Tip-leakage-flow excited unsteadiness and associated control

Tip leakage flow in turbomachinery inherently generates intense unsteady features, named self-excited unsteadiness, which has not been well understood. A Zonalised LES (ZLES) is employed around a linear cascade, with wall-modelled Large Eddy Simulation (LES) forced to be active in the tip region. The simulation is well validated and demonstrated the advantages of effectively reducing the computational effort while maintaining an equivalent prediction accuracy in the region of interest. The time-averaged and spatial-spectral characteristics of tip leakage vortex (TLV) structures are systematically discussed. The self-excited unsteady processes of TLV include unsteady vortex separation inside the tip gap, the tip leakage jet-mainstream interaction, the primary tip leakage vortex (PTLV) wandering motion and the induced separation near endwall. The Spectral Proper Orthogonal Decomposition (SPOD) is used to examine the dominant frequencies and their coherent structures. It is found that these unsteady features change from a single high-frequency mode to a multiple lower-frequencies mode due to the PTLV breakdown. The SPOD and correlation analyses reveal that the self-excited unsteadiness is mainly induced by the interactions between unsteady vortex separation, tip leakage jet, and mainstream. The associated unsteady fluctuations are convected along the tip leakage jet trajectory, causing the wandering motion of PTLV core. Based on the cause of the investigated unsteadiness, a micro-offset tip design is proposed and validated for effectively suppressing this unsteadiness, and associated turbulence generation and hence pressure fluctuations. This work improves the understanding of tip-leakage-flow dynamics and informs the control of the associated unsteady fluid oscillation and noise.

una micro-offset tip design is proposed cause have been taken to investigate the vortex structures and flow mechanics associated with the tip leakage flow.
The existing literature demonstrates that there are two types of unsteady patterns due to tip leakage flow in tur ¬ bomachinery: one is caused by the impingement between the PTLV and blades; the other is the unsteady pattern inherently excited by the tip leakage flow.The trajectory of PTLV is significantly influenced by the tip gap size 8,9 and other parameters number and blade cumber.Therefore, under specific cir ¬ cumstances such as a large tip gap size, a high rotating speed and a large number of blades 10 pingement interaction may occur between the PTLV and blades.When there is such an impingement interaction, a significant vortex oscillation occurs and causes inten ¬ sified pressure fluctuation in mixed-flow pump 11 .This unsteady oscillation has been widely studied and is re ¬ garded to be closely related to stall [12][13][14] and generates unsteady blade forces, causing flutter in compressors 15 .
However, apart from the unsteadiness caused by the PTLV-blade impingement, tip leakage flow inherently presents an unsteady mode, such as vortex wandering 16 , which is excited by the tip leakage jet itself.This selfexcited unsteadiness is found to be a potential precur ¬ sor of rotating instability and dominates the measured I. INTRODUCTION Turbomachinery plays a significant role in the ergy supply, such as pumps and compressors, in the modern world.The unsteady flows in turbomachinery result in long-standing challenges to turbomachinery performance, among which tip leakage flow is one of the most complex problems.The inevitable endwall in a rotor / impeller induces tip leakage jet (TLJ) from the blade pressure side (PS) to the blade suction side (SS), as shown in Figure 1.The tip leakage flow develops into complex TLV structures, which is regarded as an important source of unsteadiness and flow loss [1][2][3][4] in turbomachinery.The tip leakage flow generates sig ¬ nificant unsteadiness that leads to excessive noise and detrimental cavitation that have been widely reported in many types of turbomachinery, such as compressors fans 6 and pumps 7 .Consequently, considerable efforts enturbines, and consumption, such sound spectrum in a turbo fan 16 .Unfortunately, this unsteadiness is underestimated in most of the previ ¬ ous studies leakage flow using LES.Unfortunately, the demand ¬ ing grid resolution requirement of LES simulation still makes it unaffordable for high-Reynolds number indus ¬ trial flows.This fact leads to the birth of hybrid RANS-LES models 26 , such as the Detached Eddy Simulation (DES) model proposed by Spalart 27 , which depends on modelled turbulence scale and mesh size.Subsequently, improvements have been made, such as Delayed Detached Eddy Simulation (DDES) for addressing the grid-induced separation 28 problem and Improved Delayed Detached Eddy Simulation (IDDES) for improving the simulation of wall-bounded turbulent flows 29 .However, users have no control over which region is simulated by LES RANS, as it depends on the grid size and simulated flows.
The ' grey area ' between LES and RANS is also a prob ¬ lem, concerning the unphysical delays of critical flow in ¬ stabilities in sensitive regions due to a slow RANS to LES transition 30 .An alternative is a zonal approach, named ZLES, where experienced region of LES and RANS based on the prior knowledge of the flow to be simulated 31 .This approach is more suitable for our research problem, i.e. tip leakage flow, so that the expensive LES can be focused only in the tip region.
The present work is performed around a linear com ¬ pressor cascade.The paper is organized as: First, the ZLES method is introduced and applied to the cascade.The results are compared with the experiment 32,33 and LES results 2 for validation.Second, the mean and un ¬ steady tip leakage flow fields ing this, the unsteady flow features of each tip leakage flow structure are investigated to illustrate their energy- spectra features and their correlation with each other by SPOD 34,35 and correlation analysis.Subsequently, based on the revealed physics, we proposed a micro-offset tip tip leakage flow, as most of them based on Reynolds-averaged Navier-Stokes (RANS) sim ¬ ulation, which is based on the Boussinesq hypothesis.This RANS approach relies on time-averaged quantities and tends to smear out the unsteady effects.This limi ¬ tation is particularly pronounced in the context of multi ¬ scale flows that exhibit high levels of anisotropic turbu ¬ lence 17,18 , which is a distinct feature of the tip leak ¬ age flow.Some RANS approaches, like the Reynolds Stress Model (RSM), solve transport equations for in ¬ dividual Reynolds stresses and allow for a more accurate representation of anisotropic turbulence effects 19 .How ¬ ever, due to their time-averaging nature, they still can not fully capture the multi-scale nature of unsteady vor ¬ tex dynamics.The Particle Image Velocimetry (PIV) measurements 20 demonstrated that the tip leakage flow results in multiple-scale vortex structures, and the large- scale PTLV will break down and collapse into small-scale structures when developing downstream 20 is the RANS portion of the turbulence and 计严 is the LES portion of the mod-

A. Case configuration
The present research is carried out on the same GE rotor B section blade in the experimental work performed by W. Davenport 32,33 .The case configuration consists of a linear cascade with a fixed stationary topwall and a moving endwall.
generically combine RANS and LES portions, and precisely define the RANS-LES boundary.The RANS and LES configuration is adjusted to obtain the optimal balance between predic ¬ tion performance and computational effort.This method has been successfully used to predict fan tip and wake flows in a bypass engine configuration 31,39 .This zonal approach is implemented in ANSYS Fluent via User De ¬ fined Function by defining //,.The region of interest near the tip is running in LES mode (//,=0), while the rest is in the RANS mode The RANS-LES boundary is determined by a streamline marked in Figure 1  RANS is used to model the inner parts of the boundary layer and reduce the required grid resolution of streamwise Ax + and spanwise Az + compared to the wall-resolved LES.The RANS layer is applied to where a wall distance Aj +< 100, which approximately viscous sublayer and buffer layer.The grids in the tip region and testing of the independence of the grid res ¬ olution and the RANS-LES boundary are presented in Appendix B.
Taking advantage of /办 , we can The basic geometrical parameters are listed in TableI.The blade chord length c is used as the reference length.
The blade pitch Ly is 0.929 c, and the span Lz is 1.0 c.The coordinate system is adjusted based on the experimental setup, which is described in Figure 19  The Central Differencing scheme is employed for spa ¬ tial discretization, and the Second Order Implicit back ¬ wards difference method for time integration.The SIMPLEC scheme is used to decouple pressure and velocity B. Numerical methods 1. Zonal LES in solving the incompressible flow governing equations.In the transient simulation, the time step size is set as Ar = 8 x 10 _ 6 s In the present work, a ZLES method is employed to resolve the energetic turbulence of tip leakage flow (Fig- 1).To make the simulation affordable, the expensive LES is zonalised in the tip region to achieve high-fidelity prediction of tip leakage flow while the rest region is mod ¬ elled by RANS.
The SSTk -co turbulence model is used in the RANS region where the flow field is rarely influenced by the tip leakage flow.The LES mode is activated in the tip region, which is the focus of this research.The dy ¬ namic Smagorinsky-Lilly model 36 grid stresses.
x c / t/oo, which guarantees that the Courant number is below 1 in the region of primary in- r^/ 1250 ure terest.
The velocity inlet condition is applied at the inlet boundary, and the outflow condition, where a zero dif ¬ fusion flux for all flow variables, is applied at the outlet boundary.A periodic boundary condition is adopted on the sidewalls to account for the adjacent passages.The no-slip wall condition is applied on the blade surface and the moving endwall, while a free-slip boundary condition is set on the topwall.,37 is to calculate sub- In the experiment 33 , Reynolds normal stresses were mea ¬ sured on a line crossing the TLV core on X = 1.51 Ca , lo ¬ cated in the wake regions and is marked in Figure 20 of Appendix A. To validate the accuracy of the ZLES method, compar ¬ ison of Reynolds normal stresses has been made between the simulation and the experiment in Figure 3.Moreover, the LES results from You et al. 2 are also plotted as a ref ¬ erence for the validation.Generally, the ZLES method shows an equivalent accuracy to the LES but at a reduced cost due to about six times coarser mesh.Figure 4 shows that the predicted velocity power spectral density agrees well with the experimental measurements 33 .Both the experiment and simulation show the -5 / 3 slope charac ¬ teristic in the inertial subrange of turbulence.Figure 5 shows the time-averaged streamlines at the tip region from x / c=0.3 to 0.6, contoured by the time- averaged streamwise vorticity There are three dis ¬ tinct categories of vortical structures: 1) PTLV: The PTLV is generated by the tip leakage jet and is the dorminant vortical structures in the tip It detaches the blade from x / c = 0.28, breaks IV.SELF-EXCITED UNSTEADINESS OF TIP LEAKAGE FLOW region.
down at x / c ~0.5, and shows a significant wandering motion.Its unsteady features will be discussed in detail in Sections IV B and IVD.

A. Time-averaged flow patterns
This section discusses the time-averaged and spectral characteristics of tip leakage flow.Eight streamwise planes are selected at x / c=0.2, 0.3 ... and visualise the tip leakage flow features.A slice at z / c=0.0118 inside the gap, which is normal to the span- wise direction and named Clip-Gap, is extracted to inves ¬ tigate the tip leakage jet dynamics.A slice cutting ap ¬ proximately through the time-averaged vortex core tra ¬ jectory, named Clip-PTLVC, is set to investigate flow features near the vortex core.Their locations are shown in Figure 20 of Appendix A.
2) tip separation vortex (TSV): The TSV the tip gap and initiates near the blade pressure side.Strong shear occurs between them generates K-H rollers, shown by the instantaneous vortex structure defined by Q criterion in Figure 6.
3) secondary induced vortex (SIV): The SIV is induced by the primary tip leakage vortex and rotates in an oppo ¬ site direction of PTLV.It gradually grows from x / c=0.3 and disappears after x / c=0.4.This component is not extensively discussed as its influence is limited.occurs m 0.9 to monitor Uxc/Um a vortex, defined as Ro = U / rQ., where U is the ial velocity, Q is the rotation rate, and r is the vortex core radius 40 .It is a common criterion for vortex break ¬ down 41 .Based on the time-averaged results, we plotted the distribution of Ro along the PTLV core trajectory in Figure 7a.We found that the Ro at x / c=0.5 is close to the critical value of vortex breakdown found by Robinson etal.40 .The decrease of the Ro beyond this critical value reflects the significantly weakened ability of the PTLV to entrain the tip leakage flow from the gap into the vor ¬ tex core.This change caused by vortex breakdown has a dominant influence on the self-excited unsteadiness that will be discussed in Section IV C. To better understand the tip leakage flow features from a 3D and transient perspective, Figure 6 presents the tip leakage flow streamlines that roll up and form the PTLV, which is also illustrated by an iso-surface of Q criterion.The pattern of the streamlines demonstrates the rolling- up process of the PTLV, and the rotation strength of the vortex core decreases when developing downstream, in ¬ dicated by the condensed streamlines that are entrained into the vortex before the mid-chord and the dispersed streamlines after the mid-chord.A large-scale and coher ¬ ent vortex structure of PTLV is observed upstream be ¬ fore the mid-chord, and it gradually evolves to multiple small-scale structures downstream after the mid-chord.Therefore, we suspect the PTLV breaks down around the mid-chord location.The distribution of time-averaged pressure coefficient trajectory (cf. Figure 7b) fur- Cp along the PTLV ther supports the fact that the PTLV breaks down at x / c ~0.5.Cp at the vortex core first dramatically de ¬ creases in the early stage but significantly lifts up after x / c=0.5.This is caused by the increased static pressure due to the slower fluid rotational motion after the vor- core cP / / / / / Dispersed PTLV breakdown tex breakdown.The pressure at the PTLV core rapidly rises after the vortex breakdown, which causes a reduced pressure difference between the pressure side and suction side that drives the tip leakage flow from the gap to the vortex core; this may contribute to the distinct difference in unsteady features before and after the PTLV break ¬ down, as will be shown in Section IV C. Furthermore, the results imply that the pressure in the vortex rapidly increases after vortex breakdown.This is crucial to the control of vortex-induced pressure drop and cavitation.
Condensed m x/c=0.6 x/ cf 0.5 x/c=0A x/c=0.3 FIG. 6. Instantaneous streamlines of tip leakage flow, with the vortex defined by the iso-surface C. Spatio-temporal features of Qc 2 /Ul = 8 x 102.
Subsequently, we demonstrate the spatial features of turbulent kinetic energy (TKE) on the streamwise tions from JC / C=0.3 to 0.8.The tip leakage flow region is divided into the TLJ subzone and the PTLV subzone Rossby number Ro is used to quantitatively identify the vortex breakdown behaviour and location.The Ro is a ratio of the axial and circumferential momentum in sec-/s3 in the TLJ and PTLV subzones.After ;c / c=0.6, the PSD level at /si on the PTLV core significantly decreases, and a group of frequencies centred at /IMP become domi ¬ nant at x / c=0.7 and x / c=0.8 where the PTLV gets close to the adjacent blade.This is caused by the impinge ¬ ment of the PTLV on the adjacent blade, which has been studied by He et al. 15 .
(Figure 8d), as an effort to clarify their distinct features better and identify their mutual correlations in the fol ¬ lowing analyses.As shown in Figure 8, the TLJ subzone contains the highest level of TKE, and the high TKE extends along the TLJ direction towards the middle of the passage at the downstream stations.The TKE in the PTLV subzone first increases from JC / C=0.3 to 0.5, then decreases in intensity and diffuses into a larger region, which is related to the breakdown of PTLV at x / c -0.5.
Meanwhile, a significant TKE distribution near the end- wall is observed from JC / C=0.4,especially at x / c=0.4 and x / c=0.5, which is corresponding to the SIV, as shown in  To further understand the spectral characteristics of flow unsteadiness along the tip leakage flow propagation, the PSD (Power Spectral Density) of velocity is discussed along the TLJ direction and the PTLV core (PTLVC) trajectory, respectively.
Along the trajectory of maximum TKE location at each streamwise section, present in Figure 8(d) for x / c=0.6, the spectral features of the tip leakage flow are investigated by plotting the PSD of velocity.The hori ¬ zontal axis of PSD map in Figure 9 is the pitch distance, and additional 3 vertical lines are added to show the tip PS, SS and PTLV map is the normalised frequency, and the contours show the nondimensionalised PSD values.
As shown in Figure 9, the PSD map at x / c=0.3 shows single frequency at /si when the PTLV just detaches from the blade tip.This frequency is limited in the tip gap region and found to be the same as the TSV separation frequency, which will be discussed in Section IV E. When the tip leakage flow develops downstream to x / c=0.4 and x / c=0.6, the same frequency /si appears in the PTLV subzone, especially in the PTLV core and induced end- wall separated region.It indicates that this unsteady perturbation at /si could propagate from the upstream tip gap along the TLJ direction to the downstream PTLV core.Further downstream, the PTLV has already broken down and introduced several low frequencies, such as /52 , FIG. 9. PSD Map of velocity, where P -\J (E ^u+ E ^v+ E ^vw ), PTLVC denotes the mean PTLV core location.on The above discussions demonstrate the statistical char ¬ acteristics of tip leakage flow, suggesting distinct spectral features along the PTLV.So far, the underlying physics of the self-excited unsteadiness, excluding the unsteadi ¬ ness caused by vortex-blade impingement veiled.The self-excited unsteadiness is dominant at the early stage of the PTLV evolution and has a significant contribution to the flow losses 25 and noise generation 42 .Therefore, the following section will focus on the under ¬ lying physics of these unsteady flow processes, their cor ¬ relation and the cause of the unsteadiness.In the PTLV subzone, the most distinct unsteady fea ¬ ture is the PTLV wandering motion after it detaches from the blade, which is corresponding to the observed fre ¬ quencies in Section IV C. The PTLV wandering motion is shown in Figure 24 (Multimedia available online) in Appendix C, by presenting the instantaneous vortex core locations and the vector field on the streamwise section of x / c=0.6.

locations. The vertical of PSD
Figure 10(a) presents a collection of instantaneous po ¬ sitions of the PTLV core.The vortex core is wandering in the y -z plane, where y is the pitchwise direction and z is the spanwise direction.The wandering motion am ¬ plitude in the y or z directions is defined as: wandering motion.It changes from a single high- frequency wandering pattern to a more chaotic and mul ¬ tiple low-frequency oscillation.

E. Vortex separation and shear interaction the tip near
To understand the underlying flow physics in the TLJ subzone at different stages, we demonstrate in Figure 12 the instantaneous vector field on the three streamwise sections of: (a) x / c=0.2, before the PTLV detaches the blade; (b) x / c=0.4,after the PTLV detaches the blade and before the PLTV breaks down; (c) x / c=0.6, after the PTLV breaks down.The flow pattern is almost steady when the PTLV just forms and is attached to the blade surface (at x / c=0.2).After the PTLV detaches from the blade, there is an intense unsteady separation inside the tip gap and a significant shear leakage jet and mainstream outside the gap.This results in significant flow loss and unsteady fluid oscillation in ¬ side and outside the tip gap.
Furthermore, the unsteady TSV separation frequency is about /si at JC / C=0.4 (cf. Figure 12b, e, h), which is before the PTLV breakdown, and about /s3 at x / c=0.6 (cf. Figure 12c, f, i), which is after the PTLV breakdown.
These results suggest that the spectral features in the TLJ subzone follow the same trend as PTLV, and the instantaneous flow field reflects the unsteady processes corresponding to the identified frequencies.
A further SPOD computation is carried out on the section Clip-Gap, which crosses the TSV separation and TLJ-mainstream shear layer (illustrated in Figure 20).It is clearly shown in Figure 13 that the first SPOD mode at /si mainly exists before the PTLV breakdown, while that at /s2 and /53 emerge after the PTLV breakdown.These findings imply that the PTLV breakdown has the same impact on the spectral characteristics in the TLJ zone as that in the PTLV zone.

core F. Correlation of the unsteady features between the TLJ and PTLV
The unsteady features of PTLV are examined by per ¬ forming SPOD analysis in the spectral domain on the sec ¬ tion Clip-PTLVC, which roughly crosses the mean PTLV core trajectory (cf. Figure 1 and Figure 20).The tur ¬ bulence coherent structures The above discussions, especially Figures 11 and 13, imply that the unsteady features in the TLJ subzone and the PTLV subzone are related.
To validate this relatioship, the magnitude-squared co ¬ herence function 43 and the cross-correlation function 44 used to estimate their coherence (in spectral do ¬ main) and correlation (in time domain).The magnitude- squared coherence estimate is a function of frequency (Equation ( 3)), measuring the similarity of two signals Before the PTLV breakdown, the SPOD mode at /si is amplified along the PTLV core trajectory; after the PTLV breakdown, it gradually attenuates when develop ¬ ing downstream.With regard to the first SPOD mode at /s2 and /s3 , they both start after the PTLV breakdown, with increasing energy when developing downstream.
The above results suggest that the PTLV vortex break ¬ down directly influences the spectral feature of the PTLV x / c 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x / c 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x/c=0.4 x/c =0 .6 x/c=0.2 • mQcyui ??15.0 X 105 12. Instantaneous vector on the streamwise sections of x / c=0.2 (a, d, g), x / c=0A (b, e, h) and x / c=0.6 (c, f, i), contoured by the nondimensionalized Q criterion.T ,: S 1 denotes the period corresponding to the frequency /si , and T , S3 denotes the period corresponding to /s3 -The relative velocity relative to the moving endwall is applied to the vector.where Paa(/) and Pbb { f ) are the power spectral densities for signal a and signal Z?, and Pab { f ) is the cross power spectral density.Cat ,(/) denotes the magnitude-squared coherence, measuring the similarity between two signals as a function of the frequency /.
The cross-correlation estimate is a function of the time lag (Equation (4) and Equation ( 5)), measuring the sim ¬ ilarity of two signals in time.
Rab ( j) ^ab {^) coeff ⑹ y / Raa (0) Rbb (0) where Rab {^) denotes the cross-correlation function, mea ¬ suring the similarity between two signals (a and b) function of the time lag (T) of one relative to the other.
Rab {^) cotfi is the normalised cross-correlation coefficient using autocorrelations.By computing the coherence and correlation of the two signals with the above functions, we get the maps of these two coefficients present in Figure 14, respec ¬ tively.The maps provide a convincing illustration of the spatial-temporal correlation between the tip leakage jet and the PTLV cores.A served at the frequency of /si from x / c=0.3 to around x / c=0.7, which demonstrates the similarity of the steadiness in the tip gap and that at the PTLV The relation between the time lag and the streamwise lo ¬ cation is approximately tiagUoo/c = -[(x / c -0.3) + 0.012], which shows the unsteadiness, excited in the tip gap, con- vects downstream along the PTLV trajectory.The time lag of -0.012c / f/oo comes from a time delay between the signal at PTLV core at x / c=0.3 and the signal at point 1 in the gap at x / c=0.3.This further provides evidence that the PTLV wandering motion originates from the un ¬ steadiness near the tip and propagates downstream along the PTLV core trajectory at the mainstream convection speed Uoo.The above results demonstrate that the unsteady sep ¬ aration and shear interaction near the tip is the source of the self-excited unsteadiness that enhances the turbulent kinetic energy.The LES work performed by You et al. 25 also demonstrated that the significant mean velocity gradients along the spanwise direction lead to the production of turbulent kinetic energy, and the most active turbulent fluctuations are observed near the tip-SS corner.To mitigate this, one of the controlling principles is to decrease the velocity gradient, especially along the spanwise direction, in the near-tip region.Therefore, micro-offset tip design is correspondingly proposed.As shown in Figure 15, we gradually increase the blade thick ¬ ness at the suction side in the near tip region, which forces the mainstream direction to be more aligned with the tip leakage jet direction to reduce the spanwise ve ¬ locity gradient around the tip-SS corner.
In this preliminary test case, the spanwise scope of the tip offset T is set as the tip gap size S = 0.0165c, and the pitchwise scope C , is set as 25% of the original tip thick ¬ ness.The grazing between the TLJ and mainstream at the gap exit is suppressed, and thus the unsteady TSV separation is also significantly mitigated.As a result, the PTLV core wandering is remarkably suppressed.These Coherence and correlation of pitchwise velocity between points 1 in Figure 8a and the approximate PTLV the monitoring plane Clip-PTLVC: (a) magnitude-squared coherence estimate (b) cross-correlation.trajectory core on In the present work, one signal is taken as the instanta ¬ neous pitchwise velocity at point 1 at x / c=0.3, as marked in Figure 8a; the other signal is taken as the instanta ¬ neous pitchwise velocity along the approximate PTLV core trajectory determined by the TKEMAX curve on the monitoring plane Clip-PTLVC indicated in Figure 1 and Figure 20.where /?/ denotes the instantaneous pressure at the snap ¬ shot i. Figure 16 and Figure 17 show the effect of micro-offset tip on suppressing turbulence induced by tip leakage flow.By taking advantage of micro-offset tip design, the veloc ¬ ity fluctuation in the tip leakage flow region, including the PTLV core, is remarkably reduced.In particular, as shown in Figure 16, the micro-offset tip design can effec ¬ tively decrease the energy at /si , which is concluded to be the dominant frequency of PTLV core wandering motion.Figure 17 further indicates that the velocity fluctuation of tip separation and TLJ (from the SS to the PTLV core) is also significantly reduced by applying the micro-offset tip design, in addition to the PTLV core itself.
Moreover, as the PTLV core wandering motion is sup ¬ pressed, the associated pressure fluctuation around the PTLV core is significantly mitigated, as shown in The present paper provides insight into the tip- leakage-flow excited unsteadiness around a linear com ¬ pressor cascade.The following conclusions can be drawn: (1) As tip leakage flow presents a confined zonal fea ¬ ture, the ZLES approach is a promising way to resolve the interested turbulence structures with locally high fi ¬ delity for a reduced computational cost.
(2) The tip leakage flow structures present intense steady features.Along the tip leakage jet, there is a sig ¬ nificant unsteady separation for the TSV structure inside the tip gap, followed by a TLJ-mainstream shear inter ¬ action, then the PTLV core wandering motion and the un-induced separation near endwall.The PTLV breakdown has a dominant influence on the spectral features of these unsteady flow scenarios, and it leads to a change of un ¬ steady features from a single high-frequency to multiple low-frequencies.
(3) The SPOD computation and the correlation anal ¬ ysis suggest that the investigated unsteadiness is mainly - excited by the unsteady TSV separation and TLJ- mainstream shear interaction steadiness originates from the tip gap and propagates to the PTLV core.It further causes the PTLV core wan ¬ dering, which convects downstream along the PTLV core trajectory at the mainstream convection velocity.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.

VII. REFERENCE
the tip.This (4) Based on the revealed flow physics, a micro-offset tip design is proposed.It can significantly suppress this self-excited unsteadiness, by reducing the interaction be ¬ tween the tip leakage jet and mainstream near the tip suction side.This flow control method effectively mitigate the PTLV wandering motion, resulting in a re ¬ markable attenuation of the associated turbulence gen ¬ eration and pressure fluctuation.The above findings suggest that particular attention should be paid to suppressing the unsteady shear inter ¬ action near the tip suction side.We can decrease the TLJ-mainstream interaction by changing the tip leak ¬ age jet or near-tip mainstream direction through tip shape modification.The identified critical role of PTLV breakdown also suggests that advancing vortex break ¬ down active / passive fluid injection is likely to be a route to control the noise and pressure-drop due to swirling vortex.The revealed flow physics and poten ¬ tial flow control technologies will be beneficial to reduce flow losses, noise and cavitation, which are critical chal ¬ lenges to improving the performance of turbomachinery and aerial / underwater vehicles.In particular, we are very grateful for the one- year international visiting funding from the China Schol ¬ arship Council, which enables this joint work.mo-14 configuration, the statistical results of two cases with dif ¬ ferent spanwise blending boundary locations, as shown in Figure 20, are compared.One case is named BD1, and the other case with the RANS-LES boundary shifted by 4% C along z is named as BD2.As shown in Figure 23, a good agreement of the stresses curves between the two cases is observed, which verifies the independence of the RANS-LES blending boundary location.dimension and with bounded wall condition, the grid res- locally refined in partic- shown in Figure 21.The gird resolution in the region of primary interest is controlled in Ax +< 60, Ay +< 1 blade surface and Az +< 90.In the region close to the blade tip, the Az + is refined to 1 2 to accurately capture the flow details near the narrow tip gap.The mesh ar ¬ rangement in the RANS scope is much coarser, especially for the region far from the RANS-LES boundary, with Ax + and Az + increased to 50 200.Therefore, the total grid elements number is only 3,531,621 for the 4.2 tip gap, with the whole span considered.For the same more than 20 million grid elements were used in the LES studies 2,22,23,25 , with only half span simulated.
An independence study of grid resolution was also per ¬ formed to exclude the influence of grid resolution on pre ¬ dicting the turbulence field of the tip leakage flow re ¬ gion.Moreover, the experimental data was measured at X = 1.51 Ca , and a substantial amount of grid is clustered in the wake region (the region where X > \.0Ca shown in Figure 19), but this region is actually of slight inter ¬ est because this study focuses on the tip leakage flow region.Therefore, the prediction performance of the fol ¬ lowing two cases, named Case 1 and Case 2, is compared.In Case 1, the studies 2,22,25 is employed in both the tip leakage flow re ¬ gion and the wake region, and these regions are running in LES mode; In Case 2, the grid resolution is the same as described in the above paragraph, while the mesh in the wake region i Furthermore, the Reynolds normal stresses on the three lines from crossing the PTLV core are plotted to com ¬ pare the prediction performance of Case 1 and Case 2. As shown in Figure 22, the Reynolds normal stress curves obtained by the two cases show a remarkable agreement, which validates that the adopted grid resolution and LES scope in the present work can well predict the turbulence field of the tip leakage flow region.
To further validate the independence of RANS-LES Comparison of Reynolds Normal Stresses along between Case 1 and Case 2 with different grids.The line is located on the streamwise sections of x / c = 0.3 (M30), x / c = 0.6 (M60), and x / c = 0.9 (M90), respectively.Comparison of Reynolds Normal Stresses along a line crossing the vortex core between different RANS-LES boundary locations.The line is located on the streamwise sections of x / c = 0.3 (M30), x / c = 0.6 (M60), and x / c = 0.9 (M90), respectively.
Figure 27 shows the computed local SPOD energy spectra from x / c=0.3 to x / c=0.7, for the TLJ subzone and the PTLV subzone, repsectively.There is increas ¬ ing low-frequency energy in both subzones when the tip leakage flow develops downstream.Meanwhile, we ob ¬ serve multiple low-frequency peaks from x / c=0.b in the TLJ subzone and from x / c=0.6 in the PTLV subzone, which means a streamwise location delay of the spectral features between the TLJ subzone and the PTLV sub ¬ zone.This supports that the unsteadiness in the TLJ subzone convects to the PTLV subzone.

' 21 .
The obser ¬ vation shows that these structures are linked with differ- on are or able to specify the users are ent unsteady flow features, but how these unsteady flow features are excited and correlated with each other is still unknown.Although several LES works have been ducted to capture the resolved tip leakage vortices the significant influence of the tip leakage jet has been less clearly identified 25 .The role of the tip leakage jet in generating the self-excited unsteadiness and the evo ¬ lution process of the related unsteady modes remain un ¬ clear.Therefore, the present work focuses on unveiling the generation and propagation dynamics of self-exited unsteadiness.fundamental mechanisms, it is cru ¬ cial to resolve large-scale turbulence motion of the tip The ZLES is achieved by a blending function /办 38 .The turbulence stresses between RANS and LES are blended in the following way: design to control the self-excited unsteadiness.Finally, we summarise the most important conclusions and dis ¬ cuss future work.II.CASE CONFIGURATION AND NUMERICAL METHODOLOGY ^ur b = fi ^NS + ^-h) ^s ⑴ NS where stresses tensor elled turbulence stresses tensor.
(a) sep ¬ arating the tip leakage flow from the rest flow region.

Figure 1
Figure 1(b) shows the wall-modelling strategy in LES region.RANS is used to model the inner parts of the boundary layer and reduce the required grid resolution of Appendix A. The tip gap size is set as 1.65% c.The Reynolds number is around 4 x 105 based the freestream velocity and chord length.The moving endwall speed V ^dwai i is set to model the relative movement between the endwall and on the blade, and it was achieved by a moving belt in the experiment 32,33 .TABLEI.Basic parameters of the investigated cascade schemes and boundary conditions

Figure 2
Figure 2 shows a good agreement of the time-averaged between the p pressure coefficient Cp, where Cp Jpul present simulation and the experiment 32 at the mid-span.

FIG. 3 . 5 FIG. 2 .
FIG. 3. Comparison of ReynoldsNormal Stress along a line crossing the PTLV core on X = 1.5IQ between the present simulation and the experiment33 .

FIG. 4 .
FIG.  4. Comparison of velocity power spectral density at the PTLV core point on Z = 1.51 Ca between the present sim ¬ ulation and the experiment33 .
the streamwise sections of x / c=: (a) 0.3 (b) 0.4 (c) 0.5 (d) 0.6 (e) 0.7 (f) 0.8.The horizontal axis shows the spatial distribution of P along the TKEMAX curve indicated in Fig ¬ ure 8d.The vertical axis shows the spectral distribution of P.
di -d) 2 d = N ^x di (2) dw = Nt ^xwhere di denotes the instantaneous j or z at the snapshot

Figure 10 (FIG. 10 .
Figure 10(b)  shows the growth of PTLV wandering am ¬ plitude along the streamwise direction.The wandering amplitude increases almost linearly as the vortex travels downstream.Due to the end wall restriction, the ampli ¬ tude of PTLV is smaller in the spanwise direction than in the pitchwise direction.
in frequency.extracted by SPOD at three characteristic frequencies /si = 11.85 /c , /s2 = 8.65 /c and /s3 = 5.47 fc , where fc = Uoo / c.As shown in Fig ¬ ure 11, the first SPOD mode at /si shows a wave-packet pattern and starts when the PTLV detaches the blade.
CONTROL BY A MICRO-OFFSET TIP DESIGN

mam b 8 J
FIG. 14.Coherence and correlation of pitchwise velocity between points 1 in Figure8aand the approximate PTLV the monitoring plane Clip-PTLVC: (a) magnitude-squared coherence estimate (b) cross-correlation.

FIG. 17 .FIG. 18 .FIG. 16 .
FIG. 17. PSD Map of velocity, where P -\J (E ^u+ E ^v+ E ^vw ), the streamwise sections of x / c=0.6: (a) Original blade (b)Micro-offset tip.The horizontal axis shows the spatial distri ¬ bution of P along the TKEMAX curve indicated in Figure8d.The vertical axis shows the spectral distribution of P. PTLVC denotes the mean PTLV core location.
Fig ¬ ure 18.This can help suppress fluid-induced vibration VI.CONCLUSION can 4
FIG. 21.Mesh the blade and in the tip gap. on same line crossing the vortex core and running in RANS mode.
FIG. 23.Comparison of Reynolds Normal Stresses along a line crossing the vortex core between different RANS-LES boundary locations.The line is located on the streamwise

FIG. 24 .FIG. 27 .
FIG.  24.Instantaneous vector field on the streamwise sec ¬ tion of x / c=0.6 and vortex core wandering motion.Q denotes the Q criterion.The red point denotes the instantaneous pri ¬ mary tip leakage vortex core location.(Multimedia available online)