Unsteady Behavior of Tip Leakage Vortex in an Axial Compressor with Different Rotor Tip-gap Sizes Using DDES*

Lu-yang ZhongYang-wei Liu,2 Li-peng Lu,2

（1.National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics,School of Energy and Power Engineering,Beihang University,Beijing,China，2.Collaborative Innovation Center of Advanced Aero-Engine,Beihang University,Beijing,China）

Abstract：Tip leakage flow (TLF) has a large impact on compressor performance and should be accurately predicted by CFD methods.New approaches to turbulence modelling,such as delayed detached eddy simulation(DDES)have been proposed which allow for greater accuracy of the numerical predictions while computational resources can be tremendously reduced.In this paper,the numerical simulation of the rotor in a low-speed large-scale axial compressor based on DDES is performed subject to different operating conditions of different tip gap sizes.The time-averaged and instantaneous results are compared to analyze the unsteadiness in visual at different tip gap sizes.Then,the anisotropy of the TLF is analyzed through Lumley triangle.The anisotropy varies along the tip leakage vortex(TLV).The anisotropy of the small tip gap condition is weak because the flow at this condition is more stable.Last,the velocity fluctuation data along the TLV are transformed to velocity spectra using fast Fourier transformation(FFT)method to discuss the TLV unsteadiness.

Keywords：Axial Compressor Rotor,Tip Leakage Vortex,Tip-Gap Size, Unsteady Behavior,DDES

CFD Computational Fluid Dynamics

CPU Central Processing Unit

DDE SDelayed Detached Eddy Simulation

DE Design

DES Detached Eddy Simulation

FFT Fast Fourier Transformation

LES Large Eddy Simulation

NC Near Chock

NS Near Stall

RANS Reynolds-Averaged Navier-Stokes

SPIV Stereoscopic Particle Image Velocimetry

TKE Turbulent Kinetic Energy

TLF Tip Leakage Flow

TLV Tip Leakage Vortex

URANS Unsteady Reynolds-Averaged Navier-Stokes

0 Introduction

The compactness of an aircraft engine,directly related to thrust-to-weight ratio,is a highly desired design objective.As the engine size becomes smaller,fewer stages of compressors are expected,implying that the aerodynamic loading on each blade is increasing as a consequence.However,the compressor blade loading is extremely limited by many three dimensional flow losses in compressors,such as boundary layers,flow separations,tip leakage vortex,and shocks[1-4].Among these,the total flow losses caused by the tip leakage dominated annulus wall flows may account for 30%～50%inefficiency in these blade rows,and a clearance gap equal to 1%of the blade height is associated with about 2%penalty in efficiency[5].Therefore,by using conventional intrusive measurement techniques,numerous and extensive experimental studies have been conducted to investigate the three-dimensional flow field in the rotor tip region to reveal its inherent flow mechanisms and to evaluate the aerodynamic losses over the past decades,such as[6-10].

With rapid development of the computer resources,CFD codes based on RANS equations have integrated turbomachinery design processes.However,turbulence model is still a weakness in RANS for complex flows in compressors[11-12].Hence,turbulence theory[13-14],turbulence physics[3,15]and turbulence models[16-18]should be studied for complex flows.Though the cost of eddy resolving simulations such as LES and hybrid LES/RANS is still too demanding for engineering,it could provide much more promising results that can be used to study flow mechanisms and turbulence models[3].

Among the precise eddy resolving simulations,LES is an adapted method to capture the vortices from the tip-leakage flow[19].However,for high Reynolds number flows,such as those in turbomachinery,LES needs the CPU resource not much less than the direct numerical simulation to resolve the wall boundary layer.This makes the LES too expensive for engineering applications[20].To overcome the intensive CPU requirement for LES,Spalart et al.[21]developed the DES strategy,which is a hybrid RANS and LES method.Near the solid surface within the wall boundary layer,the unsteady-RANS model is realized.Away from the wall surface,the model automatically converts to LES.By using the RANS model near walls,the mesh size as well as the CPU time can be greatly reduced.The motivation of DES is that the LES is powerful in regions of massive separation and other free shear flows,such as jets.However,the grid control is inadequate in some cases when the boundary layers are thick,so DDES was proposed to solve it[22].DDES incorporates a simple modification into the initial DES by introducing kinematic eddy viscosity into the model parameterdto take the effects of both grid spacing and eddy-viscosity field into considerations.This approach narrows the grey area between RANS and LES,especially in separating cases.

To enhance our conventional understanding of tip leakage flow and investigate flow structure on it,numerical simulations are conducted under different operating conditions in a low-speed large-scale axial compressor test facility.Detailed velocity measurements were performed on cross-sections nearly perpendicular to the rotor tip chordwise direction with an interval of 5%or 10%chord length using SPIV method in this case.The purpose of this paper was to evaluate the capability of the DDES method to represent tip leakage flows in comparison with different tip gap sizes on a realistic rotor in a low-speed large-scale axial compressor.

1 Computational Method and Validation

The computational geometry model,computational meshes and simulation methods,which were used in this investigation,will be briefly introduced in this section.

The experiments were carried out in the low-speed large-scale axial compressor test facility of Beihang University by Du et al.[23].The compressor is a 1.5 stage configuration consisting with inlet guide vane,rotor and stator.The blade profile was designed with C4-series airfoil with a relatively high loading coefficientof about 0.45 at the design operating condition.The detailed parameters of the test compressor are summarized in Table 1.Detailed velocity measurements were performed on cross-sections nearly perpendicular to the rotor tip chordwise direction with an interval of 5%or 10%chord length using SPIV method[24-25].The three components of velocity in directions ofX,Y,and Z shown in Figure 1 correspond tou,vandwrespectively.These could be used to validate the simulation results.

In the simulation,the rotor was selected to investigate the tip leakage flow.Hexahedral structural meshes are generated in O4H topology,shown in Figure 2.They+adjacent to the wall is about 0.8.The inlet locates at 1.0 chord length upstream to the blade leading edge and the whole computation domain is about 4.0 chord length in axial direction.In the TLV region above 80%blade height,the refined isotropic meshes,with aspect ratio less than 2 and expansion ratio less than 1.05,were employed to resolve the TLV structure better.The total grid number is about 6.2 million for both DDES simulations,with 41 points in the tip clearance and 141 points along the radius.The grid independence has been checked.

The commercial CFD software Fluent is used to make the numerical simulation.The pressure-based implicit solver is chosen.The central-differencing scheme is used for the convection terms and the viscous terms of each governing equation to minimize the numerical diffusion.Periodic boundary conditions are set on the two sides of flow passage.The steady total pressure of the inlet flow in the simulation was measured between the IGV and the rotor using the five-hole probe.The outlet boundary condition is the same as the measurement after the rotor and before the stator leading edge.For the unsteady simulation,dual time step method is applied with an outer iteration physical-time step of3×10-6second and 30 inner iterations per physical time step in DDES method.The time step is about 1/1000 times the blade passing time and5×10-4C/U,where C is the chord length and U is the main flow velocity.This time step is appropriate to capture the unsteady flow of the tip leakage flow and meets the Cflcondition.Three thousand instantaneous results are saved,with ten physical-time steps between two samples.

Fig.1 Layout of SPIV measurement cross-sections

Tab.1 Design parameters of the test facility

A hybrid LES/RANS method,called Delayed Detached Eddy Simulation method,was used in this study to alleviate large computational mesh number requirement,compared to large-eddy simulation.The initial DES method was created to address the challenge of high-Reynolds number,massively separated flows in 1997,which must be addressed in such fields as aerospace and ground transportation,as well as in atmospheric studies.This method can accurately predict the complexity and unsteadiness naturally associated with the compressor flow.In 2006,Spalart proposed DDES method,which incorporates a simple modification into the initial DES.DDES introduces kinematic eddy viscosity into the turbulence model to take both effects of grid spacing and eddy-viscosity field into considerations.In order to study the evolvement of TLF in the rotor,DDES method based on SST model is used in this paper.

The governing equations of the DDES-SST model differ from those of the SST model.In the SST based DDES,the length scaledin the SST model is replaced by

whereCDESis a calibration constant used in the DDES model and has a value of 0.61,Δis the maximum local grid spacing(Δx,Δy,Δz).

The shielding is done by coupling the definition of DDES length scale with RANS usinglRANS.The turbulent length scale is the parameter that defines this RANS model:

Hereβ*is a constant of the SST model.kis the turbulence kinetic energy,andωis the specific dissipation rate.

The delaying functionfdis defined as

In the equationνtis the kinematic eddy viscosity,νis the molecular viscosity,Uijis the velocity gradient andκis Karman constant.this parameter equals to1 in a logarithmic layer,and falls to 0 gradually towards the edge of the boundary layer.The addition ofνin the numerator corrects the very near-wall behavior by ensuring thatrdremains away from 0.With this new formula,lDDESdepends not only on the grid,but also on the eddy-viscosity field.The new model can“refuse”LES mode if functionfdindicates that the point is well inside a boundary layer,as judged form the value ofrd.

fdis designed to be 1 in the LES region,whererd?1,and 0 elsewhere(and to be insensitive tordexceeding 1 very near the wall).

Fig.2 Computation domain and mesh of the rotor

Fig.3 Time-averaged streamwise velocity of different flow conditions using DDES

Fig.4 Time-averaged streamwise velocity of different flow conditions in experiment

The dissipation termYkof the turbulent kinetic energy is modified for the DDES turbulence model as described by[26]such that

whereFDDESis expressed as

whereF2is the blending functions of the SST model.

The result of time-averaged streamwise velocity of the representative cross-sections of Figure 1 is compared with experimental measurement.Figure 3 are contours at NC,DE and NS condition of different tip gap sizes of DDES results respectively while Figure 4 are experimental measurements.The numbers 1.75 and 1 means the ratio between rotor tip gap and blade height is 1.75%and 1%respectively.From NC to NS condition,the blue area increases because the tip leakage flow becomes strong.The tip leakage flow in big tip-gap size condition is stronger than the small tip-gap size condition.It shows that the agreement between them is quite good.The DDES simulation captures the location and track of TLV,which is the blue regions,of different flow conditions.The velocities of the other two directions also fit the experiment very well.Then we shall see the comparison quantitatively using the streamwise vorticity profile of 60%measurement plane near suction side at design condition as shown in Figure 5.The location of the profile is the black line of the figure at the bottom right corner.The line goes through the core of the TLV.The horizontal ordinate represents the distance to the blade suction side and the vertical coordinate represents the streamwise vorticity.The simulation results capture the location of TLV and show qualitatively good agreement with experimental results.That means DDES method has an advantage over the RANS method.However,both the DDES and the RANS result overestimate the positive vorticity because the numerical results simulate the strong induced vortex which is not observed in the experiment.

Fig.5 The streamwise vorticity profile of 60%measurement plane near suction side

One-point velocity spectra can not only provide insights into how the TKE distributes,i.e.,whether the TKE concentrates on large scale eddies or on small scale eddies,but also validate the simulation results by the Kolmogorov-5/3 law[27].Ten numerical probes have been put in the flow field to capture high resolution signals.They are illustrat-ed in top right corner of Figure 6.The ten points spread around theTLVregion,blade tip clearance,and main flow.

Fig.6 One-point frequency velocity spectra at point P9

The power spectral densityφuuof the fluctuating velocityu'yields

The Kolmogorov-5/3 law is so well established that numerical predictions are regarded with skepticism if they fail to reproduce it.Figure 6 shows the energy spectra of velocity fluctuations computed in the mainstream monitor point P9.Energy spectra predicts an inertial subrange that is typically described with slope of-5/3.The spectra of other monitor points accord with the same law.

2 Unsteady Flow Characteristics

Figure 7 to Figure 10 is the time-averaged and instantaneousQcontours of the TLV atr=0.495m.Qcriterion is used to identify vortices of an incompressible flow as connected fluid regions with a positive second invariant of the velocity-gradient tensor?u,whereΩis the vorticity tensor,Sis the shear strain tensor.It indicates the regions where the vorticity magnitude prevails over the strain-rate magnitude,

Time-averaged and three random instantaneous results,which ist1,t2andt3,are shown from left to right respectively in Figure 7 to Figure 10.It is found that the latter part of TLV in instantaneous results is different from the time-averaged one.Aside from the concentrated vortex,additional small eddies appear at the latter part of TLV near the pressure side of the adjacent blade trailing edge,which greatly increase the complexity of flow filed.The time-averaged concentrated vortex of NS condition is much shorter than DE condition.Meanwhile,the small vortices generate earlier in instantaneous results at NS condition.

Fig.7 Time-Averaged and Instantaneous Q Contours of the TLV in the big tip gap size at DE condition

Fig.8 Time-Averaged and Instantaneous Q Contours of the TLV in the big tip gap size at NS condition

Fig.9 Time-Averaged and Instantaneous Q Contours of the TLV in the small tip gap size at DE condition

Fig.10 Time-Averaged and Instantaneous Q Contours of the TLV in the small tip gap size at NS condition

According to the comparison between time-averaged result and the instantaneous results in the big tip gap result,we could divide the TLV into three parts.It can be seen that at the front part of TLV,there is only one vortex at each instantaneous field,whose scale is almost the same with the time-averaged TLV.That means at the beginning the TLV is quite stable,as the tip leakage vortex stays at the same place all the time.In the second part,we could also find the concentrated vortex in both time-averaged and instantaneous results.However,the shape of TLV in the instantaneous result becomes irregular and some smaller vortices appear.TLV begins to lose stability at this stage.In the third part,concentrated vortex can no longer be found in the time-averaged result.The number of smaller vortices increases and the size of the vortices decreases.TLV begins to mix with the main flow and dissipate.Hence,it can be seen that the unsteady behaviors of the tip leakage vortex do exist in the present investigated compressor.

The small tip gap result is different from the big gap one.For the DE condition,the structure is quite stable that there does not exist any small vortices at the latter part of the TLV.The contours of different time steps and the time-averaged result are almost the same.Neither TLV wandering nor rolling up small vortices happens at this condition.For the NS condition,the position where small vortices roll up is later in the TLV,at about 70%chord length.In the small tip gap condition,the numbers of the small vortices,which stands for the degree of the unsteadiness,is much fewer than those in the big tip gap.That is to say,the flow is more stable in the small tip gap condition.

3 Anisotropy of the Reynolds Stress Tensor

Further characterization of the turbulent state is achieved through the theory of[28].The properties of the isotropy tensor can be illustrated looking at the so called Lumley triangle,which involves the normalized anisotropy tensorbij.The Reynolds stress anisotropy tensor is normalized as

The normalized anisotropy tensorbijhas by defnition a zero trace as a consequence of its formulation.The traces ofare non-zero quantities and are related to the invariantsηandξof the anisotropy tensor.

Using the classical graphical representation of the turbulent states in(ξ,η)coordinates,it may be shown that the turbulent state realizable region is a triangle within which are situated all possiblebijvalues.The triangle boundaries match particular turbulent states.Invariants of the normalized Reynolds stress anisotropy tensor are related to the degree of anisotropy(η)and the characteristic shape described by the eigenvalues(ξ)[29].In Figure 11,theoretical limits are shown as vertices and edges of the triangle and represent special cases of turbulence.Special states of turbulence given on the Lumley triangle are shown in Table 2.The characteristic shapes associated with axisymmetric turbulence are either oblate or prolate spheroids for anisotropic turbulence.

Tab.2 Special States of Turbulence Given on the Lumley Triangle

Fig.11 Lumley triangle showing limits of invariantsηandξand characteristic shapes ofbij

Figure 12 and Figure 13 are theηandξcontours of the cross-sections in different flow conditions.The most obvious trend seen in Figure 12 is a region of largeηtrailing the tip region of the rotor in each case.This indicates that the turbulence state at the TLV region is the most anisotropic.From DE to NS condition,the red region expands in both 1.75%and 1%tip gap cases.This suggests that the flow of the TLV at the NS condition is more anisotropic than at DE condition.The core of the TLV is isotropic whereηis small while the flow around the vortex core is anisotropic.This anisotropic place is where the tip leakage vortex shear with the main flow.In the small tip gap results,ηshows generally the same trends as the big tip gap results at the early part of the flow passage,although the magnitude ofηis smaller.The maximum values ofη(most anisotropic turbulence)trail the rotor at tip leakage flow around the core of TLV.The third invariant ofbijis shown in Figure 13.The TLV areas evidence positive values ofξfollowing the TLV around the core and negative regions following below the positive regions.The trends of positiveξis almost the same with the largeηregion.From the Lumley triangle,positive values ofξlead to a prolate spheroid and negative values lead to an oblate shape.For the small tip gap condition,the anisotropy is weak especially at DE condition.The range of red color of the positiveηis smaller at DE condition compared to the big tip gap at the same condition.It shows that the quite sta-ble condition at this condition means weak anisotropy in the tip leakage flow.At NS condition,the bigηvalue appears at the latter part of the TLV after about 60%chord length.That means in the small tip gap condition,the anisotropy increases at a more latter part in the TLV.

Fig.13 ξon the cross-sections

Fig.12 ηon the cross-sections

4 Velocity Spectra

Figure 14 and Figure 15 are the axial velocity fluctuation at the TLV core at the 80%chord length at different operating conditions.At DE condition,the flow in the small tip gap is quite stable that the fluctuation is very small.It has almost zero scale on y-axis.In the Figure 15,the flow at 80%chord length in both tip gap sizes at NS condition is unstable,the range of the fluctuation in the small tip gap is about half of the big tip gap.Figure 16 and Figure 17 are the fluctuating axial velocity frequency spectra of 3 points along the TLV core at design and near stall condition.The locations of these 3 points are at the TLV core of 40%,60%and 80%chord length.The velocity fluctuation data of these points are transformed to velocity frequency spectra by using FFT method.The horizontal axis is the frequency and the vertical axis is the amplitude.

For the big tip gap condition,there is a conspicuous frequency at each operating condition,410Hz in DE condition and 180Hz in NS condition.The amplitude at the conspicuous frequency is higher than the other frequencies.It exists in the frequency spectra of every point from the upstream to downstream of TLV.This indicates the tip leakage flow has very obvious unsteady characters in the flow passage.At the beginning of TLV in DE condition,there are some outstanding frequencies at the left of the main frequency,at 40%chord length.Then,the amplitude of two higher frequencies,compared to the peak amplitude frequency,start to increase at 60%chord length.It is noteworthy that this is where TLV loses stability in DE condition.The amplitude of the higher frequencies becomes obvious downstream the TLV at 80%chord length.The phenomena in NS condition differs a little,in Figure 17(a).At 40%chord length,the higher frequencies amplitude is already obvious,compared to the DE condition in the big tip gap condition,as the TLV is not stable here.Along with the flow of TLV,more high-frequency amplitude increases and a broad-band hump appears,rather than a single conspicuous frequency appears in the DE condition.We could conclude that where the amplitude of higher frequencies becomesobviousisthelocationwhereTLVlosesstability.

For the small tip gap condition,the frequency spectra is not very typical because the flow at DE condition is so stable.The unsteady signal is weak that the DDES method cannot simulate the unsteadiness in this condition.For the NS condition,the energy is widely spread at many high frequencies;the amplitudes at these frequencies are almost the same.There does not exist a main frequency with the highest amplitude.

5 Conclusions

This paper tries to investigate the flow structure of tip leakage vortex in an axial compressor rotor by using DDES for a rotor blade with different tip gap sizes.Abundant flow field data are obtained and turbulent vortex structures are analyzed in detail.

Fig.14 Axial velocity fluctuation at the TLV core at the 80%chord length at DE condition

Fig.15 Axial velocity fluctuation at the TLV core at the 80%chord length at NS condition

Fig.16 Fluctuating axial velocity frequency spectra of 3 points along the TLV Core at DE condition

Fig.17 Fluctuating axial velocity frequency spectra of 3 points along the TLV core at NS condition

The unsteady flow characteristics is analyzed based on the DDES results.For the big tip gap condition,there are many complicated vortex structures in the TLV region from the instantaneous results.Large-scale vortices with relatively high vorticity are at the main structure of TLV while small scale vortices are generated when TLV mixes with the main flow.The TLV could be divide into three parts.The small tip gap result is different from the big gap one.For the DE con-dition,the structure is quite stable that there does not exist any small vortices at the latter part of the TLV.The contours of different time steps and the time-averaged result are almost the same.Neither TLV wandering nor rolling up small vortices happens at this condition.For the NS condition,the position where small vortices roll up is later in the TLV,at about 70%chord length.In the small tip gap condition,the numbers of the small vortices,which stands for the degree of the unsteadiness,is much fewer than those in the big tip gap.That is to say,the flow is more stable in the small tip gap condition.

The anisotropy invariant map has proven to be a useful and popular tool to study the structure of the turbulence.Anisotropy degree of the TLV is assessed from this method.The flow of the TLV at NS condition is more anisotropic than at NC condition.The core of the TLV is isotropic whereηis small while the flow around the vortex core is anisotropic.For the small tip gap condition,the anisotropy is weak especially at DE condition.The range of red color of the positiveηis smaller at DE condition compared to the big tip gap at the same condition.It shows that the quite stable condition at this condition means quite weak anisotropy in the tip leakage flow.At NS condition,the bigηvalue appears at the latter part of the TLV after about 60%chord length.That means in the small tip gap condition,the anisotropy increases at a latter part in the TLV.

Finally,from the spectral analysis,we find that the amplitude of high frequencies becomes obvious downstream the TLV.It is mainly because where the amplitude of the higher frequencies,compared to the peak amplitude frequency,increases is the location where TLV loses stability.In addition,the high-frequency amplitude increasing is not that obvious compared to DDES method.For the small tip gap condition,the frequency spectra is not very typical because the TLV at DE condition is smaller.The unsteady signal is weak that the DDES method cannot simulate the unsteadiness in this condition.For the NS condition,the energy is widely spread at many high frequencies;the amplitudes at these frequencies are almost the same.There does not exist a main frequency with the highest amplitude.