Fluid dynamics and motion simulation of underwater glide vehicle/Povandeninio sklendziancio irenginio skysciu dinamika ir judesio imitacija.

Xiao-xu, Du ; Bao-wei, Song ; Guang, Pan 等

1. Introduction

As a underwater vehicle with new thrust mode, the underwater glide vehicle had shown some enormous advantages. When it is moving, it changes its motion trajectory through changing the gravity, buoyancy and location of gravity, lateral wing. The underwater glide vehicle had many advantages, such as simple structure, low resistance, low noise, high utilize ratio of energy, and it can fit the need of manufacture cost, energy expend, and long time navigation, so it can be used as a kind of long time auto underwater carry platform or surveillant platform [1].

This paper presents a dynamics model of underwater glide vehicle. The underwater glide vehicle has a compressed body and a pair glide hydrofoil, and it is propelled and controlled through changing the gravity and location of gravity. Section 2 introduces the dimensional dynamics model of underwater glide vehicle. Section 3 presents the model for fluid dynamics of underwater glide vehicle. Section 4 provides the simulation results of the glide motion of the underwater glide vehicle. Finally, Section 5 draws conclusions.

2. Dimensional dynamics model

First, the kinematics equations of the underwater glide vehicle can be built.

We can commonly select two coordinate systems that are used to describe the underwater glide vehicle kinematics (Fig. 1): earth-fixed frame, which can be regarded as an inertial frame, and body-fixed frame. The position and orientation of the vehicle are described relative to the earth-fixed frame, while the linear and angular velocity and the control forces/moments are described in the body-fixed frame. The vector form of these expressions is [2 - 4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Second, the dynamical equations of the underwater glide vehicle can be built.

Using the momentum theorem and the angular momentum theorem, the dynamical equations of underwater glide vehicle can be given by [5, 6]

R = [[x.sub.0], [y.sub.0],[z.sub.0],[theta],[psi],[phi]]T position and orientation vector in earth-fixed frame.

V = [[v.sub.x], [v.sub.y], [v.sub.z], [[omega].sub.x], [[omega].sub.y], [[omega].sub.z]]T velocity vector in body-fixed frame.

f = [[F.sub.x], [F.sub.y], [F.sub.z], [M.sub.x], [M.sub.y], [M.sub.z]]T force/moment vector in body-fixed frame.

[FIGURE 1 OMITTED]

The velocity vectors in body-fixed frame and in earth-fixed frame can be described as [3, 4]

[??] = J(R)V (1)

where J(R) is the rotation matrix, which is shown in equation (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[M.sub.RB][??] + [C.sub.RB](V)V = f (3)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The right hand term in Eq. (3) is the vector of external forces and moments, such as weight, buoyant, fluid dynamics which produced by the motion of underwater glide vehicle, and disturbed force which produced by the environment. In this paper, we suppose that the underwater glide vehicle motions in the enough deep, enough big and enough quiescent water, so the disturbed force which produced by the environment is ignorable.

And the fluid dynamics which produced by the motion of underwater glide vehicle can be described by the mass forces and drag forces of the main body and the fluid dynamics of the glide hydrofoil. The mass forces and drag forces of the main body can be computed by the model which is detailed described in the literature [3] and [4]. And the model of the fluid dynamics of the glide hydrofoil is built in the next section.

3. Fluid dynamics of underwater glide vehicle

The fluid dynamics of underwater glide vehicle can be calculated by the CFD. In this part, the model of the fluid dynamics was built and the fluid dynamics of underwater glide vehicle was calculated.

The basic of the CFD model is Reynolds-Averaged Navier-Stokes equation [7, 8].

1. Continuity equation

[partial derivative][rho] / [partial derivative]t + div([rho]u) = 0 (6)

2. Momentum equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

3. Transportation equation of the variable

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The RNG k - [epsilon] viscous model which was obtained by Yakhot and Orzag, is ameliorated from the standard k - [epsilon] viscous model.

The RNG k - [epsilon] viscous model includes many assumptions whose basis in physics is not transparent, and considerable controversy surrounds the legitimacy of this formalism. But the its advantage is that the eddy flow of the average flow was included through modifying the turbulent viscosity, and the time average change ratio [E.sub.ij] of the dominating flow was also included through adding a item to the Equation e. So the adding item in the RNG k - [epsilon] viscous model not only is related with the flow, but also is a function of dimensional coordinate. And the RNG k - [epsilon] viscous model is good at computing the high change ratio and high curve flow. The equation k and equation e are

[partial derivative]([rho]k) / [partial derivative]t + [partial derivative]([rho]k[u.sub.i]) / [partial derivative][x.sub.i] = [partial derivative] / [partial derivative][x.sub.j] [[[alpha].sub.k][[mu].sub.eff][partial derivative]k / [partial derivative][x.sub.j]] + [G.sub.k] + [rho][epsilon] (9)

[partial derivative]([rho][epsilon]) / [partial derivative]t + [partial derivative]([rho][epsilon][u.sub.i]) / [partial derivative][x.sub.i] = [partial derivative / [partial derivative][x.sub.j][[[alpha].sub.k][[mu].sub.eff][partial derivative][epsilon] / [partial derivative][x.sub.j]] + + [C.sup.*.sub.l[epsilon]][epsilon] / k [G.sub.k] - [C.sub.[epsilon]2] [rho] [[epsilon].sup.2] / k (10)

where [[mu].sub.eff] = [mu] + [[mu].sub.t], [[mu].sub.t] = [rho][C.sub.mu]] [k.sup.2]/[epsilon], [C.sub.[mu]] = 0.0845, [[alph].sub.k] = [[alpha].sub.[epsilon]] = 1.39, [C.sup.*.sub.1[epsilon]] - [eta](1 - [eta]/[[eta].sub.0])/1 + [beta][[eta].sup.3], [C.sub.1[epsilon]] = 1.42, [C.sub.2[epsilon] = 1.68, [eta] = [(2[E.sub.ij] [E.sub.ij]).sup.1/2] k/[epsilon], [E.sub.ij] = 1/2 ([partial derivative][u.sub.i]/[partial derivative] [x.sub.j] + [partial derivative][u.sub.j]/[partial derivative][x.sub.i]), [[eta].sub.0] = 4.377, [beta] = 0.012.

The CFD model can be calculated by the numerical procedure, the SIMPLE.

The acronym SIMPLE stems from semi-implicit method for the pressure-linked equations. The SIMPLE algorithm uses a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field. The SIMPLE algorithm is a comprehensively used algorithm of the flow field in engineering, and it is usually used in computing the incompressible flow.

The calculated results of fluid dynamics.

For simulating the motion of the underwater glide vehicle, we calculated the fluid dynamics, which are shown in the Figs. 2, 3 and the Table.

[FIGURE 2 OMITTED]

In the Fig. 2, the velocity contour of the underwater glide vehicle was shown, and in the Fig. 3, the surface stress contour of the underwater glide vehicle was shown. Through this two Figs., we can find that the fluid field of the underwater glide vehicle is steady, so the calculation is reasonable.

In the Table, the calculate results of fluid dynamics were shown. Through the table, we can find the lift force parameter is bigger than the normal underwater vehicle. It is because the underwater glide vehicle has a big glide hydrofoil [8].

[FIGURE 3 OMITTED]

4. Simulation

For demonstrating the dynamics model of underwater glide vehicle and the model for fluid dynamics of underwater glide vehicle, a numerical example of the motion of the underwater glide vehicle is presented. The underwater glide vehicle has a compressed body and a pair glide hydrofoil, and it is propelled and controlled through changing the gravity and location of gravity.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The glide motion of the underwater glide vehicle was simulated in this paper, and the results were shown in Figs. 4, 5, 6 and 7. Fig. 4 gives the trajectory curve of the underwater glide vehicle, and Fig. 5 gives the velocity curve. Figs. 6 and 7 give the curves of pitching angle and attack angle.

The simulation results indicate that the underwater glide vehicle can steadily complete the underwater glide motion, and the underwater glide can move forward about 2293 m, when the depth change about 700 m. The results also show that the velocity, pitching angle and attack angle has definite fluctuate when the underwater glide vehicle change the glide mode, and they would get stabilization at a definite time.

5. Conclusions

The dynamics model of underwater glide vehicle and the model for fluid dynamics of glide hydrofoil were presented in this paper for analyzing the dynamics of underwater glide vehicle. The simulation results show that:

1. The underwater glide vehicle can steadily complete the underwater glide motion, and the glide radio achieved 3.27.

2. The motion parameter of the underwater glide vehicle, such as velocity, pitching angle and attack angle has definite fluctuate when the underwater glide vehicle change the glide mode, and they would get stabilization at a definite time.

References

[1.] Nina Mahmoudian, Jesse Geisbert, Craig Woolsey. 2010. Approximate analytical turning conditions for underwater gliders: Implications for motion control and path planning, IEEE Journal Of Oceanic Engineering 35(1): 131-143

[2.] Smallwood David A, Whitcomb Louis L. 2004. Model-based dynamic positioning of underwater robotic vehicles: Theory and experiment, IEEE Journal of Oceanic Engineering, vol. 29: 169-185.

[3.] Jeffery S. Riedel. 1999. Seaway Learning and Motion Compensation in Shallow Waters for Small AUVs. Ph.D Thesis.

[4.] LI Tian-sen. 1999. Torpedo Manoeuvrability. BeiJing: National Defense Industry Press.

[5.] DU Xiao-xu, Pan Guang, SONG Bao-wei, HU Haibao, and LI Jia-wang. 2007. Simulation of longdistance AUV in low speed maneuvers, Journal of System Simulation, vol 19: 470-473.

[6.] Nina Mahmoudian, Craig Woolsey. 2008. Underwater glider motion control, Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 9-11: 552-557.

[7.] Meftah, S.M.A.; Imine, B.; Imine, O.; Adjlout, L. 2011. Numerical simulation of a flow around an unmanned aerial vehicle, Mechanika 17(2): 193-196.

[8.] Rugaitis, A.; Juo?as, K.; Volkovas, V.; Deksnys. V. 2010. Modeling, experimental research and critical parameter analysis of glider's dynamic characteristics, Mechanika 6(86): 38-42.

Received February 25, 2011 Accepted August 23. 2011

DU Xiao-xu, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E mail: xiaoxu731@126.com

SONG Bao-wei, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E- mail: songbaowei@nwpu.edu.cn

PAN Guang, Northwestern Polytechnical University, Youyi Xilu 127,710072 Xi'an, China, E-mail: panguang601@163.com

Table
Calculate results of fluid dynamics

Fluid dynamics parameter     Calculate results

[C.sub.xS]                        0.2654
[C.sup.[alpha].sub.y]            12.112
[m.sup.[alpha].sub.z]            -0.5324