Numerical simulation and analysis of periodically oscillating pressure characteristics of inviscid flow in a rolling pipe

Abstract

Floating liquefied natural gas (LNG) plants are gaining increasing attention in offshore energy exploitation. The effects of the periodically oscillatory motion on the fluid flow in all processes on the offshore plant are very complicated and require detailed thermodynamic and hydrodynamic analyses. In this paper, numerical simulations are conducted by computational fluid dynamics (CFD) code combined with user defined function (UDF) in order to understand the periodically oscillating pressure characteristics of inviscid flow in the rolling pipe. The computational model of the circular pipe flow is established with the excitated rolling motion, at the excitated frequencies of 1–4 rad/s, and the excitated amplitudes of 3°–15°, respectively. The influences of flow velocities and excitated conditions on pressure characteristics, including mean pressure, frequency and amplitude are systematically investigated. It is found that the pressure fluctuation of the inviscid flow remains almost constant at different flow velocities. The amplitude of the pressure fluctuation increases with the increasing of the excitated amplitude, and decreases with the increasing of the excitated frequency. It is also found that the period of the pressure fluctuation varies with the excitated frequency. Furthermore, theoretical analyses of the flow in the rolling circular pipe are conducted and the results are found in qualitative agreement with the numerical simulations.

Keywords

pressure fluctuation / rolling / floating production storage and offloading unit for liquefied natural gas (LNG-FPSO) offshore

Introduction

In recent years, a large number of floating plants for oil production are applied in offshore energy exploitation. Compared to fixed units, floating plants have the advantages of portability, reusability, and low cost. Floating production storage and offloading unit for liquefied natural gas (LNG-FPSO) is a floating unit with the functions of production, storage and offloading for LNG, which has received wide attention from countries interested in offshore natural gas resources [1]. However, many unique characteristics and rigorous factors, induced by the periodically oscillatory platform motion originated from variable wind and dynamic ocean wave, are restricting the construction of a real floating LNG plant. The fluid flow in the processes on LNG-FPSO will be inevitably influenced by the periodically oscillatory platform motion. The induced instabilities of the fluid flow are harmful to the operation of the transport equipment and the performance of the transfer process. For example the LNG offloading process, one of the most important parts on LNG-FPSO, is carried out between two floating plants of the LNG-FPSO and LNG carrier. When the LNG is transferred between the two, the pipelines move together with the platform, inducing flow instability. As a result, the pressure fluctuations of the cryogenic fluid will cause resonance and vaporization, which are harmful to the pipeline and the LNG transport. There are high demands on the safety and economics of LNG offloading [2]. Thus the effects of the periodically oscillatory motion on the fluid flow require to be investigated systematically, to gain quantitative information for different situations.

Some research has been conducted on the pressure fluctuation of the flow and thermal hydraulic induced by such oscillatory motions. Richardson and Tyler [3] experimentally investigated the cross-sectional velocity distribution in an oscillatory pipe and found that the maximum velocity occurred near the wall, not in the center of the pipe in steady flow. Sexl [4], Wormersly [5] and Uchida [6] verified that the velocity profile was different from the parabolic distribution of steady flow by theoretical analysis of pulsating flow. Claman and Minton [7] conducted experimental investigation and measured the velocity of pulsating flow. It was found that at low Reynolds number the velocity distribution corresponded to the laminar flow theory, while at higher Reynolds number there was increasing difference between the experimental results and laminar theory. Hershey and Song [8] conducted experiments and found that the time-average friction factors of pulsating flow were greater than those of steady flow. Ohmi [9,10] investigated the flow pattern and frictional losses in laminar pulsating flow. Donovan [11] made the experimental correlations of fluid resistance for laminar pulsating flow and analyzed the one dimensional oscillatory flow. Pendyala, Jayanti and Balakrishnan [12] measured the effect of the oscillations on the flow rate and the pressure drop in the Reynolds number range of 500-6500. However, the effect of the oscillation amplitude on the flow was unclear. Ishida and Yoritsune [13], Marata [14,15], Tan [16], Pendyala [17], and Yan [18] investigated the convective heat transfer in rolling motion. Gundogdu and Carpinlioglu [19-21] reviewed the literature on the pulsating flow. They reported that for the turbulent flow there were several relationships of the classification limits of the turbulent oscillation flow proposed by different researchers. Some conclusions disagreed with others and there was a dispute about the effect of unsteadiness on the time-mean flow characteristics. The corresponding experimental measurements could not lead to significant conclusions. Since the induced oscillating characteristics of the flow are vital in design and operation of marine equipment, the influences of the flow velocity, oscillation frequency and oscillation amplitude on the flow velocity and pressures should be investigated systematically.

As the Reynolds increases, the effect of the oscillations on the flow rate becomes smaller [12], the fluctuation of the flow rate for the pumped turbulent flow in LNG offshore transfer will not be considered here. The effects of the oscillatory platform motions on the pressure characteristics of the pipe flow during the LNG offloading processes on LNG-FPSO will be investigated quantitatively in the present paper. Because the rolling motion is the typical motion of the floating unit, a rolling circular pipe is defined as the physical model. In order to understand the influences of the flow velocity, the excitated frequency and the amplitude on the pressure fluctuations of the pipe flow, a numerical model is established and detailed numerical simulations are conducted. The pressure losses in the rolling pipe may come from the forces of friction, gravity and additional inertia. However, direct solution of the pulsating viscous flow or experimental measurement of transient unsteady pressure fluctuation in experiments is extremely difficult to lead to reasonable conclusions. Therefore, the pressure fluctuations of inviscid flow, driven by the gravity and additional inertia forces as the Coriolis, tangential and the centripetal forces are investigated; the influences of the various forces and the excitated parameters are theoretically analyzed and discussed; the analytical results are also compared with the numerical simulations; and qualitatively agreement has been found between the two.

Computational formulations

The 3-D computational model is illustrated in Fig. 1. The diameter of the circular pipe is 10mm and the height is 800 mm. In order to obtain a fully developed flow field and to avoid the influence of the outflow boundary condition on the numerical results, the sampling face is set at a height of 780mm. The fluid of the upstream flow in the vertical pipe is liquid nitrogen. Because the flow velocity in the pipeline of LNG offloading is usually driven by the submerged liquid pump and the pipeline is vacuumed and multilayer insulated, the offloading process can be regarded as adiabatic. Numerical simulations are based on conservation equations. The governing equations of the computational domain are

Mass conservation

(1)\nabla •(\rho \mathit{u})=\mathrm{0},

and Momentum conservation

(2){\displaystyle \frac{\partial (\rho u_i)}{\partial t}}+\mathrm{d}\mathrm{i}\mathrm{v}(\rho \mathit{u}{u}_i)=-{\displaystyle \frac{\partial p}{\partial x_i}}+\rho f_i(i=x,y\mathrm{a}\mathrm{n}\mathrm{d}z).

The relevant variables in these equations are defined in “Notation”.

The inlet of the fluid velocity is defined as the inlet condition, by taking into account of the fact that in practical situations the velocity of the forced flow is very high. The velocity of turbulent flow is barely influenced by the motion of the rolling, and the influence decreases as the velocity increases [10]. The inlet velocity is normal to the inlet boundary face. The temperature and pressure at the inlet are set to 70 K and 10 kPa, respectively. The fluid is in a sub-cooled state at the preset pressure.

(3){\mathit{u}}_{\mathrm{n}}={\mathit{u}}_0,

(4)p=p_{\mathrm{0}}.

The outlet boundary condition is defined as an outflow for the fully developed flow at the circular pipe outlet.

(5){\displaystyle \frac{\partial u_i}{\partial \mathit{n}}}={\displaystyle \frac{\partial p}{\partial \mathit{n}}}=\mathrm{0.}

The wall boundary conditions for the circular pipe can be regarded as thermally isolated walls which are moving with the adjacent cells, and the rotational axis is x.

(6)q_{\mathrm{w}}=\mathrm{0.}

The excitated source of the circular pipe is assumed as a sinusoidal function, as described by Eq. (7). The motion equation is programmed and compiled into the present model by user defined function (UDF), and then inserted into the computational domain. The excitated frequencies and amplitudes are in the range of 1-4 rad/s and 3-15 degrees respectively, which are referenced from the practical ocean conditions.

(7)\omega ={\theta }_{\mathrm{0}}{\omega }_{\mathrm{0}}\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{\mathrm{0}}t).

Simulation results

Detailed numerical simulations have been conducted to investigate the influences of the flow velocity, oscillating frequency and amplitude on the pressure characteristics. The governing equations are discretized by the hexahedral meshes while the grid-independence has been checked to ensure the accuracy of the present simulations. The simulations in the steady state are first iterated and converged. Next, the steady-state cases are used as the initial cases for the consequent dynamic simulations, and then the motion program is uploaded and attached to the computational domain. The location and oscillation frequency of the computational zone vary in every time step. The computation does not go to the next time step until the numerical residuals reach the convergence limit of 10^-7. The time step of the iteration is very tiny, showing a little difference between the dynamic simulations according to different oscillating periods.

Effects of flow velocity

The effect of the flow velocity on the pressure fluctuation of the fluid in the vertical pipe is studied. Three inlet velocities of 2 m/s, 1 m/s and 0.25 m/s are calculated at the excitated amplitude of 15 degree and the excited frequency of 2 rad/s.

The pressures at the sampling face are compared for steady and dynamic simulations. The pressures in the steady flow are demonstrated in Fig. 2 with solid marks, while the time-averaged mean pressures of dynamic simulations are represented by hollow marks. It can be found that the pressures are higher in the motion pipe than that in the steady pipe. This indicates that the pressure drop decreases when the vertical pipe is excited by the rolling motion.

The variations of the transient pressure in the rolling pipe at different inlet velocities are depicted in Fig. 3. It is shown that the fluctuations of the pressures have the same period and amplitude. It is also indicated that the velocity of the inviscid flow has little influence on the amplitude and period of the pressure fluctuation.

Effects of excitated frequency

The excitated frequency is one of the main effects on the pressure fluctuation of the fluid flow in the vertical pipe. In this current simulation, three different excitated frequencies of 1, 2 and 4rad/s are investigated. From the simulation results, it has been found that the flow velocity has little effect on the pressure fluctuation. The inlet velocity here is kept at 1 m/s. The comparison of the pressure fluctuations excited by different frequencies at the excitated amplitude of 3 degree is displayed in Fig. 4(a) while the pressure fluctuations at the excitated amplitudes of 8 and 15 degree are exhibited in Figs. 4(b) and (c), respectively.

It can be seen in Fig. 4(a) that the periods of pressure fluctuations are different in these three cases. The periods vary with the excitated frequency. The amplitudes of pressure fluctuations also vary for different excitated frequencies. When the excitated frequencies decrease the amplitudes increase. It can be concluded that at the low excitated frequencies, the pressure fluctuations are larger and more harmful. In addition, there exist phase lags among the simulations for different excitated frequencies. The phase lag is more apparent when the excitated frequency increases. The simulation results shown in Fig. 4(b) and 4(c) for excitated amplitudes of 8 and 15 degree are similar to that shown in Fig. 4(a).

Effects of excitated amplitude

The effect of the excitated amplitude on the pressure fluctuation of fluid in the vertical pipe is also studied. The excitated amplitude of the vertical pipe varies from 3, 8 and 15 degree. The comparison of the pressure fluctuations excited by different excitation amplitudes at the excitated frequency of 1 rad/s is presented in Fig. 5(a) while the comparisons of the pressure fluctuation for the excitated frequencies of 2 and 4 rad/s are respectively given in Figs. 5(b) and 5(c).

It can be found that the excitated amplitude has little effect on the period and the phase of the pressure oscillations. But the excitated amplitude has obvious effect on the amplitude of the pressure oscillation. It can be seen that the amplitudes of the pressure fluctuations are larger when the excitation amplitude increases. The simulation results are consistent for different excitated frequencies.

Force analysis and discussion

The force analysis of the rolling pipe has been conducted. Because the pipe is in the periodically oscillatory motion of rolling, the physical model is constructed in combination with the coordinate systems of the inertial system oxyz and the kinetic system o^{\prime }{x}^{\prime }{y}^{\prime }{z}^{\prime }, as shown in Fig. 6. The kinetic coordinate system rolls together with the pipe, therefore the pipe can be regarded as static in this coordinate system.

The Euler equation of viscid flow is

(8)\rho {\displaystyle \frac{\mathrm{D}\mathit{u}}{\mathrm{D}t}}=\nabla ·\mathit{ϵ}+\rho \mathit{f},

where ϵ is the surface force (N m^-3) and is a two-order tensor, and \nabla \cdot ϵ is its divergence.

The flow velocity can be expressed as

(9)\mathit{u}={\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}}\left({\mathit{r}}_0+\mathit{r}\right),

(10){\displaystyle \frac{\mathrm{D}\mathit{u}}{\mathrm{D}t}}={\mathit{a}}_0{\left({\displaystyle \frac{\mathrm{D}{\mathit{u}}_{\mathrm{r}}}{\mathrm{D}t}}\right)}_{\mathrm{r}}+2\mathit{\omega }\times {\mathit{u}}_{\mathrm{r}}+\mathit{\omega }\times \left(\mathit{\omega }\times \mathit{r}\right)+{\displaystyle \frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}t}}\times \mathit{r}.

The variables with subscript r denote in the kinetic coordinate system.

(11)ϵ=-p{\delta }_{ij}(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}).

Equations (10) and (11) are substituted into Eq. (8), and can be rewritten as

(12){\left({\displaystyle \frac{\mathrm{D}{\mathit{u}}_{\mathrm{r}}}{\mathrm{D}t}}\right)}_{\mathrm{r}}=-{\displaystyle \frac{\nabla p}{\rho }}\mathit{+}\mathit{g}-\left[{\mathit{a}}_0+2\mathit{\omega }\times {\mathit{u}}_{\mathrm{r}}+\mathit{\omega }\times \left(\mathit{\omega }\times \mathit{r}\right)+{\displaystyle \frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}t}}\times \mathit{r}\right].

The transient flow rate of the forced pump flow does not change in the rolling pipe. This is verified by the experimental measurements conducted by Pendyala [10]. It means that the velocity normal to the pipe section, the component of BoldItalic_r in the pipe direction, is constant. On the other hand, there is no heave, sway or other motions. As a result, in the pipe direction Eq. (12) can be simplified as

(13){\displaystyle \frac{\nabla p}{\rho }}=\mathit{g}-\left[\mathit{\omega }\times (\mathit{\omega }\times \mathit{r})+{\displaystyle \frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}t}}\times \mathit{r}\right].

It can be found from Eq. (13) that the pressure loss in the rolling pipe flow is constituted by the gravity loss and the additional inertia pressure losses. The additional inertia pressure losses are caused by the centripetal and tangential force of the rolling motion. The gradient of the pressure in Eq. (13) is integrated in the direction of pipe and gives

(14)\mathrm{\Delta }p={\displaystyle \int \rho \mathit{g}\mathrm{d}{\mathit{l}}^{\prime }}-\stackrel{}{{\displaystyle \int \rho [\mathit{\omega }\times (\mathit{\omega }\times \mathit{r})]\mathrm{d}{l}^{\prime }-{\displaystyle \int \rho \left(\stackrel{•}{\mathit{\omega }}\times \mathit{r}\right)\mathrm{d}{\mathit{l}}^{\mathit{\text{'}}}}}}.

Arbitrary point M in the inertia and kinetic coordinate system is given by

{\mathit{r}}_0=\mathbf{0},

(15)M=bj+ck,

(16)M^{\prime }=(b\cos \theta -c\sin \theta )j^{\prime }+(b\sin \theta +c\cos \theta )k^{\prime },

where, \theta is the angle between the inertia and kinetic coordinate systems.

The gravity acceleration in the kinetic coordinate system can be expressed as

(17)g^{\prime }=\left|\mathit{g}\right|(\mathrm{s}\mathrm{i}\mathrm{n}\theta ·j^{\prime }-\mathrm{c}\mathrm{o}\mathrm{s}\theta ·k^{\prime }).

The pressure losses in the right side of Eq. (14) can be written as

(18)\mathrm{\Delta }{p}_{gravity}={\displaystyle {\int }_0^1\rho \left|\mathit{g}\right|}\left(\sin \theta ·j^{\prime }-\cos \theta ·k^{\prime }\right)·\left(\mathrm{d}{i}^{\prime }+\mathrm{d}{j}^{\prime }+\mathrm{d}{k}^{\prime }\right),

(19)\mathrm{\Delta }{p}_{\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}}={\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}\rho w^{\mathrm{2}}(y^{\prime }{j}^{\prime }+z^{\prime }{k}^{\prime })\mathrm{d}l}+{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}\rho \stackrel{•}{w}(z^{\prime }{j}^{\prime }-y^{\prime }{k}^{\prime })\mathrm{d}l}.

In the vertical pipe the pressure losses are integrated as

(20)\mathrm{\Delta }p=-\rho g{z}_{\mathrm{1}}\mathrm{c}\mathrm{o}\mathrm{s}\theta +{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\omega }^{\mathrm{2}}{z}_{\mathrm{1}}^{\mathrm{2}}.

From Eq. (20), it can be found that the gravity pressure loss oscillates as a function of time. The additional inertial centripetal force, that is, the second part in the right side of Eq. (20), contributes to the upstream flow.

For the in-viscid flow in the steady pipe, the pressure loss is given by

(21)\mathrm{\Delta }p=-\rho g{z}_{\mathrm{1}}.

In comparison with Eq. (20) and Eq. (21), it can be observed that the pressure loss of the rolling pipe flow is smaller than that of the steady pipe flow. This agrees with the simulation results, as illustrated in Figs. 2 and 3. The excitated conditions are listed as

(22)\theta ={\theta }_{\mathrm{0}}\sin (w_{\mathrm{0}}t),\omega ={\theta }_{\mathrm{0}}{\omega }_{\mathrm{0}}\cos ({\omega }_{\mathrm{0}}t)\stackrel{•}{\omega }=-{\theta }_{\mathrm{0}}{\omega }_{\mathrm{0}}^{\mathrm{2}}\sin ({\omega }_{\mathrm{0}}t).

When substituting them into Eq. (20), the pressure loss can be rewritten as

(23)\mathrm{\Delta }p=-\rho g{z}_{\mathrm{1}}\cos ({\theta }_{\mathrm{0}}\sin ({\omega }_{\mathrm{0}}t))+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho {\theta }_{\mathrm{0}}^{\mathrm{2}}{\omega }_{\mathrm{0}}^{\mathrm{2}}{z}_{\mathrm{1}}^{\mathrm{2}}{\cos }^{\mathrm{2}}({\omega }_{\mathrm{0}}t).

From Eq. (23), it can be conducted that the period of the pressure oscillation is \mathrm{\pi }/{\omega }_{\mathrm{0}}. The amplitude of the pressure fluctuation is given by

(24)\left|\rho g{z}_{\mathrm{1}}(\mathrm{1}-\cos {\theta }_{\mathrm{0}})-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho z_{\mathrm{1}}^{\mathrm{2}}{\omega }_{\mathrm{0}}^{\mathrm{2}}{\theta }_{\mathrm{0}}^{\mathrm{2}}\right|.

The average of pressure loss can be expressed as

(25){\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left(\rho g{z}_{\mathrm{1}}+\rho g{z}_{\mathrm{1}}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{0}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho z_{\mathrm{1}}^{\mathrm{2}}{\omega }_{\mathrm{0}}^{\mathrm{2}}{\theta }_{\mathrm{0}}^{\mathrm{2}}\right).

It can be noticed that when the excitated frequency {\omega }_{\mathrm{0}} increases, both the amplitude of the pressure fluctuation and the average pressure loss decrease. Because the influence of the first part from gravity is larger than that of the inertia force, when the excitated amplitude increases, the amplitude of the pressure fluctuation and the average pressure loss increase as well. It can be found that the simulation results are in qualitative agreement with the theoretical analysis.

Conclusions

In the present paper, numerical simulations of the effect of the flow velocity, excitated frequency and amplitude on the oscillatory pressure characteristics as a function of the period, amplitude and phase for the pipe flow have been systematically conducted. Furthermore, a physical model has been analyzed by combining the inertial and kinetic coordinate system. It is indicated that the numerical simulations are in qualitative agreement with the analytical results. The following conclusions are reached:

1) The pressure fluctuation of the in-viscid flow remains almost constant at different flow velocities.

2) When the excitated amplitude increases, the amplitude of the pressure fluctuation increases, but it has little influence on the period of pressure fluctuation.

3) When the excitated frequency increases, the amplitude of the pressure fluctuation decreases. The period of the pressure fluctuation is \mathrm{\pi }/{\omega }_{\mathrm{0}}, which is dependent on the excitated frequency. There exists a phase lag when the frequency increases, because of the hysteresis of the Reynolds stress.

The present investigation is intended to clarify the pressure characteristics of the fluid flow on the floating unit and the influences of the velocity and motion factors on the fluid flow. Furthermore, it has been demonstrated that the numerical simulations are reliable and could be used to study more complicated oscillation problems which are difficult to solve theoretically and experimentally.

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