(China Ship Scientific Research Center,Wuxi 214082,China)

1 Introduction

It is known that bubbles will form around the high speed moving objects under water.The bubbles are full of vapour and air.On the boundary of bubbles,the pressure is continuous,but the water density is changed into the gas one.The bubbles separate the object from water,which causes the reduction of friction.So cavitation can be utilized to reduce the drag of vehicles.

Although there is no cavitation on the bottom of ordinary boat,we can make bubble,for example,by blowing air to boat bottom to form a bubble.The manmade bubble will move with and be stationary relative to the boat.Because of the high speed of the boat,there will be a distinct boundary between the air and the water.The pressure in the bubble will almost be a constant.

The main study of bubble reduction is to determine the relationship of bubble profile with related variables and the effects of it on drag reduction.Once the profile of the bubble is determined,the surface overlayed by the bubble and the reduced friction can be estimated.The reduced friction should be larger than the added drag caused by introducing of bubble.This is achieved by choosing suitable parameters of the boat and the flow,including the pressure and volume speed of blowing air.The air volume consumed for maintaining the bubble and the power for air jet can also be evaluated.

There is no difference between manmade bubble and cavitation bubble in the viewpoint of flow.So we can define the bubble number σ of manmade bubble,just as cavitation:

where P0is pressure at infinite forword or atmospheric pressure,Pvis saturated steam pressure,P is pressure in manmade bubble,Vwis water speed at infinite forword or boat speed,ρ is water density.Saturated steam pressure is used to express the pressure in the bubble,that means the similarity of manmade bubble to cavitation bubble.

Bubble number is an important parameter which has decisive effect on the bubble flow.Let reference pressure P0=0,Eq.(1)can be simplified as:

2 Math modeling for flow around bubble boat

2.1 Simplifying the 3-dimensional flow into 2-dimension

Considering the high speed of the boat,the transverse flow and viscous effect can be neglected,the complicated 3-dimensional flow around the bubble can be simplified into 2-dimensional inviscous one.At the same time,because we are specially interested in the the flow around the bubble,we can further simplify the profile of the boat into a horizontal infinite plane with a wedgy air jet,see Fig.1,where L is the length of the wedgy air jet,and Lais the length of the bubble.

2.2 Analysis of the 2-dimensional flow pattern

Actually,the flow shown in Fig.1 is the closed-bubble flow pattern with a steady cavity.If the air jet pressure is large enough,or the air is jetted into the cavity continuously,then the cavity will stretch to infinite along the axis x,which is called half-open-bubble flow pattern.On the contrary,if the air jet is closed,no bubble will be formed behind the air jet(except the cavitation bubble caused by a high inflow speed),which is called as no-bubble flow pattern.

The three flow patterns have inner relationship.There is a low pressure domain after the air jet in the flow pattern of no-bubble.The pressure at the air jet is just the minimum pressure for air jetting to form a manmade bubble.If the pressure at the air jet is large enough,the pressure will drive the air flowing backward to infinite,and the flow pattern of closed-bubble will be transformed into that of half-open-bubble.

2.3 Analysis of the similarity principle for 2-dimensional flow

The similarity principle for different flow pattern is different.The simplest flow pattern is no-bubble pattern.The independent variables for this pattern are water speed Vwand length of the air jet L,so there is only one similarity number,Froude number Fw:

There are three independent variables for closed-bubble flow pattern,namely water speed Vw,length of the air jet L and the bubble pressure P.Two similarity numbers can be formed,namely Froude number Fwand bubble number σ,see Eq.(2)and Eq.(3).

It is reasonable to choose the water speed Vw,length of the air jet L and the jetting air speed Vaas the independent variables for half-open-bubble flow pattern.So the similarity numbers for this pattern are Froude number Fwbased on water speed and Froude number Fabased on air speed.Definition of Fwis the same as Eq.(3),Facan be defined as:

2.4 Numerical simulation method for 2-dimensional flow

Thin wing theory may be regarded as a suitable method to solve the flow pattern of closed-bubble.The difficulty of the method is the beforehand unknown boundary of the bubble.What’s more,the singularity of thin wing method may cause the unreal results,even give the paradoxical flow which is contrary to fact.So finite volume method is used in this paper to solve the closed-bubble flow pattern and VOF to deal with 2 phases flow.The numerical method is based on inviscous Eular equation,which is suitable to solve all the three flow patterns.

3 Analysis of the numerical results

Numerical model is shown in Fig.1,where length of air jet L=2m,and its height H=0.25m.Speed of inflow Vw=4m/s,7m/s,10m/s and 13m/s,respectively.

3.1 Calculation results of no-bubble flow pattern

The minimum pressure after the air jet is listed in Tab.1,which is linearly proportional to the square of inflow speed.

Tab.1 Minimum pressure at different inflow speed

where Rpis the pressure drag acting on the wedge.Rpand Cpare listed in Tab.2.

The coefficient Cpof pressure drag is defined as:

It can be seen that the coefficient Cpis almost a constant.Because there are differences between wedge and boat bottom,the drag acting on wedge is different from the added drag acting on boat introduced by the cavity and bubble.

Tab.2 Pressure drag acting on the wedge

3.2 Calculation results of closed-bubble flow pattern

Length of the bubble relative to different air jet pressure is shown in Fig.2.It can be seen that the length of the bubble converges to a constant at low inflow speed,but the length of the bubble increases quickly with the increase of air jet pressure at high inflow speed.

The profile of the bubble is shown in Fig.3.It can be seen that the length of the bubble will increase with the increase of inflow speed at the same bubble number,that is to say,high speed of the boat is in favor of the formation and stabilization of the bubble.

Pressure drag Rpacting on the wedge is shown in Fig.4.Rpwill linearly decrease with increase of the air jet pressure.

3.3 Calculation results of half-open-bubble flow pattern

The inflow speed is still 2m/s,5m/s,10m/s and 20m/s,respectively.

The typical profiles of the waving boundary of the bubble are shown in Fig.5.It can be seen that the boundary of the bubble is fluctuating,and the wave length increases with the increase of the inflow speed.At the same inflow speed,the speed of the air has almost no influence on the bubble.

Pressure drag Rpacting on the wedge is shown in Fig.6.It can be seen that Rpdecreases obviously with the increase of the air speed at low inflow speed.

It is also found that a larger air jet pressure is needed at a higher inflow speed when the air speed is the same.This is on the contrary with the closed-bubble.This phenomenon can be explained in the viewpoint of energy.

When the bubble is closed,the bubble only affects the local flow.Because the air volume speed is 0,no energy is needed to support the bubble.If the bubble is half open,the effect of bubble on the flow will extend to infinite.The continuous jetting of air will pull away the water.More energy is needed at a high inflow speed.This energy is provided by air jet,which can be expressed as:w

here P is pressure at air jet,and Q is air volume jetted into the bubble.Air volume is proportional to air speed.At the same air speed,if you want more energy to suppor the bubble,you have to increase the air jet pressure.That’s why larger air jet pressure is observed at higher inflow speed.

4 Concluding remarks

Calculating results show that the bubble will be longer and thinner at higher inflow speed.So bubble may be more suitable to high speed vehicles.

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