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On natural modes in moonpools with recesses
Applied Ocean Research 67 (2017) 1–8
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On natural modes in moonpools with recesses Bernard Molin a,b,∗ a b
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France Bureau Veritas Marine & Offshore SAS, 67/71 Boulevard du Château, 92571 Neuilly sur Seine, France
a r t i c l e
i n f o
Article history: Received 3 April 2017 Received in revised form 19 May 2017 Accepted 20 May 2017 Keywords: Drillship Moonpool resonance Piston mode Sloshing mode Linearized potential flow theory
a b s t r a c t The theoretical model of Molin [6] is extended to the case of rectangular moonpools with one or two recesses, as can be found in some drillships. Obtained natural frequencies and modal shapes of the piston and first sloshing modes are compared with experimental results available in literature, with good agreement. An approximation easy to implement is proposed for the natural frequency of the piston mode. Further illustrative results are presented when some geometrical parameters of the moonpool are being varied. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Moonpools are vertical openings through the hulls of some marine structures. They usually have vertical walls, from deck to keel. However some moonpools in drillships have “recesses”, some kind of sub-compartments used, for instance, for assembling drilling equipments [9]. As shown in Fig. 1, taken form [5], recesses can be located on the bow side or on the aft side of the moonpool. Moonpools are prone to bothersome resonance problems, under outer wave action or forward speed, and it is desirable, at the design stage, to be able to predict their resonant frequencies. Natural modes in moonpools consist in the so-called piston mode, up and down motion of the entrapped water, and in sloshing modes, similar to the sloshing modes in a tank. Molin [6] proposed a theoretical frame to derive the resonant frequencies for rectangular moonpools with vertical walls. His work was based on simplifying assumptions: the floating support is motionless, the waterdepth is infinite, the length and breadth of the support are infinite. The fluid domain is then decomposed into two parts: the moonpool and a semi-infinite fluid domain below the keel level. Linearized potential flow theory is used, the velocity potential being written as an eigen-function expansion in the moonpool. The matching condition with the lower fluid domain is written as an integral equation relating the potential and its vertical derivative. An eigen-value problem is then formulated and solved,
∗ Correspondence to: Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2017.05.010 0141-1187/© 2017 Elsevier Ltd. All rights reserved.
yielding the natural frequencies and associated modal shapes of the free surface. In this paper we follow the same procedure, with the moonpool being decomposed into two parts where different eigen-function expansions are used and need to be matched on the common boundary.
2. Theoretical model The geometry is illustrated in Fig. 2. The moonpool is supposed to consist in possibly two recesses. We use a rectangular coordinate system Oxyz with its origin at the keel line, at one end of the “restriction” part of the moonpool (we use here the coining “restriction” following [1], see also [10], in their study of the MONOBR platform). The restriction and the recesses are rectangular. The length of the restriction is a, its height is d. The length of the left recess is b, the length of the right recess is c, the additional water-height in the upper part is h, so that the total draft is d + h. The total length of the moonpool, at the waterline, is L = a + b + c, and its width is B (it is the same width for the restriction and for the upper part). Alike in [6], the waterdepth is assumed to be infinite, and the beam and length of the drillship are taken to the limit when they are also infinite. As a result the fluid domain consists in three parts: a semi-infinite lower subdomain (− ∞ < x <+∞; − ∞ < y <+∞; − ∞ < z ≤ 0), the restriction (0 ≤ x ≤ a; 0 ≤ y ≤ B; 0 ≤ z ≤ d), and the upper part of the moonpool (−b ≤ x ≤ a + c; 0 ≤ y ≤ B; d ≤ z ≤ d + h). Use is made of linearized potential flow theory. The flow is assumed to periodic in time at a frequency ω: (x, y, z, t) = ϕ(x, y, z) cos(ωt + ). In the moonpool the reduced potential ϕ verifies the
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B. Molin / Applied Ocean Research 67 (2017) 1–8
The bottom boundary condition writes A0 +
∞
Am cos km x
m=1
=
1 2
a
B
B0 /d +
0
0
(x
∞
B k n=1 n n
− x )2
+ (y
cos kn x − y )2
dx dy
(4)
Through integrations this can be transformed into the vectorial equation (see [6]): = MAB · B A
(5)
= (A0 , . . .An ), B = (B0 , . . .Bn ). with A In z = d we match ϕ1 and ϕ2 and their vertical derivatives. We follow Garrett’s method [2]. Consider first the equality of the potentials: z + (Am cosh km d + Bm sinh km d) cos km x d ∞
A0 + B0 = C0 +
∞
m=1
(6)
Cn cos n (x + b)
n=1
Fig. 1. Types of moonpools on drillships. Taken from [5].
which holds for 0 ≤ x ≤ a. Again we take advantage of the orthogonality of the set [coskn x] over [0 a]. Integrating each side in x over [0 a], then multiplying with coskm x and integrating again for all m, the following vectorial equation is obtained: + DB B = MC C A
(7)
where DB is the diagonal matrix (1, tanhkm d), MC is a full matrix and C = (C0 , . . .Cn ). Consider now the vertical velocities. They obey the equations Fig. 2. Geometry.
ϕ2z = ϕ1z 0 ≤ x ≤ a
Laplace equation ϕ = 0, no-flow conditions at the solid boundaries ϕn = ∂ϕ/∂n = 0, the free surface condition g ϕz − ω2 ϕ = 0 at z = d + h, and a matching condition with the flow in the lower fluid domain, written as [6]: ϕ(x, y, 0) =
1 2
a
dx 0
B
dy 0
ϕz (x , y , 0)
(x − x )2 + (y − y )2
(1)
Typical moonpool dimensions in drillships are about 40–50 m in length and 10 m in width, and we are here interested in the piston and longitudinal sloshing modes. So we make the simplifying assumption that the moonpool is narrow, so that the flow inside can be idealized as two-dimensional. Note that the flow in the lower fluid domain z ≤ 0 is three-dimensional. As in [6] we use eigen-function expansions to represent the flow in the moonpool: z ϕ1 (x, z) = A0 + B0 + (An cosh kn z + Bn sinh kn z) cos kn x d ∞
(2)
in sub-domain 1 (0 ≤ z ≤ d), with kn = n/a, and
That is D0 + m Dm cos m (x + b) h ∞
m=1 ∞
B0 = kn (An sinh kn d + Bn cosh kn d) cos kn x + d =0
0≤x≤a
(9)
n=1
− b ≤ x ≤ 0 and a ≤ x ≤ a + c
Now we make use of the orthogonality of the set [cosm (x + b)] over [−b a + c]. Integrating each side over their domains of validity, then multiplying with cosm (x + b) and integrating again for all m, gives + MDB B = MDA A D
(10)
= (D0 , . . .Dn ). with D Finally the free surface equation g ϕ2z − ω2 ϕ2 = 0 gives
C + D4 D
(11)
= (D0 , . . .Dn ) and D1 , D2 , D4 diagonal matrices: with D
z−d + (Cn cosh n (z − d) h ∞
D1 = (0, g n tanh n h) D2 = (g/h, g n )
n=1
+Dn sinh n (z − d)) cos n (x + b)
(8)
0 − b ≤ x ≤ 0 and a ≤ x ≤ a + c
= ω2 D1 C + D2 D
n=1
ϕ2 (x, z) = C0 + D0
ϕ2z =
(3)
in sub-domain 2 (d ≤ z ≤ d + h), with n = n/(a + b + c) = n/L. The Laplace equation and the no-flow conditions at the end walls are ensured.
(12)
D4 = (1, tanh n h) From (5) and (7) we get: = (MAB + DB )−1 MC C B
(13)
B. Molin / Applied Ocean Research 67 (2017) 1–8
3
Fig. 3. Base case. Moonpool dimensions. Taken from [4].
and then, from (10):
3. Illustrative results
= (MDA MAB + MDB ) (MAB + DB )−1 MC C = MDC C D
(14)
and, from (11), the following eigen-value problem is obtained (D1 + D2 MDC ) C = ω2 (I + D4 MDC ) C
(15)
with I the identity matrix. The resolution of this eigen-value problem gives the resonant frequencies ωi and the associated eigen-vectors Ci . Practically the series are truncated after 10 or 20 terms, which provides a sufficient accuracy. 2.1. Frozen restriction approximation In Molin [6] it is established that a good approximation of the natural frequency of the piston mode can be obtained by assimilating the fluid inside the moonpool as a solid body. Likewise, in the case of axisymmetric moonpools with restrictions, it has been found (see [7] or [8]) that an approximation of the natural frequency of the piston mode can be derived by ”freezing” the fluid domain inside the restriction, that is considering it as a solid body. The free heave motion of the frozen restriction obeys the equation ( a B d + Mal + Mau (ω)) Z¨ + g
B a2 Z=0 L
(16)
with a B d the mass of the restriction, Mal the added mass due to the lower fluid domain (z ≤ 0), and Mau the added mass due to the upper fluid domain (d ≤ z ≤ d + h). The added mass Mal is given in [6] as Mal = B2 a f(B/a) with f (r) = −
1 3
1
argsinh r −1 + r −1 argsinh (r) +
1 + r −2
r2 + 1
1 3
r + r −2
.
(17)
The added mass from the upper fluid domain is obtained as (see Appendix): Mau (ω) = +2
B L
B a2 h L
∞ 2 [sin n (a + b) − sin n b] g n − ω2 tanh n h n=1
3n
g n tanh n h − ω2 (18)
The natural frequency can then be obtained, in principle, from the intersection of the two curves y = ω and y = gBa2 /[L( aBd + Mal + Mau (ω))].
3.1. Comparison with the experimental results of Guo et al. [4] In two independent papers Guo et al. [3,4] present an experimental and numerical study of a drillship with a moonpool having a recess on the aft side. The moonpool is 45.6 m long and 11.2 m wide. The recess has a length (c) of 16 m. The height of the restriction (d) is 7.2 m, the waterheight in the recess (h) is 3.8 m (see Fig. 3, taken from [4]). The tests, at a scale 1:50, are performed in regular and irregular waves, at different headings from 180◦ (head waves) to 90◦ (beam waves). Experimental RAOs of the free surface elevation (relative to the moving ship) are provided at three locations along the moonpool. They exhibit four peaks, at angular frequencies (in full scale) of 0.42 rad/s, 0.82 rad/s, 1.07 rad/s and 1.36 rad/s. The authors interpret the second peak frequency (0.82 rad/s) as the piston mode and the third one (1.07 rad/s) as the first sloshing mode. This interpretation is based on wrongly using Molin’s single mode approximation formulas [6] which are valid only for a moonpool with vertical walls from deck to keel. As for the first resonant peak at 0.42 rad/s, the authors attribute it to “some coupling between the heave and pitch motions” (!). A striking feature is that, at this first resonant frequency, the measured RAOs at the aft side of the moonpool (by the end wall of the recess) are 5–10 times larger than the RAOs measured at the other end. At the second peak frequency it is the other way around: the measured RAOs at the bow side wall are about 50% larger than at the recess wall. In an other paper on drillships with recesses [9] the moonpool dimensions are similar: 38.4 m long, 12.5 m wide, the restriction has a height of 7.5 m and a length of 25.6 m, the operating draft is 12 m. Unfortunately the resonant frequencies are not given, so we take the moonpool from [3,4] as a base case. Fig. 4 shows the experimental RAOs of the relative free surface elevation at the gauges inside the moonpool, as given in [3]. They were obtained from tests in irregular waves with a rather small significant wave height (2 m) and the wave energy uniformly distributed from 0.2 rad/s up to 1.4 rad/s. The vertical bars show the first 4 natural frequencies given by our model, that is 0.414 rad/s for the piston mode and 0.778 rad/s, 1.074 rad/s and 1.339 rad/s for the first three sloshing modes. Our predictions agree well with the positions of the experimental peaks, except for the first sloshing mode where our value is about 5% lower than measured; this discrepancy could be due to the fact that the drillship is not fixed, but free to respond to the waves. As a matter of fact peaks are observed at this frequency in the heave and pitch RAOs. Hydrodynamic coupling between the moonpool and ship dynamics results in a slight shift of the resonant frequencies. Fig. 5 shows the calculated modal shapes of the free surface elevation. They are normalized with a maximum absolute value of one along the length of the moonpool. It is striking that, for the piston
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B. Molin / Applied Ocean Research 67 (2017) 1–8 6
90 dg 135 dg 150 dg 180 dg
5
4
3
2
1
0
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9 angular frequency (rad/s)
1
1.1
6
1.2
1.3
1.4
1.2
1.3
1.4
90 dg 135 dg 150 dg 180 dg
5
4
3
2
1
0
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9 angular frequency (rad/s)
1
1.1
Fig. 4. Experimental RAOs of the relative free surface elevation at the bow wall (top) and at the recess wall (bottom), for different ship headings.
10
1 7.5
0.8 0.6
5 added mass coefficient
0.4 0.2 0 -0.2 -0.4
mode 0 mode 1 mode 2 mode 3
-0.6
2.5
0
-2.5
-5
-7.5
-0.8 -1
-10
0
5
10
15
20
25
30
35
40
45
0
0.2
0.4
0.6 0.8 omega (rad/s)
1
1.2
1.4
x (m)
Fig. 5. Base case. Calculated modal shapes.
mode (mode 0), the maximum value is obtained at the end of the recess and that at the other end the free surface motion is ten times lower. This is in qualitative agreement with the experimental RAOs. Likewise, for the first sloshing mode (mode 1), the amplitude of free surface motion in the recess is about 60% of the amplitude at the bow wall. This again agrees with the experimental observations. As for the second and third sloshing modes, the amplitude of free surface motion is about the same at both ends (again in agreement with the experiments). As for applying the frozen restriction approximation, Fig. 6 shows the non dimensional added mass Mau (ω)/( B Lh) vs the frequency and Fig. 7 shows the graphical determination of the res-
Fig. 6. Base case. Non dimensional added mass Mau /( B hL) vs frequency.
onant frequency, from the intersections of the two curves y = ω and gBa2 /[L( aBd + Mal + Mau (ω))]. Due to the singular behavy= ior of the added mass Mau (ω) there are several intersection points, none of them giving a value close to the expected one. As a matter of fact a better value is obtained by taking the added mass Mau at a zero frequency, which gives a resonant frequency of 0.44 rad/s, about 6% too high. In the following paragraphs, we vary some geometric parameters of the base case moonpool. First we vary the length of the recess, then we vary the waterheight in the recess, finally we consider the case of two recesses.
B. Molin / Applied Ocean Research 67 (2017) 1–8
5
6
1 0.9
5 0.8 0.7
4 omega (rad/s)
0.6 0.5
3
0.4
2
0.3 0.2
1 0.1 0 0
Fig.
0.1
0.2
0.3
7. Intersections
of
0.4
the
0.5 0.6 omega (rad/s)
two
curves
0.7
0.8
y=ω
0.9
and
1
y=
gBa2 /[L( aBd + Mal + Mau (ω))].
0 0
4
8
12 16 20 length of right recess (m)
24
28
32
Fig. 9. Varying length of the recess. Non dimensional added mass Mau (0)/( B hL). 1.1 1
1.2 mode 0 frozen appr. mode 0 exact mode 1
0.9
1
0.8 0.7
0.8 omega (rad/s)
0.6 0.5 0.6
c=0m c=2m c=4m c=8m c = 16 m c = 32 m
0.4 0.3
0.4
0.2 0.1
0.2
0 0 0 0
4
8
12 16 20 length of right recess (m)
24
28
32
Fig. 8. Varying length of the recess. Natural frequencies of the piston and first sloshing mode.
3.2. Varying the length of the recess We keep all geometrical parameters the same as in the base case, except for the length of the recess that we vary from 0 (no recess, moonpool with vertical walls all the way from deck to keel) up to 32 m (twice the base case length). Fig. 8 shows the natural frequencies of the piston and first sloshing mode. For the piston mode the value delivered by the frozen restriction approximation (with the added mass Mau taken at zero frequency) is also shown. It can be seen that this approximation is in fair agreement with the exact value. It can also be observed that the natural frequency of the piston mode decreases strongly as the length of the recess increases. This is partly due to the decrease of the hydrostatic stiffness in Eq. (16). But it is also due to the increase of the added mass Mau : Fig. 9 shows the variation of this added mass, normalized with the mass of water in the upper part of the moonpool B hL, vs the recess length c. It is striking that, from the expected value 1 at c = 0, after an initial hollow, this non dimensional added mass steadily increases. As a matter of fact, from Eq. (18), it is easy to establish that there is an asymptotic value, as c goes to infinity, which is a2 /(3h2 ), that is 20.2 in the present case. It is non intuitive that such high values can be attained for the added mass.
10
20
30
40
50
x (m)
Fig. 10. Varying length of the recess. Modal shape of the piston mode.
Fig. 10 shows the modal shapes of the free surface elevation in the moonpool, for the piston mode. They are shown for different values of the recess length (0 m, 2 m, 4 m, 8 m, 16 m, 32 m). When there is no recess (c = 0 m), the free surface is nearly flat, with a small bump in the center. As soon as there is a recess the point of maximum elevation moves to the recess wall, and the value attained at the opposite wall steadily decreases as the recess length increases. (Note that in the case c = 32 m the value at the recess wall is also one but the figure has been truncated for the sake of clarity). Finally Fig. 11 shows the modal shapes of the first sloshing mode. They appear to be quite different depending on the recess length. At c = 0 m (no recess), due to the relatively large draft (11 m) compared with the length (29.6 m) of the moonpool, it is just a sinusoidal shape, with 1 (in absolute value) at both walls. At small recess lengths (2 m and 4 m), it drops slightly at the bow wall, when it remains equal to one (in absolute value) at the recess wall. As the recess length keeps one increasing, it goes back to one at the bow wall and steadily decreases at the recess wall, until a drastic change occurs: at c = 32 m the shape has become non monotonic, with an intermediate maximum in-between the two walls. 3.3. Varying waterheight in the recess Now, keeping again all other geometric parameters the same as in the base case, we vary the water-height h of the upper part
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B. Molin / Applied Ocean Research 67 (2017) 1–8 1.1
1 c=0m c=2m c=4m c=8m c = 16 m c = 32 m
0.8 0.6
1 0.9 0.8
0.4
0.7
0.2
0.6
0
0.5
-0.2
h=1m h=2m h=4m h=8m
0.4
-0.4
0.3
-0.6
0.2
-0.8
0.1
-1 0
10
20
30 x (m)
40
50
0
60
0
5
10
15
20
25
30
35
40
45
x (m)
Fig. 11. Varying length of the recess. Modal shape of the first sloshing mode.
Fig. 13. Varying waterheight h. Modal shape of the piston mode.
1 mode 0 frozen appr. mode 0 exact mode 1
0.9
1 h=1m h=2m h=4m h=8m
0.8 0.8
0.6 0.7
omega (rad/s)
0.4 0.6
0.2 0.5
0 0.4
-0.2
0.3
-0.4
0.2
-0.6
0.1
-0.8
0 0
1
2
3
4 5 6 7 water height in right recess (m)
8
9
10
11
-1 0
5
10
15
20
25
30
35
40
45
x (m)
Fig. 12. Varying waterheight h. Natural frequencies of the piston and first sloshing mode.
1 mode 0 frozen appr. mode 0 exact mode 1
0.9 0.8 0.7
omega (rad/s)
of the moonpool, from 0 m up to 11 m, the total draft h + d being kept equal to 11 m. This means that the height d of the restriction is 11 m−h. Fig. 12 shows the natural frequencies of the piston and first sloshing modes, together with the frozen restriction approximation of the piston mode. Again the frozen restriction approximation is doing a good job. Fig. 13 shows the modal shapes of the piston mode for 4 different h values (1 m, 2 m, 4 m, 8 m). They look similar, with, as usual, the maximum value at the recess wall, and the minimum value, at the facing wall, getting lower and lower as the waterheight h decreases. Finally Fig. 14 shows the modal shapes of the first sloshing mode, with the maximum value at the wall opposite to the recess, except in the shallowest case where an intermediate maximum appears, alike the case c = 32 m in Fig. 11.
Fig. 14. Varying waterheight h. Modal shape of the first sloshing mode.
0.6 0.5 0.4 0.3 0.2 0.1 0 0
3.4. Two recesses Finally we consider the case when two recesses are present in the moonpool. Again we start from the base case geometry, and we keep constant the total recess length: b + c = 16 m. Fig. 15 shows the natural frequencies of the piston and first sloshing modes, vs the length b of the left recess. It can be seen that, again, the frozen approximation provides a fair estimate of
2
4
6 8 10 length of left recess (m)
12
14
16
Fig. 15. Two recesses. Natural frequency of the piston and first sloshing mode.
the natural frequency of the piston mode. Depending on the relative lengths of the recesses the natural frequency varies somewhat. Note that, in the considered case, the hydrostatic stiffness in Eq. (16) remains constant but, as Fig. 16 shows, the zero frequency added
B. Molin / Applied Ocean Research 67 (2017) 1–8
7
Finally Fig. 18 shows the modal shapes of the first sloshing mode, with the maximum elevation being now attained at the wall of the shortest recess.
3
2.5
4. Final comments 2
1.5
1
0.5
0 0
2
4
6 8 10 length of left recess (m)
12
14
16
Fig. 16. Two recesses. Non dimensional added mass Mau (0)/( B Lh).
1.1 1 0.9 0.8
We have proposed an extension of Molin’s model [6] to rectangular moonpools with one or two recesses. This model delivers the natural frequencies and associated modal shapes of the piston and longitudinal sloshing modes. Good agreement has been obtained with the experimental results of [4]. A remarkable feature obtained for the piston mode is that the point of largest motion of the free surface is at the wall of the recess. A simple formula, based on assimilating the restriction part of the moonpool with a solid body, has been proposed and found to give a good approximation of the natural frequency of the piston mode. It must be stressed out, however, that not all geometric parameters of the moonpool have been covered. Therefore the formula should be used with care. We have tried to derive similar simple approximation for the natural frequency of the first sloshing mode, with no success so far. It must be emphasized that the results presented here have been obtained within the scope of linearized potential flow theory. Practically, when the waterheight in the recess is shallow, nonlinear effects are expected to quickly occur, all the more since we have obtained that, in the piston mode, the largest motion of the free surface takes place in the recess.
0.7
Acknowledgements
0.6 0.5
The author is grateful to Prof. Haining Lu, of Shanghai Jiao-Tong University, for providing the experimental RAOs.
b=0m b=2m b=4m b=6m b=8m
0.4 0.3
Appendix A. Added mass from the upper fluid domain
0.2 0.1 0 0
5
10
15
20
25
30
35
40
45
To obtain the added mass Mau (ω) we need to derive the Dn coefficients of the ϕ2 expansion that verify
x+b (m)
D0 m Dm cos m (x + b) + h ∞
ϕ2z (x, d) =
Fig. 17. Two recesses. Modal shape of the piston mode.
m=1
= 1 for 0 ≤ x ≤ a
b=0m b=2m b=4m b=6m b=8m
1 0.8
= 0 for − b ≤ x ≤ 0 and a ≤ x ≤ a + c
(19)
0.6
Taking again advantage of the orthogonality of the cosn (x + b) functions, we obtain
0.4 0.2
D0 =
0 -0.2
ah 2 [sin n (a + b) − sin n b] Dn = L 2n L
(20)
The Cn coefficients follow from the free surface condition:
-0.4 -0.6
C0 = D0
g
ω2 h
-0.8 -1
Cn = −Dn 0
5
10
15
20
25 x+b (m)
30
35
40
45
Fig. 18. Two recesses. Modal shape of the first sloshing mode.
−1
g n − ω2 tanh n h g n tanh n h − ω2
The added mass is obtained as
a
Mau (ω) = − B mass Mau (0) varies quite a lot depending on the relative lengths of the recesses. Fig. 17 shows the modal shapes of the piston mode. The maximum elevation is obtained at the wall of the longest recess.
(21)
ϕ2 (x, d) dx 0
= − B
a
[C0 + 0
∞ n=1
Cn cos n (x + b)] dx
(22)
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B. Molin / Applied Ocean Research 67 (2017) 1–8
which gives Mau (ω) = +2
B L
B a2
h
L
∞ 2 [sin n (a + b) − sin n b] g n − ω2 tanh n h n=1
3n
g n tanh n h − ω2
(23)
Note that the first term g D0 /(ω2 h) in C0 is actually an hydrostatic term (the hydrostatic stiffness that appears in Eq. (16)). References [1] R. Barreira, S.H. Sphaier, I.Q. Masetti, A.P. Costa, C. Levi, Behavior of a mono-column structure (MONOBR) in waves, in: Proc. of the International Conference on Offshore Mechanics and Arctic Engineering, OMAE2005, Halkidiki, Greece, 2005. [2] C.J.R. Garrett, Wave forces on a circular dock, J. Fluid Mech. 46 (1971) 129–139.
[3] X. Guo, H. Lu, J. Yang, T. Peng, Study on hydrodynamic performances of a deep-water drillship and water motions inside its rectangular moonpool, in: Proc. of the Twenty-sixth (2016) International Ocean and Polar Engineering Conference, ISOPE, Rhodes, 2016. [4] X. Guo, H. Lu, J. Yang, T. Peng, Resonant water motions within a recessing type moonpool in a drilling vessel, Ocean Eng. 129 (2017) 228–239. [5] E. Hammargren, J. Törnblom, Effect of the moonpool on the total resistance of a drillship, M.Sc. Thesis), Chalmers University of Technology, 2012. [6] B. Molin, On the piston and sloshing modes in moonpools, J. Fluid Mech. 430 (2001) 27–50. [7] B. Molin, I. de Vries, A. Cinello, Hydrodynamic analysis of the piston mode resonance inside a large FLNG turret, in: Proc. 29th Int. Workshop on Water Waves and Floating Bodies, Osaka, 2014. [8] B. Molin, X. Zhang, On resonant modes in moonpools with restrictions or recesses, in: Proc. 32nd Int. Workshop on Water Waves and Floating Bodies, Dalian, 2017. [9] H.-J. Son, S.-H. Choi, M.-W. Kim, S.-H. Hwangbo, Drag reduction of recess type moonpool under vessel’s forward speed, in: Proc. of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering, OMAE2008, Estoril, 2008. [10] S.H. Sphaier, F.G.S. Torres, I.Q. Masetti, A.P. Costa, C. Levi, Monocolumn in waves: experimental analysis, Ocean Eng. 34 (2007) 1724–1733.
Contents lists available at ScienceDirect
Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
On natural modes in moonpools with recesses Bernard Molin a,b,∗ a b
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France Bureau Veritas Marine & Offshore SAS, 67/71 Boulevard du Château, 92571 Neuilly sur Seine, France
a r t i c l e
i n f o
Article history: Received 3 April 2017 Received in revised form 19 May 2017 Accepted 20 May 2017 Keywords: Drillship Moonpool resonance Piston mode Sloshing mode Linearized potential flow theory
a b s t r a c t The theoretical model of Molin [6] is extended to the case of rectangular moonpools with one or two recesses, as can be found in some drillships. Obtained natural frequencies and modal shapes of the piston and first sloshing modes are compared with experimental results available in literature, with good agreement. An approximation easy to implement is proposed for the natural frequency of the piston mode. Further illustrative results are presented when some geometrical parameters of the moonpool are being varied. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Moonpools are vertical openings through the hulls of some marine structures. They usually have vertical walls, from deck to keel. However some moonpools in drillships have “recesses”, some kind of sub-compartments used, for instance, for assembling drilling equipments [9]. As shown in Fig. 1, taken form [5], recesses can be located on the bow side or on the aft side of the moonpool. Moonpools are prone to bothersome resonance problems, under outer wave action or forward speed, and it is desirable, at the design stage, to be able to predict their resonant frequencies. Natural modes in moonpools consist in the so-called piston mode, up and down motion of the entrapped water, and in sloshing modes, similar to the sloshing modes in a tank. Molin [6] proposed a theoretical frame to derive the resonant frequencies for rectangular moonpools with vertical walls. His work was based on simplifying assumptions: the floating support is motionless, the waterdepth is infinite, the length and breadth of the support are infinite. The fluid domain is then decomposed into two parts: the moonpool and a semi-infinite fluid domain below the keel level. Linearized potential flow theory is used, the velocity potential being written as an eigen-function expansion in the moonpool. The matching condition with the lower fluid domain is written as an integral equation relating the potential and its vertical derivative. An eigen-value problem is then formulated and solved,
∗ Correspondence to: Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, Marseille, France. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2017.05.010 0141-1187/© 2017 Elsevier Ltd. All rights reserved.
yielding the natural frequencies and associated modal shapes of the free surface. In this paper we follow the same procedure, with the moonpool being decomposed into two parts where different eigen-function expansions are used and need to be matched on the common boundary.
2. Theoretical model The geometry is illustrated in Fig. 2. The moonpool is supposed to consist in possibly two recesses. We use a rectangular coordinate system Oxyz with its origin at the keel line, at one end of the “restriction” part of the moonpool (we use here the coining “restriction” following [1], see also [10], in their study of the MONOBR platform). The restriction and the recesses are rectangular. The length of the restriction is a, its height is d. The length of the left recess is b, the length of the right recess is c, the additional water-height in the upper part is h, so that the total draft is d + h. The total length of the moonpool, at the waterline, is L = a + b + c, and its width is B (it is the same width for the restriction and for the upper part). Alike in [6], the waterdepth is assumed to be infinite, and the beam and length of the drillship are taken to the limit when they are also infinite. As a result the fluid domain consists in three parts: a semi-infinite lower subdomain (− ∞ < x <+∞; − ∞ < y <+∞; − ∞ < z ≤ 0), the restriction (0 ≤ x ≤ a; 0 ≤ y ≤ B; 0 ≤ z ≤ d), and the upper part of the moonpool (−b ≤ x ≤ a + c; 0 ≤ y ≤ B; d ≤ z ≤ d + h). Use is made of linearized potential flow theory. The flow is assumed to periodic in time at a frequency ω: (x, y, z, t) = ϕ(x, y, z) cos(ωt + ). In the moonpool the reduced potential ϕ verifies the
2
B. Molin / Applied Ocean Research 67 (2017) 1–8
The bottom boundary condition writes A0 +
∞
Am cos km x
m=1
=
1 2
a
B
B0 /d +
0
0
(x
∞
B k n=1 n n
− x )2
+ (y
cos kn x − y )2
dx dy
(4)
Through integrations this can be transformed into the vectorial equation (see [6]): = MAB · B A
(5)
= (A0 , . . .An ), B = (B0 , . . .Bn ). with A In z = d we match ϕ1 and ϕ2 and their vertical derivatives. We follow Garrett’s method [2]. Consider first the equality of the potentials: z + (Am cosh km d + Bm sinh km d) cos km x d ∞
A0 + B0 = C0 +
∞
m=1
(6)
Cn cos n (x + b)
n=1
Fig. 1. Types of moonpools on drillships. Taken from [5].
which holds for 0 ≤ x ≤ a. Again we take advantage of the orthogonality of the set [coskn x] over [0 a]. Integrating each side in x over [0 a], then multiplying with coskm x and integrating again for all m, the following vectorial equation is obtained: + DB B = MC C A
(7)
where DB is the diagonal matrix (1, tanhkm d), MC is a full matrix and C = (C0 , . . .Cn ). Consider now the vertical velocities. They obey the equations Fig. 2. Geometry.
ϕ2z = ϕ1z 0 ≤ x ≤ a
Laplace equation ϕ = 0, no-flow conditions at the solid boundaries ϕn = ∂ϕ/∂n = 0, the free surface condition g ϕz − ω2 ϕ = 0 at z = d + h, and a matching condition with the flow in the lower fluid domain, written as [6]: ϕ(x, y, 0) =
1 2
a
dx 0
B
dy 0
ϕz (x , y , 0)
(x − x )2 + (y − y )2
(1)
Typical moonpool dimensions in drillships are about 40–50 m in length and 10 m in width, and we are here interested in the piston and longitudinal sloshing modes. So we make the simplifying assumption that the moonpool is narrow, so that the flow inside can be idealized as two-dimensional. Note that the flow in the lower fluid domain z ≤ 0 is three-dimensional. As in [6] we use eigen-function expansions to represent the flow in the moonpool: z ϕ1 (x, z) = A0 + B0 + (An cosh kn z + Bn sinh kn z) cos kn x d ∞
(2)
in sub-domain 1 (0 ≤ z ≤ d), with kn = n/a, and
That is D0 + m Dm cos m (x + b) h ∞
m=1 ∞
B0 = kn (An sinh kn d + Bn cosh kn d) cos kn x + d =0
0≤x≤a
(9)
n=1
− b ≤ x ≤ 0 and a ≤ x ≤ a + c
Now we make use of the orthogonality of the set [cosm (x + b)] over [−b a + c]. Integrating each side over their domains of validity, then multiplying with cosm (x + b) and integrating again for all m, gives + MDB B = MDA A D
(10)
= (D0 , . . .Dn ). with D Finally the free surface equation g ϕ2z − ω2 ϕ2 = 0 gives
C + D4 D
(11)
= (D0 , . . .Dn ) and D1 , D2 , D4 diagonal matrices: with D
z−d + (Cn cosh n (z − d) h ∞
D1 = (0, g n tanh n h) D2 = (g/h, g n )
n=1
+Dn sinh n (z − d)) cos n (x + b)
(8)
0 − b ≤ x ≤ 0 and a ≤ x ≤ a + c
= ω2 D1 C + D2 D
n=1
ϕ2 (x, z) = C0 + D0
ϕ2z =
(3)
in sub-domain 2 (d ≤ z ≤ d + h), with n = n/(a + b + c) = n/L. The Laplace equation and the no-flow conditions at the end walls are ensured.
(12)
D4 = (1, tanh n h) From (5) and (7) we get: = (MAB + DB )−1 MC C B
(13)
B. Molin / Applied Ocean Research 67 (2017) 1–8
3
Fig. 3. Base case. Moonpool dimensions. Taken from [4].
and then, from (10):
3. Illustrative results
= (MDA MAB + MDB ) (MAB + DB )−1 MC C = MDC C D
(14)
and, from (11), the following eigen-value problem is obtained (D1 + D2 MDC ) C = ω2 (I + D4 MDC ) C
(15)
with I the identity matrix. The resolution of this eigen-value problem gives the resonant frequencies ωi and the associated eigen-vectors Ci . Practically the series are truncated after 10 or 20 terms, which provides a sufficient accuracy. 2.1. Frozen restriction approximation In Molin [6] it is established that a good approximation of the natural frequency of the piston mode can be obtained by assimilating the fluid inside the moonpool as a solid body. Likewise, in the case of axisymmetric moonpools with restrictions, it has been found (see [7] or [8]) that an approximation of the natural frequency of the piston mode can be derived by ”freezing” the fluid domain inside the restriction, that is considering it as a solid body. The free heave motion of the frozen restriction obeys the equation ( a B d + Mal + Mau (ω)) Z¨ + g
B a2 Z=0 L
(16)
with a B d the mass of the restriction, Mal the added mass due to the lower fluid domain (z ≤ 0), and Mau the added mass due to the upper fluid domain (d ≤ z ≤ d + h). The added mass Mal is given in [6] as Mal = B2 a f(B/a) with f (r) = −
1 3
1
argsinh r −1 + r −1 argsinh (r) +
1 + r −2
r2 + 1
1 3
r + r −2
.
(17)
The added mass from the upper fluid domain is obtained as (see Appendix): Mau (ω) = +2
B L
B a2 h L
∞ 2 [sin n (a + b) − sin n b] g n − ω2 tanh n h n=1
3n
g n tanh n h − ω2 (18)
The natural frequency can then be obtained, in principle, from the intersection of the two curves y = ω and y = gBa2 /[L( aBd + Mal + Mau (ω))].
3.1. Comparison with the experimental results of Guo et al. [4] In two independent papers Guo et al. [3,4] present an experimental and numerical study of a drillship with a moonpool having a recess on the aft side. The moonpool is 45.6 m long and 11.2 m wide. The recess has a length (c) of 16 m. The height of the restriction (d) is 7.2 m, the waterheight in the recess (h) is 3.8 m (see Fig. 3, taken from [4]). The tests, at a scale 1:50, are performed in regular and irregular waves, at different headings from 180◦ (head waves) to 90◦ (beam waves). Experimental RAOs of the free surface elevation (relative to the moving ship) are provided at three locations along the moonpool. They exhibit four peaks, at angular frequencies (in full scale) of 0.42 rad/s, 0.82 rad/s, 1.07 rad/s and 1.36 rad/s. The authors interpret the second peak frequency (0.82 rad/s) as the piston mode and the third one (1.07 rad/s) as the first sloshing mode. This interpretation is based on wrongly using Molin’s single mode approximation formulas [6] which are valid only for a moonpool with vertical walls from deck to keel. As for the first resonant peak at 0.42 rad/s, the authors attribute it to “some coupling between the heave and pitch motions” (!). A striking feature is that, at this first resonant frequency, the measured RAOs at the aft side of the moonpool (by the end wall of the recess) are 5–10 times larger than the RAOs measured at the other end. At the second peak frequency it is the other way around: the measured RAOs at the bow side wall are about 50% larger than at the recess wall. In an other paper on drillships with recesses [9] the moonpool dimensions are similar: 38.4 m long, 12.5 m wide, the restriction has a height of 7.5 m and a length of 25.6 m, the operating draft is 12 m. Unfortunately the resonant frequencies are not given, so we take the moonpool from [3,4] as a base case. Fig. 4 shows the experimental RAOs of the relative free surface elevation at the gauges inside the moonpool, as given in [3]. They were obtained from tests in irregular waves with a rather small significant wave height (2 m) and the wave energy uniformly distributed from 0.2 rad/s up to 1.4 rad/s. The vertical bars show the first 4 natural frequencies given by our model, that is 0.414 rad/s for the piston mode and 0.778 rad/s, 1.074 rad/s and 1.339 rad/s for the first three sloshing modes. Our predictions agree well with the positions of the experimental peaks, except for the first sloshing mode where our value is about 5% lower than measured; this discrepancy could be due to the fact that the drillship is not fixed, but free to respond to the waves. As a matter of fact peaks are observed at this frequency in the heave and pitch RAOs. Hydrodynamic coupling between the moonpool and ship dynamics results in a slight shift of the resonant frequencies. Fig. 5 shows the calculated modal shapes of the free surface elevation. They are normalized with a maximum absolute value of one along the length of the moonpool. It is striking that, for the piston
4
B. Molin / Applied Ocean Research 67 (2017) 1–8 6
90 dg 135 dg 150 dg 180 dg
5
4
3
2
1
0
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9 angular frequency (rad/s)
1
1.1
6
1.2
1.3
1.4
1.2
1.3
1.4
90 dg 135 dg 150 dg 180 dg
5
4
3
2
1
0
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9 angular frequency (rad/s)
1
1.1
Fig. 4. Experimental RAOs of the relative free surface elevation at the bow wall (top) and at the recess wall (bottom), for different ship headings.
10
1 7.5
0.8 0.6
5 added mass coefficient
0.4 0.2 0 -0.2 -0.4
mode 0 mode 1 mode 2 mode 3
-0.6
2.5
0
-2.5
-5
-7.5
-0.8 -1
-10
0
5
10
15
20
25
30
35
40
45
0
0.2
0.4
0.6 0.8 omega (rad/s)
1
1.2
1.4
x (m)
Fig. 5. Base case. Calculated modal shapes.
mode (mode 0), the maximum value is obtained at the end of the recess and that at the other end the free surface motion is ten times lower. This is in qualitative agreement with the experimental RAOs. Likewise, for the first sloshing mode (mode 1), the amplitude of free surface motion in the recess is about 60% of the amplitude at the bow wall. This again agrees with the experimental observations. As for the second and third sloshing modes, the amplitude of free surface motion is about the same at both ends (again in agreement with the experiments). As for applying the frozen restriction approximation, Fig. 6 shows the non dimensional added mass Mau (ω)/( B Lh) vs the frequency and Fig. 7 shows the graphical determination of the res-
Fig. 6. Base case. Non dimensional added mass Mau /( B hL) vs frequency.
onant frequency, from the intersections of the two curves y = ω and gBa2 /[L( aBd + Mal + Mau (ω))]. Due to the singular behavy= ior of the added mass Mau (ω) there are several intersection points, none of them giving a value close to the expected one. As a matter of fact a better value is obtained by taking the added mass Mau at a zero frequency, which gives a resonant frequency of 0.44 rad/s, about 6% too high. In the following paragraphs, we vary some geometric parameters of the base case moonpool. First we vary the length of the recess, then we vary the waterheight in the recess, finally we consider the case of two recesses.
B. Molin / Applied Ocean Research 67 (2017) 1–8
5
6
1 0.9
5 0.8 0.7
4 omega (rad/s)
0.6 0.5
3
0.4
2
0.3 0.2
1 0.1 0 0
Fig.
0.1
0.2
0.3
7. Intersections
of
0.4
the
0.5 0.6 omega (rad/s)
two
curves
0.7
0.8
y=ω
0.9
and
1
y=
gBa2 /[L( aBd + Mal + Mau (ω))].
0 0
4
8
12 16 20 length of right recess (m)
24
28
32
Fig. 9. Varying length of the recess. Non dimensional added mass Mau (0)/( B hL). 1.1 1
1.2 mode 0 frozen appr. mode 0 exact mode 1
0.9
1
0.8 0.7
0.8 omega (rad/s)
0.6 0.5 0.6
c=0m c=2m c=4m c=8m c = 16 m c = 32 m
0.4 0.3
0.4
0.2 0.1
0.2
0 0 0 0
4
8
12 16 20 length of right recess (m)
24
28
32
Fig. 8. Varying length of the recess. Natural frequencies of the piston and first sloshing mode.
3.2. Varying the length of the recess We keep all geometrical parameters the same as in the base case, except for the length of the recess that we vary from 0 (no recess, moonpool with vertical walls all the way from deck to keel) up to 32 m (twice the base case length). Fig. 8 shows the natural frequencies of the piston and first sloshing mode. For the piston mode the value delivered by the frozen restriction approximation (with the added mass Mau taken at zero frequency) is also shown. It can be seen that this approximation is in fair agreement with the exact value. It can also be observed that the natural frequency of the piston mode decreases strongly as the length of the recess increases. This is partly due to the decrease of the hydrostatic stiffness in Eq. (16). But it is also due to the increase of the added mass Mau : Fig. 9 shows the variation of this added mass, normalized with the mass of water in the upper part of the moonpool B hL, vs the recess length c. It is striking that, from the expected value 1 at c = 0, after an initial hollow, this non dimensional added mass steadily increases. As a matter of fact, from Eq. (18), it is easy to establish that there is an asymptotic value, as c goes to infinity, which is a2 /(3h2 ), that is 20.2 in the present case. It is non intuitive that such high values can be attained for the added mass.
10
20
30
40
50
x (m)
Fig. 10. Varying length of the recess. Modal shape of the piston mode.
Fig. 10 shows the modal shapes of the free surface elevation in the moonpool, for the piston mode. They are shown for different values of the recess length (0 m, 2 m, 4 m, 8 m, 16 m, 32 m). When there is no recess (c = 0 m), the free surface is nearly flat, with a small bump in the center. As soon as there is a recess the point of maximum elevation moves to the recess wall, and the value attained at the opposite wall steadily decreases as the recess length increases. (Note that in the case c = 32 m the value at the recess wall is also one but the figure has been truncated for the sake of clarity). Finally Fig. 11 shows the modal shapes of the first sloshing mode. They appear to be quite different depending on the recess length. At c = 0 m (no recess), due to the relatively large draft (11 m) compared with the length (29.6 m) of the moonpool, it is just a sinusoidal shape, with 1 (in absolute value) at both walls. At small recess lengths (2 m and 4 m), it drops slightly at the bow wall, when it remains equal to one (in absolute value) at the recess wall. As the recess length keeps one increasing, it goes back to one at the bow wall and steadily decreases at the recess wall, until a drastic change occurs: at c = 32 m the shape has become non monotonic, with an intermediate maximum in-between the two walls. 3.3. Varying waterheight in the recess Now, keeping again all other geometric parameters the same as in the base case, we vary the water-height h of the upper part
6
B. Molin / Applied Ocean Research 67 (2017) 1–8 1.1
1 c=0m c=2m c=4m c=8m c = 16 m c = 32 m
0.8 0.6
1 0.9 0.8
0.4
0.7
0.2
0.6
0
0.5
-0.2
h=1m h=2m h=4m h=8m
0.4
-0.4
0.3
-0.6
0.2
-0.8
0.1
-1 0
10
20
30 x (m)
40
50
0
60
0
5
10
15
20
25
30
35
40
45
x (m)
Fig. 11. Varying length of the recess. Modal shape of the first sloshing mode.
Fig. 13. Varying waterheight h. Modal shape of the piston mode.
1 mode 0 frozen appr. mode 0 exact mode 1
0.9
1 h=1m h=2m h=4m h=8m
0.8 0.8
0.6 0.7
omega (rad/s)
0.4 0.6
0.2 0.5
0 0.4
-0.2
0.3
-0.4
0.2
-0.6
0.1
-0.8
0 0
1
2
3
4 5 6 7 water height in right recess (m)
8
9
10
11
-1 0
5
10
15
20
25
30
35
40
45
x (m)
Fig. 12. Varying waterheight h. Natural frequencies of the piston and first sloshing mode.
1 mode 0 frozen appr. mode 0 exact mode 1
0.9 0.8 0.7
omega (rad/s)
of the moonpool, from 0 m up to 11 m, the total draft h + d being kept equal to 11 m. This means that the height d of the restriction is 11 m−h. Fig. 12 shows the natural frequencies of the piston and first sloshing modes, together with the frozen restriction approximation of the piston mode. Again the frozen restriction approximation is doing a good job. Fig. 13 shows the modal shapes of the piston mode for 4 different h values (1 m, 2 m, 4 m, 8 m). They look similar, with, as usual, the maximum value at the recess wall, and the minimum value, at the facing wall, getting lower and lower as the waterheight h decreases. Finally Fig. 14 shows the modal shapes of the first sloshing mode, with the maximum value at the wall opposite to the recess, except in the shallowest case where an intermediate maximum appears, alike the case c = 32 m in Fig. 11.
Fig. 14. Varying waterheight h. Modal shape of the first sloshing mode.
0.6 0.5 0.4 0.3 0.2 0.1 0 0
3.4. Two recesses Finally we consider the case when two recesses are present in the moonpool. Again we start from the base case geometry, and we keep constant the total recess length: b + c = 16 m. Fig. 15 shows the natural frequencies of the piston and first sloshing modes, vs the length b of the left recess. It can be seen that, again, the frozen approximation provides a fair estimate of
2
4
6 8 10 length of left recess (m)
12
14
16
Fig. 15. Two recesses. Natural frequency of the piston and first sloshing mode.
the natural frequency of the piston mode. Depending on the relative lengths of the recesses the natural frequency varies somewhat. Note that, in the considered case, the hydrostatic stiffness in Eq. (16) remains constant but, as Fig. 16 shows, the zero frequency added
B. Molin / Applied Ocean Research 67 (2017) 1–8
7
Finally Fig. 18 shows the modal shapes of the first sloshing mode, with the maximum elevation being now attained at the wall of the shortest recess.
3
2.5
4. Final comments 2
1.5
1
0.5
0 0
2
4
6 8 10 length of left recess (m)
12
14
16
Fig. 16. Two recesses. Non dimensional added mass Mau (0)/( B Lh).
1.1 1 0.9 0.8
We have proposed an extension of Molin’s model [6] to rectangular moonpools with one or two recesses. This model delivers the natural frequencies and associated modal shapes of the piston and longitudinal sloshing modes. Good agreement has been obtained with the experimental results of [4]. A remarkable feature obtained for the piston mode is that the point of largest motion of the free surface is at the wall of the recess. A simple formula, based on assimilating the restriction part of the moonpool with a solid body, has been proposed and found to give a good approximation of the natural frequency of the piston mode. It must be stressed out, however, that not all geometric parameters of the moonpool have been covered. Therefore the formula should be used with care. We have tried to derive similar simple approximation for the natural frequency of the first sloshing mode, with no success so far. It must be emphasized that the results presented here have been obtained within the scope of linearized potential flow theory. Practically, when the waterheight in the recess is shallow, nonlinear effects are expected to quickly occur, all the more since we have obtained that, in the piston mode, the largest motion of the free surface takes place in the recess.
0.7
Acknowledgements
0.6 0.5
The author is grateful to Prof. Haining Lu, of Shanghai Jiao-Tong University, for providing the experimental RAOs.
b=0m b=2m b=4m b=6m b=8m
0.4 0.3
Appendix A. Added mass from the upper fluid domain
0.2 0.1 0 0
5
10
15
20
25
30
35
40
45
To obtain the added mass Mau (ω) we need to derive the Dn coefficients of the ϕ2 expansion that verify
x+b (m)
D0 m Dm cos m (x + b) + h ∞
ϕ2z (x, d) =
Fig. 17. Two recesses. Modal shape of the piston mode.
m=1
= 1 for 0 ≤ x ≤ a
b=0m b=2m b=4m b=6m b=8m
1 0.8
= 0 for − b ≤ x ≤ 0 and a ≤ x ≤ a + c
(19)
0.6
Taking again advantage of the orthogonality of the cosn (x + b) functions, we obtain
0.4 0.2
D0 =
0 -0.2
ah 2 [sin n (a + b) − sin n b] Dn = L 2n L
(20)
The Cn coefficients follow from the free surface condition:
-0.4 -0.6
C0 = D0
g
ω2 h
-0.8 -1
Cn = −Dn 0
5
10
15
20
25 x+b (m)
30
35
40
45
Fig. 18. Two recesses. Modal shape of the first sloshing mode.
−1
g n − ω2 tanh n h g n tanh n h − ω2
The added mass is obtained as
a
Mau (ω) = − B mass Mau (0) varies quite a lot depending on the relative lengths of the recesses. Fig. 17 shows the modal shapes of the piston mode. The maximum elevation is obtained at the wall of the longest recess.
(21)
ϕ2 (x, d) dx 0
= − B
a
[C0 + 0
∞ n=1
Cn cos n (x + b)] dx
(22)
8
B. Molin / Applied Ocean Research 67 (2017) 1–8
which gives Mau (ω) = +2
B L
B a2
h
L
∞ 2 [sin n (a + b) − sin n b] g n − ω2 tanh n h n=1
3n
g n tanh n h − ω2
(23)
Note that the first term g D0 /(ω2 h) in C0 is actually an hydrostatic term (the hydrostatic stiffness that appears in Eq. (16)). References [1] R. Barreira, S.H. Sphaier, I.Q. Masetti, A.P. Costa, C. Levi, Behavior of a mono-column structure (MONOBR) in waves, in: Proc. of the International Conference on Offshore Mechanics and Arctic Engineering, OMAE2005, Halkidiki, Greece, 2005. [2] C.J.R. Garrett, Wave forces on a circular dock, J. Fluid Mech. 46 (1971) 129–139.
[3] X. Guo, H. Lu, J. Yang, T. Peng, Study on hydrodynamic performances of a deep-water drillship and water motions inside its rectangular moonpool, in: Proc. of the Twenty-sixth (2016) International Ocean and Polar Engineering Conference, ISOPE, Rhodes, 2016. [4] X. Guo, H. Lu, J. Yang, T. Peng, Resonant water motions within a recessing type moonpool in a drilling vessel, Ocean Eng. 129 (2017) 228–239. [5] E. Hammargren, J. Törnblom, Effect of the moonpool on the total resistance of a drillship, M.Sc. Thesis), Chalmers University of Technology, 2012. [6] B. Molin, On the piston and sloshing modes in moonpools, J. Fluid Mech. 430 (2001) 27–50. [7] B. Molin, I. de Vries, A. Cinello, Hydrodynamic analysis of the piston mode resonance inside a large FLNG turret, in: Proc. 29th Int. Workshop on Water Waves and Floating Bodies, Osaka, 2014. [8] B. Molin, X. Zhang, On resonant modes in moonpools with restrictions or recesses, in: Proc. 32nd Int. Workshop on Water Waves and Floating Bodies, Dalian, 2017. [9] H.-J. Son, S.-H. Choi, M.-W. Kim, S.-H. Hwangbo, Drag reduction of recess type moonpool under vessel’s forward speed, in: Proc. of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering, OMAE2008, Estoril, 2008. [10] S.H. Sphaier, F.G.S. Torres, I.Q. Masetti, A.P. Costa, C. Levi, Monocolumn in waves: experimental analysis, Ocean Eng. 34 (2007) 1724–1733.