## Abstract

Direct numerical simulations (DNS) are performed of a cantilevered stator blade to identify the unsteady and turbulent flow structure within compressor tip flows. The simulations were performed with clearances of 1.6% and 3.2% of chord. The results show that the flow both within the gap and at the exit on the suction side highly unsteady phenomena controlled by fine-scale turbulent structures. The signature of the classical tip-leakage vortex is a consequence of time-averaging and does not exist in the true unsteady flow. Despite the complexity, we are able to replicate the flow within the tip gap using a validated quasi-three-dimensional (Q3D) model. This enables a wide range of Q3D DNS simulations to study the effects of blade tip corner radius and Reynolds number. Tip corner radius is found to radically alter the unsteady flow in the tip; it affects both separation bubble size and shape, as well as transition mechanisms in the tip flow. These effects can lead to variations in tip mass flow of up to 10% and a factor of 2 variation in dissipation within the tip gap.

## 1 Introduction

Tip flows are a significant contribution to loss in axial compressors, accounting for up to a third of overall loss [1]. The schematic in Fig. 1 shows the detail of the flow over the tip of a blade as it leaks from the pressure side to the suction side. The total loss generated scales with the mass flow that leaks through the gap, this is in turn set by the size of the separation that typically originates from the pressure surface corner. Prediction and control of the size of this separation are therefore of primary importance in determining the loss generated by the leakage flow and the performance of the whole machine.

Fig. 1
Fig. 1
Close modal

There are two possible routes to affect the size of the separation and therefore the leakage mass flowrate. Either control of the gap size and therefore aspect ratio, or local modifications to the blade tip itself, both are investigated within this paper.

While the separation point is generally fixed, whether it re-attaches or not depends upon the size of the gap relative to the blade thickness. If the tip width w > 2.5τ the separation can re-attach through turbulent mixing and pressure recovery within the gap [2]. Storer and Cumpsty [2,3] showed that loss in the passage was dominated by mixing of the mismatched leakage flow and suction-side passage velocities and this in turn was affected by whether the flow re-attached on the tip of the blade or not. For the majority of compressors currently in service closing the height of the gap always reduces the mass flow through it and therefore the loss. However, this is often limited by the mechanical design of the compressor and its ability to tolerate rubbing during transient operation.

Geometry modifications local to the blade tip can be used to control the tip-leakage flow. Early experiments showed that rounding the pressure side corner led to smaller separations over the tip [4,5]. Aside from this early work, research into compressor tip geometry has focused on its effects on the passage flow rather than flow in the clearance gap itself. There is still much debate over the optimal shape of blade tips since it depends upon details of the compressor design as well as geometry changes caused by wear during service. This uncertainty is also likely to be caused by mis-prediction of the details of the tip flow by current CFD methods that often do not align well with experimental test data.

In order to assess actual tip shapes, micrographs were taken of blades as-manufactured. Figure 2 shows an example section through a stage 10 blade at 50% chord before entry into service. The tip is ground, removing a coating layer of around 20 μm thickness and leaving a bare metal surface. The resulting tip surface has a measured roughness of Ra ∼ 2−4 μm. The figure also shows an example of a typical pressure-side edge extracted from the micrograph. De-burring of the pressure-side edge leaves a residual chamfer, reducing the sharpness of the edge. Micrograph measurements of several sections on separate blades indicated edge radii of between 2% and 6% of the tip width. As the edge radii are an order of magnitude larger than the surface roughness, they are thought to dominate the effects on the flow and are a main focus of the simulations discussed later in this paper.

Fig. 2
Fig. 2
Close modal

The true turbulent nature of the tip flow and how this is affected by tip shape is not well understood. Locally, the tip flow Reynolds number will be in the range 103−105, based on tip width, meaning that significant laminar flow could occur. However, the tip is subjected to flow separation and incoming turbulence from the passage flow. High-fidelity simulations are required to understand and capture the correct physical mechanisms. Previous high-fidelity simulations have focused on the tip-leakage vortex [6,7], its interaction with the passage flow [8] and the effect of gap size on this physics [9,10]. Separation bubbles in turbine gaps are found to break down to turbulence through the development of spanwise streaks 15–20% of the tip gap height [11] and this breakdown alters the tip gap flow significantly. Compressor tips are narrower and Mach numbers lower, and therefore the separation bubble physics is likely to be different. There is a lack of physical understanding of mechanisms within compressor gaps and capturing the correct bubble re-attachment mechanisms is key to accurately capturing the tip gap mass flow.

In this paper, high-fidelity simulations are used to identify the unsteady flow physics within the tip gap. Three sections are presented that aim to answer three research questions:

1. What is the unsteady flow structure within the tip gap of a cantilevered stator blade?

2. How is the unsteady flow structure affected by tip gap height?

3. How do the tip corner radii affect the unsteady flow structure?

## 2 Solver Details

In order to capture the correct turbulent flow within the tip gap, direct numerical simulations are performed. The high-order compressible Navier–Stokes solver 3DNS [12] is used. Spatial discretization is achieved with a fourth-order DRP finite difference central scheme by Tam and Webb [13] coupled with an eighth-order standard filter. Explicit time stepping is implemented with a four-stage fractional step Runge–Kutta. Characteristic boundary conditions outlined in Ref. [14] are used to minimize the effect of reflecting pressure waves.

In this paper, we conduct two different types of simulations, their domains are shown in Fig. 3. First, full 3D simulations of a cantilever stator with tip gap, and second, quasi-three-dimensional (Q3D) cases that are locally accurate in the tip region. Using these two different types of cases in combination maximizes understanding of the flow with current computational resources.

Fig. 3
Fig. 3
Close modal

By restricting the solver to prismatic geometry in the z-direction significant savings can be made. However, this means that the full 3D simulations are restricted to a cascade of 2D prismatic blades with a moving endwall. While all of the important flow phenomena are represented, it is limited in that it is impossible to create a valid experiment to compare this case against and it is not possible to model the local tip geometry with rounded corners. For this it is necessary to use the quasi-three-dimensional simulations, now the z-direction is aligned with the chord of the blade and it is possible to run many different configurations with current resources.

## 3 Cantilever Stator Simulation

In this section of the paper, the full 3D cases are used to identify the unsteady flow structures that are driven by the leakage flow.

### 3.1 Simulation Setup.

Simulations of axial compressor cantilever stator geometries were performed. The simulations are set up to match the Gibbons research compressor at the Whittle Laboratory. This is a medium speed (M < 0.5) single-stage research compressor that is used most recently in Ref. [15] and is fully described in Refs. [16,17]. The blade geometry is a new cantilevered design representative of a modern aero-engine compressor with controlled diffusion blading.

The DNS cases are computationally challenging and are limited by computing costs that scale strongly on domain size. In order to address this, the spanwise extent of the domain was reduced. However, it is important that this reduction in domain size does not affect the accuracy of simulation of the local flow around the tip gap and so a study using RANS was performed to determine the validity of this approach. This showed that the tip-leakage mass flow and loss were unaffected when the span was reduced to 30% of the original value and the top of the domain replaced with an inviscid wall. At this spanwise location, the flow can be considered relatively 2D, Fig. 4 shows the relative spanwise displacements of streamsurfaces tracked through the row. At 30% of the span, the maximum radial movement of a streamline is 3% of the total span of the stator blade.

Fig. 4
Fig. 4
Close modal

At the hub end of the DNS, the boundary is set to a no-slip wall that is moving in the tangential direction. The velocity of this endwall was set to match the flow coefficient of the compressor when the stator is operating at its design point.

Inlet profiles of velocity, yaw angle, and turbulence are matched to experimental measurements from the test rig. A single hot-wire probe was used to survey the flowfield midway between two stator leading edges. By recording measurements at multiple radial locations and yaw angles of the probe it is possible to compute both direction and velocity of the flow. The time-averaged velocity normalized by freestream velocity for both DNS and experiment is shown in the first plot of Fig. 5. The time-averaged yaw angle is shown in the second, and the skew in the hub endwall boundary layer is accurately replicated.

Fig. 5
Fig. 5
Close modal

The inlet turbulence is prescribed using the method of Ref. [18], it is precomputed and fed into the inlet during run time. The magnitude of velocity perturbations and their lengthscales are varied radially to match those observed in the test rig. The final plot of Fig. 5 shows the spanwise distribution of turbulence intensity while Fig. 6 shows the power spectra at 10% span for both DNS and experiment. The intensity of the lengthscales are well matched in the inertial region as is the frequency at which the roll off begins. Note, at a reduced frequency of 7000 the experiment starts to roll off faster as the structures become smaller than the miniature hot-wire used.

Fig. 6
Fig. 6
Close modal

Figure 7 shows the computational mesh. A summary of the cases is shown in Table 1. Small and large gap cases are run at 1.6% and 3.2% tip gap to chord. The blade Reynolds number, based on axial chord, is 120,000 and the inlet Mach number is 0.29. A coarse mesh case is run with 45% total mesh cell count for the large gap case to test mesh sensitivity.

Fig. 7
Fig. 7
Close modal
Table 1

Cantilever stator testcases

Tip gap (% chord)Mesh cell count
Small gap1.61.6 × 109
Large gap3.21.8 × 109
Large gap coarse3.20.76 × 109
Tip gap (% chord)Mesh cell count
Small gap1.61.6 × 109
Large gap3.21.8 × 109
Large gap coarse3.20.76 × 109

### 3.2 Mesh Resolution.

Three steps are taken to determine that the simulation has reached the required level of fidelity. First, the choice of mesh refinement ensured that the wall normal wall units are small enough for high-fidelity CFD. Table 2 shows these values, note that z indicates the direction normal to the wall in these simulations.

Table 2

Near-wall mesh sizes for the hub and tip surfaces

Wall unitMeanMinMaxMeanMinMax
z+0.500.270.770.200.130.32
y+6.20.308.93.93.46.1
x+6.23.39.13.01.84.6
Wall unitMeanMinMaxMeanMinMax
z+0.500.270.770.200.130.32
y+6.20.308.93.93.46.1
x+6.23.39.13.01.84.6

Second, the entropy transport equation was used to determine the unresolved dissipation. Two subvolumes of data are extracted from the tip gap and an entropy budget calculation conducted in the same manner as Ref. [19]. The subvolumes are shown in Fig. 8. They run parallel to the camberline and extend over the entire height of the tip. The volume is three nodes thick in the camber-normal direction, so that a centered difference can be used to compute the gradient terms in the entropy transport equation and evaluation of volume and surface integral terms are possible.

Fig. 8
Fig. 8
Close modal

Discretization and filter errors add entropy to the flow (ɛN). As long as this remaining amount of entropy is small relative to the entropy generated by the flow, it is assumed that the simulation is accurately resolving the entire range of scales.

The entropy budgets were computed for two levels of mesh refinement. A mesh of 1.8 billion points was compared to a case with 0.76 billion points. The coarser mesh had 25% fewer points in each direction compared to the fine mesh. Figure 9 shows the comparison of the entropy budgets for the coarse and fine meshes. In the coarse mesh, the size of the numerical dissipation term is 5% of the size of the viscous dissipation. In the fine mesh budgets, it is less than 1% and is not visible on the same scale, in this case, it was concluded that all scales in the flow were adequately captured.

Fig. 9
Fig. 9
Close modal

Finally, capturing the full range of timescales in the flow is confirmed by frequency spectra computed from probes taken at the highest sample rate of the simulation. Figure 10 shows the power spectral density of probes located at 50% gap height on a slice normal to the camberline at 33% axial chord. This slice location contains mechanisms responsible for creating unsteady flow in the domain. The figure shows the development of the flow from the pressure-side edge to the suction-side edge, showing a convergence towards the Kolmogorov −5/3 cascade at the exit of the gap. At the exit of the gap, the spectrum shows over two orders of magnitude variation in power across the inertial subrange (f = 40−600 kHz), and the rapid drop in energy in the dissipation range for f > 600 kHz.

Fig. 10
Fig. 10
Close modal

### 3.3 Data Processing.

Data are collected from the stator simulations for 2.1 passage throughflows after the transients have been flushed through the domain. This is equivalent to 40 tip throughflows. This is sufficient time for turbulent statistics of the tip flow to converge. These simulations used 5500 cores on ARCHER and ARCHER2 taking 50 days of wall time to gather statistics. Simulation time-step is set using the CFL condition equal to 1.

## 4 Time-Averaged Flow Field

Figure 11 shows the variation in stagnation pressure loss and tip mass flowrate with tip gap. The exchange rate (ΔYPo/τ) indicated by the present results is 1.25% (where YPo is given in Eq. (1)). This is between quoted literature values for exchange rate of 1% [20] and 1.5% [21]. The mass flow through the gap increases as a proportion of total mass flow by 6% from the small to the large gap
$YPo=Po2−Po1Po1−P1$
(1)
Fig. 11
Fig. 11
Close modal

Figure 12 shows the time-averaged radial velocity contours on radially constant cuts at 50% gap height. The region of negative radial velocity on the pressure side of the blade indicates the fluid entering the gap. After the flow exits the gap, the signature of the tip-leakage vortex is indicated by high positive and negative radial velocity along a line crossing the passage (labeled “TLV”). It is important to note that this phenomenon appears only in the time average of the flow solution. In the next section, instantaneous snapshots show that it is in fact the fine-scale unsteady flow that dominates the structure.

Fig. 12
Fig. 12
Close modal

Gap size also influences the flow within the gap. Figure 13 shows the difference in shear stress in the camber-normal direction on the blade tip surface. The region of separation is indicated by the region of positive shear stress, due to flow reversal in the separation. The re-attachment of the bubble is indicated by the region where the shear stress is zero. The figure shows a shift in the re-attachment position and lengthening of the bubble as the gap height is increased. The flow in the tip is observed to change slowly for much of the chord. This quasi-three-dimensional region is estimated and labeled in the figure. These estimates are verified later.

Fig. 13
Fig. 13
Close modal

Figure 14 shows the separation height as a proportion of the tip gap height over the blade for both the large and small gap cases. With this calculation, it can be seen that the data collapse for both gap sizes where 0.2 < Cax < 0.8. This means that the separation height is only a function of gap height. The length of the separation l (measured in the camber-normal direction) is always less than the full gap width, implying re-attachment of the tip separation. This is surprising for the large gap as the gap aspect ratio is smaller than the re-attachment limit of w/τ = 2.5 set out for re-attachment by Storer and Cumpsty [3].

Fig. 14
Fig. 14
Close modal

Figure 14 suggests a linear scaling of the separation height within the gap from 0.2 < Cax < 0.8, which indicates self-similarity of the flow in this region. Figure 15 shows the angle α = tan−1 (Vnorm/Vtang) of the flow along the camberline of the blade S. For regions where ∂α/∂S < 1, the flow is parallel and changes only gradually in the direction of the camberline. The region between these limits is defined therefore as a Q3D region of flow. This is larger for the small tip gap stretching from 0.27 to 0.9 whereas the fraction of true chord where the flow is Q3D for the large gap is between 0.33 and 0.83.

Fig. 15
Fig. 15
Close modal

## 5 Unsteady Flow Structure

Figure 16 shows instantaneous snapshots of the flow on a slice at 50% of the gap height for both clearances tested. In both cases, the flow in the gap is unsteady; shedding of the separation from the pressure side corner generates fine-scale structures. These structures continue into the passage where they interact with the main passage flow, including both the endwall boundary layer and leakage flow from further upstream on the blade tip. In the case of the large gap (b), the structures shed from the separation at the gap inlet are increased. It is also interesting to note the unsteady perturbations entering the gap from the pressure side due to the near-wall turbulence on the endwall; these perturbations are elongated due to the acceleration of the flow into the gap.

Fig. 16
Fig. 16
Close modal

Figure 17 shows axial vorticity on camber-normal planes at 33% axial chord. The structures on the edge of the separation highlighted by this plot of axial vorticity are responsible for the structures highlighted by radial velocity in Fig. 16. In this view, the taller separation bubble in the large gap case sheds the larger structures.

Fig. 17
Fig. 17
Close modal

Figure 17 also shows a second mechanism for the unsteady shedding of turbulent structures. As the flow exits the gap on the suction side, a fully turbulent shear layer is formed, and both positive and negative axial vorticity are seen, these structures are the primary mechanism responsible for mixing and loss generation.

The center of the tip-leakage vortex, where radial velocity is zero in the time-mean flow, is marked in Fig. 17 with a “+” symbol. The purpose of marking the TLV center is to show that no such vortex exists in the instantaneous flow. Instead, the flow in this region is made up of a series of unsteady structures with a range of scales that depend on the instability mechanisms both within the gap and in the flow leaving the gap.

Probes upstream of the gap and at the gap inlet are used to track the turbulent kinetic energy variation of the flow entering the gap (marked in Fig. 17). Figure 18 shows that for both gap heights the turbulent kinetic energy increases from passage to gap inlet. The turbulence levels at gap inlet are equivalent to an intensity based on the blade inlet velocity of 5.5%. It is striking that, although the flow experiences a strong acceleration into the gap, there is a rise in turbulent kinetic energy; results (not shown here) confirm that the turbulence production is indeed positive in this region, driven by the perturbations in velocity and the local strain. This is also observed in the Q3D model described later.

Fig. 18
Fig. 18
Close modal

The Reynolds number based on the gap width and hub speed is Re = 7500 (based on camber-normal velocity this is 6900). At this Reynolds number, laminar flow would normally be expected within the gap; however, Fig. 17 clearly shows a highly unsteady, turbulent flow in the gap. The results show that the separated shear layer is both highly unstable and subject to perturbations entering the gap from the endwall flow. This drives the rapid breakdown to turbulence within the gap.

Transition of the pressure side separation bubble is also affected by gap height. Normalized turbulence production $P^=Pτ/U3$ is shown in Fig. 19. The figure shows that turbulence production peaks close to the re-attachment line previously shown in Fig. 13. For the $1.6%c$ gap production peaks from 15% to 50% axial chord, whereas in the $3.2%c$ gap case the peak is delayed to 35–65% axial chord and is less intense.

Fig. 19
Fig. 19
Close modal

## 6 Development of Quasi-Three-Dimensional Tip Model

In order to investigate local tip geometry modifications in DNS, a Q3D model is developed. This is done for two reasons: First, the solver is restricted to prismatic geometry in one dimension, to investigate local tip geometries, for example, radii on the PS and SS corners, it is necessary to reorient the solution so that the geometry is prismatic in the blade chord direction. This section will show that this is a valid approach as much of the flow in the tip gap is in fact Q3D. Second, restricting the simulations to this Q3D approach vastly reduces the required computational resources and many different configurations can be explored.

The Q3D geometry is extracted from the 3D domain of the large tip gap case by taking a camber-normal slice through the blade. Figure 3 shows the shape of the new domain, now the span of the simulation is aligned with the local camberline of the blade.

The slice is taken at 33% axial chord as at this point the pressure gradient across the tip is large, while the pressure gradient along the chord of the blade is small. This allows the assumption of 2D flow with spanwise motion [22]. This position is also within the Q3D region of the blade observed in the stator simulations, as confirmed by the parallel flow in Fig. 15. On this slice the Reynolds number of the tip, based on the gap width and hub velocity, is 7500. The tip gap exit Mach number on this slice is 0.3.

### 6.1 Simulation Setup.

Figure 20 shows the boundary conditions labeled in Table 3. The spanwise boundary conditions are set to be periodic. Figure 20 also shows the mesh topology, a 10-block O-H mesh is used around the tip consisting of 100,000 nodes in a single plane. The spanwise direction is 61 nodes deep, giving a total mesh count of 6.1 million nodes per simulation.

Fig. 20
Fig. 20
Close modal
Table 3

Q3D boundary conditions

LabelNameBoundary condition
AInletPressure inlet
B,CPS top, SS topFree-shear wall
DOutletPressure outlet
EHubNo-slip wall
F,G,HPS, SS, and Tip surfaceNo-slip wall
LabelNameBoundary condition
AInletPressure inlet
B,CPS top, SS topFree-shear wall
DOutletPressure outlet
EHubNo-slip wall
F,G,HPS, SS, and Tip surfaceNo-slip wall

The meshing tool is capable of independently varying the PS and SS corner radii from 0.1% to 50% of the tip gap width, one of these cases with symmetrically rounded corners is shown in Fig. 20. Across all geometries typical values for the wall normal wall units (y+) is 0.5. The wall parallel wall units in the camber parallel (x+) and spanwise (z+) directions are typically 2.5 and 2. In order to verify mesh sensitivity, a mesh refinement factor of 1.2 in each direction was applied, creating a mesh of 100 million nodes; the resulting predicted mass flow was within 1% of the baseline mesh.

The boundary conditions and geometry are set in order to match the flow to the full 3D cantilever stator blade simulations. This is achieved in five parts.

#### 6.1.1 Inlet and Exit Conditions.

The inlet and outlet boundary conditions are used to match the Reynolds number and Mach number of the tip flow. The inlet pitch and yaw angles, relative to the blade, are extracted from the 3D stator cases and applied at the Q3D inlet. The height of the PS plenum is used to adjust the incidence of the flow onto the tip. This allows the matching of the radial velocity profiles. Figure 21 shows the profiles of velocity at the gap inlet. While the Q3D simulation has some difference in the out-of-plane component w, the agreement near the tip is very close. Crucially, a close agreement is achieved with the camber-normal velocity component (u) which controls the mass flow, and the radial component (v) which controls the incidence at the pressure-side edge.

Fig. 21
Fig. 21
Close modal

#### 6.1.2 Wall Motion.

Particular attention is paid to wall motion such that boundary layers are matched from 3D to Q3D. The wall motion of the hub is at an inclined angle to the blade chord, with components in both parallel and normal directions. A reference velocity equal to that of VS/U, at the slice location, is applied to all viscous surfaces (E, F, G, and H) to replicate the relative speed of the flow past the blade. This allows the matching of boundary layers on viscous surfaces. Figure 22 shows the boundary layer profiles for the pressure side and suction side of the blade. There is good agreement between the 3D and Q3D profiles.

Fig. 22
Fig. 22
Close modal

#### 6.1.3 Suction-Side Passage Flow.

A body force is necessary to drive the spanwise motion in the plenums. This creates the difference in velocities of the tip flow and the suction-side passage flow that is present in the stator calculations. Viscous forces are calculated on the 3D slice and used to define an average skin friction coefficient. The target average shear force on the Q3D domain is calculated from this nondimensional number. Through a force balance calculation on the Q3D domain the spanwise body force is specified. This force is applied across the whole domain and so the yaw angle at the domain inlet is adjusted to compensate. Velocity triangles are computed at the gap inlet plane and gap exit plane for both the 3D and Q3D cases. Figure 23 shows good agreement in the velocity triangles at gap inlet and exit between the 3D and Q3D cases.

Fig. 23
Fig. 23
Close modal

Suction-side plenum size was varied from 2.5 to 12 times the gap width to ascertain the effect on the tip outlet flow. This had only a minor impact on the flow within the gap, giving rise to a variation in the mass flow ratio $m˙/m˙ref$ of 1.5%. A size of 2.5 gap widths was selected.

#### 6.1.4 Tip Gap Separation.

The depth of the domain was varied between 10% and 60% of the tip gap width. A value of 33% was found to allow the breakdown to turbulence of the pressure side separation shear layer whilst remaining computationally competitive. Note that the value of mass flow through the tip varied by less than 1% across all spanwise extents studied.

#### 6.1.5 Gap Inlet Unsteadiness.

In order to match the perturbations entering the gap observed in the 3D calculations, the method of Touber [23] was used to impose unsteady fluctuations at the inlet to the Q3D domain. The perturbation amplitude and wavelength were set to match the 3D simulation. Figure 24 shows a comparison of the velocity perturbations at a slice close to the pressure side entrance to the gap at half of the gap height. The inflow unsteadiness is well matched to the 3D simulation. Importantly the turbulent kinetic energy at the gap inlet observed in the 3D simulation is also matched, as shown in Fig. 18. This shows that both the absolute levels of turbulent kinetic energy and the increase in turbulent kinetic energy from the passage to the gap inlet in the Q3D model match that of the 3D case.

Fig. 24
Fig. 24
Close modal

### 6.2 Validation of Quasi-Three-Dimensional Model.

Figure 25 shows the comparison of time-averaged camber-normal velocity between the 3D and Q3D cases. The shape in terms of both height and length of the separation bubble is accurately captured in the Q3D case. The height and length of the Q3D bubble are within 0.5% and 5%, respectively, of the 3D simulation.

Fig. 25
Fig. 25
Close modal

Differences between the stator blade and Q3D simulations can be observed in the suction-side plenum downstream of the gap (see Fig. 25). Although the local shear layer behavior at gap exit is Q3D, the later mixing of the leakage flow with the main passage flow becomes three-dimensional and the assumption of similarity breaks down.

Figure 26 shows comparisons of normalized turbulence production for the 3D and Q3D simulations. The figure confirms that the levels in both cases are well matched. Overall the Q3D simulation is able to capture the flow local to the gap, both in terms of the bubble shape and size, and in terms of the turbulent transition mechanism.

Fig. 26
Fig. 26
Close modal

## 7 Effect of Tip Corner Radii

Tip corner radii have a powerful effect on the structure of the tip flow and therefore the performance of the compressor blade row. In this section, the effects of corner radii are investigated using the validated Q3D model. Data from 12 simulations are presented: six different corner radii from 0% to 17.5% of the tip width in a 1:1 ratio on both pressure and suction sides and two different Reynolds numbers, 7500 and 11,200 based upon gap width. As discussed previously, measurements of manufactured blades feature tip corner radii in the range of 2–6 of tip width. Rubbing wear during service may increase this. Therefore, the range of corner radii tested here represents both as-manufactured and in-service tips.

Figure 27 shows how the structure of the flow varies with changing radius, there are large differences in the unsteady flow mechanisms both within the gap and as the flow exits and begins to mix with the mainstream flow on the suction side. The impact of these changes is discussed in five parts.

Fig. 27
Fig. 27
Close modal

### 7.1 Separation Bubble.

The effect of corner radius on the time-mean separation bubble shape can be seen in the velocity contours plotted in Fig. 27. As tip radius is increased from (a) to (f) the separation bubble decreases in height whilst maintaining a similar length. The magnitude of reversed flow in the bubble is also observed to decrease with the increasing tip radius and decreasing bubble height.

Figure 28 quantifies these changes, the decrease in height is broadly linear, reducing by a factor of 3, from 16% of gap height down to 5%. Separation length also begins linear but saturates at a length of 1.1 times the gap height for tip radii larger than 12.5%. These changes in separation bubble size have a significant impact for the discharge coefficient through the gap and the quantity of leakage flow.

Fig. 28
Fig. 28
Close modal

### 7.2 Leakage Mass Flow.

Inspecting the camber-normal velocity contours of Fig. 27 it can be seen that the peak normal velocity leaking through the gap is reducing as tip corner radius is increased from cases (a) to (f). However, it also significantly affects the velocity profile across the height of the gap as the flow exits on the suction side. With increasing tip radii, the velocity profile is more uniform from the tip of the blade up to the endwall surface.

To quantify the performance of these different Q3D cases it is essential to define a nondimensional metric of tip mass flowrate. In this section, it is defined as the ratio of the actual tip mass flow relative to an isentropic mass flow $m˙*=m˙/m˙ref$. The isentropic mass flow $m˙ref$ is calculated by defining an isentropic Mach number using the ratio of mass-averaged inlet total pressure and area-averaged suction-side plenum static pressure. This isentropic Mach number is used in Eq. (2) to compute the reference mass flow
$m˙ref=ApoCpToγγ−1(1+γ−12Ma2)−(1/2)((γ+1)/(γ−1))$
(2)

Figure 29 shows that mass flow increases with increasing tip radii up to $r/w<10%$. At radii where $r>10%$ the effect flattens off. In these cases although a separation bubble still exists and is shrinking with increasing radius, it has reached such a small size that it provides little restriction to the total leakage flow. Over the range studied the leakage mass flow varies by 10.4%.

Fig. 29
Fig. 29
Close modal

### 7.3 Reynolds Number Sensitivity.

The flows shown in Fig. 27 show separation, re-attachment, and breakdown to turbulence, these structures could be sensitive to Reynolds number. Due to current computational limits, these Q3D DNS cases are run at an equivalent chord-based Reynolds number of 120,000, this is a Reynolds number (based on tip width and hub velocity) of 7500. To investigate the sensitivity all cases were run at an increased tip Reynolds number of 11,200 and the results are plotted in Fig. 29.

The results show that there is very little sensitivity to Reynolds number, and the largest difference in calculated leakage mass flow is only 1.8%. Complete insensitivity at smaller tip radii is due to the initial point of separation being locked to the pressure-side edge. Sensitivity then increases at large radii as a small laminar boundary layer begins on the radius and then later separates as shown in Fig. 27 case (f). These results indicate that the reduced Reynolds number cases considered here are representative of the real tip flows within an engine.

### 7.4 Within Gap Mixing.

Figure 27 shows the time-averaged turbulence production within and just outside of the tip gap. The peak value within the gap is located on the edge of the separation bubble where the relative velocities are the greatest, marked α. As tip corner radius is increased to 17.5% of tip width this value falls by an entire order of magnitude.

Figure 30 shows the calculated dissipation in the tip decomposed into contributions from the steady strain (ϕs) and the turbulent dissipation (ɛ) as defined in Eqs. (3) and (4)
$ϕ=ϕs+ε$
(3)
$ϕs=ν∂ui¯∂xj∂ui¯∂xj,ε=ν∂ui′∂xj∂ui′∂xj¯$
(4)
Fig. 30
Fig. 30
Close modal

The variation in dissipation shows that a minimum dissipation occurs due to the combined effects of turbulent dissipation and dissipation due to the time-mean strain. As tip corner radius increases the contribution from the time-mean strain ϕs rises slowly, while the contribution from turbulence decreases.

### 7.5 Transitional Behavior.

Returning to Fig. 27 it is also possible to infer the transition mechanisms from regions of turbulence production. Two different transition mechanisms occur in the tip flow, driving the variation in turbulent dissipation. For small pressure-side radii $r<10%$ the flow undergoes transition in the re-attachment region of the separation bubble and the production peaks in locations marked by α.

This leads to pressure recovery in the gap and lower mass flow. However, as r increases the transition shifts to locations outside of the gap, marked as β. For all points, both mechanisms are present but increasing tip radius increases the dominance of the β mechanism over the α mechanism.

Figure 27 shows instantaneous snapshots of vorticity in the tip gap flow. For small r the transition occurs due to the breakdown of the tip separation shear layer, again denoted by α. For large r the flow re-attaches on the tip of the blade. It transitions to turbulence just as the flow is exiting on the rounded suction surface corner. In these cases, turbulence production ramps up outside of the gap itself and peak production moves further away from the suction surface of the blade, marked β. Larger structures can also be observed in the instantaneous snapshots of vorticity plotted for cases (e) and (f),

### 7.6 Summary.

This section has shown two unsteady flow mechanisms, mixing within the gap and mixing as the flow exits and joins the mainstream. Varying the corner radius significantly alters the balance between the two. Sharp corners separate immediately and have relatively tall bubbles that breakdown to turbulence and mix within the gap, this was shown schematically in Fig. 1. Rounded tip corners have a low separation that begins on the flat part of the tip, this forms a more stable shear layer that breaks down just as the flow exits the gap, this is now shown in Fig. 31. In these cases, the majority of mixing now occurs within the passage.

Fig. 31
Fig. 31
Close modal

Only with these DNS cases, it was possible to get to the truth of the flow and the real unsteady mechanisms that dominate the flow. Under-resolved simulations and the nonsmooth nature of this design space are likely to have misled our understanding for some time.

## 8 Conclusion

Two direct numerical simulations are performed of a cantilevered compressor stator blade with tip clearances of $τ/C=1.6%and3.2%$, at Re = 120,000 and inlet Mach = 0.3. Time-averaged boundary conditions as well as inflow turbulence are matched to experimental measurements of a modern, representative compressor.

Despite the complexity of the tip-leakage flow, the flow local to the tip itself is found to behave in a quasi-three-dimensional way over a large fraction of the chord. In this region, the tip gap separation height scales only with tip gap height, in both cases studied the flow re-attaches on the tip of the blade itself. This is expected for the small gap but re-attachment in the large gap implies that the Storer and Cumpsty [3] gap aspect ratio limit for re-attachment may be too high.

The simulations show that the unsteady shedding of the pressure side separation bubble and immediate breakdown to turbulence of the shear layer at gap exit on the suction side are the most important features of the flow. Although identifiable in the time average, there is no evidence of the tip-leakage vortex in the instantaneous flow. The size of the gap affects the nature of these structures. The smaller 1.6% τ/C gap has significantly finer scale structures in the tip gap and transition of the tip flow also occurs closer to the pressure-side edge.

A Q3D DNS model is used to investigate the effects of local tip geometry as it is significantly less intensive to run than the full 3D stator cases. Boundary conditions were extracted from the full 3D cases and results are also successfully validated against them.

The effect of tip corner radius was investigated using the Q3D simulations, it is found that mass flow through the tip increases by more than 10% as corner radius was increased up to 17.5% of tip width. This is equivalent to a change in clearance height of between 0.1% and 0.3% of chord for typical clearances. The balance between the turbulence production mechanisms was also changed by corner radius. For sharp tips with corner radii less than 10% of tip width, the shear layer breakdown of the pressure side separation bubble dominates the turbulence production. Micrographs of actual tip sections indicate this to be the primary mechanism for as-manufactured tips which were measured to have corner radii of typically 2–6% of width. In the case of rounded tips with corner radii larger than 10% of tip width, the shedding of the tip surface boundary layer as it mixes with the main passage flow is more important as dissipation within the gap itself drops by half. The results suggest that in-service deterioration is likely to significantly alter the flow mechanisms within the gap.

The flow through compressor tip gaps and the subsequent mixing outside is driven by multiple unsteady turbulence producing mechanisms. This region is difficult to access experimentally and yet is critical for overall compressor performance. Micrograph data suggest that the effects of manufacturing processes can greatly alter the unsteady flow structure within the gap. As yet, it remains to be understood whether the roughness resulting from manufacturing can also alter the flow structure in the gap. This paper shows that DNS is an ideal tool to address this problem. With current computational resources, it is possible to run a small number of full 3D simulations at moderate Reynolds numbers. With the validated Q3D method local tip geometries can be explored rapidly and accurately to understand the balance of mechanisms.

## Acknowledgment

The authors would like to thank EPSRC for the computational time made available on the UK supercomputing facility ARCHER via the UK Turbulence Consortium (EP/R029326/1). The authors are also grateful to Siemens Energy and Rolls-Royce for the geometry and permission to publish. The content of this paper is copyrighted by Siemens Energy Global GmbH & Co. KG and is licensed to ASME for publication and distribution only. Any inquiries regarding permission to use the content of this paper, in whole or in part, for any purpose must be addressed to Siemens Energy Industrial Turbomachinery Limited, directly.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

• f =

frequency

•
• h =

separation height

•
• k =

reduced frequency

•
• l =

separation length (camber normal)

•
• $m˙$ =

mass flow

•
• r =

•
• s =

entropy

•
• u =

camber-normal velocity component

•
• v =

•
• w =

spanwise velocity component

•
• C =

•
• M =

Mach number

•
• P =

static pressure

•
• T =

temperature

•
• V =

velocity

•
• FS =

freestream

•
• PS =

•
• Re =

Reynolds number

•
• SS =

•
• Cp =

specific heat at constant pressure

•
• Po =

total pressure

•
• To =

total temperature

•
• α =

transition mechanism in tip gap

•
• β =

transition mechanism in suction-side plenum

•
• γ =

ratio of specific heats

•
• ɛ =

turbulent dissipation

•
• η =

span

•
• θ =

yaw angle

•
• κ =

conduction coefficient

•
• ρ =

density

•
• τ =

tip gap height

•
• ϕ =

dissipation

•
• Ω =

control volume

•
• ω =

vorticity

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