## Abstract

Stereo-PIV data are used for investigating the effect of axial casing groove (ACG) geometry on the distribution, evolution, and production rates of turbulent kinetic energy (TKE) and Reynolds stresses near a rotor tip. The ACGs delay the onset of stall by entraining the tip leakage vortex (TLV) and cause periodic changes to incidence angle. These effects are decoupled using semicircular, U-shaped, and S-shaped grooves that have similar inlets, but different outflow directions. Most TKE distribution trends can be explained by the local turbulence production rates, elucidating the different mechanisms involved and providing a unique database for turbulence modeling. Interaction of the tip flow with the ACGs modifies the highly anisotropic and inhomogeneous passage turbulence. In all cases, the TKE is high in the TLV center and shear layer connecting the TLV to the rotor tip. At prestall flowrate, TLV entrainment reduces the passage turbulence level, but introduces elevated turbulence in the corner vortex formed at the downstream corner of grooves, and in shear layers developing at the exit from grooves. The location of peaks and the dominant components vary among grooves. Near the best efficiency point, interactions of the TLV with the circumferentially negative outflow from the U and semicircular ACGs generate high turbulence levels, which extend deep into the passage. In contrast, interactions with S grooves are limited, resulting in a substantially lower turbulence level. Accordingly, the S groove maintains the untreated endwall efficiency, while the U and semicircular grooves reduce the peak efficiency.

## 1 Introduction

Most of numerical simulations of turbomachinery flows are based on solving Reynolds-averaged Navier–Stokes (RANS) equations with a plethora of closure models to estimate the Reynolds stresses [14]. However, the complex flow structures and rapidly changing strain rates in turbomachines, which cause nonequilibrium conditions, introduce unique challenges for RANS modeling. Consequently, the inhomogeneous and anisotropic turbulence [5,6] displays anomalies [7,8] and shows poor correlation between the magnitude and even the sign of the mean strain rate field and that of the Reynolds stresses [911]. Applications of large-eddy simulations, which simulate the large/resolved scales of the turbulence, have shown promising results [12,13], but are inherently more computationally intensive and still require modeling of subgrid stresses. Hence, RANS will continue to be a primary tool for characterizing turbomachinery flows in the foreseeable future. High-resolution experimental data on the flow structure and turbulence, which are difficult to obtain owing to the complex geometry of turbomachines, are essential for validating the RANS predictions and for understanding causes and effects when results do not agree [14,15]. The refractive index-matched facility at Johns Hopkins University (JHU) has been constructed to address this challenge by allowing unobstructed access to optical sensors anywhere within the machine. Results of measurements performed in this facility have already been used for characterizing the flow structure [1618] and correlating it with the Reynolds and subgrid stresses in several turbomachines [911,19].

Fig. 1
Fig. 1
Close modal

Another recent study [11] examines the impact of the semicircular ACGs on the production and evolution of turbulence in the passage. The present article broadens this scope of this analysis substantially by comparing the structure of turbulence in the tip region for the three aforementioned grooves at prestall and BEP conditions. The discussion involves the distribution and evolution of turbulent kinetic energy (TKE) and individual normal Reynolds stress components contributing to it. Flow mechanisms causing the drastic differences in the anisotropic structure of turbulence in the passage and within the grooves are elucidated by examining the corresponding primary contributors to turbulence production rates. For example, circumferential flow contraction caused by the outflow from the grooves plays a major role in generating circumferential velocity fluctuations for the cases with backward outflow, i.e., the U and semicircular grooves. In contrast, shear production is a primary contributor to generation of circumferential velocity fluctuations within the S groove. Also, high TKE production caused by undesirable secondary flows generated by groove–passage interactions at the BEP for the U groove is likely contributors to the degradation in performance. The experimental setup is described briefly in the following section. The results presented in Sec. 3 provide sample distributions of the mean flow structure, followed by detailed data comparing the turbulence and its production in the passage. The accompanying discussion elucidates the mechanisms involved, leading to the conclusions presented in Sec. 4.

## 2 Experimental Setup

The Stereo-PIV measurements detailing the flow in the tip region have been conducted at the JHU refractive index-matched facility. Details about this facility have been discussed in several articles [27,28,3034]. The experiments are conducted on a one and a half stage compressor (Fig. 1(a)) whose blades are designed based on the first one and a half stages of the low-speed axial compressor at NASA Glenn Research Center [35]. Relevant scales are provided in Table 1.

Table 1

Geometric parameters and flow parameters

 Casing diameter (D) (mm) 457.2 Hub radius (rhub) (mm) 182.9 Rotor passage height (L) (mm) 45.7 Rotor diameter (DR) (mm) 453.6 Rotor blade chord (c) (mm) 102.6 Rotor blade span (H) (mm) 43.9 Rotor blade stagger angle (γ) (deg) 58.6 Rotor blade axial chord (cA) (mm) 53.5 Measured tip clearance (h) (mm) 1.8 (0.0175c or 0.041H) Axial casing groove diameter (mm) 34.8 Groove skew angle (deg) 45 Total number of grooves 60 Shaft speed (Ω) (rad/s) (rpm) 50.27 (480) Rotor blade tip speed (UT) (m/s) 11.47 Reynolds number (UTc/ν)) 1.07 × 106
 Casing diameter (D) (mm) 457.2 Hub radius (rhub) (mm) 182.9 Rotor passage height (L) (mm) 45.7 Rotor diameter (DR) (mm) 453.6 Rotor blade chord (c) (mm) 102.6 Rotor blade span (H) (mm) 43.9 Rotor blade stagger angle (γ) (deg) 58.6 Rotor blade axial chord (cA) (mm) 53.5 Measured tip clearance (h) (mm) 1.8 (0.0175c or 0.041H) Axial casing groove diameter (mm) 34.8 Groove skew angle (deg) 45 Total number of grooves 60 Shaft speed (Ω) (rad/s) (rpm) 50.27 (480) Rotor blade tip speed (UT) (m/s) 11.47 Reynolds number (UTc/ν)) 1.07 × 106

The blades and the casing are made with transparent acrylic having the same refractive index as the working fluid, a concentrated aqueous solution of NaI, enabling unobstructed optical measurements. The grooves sketched in Figs. 1(b)1(e) are machined into transparent acrylic rings that surround the machine. The axial width of the three grooves and their location relative to the rotor leading edge are the same for all cases. The rotor leading edge overlaps with 33% of the groove axial width measured from its downstream end. All the grooves are uniformly distributed along the circumference, with each rotor passage having four grooves. Stereo-PIV measurements are performed at different meridional and radial planes to completely elucidate the groove–passage flow interactions, including the space within the groove. In this article, the discussion is restricted to the structure of turbulence in the meridional plane, denoted as M1 in Fig. 1. The experimental setup showing the optical setup along with the procedures for calibration and data processing has been discussed in detail in Refs. [27,28,32]. The field of view of 44.57 × 45.13 mm covers the entire axial extent of the casing grooves and covers the outer 25% of the rotor passage, including the tip leakage flow, tip vortex, as well as the inflow and outflow from the grooves with a resolution of 0.14 mm. Data have been acquired at 14 different rotor orientations relative to the groove, covering an entire blade passage. The timing for data acquisition is controlled by varying the delay between the image acquisition and the rotor shaft encoder. Each orientation/phase is defined by the chord fraction (s/c), where the blade chord intersects with the M1 plane, with s/c = 0 corresponding to the rotor leading edge and at s/c = 1 corresponding to the trailing edge. The turbulence statistics are calculated for four orientations, three of which corresponding to the mid-chord located near the downstream edge of the groove (s/c = 0.33, 0.44, and 0.55), and the fourth, with the blade located away from the M1 plane (s/c = −0.11). For each orientation, the ensemble-averaged flow and turbulence statistics are calculated from 1000 independent instantaneous samples for the U and S grooves and 2500 realizations for the semicircular grooves. These numbers are required to obtain converged turbulence statistics. As quantified in Ref. [34], for interrogation windows containing at least four to five particles, the uncertainty in the instantaneous displacement between exposures is about 0.1 pixel, corresponding to 0.4–0.8% of the tip speed. Ensemble averaging reduces the uncertainty in the mean velocity by at least an order of magnitude. The data are presented in a cylindrical coordinate system (r, θ, z). The origin of this system is located at the center of the machine, and θ = 0 coincides with the M1 planes of the three grooves. The corresponding instantaneous radial, circumferential, and axial velocity components are denoted as ur, uθ, and uz, respectively. The ensembled averaged velocity components are denoted as Ui, and the fluctuating component are denoted as ui′ = ui− Ui. The Reynolds stresses are denoted as $⟨u′iu′j⟩$, and the TKE as $k=0.5×(⟨u′ru′r⟩+⟨u′qu′q⟩+⟨uz′uz′⟩)$. The Reynolds stresses and TKE are normalized by UT2, and their production rates, which are specified later, are normalized by the ΩUT2.

Turbulence data are provided for two flowrates, φ = 0.25 and φ = 0.38, where the flow coefficient φ is defined as VZ/UT, VZ being the spatially averaged axial velocity in the rotor passage calculated by dividing the flowrate with the through area. As shown and discussed in Refs. [27,28,32], φ = 0.25 corresponds to the prestall flowrate of the untreated endwall, and φ = 0.38 corresponds to the BEP of the untreated machine.

## 3 Results and Discussion

This section provides a brief review of some of the main/relevant flow features followed by detailed data on the turbulence in the tip region under the influence of the different casing grooves. The trends are compared to those of the untreated endwall. Detailed descriptions of the mean flow features can be found in several recent papers [27,28,32] along with the performance and efficiency curves for this machine.

### 3.1 Turbulence at φ = 0.25 (Prestall Flowrate for Untreated Endwall).

Figure 2 compares several important flow structures observed at the low flowrate with the grooves to those occurring near the untreated endwall. In the latter case (Fig. 2(a)), the flow from the blade pressure side (PS) leaks across the tip gap and rolls up to form the TLV in the SS of the blade. The TLV is linked to the SS tip corner by a shear layer located at the interface between the leakage flow and the main passage flow. Upon introduction of the semicircular casing grooves (Fig. 2(b)), part of the flow from the blade pressure side along with the TLV, which has positive vorticity, is entrained into the grooves. Separation of the leakage flow occurring at the downstream corner of the groove entrains negative vorticity away from the endwall, creating a counter rotating “corner vortex.” In the displayed phase, the TLV is located near the entrance to the U and S grooves with a corner vortex forming to the right of it (Figs. 2(c) and 2(d), respectively), but in subsequent phases, the TLV migrates completely inward [32]. In all cases, entrainment of at least part of the TLV into the groove reduces the strength of the remaining part. Furthermore, the flow entrained into the grooves is reinjected into the passage from their upstream end and generates a shear layer of negative vorticity, which is oriented radially inward. In the shown samples, this shear layer is particularly visible for the S groove (Fig. 2(d)). By design, the outflows from the semicircular and U grooves are directed in the negative circumferential direction and that from the S grooves in the positive direction. These outflows cause periodic variation of flow angle near the leading edge of the blade. The combined effect of entraining the TLV and backflow vortices as well as the periodic variations in flow angle appears to be primary contributors to the delay in stall.

Fig. 2
Fig. 2
Close modal

#### 3.1.1 Turbulent Kinetic Energy Distribution.

Figure 3 compares the evolution of the turbulent kinetic energy for the untreated endwall (left column—Figs. 3(a), 3(e), and 3(i)) to those of the grooved casings as the blade clears the groove. For the smooth endwall, the turbulent area and peak magnitude of TKE increase with the phase owing to local production, TLV breakup, and entrainment of turbulent leakage flow from the PS [9]. For the grooved endwalls, there are persistent TKE peaks near the TLV center, the corner vortex, and the shear layer connecting the TLV to SS tip. In general, the area of elevated TKE and its magnitude in the U and S grooves are higher than those of the semicircular groove. Furthermore, while the peaks remain localized in the semicircular groove, they broaden and become elevated over the entire inlet area as the TLV is entrained into the S and U grooves. In all cases, the TKE gradually decreases as the blade moves away from the grooves, suggesting, as will be demonstrated, that blade–groove interactions play a major role in the turbulence generation. This decay is in contrast with the increase in the turbulence for the smooth endwall case. However, the decay rates are different, appearing to be fastest for the semicircular groove and quite slow for the S groove. Another broad area of elevated TKE can be seen near the exit (upstream end) from the grooves, especially in the shear layer developing as the outflow interact with the passage flow.

Fig. 3
Fig. 3
Close modal

#### 3.1.2 Distribution of Normal Reynolds Stresses and Associated Dominant Production Terms.

Figure 4 shows the distribution of normal Reynolds stresses contributing to the TKE at s/c = 0.44, with the radial component in the top row, the circumferential component in the middle row, and the axial component in the bottom row. The inherent anisotropy and the spatial inhomogeneity of the distribution of turbulence are highlighted by the differences in the magnitude and the location of normal Reynolds stress peaks for all the cases. The locations of peaks and the relative significance of the components vary among the cases. For the smooth endwall, $⟨u′ru′r⟩$ is the dominant contributor to the TKE near the TLV center, whereas $⟨u′zu′z⟩$ peaks in the shear layer separating the passage flow from the leakage flow. The distribution of $⟨u′θu′θ⟩$ shows a peak in the TLV center and a broad area with the elevated level in the leakage flow. For the semicircular grooves, $⟨u′zu′z⟩$ is by far the dominant contributor, with the main peak almost coinciding with the TLV center but extending slightly upstream of it. Elevated $⟨u′ru′r⟩$ can be seen in the corner vortex but not near the TLV center, and $⟨u′θu′θ⟩$ has several low magnitude peaks. In contrast, for both the U and S grooves, $⟨u′ru′r⟩$ is the largest component, but its peaks are located in different places. For the U groove, it is centered in the corner vortex and extends to the TLV at a lower magnitude, while for the S groove, a broad peak covers the cortex vortex, TLV, and space between them. The $⟨u′θu′θ⟩$ distribution also peaks in different places, namely, below the TLV for the U groove and above the TLV inside the S groove. Finally, for both, $⟨u′zu′z⟩$ is elevated high in the TLV and shear layer connecting it to the SS corner.

Fig. 4
Fig. 4
Close modal
To understand the mechanism causing differences in the distributions of normal Reynolds stresses and TKE, one has to look at their local production rates. They typically involve products of different Reynolds stress components and the mean velocity gradient. By identifying the dominant terms, one could determine which features of the ensemble-averaged flow structure contribute most significantly to the turbulence generation. The TKE production rate is expressed as follows [1]:
$P=0.5*(Prr+Pθθ+Pzz)$
(1)
where Prr, Pθθ, and Pzz are the production rates of the radial, circumferential, and axial normal Reynolds stress components, respectively. Their complete formulations are given as follows:
$Prr=−2[⟨ur′uz′⟩∂Ur∂z+⟨ur′2⟩∂Ur∂r−2⟨ur′uθ′⟩r∂Ur∂θ+⟨ur′uθ′⟩∂Urr∂θ]$
(2)
$Pθθ=−2[⟨uz′uθ′⟩∂Uθ∂z+⟨ur′uθ′⟩∂Uθ∂r+⟨uθ′2⟩r(∂Uθ∂θ+Ur)+⟨ur′uθ′⟩r(Uθ)]$
(3)
$Pzz=−2[⟨uz′2⟩∂Uz∂z+⟨ur′uz′⟩∂Uz∂r+⟨uz′uθ′⟩∂Uzr∂θ]$
(4)

The planar distributions of all velocity components enable us to calculate most, but not all of the terms in these equations. The only missing terms are those involving circumferential gradient of Ur and Uz. The circumferential gradients of Uθ can be calculated from the in-plane gradients using the continuity equation. While it is not possible to show the distributions of all these terms in a single paper, several of the dominant ones have been selected to highlight specific trends. The selected production terms for the normal radial, circumferential, and axial stress components are presented in Figs. 5, 6, and 7, respectively. It should be noted that presenting a limited number of terms does not provide a complete picture, and that the other terms also affect the distributions. Furthermore, one has to account for the advection by mean flow, turbulent transport, dissipation, and pressure on the local distribution of Reynolds stresses [9,10]. Yet, some of the observed differences in turbulence within the grooves can be readily explained by examining the dominant production terms.

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

The radial contraction term, namely, $−2⟨ur′2⟩(∂Ur/∂r)$, is the dominant contributor to production of $⟨u′ru′r⟩$ for all the grooves (Fig. 5). For the semicircular groove, entrainment of flow into the groove causes radial extension (∂Ur/∂r > 0), resulting in the negative contribution to production near the TLV center. The corner vortex on the other hand undergoes radial contraction resulting in the high positive production rate. In the U groove, the radial contraction is very high in the corner vortex and to the left of it, and in the S groove, it is high in the corner vortex, the top-right quadrant of TLV, and in the region between them. For the latter two cases, the high radial contraction occurs as the flow entering the groove is slowed down radially along the ramps and once it reaches the flat top of the grooves (see Fig. 2). This contraction does not occur in the semicircular groove, where there is a continuous/smooth path between the inlet and the outlet. The higher outflow from the upstream end of the S groove in this plane (Fig. 2(d) [32]) also causes an elevated radial contraction term as this outflow impinges and is slowed down by the passage flow (Fig. 5(c)). The magnitudes involved are not as high as those occurring at the entrance. A comparison to the distributions of $⟨u′ru′r⟩$ (Fig. 4) clearly shows the correspondence of the high production to the regions with peak radial velocity fluctuations in the three grooves.

The circumferential contraction, namely, $−2(uθ′2/r)(∂Uθ/∂θ)$, is the dominant contributor to $⟨u′θu′θ⟩$ for the semicircular and U grooves. In both cases, injection of the outflow from the grooves in the negative circumferential direction, by design, causes flow contraction in the tip region of the passage, hence elevated production of $⟨u′θu′θ⟩$. The circumferential flow is also contracted within the skewed semicircular groove, resulting in elevated production there as well. In contrast, circumferential extension of the flow within the U groove results in negative production there. At the upstream end of the groove, the elevated production of $⟨u′θu′θ⟩$ can also be attributed to circumferential contraction resulting from the impingement of the circumferentially negative outflow on the positive passage flow. As for the S groove, the outflow in the positive circumferential direction results in extremely mild contraction in the passage, and hence, this term has little influence on the turbulence production. Instead, within the groove, very high shear production, $−2⟨u′ru′θ⟩(∂Uθ/∂r)$, is the dominant contributor due to the high radial gradients of circumferential velocity there $(∂Uθ/∂r<0)$, owing to the separation of flow at the corners and near the wall of the inlet ramp. The elevated $⟨u′θu′θ⟩$ inside the S groove (Fig. 4) is directly attributable to this term.

For all the grooves, both axial contraction, $−2⟨uz′2⟩(∂Uz/∂z)$, and shear production, $−2⟨u′ru′z⟩(∂Uz/∂r)$, have comparable magnitude. The shear production (not shown) is high in the shear layer connecting the TLV to the SS corner, similar to phenomena observed for the untreated endwall [9]. For the semicircular groove, it is high also in the separated shear layer leading to the corner vortex. The levels of $−2⟨uz′2⟩(∂Uz/∂z)$ are high upstream of the TLV for the semicircular grooves owing to impingement of the groove outflow on the TLV. High axial contraction is also observed in the tip leakage zone of the U and S grooves as the leakage flow is slowed down axially. Elevated levels can also be observed within the U and S groves around the TLV. Finally, for all the grooves, the corner vortex undergoes axial extension, resulting in negative contribution to the turbulence production. The correspondence of the axial contraction terms to the distribution of $⟨u′zu′z⟩$ in Fig. 4 is evident.

Finally, for modeling purposes, it would be of interest to compare the structure of turbulence anisotropy among the different cases. A convenient method is to compare the anisotropy invariant maps [36,37], where the values at each point are evaluated by calculating the eigenvalues and the invariants of the anisotropy tensor (bij = τij/τkkδij/3, where τij is the Reynolds stress tensor) at the corresponding points. The spatial distribution of the anisotropy is determined by their respective position in the “Lumley triangle” [3638]. Following Ref. [39], the vertices of the equilateral triangle (Fig. 8), which represents the number of dominant components of Reynolds stress tensor eigenvalues, are color coded with red, green, and blue. Based on the relative magnitude of the Reynolds stress tensor eigenvalues, this color map shows whether the local turbulence is dominated by one (X1c, red), two (X2c), or three (X3c) components. Results for all the grooves and smooth endwall are presented in Fig. 8. As is evident, while the magnitudes and distributions vary, one can still find some similarities in the anisotropy structure for all cases. The turbulence, dominated by the $⟨u′zu′z⟩$ term, is nearly one dimensional in the shear layer connecting the TLV to the SS tip for all the cases. The outflow from the grooves at the upstream end also generates near one-dimensional turbulence, although different components of the TKE dominate for each groove (Fig. 4). As discussed earlier, the orientation of the mean outflow influences the local production rates and determines the dominant component of the TKE. For the semicircular grooves, $⟨u′zu′z⟩$ is the largest component. In contrast, high circumferential contraction at the exit from the U grooves makes $⟨u′θu′θ⟩$ the largest component, and high radial contraction at the exit from S grooves makes $⟨u′ru′r⟩$ the largest. One-dimensional turbulence is also observed at the entrance to the U and S grooves owing to the aforementioned strong radial contraction and resulting high $⟨u′ru′r⟩$ there. Two-dimensional turbulence dominated by the circumferential and radial components of the normal Reynolds stress can be seen under the TLV for the smooth endwall, as well as the semicircular and U grooves. Finally, there are also broad areas with nearly isotropic turbulence in parts of all the grooves.

Fig. 8
Fig. 8
Close modal

### 3.2 Turbulence at φ = 0.38 (Best Efficiency Point of Untreated Endwall).

The flow structures and associated turbulence for all the grooves at the BEP are elucidated in the current section. Reference corresponding data for a smooth endwall close to the BEP are provided in Refs. [911]. Figure 9 shows samples of the mean flow structure in the tip region as the blade clears the grooves. As is evident, the interactions of the passage flow with the grooves are considerably milder than those at the low flowrate, and there is no evidence of entrainment of TLV by any of the grooves. A small corner vortex entraining negative vorticity away from the endwall can still be detected at the downstream end of the grooves. At a later stage [11], this structure is entrained by the TLV into the passage. At the upstream end, the semicircular groove still has a mild outflow in the negative circumferential direction, which is separated from the passage flow by a slightly inclined shear layer, extending up to 10% of the blade passage into the tip region. The upstream ends of the U and S groove have little interaction with passage flow; hence, the shear layers are nearly horizontal. However, flow does jet put of the U groove from the downstream end to the left of the TLV [31]. This jet hits the TLV and scrambles parts of it, resulting in a relatively large region of elevated vorticity. In contrast, for the S groove, the groove–passage interactions are very limited, resulting in the formation of a nearly horizontal shear layer along most of the downstream end of the groove as well. The size of the TLV is also the smallest of the three cases. These trends are consistent with the fact that the S groove does not degrade the machine performance at high flowrates.

Fig. 9
Fig. 9
Close modal

#### 3.2.1 Turbulent Kinetic Energy Distribution.

Figure 10 shows the evolution of the TKE as the blade clears the groove. As a general observation, the turbulence level everywhere is much lower than that at low flowrate (note the difference in color scales). At s/c = 0.33 (top row), when the blade partially overlaps with the groove, the TKE is high in the TLV, which in the semicircular and S groove just begins to roll up, and in the corner vortex. The locations of these peaks are different owing to variations in the rate of the TLV development. Several other milder peaks as well as broad areas with elevated turbulence can be seen distributed near the entrance/downstream part of the U and S grooves as well as above the blade tip. As the TLV grows and begin to entrain the corner vortex at later phases (second and third rows of Fig. 10), the area with elevated turbulence increases for all the grooves, but especially for the U groove. It encompasses the TLV and its vicinity, the corner vortex, and the shear connecting the TLV to the SS corner. The radial extent of these highly turbulent regions is the shallowest for the S groove and deepest for the U groove. The bottom boundaries of these regions roughly define the extent of interaction between the outflow from the groove with the passage flow. The observed trends highlight the limited outflow from the S groove, and the impact of the flow jetting out of the (downstream end of the) U groove at high flowrates.

Fig. 10
Fig. 10
Close modal

Turbulent layers with lower magnitude than those around the blade tip also extend from the upstream end of all the grooves. These layers are nearly horizontal and relatively thin for the U and S grooves but extends deeper into the passage for the semicircular groove. These layers coincide with the elevated vorticity layers at the interface between the passage flow and the grooves (Fig. 9). Their shapes and magnitude of turbulence in them do not change substantially as the blade clears the groove.

#### 3.2.2 Distribution of Normal Reynolds Stresses Near the Blades.

Figure 11 shows the distribution of normal Reynolds stresses at s/c = 0.44 for all the grooves. The spatial inhomogeneity and anisotropy persist, similar to the observations at low flowrates. For both semicircular grooves and S grooves, $⟨u′zu′z⟩$ is the dominant contributor, whereas $⟨u′ru′r⟩$ is the largest for the U groove. In all cases, $⟨u′θu′θ⟩$ is the smallest component. For the semicircular grooves (left column), the peaks are centered around the tip and corner vortex. All components contribute to the turbulence in the shear layer extending from the upstream/exit corner of the groove, but $⟨u′zu′z⟩$ is still the largest. The production rate of this component (not shown) is affected both by axial contraction $−2⟨uz′2⟩(∂Uz/∂z)$ just upstream of the TLV and corner vortex where the leakage flow meets the passage flow, and shear production $−2⟨u′ru′z⟩(∂Uz/∂r)$, both in the vicinity of the TLV and in the long shear layer extending from the upstream end. For the U grooves, in addition to the peaks in the vicinity of the TLV and corner vortex, $⟨u′ru′r⟩$ and $⟨u′zu′z⟩$ are elevated in the regions, where flow jets out of this groove deep into the passage. The production rate of $⟨u′ru′r⟩$ is mostly affected by radial contraction, especially near the TLV, and the production of $⟨u′zu′z⟩$ is dominated by $−2⟨u′ru′z⟩(∂Uz/∂r)$ (not shown). All the normal stresses contribute to a mild horizontal shear layer extending from the upstream corner. For the S groove, $⟨u′zu′z⟩$ is the largest component in the relatively mild shear layer extending from the upstream corner and along the interface between the downstream side of this groove and the passage. The latter is associated with interaction of intermittent outflow from this groove with the passage flow. Shear production $−2⟨u′ru′z⟩(∂Uz/∂r)$ is the dominant contributor.

Fig. 11
Fig. 11
Close modal

#### 3.2.3 Turbulence When the Blade Is Away From the Groove.

The turbulence generated by groove–passage interactions changes significantly when the blade is located far from the groove. Owing to their very different interactions of the U and S grooves with the passage flow, the discussion focuses on the differences between the turbulence generated by them. Data for the semicircular groove can be found in Refs. [11,28]. To elucidate the observed trends, Fig. 12 shows the distribution of the radial and circumferential velocity components in the tip region at φ = 0.38, when the flow is not dominated locally by the tip flow for the U and S grooves. For the S grooves, the velocity distributions demonstrate that there are very limited interactions with the passage flow, and that the flow in the groove behaves like a cavity flow. In contrast, there is a strong outflow (Ur < 0) with Uθ < 0 from the downstream end of the U groove, which disrupts the passage flow, and which increases the turbulence in the tip region.

Fig. 12
Fig. 12
Close modal

Figure 13 compares the normal Reynolds stress components of tip flow in the U and the S grooves when the blade tip is located away from these grooves. The magnitudes of all the components are much higher and extended deeper into the passage for the U grooves. In particular, $⟨u′zu′z⟩$, and to a lesser extent $⟨u′ru′r⟩$, is high along the boundary of the flow jetting out from the groove. The magnitude of $⟨u′θu′θ⟩$ peaks in the TLV, which at this phase is located mostly outside of our field of view. The main contributors to the production rates of $⟨u′zu′z⟩$ for the U groove are presented in Fig. 14. They show that axial contraction $−2⟨uz′2⟩(∂Uz/∂z)$ on both sides of the jet and shear production $−2⟨u′ru′z⟩(∂Uz/∂r)$ on the upstream side are responsible for the high $⟨u′zu′z⟩$. The contraction is caused by the jet-induced blockage to the passage flow. The shear production rate $−2⟨u′ru′z⟩(∂Uz/∂r)$ is also high in the shear layer with ∂UZ/∂r < 0 developing along the interface between the jet and the passage flow.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

## 4 Conclusions

This study compares the structure and evolution of turbulence in the tip region of an axial turbomachine under the influence of three different axial casing grooves, namely, semicircular, U-shaped, and S-shaped grooves. Differences in the distributions of TKE and normal Reynolds stresses are highlighted, and the mechanisms involved are elucidated by examining the dominant production rate terms. This comparison is performed for two flowrates corresponding to prestall and best efficiency point for the untreated end wall. The flow in the tip region is modified by the groove–passage flow interactions, which in turn influence the structure of the turbulence. All the grooves introduce high TKE in the TLV, corner vortex generated as the endwall boundary layer separates at the entrance to the groove, and in the shear layer connecting the TLV to the SS tip corner. Regions of high turbulence also develop within the grooves, but their locations vary with the groove geometry. The size of regions with elevated TKE is also different among the grooves. It is localized in the vicinity of TLV and corner vortex for semicircular grooves but extends over the entire groove inlet in the U and S grooves as they entrain the entire TLV. In contrast to the smooth endwall, where the turbulence increases as the TLV grows, the TKE levels gradually decrease as the blade clears the grooves. However, the decay rates vary, being the fastest for semicircular grooves and the slowest in S grooves. Hence, it appears that by ingesting the TLV and secondary structures associated with it, the ACGs reduce the turbulence in the tip region of the rotor passage. The outflow from the upstream end of the grooves also introduces a region of elevated turbulence, but its magnitude is lower than that associated with interactions with the blade tip. Differences in the groove geometry, e.g., the flat tops of the U and S grooves, and directions of outflow from the grooves, affect the distribution of mean strain rates, hence the local production rate of Reynolds stress components. The different terms of the local production rate are used to explain the substantial differences in the relative magnitudes of the normal Reynolds stress components for the three grooves. The inherent anisotropy and inhomogeneity of the structure of turbulence is evident for all the cases and quantified by plotting the anisotropy map. However, there are also similarities in the anisotropy map, e.g., the one-dimensional turbulence at the upstream end of the grooves as the outflow from it interact with the passage flow, although the dominant Reynolds stress components are not necessarily the same.

Near the best efficiency point, the radial extent of regions with elevated TKE mirror the extent of interaction of the grooves with the passage flow. The area of elevated TKE around the TLV and corner vortex expand as these vortices grow for all the grooves. Owing to the stronger interactions of the U groove with the passage flow, it also generates more turbulence than the other grooves, which penetrates deeper into the passage. Specific mechanisms include contraction and shear production as the flow jetting out of the groove blocks the passage flow. In contrast, the area with elevated TKE around the S grooves is the smallest, also mirroring its limited interaction with the passage at high flowrates. Accordingly, the S grooves do not degrade the performance and efficiency of the compressor in comparison to the untreated endwall at high flowrate. Conversely, as shown in Refs. [28,31,32], the U and semicircular grooves degrade the BEP performance.

## Acknowledgment

This project along with the facilities and instrumentation involved has been funded by NASA and in part by the Office of Naval Research. The authors would like to thank Chunill Hah, Michael Hathaway, and Ken Suder from NASA Glenn for valuable discussions and guidance as well as for modifying the LSAC blade geometries to match the constraints of the JHU index-matched facility. The authors would also like to express their gratitude to Yury Ronzhes who designed all the mechanical components of the test facility.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this paper are obtainable from the corresponding author upon reasonable request. The data and information that support the findings of this paper are freely available at: katz@jhu.edu. Data provided by a third party listed in Acknowledgment.

## Nomenclature

c =

h =

width of the rotor blade tip gap

k =

turbulent kinetic energy

s =

H =

L =

nominal distance from the hub to the inner casing endwall

P =

production rate of TKE, production rate of Reynolds stress

UT =

Vz =

average axial velocity in the rotor passage

r* =

uʹ =

velocity fluctuation

r, z, θ =

ur, uz, uθ =

〈·〉 =

ensemble-averaged quantity

ρ =

NaI solution density

φ =

flow coefficient

ωθ =

circumferential vorticity component

Ω =

rotor angular velocity

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