A numerical flat plate friction line and its application*

WANG Zhan-zhi （王展智）, XIONG Ying （熊鹰）, SHI Li-pan （时立攀）, LIU Zhi-hua （刘志华）

Department of Naval Architecture, Naval University of Engineering, Wuhan 430033, China,

E-mail: wzz200425@126.com

A numerical flat plate friction line and its application*

WANG Zhan-zhi （王展智）, XIONG Ying （熊鹰）, SHI Li-pan （时立攀）, LIU Zhi-hua （刘志华）

Department of Naval Architecture, Naval University of Engineering, Wuhan 430033, China,

E-mail: wzz200425@126.com

(Received January 6, 2014, Revised May 12, 2014)

This paper studies the regression of a numerical two-dimensional flat plate friction line using the RANS method with the SST k-wturbulence model. Numerical simulations with different inlet turbulence kinetic energies are first conducted. Comparing with the experimental data, the finest grid and the appropriate inlet turbulence kinetic energy are selected to compute the flat plate friction resistance at 14 Reynolds numbers. Two numerical friction lines are obtained by the least squares root fitting method and one similar to that of the ITTC-1957 line and the cubic polynomials in logarithmic scales, and the results are compared to the friction line proposals available in the open literature. Finally, the full scale viscous resistance predictions of DTMB5415, KVLCC2, SUBOFF are compared between the numerical friction line and the friction line proposals available in the open literature based on the form factor approach. It is shown that the form factor keeps relatively constant via the numerical friction line for the bare hull, but the form factor concept in extrapolating the model test results is not appropriate for appended hulls. It is suggested that for computing a form factor numerically, it is best to use a numerical friction line.

friction line, form factor, flat plate, RANS, uncertainty analysis

Introduction

The full scale ship resistance prediction is a fundamental issue in the ship design. At present, the model testing is an effective way to determine the resistance of a ship. The forces measured in the model scale are “extrapolated” to the full scale ship by a special procedure. The so-called form factor approach suggested by the International Towing Tank Conference (ITTC) is a popular procedure used by many modern basins. In this method, the drag of a ship is split into two independent components, the viscous resistance and the wave resistance. The viscous resistance is a function of the Reynolds number,Re , while the wave resistance is affected by the Froude number, Fr.

The viscous resistance coefficient Cvis assumed to be proportional to the frictional resistance coefficient given by a ship-model correlation line (based on equivalent smooth flat plate flow experiments) where(1+k )is the form factor and CFis the equivalent smooth flat plate frictional resistance coefficient.

In practice, the form factor(1+k )is assumed to be scale-independent, and the equivalent smooth flat plate resistance is computed from the ITTC-1957 friction line, then the full scale ship resistance is determined. Therefore, the validity of this approach heavily relies on the independence of the form factor on the Reynolds numberRe and the accuracy of the equivalent flat plate resistance. Unfortunately, the form factor does depend on the Reynolds number if the ITTC-1957 correlation line is chosen. Grigson[1]reanalyzed the resistance tests of the Lucy Ashton and Victory and found that the form factor is obviously scale-dependent when the ITTC-1957 friction line is adopted. Garcia[2]analyzed the experimental data of four geosim models and concluded that one sees a significant scale effect by using the ITTC-1957 frictionline, and proposed an empirical formula to correlate the form factor from the model scale to the full scale. Kouh et al.[3]investigated the scale effect of the form factor via a numerical method and found that the form factor has a near linear and increasing dependence on the Reynolds number. During the last decades, the ITTC-1957 correlation line was put into doubt and new ones were proposed[4,5]. Through the two-dimensional boundary-layer equations in zero pressure gradients using the boundary-layer velocity distribution based on Coles, the plate friction resistance could be obtained.

With the rapid development of the CFD, it is possible to derive a plate friction line by the RANS method ranging from the model scale Reynolds number to the full scale Reynolds number. Eca and Hoekstra[6,7]presented a numerical study of the twodimensional flat plate friction resistance as a function of the Reynolds number via seven eddy-viscosity models. They found that the differences between the numerical friction lines are smaller than the difference between the four lines proposed in the open literature. However because of the low level inlet turbulence kinetic energy, the calculations show that in a region at the leading edge of the plate, one sees a transition from laminar to turbulent flows, so the predicted friction resistance is lower than that in the open literature, and the model test should be carried out in a turbulent flow condition.

The present paper studies the regression of a numerical two-dimensional flat plate friction line using the RANS method with the SST k-wturbulence model. Numerical simulations with different inlet turbulence kinetic energies and numerical uncertainty studies are carried out. Comparing with the experimental data, the finest grid and the appropriate inlet turbulence kinetic energy are selected to compute the flat plate friction resistance at 14 Reynolds numbers. Two numerical friction lines are obtained using the least squares root fitting method: from one similar to that of the ITTC-1957 friction line and cubic polynomials in logarithmic scales. The results are compared to the friction line proposals available in the open literature. The full scale viscous resistance predictions of DTMB5415, KVLCC2, SUBOFF are compared between the numerical friction line and the friction line proposals available in the open literature based on the form factor approach, and the direct CFD result. It is shown that the form factor keeps relatively constant with respect to the numerical friction line for a bare hull.

1. Numerical model

1.1 Governing Equations field over a two-dimensional flat plate are the instantaneous conservation of mass (continuity equation) and momentum (Reynolds averaged Navier-Stokes equation, RANS). These equations can be expressed as follows:

where, all variables are time-averaged,t,ui,ρ,p,are the time, the velocity, the fluid density, the static pressure, the dynamic viscosity, the body force per unit volume and the Reynolds stress, respectively.

1.2 Turbulence model

An additional equation is needed in order to obtain the unknown Reynolds stress. The turbulence model is the closure equation which combines the fluctuating quantities and the time averaged ones. Here, the SST k-w turbulence model is selected. The SST k-wturbulence model was developed by Menter[8]to effectively blend the robust and accurate formulation of the standardk-wmodel in the near-wall region with the free-stream independence of the standardkεmodel in the far field. The SST k-wturbulence model refines the standardk-wmodel in the following manner: firstly, in the SSTk-wturbulence model, a damped cross-diffusion derivative term is incorporated in thewequation, secondly, the definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress. These features make the SST k-wturbulence model more accurate and reliable for a wider class of flows than the standardk-wmodel.

The turbulence kinetic energykand the specific dissipation rateware obtained from the following transport equations:

The governing equations for the turbulent flow

and

where

1.3 Computational domain and boundary conditions

The plate length is Lp, the undisturbed inlet flow velocity isU∞, and the Reynolds number is defined as Re=U∞Lp/ν, andνis the kinematic viscosity.

The computational domain is a rectangle of 4Lpin length with the inlet boundary located 0.5Lpupstream of the leading edge of the plate, the outlet boundary2.5Lpdownstream of the trailing edge and the top boundary0.5Lpaway from the plate. The structured grids are generated to discretize the flow domain. The grid node distribution in the longitudinal(x)direction follows an uneven distribution along the plate where the leading edge of the plate is refined and a one-sided geometric law is applied upstream and downstream of the plate. In the normal(y)direction, a one-sided geometric law is applied. On the plate, the number of grid nodes in the longitudinal direction, Nx=2800, and that in the normal direction,Ny= 200, 89% of that in the longitudinal direction.

The medium is water with the density of 998.2 kg·m-3and the fluid viscosity of 0.001003 kg·m-1·s-1. The fluid control and turbulence equations are solved numerically using the finite volume method, which is realized by the Ansys Fluent code. The inlet boundary is set as the velocity inlet, the velocity varies according to the Reynolds number, the outlet boundary is set as the pressure outlet, the top field boundary is set as the symmetry plane. The plate is a wall assumed to be hydrodynamically smooth. The governing equations and the turbulence model are discretized by the finite volume method with a second order upwind scheme and the pressure-velocity coupling is realized by the SIMPLEC method.

2. Results and discussions

2.1 Influence of inlet turbulence kinetic energy

In practical turbulent flows, the transition from laminar to turbulent flows may be promoted by the turbulence kinetic energy of the undisturbed flow,k∞. Experimental data are available for the flat plate flow field with different levels of inlet turbulence kinetic energies according to the ERCOFTAC Classic Database.

In the transition region from laminar to turbulent flows, there is a bouncing of the local skin friction coefficient,Cf, along the plate. The local skin friction coefficient,Cf, is defined as

Figure 1 presents Cfalong the flat plate for the calculations atlg(Re)=6.25. The plot includes the experimental results, the numerical predictions with different inlet turbulence kinetic energies (k∞=and the Blasius formula for laminar flow,

Fig.1 Local friction coefficient,Cf, along the flat plate at lg(Re)=6.25

From Fig.1, it is found that:

(1) both the experimental and CFD results agree well with the Blasisus formula in the laminar region,

(2) the numerical simulations for all different inlet turbulence kinetic energies produce a smallerRexthan where such departure appears in the experiment with k∞=at which transition occurs,Rex= U∞x/ν, andxis the distance to the leading edge of the plate,

(3) when the inlet turbulence kinetic energy increases, the transition occurs for lowerRex,

(4) with the increase of the inlet turbulence kinetic energy, the amplitudes and the range of the local skin friction in the transition region are reduced,

(5) the value ofCfis nearly independent of the inlet turbulence kinetic energy for high Rex.

It must be noticed that the model test is carried out in a relatively fully developed turbulent condition via the the turbulence stimulator, and it is appropriate to set a higher inlet turbulence kinetic energy for an earlier transition at lower Reynolds numbers to obtain the flat plate friction resistance.

Fig.2 Local friction coefficient,Cf, along the flat plate at lg(Re)=9

2.2 Influence of y+at high Reynolds number

Under the high Reynolds number flow condition, the boundary layer becomes much thinner compared to the low Reynolds number flow, thus it is not appropriate to place certain cells in the viscous sublayer because too large cell aspect ratios to the wall may lead to numerical problems, and the wall function approach is a good choice to resolve the problem. The wall functions are a set of semi-empirical formulae and functions that in effect “bridge” or “link” the solution variables at the near-wall cells and the corresponding quantities on the wall. The nondimensional spacing, y+, is defined by

where∆yis the distance of the first grid node to the wall,uτis the friction velocity, and uτ=τwis the shear stress at the wall.

The wall function works well when the first grid node is located in the logarithmic boundary layer, 11.225≤y+≤300.

In order to study the influence of the nondimensional spacing,y+, on the flat plate friction resistance, 4 sets of grids are designed to approach the target of y+=50,y+=100,y+=200,y+=300. Figure 2 is the comparison of the local friction coefficient along the plate between theexperimentaldata[9]and the results of Karman-Schoenherr equation,

where

From Fig.2, it is found that:

(1) the numerical results agree well with the results of the Karman-Schoenherr equation and the experimental data, indicating that the wall function approach is capable of simulating high Reynolds number flows equivalent to the full-scale ship Reynolds number,

(2) the local friction coefficient along the flat plate is nearly independent of y+when the first grid node is located in the logarithmic boundary layer, in which the wall function works well, indicating that the wall function approach is not sensitive to the grid distribution in calculating the friction resistance.

2.3 Numerical uncertainty analysis

In the numerical simulation of the two-dimensional flat plate flow field using the RANS method, the uncertainty analysis must be carried out for the solution and the computational grid. The RANS method is a steady simulation method, thus the simulation uncertaintyUSNis composed of the grid uncertainty UGand the iterative uncertainty

The assessment of UIis made by observing the oscillatory amplitude of the calculated value with time Zhang et al.[10], the iterative uncertainty is smaller than the grid uncertainty by 2 orders of magnitude, than the grid uncertainty by 2 orders of magnitude, and can be neglected. Thus, the numerical simulation uncertainty,USN, is equal to the grid uncertainty,UG.

The uncertainty analysis is carried out for the flat plate friction resistance coefficient. Many studies made CFD uncertainty analyses[11-14]. Based on the recommended Procedure 7.5-03-01-01 from the International Towing Tank Conference, three sets of computational grids with different grid scales (i.e., fine, medium, and coarse grids) could be built for the numerical simulation of the flat plate flow field. The refinement ratio,rG, is2in each direction of the coordinate. The cell numbers of the grids are 560 000, 281 160 and 140 000.

Table1 The flat plate friction coefficient for three grids

Table1 shows the calculated flat plate friction coefficients for the three grids from lg(Re) =6.25to lg(Re)=9.5, and the uncertainty analysis procedure is illustrated below.

The values calculated by the coarse, medium, fine grids are denoted as SGC,SGMand SGF, respectively. Grid changes(ε)in the coarse-to-medium and medium-to-fine grids at every Reynolds number are defined by

The grid convergence ratio,PG, is defined by

The order of accuracy,PG, is defined by

The correction factor,CG, is defined by

The grid uncertainty at every Reynolds number, UG, is defined by

The parameters used for the uncertainty verification of the flat plate friction coefficient at every Reynolds number are shown in Table 2. The grid convergence ratio,RG, for every Reynolds number is less than 1, indicating that the computational grids ensure a monotonic convergence.

Table2 The flat plate friction coefficient for three grids

2.4 Curve fit of the numerical data

In order to obtain a numerical two-dimensional flat plate line, a curve fit must be performed to the numerical data for the 14 Reynolds numbers. Here, the cubic polynomial in the logarithmic scales and a formula like that of the ITTC-1957 are tested.

The parameters,a0,a1,a2,a3,b1,b2,b3, in the two equations are estimated using a weighed least squares method, and they are listed in Table 3.

Table3 The parameters in Eqs.(14) and (15)

There are many analytical equations proposed for the friction resistance coefficient of a flat plate as a function of the Reynolds number. In the present study, 5 alternative proposals from the open literature are selected for comparison of the numerical two-dimensional flat plate friction lines.

Fig.3 Comparison of the numerical friction lines and proposals from the open literature

The first is the ITTC-1957 correction line, the second is the line derived by Katsui et al.[5], the third is the line proposed by Grigson[4], the fourth is the Schoenherr line, which is very popular in the ATTC, and the last is recently proposed by Eca et al.[8], which is also a numerical friction line obtained via the RANS method coupled with the SST k-wturbulence model.

Figure 3 shows the comparison of the numerical two-dimensional flat plate friction lines and proposals from the open literature, part (b) is the zoom-in in the case when lg(Re)≤7.75.

From Fig.3, we found that:

(1) at high Reynolds numbers, there is an excellent agreement between the numerical line and the proposals of ITTC-1957, Katsui, Schoenherr and Eca, while the line derived by Grigson is slightly higher than the others.

(2) at low Reynolds numbers, the differences between these lines become much larger. Obviously, the ITTC-1957 line has the largest slope and makes the values of CFlarger than those of the others because it is not a merely turbulent flat plate friction line, and it is modified for the purpose of the resistance extrapolation from the model scale to the full scale. Eca’s line has the least slope and gives the lowest value of CFbecause there is a laminar flow region at the edge of the plate in the numerical simulation. The numerical result is very close to the Grigson line when lg(Re)<7.

(3) the line derived by Grigson has a discontinuity of slope atlg(Re)=7, while the others are continuous.

3. Application of the numerical friction line

In the towing-tank test, the form factor is determined by the Prohaska’s method, measuring the total resistance at low speed, where the wave resistance contribution vanishes or can be estimated. This method might be quite inaccurate. Compared with the experimental procedure, the RANS method is more straightforward for determining the form factor: the deformation of the water surface is simply not taken into account, and a “double-body flow” computation is done. The scale effect on the form factor using different friction lines can be simply determined if making such computations in both the model scale and the full scale.

3.1 DTMB5415

The DTMB 5415 is first tested for determining whether the numerical friction line is suitable for the model-ship extrapolation method. It was conceived as a preliminary design for a navy surface combatant. The hull geometry includes both a sonar dome and a transom stern. The propulsion is provided by twinopen-water propellers driven by shafts supported by struts. Since 1996, the DTMB 5415 model has been adopted by the ITTC as a recommended benchmark for the CFD validation for resistance and propulsion[15,16]. The hull’s geometry is shown in Fig.4.

Fig.4 Geometry of DTMB 5415

The model speed is 2.097 m/s, and the Reynolds number based on the model length and the model speed is 1.19× 107and the Froude number is Fr= 0.28. The full scale ship speed is 10.288 m/s, and the Reynolds number is 1.45× 109. This is one of test cases in the Gothenburg 2010 Workshop on Numerical Ship Hydrodynamics.

In the “double-body flow” computation at the model scale, the near-wall spacing is set to ensure the nondimensional spacing being at a target of y+=100 at the middle of the hull. The computational domain has 300×60×90nodes (1.62× 106cells) in the longitudinal, wall-normal and girthwise directions. In the full scale, the near-wall spacing is set to ensure the nondimensional spacing being at a target ofy+=300 at the middle of the hull. The computational domain has 750×200× 140nodes (2.1× 107cells) in the longitudinal, wall-normal and girthwise directions. All computational grids are constructed of the HO topology, refined longitudinally towards the bow and the stern to resolve the large velocity gradients. The computational grids are shown in Fig.5.

Fig.5 Computational grids of DTMB 5415

The inlet boundary is set as the velocity inlet, the outlet boundary is set as the pressure outlet, and the top field boundary is set as a symmetry plane. The ship‘s wall is assumed to be hydrodynamically smooth. The governing equations and the turbulence model are discretized by the finite volume method with a second order upwind scheme and the pressure-velocity coupling is achieved by the SIMPLEC method. The full scale computation is carried out in parallel processing in 32 cores (Intel Xeon E5-2670, 2.6 GHz), realized in the Dawning TC4600 high performance computer.

The computed resistance components in the model scale and full scale ships are summarized in Table 4, where,CVis the viscous resistance coefficient,CFis the frictional resistance coefficient and CVPis the viscous pressure resistance coefficient.

In the form factor approach, the viscous resistance coefficient of a full scale ship is assumed to be proportional to the friction coefficient of a flat plat. In the so-called ITTC-1978 method, the ITTC-1957 model-ship correlation line is chosen for the plate friction line. However, it must be noticed that the ITTC-1957 model-ship correlation line is not a real friction line, it is a modified one for better prediction of the ship friction resistance if no form factor method is used.

Evidently, the form factors depend on the plate friction coefficients assumed. In the last section, Fig.4 shows the numerical two-dimensional flat plate friction lines and other proposals from the open literature. Although the difference in the Re-dependence is limited but it is sufficient for a very different performance predictions. The form factors for model scale and full scale ships based on these lines are listed in Table 5, where, the ratio is the full scale ship form factor to the model scale ship form factor.

From Table 5, the following observations can be made.

(1) The form form factor(1+k )increases substantially from the model scale to the full scale ships if the ITTC-1957 line is chosen. For the DTMB 5415, an extrapolation using a fixed form factor would underestimate the full scale viscous resistance by 4.86%.

(2) The form factor tends to be constant if the numerical line is chosen.

Table5 Form factors of DTMB5415 for model scale and full scale ships based on different friction lines

(3)The proportionality between the ship viscous resistance and the flat plate resistance is well reproduced if one of the modern friction lines is used (Katsui line and Grigson line).

3.2 KVLCC 2 tanker

The KVLCC2 is the second test case and is the second variant of the MOERI tanker with more U-shaped stern frame-lines. The local flow characteristics around the VLCC hull were extensively studied experimentally and the results were documented in detail and used as the validation resource for numerical solutions[17]. The hull’s geometry is shown in Fig.6.

Fig.6 Geometry of KVLCC2

The model speed is 1.047 m/s, the Reynolds number based on the model length and the model speed is 4.6× 106and the Froude number is Fr= 0.142. The full scale KVLCC2 speed is 7.973744 m/s, and the Reynolds number is 2.54× 109. This is also one of test cases in the Gothenburg 2010 Workshop on Numerical Ship Hydrodynamics.

Fig.7 Computational grid of KVLCC2

In the model scale, the computational domain is meshed with 400×60× 150nodes (3.6× 106cells) in the longitudinal, wall-normal and girthwise directions. While in the full scale, it is meshed with800×200× 300 nodes (4.8× 107cells) in the longitudinal, wallnormal and girthwise directions. All computational grids are constructed in the HO topology, refined longitudinally towards the bow and the stern to resolve the large velocity gradients. The computational grid is shown in Fig.7.

The boundary condition is the same as the DTMB5415.The full scale computation is carried out in parallel processing in 64 cores (Intel Xeon E5-2670, 2.6 GHz) in the Dawning TC4600 high performance computer.

Table6 Computed resistance components of KVLCC2 for model scale and full scale ships

The computed resistance components of the KVLCC2 for the model scale and full scale ships are summarized in Table 6, which shows that the viscous resistance coefficient decreases by a factor of 0.4276 from the model scale to full scale ships.

The form factors of the KVLCC2 for the model scale and full scale ships based on different friction lines are listed in Table 7.

Table7 shows that the Schoenherr line gives the best prediction of the full scale ship viscous resistance if a constant form factor is used, the numerical line and the Katsui line give a fairly constant form factor（-1.2%， -2.9%）， although not as precise as that of the DTMB5415, while the ITTC-1957 line and the Grigson line yield a significant increase (3.5%, 4.5%) of the from factor1+k from the model scale to the full scale. This phenomenon is encouraging.

3.3 Darpa suboff

The Suboff model was built by the David Taylor Research Center as a recommended benchmark for the CFD validation for the flow field and the resistance[18]. It has an overall length of 4.356 m. The hull is composed of a fore-body of 1.016 m in length, a parallel mid-body of 2.229 m in length, an aft-body of 1.111 m in length, and an end cap of 0.095 m in length.The scale ratio is 24.

Table7 Form factors of KVLCC2 in model scale and full scale based on different friction lines

The model speed is 3.0452 m/s, and the Reynolds number based on the model length and the model speed is 1.32× 107. The full scale Suboff speed is 14.9186 m/s, and the Reynolds number is1.55× 109.

For the bare hull in the model scale, the computational domain is meshed with350×60×50nodes (1.05× 106cells) in the longitudinal, wall-normal and girthwise directions. While for the bare hull in the full scale, it is meshed with700×200× 100nodes (1.4× 107cells) in the longitudinal, wall-normal and girthwise directions. For the hull with full appendages in the model scale, the computational domain is meshed with480×70× 100nodes (3.6× 106cells) in the longitudinal, wall-normal and girthwise directions. While for the hull with full appendages in the full scale, it is meshed with850×200×200nodes (3.2× 107cells) in the longitudinal, wall-normal and girthwise directions. All the computational grids are constructed of the HO topology, refined longitudinally towards the bow, the stern and the appendages to resolve the large velocity gradients. The computational grids are shown in Figs.8 and 9.

Fig.8 Computational grids of Suboff bare hull

Fig.9 Computational grids of Suboff with full appendages

The boundary condition is the same as the DTMB5415. The full scale computation is carried out in parallel processing in 32 cores (Intel Xeon E5-2670, 2.6 GHz) in the Dawning TC4600 high performance computer.

The computed resistance components of the Suboff bare hull and those with full appendages for model scale and full scale ships are summarized in Table 8.

Table8 shows the numerical results of viscous resistance coefficients, which agree quite well with the experimental ones for both the bare hull and those with full appendages. For the bare hull, the viscous resistance coefficient decreases by a factor of 0.5276 from the model scale to the full scale ships; while the factor is 0.5591 for the full appendage Suboff.

The form factors of the Suboff bare hull and those with full appendages for model scale and full scale ships based on different friction lines are listed in Table 9.

Table9 shows that for the bare hull, the numerical line gives almost constant form factor （-0.54%），the Katsui and Grigson lines give a fairly constant form factor （-1.12% and -1.17%， respectively）. Again, the ITTC-1957 line will yield a significant increase (4.04%) of the form factor(1+k )from the model scale to the full scale. While for the Suboff with full appendages, the form factor increases substantially from the model scale to the full scale ships for all friction lines, the ITTC-1957 line suffers a great scale effect. The assumption of a constant form factor seems not reasonable for the hull with appendages, which may be due to the different local Reynolds number of the appendages in the full scale. Anyway, the extrapolation method from the model scale to the full scale ships with appendages needs to be studied furthers.

4. Conclusions

The present work is devoted to a numerical plate friction line via the RANS method with the SST k-w turbulence model. The inlet turbulence kinetic energy and numerical uncertainty analyses are conducted. Comparing with the experimental data, the finest grid and the appropriate inlet turbulence kinet ic energy a re selected to compute the flat plate frictionresistanceat14 Reynolds numbers. Two numerical friction lines are obtained using the least squares root fitting method, from one similar to that of the ITTC-1957 line and cubic polynomials in logarithmic scales, and are compared to the friction line proposals available in the open literature.

Table8 Computed resistance components of Suboff for model scale and full scale ships

Table9 Form factors of Suboff for model scale and full scale ships based on different friction lines

The form factor of DTMB5415, KVLCC2, SUBOFF are compared between the model scale and the full scale ships using different friction lines. It is shown that the numerical line sees little scale effect from the model scale to the full scale ships, while the assumption of a constant form factor using the ITTC-1957 line is not reasonable. It is suggested that when computing a form factor numerically, it is best to use a numerical friction line. Several more ships need to be studied to validate the numerical friction line for a fundamental support to the form factor concept.

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* Project supported by the National Natural Science Foundation of China (Grant No. 51179198, 51479207), the High Technology Marine Scientific Research Project of Ministry of Industry and Information Technology of China (Grant No. [2012]534).

Biography: WANG Zhan-zhi (1986- ), Male, Ph. D., Lecturer

Correspondindg author: XIONG Ying,

E-mail: ying_xiong28@126.com