Multi-physics Coupling of Hydraulic System

ZHANG Jian， LUO Nianning， JIANG Jihai*

1.School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150080, China；2.School of Automobile Engineering, Harbin Institute of Technology, Weihai 264209, China

Multi-physicsCouplingofHydraulicSystem

ZHANG Jian1， LUO Nianning2， JIANG Jihai1*

1.SchoolofMechatronicsEngineering,HarbinInstituteofTechnology,Harbin150080,China；2.SchoolofAutomobileEngineering,HarbinInstituteofTechnology,Weihai264209,China

Inthispaper,asummaryofmulti-physicscouplingofhydraulicsystemisgiven.Thepaperintroducesthetaxonomyofvariouscouplingrelationshipandthemainlyphysicsfieldsinthehydraulicsystem.Andthispaperintroducesasummaryofmulti-physicscouplingofhydraulicsystembothwithinChinaandabroad.Accordingtosomecommonproblemsofthermal,cavitation,pressurerippleandflowrippleonhydraulicsystem,thispaperanalyzeswhichfieldsinfluenceontheseproblems,andbrieflyintroducestheexistingmulti-physicscouplingphenomenaofproblems.Thispaperpredictstheresearchtrendoftheseproblems,andpointsoutthatthemathematicalmodelofmulti-physicscouplingneedstobeestablishedinthefutureresearchwork.studiesonthemainlyphysicsfieldsareinvolvedinhydraulicsystemwhentheseproblemsinfluencehydraulicsystem.Finally,thispaperputsforwardtheresearchmethodofmulti-physicscouplingofhydraulicsystem.

hydraulicsystem,thermal,cavitation,pressureripple,flowripple,multi-physicscoupling

1.Introduction

Multi-physics coupling refers to the interaction of two or more than two fields’ formation of physical phenomena[1], which exists widely in nature and engineering application. Physics holds that field is interaction of substances[2]. However, the fields in hydraulic system cite the conception of physics, mainly refer to flow field, pressure field, temperature field, noise field, structure field, and so on. Multi-physics coupling of hydraulic system mainly refers to the relationship of interaction of different kinds of fields.

In 1648, Pascal put forward the principle of hydrostatic transmission, but this principle was first applied to hydraulic crane by England until 1850[3]. After this, hydraulic transmission has been widely applied in various machinery equipments, because of these series of advantages. The advantages of hydraulic transmission include high power density, components flexible disposal, convenient control, good dynamic performance, and so on. With the development of science and technology, hydraulic system constantly develops towards high pressure, high velocity and high power which put forward higher requirements for structure, friction, vibration, noise, seal and thermal effect of hydraulic system and hydraulic components which constitute the hydraulic system. However, because the coupled relationship of interaction of several physical factors which are flow field, temperature field, noise field, pressure field and structure field of hydraulic system is complicated, it is difficult to make breakthrough on single performance of hydraulic component or hydraulic system for us, which seriously restricts the application range of hydraulic transmission technology.

This paper mainly analyzes the properties of kinds of fields of hydraulic system and interaction relationship between these fields, and makes classification of multi-physics coupling from several perspectives. And this paper introduces a summary of multi-physics coupling of hydraulic system both within China and abroad. Through the theory analysis, this paper reveals the action characters and the interactional relationship of physical fields of hydraulic system. Moreover, this paper seeks the ways and the methods to solve the present problem of hydraulic system, from the view of improving the reliability and the integrated transmission efficiency of hydraulic system.

2.Classification of multi-physics coupling and fields of hydraulic system

There are many classification methods of multi-physics coupling. According to the region of coupling occurrence, multi-physics coupling is divided into boundary coupling and domain coupling. According to the interaction of coupling, multi-physics coupling is divided into two-way coupling and one-way coupling. According to the action pathway of coupling, multi-physics coupling is divided into direct coupling and indirect coupling. According to the form of coupling equation, multi-physics coupling is divided into differential coupling and algebra coupling. According to the occurrence mechanism of coupling, multi-physics coupling is divided into source coupling, fluid coupling, attribute coupling and geometry coupling[4]. According to the coupling relationship, multi-physics coupling is divided into one-to-one coupling, one-to-many coupling, many-to-one coupling and many-to-many coupling.

Multi-physics coupling problem of hydraulic system mainly involves in the coupling of structure field, flow field, noise field and temperature field. There are kinds of coupling relationship between related fields. Moreover, with the development of the electronic technology, the trend of electronic control of hydraulic component is more and more obvious in recent years. So based on above several fields, electromagnetic field is also introduced into hydraulic system. And because of the electromagnetic field, the analysis of multi-physics coupling problem of hydraulic system becomes more and more complex.

People mainly concern system efficiency, system controllability, system noise and reliability of hydraulic system in engineering application. The coupling relationship which has great influence on above factors mainly includes heat-fluid coupling, thermal-structure coupling, fluid-structure interaction, magnetism and solid coupling, electromagnetic coupling, fluid-pressure coupling, solid-pressure coupling, fluid-noise coupling, and so on.

Heat-fluid coupling: temperature field couples with fluid field.

Thermal-structure coupling: temperature field couples with structure field.

Fluid-structure interaction: fluid field couples with structure field.

Magnetism-solid coupling: magnetism field couples with structure field.

Electromagnetic coupling: electric field couples with magnetic field.

Fluid-pressure coupling: fluid field couples with pressure field.

Solid-pressure coupling: structure field couples with pressure field.

Fluid-noise coupling: fluid field couples with noise field.

Temperature field is the most influential physical fields for the performance of hydraulic system. All other physical fields are more or less affected by temperature. Meanwhile, most of the final results of interaction of physical fields in hydraulic system are displayed through heating in the hydraulic system.

The route loss, local pressure loss, throttle loss, and mechanical loss and volumetric loss of each element in hydraulic system are the main reasons of heating in the hydraulic system[5]. Except for driving load and outputting the useful work, other powers produced by hydraulic system are carried away by the oil, and the temperature of hydraulic system will increase in the form of heat during hydraulic system operation. The route pressure loss is caused by fluid field of oil. The mechanical loss is caused by structure field. Throttle loss, local pressure loss and volumetric loss are not only caused by fluid field but also by structure field. Conversely, the increase of temperature of oil influences directly on oil flow and properties of oil. Meanwhile, the increase of temperature of oil makes the element of hydraulic system to produce volume change, which will influence the tolerance clearance of system, and has an indirect effect on structure field of system. At the same, the system or structure change will directly influence on the fluid field and the pressure field of hydraulic system, and also has an indirect effect on temperature field and noise field.

3.Review on research status of multi-physics coupling

The researches on multi-physics coupling of hydraulic system within China and abroad mainly focus on fluid-solid coupling and thermal performance of system at present.

3.1.Theresearchonthecharacteristicsoffluid-solidcoupling

Fluid-solid coupling mainly studies the coupling relationship between fluid field and structure field.

In 1968, Wood[6] considered pipeline as spring-mass system, and he researched on the vibration response of simple pipe system which was carried out under the periodic excitation and sudden valve closure, respectively. In 1977, Walker & Phillips[7] considered the stress waves in the pipe caused by the pressure wave of the pipeline fluid and the poisson coupling between pipelines. Walker & Phillips put forward four governing equations model, then used this model to calculate the reepense of fluid pressure at a valve to sudden valve closure in RPV system. Regetz[8] studied the pressure and flow rate fluctuation of a straight pipe in rocket fuel supply system. And the end of the straight pipe was free closed. Meanwhile, Regetz measured the vibration velocity of the free end of the pipeline through experimental method. Then the measurement results were compared with calculation results in frequency area. The comparison results prove that the influence pipes moving on motion characteristics of fluid is very notabale. In 1980, El1is[9] studied the coupling between valve and joint part of bifurcated pipe in a practical piping. Ellis considered the influence of the pressure wave of fluid, axial stress waves and moment wave in the pipeline. And Ellis used characteristic method to solve the motion equation of the coupling. In 1995, Lee et al[10] combined Paidoussis’s kinetic equation of pipeline and Wiggert’s kinetic equation of pipeline, and ignored the effect of poisson coupling. Lee et al obtained the first nonlinear model which describes the fluid-solid coupled motion. In 1998, WANG Zhongmin et al[11] studied the pressure ripple and dynamic response characteristics of pipeline of hydraulic system through establishing fully fluid-solid coupling model. ZHONG-MIN WANG et al drew the conclusion that fluid-solid coupling model is very important in studying the dynamic characteristics of flexible pipe system. In 2010, according to different types of link coupling, A. Ahmadi and A. Keramat[12] studied the fluid-solid coupling, and they extended the conception of link coupling. In 2011, R. Jaiman et al[13] studied the instable fluid-solid coupling phenomenon used in combined interface boundary condition method, and they proposed and applied a kind of weak coupling method to fluid-solid coupling problem so as to overcome potential instability in coupling interface. In 2012, A. Keramat et al[14] studied the viscoelastic fluid-solid coupling function of tube wall when water hammer happened, and they carried out theoretical analysis and numerical simulation on fluid-solid coupling and viscoelastic of axial movable straight pipeline.

Research on fluid-solid coupling in China include: In 1989, ZHU Geqi et al[15] introduced an approach for the mathematical modeling of piping system with fluid-solid coupling. The approach which was introduced by Zhu Geqi et al combined complex transfer matrix of fluid and complex mode matrix of structure to establish the fluid-solid coupling model. In 1997, FEI Wenping et al[16] set up a fluid-solid coupling general mathematical model of pipeline system with complex boundary conditions from the two angles of that the free body was never taken and the free body was taken, and they made a theoretical investigation. In the same year, XU Mubing et al[17] from Huazhong University of Science and Technology studied the energy flow in fluid-filled cylindrical shells. In 1999, Jiao Zongxia, Hua Qing et al[18] carried out a modal analysis on the fluid-solid coupling vibration in transmission pipeline. Jiao Zongxia et al had successful carried out a node coupling simulation by using the method of twice coordinate transfers, and deduced a high precision fluid-solid coupling vibration model. In 2008, Tao Yuhua et al[19] carried out a fluid-solid coupling simulation of aircraft hydraulic system, and preliminarily studied the pulsation of hydraulic pipeline. In 2011, according to the clip problem of hydraulic spool valve, WANG Anlin et al[20] studied the fluid solid heat couple of slide valve structure. In 2012, ZHANG Huixian et al[21] studied the fluid-solid coupling of pipeline under the hydraulic exciting wave, and they set up a fluid-solid coupling dynamic model.

3.2.Researchonthethermalcharacteristicsofhydraulicsystem

Research on the thermal characteristics of hydraulic system mainly refers to research on the relationship of among temperature, pressure, flow rate and structure of hydraulic system.

In 1996, J A sidders，D G Tilley，P J ChapPle et al[22] from the British university of Bath studied the thermal characteristics of open circuit hydraulic system, set up the thermodynamic model of pump, valve, oil tank, radiator, and so on, by using the first order differential equation which was established based on a series of control volume energy conservation equations. According to different conditions, J A sidders et al simulated the characteristics of system pressure, flow rate and temperature. In 1998, Dominique Legendre et al[23] studied the thermal and dynamics development of spherical bubble in superheated or supercooling fluid. In their research, they assumed that the development of bubble was controlled by heat conduction, and they carried out a direct numerical simulation on the growth and collapse of spherical steam bubble in the fluid. Meanwhile, they obtained the fluid force acted on bubble when there is mean flow. In 2000, the thermodynamic model based on aircraft hydraulic system was developed by Engelhardr J[24]. Then Engelhardr J set up a heat transfer mathematical model of fluid and solid, and combined heat generation of hydraulic oil with heat-transfer of contact solid.

In 2003, C. Vortmann et al[25] set up a new cavitation model. The phase change process of memory alloy was extended to the phase change of water in this new cavitation model. And the thermodynamic description of cavitation phase change process of water was made by this new cavitation model, where involved density, pressure and temperature of water. In 2006, the thermodynamic model of gear pump based on the first law of thermodynamics was established by Eduardo Dalla Lana et al[26]. And this model considered the power loss when the hydraulic oil passes through the gear pump, and the heat transfer occurs between the hydraulic oil and the ambient air. Then Eduardo Dalla Lana et al obtained a new kind of gear pump efficiency formula. In 2008, the thermodynamic model of vehicle one-dimensional transient power steering hydraulic system based on the first law of thermodynamics was established by Timothv C Scott[27]. Timothv C Scott set up the thermodynamic model of ordinary hydraulic component, pipeline and radiator. Then Timothv C Scott considered the effect of the distance between radiators on cooling air distribution in thermal module by using one-dimensional method, and used system model to simulate system temperature characteristics. In 2010, Maria Grazia De Giorgi et al[28] studied the effect of temperature and heat transfer on cavitation in vapor-liquid conversion process when the cavitation occurs in fluid. Maria Grazia De Giorgi et al mainly studied the water and cryogenic fluid, and concluded under the condition of that the same cavitation number cavitation is increased with temperature increasing. In 2011, D. T. Frate et al[29] studied the flow and heat transfer in the hydraulic reservoir of thrust vector control system. They obtained that for the laminar flows, when the inlet temperature of the working fluid is maintained at a lowlevel, the buoyancy effects at both sides of the reservoir walls are not dominant in the flows. However, at a higher inlet temperature of the working fluid, this greatly changes both flow patterns and heat transfer performance of the working fluid and the cooling air around the reservoir due to the stronger buoyancy effects.

In China: In 1994, LU Tinghai[30] studied the effect of temperature on main performance parameters of hydraulic pump, and set up a simple mathematical model for the relationship between some of performance parameters of hydraulic pump and temperature, then pointed out that only simultaneous studing the pressure and temperature characteristics of hydraulic pump, could we comparatively detailed assess the performance of pump. In 2001, Yang Yung-Kuang and Jeng Ming-Chang[31] from Taiwan studied the thermal effect of asymmetrical hydraulic servo cylinder. Yang Yung-Kuang and Jeng Ming-Chang analyzed the influence of eccentricity and misalignment factor between hydraulic bearing and piston rod using numerical method. In 2006, according to the thermodynamic properties of axial piston pump, LI Chenggong et al[32] set up a thermodynamics model of aviation piston pump, and carried out a detail heat transfer analysis for internal structure of piston pump. In 2008, LI Chenggong et al[33] put forward the basic function of establishing simple model of thermal-hydraulic elements, and introduced the association rule and method of thermal- hydraulic system model. With the above methods, it can fully automatic generate the thermal-hydraulic system. In 2009, based on energy conservation principle, LI Yonglin et al[34] deduced the calculation formula for temperature variation in control volume, and combined with the pressure flow characteristic of servo valve, they put forward the method of establishing thermodynamics model of servo valve. Meanwhile, LI Yonglin et al carried out some modelings and simulations for hydraulic system which includes a four-through slide-valve. In 2010, LI Yonglin et al[35] applied the control volume method to set up the thermodynamics model of hydraulic poppet valve. LI Yonglin et al considered the effect of the change of thermophysical characteristics of hydraulic oil on the pressure loss of valve in model, then carried out a simulation study on hydraulic system which includes a poppet valve. In 2011, XI Renguo et al[36] studied the thermal analysis of aircraft hydraulic system. They used steady analysis method to estimate the equilibrium temperature of hydraulic system, used transient analysis to predict the law of system temperature change, and introduced a method of neural network based on analyzing.

4.Thermal field analysis of hydraulic system

The thermal-fluid coupling and the thermal-structure coupling are eqaul to temperature field couples with fliud field and structure field.

The hydraulic system heating is the concentrated expression of interaction of multi-physics fields of hydraulic system which includes fluid field, pressure field, structure field and temperature field. Setting the throttle loss caused by various hydraulic valves as an example, when the hydraulic oil flows through hydraulic valve, for the opening size of hydraulic valve compares with hydraulic pipeline has a big change, it would cause the sudden change of oil flow velocity, and then lead to the sudden change of oil pressure, which causes the power loss of hydraulic system and the system heating. Meanwhile, the sudden change of pressure causes structural vibration, and produces noise.

4.1.Basicthermodynamicsmodelofhydraulic

system

To evaluate the impact relation of physics fields in hydraulic system which include thermal field with fluid field, pressure field and temperature field, this paper analyses a typical thermodynamics model of hydraulic system which on the law of conservation of energy.

This paper chooses the control volume as shown in Fig.1 for one-dimensional flow fluid.

Fig.1 Control volume

The energy conservation equation of control volume for

(1)

If the kinetic and potential energies are neglected, the time rate of change of the energy can be expressed according to

(2)

where,uis the specific internal energy of control volume.

The time derivative ofhcan be expressed as

(3)

where,αpis the cubical expansion coefficient，cpis the specific heat at constant pressure.cpis expressed as

(4)

The specific enthalpy is defined as

h=u+pv

(5)

After introducing Eq. (3) and (5) into Eq. (2), it is obtained that

(6)

where,Vis the fluid volume.

The continuity equation for one-dimensional flow is given as

(7)

Combining Eq. (6), (7) and (1), it is obtained that

(8)

(9)

(10)

Introducing Eq. (9), (10) into Eq. (8), it is given

(11)

Assuming that the average enthalpy within the control volume equates to the leaving enthalpy regardless of the inlet conditions, Eq. (11) can be expressed as

(12)

The change in specific enthalpy within the control volume is related to the change in pressure and temperature by

(13)

According to above thermodynamics model, it can be gotten that the thermal field of hydraulic system has a close relation with temperature field, pressure field and fluid field of hydraulic system. The change of temperature, pressure and fluid flowing situation can cause the change of thermal field in hydraulic system. The change of thermal field of hydraulic system is a kind of multi-physics coupling problem.

This mathematical model is the classic one-dimensional flow thermodynamics model at present. More perfect multi-dimensional flow thermodynamics model will be described in the future research.

4.2.Asummaryofthermalfieldcoupleswithotherphysicsfieldsphenomenoninhydraulicsystem

The hydraulic system heating will cause physical change in hydraulic oil. The viscosity of hydraulic oil is decreased as the temperature increases. The viscosity decrease of hydraulic oil causes the leakage increase of hydraulic system, and results in the reduction of the volume efficiency of system. Meanwhile, the increase of oil temperature also affects the Reynolds number of oil. The temperature increases causes the increase of the Reynolds number of oil and the flow instability of oil increases, easy leads to turbulent, and induces greatly pressure ripple. The process of change in physical propertics of oil is caused by hydraulic system heating including thermal field, fluid field and pressure field. The coupling relationships in these fields including thermal field causes the change of fluid field, then the change of fluid field causes the pressure field ripple. Conversely, the changes of fluid field and pressure field will decrease the efficiency of hydraulic system, cause the power loss and the system heating, and make the continual system temperature rise.

There is another impact of the change of thermal field of hydraulic system on physical properties of hydraulic oil. That is the bulk modulus of elasticity of hydraulic oil. The bulk modulus of elasticity of hydraulic oil is an important physical parameter for hydraulic oil. It has significant effect on position precision, power level, response time and stability of hydraulic system[37-38]. The influencing factors of bulk modulus of elasticity of hydraulic oil mainly include the pressure, temperature and gas content of hydraulic oil. The effect of thermal field on bulk modulus of elasticity of hydraulic oil mainly includes two respects. On the one hand, the temperature increase causes the decrease of oil viscosity, then reduces the bulk modulus of elasticity of hydraulic oil. On the other hand, the temperature increase causes the change of gas content of hydraulic oil, then impacts the bulk modulus of elasticity of hydraulic oil. The effects of the change of thermal field on gas content and bulk modulus of elasticity of hydraulic oil are expressed as follows:the gas solubility decreases with increasing the temperature under the same gaseous phase partial pressure, and which causes the increase of air bubble, then decreases the effective bulk modulus of elasticity of hydraulic oil. The decrease of effective bulk modulus of elasticity of hydraulic oil causes the decrease of the efficiency of hydraulic system. And the loss of power of hydraulic system will transfer into the quantity of heat. This causes the increase of the system heating. The increase of the system heating makes the temperature of hydraulic system continue increase, causes the dissolved gas continue to precipitate, and which impacts the flowability of hydraulic oil in hydraulic system. Meanwhile, the increase of the system heating causes the change of pressure of hydraulic oil along the way. This also shows the characteristics of coupling of multi-physics of hydraulic system.

The thermal-structure coupling which caused by the change of thermal field displays that the temperature field forms the temperature difference, and leads to the expansion or reduction of the structure, which generates thermal stress[39]. The effect of temperature on element structure of hydraulic system is manifested as the element of hydraulic system expansion deformation caused by hydraulic system superheat, and it decreases the tolerance clearance between different parts of the kinematic pair, then causes the increase of frictional resistance, even the clamping stagnation, and eventually the hydraulic control element failure. Because of the structural deformation, it causes the increase of frictional resistance of kinematic pair, which will cause the power loss, then cause the increase of hydraulic system heating.

Therefore, the change of thermal field of hydraulic system is not only manifested as increasing or decreasing the hydraulic system temperature, but can also produce the influence on fluid field, structure field and pressure field of hydraulic system, meanwhile, fluid field, structure field and pressure field react on temperature field. Hydraulic system heating phenomenon is a multi-physics coupling problem which involves temperature field, fluid field, structure field, pressure field, and so on.

4.3.Theforecastofresearchonthermalproblemofhydraulicsystem

For the research on thermal problem of hydraulic system, researchers should consider this problem from the view of multi-physics coupling. Firstly, researchers should make a thorough study on the heating mechanism of hydraulic system, optimize the existing research results, through comprehensive consideration of the coupling effect among multi-physics fields, find the coupling relationship among the multi-physics fields effecting on hydraulic system heating; Secondly, researchers should set up the mathematical model of thermal fields and other physics fields, and carry out simulation and experiment researches on establishing and optimizing the mathematical model, so as to achieve twice the result with half the effort for research on thermal of hydraulic system.

5.The multi-physics coupling of cavitation of hydraulic system

5.1.Briefintroductiononcapitationphenomenon

Cavitation was defined earliest by Knapp et al[40]. The solution behavior of gases in hydraulic oil follows Henry law, that the volume of gases which dissolve in hydraulic oil is proportional to the absolute pressure of hydraulic oil under certain temperature, and increases with the increase of time. When the hydraulic system appears local low pressure, the volume of original micro-bubble dropping in hydraulic oil expands continuously, and these micro-bubbles polymerize each other, finally form certain volume bubbles and dissociate out. In this case, slight cavitation is formed. If the pressure continuously decreases to lower than the air apart pressure, the gases dissolving in hydraulic oil would separate out the hydraulic oil rapidly and form numerous bubbles. In this case, serious cavitation is formed. If the pressure continuously decreases to lower than the saturated vapor pressure of hydraulic oil, the hydraulic oil would vaporize and form numerous bubbles. In this case, intense cavition is formed. Cavitation often occurs the pressure of hydraulic oil lower than vapor pressure of hydraulic oil under current thermodynamics conditions[41]. The formation of low pressure usually relates to the pressure ripple of hydraulic system caused by flow ripple of hydraulic system.

According to the research results of cavitation problem, it is one of the necessary conditions of hydraulic oil generating cavitation that there is micro gas nucleus in hydraulic oil. In order that the hydraulic oil does not generate cavitation, the free gas nucleus and the circle flowing liquid should satisfy the equilibrium condition. Besides satisfying the force equilbrium condition, the free gas nucleus and the circle flowing liquid need to meet the thermal equilibrium condition and the phase equilibrium condition. Namely when gas-liquid two-phase coexists in the equilibrium state, the second law of thermodynamics requires that both of the temperatue and the chemical potential must keep continuity at phase interface. The process of cavitation has intense thermodynamics imbalance[42]. It is showed that the generation and development of cavitation relate to multi-physics.

When the cavitation occurs, the produced bubbles flow to the high pressure region with hydraulic oil, and the bubbles are compressed by the shock of ambient high pressure oil, the volume of bubble reduction rapidly then collapse, finally bubbles condense fluid. In this case, the space occupied by original bubbles forms a vacuum, then the nearby high pressure oil fills the space with a high flow speed. Because the collapse of bubble occurs in transient, it will cause a strong pressure shock, and both the pressure and the temperature increase drastically at the occurrence station of the bubble collapse. According to literature[43], the continuous bubble collapse will cause the increase of pressure up to 1 000 MPa. The pressure rised instantly spreads as pressure waves, and it makes the hydraulic system produce a strong vibration and noise. Partial pressure energy can be transformed into thermal energy by particle oscillation, and it makes the temperature of partial high pressure region be up to more than 1 000℃. The structure surface of hydraulic system at the bubbles accumulative station endures the hydraulic impact and the high temperature. The erosion and spalling appears, and this phenomenon is named cavitation erosion.

People are not willing to see the phenomenon of cavitation in hydraulic system. The cavitation will cause the decrease of flow rate in hydraulic system, the increase of low pressure in pump, asymmetric load, vibration, noise, cavitation erosion, and so on[44]. It is the result of multi-physics coupling effect for the occurrence, development and collapse of cavitation. The cavitation relates to fluid field, pressure field, temperature field, noise field and structure field.

5.2.Bubbledynamicequation

The micro gas nucleus in hydraulic oil is one of the necessary preconditions of hydraulic oil generating cavitation. When free gas nucleus which contains gas and vapor, circle flowing liquid keep a dynamic balance, the force equilibrium equation is

p=pv+pg-2S/R

(14)

where,pis the pressure of liquid aroud gas nucleus (Pa),pvis the vapor pressure (Pa),pgis the gas pressure inside the bubble;Sis surface tension of liquid,Ris gas nucleus radius under equilibrium state (m).

According to Eq. (14), the condition of gas nucleus expansion is that the right side of the equation is larger than the left side of the equation, which means that the pressure inside gas nucleus is larger than the pressure of liquid aroud gas nucleus. When the balance is broken, the bubble dynamic equation is written as the Rayleigh-Plesset equation[45] as

(15)

where,Rbis the bubble radius (m),Sis the surface tension of liquid,Pis the pressure of liquid aroud bubble (Pa),vlis the kinematic viscosity of liquid;ρlis the liquid density (kg/m3),pbis the bubble surface pressure.pbis expressed as

(16)

whereTcis the temperature inside the bubble (K),T∞is the temperature of liquid (K),pvis the vapor pressure (Pa),pg0is the initial gas pressure;Ris the initial gas radius (m),Rbis the bubble radius (m).

Assuming that the behavior of the gas in the bubble is polytropic,

(17)

whereγis the polytropic exponent,pgis the gas pressure at arbitrary time.

According to Eq. (15)～(17), the occurrence and development of cavitation in hydraulic oil have intimate relationship with the pressure, temperature, viscosity and density of hydraulic oil. Meanwhile, the gas pressure decreases gradually with the development of bubble, but when the volume of bubble decreases rapidly, the pressure inside bubble will increase greatly. So it will produce very high pressure at the centre of collapse when the bubble collapses, then cause vibration and noise in hydraulic system.

5.3.Asummaryofthephenomenonofmulti-physicscouplingincavitation

After the occurrence of cavitation, the bubbles collapse. Besides producing vibration and noise, because the hydraulic oil around bubbles occupies the space,where formed bubbles collapse when bubbles collapse, it will cause the continuity of fluid to be broken, decrease the ability of through oil of oil suction pipe, then decrease the ability of through oil of hydraulic system.

There is gas nucleus at the hydraulic system structure solid surface. The surface roughness of solid surface is a key factor for the influence on the size and quantity of gas nucleus. Besides, the solid surface irregularity has an effect on fluid flow, then produces the change of pressure. Commonly, the higher surface roughness is, the easier occurrence of cavitation will be[46]. The occurrence of cavitation is influenced by the structure surface quality. The structure surface quality has an effect on the fluid field and pressure field of hydraulic system. When cavitation occurs, there are the effects among structure field, fluid field and pressure field.

The cavitation erosion is the cavitation effect on the hydraulic system structure surface. The occurrence of cavitation erosion not only damages the hydraulic system structure surface and decreases the service life of hydraulic element, but also produces the flow rate ripple and the pressure ripple in hydraulic pipe, then causes the power loss, leads to the system heating. So there are interactions among structure field, fluid field, pressure field and temperature field in cavitation erosion.

5.4.Theforecastofresearchoncavitation

According to the analysis of cavitation in hydraulic system, it can be known that the cavitation relates to several physics fields, but not an independent action by a certain field. There are multi-physics interactions among temperature field, pressure field, fluid field, structure field and noise field in cavitation. The occurrence and development of cavitation are typical multi-physics coupling problems.

In the past study on cavitation, researchers often ignore the multi-physics coupling problem, and more studies of the cavitation problem is from a single angle, so research results mostly have limitation. So, according to the multi-physics coupling characteristic of cavitation, people study cavitation should apply multi-physics coupling ideological which contains thermal, pressure and fluid fields, and so on. It is need to inquiry into the change of thermal, pressure and oil flow condition effect on cavitation, and more deeply seek for the mechanism of occurrence, development and disappearance of cavitation. Specially, it is a trend that when the mathematical models of occurrence, development and disappearance of cavitation are built, researchers should consider more effect factors to model, and get more detailed mathematical models.

6.Multi-physics coupling of hydraulic system pressure and flow rate ripple

The hydraulic pump output pressure and the flow rate are not absolute stable in the hydraulic system practical work. Because of pump’s structural characteristic, the pump output flow rate changes periodically, and it is the inherent attribute of pump. The periodic flow rate of pump causes the pressure ripple of pump outlet and pipeline, and spreads to the whole system.

Taking the axial piston pump as an example, as Fig.2 shows, the displacement of axial piston pumpVis

(18)

where,dis the piston diameter (m),lis the piston stroke (m),Ris the pitch circle diameter of piston centre (m),γis the theswashplate angle (°),Zis piston number.

The flow rate of axial piston pump is

(19)

where,nBis the rotation speed of pump (r/min),ηwis the volume efficiency of pump.

When the cylinder block rotatesφangle, the displacement of piston ball-end centre relative to cylinder block centre is

s=R(1-cosφ)tanγ

(20)

where,φis the rotated angle of piston relative to vertical centre line (°).

The piston axial movement is expressed as

(21)

So, the instantaneous flow rate of every pistonQ′ is expressed as

(22)

The instantaneous flow rate of pumpQis expressed as

(23)

According to equations (18)～(23), the instantaneous flow rate of piston pump is ripple and changes periodically. So with the periodical change of the flow rate, the pressure of pump changes periodically.

Fig.2 The nomogram of instantaneous flow on piston-pump

The hydraulic pump is the main noise source in hydraulic system. The noise of hydraulic pump can be classified into fluid-borne noise and structure-borne noise. The fluid-borne noise includes the flow rate ripple, the pressure shock and the cavitation noise. The same, taking the axial piston pump as an example, the key root cause of the fluid-borne noise is the flow rate ripple which then causes the pressure ripple[47]. And the fluid-borne noise will be transferred to the whole hydraulic system by hydraulic pipeline. Study on the generation mechanism of the fluid-borne noise of piston pump needs to begin with the flow rate ripple of piston pump, then study on pressure ripple, and they are complementing each other.

Because of the flow rate ripple of piston pump, when there is no enough hydraulic oil in piston chamber, and which causes the pressure of piston chamber is lower than the oil vapor pressure, the piston pump will generate cavitation[48-49]. The occurrence of cavitation will bring a series of multi-physics coupling problems as noted in the previous section for hydraulic system.

Due to the change of the flow velocity, the flow hydraulic oil in hydraulic system causes the change of pressure, and it sometimes forms a local low pressure region in hydraulic system. When the pressure of local low pressure region is lower than air apart pressure of gases which is dissolved into hydraulic oil, the dissolved gases will be isolated from hydraulic oil, then generates cavitation. It forms the complicated multi-physics coupling problem and produces a series of adverse effects on hydraulic system.

When pump output fluid units which pipeline, hydraulic valves, hydraulic cylinder, hydraulic motors, if the pipeline or the element has great stiffness, great oscillating friction force and zero defect, the pressure ripple would be gradually damped and tend to stable the pressure flow; when the frequency of pressure oscillations of pulsating fluid is consistent or close to the inherent frequency of element or system, the resonance will occur, which makes the system can not work normally, even cause the element damage[50]. This embodies the coupling relationships among fluid field, pressure field and structure field in hydraulic system which flow rate ripple causes pressure ripple, then causes structure vibration.

Due to the sudden closing of valve, the flow hydraulic oil in pipeline will form a high peak pressure in pipeline. The peak pressure will sometimes enhance with source pressure ripple or sutained oscillation in pipeline. It is the hydraulic shock[50].The hydraulic shock not only causes great vibration, but also can cause the damage of hydraulic system. The hydraulic shock also causes the hydraulic system power loss and the hydraulic system heating. The hydraulic shock relates to multi-physics coupling problem which contains fluid field, pressure field, structure field and temperature field.

In a word, the pressure ripple and the flow rate ripple of hydraulic system have many-sided effects on hydraulic system working performance, it relates to multi-physics coupling problem which contains fluid field, pressure field, noise field, temperature field and structure field, and is not simple pressure field or fluid field problem.

The effect pressure ripple and flow rate ripple of hydraulic system on hydraulic system working performance is the process of multi-physics coupling. When the pressure and the flow rate ripple of hydraulic system are studied, researchers apply the multi-physics coupling theory to study the generation mechanism of them, and build the detailed mathematical model of pressure and flow rate ripple couple with other physics fields, then find that the methods and the measures of decreasing pressure and flow rate ripple of hydraulic system is one of the research topics in pressure and flow rate ripple of hydraulic system.

7.Research method of multi-physics coupling problem

At present, little investigation has been made on multi-physics coupling characteristics of hydraulic system. Because the performance of hydraulic system is required to be increasingly improved, people should develop to study the multi-physics coupling characteristics of hydraulic system. The multi-physics coupling problem of hydraulic system is a complicate physics phenomenon, and it relates to multi-physics fields interaction. The multi-physics fields contain temperature field, pressure field, noise field, fluid field and structure field. The action mechanism between physics fields is complex. Based on the complicated of multi-physics coupling of hydraulic system, studying multi-physics coupling should concentrate on the following aspects:

1) Set up mathematical model. According to thermal, cavitation, pressure ripple and flow rate ripple, these problems involved multi-physics coupling characteristics are theoretically analyzed. Researchers set up the interaction of parameters relationship between physics fields, establish and optimize the mathematical model of multi-physics coupling, then understand the interaction mechanism of physics fields.

2) Numerical simulation. With the development of computer technology, numerical simulation has become a very practical research method. Researchers apply numerical simulation technology and combine with specific problem to carry out the simulation on multi-physics coupling characteristics of hydraulic system, verify the correctness of mathematical model, find the interaction characteristics between physics fields, and get the optimal solution of multi-physics characteristics through improving the simulation parameters.

3) Experimental research. According to the numerical simulation results of multi-physics coupling characteristics of hydraulic system, researchers design and build corresponding experimental platform, carry out verification test on the numerical simulation, then get compellent research results.

4) Design theory and design method of multi-physics coupling of hydraulic system. Researchers improve the design level of hydraulic system through theoretical analysis and summarizing experimental research, and find design theory and design method of hydraulic system suitable for the requirement of modern industry.

8.Conclusions

The working process of hydraulic system relates to fluid field, pressure field, noise field, temperature field, structure field, and so on. Through the analyses of thermal, cavitation, pressure ripple and flow rate ripple, this paper can obtain that the working process of hydraulic system not only relates to single physics field, related physics fields not isolated effect on working performance of hydraulic system, between fields interaction and mutual influence. It is a trend that people research on and design the hydraulic system in the view of multi-physics coupling.

Based on that the multi-physics coupling characteristics of hydraulic system has important effect on performance of hydraulic system, this paper predicts the research direction on thermal, cavitation, pressure ripple and flow rate ripple. The research direction includes studying the interaction relationship between physics fields, setting up the mathematical model of coupling relationship between physics fields, carrying out the simulation and experimental verification on mathematical model, finally finding methods and measures to optimize the performance of hydraulic system, and improving the reliability and the integrated transmission efficiency of hydraulic system.

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液压系统多场耦合

张 健1，罗念宁2，姜继海1*

1.哈尔滨工业大学 机电工程学院，哈尔滨 150080；2.哈尔滨工业大学 汽车工程学院，山东 威海 264209

概述了液压系统多物理场耦合问题，介绍了各种耦合关系的分类方法以及液压系统中所涉及的主要物理场，并介绍了液压系统多场耦合问题的国内外研究现状。针对液压系统中常见的热、气穴、压力与流量脉动问题，分析了这些常见问题主要受到了哪些物理场的影响，并简要介绍了这些问题中存在的多物理场耦合现象，对这些问题的研究趋势进行了预测，指出在日后的研究工作中应建立多场耦合问题数学模型。最后提出了针对液压系统多场耦合特性的研究方法。

液压系统；热；气穴；压力脉动；流量脉动；多场耦合

TH137.1

2012-08-25

National Natural Science Key Foundation of China (50875054), Science Key Foundation of Zhejiang University (GZKF-2008003), Natural Science Key Foundation of Shandong(ZR2009FL002)*JIANG Jihai, Professor. E-mail: jjhlxw@hit.edu.cn

10.3969/j.issn.1001-3881.2013.06.003