1.Introduction

The past decade has seen that the emission standards for various types of vehicles have been becoming more stringent.To achieve less emission,car manufacturers are attempting to downsize an internal combustion engine that can provide the same power of a large engine by adding a boosting device,e.g.turbochargers.1To better acceleration and deliver more power across a wider range,a Variable Nozzle Turbine(VNT)has been developing and applied.2A common method of variable nozzle in a radial turbine is the use of the pivoting vanes that are mounted on a flat plate.These vanes are connected with spindles,sometimes together with handles,to achieve the function of rotating axially.Between the flat plate and nozzle vanes,nozzle end-clearances have to be designed.

Efforts to reduce the nozzle end-clearances’size have been doing,and so far the size has been successfully decreased in advanced VNT turbos.But the unfavorable effects of the clearances are still significant and further decreasing the size will be very hard,since the nozzle vanes would get stuck due to thermal expansion.The temperature of exhaust gas is often more than 900 °C for a diesel engine and beyond 1000 °C for a gasoline engine.In addition,complicated sulfide from exhaust gas moves outward from wheel inducer to nozzles under the action of centrifugal force,and then is possible to stick the nozzle vanes as the clearances are too small.

It has been reported that the presence of the nozzle endclearances usually has unfavorable effects on aerodynamic performance of a VNT turbine.Hu et al.3performed numerical simulations on a VNT turbine to study the nozzle endclearances’effect on turbine performances and revealed that the presence of nozzle end-clearances deteriorated turbine performances,especially at small open condition.Walking shaw et al.4carried out numerical simulations of the flow fields in a highly off-design VNT turbine and indicated that the leakage vortices changed the circumferential distribution of the approaching flow to the rotor and then deteriorated aerodynamic performance.The relationship between the nozzle end clearances and the turbine performance was also focused by Tamaki et al.5,6Besides the aerodynamic performance,the high cycle fatigue is one of the critical issues that the development of an advanced VNT turbine has to address.Shock waves,one of the contributors to the high cycle fatigue failure,weigh heavily on the turbine wheel blade vibration and,sometimes,could seriously damage a turbine wheel,as shown by Chen.7The intensity of the shock wave was proved to be closely related with nozzle end-clearances in a VNT turbine.8Therefore,the nozzle end-clearances must have an effect on turbine wheel reliability.In fact,the nozzle clearance leakage flow is also able to directly interact with a turbine wheel and then causes it to vibrate.9,10

Though it has been reported that the nozzle end-clearances are closely related to the turbine performance,the relationship between the forced response of turbine wheel and the changed clearances,i.e.increased clearance size and different distribution,is not fully clear.If the nozzle shroud clearance has a significantly different effect on a VNT turbine from the hub clearance,it will provide a possibility to further improve a VNT turbine in both the aerodynamic performance and the reliability by optimizing the more sensitive clearance,instead of the hard effort to further decrease the size of total nozzle end-clearances.

In this paper,a VNT turbine was first modeled and simulated by a 3D steady Computational Fluid Dynamics(CFD)method to validate computational mesh,and then the VNT turbine model with various nozzle end-clearances were predicted by both steady and unsteady CFD methods.After CFD simulations,the information about aerodynamic excitation pressure on a wheel blade surface was transferred from the CFD results to the Finite Element Analysis(FEA)calculation by interpolation so that the harmonic response could be predicted and analyzed.Finally,an optimized nozzle end clearances’distribution with high turbine efficiency and low forced response was suggested.

2.Research model

The research model is a VNT turbine.In order to reduce the requirement of CFD simulations,the volute was neglected.This simplification should have little influence on the conclusions stated in this paper,since the flow is guided and redistributed in crossing the nozzles.Therefore,the model used just consists of variable nozzles vanes and a mixed- flow rotor.

Fig.1 shows the meridional view of a VNT turbine with detailed illustrations of nozzle end-clearances,and two types of nozzle vanes.Inside the clearances,the spindle and the handle of a nozzle vane are not illustrated.

The pivoting nozzle vanes are often fixed and controlled by a spindle or a combination of a spindle and a handle.If the spindle is located round the center of a vane,the handle will be not needed,whereas if it is located near trailing edge of a vane,the handle is often designed and installed near the vane’s leading edge to well control the vanes’setting angle,as shown in Fig.1.

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The above analysis on the effect of nozzle vanes’spindles proves that the nozzle end-clearance can cause significant deviations in turbine performance at small opening,but the spindle and handle are still able to be neglected in this study,because those curves in Fig.4 have very similar trends.Consequently,the nozzle vanes’spindle and handle are neglected in Cases 3–12.Besides,it is only at the small opening that the following analyses are conducted,since the nozzle clearances’impact becomes significant at such an opening.

3.Numerical schemes

3.1.CFD model

A fully automatic hexahedral grid generator was used to generate the computational mesh for the whole flow field.Multi block Structured Meshing with an O4H topology including 5 blocks was used to discrete the channels,as shown in Fig.2.The skin block was an O-mesh surrounding the blade and the H-mesh was applied in inlet,outlet,up and down blocks.The mesh inside the hub and shroud clearances was created using a butter fly topology.The number of grid points had to be adjusted for various clearance sizes to keep a good mesh quality under the condition of the same first cell length.

The subsonic inlet condition with total pressure,total temperature,velocity angles and turbulent viscosity were specified at the inlet boundary of the computational domain.At outlet boundary,a radial equilibrium technology was used.With this,the outlet static pressure was imposed on the given radius and integration of the radial equilibrium law along the span wise direction permitted to calculate the hub-to-shroud pro file of the static pressure.Therefore,a constant static pressure would be imposed along the circumferential direction.On the solid surface,the adiabatic walls were used and the velocity vector vanished.The rotational speed of the unsteady simulations was selected according to the results of the Campbell Diagram,and other boundary conditions of the unsteady simulations were obtained by running GT-Power software.

Fig.1 Illustration of nozzle end-clearances(at meridional view)and two types of nozzle vane.

Table 1 Illustration of research models.

Model CshroudChubSpindles Case 1 100%h 100%h Spindle Case 2 100%h 100%h Spindle&Handle Case 3 0%h 0%h –Case 4 50%h 50%h –Case 5 100%h 100%h –Case 6 150%h 150%h –Case 7 0%h 200%h –Case 8 20%h 180%h –Case 9 40%h 160%h –Case 10 60%h 140%h –Case 11 160%h 40%h –Case 12 200%h 0%h –

There is one rotor/stator interface in the whole flow field.For steady simulations,the conservative coupling by pitch wise rows approach was used due to its capacity to provide an exact conservation of mass flow,momentum and energy through the interface.And the full non-matching boundary connecting algorithm was used in order to exchange the flow variables and fluxes between the left and right image.For unsteady simulations,the sliding mesh technology was applied on the rotor/stator interface.

The perfect gas relation and the Spalart-Allmaras turbulence model were employed to close the RANS/URANS equations.The discretization in space was based on a cell centered control volume approach.An explicit 4-stage Runge-Kutta scheme was applied to solve the equations mentioned above.

《宋词互见考》：“案此黄庭坚词，见《彊村丛书》本《山谷琴趣外编》。《能改斋漫录》云……即此词也。并有陈后山同韵和词。据此则为黄词明甚，《淮海词》收之，毛本《山谷词》删之，并误。”[7]271

Fig.2 Computational grids.

For all the CFD simulations,the Courant-Friedrichs-Lewy(CFL)number was 3.And after calculating a set of computational meshes with the same topology and different number of grid points,the relationship of the mesh density and the changed performance suggested the current computational mesh which was within affordable computation source limit.The y plus value was less than 3 in most region and,in nozzle end clearances,the average value was less than 2.68.For unsteady simulation,the total number of the grid nodes was about 4.53 million and the physical time step was about 1.15×10-8s.Fig.3 shows the validation of CFD methods which were performed on a similar VNT turbine.Although the deviation exists between the numerical and experimental data,they are still within an acceptable range.

3.2.FEA model

Commercial FEA software with linear analysis method was employed to predict the natural frequencies and harmonic response of a turbine wheel in fluenced by the unsteady aerodynamic excitation source.

In order to reduce the requirement of FEA calculations,only one wheel blade was modelled.The tetrahedral elements were used to discrete the solid field,in which Inconel was assumed as an isotropic elastic material with a Young’s modulus and a Poisson ratio.The detailed material properties were selected from a composite material database.

Fig.3 Validation of the CFD method performed on a similar VNT turbine.

The centrifugal force was applied to the FEA analysis,which was calculated according to angular velocity obtained from the Camp bell diagram.An assumption of nod is placement is applied at the center line of the wheel shaft.And a harmonic loading was applied on the wheel blade,which was calculated based on the unsteady CFD method.

4.2.3.Loading variations with nozzle end-clearances’distribution

The damping parameters significantly affect the accuracy of the vibration solution and the only way to obtain the precise damping parameters is through experimentation.Lu et al.11reported that the maximum relative error for a computational method with a constant damping ratio was lower than 10 percent.A constant damping ratio,therefore,is used in this research to compute the structure dynamics.The numerical prediction of the investigated model will be compared against experimental results of a similar VNT turbine to verify current numerical methods.

4.Results and analysis

4.1.Turbine performance

4.1.1.Effects of nozzle vane’s spindles

Beside the flow capacity,the turbine efficiency is always higher as small part of the total nozzle end-clearance is located at shroud side.Among those cases studied,no matter which point the turbine operates at,the peak efficiency always occurs when 20%of the total nozzle end-clearance size is located at shroud side(Case 8).In contract,the turbine efficiency is always deteriorated seriously as most clearances exist only at shroud side(Case 12).And the maximum deviation in turbine efficiency caused by varying nozzle end-clearances’distribution is about 5 percentage points.

Fig.4 compares turbine performance of a VNT turbine with and without spindles at both full and small openings.The top two plots compares the flow rate-pressure ratio performance and the bottom two plots represent the com-par is ons of the efficiency-speed ratio(U/C)characteristics.The very small deviations among curves at the full opening reveal that the nozzle’s spindles have no significant influence on the turbine performance.At the small opening,however,significant deviations indicate that the presence of spindles can improve slightly the turbine efficiency,but significantly reduce turbine flow capacity.Hu et al.12studied the effect of the nozzle end-clearances on a VNT turbine and pointed out that,at fully open position,the loading acting on vanes was so small that it was not enough to drive much fluid over the narrow clearance to form strong leakage flow and to significantly deteriorate turbine performance.The reported conclusions well support the current numerical findings.Besides the loading,the ratio of the clearance area and the nozzle throat area becomes gradually larger with the decreased nozzle opening.This trend also takes responsibility for the larger deviations in turbine performance at small opening.

共享经济的理论基础是经济剩余。所谓经济剩余，即为社会发展中过度生产的产物。此概念最早由美国激进经济学派代表人物保罗·巴兰于1966年提出，他对经济剩余的定义是一个社会所生产的产品与生产它的成本之间的差额，主要有两方面的表现：一方面表现为财富积累，另一方面表现为价值增值。共享经济也正是基于这两方面表现得以产生与发展。

With the controlling structure consisting of a spindle and a handle,a VNT turbine always operates with the best aerodynamic efficiency and the lowest flow capacity.The first reason is that for nozzle vanes with a spindle and a handle,the clearance passage area is further decreased so that the nozzle clearance leakage flow is reduced.And the second one is that the spindle is usually located near the vane’s trailing edge where the leakage flow is very strong.Zhao et al.8investigated the nozzle clearance flow behavior and stated that the flow in the second part of the clearance was stronger because of relatively small vane thickness,high momentum of fluid and relatively large static pressure differential between two sides of a nozzle vane.

4.1.2.Effect of nozzle clearance height

In order to clearly identify the research models used,the investigated VNT turbine with various nozzle end-clearances is numbered and listed in Table 1.The letter,h,stands for the normal height of the nozzle end-clearance at each side of a VNT turbine product,and is about 2 percent of the span of flow passage from the hub wall to the shroud one.

事实上，正是在贵州省委推进磷石膏资源综合利用的背景下，何光亮被同时任命为开磷集团的董事长。也就是说，实现“守好发展和生态两条底线，大力推动经济高质量发展”的目标才是这次任命的真实目的。

Fig.4 Comparison of performance of numerical model with and without spindles.

Fig.5 shows turbine efficiency variations with the increased nozzle end-clearance size.It can be seen that the maximum flow rate through the turbine gradually increases with the increased size of nozzle end-clearance,but the turbine efficiency deteriorates significantly.The maximum deviation in the flow rate is up to 21%,and the efficiency drop is about 16 percentage points as the nozzle end-clearance increases to the largest size.It should be noted that the reason for such significant deviations is the use of the large nozzle clearance size in Case 6.This clearance size is slightly larger than that in current advanced VNT turbines.Meitner and Glassman13proposed a clearance- flow model and stated that the changes in both turbine efficiency and flow rate became significantly larger with decreasing stator area and varied nearly linearly with clearance.

The trend of the efficiency degradation increasing with the clearance size suggests that the nozzle end-clearance is unfavorable to the aerodynamic performance of a VNT turbine.This is why the mitigation of the nozzle end-clearance is still one of the main objects in both academic and industrial fields.But the thermal expansion caused by the high temperature environment makes it hard to further decrease the clearance size.

Fig.5 Turbine efficiency variations with increased nozzle end clearance,at small opening.

Instead of directly decreasing the nozzle end-clearances,Sun et al.14developed a VNT turbocharger with sliding vanes,and the totally new nozzle vanes successfully reduced the nozzle end-clearance leakage flow and improved the VNT turbine performance,especially at small opening.Yang et al.15proposed a VNT turbine with a forepart rotation vane and numerically predicted results showed that the turbine efficiency was increased by 6%at small opening by removing partial nozzle end-clearances.These results prove that it is worth studying the nozzle end-clearances to find out more effective ways to improve the aerodynamic performance of a VNT turbine.

在现场作业工作完成后，要对现场进行仔细清理，并且注意要恢复现场的有关安全防护措施，还原原有的设备定置，防止由于作业而给现场留下日后的安全隐患。

Regardless of the control styles,VNT turbine always requires the presence of nozzle end-clearances to achieve the relative movement between nozzle vanes and the flat plates(endwalls).Herein,the nozzle shroud clearance is marked with Cshroudand the clearance on another side is represented by Chub,as shown in Fig.1.

4.1.3.Effect of nozzle end-clearances’distribution

Fig.6 shows the turbine efficiency variations with the nozzle end-clearances’distribution at several operation points within turbine map at the small opening.It is fairly clear that varying the nozzle end-clearances’distribution not only affects the turbine flow capacity,but also changes the turbine efficiency.For instance,for the turbine with either nozzle shroud clearance(Case 12)or hub clearance(Case 7),the turbine flow capacity is obviously increased.Between those two cases,the maximum flow capacity occurs when all clearances are located at the hub side(Case 7),because of the relatively large distance between two rows at the hub side.Zhao et al.16studied the rotor–stator interaction and proved that the blocking effect was closely related with the gap between two adjacent rows.However,when about 80%of the total nozzle end-clearance is located at shroud side(Case 11),the turbine flow capacity is the lowest.The maximum deviation in flow rate induced by nozzle end-clearances’distribution is less than 5%.

Fig.6 Turbine performance responses to various nozzle end-clearances’distributions.

It is well known that nozzle vanes can rotate round their spindles to achieve the function of changing the nozzle opening.This function requires the nozzle end-clearances to be designed and to exist between nozzle vane ends and hub/shroud walls.Meanwhile,the clearance provides an unexpected passage for fluid to cross over nozzles.After exiting the nozzle clearances,the fluid called clearance leakage flow interacts with main flow in nozzle channels,thus generating extra flow loss.Inside the clearances,the presence of spindles reduces the clearance area and therefore,has a contribution to the reduction of the leakage flow.But in lots of research work,the spindles were often neglected during modeling a research model in order to simplify geometry.In this section,the spindles’effect will be quantified and the simplification will be evaluated.

It should be noticed that the use of the mixing plane approach at rotor/stator interface may show higher influence of nozzle end-clearance leakage flow on the turbine efficiency.Zhao et al.17studied a VNT turbine using commercial code EURANUS and reported that compared with the unsteady simulation with direct interpolation at rotor/stator interface,steady simulations with the mixing plane method showed larger turbine efficiency variations in response to the different nozzle end-clearances.

（1）依据阶段梯度变化所获得爆破效果，对不同类型岩体爆破炸药单次损耗量进行分析。以爆破对岩体影响深度最小、爆破炸药量使用最少为原则，得出最佳高拱坝拱肩槽开挖爆破数值如表1所示。

A FEA calculation of a single blade was performed to predict the natural frequencies of turbine wheel.Since the critical modes for blade fatigue are mainly low order modes20,no higher order modes are analyzed in this paper.The numerically predicted frequencies are used as input for the Campbell diagram shown in Fig.7.The aerodynamic excitation frequencies are plotted as a function of rotational speed of a turbine wheel.The two horizontal lines drawn in Fig.7 represent the natural frequencies of Mode 1 and Mode 2,respectively.In the plot,the rotational speed and frequency was normalized by the turbine top speed and the natural frequency at Mode 1,respectively.The intersection points of natural frequencies with rotational speed orders are identified as potential operation points at which the blade resonance may occur.For the research turbine,resonances of the 1st speed order may excite blade Mode 1 whereas the 2nd speed order may excite Model 2 within a realistic operating range from the rotational speed of 0 to the top speed limit.

4.2.Aerodynamic excitation

4.2.1.Turbine wheel resonant

1．3 统计学方法 用Microsoft Excell 2003软件对数据进行录入；用SPSS 13．0统计软件对数据进行统计分析。

Roumeas and Cros18tested two con figurations:the nozzle gap divided equally at two ends of the nozzle vane and the whole gap located on the hub side.A significant deviation in turbine performance was observed.Walking shaw and Spence19tested a radial turbine with either hub side or shroud side stator vane clearance at full,middle and small openings,and revealed that the turbine performance differed for different nozzle clearances’distribution.Those literatures provide evidences for that the nozzle end-clearances’distribution significantly change the aerodynamic performance of a VNT turbine.In spite of that the current steady simulations may show higher influence of nozzle end-clearances,the effect on the turbine efficiency of the various nozzle end-clearances is still worth focusing.

According to the Campbell diagram,the critical rotational speed is easily known.And other parameters of the critical operation point can be obtained by running GT-Power software.In consequence,unsteady CFD simulations with those boundary conditions were carried out on a VNT turbine with various nozzle end-clearances to predict the aerodynamic excitation pressure.

4.2.2.Loading variations with increased nozzle end-clearance size

Fig.7 Campbell diagram with Modes 1 and 2.

The aerodynamic loading,f(x，y，z，t),is the differential of static pressure on both surfaces of a turbine blade,and represents the aerodynamic force that acts on blade surfaces to transfer energy from the fluid to the turbine wheel.where f and p stand for the loading and pressure at an arbitrary point on blade surface,respectively;x,y,z represent the coordinate point at one surface of turbine blade and the x′,y′,z′represent the coordinate points at another surface of turbine blade;the t is time.

老贾看到钱币之后，依旧面无表情。虽然手上又是摸又是掂量，但是脸上却纹丝不动，孟导看不出老贾的意思，心里焦躁得不行，只得离开座位绕着老贾和他的那盒子钱踱起步来，不时还偷偷观察老贾的神色。过了好一阵子，老贾才把目光从这盒子钱上移开，陷入了短暂的思考。孟导实在是受不了这种气氛，看老贾一闲下来马上就问如何，急切的心情可以赶上产房外新生儿的父亲了。

If the aerodynamic loading is steady,the turbine wheel will operate smoothly.In fact,the flow inside a VNT turbine is always highly unsteady,and therefore,there is serious aerodynamic interaction between the nozzle vanes and turbine wheel.This rotor–stator interaction must cause the aerodynamic loading to be high unsteadiness.The unsteady loading is a dominant excitation source to the vibration of a VNT turbine wheel.

The aerodynamic loading can be divided into two parts,steady and unsteady parts.The steady part represents the time-average loading.And the unsteady part is corresponding to the fluctuating loading caused by the unsteady interaction and other complicated flow,which transfers exciting energy from the fluid to the turbine wheel to have the wheel deformed repeatedly and vibrated.They are expressed as follows:

wherestands for the time-average value of the loading andrepresents for the unsteady part of the loading.

The time-average loading is calculated as follows:

除星光效应外，只有半透明或透明且色彩鲜艳的刚玉才能称做宝石。红色并含铬元素的刚玉呈红色调，故被称为红宝石；蓝色的蓝宝石则是因为含有了微量的钛和铁元素。

where T is the time period that the pressure was monitored and recorded.Fig.8 shows the distribution of the steady part of aerodynamic loading on wheel blade surface for four nozzle end-clearance sizes.It is fairly clear that high loading is centrally located in inducer.The reason is that at small opening the nozzle vanes are pivoted at the large setting angle so that the positive incidence is formed at inducer inlet.As the nozzle end-clearance size increases,the loading density near inducer blade tip and root obviously increases,but it decreases in middle span wise section in inducer.In exduer,the steady loading also increases with the increased clearance size.

It is well known that the distribution of the steady loading is closely related with turbine efficiency and torque.When a turbine operates at small opening,the high loading density in inducer is in favor of recovering energy from exhaust gas,because an internal combustion engine is often to operate at part loads.Though the increased nozzle end-clearance size promotes the loading density in exducer,the decreased loading density in whole inducer deteriorates the turbine efficiency.

Fig.9 shows the variations of aerodynamic excitation intensity at both the 1st and the 2nd orders with the increased size of nozzle end-clearances.They are obtained by using Fast Fourier Transform(FFT)algorithm and expressed as follows:

where Force is for the excitation force at the 1st order and/or the 2nd order.For the model without nozzle end-clearances,the aerodynamic excitation acted on the wheel blade is very weak in most regions.After adding nozzle end-clearances,however,the aerodynamic excitation intensities including the 1st and 2nd orders is reinforced not only in inducer tip and root regions but in exducer as well.Compared with the 2nd order aerodynamic excitation,the excitation intensity of the 1st order is stronger.Therefore,the 1st order excitation should transfer more energy to the wheel blade to have it vibrated and do much for the turbine wheel high cycle fatigue failure.And with the increased nozzle end-clearance size,the intensity of the 1st order excitation increases in the region near the blade tip round 70%of blade chord.It should be noticed that the span from blade tip to root is so large in exducer that the high excitation intensity at the 1st order mode greatly increases the risk of the turbine wheel suffering from high cycle fatigue failure.

在制定项目预算时要安排科研项目负责人与相关财务部门人员共同依据预算科学性、效益性、经济性、效率性等标准来制定。在实际情况中，科研项目不同，预算标准也会有差异，要依据不同的科研项目进行合理化的预算标准调整。预算编制的质量对后期项目经费的实际使用以及项目结题都会有很大的影响。在项目开展的过程中，项目负责人要及时、客观地向财务部门反映预算的合理性与否，如项目支出明细的设置和支出金额是否合理，这样才能为日后制定更加合理、完善的预算标准提供依据。完善的科研项目经费预算及调整机制，是科研项目的正常运行，科研项目经费效益最大化的保障。

Once the natural frequencies of the turbine wheel were predicted by the modal analysis and the unsteady CFD simulations were carried out, the information about the aerodynamic excitation acting on the wheel blade was added from the CFD results to FEA analysis by interpolation.The detailed data transmission will be introduced in section of the forced response.

Fig.10 shows the variations of the steady loading distribution on a wheel blade surface with the nozzle end-clearances’distributions.When most nozzle end-clearances,or all,are located at the hub side,the obviously high loading occurs near inducer blade root.And,with the decreased nozzle hub clearance and increased nozzle shroud clearance,the steady loading in the region near the exducer blade tip gradually increases.Meanwhile,the average loading density in the whole inducer decreases.The loading style in Case 12 is unfavorable to the recovery rate of energy from exhaust gas.Therefore,it can be drawn that the nozzle end-clearances’distribution is able to change the distribution of the loading on a wheel blade.

Fig.11 shows the variations of aerodynamic excitation intensity at both the 1st and 2nd orders on the wheel blade surface with various nozzle end-clearances’distribution.It is fairly clear that the aerodynamic excitation intensity in the region near blade tip around 70%of chord is very weak when all nozzle end-clearances are located at the hub side,as shown in Case 7 in Fig.11.However,as the nozzle shroud clearance increases,the 1st order aerodynamic excitation intensity in that region gradually increases and is up to the maximum when all nozzle clearances are located at shroud side.Therefore,it can be concluded that the nozzle shroud clearance leakage flow is very sensitive to the 1st aerodynamic excitation intensity.

Fig.8 Distribution of time-averaged loading on wheel blade.

Fig.9 Variations of aerodynamic excitation at both the 1st and 2nd orders on wheel blade with increased size of nozzle end-clearances.

Fig.10 Variations of steady loading acted on wheel blade with nozzle end-clearances’distribution.

Fig.11 Variations of aerodynamic excitation at both the 1st and 2nd orders on wheel blade surface with nozzle end-clearances’distribution.

Beside the excitation at the 1st order,the aerodynamic excitation with the 2nd order frequency also varies with the nozzle end-clearances’distribution.For instance,the excitation intensity at the 2nd order at inducer blade root gradually decreases with the decreased nozzle hub clearance.Meanwhile,it increases in the inducer blade tip region.But,compared with the 1st order excitation,the excitation intensity of the 2nd order is far weaker.

4.3.Forced response

4.3.1.Turbine wheel dynamic stress distribution

显示的名字是程颐。粒粒心里一动，忽然想起那副温暖的羊皮手套，腾冲的大青石还有那一晚月色中的闲聊。人世间，熙熙攘攘，大多是陌生和冰冷，温暖也是一种缘分。她忽然很想见他。

Independent calculations of the internal flow field and the harmonic response are presented in this paper,respectively.The calculations were carried out with two different computational meshes.The length scale of grid cells used in the CFD simulations is fully different on the wheel blade surface due to the use of clustering technology.However,the mesh used in the harmonic response calculation is almost uniform on the blade surface except the fillet region.With the Matlab code based on the cubic spline function interpolation,the data transmission was performed to transfer the information of the loading fluctuation on the grid nodes of the flow field to the grid points of the solid field.

The evaluation of the accuracy of the interpolation is illustrated in Fig.12.The comparison of two plots shows very small deviations,so that the interpolation method used in this paper could be accurate enough to meet the requirements of current research work.After the interpolation,the aerodynamic loading was further analyzed by FFT Algorithm to obtain the real and imaginary components which would be used for the FEA calculations.

闸室为分离式钢筋混凝土结构，顺水流方向主槽闸室长8 m，底板高程1 004 m；滩地闸室长8 m，底板高程1 005 m。闸门为可升降的钢坝闸门。主槽拦河闸为1孔，闸板总长40 m，挡水高度3.0 m；左岸滩地拦河闸为3孔，闸板总长120 m，单扇闸板宽度40 m，挡水高度2.0 m；右岸滩地拦河闸为3孔，闸板总长120 m，单扇闸板宽度40 m，挡水高度2.0 m；拦河闸由设在左岸的液压控制室控制启闭。拦河闸闸墩为钢筋混凝土箱式结构，根据超高计算，闸墩顶高程1 008.2 m。下游连接段，包括消力池、海漫以及两岸翼墙和护坡。引导出闸水流均匀扩散，下游连接段总长18 m。

Based on the analysis of the harmonic response,the stress distributions at both the 1st and the 2nd orders are illustrated in Fig.13,respectively.The magnitude is normalized by dividing the vibration stress value by the peak value in the case with normal nozzle end-clearances(Case 5).From the contour distribution in two plots in Fig.13,it can be seen that the maximum stress at the 1st order mode is about 2.5 times than the peak stress at the 2nd order and occurs at exducer blade root.This numerical result points out that the vibration stress of the 1st order induced by the 1st harmonic loading can be considered as the major alternating stress on the turbine wheel and the exducer blade root region may suffer from high cycle fatigue failure.The numerically predicted region with high risk is in agreement with the experimental result of a similar turbine reported by Hu et al.12and illustrated Fig.14.

4.3.2.Forced response of turbine wheelFig.15 shows the variations of the peak vibration stress at both the 1st and 2nd orders with the increased height of the nozzle end-clearances which were equally divided at two ends of nozzle vanes.It is fairly clear that the presence of nozzle end-clearances does increase the peak stress of the 1st order vibration.And,with the increased size of nozzle clearances,the peak value of the stress at the 1st order mode gradually increases.Compared with the model without nozzle end clearances(Case 3),the 1st order stress is increased by twice as much.This trend means that the nozzle clearance leakage flow is one of the major resources contributing to the damage of a turbine wheel due to the high cycle fatigue.

Fig.12 Validation of the interpolation approach.

Fig.13 Distribution of stress on wheel blade(Case 5).

Fig.14 Example of high cycle fatigue failure at exducer blade root,observed on a similar turbine wheel.

Fig.15 Turbine wheel maximum stress variations with increased nozzle end-clearances.

Fig.16 Turbine wheel maximum stress variation with nozzle end-clearance distribution.

Besides the 1st order excitation,the peak stress of the 2nd order vibration is slightly changed with the increased nozzle end-clearances.This decreased trend reveals that the nozzle clearance leakage flow acts as the 1st order aerodynamic excitation source for the current turbine wheel.

Fig.16 shows the variation of the peak stress at both the 1st and 2nd orders with the various nozzle end-clearances’distribution.As was analyzed above,with the decreased nozzle end-clearance on the shroud side the turbine efficiency significantly increases.Herein,the curve of the peak stress at the 1st order mode clearly shows that the peak stress obviously decreases with the decreased nozzle shroud clearance,and the maximum deviation is about 30%.These results reveal that the nozzle shroud end-clearance is more sensitive not only to the turbine efficiency but to the forced response of the current turbine wheel at the 1st order mode as well,compared with the nozzle hub clearance.In contrast,the forced response at the 2nd order mode firstly increases and then slightly decreases with the decreased nozzle shroud clearance.The narrow variation range of the peak stress at the 2nd order means that the effect of the nozzle end-clearances’distribution on the wheel vibration at the 2nd order mode is insignificant.

According to the above analysis on the harmonic response,not only the presence of the nozzle end-clearances but also the increased size of nozzle shroud clearance does increase the risk of the turbine wheel suffering from high cycle fatigue failure and,between the shroud and hub clearances,the nozzle shroud clearance has more contribution for the vibration of the turbine wheel at the 1st order mode.These results,therefore,suggest a possibility to improve both the turbine efficiency and the reliability of a turbine wheel by reducing the unfavorable effect of the nozzle shroud clearance.For instance,Wang et al.21proposed a controlling structure to decrease the nozzle shroud clearance to improve the aerodynamic performance of a VNT turbine.This method,or other approaches with similar principles,may also benefit the turbine performance,while also solving the high cycle fatigue issue that the design and development of an automotive turbocharger is trying to address.Besides,an introduction of 3D nozzle vane with oblique trailing edge may be possible to mitigate the unfavorable impact of the nozzle shroud clearance.The differential of pressure on both sides of a nozzle vane is the force to drive the air over the narrow nozzle end-clearances.Decreasing the pressure difference at shroud side by increasing the nozzle opening only at shroud side is likely to reduce the nozzle shroud clearance leakage flow;in the meantime,the opening of the nozzle vanes at the hub side needs to be decreased in order to keep the same nozzle throat.

5.Conclusions

This paper numerically investigated the effects of various nozzle end-clearances on both the aerodynamic performance and the forced response of a VNT turbine in order to find out the possibility to improve both the stage efficiency and the reliability of a turbine wheel via optimizing nozzle end-clearances.Research models are a VNT turbine with different total sizes of nozzle end-clearances which are equally divided at two ends of nozzle vanes or with various distributions of the clearances at the hub and shroud sides which have the same total size.

(1)As the total nozzle end-clearances remain the same,the nozzle end-clearances’distribution is closely related to the turbine efficiency,the flow capacity,and the vibration stress at the 1st order mode at small openings.When most nozzle end-clearances are located at shroud side,a VNT turbine has the minimum flow capacity while the flow capacity is the maximum when no nozzle clearance is located at shroud side.When small part of the total nozzle end-clearances is located at shroud side,the peak efficiency always occurs and the peak stress of the 1st order vibration at exducer blade root can be decreased greatly.

(2)The nozzle end-clearance leakage flow has stronger effects on vibration stress at the 1st order mode for the current model than that of the 2nd order.Compared with the nozzle hub clearance,the nozzle shroud clear-ance is more sensitive to the turbine efficiency degradation and the forced response of a turbine wheel at the 1st order,but has less contribution to increase the turbine lf ow capacity.

(3)It has reached broad agreement that decreasing total size of the nozzle end-clearances can bene fit the aerodynamic performance of a VNT turbine.But due to thermal expansion of the material,it is very hard to further decrease the size in the development of an advanced VNT turbine.According to the results reported in this paper,controlling the nozzle end-clearances’distribution,or decreasing the nozzle shroud clearance’s effect,may be possible methods to design an advance VNT turbine with higher efficiency and lower forced response in the future,instead of the hard efforts to further decrease the total size of the nozzle end-clearances.

职业倦怠作为目前困扰高校基层行政管理人员日常工作和生活的问题，已有学者从外部归因和内部归因对其展开分析，主要有以下几种解释：

Acknowledgements

This study was co-supported by the Natural Science Foundation of Hebei Province of China(No.E2017402135)and the Program of Science and Technology Research and Development of Handan of China(No.1621212047-2).

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