INTERFERENCE OF SIDE STRUT W ITH THE NATURAL CAVITATING FLOWS AROUND A SUBMERGED VEHICLE IN WATER TUNNEL EXPERIMENTS*

CHEN Xin, YAO Yan

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: xinchen@sjtu.edu.cn

LU Chuan-jing

Department of Engineering Mechanics and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

CHEN Ying, CAO Jia-yi

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

INTERFERENCE OF SIDE STRUT W ITH THE NATURAL CAVITATING FLOWS AROUND A SUBMERGED VEHICLE IN WATER TUNNEL EXPERIMENTS*

CHEN Xin, YAO Yan

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: xinchen@sjtu.edu.cn

LU Chuan-jing

Department of Engineering Mechanics and State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

CHEN Ying, CAO Jia-yi

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

To apply themeasurements ofmodel experiment in water tunnel to the actual sailing condition, it is necessary to know accurately the strut effect and its rule. In the present work, the corresponding interferences of one-side strut and two-side strut on the natural cavitating flows around a submerged vehicle in water tunnel were investigated numerically, using the homogeneous equilibrium two-phasemodel coupled with a natural cavitationmodel. The numerical simulation results show that the strut types have distinct effects on the hydrodynamic properties. For the same given upstream velocity and downstream pressure, the existence of the strut leads to an increment of natural cavitation number, reduces the low-pressure region and depresses the pressure on the vehicle surface near the sides of strut. In the case of given cavitaiton number, the influences of the two-side strut on the drag and lift coefficients are both enhanced along with the increment of attack angle, however the influence of the one-side strut gradually gets stronger on the drag coefficient but weaker on the lift coefficient contrarily. In addition, based on the present numerical results, a correctionmethod by introducing the sigmoidal logistic function is proposed to eliminate the interference from the foil-shaped strut.

strut effect, natural cavitating flow, submerged vehicle, water tunnel experiment

Introduction

Hydrofoils are usually adopted as side-support strut to fix a vehiclemodel in water tunnel experiments. Even if a carefully designed foil w ill have rather low incipient cavitation number, it can not avoid the problem of disturbing the surrounding flow. In some specific situations, its interference on the flow field and hydrodynam ic properties is even stronger than that of the tunnel-wall effect. To apply themeasurements ofmodel experiment in water tunnel to the actual sailing condition, it is necessary to know accurately the strut effect and its rule, and to find a suitablemethod tomodify the strut interferences, so as to guarantee accurate and reliable experimental results in water tunnel tests.

The researches on strut interference aremainly focused on themethods of reducing and eliminating such interferences. The primarymethods have been usually divided into two types. The first one is entirely based on pure experimental approach. For example, Jiang et al.[1]carried out themodel experiments of three types of strut, namely tail-support, belly-support and head-support, and analyzed the results by contrast. The results indicated that the head-support strut ismost applicable to research the hydrodynamic properties around the tail of supercavitating vehicle. A ll the works of the first type have the shortcomings of great cost and capability restriction in experiments. Further-more, an assistant strut coupled with the original strut is generally used for the correction of strut interference, thus an additional effect cannot be neglected. Therefore, a secondmethod[2]combining experiment and numerical simulation appeared, in which numerical computation was used to analyze the difference between the working cases with and without a strut, and such a difference would be eliminated in routine experiments by using an additional strut. He et al.[3]studied themulti-influence of the uniform-diameter segment on the aerodynam ic performance of airplanemodel and the disturbance of strut, from various aspects such as length, diameter, conical degree and attack angle. They also conducted a primary correction for zero-drag of the airplanemodel concerning the tailstrut influence. This approach can effectively avoid the disadvantage of pure experimental approach and is suitable for variousmodels and strut types. By this approach one can predict the interference of the strut beforehand and analyze the influences of various strut parameters on the quantity of interference.

The above-mentioned works weremostly carried out based on the vehicles in w ind tunnel, whereas the works regarding the analysis of strut interference focused on the vehicles in water tunnel are not so common. Therefore, in the present work, numerical approach w ill be applied to give an elementary study about the interference ant its correction of a foilshaped side-strut on the submerged vehicle with natural cavitation.

1. Mathematicalmodel and numericalmethod

In this work, amixture-typemultiphase flowmodel[4-7]based on the isotropic hypothesis for fluid and the Reynolds averaged Navier-Stokes equations is employed. Them ixturemedium composed of liquid and vapor is regarded as a kind of single phase fluid with variable density. Each component phase shares the same physical fields, viz., pressure, and velocity and so on. The gravity and slip velocity between phases are not considered here. The volume fractions of liquid phase and vapor phase, denoted aslα and vα respectively, are introduced to obtain a set of governing equations describing the flow of the vapor and the liquid phases.

The continuity equation for themixture is as follows

Themomentum theorem for them ixture phase holds as

The vapor phase should also satisfies continuity during phase-transition process[8]

where ρ is the density, t the time, uithe velocity component, p the pressure, the subscriptsm, l and v respectively indicate themixture, liquid phase and vapor phase, i and j indicate the Cartesian coordinates which can be 1, 2 or 3.

One compatible equation lies in the phases

Additionally, two separate transport equations were employed to describe the phase-transition process between the vapor and the liquid.

σ and k denote the saturated pressure of liquid, surface tension and turbulence energy, respectively.

The density and viscosity of themixture were calculated as weighted average of the volume fraction of each phase:

Moreover, the RNG k?ε turbulencemodel[9,10]together with the standard wall function[11]were included tomake the governing equations closed.

The finite volumemethod was utilized to discretize the set of governing equation. A SIMPLE-type algorithm[12]

was adopted to consider the pressure-velocity-density coupling process. The pressure term in the RANS equations was computed with the PRESTO! Scheme[13], and a second-order upw ind scheme was employed to treat the convection. The ultimately linearized algebraic equation was solved with the Gauss-Seidel iterationmethod[14,15]and accelerated using algebraicmultigridmethod[16]. The establishment and solving of the presentmathematicalmodel was totally grounded on the framework of the commercial CFD code Fluent.

Fig.1 Vehicle body, strutmodes, and surfacemesh

Fig.2 Cavity shapes at different combination of strut type and cavitation number, α=0o, top view

2. Computational domain and boundary conditions

Figures 1(a)-1(c) illustrate the configuration of the vehicle for the three types of strutmodes and the correspondingmeshes distributed on the vehicle surface. The total length and themaximal diameter of the vehicle are indicated by L and D respectively. Figure 1(d) presents the cross section of the foilshaped strut with a length of 0.25L and amaximum thickness of 0.30D. The origin of the coordinate system is located at the peak of the vehicle head when the angle of attack equals zero. The positive direction of abscissa and ordinate axes points rightward and upward respectively. The angle of attack is defined as the included angle generated by the x-axis and the axis of the vehicle (<90o). The cross section area of the water tunnel is 10D×10D. The upstream boundary and the downstream boundary are 1.5L and 4L respectively from the origin of the coordinate system.

The boundary conditions for computation were disposed as follows: upstream velocity (u1=10m/s and u2= u3=0m/s ) for the upstream boundary, fixed static pressure (Poutlet=constant) for the downstream boundary, and non-slip condition for the other boundaries.

The angle of attack of the vehicle body was set as 0o, 2oor 4ofor three different working conditions. The position and zero degree angle of attack of the strut keep constant for all the cases. The cavitation number, which is defined as σv=(P∞? pv)/0.5ρ, was achieved by adjusting the static pressure at the downstream boundary. pvindicates the saturated vapor pressure of water at specific temperature, and P∞and V∞denote the upstream velocity and pressure.

3. Results and discussion

3.1 Strut effect on natural cavitation number

Figure 2 presents an overview of the cavity shapes for various combinations of strut types and cavitation number at the angle of zero degree. Here, the cavity outlines were displayed as the iso-surface of αv=0.1.

It seems evident that for one specific type of strut the cavities attached on the shoulder and the tail of vehicle both shrinks when the cavitation number increases. Besides, the strut type also has rather notable effect on the value of cavitation number. For the same upstream velocity and downsteam pressure, all the cavitation numbers in working conditions with a strut installed are greater than that in non-strut conditions. Additionally, the cavitation number for the C-type strut is greater than that for the B-type one. Consequently, the influence of strut type on cavity configuration should be concerned for gravity driving water tunnel due to its constant pressure at the tunnel exit which is opened to atmosphere.

Fig.3 Shoulder cavity length versus cavitation number for different types of strut, α=0o

The relation of the cavitation number and the length of shoulder cavity at 0oangle of attack are expressed as the curves in Fig.3 for various types of struts. It indicates that the difference between the cavity lengths of different strut types is not so obvious at identical cavitation number, even though it exists. The relations for the working condition with angle of attack of 2oand 4oare similar to that of 0o.

3.2 Strut effect on surface pressure distribution

Figure 4 presents the distributions of pressure coefficient, cp=( p? P∞)/0.5, along the axial direction of the vehicle on the lateral side, pressure side and suction side, taking α=2ofor example.

Generally, the axial pressure coefficient variation has the follow ing threemain features: (1) themaximal pressure on the stagnation point of the vehicle head, (2) theminimal pressure approaching the saturated vapor pressure in the cavitated region at the shoulder of the vehicle, (3) the gradual drop of pressure due to friction along the axial direction. The pressure distribution along the vehicle surface ismainly influenced by strut type in the follow ing two aspects.

Fig.4 Pressure coefficient distributions on vehicle surface for various strut type, α=2o

On the one hand, as the upstream velocity and the downstream pressure are fixed, the region where relative low pressure exists on the vehicle shoulder is smaller for the C-type strut than for the B-type one, and both of these two types are also smaller than nonstrut one. Such a phenomenon results from the follow ing two points. Firstly, the existence of strut causes an increment in local pressure loss, whichmakes the upstream pressure of water tunnel with a strut to be relatively higher, given the fixed downstream pressure. Secondly, the existence of strut causes the incoming flow to cease near the foil’s leading edge tomake the local pressure enhance and transm it toward the up-stream, which can be noticed in Figs.4(c) and 4(d) where a pressure peak is present in the vicinity of the leading edge.

On the other hand, the pressure coefficient on the suction side of the vehicle ismuch smaller in the B-type strut condition than in non-strut condition, and such a pressure drop is evenmore remarkable for the C-type one. This indicates that the surface pressure of the vehicle could be influenced by the low-pressure region on the foil surface.

Fig.5 Drag coefficient in variation with cavitation number for various types of struts

3.3 Strut effect on drag coefficient

Figure 5 describes the relations between the cavitation number and the computed drag coefficient, c= F/0.5, for the various types of struts,

d x specified for three different angles of attack, where Fxdenotes the streamw ise component of the resultant force acting on the vehicle body, SDdenotes themaximal cross section area on the vehicle.

For one identical cavitation number, the drag coefficien of the vehicle enhances slightly as the angle of attack increases, andmaintains nearly constant in non-cavitating conditions. The drag coefficient obtained in with-strut condition is larger compared with non-strut condition. The strut effect on the deviation of drag coefficient gradually dim inishes between the B-type strut condition and the non-strut one, but strengthens a little between the C-type strut condition and the non-strut one on the contrary as the angle of attack increases. It is considered that, when zero degree angle of attack is concerned, the B-type strut produces an asymmetric flow structure, which induces the lowpressure region on the bottom of the vehicle to be w ider than the other two types of struts, and accordingly produces larger pressure-difference drag. However, with the increase of the angle, the effect from the angle on the pressure overweighs that from the asymmetry caused by the strut. Therefore, after the cavities on the shoulder and the tail of the vehicle appear by reducing cavitation number, the influence of strut type on the decreasing drag coefficient becomes weaker and weaker. This is because nomatter what type of strut is employed, the vehicle is exposed to nearly equal stagnation pressure in front of the head and equal saturated vapor pressure on the bottom. Therefore, the pressure drag acting on the vehicle body is almost the same.

Next, amethod of correction was proposed to revise the drag coefficient obtained with a strut to that without a strut, to remove adequately the interference of the strut on the drag of the vehicle.

A fter several times of combination and comparison, correction formulas showed in Eqs.(9) for parameters in the fitting function under the non-strut condition are determined at small angle of attack, where the subscript d stands for the drag coefficient, superscripts Non, B and C refer to the non-strut type, B-type and C-type strut conditions respectively.

Tables 1-3 list the parameters in the fitting function of the drag coefficient with different kind of strutmode, corresponding to 0o, 2oand 4oangle of attack respectively.

Table 1 Parameters in the fitting function of the drag coefficient by different type of strutmode, α=0o

Table 2 Parameters in the fitting function of the drag coefficient by different type of strutmode, α=2o

Table 3 Parameters in the fitting function of the drag coefficient by different type of strutmode, α=4o

Table 4 Relation errors between the drag coefficients obtained by using the correction formula and simulation

Table 4 gives the relative errors between the drag coefficients obtained by using the correction formula and simulation, including the average error,maximum error and standard deviation, for the range of small attack angles. It turns out that the proposed correction formula of the drag coefficient is of a good accuracy. 3.4 Strut effect on lift coefficient

Figure 6 further presents the relations between the cavitation number and the lift coefficient, cl= F/0.5ρ, for the different types of strut, where

y l Fydenotes the vertical component of the resultant force acting on the vehicle.

Fig.6 Lift coefficient in variation with the cavitation number for various types of strut

The lift coefficient exhibits a variationmode sim ilar to that of the drag coefficient. That is, the lift coefficient increase along with the increment of angle of attack, andmaintains nearly constant in non-cavitating conditions, and the lift obtained in with-strut condition is larger comparing with non-strut condition. A fter the cavities on the shoulder and the tail of the vehicle begin to appear by reducing cavitation number, the lift coefficient dim inishes gradually. Furthermore, the lift obtained at zero degree attack angle for the three different types of strut are all close to zero. Accompanying with the improvement of the angle, the strut effect of the B-type and the C-type struts both strengthens, which ismainly due to the physical fact that the peak of pressure near the leading edge of the strutmay also put an action on the belly of the vehicle and extend its influence region along with the increment of angle of attack, given the condition of fixedposition and angle of attack of the strut.

The trend of the lift coefficient variation is similar to that of the drag coefficient with the cavitation number. Thus, the same fitting function and correctionmethod are adopted for the lift coefficient. Consequently correction formulas showed in Eqs.(10) for parameters in the fitting function under the non-strut condition are found out at small angle of attack, where the subscript l stands for the lift coefficient, three superscripts have the samemeaning as those in Eqs.(9).

Tables 5-7 list parameters in the fitting function of the lift coefficient with different kind of strutmode, corresponding to 0o, 2oand 4oangle of attack respectively.

Table 5 Parameters in the fitting function of the lift coefficient by different type of strutmode, α=0o

Table 6 Parameters in the fitting function of the lift coefficient by different type of strutmode, α=2o

Table 7 Parameters in the fitting function of the lift coefficient by different type of strutm ode, α=4o

Table 8 gives the relative errors between the lift coefficients obtained by using the correction formula and simulation in the range of small attack angles. The average error,maximum error and standard deviation are all lager than those for the drag coefficient except at zero degree angle of attack.

Table 8 Relative errors between the lift coefficients obtainned by using the correction formula and simulation

4. Concluding remarks

In the present work, the interference of side struts with the natural cavitating flow around a submerged vehicle in water tunnel has been numerically investigated, using the homogeneous equilibrium two-phasemodel coupled with a natural cavitationmodel. The corresponding effects of the three conditions of nonstrut, one-side strut and two-side strut, on the cavity shape, the pressure distribution on vehicle surface and the hydrodynam ic properties have been discussed, and the influencemechanism explored. Since the absence of experimentalmeasurement, only have the numerical results been qualitatively analyzed to draw some conclusions as follows:

(1) When identical upstream velocity and downstream pressure are specified, the existence of the foilshaped strut could heighten the upstream pressure in a water tunnel, and consequentlymake the cavitation number increased and the cavity size diminished.

(2) The foil-shaped strut has remarkable effect on the pressure distribution of vehicle surface. Under the conditions of the same upstream velocity and downstream pressure, on the one hand the strut w illmake the low-pressure region near the shoulder of a vehicle shrink, and on the other hand the pressure on the vehicle surface near the strut reduce by the low-pressured on both sides of the foil.

(3) Given the cavitation number, when the angle of attack is increased, the two-side strut has gradually strengthened influence on both lift coefficient and drag coefficient, whereas the one-side strut has strengthened influence on drag but weakened influence on lift.

(4) In the range of small attack angle (0o-4o), the influence of one-side or two-side type of strut and the angle of attack on the hydrodynamics is taken into account. The correctionmethod for lift and drag coefficients are introduced to eliminate the interference from the foil-shaped strut, and the correction formulas of parameters in the sigmoidal logistic function are proposed. The results of error analysis imply that the accuracy of correction formula for drag coefficient is higher than that for lift coefficient.

More in-depth works to be carried out in future w ill involve the improvement of numerical simulationmethods and the inclusion of gravity effect in simulation so as to provide amore accurate analysis of the cavitating flows in water tunnel, especially in the aspects of lift and cavity shape.

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December 26, 2010, Revised July 10, 2011)

10.1016/S1001-6058(10)60149-7

* Project supported by the National Natural Science Foundation of China (Grant Nos. 11002089, 10832007), the Shanghai Leading Academ ic Discipline Project (Grant No. B206).

Biography: CHEN Xin (1976-), Male, Ph. D., Lecturer