Kang Renand Shili Sun

1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Free Surface Flow Generated by Submerged Twin-cylinders in Forward Motion Using a Fully Nonlinear Method

Kang Ren1and Shili Sun2*

1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

The free surface flow generated by twin-cylinders in forced motion submerged beneath the free surface is studied based on the boundary element method. Two relative locations, namely, horizontal and vertical, are examined for the twin cylinders. In both cases, the twin cylinders are starting from rest and ultimately moving with the same constant speed through an accelerating process. Assuming that the fluid is inviscid and incompressible and the flow to be irrotational, the integral Laplace equation can be discretized based on the boundary element method. Fully-nonlinear boundary conditions are satisfied on the unknown free surface and the moving body surface. The free surface is traced by a Lagrangian technique. Regriding and remeshing are applied, which is crucial to quality of the numerical results. Single circular cylinder and elliptical cylinder are calculated by linear method and fully nonlinear method for accuracy checking and then fully nonlinear method is conducted on the twin cylinder cases, respectively. The generated wave elevation and the resultant force are analysed to discuss the influence of the gap between the two cylinders as well as the water depth. It is found that no matter the kind of distribution, when the moving cylinders are close to each other, they suffer hydrodynamic force with large absolute value in the direction of motion. The trend of force varying with the increase of gap can be clearly seen from numerical analysis. The vertically distributed twin cylinders seem to attract with each other while the horizontally distributed twin cylinders are opposite when they are close to each other.

free surface flow; submerged twin cylinders; fully nonlinear method; forced steady motion; boundary element method

1 Introduction1

The two-dimensional surface flow generated by twin cylinders with a radius submerged at a certain depth starting from rest and ultimately moving with the same uniform speed U is considered in present work. The hydrodynamic forces and wave profile are the main points we examine in the numerical simulation. The sketch of this problem can beseen in Fig. 1. The gap between the centers of the two cylinders is defined as L and the submergence h is defined as the vertical distance between the undisturbed free surface and the center of the cylinder. The dimensionless parameter a/h can be used to represent the submergence.

The surface flow generated by submerged body has been widely studied. However, the existence of the free surface adds the difficulty of this problem and the analytical solution can only be obtained for some basic and simple 2D or 3D shapes. For example, the motion of the circular cylinder was studied in Havelock (1928; 1936; 1949a; 1949b) for two-dimensional flow and spheroid in Havelock (1931) for three-dimensional flow. Apart from above, numerical simulation has been conducted in corresponding research. Haussling and Coleman (1977, 1979) investigated the surface waves generated by a submerged circular cylinder by finite-difference method with boundary-fixed coordinate system. Campana et al. (1990) adopted an iteration procedure for the similar problem, and the iteration method is also used in Scullen and Tuck (1995). Actually, many engineering problems related to ships and other marine structures can be simplified by modeling as twin cylinders or multi-cylinders. For instance, the study of wave making interactions between submerged bodies and the accompanying drag reducing research. What is more, when two submerged structures in motion are close to each other, the possibility of collision might exist and bring disasters to lives and goods. The related researches about the twin cylinders based on potential theory are Koo et al. (2004) and Wang et al. (2011). Koo et al. (2004) once studied the interaction between two submerged cylinders fixed in water interacting with propagating incident wave by modeling a fully nonlinear wave tank. Wang et al. (2011) studied the resonant phenomenon between two half-floating cylinders by finite element method. The present work is about cylinders moving in initially rest water and it also can be equally seen as interaction between twin fixed cylinders with a steady current, which is different from the works mentioned above.

In present work, the single cylinder case is calculated and compared with the existing analytical or numerical results in order to check the accuracy of the numerical procedure and the codes. In detail, the linear code is running for a singlecircular cylinder case and the numerical solution is compared with the analytical solution by Havelock (1928; 1936). Furthermore, the fully-nonlinear code is running for a single elliptical cylinder case to compare with the numerical solution obtained by Campana et al. (1990) as well as the experimental data from Maruo and Ogiwara (1985). Once the checking finished, fully-nonlinear numerical simulations are conducted on twin-cylinder cases, which are horizontally distributed or vertically distributed and the free surface and the consultant force are fully analysed, respectively. The study of horizontally-distributed cylinders mainly focuses on the wave making resistance and the wave making interaction of the two cylinders with different gaps, which is meaningful for reducing the resistance. As for vertically-distributed cylinders, the study for the vertical force on them and the attraction between them can be useful to avoid collision.

Fig.1 Sketch of the twin circular cylinders

2 Mathematical formulation and numerical procedure

2.1 Initial-boundary-value problem

The fluid is inviscid and incompressible and the flow to be irrotational is assumed in this study and the velocity potential φ is introduced for ideal fluid whose gradient is the velocity, orU=▽φ. The fluid domain satisfies the Laplace equation

The fully-nonlinear free surface boundary conditions consist of the kinematic free surface condition

as well as the dynamic free surface condition

where g is the acceleration due to gravity. On the moving body boundary, the boundary condition can be written as

where n is the normal vector with positive direction pointing out of the fluid domain. While on the far field boundary, the condition is

It is important to realize Eq. (5) for t＜∞. When t＜∞, although there may be no wave far ahead of the body, wave will exist behind the body (Wehausen and Laitone, 1960). This radiation condition was also used in Haussling and Coleman (1979).

Once the nonlinear terms are removed, the linear free surface conditions are obtained, which are satisfied on the undisturbed free surface

Assuming the fluid is initially calm and the initial free surface condition is

where0ηand0φare the initial values of the elevation and the velocity potential on the free surface.

2.2 The discretization of the boundary integral equation

The Laplace equation in the fluid domain can be converted into an integral equation over its boundary through Green’s identity,

where p and q are source point and filed point and rpqis the distance between p and q, respectively. The two-dimensional boundary element method is used to deal with the integral equation. The whole boundary is discretized into N elements and in each element the linear shape functions are adopted to represent distribution of physical values on the element. The integral equation can ultimately be transformed to a system of linear equation,

More details about such as the definition of matrix H and G and the method for computing the influential coefficients can be found in Brebbia (1982) and Lu et al. (2000). In this system of linear equations, N unknown variables are require to be found. For a certain time step, the potential normal derivations on the body surface, far field control surface） as well as the velocity potentials on the free surface（φF） are known. The unknown terms consist of the velocity potentials on the body surface（φB）and far fieldcontrol surface（φC） and the potential normal derivations on the free surface）as well. Moving the unknown terms to the left hand side and the known terms to right, the equation becomes

Once the system of above linear equations was solved within each time step, the velocity potential on the body surface and the potential normal derivation on the free surface would be found. The former will be used to calculate the dynamic pressure while the latter to update the position of the nodes on the free surface.

2.3 Time-stepping methods

For the linear method, the computation domain is fixed and the nodal points will move with the body, the Eulerian framework is used to update the wave elevationη. We have

For the fully-nonlinear method, we adopt the Lagrangian framework to update the location x of the nodal points as well as the value of the velocity potential on the nodal points

The single step method is used to update the nodal information

It is worth mentioning that even though a first-order scheme is used here for updating, good results can be obtained as long as a suitable and relative smaller time interval is chosen. Through convergence study in section 4.1, the possibility of numerical diffusion caused by lower-order scheme can be excluded.

2.4 Hydrodynamic pressure and force

Hydrodynamic pressure on the body surface can be obtained from Bernoulli equation：

whereρis the density of fluid.tφcan be obtained by using the equation below：

Then the equation for hydrodynamic pressure can be written as

In addition, the termtφ in Eq. (19) can also be obtained by using the auxiliary function technique (Wu and Taylor, 1996, 2003), the details can be found in Wu and Hu (2005) and Sun & Wu (2013, 2014). The hydrodynamic force can be obtained through integrating the pressure along the body surface.

The resistance R on the cylinder is the horizontal force in Eq. (22) with a negative sign and its positive vector is along the negative x-axis. The vertical force on the cylinder is the vertical component of the wave hydrodynamic force and its direction is along the positive z-axis.

2.5 Numerical procedure

The initial-boundary-value problem is solved by the boundary element method in each time step, while the time stepping method is used to update the free surface based on Euler or Lagrangian framework. In addition, regriding and smoothing procedure are also used to avoid the numerical instabilities. The size of elements on the free surface may become irregular due to the updating procedure. Through re-discretizing of the free surface every few time steps, a series of new nodes on the free surface are re-set and the nodal values can be found by a cubic spline interpolation algorithm (Sun, 2007). In order to refrain from the saw-tooth behavior appearing on the free surface during the time domain simulation process, a five-point smoothing algorithm in Moruo and Song (1994) and Sun (2007) is adopted to eliminate such instability phenomenon in present work every several time steps.

3 Numerical comparison for accuracy checking

3.1 Single circular cylinder case for comparison

For the two-dimensional case of single submerged circular cylinder, Havelock (1928) investigated the surface elevations generated by a doublet in a uniform stream by giving analytical expressions of the steady surface elevation Eqs. (1) and (2) in Havelock (1928). Later，Havelock (1936) obtained the exact analytical solution for steady resistance (Eq. (29) in Havelock, 1936). Analytical solution in terms of the wave elevation and wave resistance can be obtained by the usage of Mathematica according to the analytical expressions. Meanwhile, numerical simulation by linear version of the codes is conducted. The comparison between the two results can be used to check the accuracy of our codes. The checking case adopted in Campana et al. (1990) is used in this study, that is, h/2a=1.8,. Inorder to avoid the disturbance brought by the instant starting, the body starts with a constant acceleration in order to make the starting process more smoothly. The speed U is set as

The comparison between the numerical solution and analytical steady solution of wave elevation and resistance can be seen in Fig. 2. The time history of the resistance is an oscillating curve which decays with time (Havelock, 1949a). The solid line in Fig. 2(a) is the numerical time domain solution of the resistance, while the dashed dot line is the analytical solution obtained by Havelock (1936). As can be seen from Fig. 2(a), the analytical solution is coincident with the average resistance of the time domain solution. The moments corresponding to the mean resistance in Fig. 2(a), was selected and then compare the wave elevation profile at that moment with the analytical steady solution (See Fig. 2(b)), and a good agreement can be also obtained.

Fig. 2 Comparison with the analytical steady solution

3.2 Single elliptical cylinder case for comparison

For the two-dimensional case of single submerged elliptical cylinder, Campana et al. (1990) studied it by an iterative nonlinear method. Through iteration, an approximate steady wave profile can be obtained. They compare their numerical results with the experimental results (Maruo and Ogiwara, 1985) and a good agreement was achieved. The fully nonlinear codes for the same case used in Campana et al. (1990) to obtain the time domain solution was used in this study. The case calculated in this study was a = 4b, h = 1.7a,where a and b are the semi-axes of the elliptical cylinder. LA moment corresponding to mean resistance was selected to see the wave profile at that moment as an approximate steady state and to compare the wave file around the body with the existed ones. Free surface profile is set in the relative position to the moving cylinder. (See Fig. 3)

Fig. 3 Comparison of the wave profile around the submerged cylinder

4 Twin circular cylinders

4.1 Convergence study

The convergence study is based on both time interval and element size on the free surface. According to the linear analysis (Havelock, 1949a; Liu & Yue, 1996), the time period and the wavelength of the steady wave are T = 8π U/g and λ= 2πU2/g, respectively. Four grid systems are chosen to make the convergence study, whose detailed information is shown in Table 1.

Table 1 Information of the grid systems

The case for the convergence study is set as twin circular cylinders which are horizontally distributed in the same depth below the free surface where a/h=0.1, L/a=3 and,which corresponds to= U2/gh = 0.3183. In order to make the starting process more smoothly from the rest to the ultimate uniform speed, a constant acceleration is used for accelerating. In the convergence study it can be seen from the calculations that one fifth of the period’s time is used for accelerating. In this case, the acceleration ac is set as ac=0.1989g. The Mesh 1 is set as the basic grid. Mesh 2 has a smaller element size and Mesh3 has a smaller time interval than the basic grid. In addition, a denser grid in both time interval and element size is set as Mesh4. The time domain solutions of the components ofhydrodynamic force and the wave profile around the submerged body at certain moments are shown in Fig. 4 and Fig. 5 in terms of the different grid systems, respectively. In Fig. 4, the horizontal axis ξ/d is the dimensionless parameter in terms of time, where ξ is the horizontal displacement the cylinders had passed from starting to a certain moment while d is the diameter of the cylinder. The time domain solutions corresponds to different grid systems are displayed and good agreement has been achieved. As for Fig. 5, the coordinates are all non-dimensionlized by the diameter d and the wave profile, especially in the area above the cylinders where uniform elements are used, achieved a good coincidence.

Fig. 4 Time domain solution of the forces in terms of different gird systems

Fig. 5 Wave profile around the submerged bodies

4.2 Horizontally-distributed twin circular cylinders

Two identical circular cylinders that are horizontally distributed beneath the same depth of the free surface suffering forced motion are discussed in this section. A non-dimensional parameter is introduced as L/a in order to indicate the horizontal gap between the two centers. In order to find out the impact of the horizontal gap on the free surface profile and the wave resistances on both two cylinders, in the following work, a series of L/a are chosen in the simulation corresponding to the fixed speed. The analysis focuses on the variety of hydrodynamic force on each cylinder and on the twin cylinders as a whole system against different gaps. The Froude number is defined as, thus we have

The first case we examined is a/ h= 0.1 and what we concern about is the interaction between the twin cylinders. In the numerical simulation, the Froude number is set as

The mean value of the force can be calculated by averaging the peak values of the corresponding time history curve. The mean resistances on the twin cylinders are shown in Table 2 and non-dimensionlized by the unit force πρga2.

Table 2 The mean resistances on the twin cylinders

Fig. 6 The mean resistance on each cylinder against gap (a/h=0.1 & Fn2=0.3183)

Fig. 7 The joint mean resistance on twin cylinders against gap (a/h=0.1 & Fn2=0.3183)

The interaction between the horizontally distributed twin cylinders can be divided into two parts according to the value of gap. For the first part, the gap between the twin cylinders is not large enough and the front cylinder is still located in the influential area of the rear one. Since the wave generated by the rear cylinder is located not far behind the front cylinder, the superposition of the wave trains can still pose an obvious impact on the wave force on the front cylinder. Once the gap reaches or exceeds a critical value, the influence of the rear cylinder’s flow field on the front one can be ignored. It is noticeable that for the rear cylinder, once the downstream wave trains generated by the front moving cylinder transfer to its near field, the influence from the front cylinder starts to immerge and will last from then on. When the gap gets large enough, the resistance on the front cylinder is coincident with single cylinder case, which can be seen in Table 2.

The wave-making disturbance between the twin cylinders is of vital importance when considering the wave making resistance. In addition to the last case corresponding to λ=10d, another case that corresponds to a smaller ratio of linear steady wavelength against body’s scale is also considered. The parameters are set to a/h=0.2, λ=2πU2/g=3d=6a, which corresponds to. By comparing the cases corresponding to different linear wavelength against body’s scale, it can be seen that the trend of the curve in Fig. 6 and Fig. 8 seems alike. When the L/a value are not large enough, the front cylinder suffers a negative resistance, which means that the horizontal force is along the positive direction of x-axis. However, when L/a reaches and exceeds a relative large value, the resistance on the front cylinder turns to be positive one. As for the rear cylinder, it is always suffering positive resistance in the above simulations.

Another point of special interest is the mean joint resistance on the twin cylinders. The wave-making disturbance of the twin cylinders not only affects the resistance on each cylinder, but also exerts a significant impact on the whole system that sees the twin cylinder as integration. The joint mean resistance against the gap between twin cylinders is shown in Fig. 7. It witnesses a downward trend when the L/a value is small before starting to increase after a critical gap (roughly bigger than a half linear wavelength).

Table 3 The mean resistances on the twin cylinders (a/h=0.2 & Fn2=0.1910)

Fig. 8 The mean resistance on each cylinder against gap (a/h=0.2 & Fn2=0.1910)

The minimal value is present even though it is hard to be exactly sure by numerical simulation. In the cases examined, the joint resistances were all positive. However, since the joint mean resistance has the minimal value, optimizing the gap for reducing the wave making resistance becomes possible.

Snapshots of the wave elevation for horizontally-distributed twin cylinders are displayed for different parameters L/a at the same moment in Fig. 9. The moment corresponding to ξ/d=28.8 was selected. It can be seen that the wave distortion is obvious, which reflects the wave disturbance of the twin cylinders.

Fig. 9 Snapshots of the wave elevation for horizontallydistributed twin cylinders(a/h=0.2, Fn2=0.1910, ac=0.1989g)

4.3 Vertically-distributed twin circular cylinders

Another problem considered in this study is the horizontal forced motion of the twin circular cylinders, which are vertically distributed beneath the free surface. The sketch of this problem can be seen in the Fig.1(b), where the vertical distance between the center of the upper cylinder and the undisturbed free surface is defined as h and the vertical gap between the two centers of cylinders is also defined as L. The case where a/h=0.1 and Fn2=0.3183 is still chosen for this problem. The numerical results are displayed in Table 4 and the mean value of different components of hydrodynamic force against the parameter L/a can be seen in Fig. 10.

Table 4 The mean resistances on the twin cylinders (a/h=0.1 & Fn2=0.3183)

Fig. 10 The mean hydrodynamic forces on the twin cylinders against different L/a (a/h=0.1 & Fn2=0.3183)

The vertical force is the main point discussed when the vertically-distributed twin cylinders are investigated. For one thing, the twin cylinder’s relative location determines that the main interaction happens in the vertical direction. It can also be seen in Fig. 10 that when the two cylinders are close to each other the horizontal force is relatively small compared with the vertical force. As for the relationship between the force and gap between the twin cylinders, it is clearly shown that with the gap increasing, the vertical force on the upper cylinder is negative with the absolute value increasing. On the other hand, the vertical force on the lower cylinder remains positive, but with the absolute value decreasing.

With the vertical gap between the twin cylinders getting larger, say, h=100a, the lower cylinder is far from the free surface and can be seen as moving in the unbounded fluid domain and its existence also exerts little impact on the upper cylinder. So in this case, the time history of the force on the upper cylinder is coincident with the corresponding single-cylinder case, which can be seen in Fig. 11. Fzis used to refer to the vertical force.

Fig. 11 Comparison between the vertical force on the upper cylinder and on single cylinder

4.4 The effects of the submergence

With cylinders approaching to the free surface, the disturbance on the free surface becomes much more obvious than the case with deeper submergence and so as the nonlinear effects. For fully nonlinear simulation, once the free surface becomes steep enough and wave breaking happens then the time stepping procedure cannot carry on. As the wave breaking remains to be troublesome to tackle with, the cases without wave breaking are only taken into consideration. In order to minimize the steep, the wavelength should be set larger properly to counteract the growth of the wave elevation. This study considers a shallower submergence case, ie, h=2a. A larger velocity is set as, which corresponds to a relatively larger Froude number. Different with above simulation, here we adopt a larger constant acceleration ac= 0.7958g for accelerating in this case. The free surface profile at the moment corresponding to ξ/d=298 is seen in Fig. 12. However, the free surface distortion is noticeable compared with the cylinders’ scale.

Fig. 12 Free surface profile at ξ/d=298

To examine the effects of the submergence on the wave profile as well as the hydrodynamic force, different cases are calculated with same parameters of U，L and a, but different h. The free surface profile at the same moment and the time history of the vertical force for these cases are shown in Fig. 13 and Fig. 14, respectively. The case where L/a=3, a/h=0.1 andhas already been conducted in last section. The other two cases corresponding to a/h=0.12, 0.15 are simulated. Fig. 13 shows that the surface wave becomes steeper when the submergence gets smaller. This means a higher possibility of wave breaking. The vertical force on each cylinder is also displayed in Fig. 14. The cylinders suffer a force with a larger absolute value when they approach the free surface that can be seen in Fig. 14.

Fig. 13 Free surface profiles for three different submergences at ξ/d=96

Fig. 14 Vertical Force on cylinders for different submergence

5 Conclusions

A two dimensional fully nonlinear method based on the boundary element method is applied to study the free surface flow generated by the twin cylinders’ motion as well as the resultant wave forces.

1) For the horizontally distributed both twin cylinders suffer horizontal hydrodynamic forces with relative large absolute values when they are close to each other. Before the gap between them reaches a critical value, the resistance on the front cylinder is negative and with the increase of the gap its absolute value gets smaller. The rear cylinder always suffers a positive resistance and its value is also getting smaller with the increase of the gap. Once the gap exceeds to the critical value, the resistance on both twin cylinders becomes positive. This trend is also suitable for other cases with a different ratio of wavelength to body’s scale. The joint mean resistance on the twin cylinders in terms of different gaps is analysed and the minimal value is existed.

2) For the twin vertically distributed cylinders, the relationship between the vertical force on cylinders and the gap is similar with the horizontally distributed twin cylinders. However, the vertically distributed twin cylinders seem to attract with each other while the twin horizontally distributed cylinders are opposite. When two structures are vertically distributed with a small gap beneath the free surface, the large attraction between them might add the possibility of collision.

3) The impact of the submergence on surface wave profile and cylinders’ load conditions are also analysed. Vertically distributed twin cylinders with different submergence are simulated and compared. With submergence getting smaller, the free surface profile becomes steeper and the absolute value of the vertical force on each cylinder also becomes larger.

The conclusions obtained from studying two-dimensional cases to some extent can reflect and help understand the real three dimensional engineering problems that exist in ship motion and ocean structures, such as the possibility of reducing the wave resistance and enhancing the stealth ability of multiple ships or submarines. Lastly, the interaction analysis can also help avoid collisions in the navigation.

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1671-9433(2015)02-0146-10

10.1007/BF01559690

DOI： 10.1007/s11804-015-1300-3

Received date： 2014-10-21.

Accepted date： 2014-12-10.

Foundation item： Supported by the Lloyd's Register Foundation, the Fundamental Research Funds for the Central Universities (Grant No. HEUCF140115), the National Natural Science Foundation of China (11102048, 11302057)，the Research Funds for State Key Laboratory of Ocean Engineering in Shanghai Jiao Tong University (Grant No. 1310), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20132304120028).

*Corresponding author Email： shili_sun@163.com

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2015