系數(shù)進行了計算，計算結(jié)果與國外學(xué)者的研究結(jié)論吻合得較好，說明了積分法預(yù)測污損對船體摩擦阻力的影響具有可行性，且準(zhǔn)確性較高。最后將積分法在FFG-7艦的計算結(jié)果與19 57-ITTC公式及其粗糙度補貼系數(shù)ΔCf進行了對比，結(jié)果表明ITTC提出的ΔCf公式不適于表征污損對船體阻力的影響。

海洋污損；摩擦阻力；船體粗糙度；邊界積分法

U661.3

A

國家自然科學(xué)基金資助（51479207）；海洋工程國家重點實驗室基金資助（1514）

閔少松（1978-），男，博士，海軍工程大學(xué)艦船工程系講師；

彭 飛（1975-），男，博士，海軍工程大學(xué)艦船工程系副教授；

王展智（1986-），男，博士，海軍工程大學(xué)艦船工程系講師；

張 濤（1987-），男，博士，中國人民解放軍91872部隊工程師。

U661.3 Document code:A

10.3969/j.issn.1007-7294.2017.12.002

date：2017-09-03

Supported by the National Natural Science Foundation of China(Grant No.51479207);the State Key Laboratory of Ocean Engineering(Grant No.1514)

Biography：MIN Shao-song(1978-),male,Ph.D.,lecturer of Naval University of Engineering,

E-mail:minshaosong@163.com;PENG Fei(1975-),male,Ph.D.,associate professor of

Naval University of Engineering.

1007-7294（2017）12-1460-08