不规则波和流下的重力式网箱水弹性响应研究

第22卷第3期 2018年3月
船舶力学
Journal of Ship Mechanics
Vol.22 No.3
Mar. 2018
Article ID：1007-7294(2018)03-0260-16
Hydroelasitic Analysis of the Gravity Cage Subjected to
Irregular Waves and Current
HU K e1,2, FU S h i-x ia o1,2
(1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;
2. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China)
Abstract: In this paper, the hydroelastic response of a gravity cage exposed to irregular waves and current was analyzed. A full scale net cage bulit by the FEM was introduced to further study the cag e's motion and deformation. In the numerical model, nonlinear spring elements and truss elements were used to simulate the mooring lines and the net, respectively. Furthermore, the 耶 buoyancy distribu­tion' method was adopted by simulating the floating collar with several coupled beams in order to calculate the instantaneous buoyancy force acting on the collar. On this basis, when the net cage is subjected to wave-current flows, the dynamic response of the floating collar, the modal contribution to the collar's deformation from each mode shape were carefully studied. The results show that more flexible modes will be aroused in the vertical direction when the significant wave height increases;
while the current will have a great contribution to the rigid-body motion in the horizontal direction.
Key words: gravity net cage; finite element analysis; irregular waves and current flow;
hydroelastic response
CLC number: O357 Document code: A doi: 10.3969/j.issn.1007-7294.2018.03.002
0 Introduction
The offshore environmental problem is becoming a crucial issue for human society and the development of the nearshore aquaculture industry. Moreover, the demand for more sea foods high in protein is pushing engineers to design cost-effective fish cages that can withstand ex­treme environmental loads in deeper ocean conditions. Therefore, accurate prediction of a cage 爷 s hydroelastic response has become a key focus in aquaculture engineering.
Previous investigations into sea loads on gravity net cages normally considered the im­pact of waves and currents separately.
First of all, the investigations into the wave loads exerted on the net cages focused on three main aspects:the loads on the net, the loads on the collar and the dynamic response of the fish
Received date：2017-12-07
Foundation item: Supportted by the National Natural Science Foundation of China (Grant No. 51279101, 51490674 and 51490675); National Basic Research Program of China (973 Program-2013CB036103); the High-Tech
Ship Research Projects of the Ministry of Industry and Information Technology (special topic: Mooring
positioning technology research of floating structures)
Biography：HU K e(1986-), male, Ph. D., student of Shanghai Jiao Tong University;
FU Shi-xiao(1976-), male, professor, corresponding author, E-m ail: shixiao.fu@.
第3期HU Ke et al: Hydroelasitic Analysis of the Gravity Cage 噎261
cage system.
Concerning wave loads on the net, Lader[1]compared the changes in the wave height and energy before and after the wave passed through the net. Song[2]successfully predicted wave loads of the net by calculating the cubic net cage?s hydrodynamic response based on sinu­soidal wave theory and the Morison equation, and he claimed that the relative error between the numerical prediction and test result was under 15%. Ito[3-4]simplified the wave condition into- forms of oscillating flow, under which the hydrodynamic forces on the net with different solidi­ty ratios and pretensioning forces were studied.
In order to conduct detailed research into the wave loads on the collar, Krassimi[5]calcu­lated the damping coefficient and added mass coefficient for the forced oscillating collar based on the potential flow theory. Kristiansen[6]later conducted a model test in a wave tank with a cylinder fixed on the free surface. He further investigated the nonlinearities in the wave forces on the collar caused by the influence of the free surface.
To quantify the response of the fish cage, Colbourne[7]conducted an experiment on multi­ple cages to compare the mooring forces under different kinds of wave loads. Fredriksson[8]and Fredriksson, et al[9-10]carried out a serious of experiments to investigate the mooring line forces and motion of the realistic fish cage. Besides, numerical simulations were also performed, and comparisons between experimental and simulated results indicated good agreements. By using the lumped mass point method and rigid body kinematics theory, Dong[11]and Xu[12]predicted the response of a net cage under irregular wave loads.
Secondly, in the case with current only (no waves), Aarnses[13]studied the drag force on the net cage, the changes in the cage?s volume and the reduction in current speed by towing a gravity cage model in calm water. Lader[14]conducted an experiment with a full-scale cage to specifically study the relationship between the current speed and cage 爷 s volume. Huang[15]and Zhao[16-17]studied the hydrodynamic response based on the 耶 lumped mass point method 爷.Huang found that the total force of the numerical model was lower than the experimental data when the Reynolds number was lower than the range of 1400-1800, while Zhao noticed that the volume of a net cage with diamond grids was larger than that with square grids. Berstad[18]cal­culated the mooring forces and the volume changes of a net cage by using finite element soft­ware (AquaSim). Moe[19]used ABAQUS to analyze the deformation of the net cage in currents with different speeds. Kristiansen[20]estimated the drag force and the volume changes of a net cage by replacing the twine5s drag and lift forces with each plane5s tangential forces and nor­mal forces. The numerical results were in good agreement with the experimental data.
Based on the previous study, the hydrodynamics of the fish cage under wave or current loads has been researched extensively. Because irregular waves and currents normally co-ex­ist in the real ocean environment, the dynamic response of the gravity net cage under the com­bined effects of irregular waves and currents needs to be further studied. Besides, because ge­ometric nonlinearity due to net cage?s large deformation under the wave-current loads is evi­dent, research on the full scale model should be conducted.
262船舶力学第22卷第3期
In this paper, a full-scale numerical model of a gravity net cage under irregular waves and currents was studied by using FEM. The irregular waves were simulated based on the JON- SWAP wave spectrum. On this basis, the dynamic response of the floating collar, the modal contribution from each mode shape to the collar in the combined wave-current flows together and the changes in the mooring-line tension were analyzed.
1 Basic theory
1.1 Equations of motion
When the whole gravity cage is exposed to irregular waves and current, the dynamic equi­librium equations of the structure can be expressed as:
[m][蓘]+ [c] [x] + [k] [x]=F(1)
e x t
F=G+f b+f w+f c(2) where 蓘m蓡is the mass matrix,蓘c蓡is the damping matrix and 蓘k蓡is the stiffness matrix of the
e x t
system. The total external loads on the structure represented by F include four parts, which are the gravity force G, the buoyancy force fb,the wave forces fw and and the current forces f c. In Eq.(1) and Eq.(2), the gravity force G can be calculated by the static buoyancy force. The buoyancy force fb is calculated according to the volume of water displaced by the structure, and will be described further in Chapter 3.
Both the wave forces f and the current forces f can be estimated by the modified Mori-
w c
son equation[21], where the velocity of the current and wave is superposed linearly, as shown in Eq.(3),
F=fw+fc =C d\p D\U-Vp±U\〈U-Vp±U+C M P：4D u-C m P：4D a p⑶where C m represents the inertia coefficient, CM=Cm+1, C d is the drag coefficient, D is the ef­fective diameter of the beam elements and the truss elements, u and u represent velocity and acceleration of water particles in the wave-only condition. U is the current velocity and p is the water density. In the dynamic analysis where the motion of the structure must be taken into ac­count, Vp and Up represent the velocity and acceleration of an element forming the structure. In this case, the influence of the mutual interference of the velocity field in the combined wave- current condition is not considered, see Lee[22]. Based on the equation, the dynamic reponse of the net cage will be studied under wave-current combined condition, and the results will be further compared against those in wave-only condition.
1.2 Description of irregular waves
Several linear waves with random phase angles can be combined to generate an irregular wave.
Firstly, the elevation of water surface in an irregular wave can be written as:
第3期HU Ke et al: Hydroelasitic Analysis of the Gravity Cage 噎263
浊(x, z, t )=移 A n sin (k nx -M nt+£n )
(4)
n=1while the horizontal and vertical velocities of a water particle can be expressed as:
U ( x, z , t 蔀=移 A n 棕 n cosh^ ^ sin ( k n x -棕 n t+着 n 蔀 (5)
n =1 sinh(k nd )
肄 T sinh (k n (z+d 蔀）W ( X , Z , t 蔀=^A
n 棕 n —• A t i +、 c o s ( k n X -棕 n t +着 n 蔀 (6)
n =1 sinh (k n d )where A can be further denoted as:n
A n =姨2S 浊(棕)驻棕 (7)In the four equations above, A n , k n , ^n and 棕n represent the wave amplitude, the wave number, the random phase angle and the circular fre­
quency of the nth regular wave component, repective-
ly. z is the vertical position of a water particle, d is the
depth of the water, S 浊(棕)is the wave spectrum and
驻棕 denotes the difference between the circular frequen­
cies of the measured components. The irregular wave
was formed by choosing appropriate input parameters
based on the JONSWAP wave spectrum, moreover, the
significant wave height defined as the mean of the one Fig.1 JONSWAP wave spectrum with different significant wave height
third highest waves (H "3 ) and the mean wave period (T j in this paper were chosen a s 3m and 5 s, respectively. The corresponding wave spectrum is shown in Fig.1.
Based on the wave spectrum presented above, the time series at point (0, 0, 0) is shown in Fig.2.
1.3 Geometric nonlinearity
According to the small deformation
hypothesis, the strain in a certain direction
at an arbitrary point can be derived by cal­
culating the first-order partial derivative
of the corresponding displacement. Under
this hypothesis, the large deflection and ro­
tation of the element can be ignored when
formulating the equilibrium. Nevertheless, due to the large deformation experienced by the structure, the geometric nonlinearities in the finite element analysis should be focused on in this study.
1.3.1 Strain-displacement relationship
When geometric nonlinearities are considered, the relationship between the stress and the strain can be expressed as:
Fig.2 Wave elevation time series at point (0, 0,
0)
264船舶力学第22卷第3期
{着}=蓘]{啄}
[軍j is a strain matrix, which can be further decomposed into:蓘 1 = [B0] + [B l](8)
(9)
where [B〇]isaconstan tstrain m atrix，and [B L j is the nonlinear strain matrix related to the displacement of the node considered.
1.3.2 Stress-displacement relationship
The relationship between the increment of stress and that of strain can be expressed as:
d {滓r=[ d j d {着r(i〇) wher
e [D j is the constitutive matrix for the material. Combining Eqs.(8)-(10) can lead to Eq.
(11)，
d {滓r=蓘d蓡蓸蓘b〇蓡+蓘B l j) d {啄r(11)
1.3.3 Equilibrium equation
Based on the principle of virtual work，the equilibrium equation can be expressed as: |f e T e e T e
J{着*瑟{滓瑟d v-{啄*瑟{F}=0 (12)
V
e e e where {着* 瑟and {滓瑟represent virtual strain and stress o
f an element，respectively. {啄* }
e
is the virtual nodal displacement and {F瑟represents the nodal forces.
A combination of Eq.(8) and Eq.(12) results in，
J蓘]V}e d v-{F}e=0 (13)
V
Differentiating this equation can lead to，
J d蓘j {滓} dv+ J 蓘]d{^} dv=d{F} (14)
V V
Eq.(14) can be rewritten as:
([k〇j + [k滓j + [kLj)d{啄瑟e=d{F瑟e(15) Eq.(15)isthebasisforsolvin ggeom etricnon lin earp roblem s.Inthisequ ation，[k〇j is a standard linear stiffness matrix，[k滓]is the initial-stress matrix for nonlinear conditions，and [&] is the initial displacement matrix under large deformation. The three matrices can be ex­pressed as:
[koj= J d[B0j[D j [B0j dv [k滓]d{滓瑟=J d[^j {滓瑟dv (16)
(17)
第3期HU Ke et al: Hydroelasitic Analysis of the Gravity Cage 噎265
[kL ] = | ( [B0]T[D] [B l ] + [b /[D ] [B0] + [b /[D ] [B l ] )dv (18)
V 1.4 Modal superposition method
In order to have an overview of the motion and deformation of the system, it is important to study the global deformation of the floating collar at first. Even though it is impossible to use modal superposition method to predict the nonlinear hydrodynamic response of the floating collar, the method can still be considered as a 4data post-processing5 procedure. Based on the predicted nonlinear hydroelastic results, the method can be applied to analyze the weight of the participation of each mode at any instant.
In this method, the hydrodynamic response of the floating collar can be described as a lin­ear superposition of all the possible motion and deformation modes:
肄
w (t , x ')=移 Pi (t)渍;(x 蔀，x e 蓘 0, l 蓡i =1Replacing 肄 by n in Eq.(19) yields:
n
w (t, x 蔀=移Pi (t 蔀渍;(x 蔀，x e 蓘0, l 蓡i =1where p { (t ) represents the weight of the ith mode in the global response at time t and 渍i (x ) represents the ith-order mode shape. Eq.(20) can be further expressed in matrix form:
蓘w 蓡(t)=蓘渍蓡[p 蓡(t) (21)
. . . T ...After multiplying both sides of Eq.(21) by [渍],the modal-weight matrix at time t can
be rewritten as:
(19)(20)T -1 t
L p ]u)=蓸渍][渍])[渍][w ]w Therefore, the standard deviation of the modal weight can be derived as follows:
(22)03)
where T is the total time length and Pi (t ) symbolizes the time-averaged modal weight.2 Finite element model descriptions
Numerical and experimental results in the previous work by Lader[14] and Berstad[18] have shown that the bottom nets normally have a negligible effect on the global dynamic response of fish cages. Therefore, in the finite element model, the bottom net and its knots were ex­cluded. The numerical model of the whole gravity net cage is shown in Fig.3. The numerical model is composed of 4 main parts: the floating collar, containment net, mooring lines and bottom ring. To avoid unwanted friction caused by chains and ropes, the bottom ring is attached to the net directly, see Lader[23]. The original solidity of net panel is 0.32. The materials
used
266船舶力学第22卷第3期and their relative properties are also listed
in Tab.1. Due to the limitation of the com­
putational capability, the mesh size of the
net is generally enlarged in order to reduce
the computational time. The validation of the
simplified method is shown in the section 3.
1. Four points on the floating collar (A, B,
C and D), marked as chief indications for
the the dynamic response of the floating
collar, will be further investigated in Chap­
ter 3.Tab.1 Properties of fish cage system
Diameter of fish cage (m)
20Depth of fish cage (m)
20Floating collar
sinker Net Outer diameter (m)
0.30.10.05Thickness (m)
0.02--Elastic modules (MPa)
950950350• 3Density (kg/m )9532 0001 120
To begin with, the collar and bottom ring were simulated by the beam elements. Consid­ering the non-bending properties of net twines, the truss elements were adopted to simulate the twines.
The instantaneous buoyancy acting on the collar often makes it difficult to calculate the hydrodynamic forces accurately. Therefore, the 耶Buoyancy Distribution5method, see Li[24], was adopted to solve this problem by replacing the partly immerged floating collar with 11 dis­tributed coupled beams as shown in Fig.4.
^^p istirib u ted coupled section
扭41 1
B s o i e 2
B it 柳 3
B e 拉M B e a n 5 B e a u G 7
S (issri ： S f g e d )
Bean 9 fiaaQr*e：d)
O f l &E L I (ia m g ru e d ]
Fig.4 Illustration of the distributed coupled beam section
The instantaneous buoyancy of the whole section fB s e c t i 〇n equals the sum of the buoyancy of each immerged beam ( f B _i m m e r g e d _b e a m ) ^, which can be expressed as:
Fig.3 The complete fish cage system
model
第3期HU Ke et al: Hydroelasitic Analysis of the Gravity Cage 噎267
fB_s e c t i o n移(f i m m e r g e d_b e(24) In order to ensure that the distributed beam sections move and deform simultaneously, the six degrees of freedom for each pair of nearby nodes on the neighbouring beams should satisfy the linear constraints.
Meanwhile, the mass and bending-stiffness properties of the floating collar and the beams must also be equivalent, as described in the following equations:
m s e c t i o n^m i05)
(E I)se-=移E i l(26)
where m section ™d (E/)sectio n arethemassdensity andthebending stiffnessofthesectioninthe floating collar, respectively, while m i and (E l)i are the mass density and the bending stiffness of the ith distributed beam.
Secondly, in the simulation of the mooring lines, four spring elements with 6 000 N/m lin­ear tensional stiffness, were employed. The spring elements were attached horizontally to the floating collar in the xoy plane in Fig.3.
3 Results and discussions
ABAQUS/Standard, a software for finite element analysis, was used to simulate the model under the combined effect of current and irregular waves. Both the wave load and the current load were calculated based on the Morison equations, the hydrodynamic coefficients C M and C d should be chosen according to the Re and K C numbers. In this paper, the R e number was pretty low and the K C number was very high, hence C m and Cd were chosen as 2.0 and 1.2, respectively^251. Moreover, the geometrical nonlinearities associated with the nets 爷large defor­mation and motion were also taken into account.
3.1 Validation of the numerical model
Owing to the large number of meshes in a full-scale net, it is hard to conduct calculations on a model with detailed mesh. Thus, the full-scale model was simplified. The hydrodynamic force, tensile stiffness and mass in the simplified model should be equivalent in the numerical models before and after simplification. This can be described in the following equations:
A t-= ^A k(27)
(s e c t i o n. )t r u s s•^移E k A s e c t i o n_k08) M t r u s s=M(29)
268船舶力学第22卷第3期
where A and A are the proiected area and cross-sectional area of the twine; M is the mass section
of the net, and E represents the elastic modulus. Moe[19]validated their numerical models by comparing predicted deformation to that of a real model. Similar deformation to that observed by Moe[19]was observed in this model. Moreover, the deformation of the model with detailed mesh agrees well with that of the simplified model. The comparison also indicates that the model with coarse mesh was sufficiently accurate to study the motion and deformation of the gravity cage. The result is shown in Fig.5.
Comparison of models Companion ot models
with dettiiled mesh with applied (coiiise) mesh
(h i)Dctormaiion picdi.LH.-Ll(cJ) JJemrinanun，prcdicicd
by Moc ct a J 12010)b y M oecU ili：2U]〇)
(<l)Model E C.S I
化戒T e L tiim
(b2) DcJornialion predfcicd le i) Petomialion prcdicicd
m ill i s s i u d v m ih is sti]d v
Fig.5 Validation and verification of the numerical model
3.2 Modal analysis of floating collar under combined effects of irregular waves
and current
As the modal superposition method mentioned in the section 1.4, the mode shapes of the 1st to the 20th modes of the floating collar calculated by the modal analysis were shown in Fig.6. The modes numbered from 1t o6correspond to the six rigid-body-motion modes, while the rest correspond to the flexural deformation modes of the structure. In this analysis, the non­linearities in the mooring-line are ignored.
第3期HU Ke et al: Hydroelasitic Analysis of the Gravity Cage 噎269
0 Hz (Rigid-body mode,/>-9.54 «10* 'Hz (Rigid-body
( Rigiii-hoiJy mode, vertical rotanon in plane ) mode, roll iti in pltme O'r-z)
motion in the plane O-x-z)
/&-3.1〇x l〇--Hz
\\z(Rimd-hodw
_/4-1.22^]r- Hz(Rigid-badv(Kiyiu-bodv mcxie.
mode, trai]sial[omn the plane
H id e, roll m the plane O-.t-j)transJatiomn the plane
O-x-v)
J7 =0.55213 Ilz/s =0,552 13 H/.
(Flexural mode in U’lexuraJ mode i n sitk(He s ury I mode iji(Hm jrd mode in O-.^-r view)view)O-x-z view)yidtr view)
/g =0.569 25 Ux/,〇 =0,571 95 Hz/,i=].584 4Hz
(Flexunil rm)de m the iTIcxuml mode in the(Hcxural mode in(J Jcxuial m ode
a tlL-
plane O'x-y)O-x-z view)e ii side view)
Fig.6 shows that the deformation of the 1st, 5th, 6th, 9th, 10th, 13th, 14th, 17th and 18th modes appears in the O -x -y plane, while that of the 2nd, 3rd, 4th, 7th, 8th, 11th, 12th, 15th, 16th, 19th and 20th modes occurs in the O-x-z plane in Fig.3.
In order to investigate the modal contribution, the numerical model operated in conditions where the current speed was set as0m/s, 0.5 m /s and 1m/s, coupled with i^egular waves with /『]=1.584 4 Hz yij = !.ftll 2 Hz fu = l£i \ 2llz
\ /t 〇
k ' / / 飞咖 r '!K i / \'I .丨一':'1
^s.J \ ■ll
}:■(Flexural mode in
O-x-z view)(Flexural mode in side view)
(Flexuralmode in the plane (l lcxural mode in tlic plane C i-.v -v )057 2 11/./ifi^3.057 2 Hz
■
_ r
】node h i
P-.v-z view)
(Flexural mode h i side view)(Fkxural mode in view)(Flexural mode in side view)/7-3.0S 9 7IIZ H O W 3 M z /a ，=4,961 4 Hz (〕
(Klexunil mode in the plane O-r-v)(Flexural mode in the pfane O 'w )(Flexural mode in O u view)(Flexural mode in side view)/m =4.961 4 Hz ,,jL //\
^Y '■n .
(FlexuraJ mode in O-r-j view)(HeKural mode in side view)
Fig .6 The first 20 mode shapes of the floating collar
significant heights set as 0.3 m,1m and 3 m. Time histories of the modal weight in the hori­zontal and vertical directions are shown in the figures below (Figs.7-16), and their correspond­ing standard deviations are depicted in the four following figures (Figs.17-20): Analysis of modal weights in the horizontal response revealed that the 5th, 6th, 9th and 14th modes were the dominant modes. This means that the translational rigid-body-motion modes as well as the in-plane flexural structural-deformation modes dominated the response of the floating collar. However, as the current speed increased, the modal weights of the 5th and 6th modes experienced a steeper increase than the flexural structural-deformation modes, indicat­ing that the current had a stronger influence on the translational rigid-body-motion modes.
电 〇
&0 90 10G M s)Fig.16 Modal weight (vertical motion when H 1/3=3 m, Tx =5 s, C=1.0 m/s)
Fig.14 Modal weight (vertical motion when H 1/3=3 m, T 1=5 s, C=0.5 m/s)
Fig.15 Modal weight (horizontal motion when
H "3=3m , (=5s ，C=1.0m /s)Fig.17 Standard deviation of the horizontal response of each mode for different significant wave heights
With regard to the vertical response, the 2nd, 3rd, 4th and 8th modes participated most actively, as can be observed in Figs.8, 10, 12, 14 and 16. Each modal weight increased with the significant wave height. On the other hand, Figs.12, 14, and 16 show that the current had a smaller influence on the modal weight in the vertical direction compared to that in the hori­zontal direction.
7C 孩分 90 1Q 〇
Modal weight (horizontal motion when
H "3=3m , T !=5s, C=0.5m/s)-s j
s l -l
l ft f
x
s
:l l f .^^n l
o
p
o
s
1]
Mode nymber
Fig.18 Standard deviation of the horizontal response of each mode for different current speeds
Fig.19 Standard deviation of the vertical response of each mode for different significant wave heights
Fig.20 Standard deviation of the vertical response of each mode for different current speeds
Besides, the comparison of the horizontal and vertical standard deviations of each mode above shows that much higher modes (flexural structural deformation modes) were excited ver­tically. The current had a stronger impact on the standard deviation of the 5th and 6th modes in the horizontal direction. In addition, the standard deviation of modal weight increased with significant wave height in both directions, which indicates that higher waves may induce higher l t o p fi >3p
-H E
P
U
E
^
order modes.
From the discussion in this section, it can be seen that compared with the wave-only-con­dition, the combination of current and wave has a greater influence on the translational rigid- body-motion in the horizontal direction. This indicates that the rigid-body motion of the float­ing collar should be paid more attention in the design of mooring systems attached to the fish cage in the wave and current combined condition. It has also been suggested that higher wave will arouse more flexible modes, while current contributes little to the flexible modes.
4 Conclusions
This paper presents an analysis based on the FEM in predicting the dynamic response of the gravity net cage system under the combined effects of irregular waves and current. The fol­lowing conclusions are derived: the modal weight in both the horizontal and vertical directions becomes larger as the significant wave height increases, which can be found from the modal analysis of the floating collar under the combination of irregular wave and current. Meanwhile, the modal weight of the rigid-body-motion mode in the horizontal direction grows with the current speed, while the modal weight in the vertical direction is only slightly in^uenced by the variation of the speed. Moreover, it can be seen from the standard deviation of modal weight that much higher order modes will be excited with significant wave height increased. This in­dicates that when analyzing the total dynamic response under larger wave height, more atten­tion should be paid on deformation.
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