Quadratic dissipation effect on the moonpool resonance

This paper adopted a semi-analytical method based on eigenfunction matching to solve the problem of sharp resonance of cylindrical structures with a moonpool that has a restricted entrance. To eliminate the sharp resonance and to measure the viscous effect, a quadratic dissipation is introduced by assuming an additional dissipative disk at the moonpool entrance. The fluid domain is divided into five cylindrical subdomains, and the velocity potential in each subdomain is obtained by meeting the Laplace equation as well as the boundary conditions. The free-surface elevation at the center of the moonpool, along with the pressure and velocity at the restricted entrance for first-order wave are evaluated. By choosing appropriate dissipation coefficients, the free-surface elevation calculated at the center of the moonpool is in coincidence with the measurements in model tests both at the peak period and amplitude at resonance. It is shown that the sharp resonance in the potential flow theory can be eliminated and the viscous effect can be estimated with a simple method in some provided hydrodynamic models.


Introduction
A vertical cylindrical structure-due to its simple geometric form-is not only regarded as an academic case in hydrodynamic analysis, but also considered to be fundamental in practical applications for offshore structures like Spars, buoys, drilling rigs, TLPs and Sevan FPSOs. Interaction between cylindrical structures and waves has been studied by researchers in the past decades.
The semi-analytical method that is based on eigenfunction matching was thus proposed. This pioneer work can be traced to MacCamy and Fuchs (1954) on the simplest case with a vertical cylinder standing on the seabed. Miles and Gilbert (1968) formulated the scattering problem of the surface waves with a circular dock, and they obtained a variational approximation to the far field. Since then, many researchers have studied various types of vertical cylinders (truncated and with different diameters) by using the socalled eigenfunction matching method that divides the fluid domain into several cylindrical subdomains and matches the eigenfunction expansions in different subdomains. Many researches have been carried out on this topic such as a series of different configurations of a single cylinder by Liu et al. (2012). The key issue in all the above studies is the analytic-al method based on the series expansion of eigenfunctions which meets the Laplaces equation, so the velocity potentials around cylindrical structures can be evaluated and the wave loads can be obtained. The fluid domain around the complex structure hull is divided into an array of cylindrical subdomains. In each domain, the eigenfunctions are expanded and the unknown coefficients in expansion series are determined by the boundary condition on the cylinder hull as well as by the continuous conditions through the control surfaces that separate the subdomains.
The moonpool equipped cylindrical structure which contains dissipation effect was developed by Garret (1970), who took a bottomless cylinder equivalent to a cylinder with a fully open moonpool. Miloh (1983) presented a method to estimate the wave loads, wave motions and the hydrodynamic coefficients of the sway, roll as well as the coupling terms. By using the first-order potential, he calculated the first-order force, elevation at free surface, and added masses and damping coefficients to them. Mavrakos (1985) solved the problem of wave loads on a stationary floating bottomless cylinder within a finite wall thickness. Cylindrical moonpool equipped structures with restricted entrance was presented by Sphaier et al. (2007), who measured the water column motion in the fixed moonpool in a model test. Various diameters of the moonpool entrance at the bottom of mono column were tested in the same wave conditions, and it was found that large vertical motions happened at the resonant period to a cylindrical structure without moonpool and to a structure with a fully open moonpool.
However, the classical potential flow theory does not resolve the tip resonance of the floating moonpool structures. Hence a refined model was proposed by Chen et al. (2011) to reduce the resonant amplitude by taking dissipation into consideration. He applied the linear cylindrical dissipative surface to the case of a bottomless cylinder with zero thickness walls and compared the numerical results with the model test done by Miloh (1983).  improved Linton and Evans' method (1990) with consideration of a group of vertical cylinders standing on the seabed under the action of prescribed regular heading waves and introduced a dissipation term in the boundary condition on the free surface. Recently, Chen et al. (2015) have introduced the free-surface dissipation for wave diffraction of moonpool equipped cylindrical structures with restricted entrance by using the semi-analytical solution, and the free-surface elevation on the center of the moonpool is positive in coincidence with measurements in model test.
To better understand dissipation effect, some researchers adopted the method of in-fluid dissipation. Qin et al. (2011) adopted this method to introduce dissipation to study the potential flow generated by a source in a wave tank, in which the viscous effect and the partial reflection from side walls were taken into account. In the development of a multi-domain boundary element method raised by Chen and Duan (2012), the dissipation is introduced in the same way, and the motion of the water column in a fully open moonpool inside a very thin hull wall structure is successfully anticipated.
As approaches to estimate the overall fluid viscous effect, it seems that the free-surface and linear in-fluid dissipation attracted too much attention while few researchers paid their attention to the quadratic in-fluid dissipation. In fact, neither the parametric form nor location selection of dissipation can be confirmed from current researches. A series of quadratic dissipations are introduced in the present work at the entrance of the moonpool based on a tested theoretical model. The updated division of fluid with imaginary dissipative disk was conducted and the velocity potential in each subdomain was obtained by using eigenfunction expansion matching method. The free-surface elevations in the moonpool center were calculated with various quadratic dissipations and entrance radius of the moonpool under linear regular wave excitation. The numerical results were compared with those in the model test results.

Mathematical formulations
Large quantity of the water column in a moonpool oc-curs in a wave period close to that of moonpool resonance, as demonstrated in MonoBr, which is presented by Sphaier et al. (2007) in the model tests. However, numerical predictions based on potential flow exaggerate the motion magnitude since no damping due to fluid viscosity is taken into account. In order to eliminate the sharp resonance and measure the viscous effect, an in-fluid quadratic dissipation is introduced in MonoBr by assuming that there is an additional dissipative disk at the restricted entrance of the moonpool. Fig. 1a shows a assumption diagram of MonoBr and dissipative disk with uniform opening (open porosity is 50%) located at the restricted entrance of the moonpool. As depicted in Fig. 1b, we defined the cylindrical coordinated system (r, θ, z) by its origin location at the center of the MonoBr and on the mean plane of the free surface. The axis oz is vertically upward.
2.1 Boundary element method with in-fluid quadratic dissipation By assuming that the fluid is perfect and the flow is irrotational, fluid velocity V(r, θ, z, t) can be represented by the gradient of a potential function Ψ(r, θ, z, t) which satisfies the Laplace's equation in the fluid domain: (1) Under the infinitesimal wave assumption, we consider that only the linear potential is proportional to the wave steepness. Furthermore, an analysis in the frequency domain is performed so that the factor exp(-iωt) representing the time-harmonic variation will be omitted in the following for the sake of simplicity.
On the mean free surface, the combined kinematic and dynamic boundary condition is provided by: with ω being the wave frequency, and g, the acceleration due to gravity.
On the cylinder hull (cylinder surface and base of the bottom or top) is expressed by: where V n is the cylinder's velocity in the direction that is normal to the hull. As for the diffraction problem, we have V n =0.
In the same way, the boundary condition on the sea bed is: In addition, we have a radiation condition requiring all perturbations for cylinders disappear at infinity, i.e.
to ensure the uniqueness of the solution and it is called the Sommerfeld radiation condition where Φ 0 represents the velocity of incoming waves. Indeed, the potential in Eq. (4) of incoming waves can be given as: and the known coefficients are where (A, ω, k 0 ) denote the amplitude, frequency and wavenumber of incoming waves propagating along the positive x-axis respectively. The wave number k 0 is defined in the dispersion equation k 0 tanh(k 0 h)=f 2 which is obtained by satisfying the boundary condition Eq.
(2) at the free surface and Eq. (4) on the seabed. In order to further introduce the dissipation effect, we introduce dissipative surface at the entrance of the moonpool where the large dissipation occurs in a real fluid. Across the dissipative surface, there is a difference between dynamic pressures: (9) The function f(•) that is dependent on the normal ∂ n Φ can be linear or quadratic while ∂ n Φ is continuously across the dissipative surfaces.
Derived from Eq. (9) by assuming a quadratic dependence of dynamic pressure change with respect to the fluid velocity, in-fluid quadratic disk dissipation condition can be written as: where [Φ] and [∂ n Φ] stand for the difference between Φ and ∂ n Φ along the positive direction of the normal vector, respectively. The coefficient μ is a positive constant to characterize the dissipation effect.

Domain division and potential expansions in each subdomain
The fluid domain surrounding the monocolumn structure with the moonpool of restricted entrance is divided into five cylindrical subdomains, as shown in Fig. 1. They are listed as follows: • E (R E ≤r≤∞, -h≤z≤0): External subdomain exterior to the structure; • B (R I ≤r≤R E , -h≤z≤-d): Below subdomain underneath the structure; • T (R I ≤r≤R T , -d≤z≤0): Top subdomain due to the restricted entrance; • L (0≤r≤R I , -h≤z≤-d): Lower interior subdomain of the moonpool below the dissipative disk; • U (0≤r≤R I , -d≤z≤0): Upper interior subdomain of the moonpool above the dissipative disk.
In the same way of incoming wave Eq. (6) propagating along the positive x-axis, the velocity potential due to diffraction in each above subdomain, namely, Φ E in the exterior subdomain E, Φ B in the below subdomain B, Φ T in the top subdomain T, Φ L in the lower interior subdomain L, and Φ U in the upper interior subdomain U can be expressed in a generic manner as: Following Garrett (1970), the velocity potential in the exterior subdomain E is written as: ϕ l E and k n defined by k n tan(k n h)=-f 2 for n≥1. The function Z 0 (k 0 , z, h) is defined in the incoming waves and ensures to satisfy the boundary condition Eq. (2) at the mean free surface and Eq. (4) on the sea bed. Furthermore, the radiation condition (5) is well respected according to the asymptotic property of the Hankel function H l (•) = J l (•)+iY l (•) and that of the modified Bessel function K l (•). Indeed, the first term on the right-hand side of Eq. (12) involving the wavenumber k 0 represents the propagating wave mode, while the second term involving the wavenumber k n for n>1 represents the evanescent modes disappearing exponentially for large k n r.
The velocity potential in the below subdomain B can be expressed as: in which the function and are defined by P 0

The constants
are given by: Furthermore, the wavenumber λ n is defined by λ n =nπ/ (h-d) to satisfy the boundary condition Eq. (3) on the cylinder hull (R I ≤r ≤R E , z=-d) and Eq. (4) on the seabed. The velocity potential in the top subdomain T is written as: with the wave number defined by γ 0 tanh(γ 0 d)=f 2 and γ n tan(γ n d)=-f 2 for n≥1 which is derived from the fact that the potential satisfies the boundary condition Eq.
(2) at the mean free surface. Furthermore, the boundary condition Eq.
(3) on the hull at the top of the moonpool restriction (R I ≤r ≤R T , z=-d) and cylinder hull (r=R T , -d ≤z ≤0) is also satisfied.
ϕ l T The velocity potential in the lower interior subdomain L below the dissipative disk can be expressed with the sum of two expansions: (18) The first expansion is assumed to be a transparent circular disk of zero thickness and is written as: It has the same function Z n (k n , z, h) and wavenumber k n for n≥0 as those for since satisfies the same boundary condition as at the mean free surface and the sea bed. The second expansion is written as the so-called Fourier-Bessel series that is composed of eigenfunctions: . (20) The wavenumber is the root of the positive zero of Bessel function: ϕ l L which is obtained by making satisfy the boundary condition Eq. (4) at the seabed. ϕ l L ϕ l U Also, in a similar way of velocity potential , the velocity potential in the upper interior domain U above the dissipative disk is expressed by the sum of two expansions: ϕ l U The Fourier-Bessel series for is: E l n in which, the constant is given by: ϕ l U which is obtained by making satisfy the boundary condition Eq. (2) on the free surface.
In the eigenfunction expansion for the velocity potentials and , the unknown coefficients , , , , , and for n≥1 and l≥0 are to be determined by matching the potential and its normal derivative on the juncture boundaries surface associated with dissipation surface.

Matching equation on the juncture boundaries surface associated with dissipation surface
There are three matching surfaces associated with one dissipative surface between the five subdomains as shown in Fig. 1. They are listed as follows: • S EB (r=R E , -h≤z≤-d): matching the surface shared by domain E and domain L; • S BL (r=R I , -h≤z≤-d): matching the surface shared by domain B and domain L; • S TU (r=R I , -d≤z≤0): matching the surface shared by domain T and domain U; • S UL (0≤r≤R I , z=-d): the dissipative surface connected by domain U and domain L.
The continuous condition of velocity potentials and normal derivative of velocity potential provide: on matching surfaces S EB , S BL , and S TU , respectively. v l D (r) On the dissipative disk surface for 0≤r≤R I , z=-d, it can be easily checked that the normal velocity across the dissipative disk is continuous and is used to demonstrate the velocity: v l .
(26) ∆ϕ l D (r) However, the velocity potential on the two sides of the dissipative disk surface is different and is used to demonstrate the drop of the velocity potential: With Eq. (10), identity and are used to express the quadratic dissipation boundary condition: in which μ is the quadratic dissipation coefficients.
Taking the advantage of the orthogonality, in the socalled Garrett's method (Garrett, 1970), multiplying Eqs. (25) and (28) by its orthogonality function respectively, making use of the fact that =0 at r=R E , -d≤z≤0, for n=0, …N, we obtain six sets of equations in Galerkin's form: The index nl≥1 in Eq.(26). In this way, six sets of equations above can determine fully the five sets of the unknowns coefficients , , , , , and for n≥1 and l≥0 in the eigenfunction expansion for the velocity potentials.

Explanation of the dissipation
As stated in Section 2, the introduced dissipation can be explained by the velocity continuity and the pressure gradient at the restricted entrance. The numerical work is involved in the choice of the number of terms used in infinite simulations. The former 100 terms (N=100) are adopted in the infinite summations to compute numerical results because the infinite summations have excellent truncation characteristics as in Chen et al.'s (2015). The pressure and velocity at the restricted entrance can be obtained according to the velocity potentials by using the Bernoullis Equation and differentiation, and it is given by: for the pressure below the dissipative disk; for the pressure above the dissipative disk; ) for the velocity at the dissipative disk. Fig. 2 depicts the pressure above and below the dissipative disk with θ=0 and θ=π for various quadratic dissipative coefficients. With the enlargement of the dissipative coefficients, the pressure decreases rapidly while the difference between the lower and upper pressure increases. More interestingly, there are pressure fluctuations along the entrance radius for μ>0. In addition, the difference between the lower and upper pressure at the negative part of the x-axis is higher than that at the positive part when the dissipative coefficients increase.
As a measure of input energy obtained by the mono column, the continuous velocity without dissipation at the restricted entrance is much larger than that with dissipation, as shown in Fig. 3. This means that the dissipative disk dissipates so much energy offered by the incoming wave. Similarly, there are fluctuations of the velocity along the entrance radius for μ>0 and more obvious for larger coefficients.

Water motion in the moonpool
In this section, the mono-column with a moonpool of a restricted entrance is considered based on the theoretical formulations development. Indeed, Sphaier et al. (2007) have done this model through the model test. Their work consists of measuring the water column in the moonpool with a restricted entrance at a fixed position. Various diameters of the moonpool entrance at the bottom of a monocolumn are tested under the same wave conditions. The main dimensions of the structures employed here are the same as those in Sphaier et al. (2007) with R E =47.5 m for the external radius of the main hull, the draft d=38 m, and the size of the moonpool main hull R T =34.5 m. The radius of the openness at the bottom of the moonpool varies from R I =34.5 m (moonpool without restriction), 32 m, 29 m, 23.5 m, 16.5 m to 11.5 m and the water depth h=200 m (with reference to Fig. 1) is adopted.
In the presence of regular waves of periods from 5 to 35 s, as the measurement of viscous effect, the parametric nonlinear dissipative coefficient μ in Eq. (28) is introduced at the entrance of the moonpool. Comparisons are made between the values of the water column motions from three operations: model tests and numerical calculations with and without nonlinear dissipation. The dimensionless vertical motion of the water column in the moonpool can be represented by the free-surface elevation at the moonpool center given by: The case of a fully open moonpool with an entrance radius R I =R T =34.5 m is first considered and the values of the water-column motions obtained by three methods are depicted in Fig. 2.
The values from the form of model tests are represented by square symbols. The numerical results using Eq. (30) without dissipation are depicted by dotted line and solid line for the one with nonlinear dissipation associated with coefficient μ=0.002 in Eq. (28).
The numerical results with nonlinear dissipation effects are in coincidence with the measurement of the model test. As illustrated in Fig. 4, there is a peak at the resonant peri-  od T 0 ≈15.6 s in all the forms. In addition, near the resonant period, numerical computations without dissipation give much higher values than those results with nonlinear dissipation, which are very close to those of the model tests. To estimate the low-frequency resonance period of moonpools, a number of analyses were performed by researchers including Molin (2001). Assuming that the water mass inside a moonpool is a rigid body, there will be a balance between the hydrostatic restoring force and inertial force which occurs at the resonance of the so-called piston mode. Thus the reaction of fluid at the moonpool bottom plays the role of adding mass and then the first resonant period of the piston mode can then be written as: in which the added-mass coefficient α should be a value larger than 0, assuming that there is no reaction of fluid below the moonpool bottom to the water-column motion and it should be smaller than α M = (r M /d)8/(3π).
which is caused by the maximum reaction with the infinity thickness of moonpools (R E →∞) according to the results obtained by Molin (2011) andNewman (2003). As depicted in Fig. 4, the corresponding α=0.591 for the peak period of T 0 =15.6 s observed in both the numerical results and those in the model test are smaller than α M =0.771 for R T /d=0.908. The cases of a moonpool with a restricted entrance are presented in Figs. 5, 6, 7, 8 and 9 for R I =32 m, 29 m, 23.5 m, 16.5 m and 11.5 m, respectively. With similar manner, the measurements of the model tests are represented by square symbols, while the numerical results without dissipation are denoted by dotted lines and those with quadratic dissipation are denoted by solid lines. Several important features can be observed from these figures.
Firstly, the resonant period (where the peak of the water-column motion occurs) increases when the moonpool entrance radius decreases, from 15.6 s for R I /R T =1 to 15.8 s, 16.4 s, 17.9 s, 21.4 s, 26 s for R I /R T =0.928, 0.841, 0.681, 0.478, 0.333, respectively. This means that the restriction of the moonpool entrance plays an important role in increasing the reaction from the moonpool bottom, including the fluid interaction through the opening and the reaction from the main hull bottom. The corresponding added mass coefficients can be calculated according to Eq. (31) and α=0.632, 0.759, 1.095, 1.995, 3.421, respectively. Thus it can be seen that the added mass coefficient α spills over (0, α M ) when R I /R T <0.681. This means that the derivation of the pistonmode theoretical value for the fully open moonpool cannot be applied to the present cases of a moonpool with a restric-    ted entrance.
Secondly, with the entrance size reduction of the moonpool, the peak values of water column motion from the measurements in the model tests decrease dramatically. This shows that the damping due to in-fluid dissipation increases greatly in the region through the reduced entrance. In particular, the water column motions are over-damped in the smallest openings R I /R T =(0.478, 0.333), as shown in Figs. 6 and 7. The peak values and the peak positions (the resonance period) are not obvious and difficult to be observed in the model tests.
Thirdly, the numerical computations without dissipation give very large responses to wave periods around the resonant period. The introduction of quadratic dissipation reduces the resonant response and provides results that are very close to those of the model tests. The associated dissipation coefficient μ in Eq. (28) increases from μ=0.002 for R I /R T =1 to μ=0.009, 0.015, 0.028, 0.041, 0.052 for R I /R T =0.928, 0.841, 0.681, 0.478, 0.333, respectively. This confirms that the fluid dissipation increases when the moonpool entrance decreases.

Conclusions
In this paper, the diffraction of linear water waves by cylindrical structures with a moonpool that has a restricted entrance in water with constant depth is examined by using the method of eigenfunction expansion. Analytical expressions for diffracted potentials in different subdomains are given and confirmed by matching (both potential function and its normal derivative) with the juncture boundaries. The derived expressions are updated based upon that the in-fluid quadratic dissipation is introduced by assuming the velocity continuity and the pressure gradient is at the restricted entrance.
Both the distributed pressure and velocity above and below the virtual dissipative disk for various dissipation coefficients are calculated. It is shown that the pressure and velocity decrease rapidly when the dissipative coefficients increase. This means that the introduced dissipation at the restricted entrance can promote energy degradation from in-cident waves effectively. The free-surface elevation at the moonpool center is computed to represent the motion of the water column inside a moonpool. Good agreement of the free-surface elevation with measurements in model tests is obtained from both the peak period and the amplitude at resonance by choosing appropriate dissipation coefficients. It is shown and confirmed by model tests that the resonant period of water motion in a moonpool increases when the size of the moonpool entrance decreases, and that the motion amplitude decreases because of greater in-fluid dissipation through a smaller moonpool entrance.
The investigation work of the general expression form of the dissipation as well as the specific relationship with related variables is currently under way.