Joshi, S.P. (2014) Hydrodynamic analysis of pressure-equalized buoy using hyper-singular boundary element approach, https://drive.google.com/file/d/0BziIRaDEfaNnYWJDN1ViR0pqTkk

HYDRODYNAMIC ANALYSIS OF PRESSURE-EQUALIZED BUOY USING HYPERSINGULAR BOUNDARY ELEMENT APPROACH Sanjay Joshi, PhD, PE 21915 Fieldvine Ct Katy, Texas 77450

ABSTRACT A procedure is presented for hydrodynamic analysis of a buoy submerged in sea water and partially filled with Nitrogen. The procedure uses hyper-singular boundary element approach. It is general enough to consider effects of sloshing of the Nitrogen-sea water interface on the magnitude and distribution of hydrodynamic forces. Utilizing the procedure presented, hydrodynamic effects on the open-bottom buoy including the natural vibration (sloshing) of the interface between sea water and Nitrogen have been studied.

Under pressure of approximately 3000 ft of sea water, the Nitrogen is roughly 100 times denser than at the water surface. Having accounted for the change in density due to hydrostatic pressure, the Nitrogen has been assumed incompressible for hydrodynamic computations. Fundamental acoustic natural frequency of Nitrogen in a cylindrical cavity of 15ft diameter and 30.5ft height is 15.4 Hz [1]. This value is high in comparison with the wave frequencies and the fundamental sloshing frequency. Therefore,

nA

INTRODUCTION Pressure equalized buoys are used to increase overall buoyancy and thus the capacity of a permanent floating unit. The buoys are tethered to the mooring lines at a depth well below the wave activity zone. The buoy is free to swing like an inverted pendulum. It is partially filled with a gas, usually Nitrogen. The interface between Nitrogen, which is under the pressure of thousands of feet of sea water, and sea water is free to slosh. The magnitude and distribution of hydrodynamic forces on the buoy thus depends on the extent of tuning between excitation frequency and sloshing frequencies. In order to understand the dynamic behavior of the buoy-mooring line system, it is important to first understand the hydrodynamic effects on the open-bottomed buoy including the sloshing of the interface between sea water and Nitrogen. SYSTEM The idealized system consists of a thin-walled buoy shell submerged in infinite domain of sea water. In part of the buoy, the sea water has been displaced by Nitrogen gas forming an interface with sea water (Figure 1).

Nitrogen

ΓA

ΩN

nB ΓB Inside sea water

Ω iw nC

ΓD nD

ΓC Ω ow Outside sea water

Figure 1. Domain definitions

neglecting the compressibility of Nitrogen will not significantly affect the sloshing behavior. The buoy shell has been assumed rigid in the formulation presented here. The buoy shell natural frequencies are expected to be high in comparison with sloshing or wave frequencies. Therefore, the flexibility of buoy shell will not significantly affect the sloshing frequencies or overall hydrodynamic behavior. The only source of excitation considered is acceleration of the buoy shell.

The frequency response functions of unknown hydrodynamic pressures (in excess of hydrostatic pressure) on Nitrogen-buoy shell interface because of excitation of buoy along x, y and z axes can be expressed in terms of accelerations of buoy by analysis of the Nitrogen domain. The small amplitude, irrotational motion of Nitrogen assumed to be inviscid incompressible is governed by the three-dimensional Laplace equation [2, 3]:

(2)

where ρ N is the mass density of compressed Nitrogen, n A (x ) and a A (x ) are the normal direction vector and acceleration vector at point x on ΓA (Figure 1). Similarly, the motion of inside sea water is governed by the three-dimensional Laplace equation:

∇ 2 piw (x, ω) = 0

(4)

where ρ sw is the mass density of sea water, n C (x ) and a C (x ) are the normal direction vector and acceleration vector at point x on ΓC (Figure 1).

∇ 2 pow (x, ω ) = 0

(3)

where p iw ( x , ω ) is the frequency response function for hydrodynamic pressure at point x in inside sea water domain. This pressure is related to the motion

(5)

∂ p ow = − ρ sw [n C ( x ) ⋅ a C ( x ) ] ∂n C

On ΓC

(6)

∂ pow = − ρ sw [n A (x) ⋅ a A (x)] ∂n A

On Γ A

(7)

Boundary condition at far field is

(1)

where p N 2 ( x , ω ) is the frequency response function for hydrodynamic pressure at point x in Nitrogen domain. The hydrodynamic pressures in Nitrogen are related to the motion of buoy shell by the following boundary condition on the Nitrogen-buoy shell interface ΓA : ∂ p N = − ρ N [n A ( x) ⋅ a A ( x)] ∂n A

∂ piw = − ρ sw [n C (x) ⋅ a C ( x)] ∂nC

Similar equations are written for outside sea water domain:

BOUNDARY VALUE PROBLEM

∇ 2 p N (x, ω) = 0

of buoy shell by the following boundary condition on the inside sea water-buoy shell interface ΓC :

(8)

p ow = 0

where p ow ( x , ω ) is the frequency response function for hydrodynamic pressure at point x in outside sea water domain. On the fictitious boundary ΓD continuity of hydrodynamic pressures p iw ( x , ω ) and p ow ( x , ω ) and the normal gradients is imposed by the following equations:

∂ ∂ piw = pow ∂nD ∂nD piw = pow

 (a )     (b) 

(9)

On interface between Nitrogen and inside sea water ΓB , the following conditions hold [4, 5]: 1

ρN 1

ρ sw

ω 2 piw − p N ∂ pN = ∂n B g ρ sw − ρ N ω 2 piw − p N ∂ piw = ∂n B g ρ sw − ρ N

 (a )    On ΓB (10)  (b )  

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NUMERICAL EVALUATION PROCEDURES

Source point on ΓC

The boundary value problems posed by the Equations (1)-(9) are solved using boundary integral equation approach. Since the buoy shell is thin compared with the size of the domains, numerical difficulties arise with conventional boundary integral approach [6-9]. One of the approaches to circumvent these difficulties is to use hyper-singular boundary integral equation approach. In this approach it is assumed that the walls or interfaces are infinitesimally thin and the above equations are re-written in terms of pressure jump across the interfaces ΓA , ΓB and ΓC as functions of acceleration of interfaces ΓA and ΓC .

 ∂ 2G  ∂ 2G  ∂ 2G = p d p d pC dΓ + Γ + Γ  ∂n ∂n A  ∂n ∂n B   s x  s x  ∂n s ∂n x ΓA

−

ΓB

ΓC

ω  ∂G 2

p B dΓ g   ∂n s

(13)

ΓB

 ∂G ( ρ N − ρ sw )a A dΓ = 4πρ swaC −   ∂n s ΓA

the differential pressure [ p N ( x, ω ) − p ow ( x, ω ) ] on boundary ΓA . Similarly let and pC be the differential pressures pB [ p iw ( x , ω ) − p N ( x, ω ) ] and [ p iw ( x, ω ) − p ow ( x, ω ) ] on interfaces ΓB and ΓC , respectively.

In the above equations, G, the free-space Green’s function for Laplace equation is given below:

Hyper-singular integral equations for hydrodynamic pressures that satisfy Equations (1-10) are given below.

The two horizontal bars across the integration sign in Equations (11)-(13) denote integration in the sense of Hadamard Finite Parts [10-14].

Let

pA

be

 ∂ 2G  ∂ 2G  ∂ 2G = p d p d + + Γ Γ A B  ∂n ∂n pC dΓ  ∂n ∂n  ∂n ∂n  s x  s x  s x −

ΓB

ω  ∂G 2

g   ∂n s

ΓC

p B dΓ

1 x−s

(14)

2

The unknown hydrodynamic pressures pA , pB , and pC for known accelerations a A and aC are evaluated by utilizing Equations (11)-(13) with discretized boundaries as shown in figure 2. The discretized equations take the form:

Source point on ΓA

ΓA

G(s, x) =

Kp − ω 2Mp = f

(15)

 ∂ 2Gij Kij =  dΓ =  ∂ni ∂n j

(16)

(11) where

ΓB

 ∂G ( ρ N − ρ sw )a A dΓ = 2π ( ρ N + ρ sw )a A −   ∂n s ΓA

Γj

Source point on ΓB

M ij =

−

ΓB

ΓB

 ∂G = − ( ρ N − ρ sw )a A dΓ  ∂n s ΓA

2π ρ sw + ρ N =− g ρ sw − ρ N

ΓC

ρ + ρN ω2 p B dΓ + 2π p B sw  g  ∂n s g ρ sw − ρ N

ω 2  ∂G

if Γ j ∈ ΓB (17)

Γj

 ∂ 2G  ∂ 2G  ∂ 2G = p d p d Γ + Γ + A B  ∂n ∂n  ∂n ∂n  ∂n ∂n pC dΓ  s x  s x  s x ΓA

1  ∂Gij dΓ g  ∂n i

(12)

if Γi ∈ ΓB

N  ∂Gij f i = −  ( ρ N − ρ sw )a j dΓ j =1  ∂n i el

if Γ j ∈ ΓA

Γj

= +2π ( ρ N + ρ sw )a A = +4πρ swa C

if Γi ∈ ΓA (18) if Γi ∈ ΓC Page 3 of 6

RESULTS

Analytical procedures and numerical evaluation technique presented above have been implemented in a computer program. The analysis procedure has been utilized to develop a fundamental understanding of hydrodynamic effects on buoy shell including sloshing of the interface. To this end hydrodynamic forces have been evaluated for a wide range of excitation frequencies. To understand the sloshing phenomenon, eigenvalue analysis has been performed and natural vibration modes and frequencies of the Nitrogen-sea water interface have been evaluated. Figure 2 shows the discretization used for the numerical computations. Equation (15) is used to compute hydrodynamic pressures exerted on the buoy shell in frequency domain by the internal and external fluids due to known excitation frequency and accelerations at interfaces. The distributed pressures are then

Figure 3. Hydrodynamic response to horizontal and vertical accelerations of buoy

integrated to obtain total force on the buoy. The buoy swings like an inverted pendulum about the tether point on a mooring line. The tether point itself moves up and down. Figure 3 shows added mass for horizontal and vertical motion as a function of excitation frequency. It can be observed that the plot shows several spikes. These are sloshing frequencies and when excitation frequency coincides with sloshing frequency, the hydrodynamic force is amplified. It should be noted that the peaks are sharp because the calculations do not consider any damping. In reality there is always some damping available. As expected, the sloshing activity reduces as excitation frequency is increased. In the limit the sloshing is completely suppressed and added mass converges to a constant value. Table 1 shows such limiting values for horizontal and vertical acceleration of the buoy shell. The eigenvalue problem posed by equating right hand side of Equation (15) to zero has been solved to evaluate natural vibration properties of the interface between Nitrogen and sea water. LAPACK routines have been used for all the matrix computations [15]. Table 2 shows lowest few natural vibration frequencies and mode shapes. As expected the modes associated with repeated frequencies are anti-symmetric about two orthogonal horizontal axes. Truly speaking the mode shape associated with a repeat frequency is an eigenspace. Any shape that is a linear combination the two orthogonal modes is also a valid mode shape. These are the modes that contribute to response if excitation has horizontal component.

Figure 2. Discretization of Interfaces

The fundamental sloshing period is 2.8 sec. Since this period is far away from periods of both large and small (fatigue) waves, there is no possibility of spilling of Nitrogen due to sloshing.

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Table 1. Hydrodynamic Added Mass Limiting Excitation Frequency

Direction of Acceleration Horizontal

Vertical

348 kip

172 kip

Table 2. Natural vibration properties of the Nitrogen-sea water interface

Mode No.

Mode Shape

Nat. Freq (Hz)

ω→ 0

(constant

acceleration)

ω→ ∞

328 kip

1, 2

0.36

3, 4

0.49

5

0.58

6, 7

0.60

8, 9

0.69

172 kip

CONCLUSIONS

Hypersingular boundary integral equation approach has been successfully used for investigation of hydrodynamic properties of buoy shell submerged in sea water and partially filled with Nitrogen. REFERENCES

1.

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2.

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3.

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4.

Tang, Y., 1993, “Sloshing Displacements in a tank containing two liquids,” Proc. ASME Pressure Vessel Piping Conf., PVP-258, pp. 143149.

5.

Tang, Y., 1994, “Effect of Gravitation on Dynamic Response of Tanks Containing two liquids,” Report No. ANL/RE/CP—79448, Argonne National Lab.

6.

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7.

Krishnasamy, G., Rizzo, F. J., Liu, Y. J., 1991, “Scattering of acoustic and elastic waves by crack-like objects: the role of hypersingular integral equations,” Review of Progress in Quantitative Nondestructive Evaluation, 11A, Plenum Press.

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9.

Wu, T. W., 1995, “A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies,” J. Acoust. Soc. Am., 97, pp. 84-91.

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