## Abstract

Challenges in turbulence modeling in the tip region of turbomachines include anisotropy, inhomogeneity, and non-equilibrium conditions, resulting in poor correlations between Reynolds stresses and the corresponding mean strain-rate components. The geometric complexity introduced by casing grooves exacerbates this problem. Taking advantage of a large database collected in the refractive index-matched liquid facility at Johns Hopkins University (JHU), this paper examines the effect of axial casing grooves on the distributions of turbulent kinetic energy (TKE), Reynolds stresses, anisotropy tensor, and TKE production rate in the tip region of an axial turbomachine. The comparisons are performed at flowrates corresponding to prestall and best efficiency points of the untreated machine. Common features include high TKE near the tip leakage vortex center and in the shear layer connecting it to the blade suction side tip corner. The turbulence is highly anisotropic and inhomogeneous, with the anisotropy tensor shifting from one-dimensional (1D) to 2D and 3D structures over small distances. With the grooves, the flow structure, hence the distribution of Reynolds stresses, becomes more complex. Additional sites with elevated turbulence include the corner vortex that develops at the entrance to the grooves, and in the flow jetting out of the grooves into the passage. Consistent with trends of the production rates of normal Reynolds stress components, the grooves increase the axial but reduce the radial velocity fluctuations as the inflow and outflow from the groove interact with the passage flow. These findings might assist the development of Reynolds stress models suitable for tip flows.

## Introduction

Computational fluid dynamics (CFD) is being widely used to simulate the flow in turbomachines [14]. Reynolds-averaged Navier–Stokes (RANS) simulations with different closure models for the Reynolds stresses are the most popular due to the low computational cost. Questions arise on the applicability of common Reynolds stress models for applications of RANS in turbomachines, which is inherently complicated with varying flow structure scales, and fast-changing strain-rate fields. Issues associated with RANS turbulence models such as the generation of negative normal stresses [5], stagnation point anomaly [6], and anisotropy of turbulence [7,8] have been discussed extensively in previous publications. A comparison between different models [9] has shown that using full Reynolds stress transport models could improve the results by introducing anisotropy to the simulation. Large-eddy simulation (LES) provides a better prediction of turbulence in turbomachines, but its applicability is limited at present by high computational cost and issues associated with modeling the sub-grid stresses. Hence, currently, RANS simulation supported by calibrations of models by experimental data [10,11] is still the first choice for practical applications. Further advances in the development of turbulence models require high-resolution experimental data, which is challenging in the complex environment of turbomachines. Our past studies in the refractive index-matched facility at Johns Hopkins University (JHU) using stereoscopic-particle image velocimetry (SPIV) have examined the flow and turbulence in the tip region of turbomachines in great detail [1214] and have addressed a series of issues in modeling them [15,16]. The prevailing non-equilibrium condition in the tip region has challenged the popular eddy-viscosity-based turbulence models, which assume isotropy of turbulence.

The applications of casing treatments to delay the onset of a stall in turbomachines [1723] further complicate the situation, introducing more challenges for popular turbulence models. The increased flow complexity involves the interaction of the tip flow with the groove, the re-orientation of the tip leakage vortex (TLV), and secondary flow structures generated by these interactions. Out past studies have examined the dynamics of the fluid–groove interactions for skewed semicircular casing grooves [24] (ACGs) at different flowrates [25,26]. Results have shown that when the blade is passing the groove, part of the TLV and the circulation carried with it is entrained into the groove at a low flowrate. Flow separation at the groove corner generates a corner vortex rotating in the opposite direction of the TLV. At a high flowrate, the TLV entrains the corner vortex into the main passage, contributing to blockage effects. The reinjected flow from upstream of the groove enhances the mixing and generates a layer with elevated turbulence.

In this study, we utilize the data set for the flow within and around an axial casing groove to examine the spatial distributions of turbulent kinetic energy, individual Reynolds stress components, and associated dominant turbulent kinetic energy (TKE) production terms at two different flowrates. The results are compared with data obtained for the same machine with a smooth endwall. This comparison reveals several differences, such as a shift of the dominant Reynolds normal stress term from the radial direction for the smooth endwall to the axial direction. This change is caused by axial contraction of the mean flow when the leakage flow enters the groove. Sample eddy-viscosity distributions calculated from individual Reynolds stress and corresponding averaged strain-rate components show spatial variations from negative to large positive numbers, confirming that assumptions of isotropy and eddy-viscosity-based Reynold stress models are inherently flawed. Furthermore, anisotropy maps show the structure of turbulence varies spatially from one dominant direction (1D) to 2D and 3D over small distances. Yet, despite the differences, the spatial distributions of anisotropy also show some similar trends for different flowrates, with or without grooves. This observation opens possibilities to exploit the anisotropy structure for the development of data-driven models.

## Experimental Setup

Detailed descriptions of the JHU refractive index-matched facility and the axial casing grooves can be found in Refs. [15,27,28] and in Refs. [25,26], respectively. Relevant geometric parameters for the compressor model and the grooves are included in Table 1. The blades of the one-and-half-stage compressor model, which is illustrated in Fig. 1(a), are based on the low-speed axial compressor facility at National Aeronautics and Space Administration (NASA) Glenn Research Center. The rotor blades and casing are made of transparent acrylic, whose refractive index matches that of the working fluid, a concentrated aqueous solution of NaI. The resulting unobstructed optical access allows the implementation of optical measurement techniques. The semicircular axial casing grooves, with diameters of 65% of the blade axial chord, are located around the rotor leading edge, with 33% of the groove overlapping with the rotor, and the rest, extending upstream of the rotor (Fig. 1(b)). The groove geometry is based on the numerical study described by Müller et al. [1]. The evenly spaced, four-per-passage grooves are skewed by 45 deg in the direction of rotor rotation (Fig. 1(c)). SPIV has been used to quantify the flow field in meridional and radial planes, but only the former is discussed in this paper.

Fig. 1
Fig. 1
Close modal
Table 1

Stage-relevant geometrical 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 tip 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−1) (RPM) 50.27 {480} Rotor blade tip speed (UT) (m s−1) 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 tip 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−1) (RPM) 50.27 {480} Rotor blade tip speed (UT) (m s−1) 11.47 Reynolds number (UTc/ν) 1.07 × 106

Previously published performance and efficiency curves are shown in Figs. 1(d) and 1(e), respectively. The measurement procedures are discussed in detail in Refs. [15,26]. The flow coefficient φ is defined by φ = VZ/UT, where is VZ is the spatially averaged axial velocity in the rotor passage (flowrate divided by the throughflow area). The static-to-static head rise coefficient is defined as ψSS = (PexitPin)/0.5ρUT2, where PexitPin represents the pressure rise across the entire machine. The efficiency of the machine is calculated from η = (PexitPin)Q/, where Q is the volumetric flowrate and T is the input torque measured at the shaft. As is evident, without casing grooves, the machine stalls at around φ = 0.25. In contrast, the stall flowrate is reduced by 40%, to about φ = 0.14, when the casing grooves are installed. However, at high flowrates, namely, φ > 0.35, the ACGs cause a slight decrease in head rise accompanied by a 2.8% loss of peak efficiency. Most of the data and discussions in this paper focused on the Reynolds stresses at φ = 0.25 and φ = 0.35, the former corresponding to prestall flowrate of the smooth endwall, and the latter representing conditions near the best efficiency point of the machine.

## Results and Discussions

The discussion begins with a brief review of the tip flow–groove interactions at two different flowrates (Figs. 2(a) and 2(c)), and a comparison with the tip flow structure of the smooth endwall case (Figs. 2(b) and 2(d)). Detailed analyses for the smooth end wall can be found in Refs. [14,15,27,28], and the <uiuj′> results for the machine with ACGs are presented in Ref. [25] for φ = 0.25 and in Ref. [26] for φ = 0.35. In the selected sample blade phase (s/c = 0.44), the blade suction side (SS) just clears the downstream end of the groove. At φ = 0.25 (Figs. 2(a) and 2(b)), both the smooth and treated endwall flows feature a tip leakage flow, a tip leakage vortex (TLV), and a shear layer connecting the TLV to the suction side (SS) tip corner. However, a secondary corner vortex with an opposite vorticity sign forms at the downstream corner of the groove, owing to flow separation as the leakage flow enters the groove. As discussed in Ref. [25], suction of the tip flow into the groove and the resulting reduction in the TLV strength is one of the main causes for stall suppression by the casing groove. The entrained fluid then circulates around and is reinjected into the main passage flow at the upstream end of the groove. This reinjected flow generates a radially oriented shear layer with negative vorticity at the upstream corner and a horizontal shear layer with positive vorticity at the transition between the groove and main passage flow. This injection also causes periodic changes to the flow angle near the blade leading edge, which is another likely contributor to the delay of stall [25]. The TLV, corner vortex, and a horizontal shear layer with positive vorticity at the transition between the (much slower) reinjected fluid and the main passage flow can also be seen at φ = 0.35 (Fig. 2(c)). However, in this case, the corner vortex and some of the fluid associated with it are entrained into the main passage by the TLV at the downstream side of the groove. As shown later, this phenomenon contributes significantly to the turbulence production rate at high flowrates. Owing to these interactions, the peak vorticity in the TLV center is much lower for the grooved case compared with that of the smooth endwall (Fig. 2(d)). In contrast, despite the effect of TLV suction on the groove at a low flowrate, the peak vorticity and the TLV circulation of the grooved endwall are still higher than those of the smooth endwall (Figs. 2(a) and 2(b)). This trend is caused by a shift in blade loading toward the leading edge, which increases the TLV strength of the grooved endwall at low s/c. However, as shown in Ref. [25], entrainment of the TLV into the groove decreases the TLV circulation rapidly further along the blade.

Fig. 2
Fig. 2
Close modal

### Turbulent Kinetic Energy Distribution.

Figure 3 compares the turbulence kinetic energy at φ = 0.25 for three different blade orientations. In Figs. 3(a) and 3(b), the laser sheet cuts the blade at s/c = 0.33 when the blade tip is partially overlapped with the downstream corner of the groove. Prominent TKE peaks can be identified at the centers of the TLV and the corner vortex as well as in the high shear region between them (Fig. 3(a)). The reinjected flow exiting from the groove at the upstream end of the groove also generates a layer of elevated turbulence, but its strength is still one order of magnitude lower than the peaks appearing in the downstream corner. As the outflow with elevated turbulence interacts with the flow induced by the tip and corner vortices, it creates a broad turbulent region that surrounds these vortices. In comparison, near the smooth endwall (Fig. 3(b)), the turbulence is high in the vortex center and in the shear layer connecting it to the SS tip corner. Furthermore, the TKE level on the pressure side (PS) of the blade is high over a broad area owing to the presence of remnant of a previous TLV and related unsteady secondary structures, including the backflow vortices [25]. This elevated turbulence propagates through the tip gap to the SS of the blade and is entrained into the flow surrounding the TLV. Because of this entrainment under prestall conditions of the smooth endwall, the TKE in the vicinity of the TLV is significantly higher than that observed at a high flowrate (Fig. 4(a)). Such turbulent remnants of a previous TLV do not exist in the PS of the grooved endwall (Fig. 3(a)), presumably since large fractions of the TLV are entrained into the ACGs, as discussed before.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

At s/c = 0.44, the blade is located just downstream of the groove (Fig. 3(c)). The TKE peaks in the vortex centers and the shear layer are lower than those in the previous phase, but the region with elevated turbulence is spread over a substantially broader area. The shear layer with (slightly) elevated turbulence between the reinjected flow and the main passage flow persists. In contrast, near the smooth endwall (Fig. 3(d)), the peak of the TKE remains very high, but the area with a high TKE level expands dramatically. The latter trend has been attributed to TLV breakup, as discussed in previous papers [12,15], but it might also be associated with persistent entrainment of high PS turbulence across the tip gap. At s/c = 0.55 (Fig. 3(e)), when the blade section is located downstream of the groove, the TKE peaks in the TLV and the corner vortex decrease further, presumably since significant parts of these vortices are entrained into the groove [25] (Fig. 3(e)). In contrast, the peak TKE and the region occupied by it still grow near the smooth endwall (Fig. 3(f)). In summary, when the blade passes by the groove, the TKE peaks are higher than those observed for the smooth endwall, due to the interaction of the tip flow with the groove. However, when the blade section is located downstream of the groove, entrainment of the tip flow into the groove, which also causes a lower turbulence level on the PS of the blade (owing to the reduced presence of previous TLV remnants there), reduces the turbulence level near the downstream corner of the groove.

For the high flowrate data shown in Fig. 4, since data for the flow downstream of the groove are also available, we have patched the two axial views together to show a more comprehensive picture of the tip region turbulence. The resulting field of view extends from upstream of the groove (z/cA = −0.45) to 80% of the blade chord (z/cA = 0.8). To highlight the turbulence in the shear layer extending from the upstream corner of the groove, the results are presented in insets with different scales (4:1 smaller). At s/c = 0.33 (Fig. 4(a)), the TKE peaks in the TLV and the corner vortex, as well as in the shear layer connecting the TLV to the blade, all trends appearing at low flowrates as well. The layer with elevated TKE at the interface between the main passage flow and the outflow from the groove is also evident. In addition, a broad region with elevated turbulence, which is associated with the TLV and secondary vortices generated by the previous blade, can be seen downstream of the blade PS. As discussed in Ref. [26], due to interactions of the TLV with the flow within the groove, particularly the corner vortex, the TLV lingers around the downstream corner of the groove instead of being convecting downstream by the mean flow. As a result, the elevated TKE region associated with the previous TLV is larger and located closer to the PS of the current blade at this early phase. In contrast, for the smooth endwall case (Fig. 4(b)), the TLV just starts to roll up and detach from the blade SS corner. Hence, the associated TKE is much lower than that of the grooved.

When the blade section is already located downstream of the groove, at s/c = 0.44 (Fig. 4(c)), the TKE peak in the vortex center increases slightly, and the areas with high TKE expands. The corner vortex begins migrating toward the passage with a lower TKE peak in its center than that in the TLV. The interaction between these two vortices increases the turbulence level in the region between them as well. In addition, the blade PS is now approaching the remnant of the previous TLV, causing a slightly higher turbulence level at the PS tip corner. Compared with the smooth endwall case shown in Fig. 4(d), the TKE peak in the vortex center is much higher, and the associated area is much broader with a casing groove. The reason for these trends can be inferred from our previous publication [26], where we show that in instantaneous realizations, both the TLV and the corner vortex consist of multiple vortex filaments and that interaction of the TLV with the corner vortex fragments the TLV. As discussed in an earlier work involving a different axial pump with a smooth endwall [29], interactions between the vortical filaments account for more than 40% of the TKE budget with the TLV. At a later phase (s/c = 0.55), the TKE peak decreases significantly for the grooved case (Fig. 4(e)), but the area occupied by it increases. In contrast, the peak TKE for the smooth endwall (Fig. 4(f)) keeps on increasing, reaching a higher level than that of the grooved endwall, but the area occupied by high TKE is still concentrated around the vortex center.

### The Reynolds Normal Stress Distributions and Turbulence Production.

This section examines and compares the distributions of normal Reynolds stress components and flow mechanisms affecting them. The phase where the blade section is located downstream of the groove (s/c = 0.44) is selected as the basis for this discussion. Figure 5 compares the three Reynolds normal stresses at φ = 0.25 for the smooth and grooved endwall cases. A detailed discussion of the turbulence for the smooth endwall is included in Ref. [15], so the present discussion focuses on the impact of the ACG. The first impression is the differences in magnitudes and spatial distributions of the stresses, clearly showing the anisotropy in the turbulence for both cases. Furthermore, the dominant contributors to the TKE peak in the TLV center are different. For the grooved endwall, the main contributor is the normal axial stress, <uzuz′> (Fig. 5(a)), while the radial normal stress, <urur′>, is the highest component for the smooth endwall (Fig. 5(d)). For the grooved endwall, <uzuz′> is also high in the corner vortex, and in the region located between the two vortices. Also, <uzuz′> is the main contributor to the elevated turbulence in the shear layer extending from the upstream corner and in the shear layers connecting the TLV to the blade SS corner. In contrast, for the smooth endwall (Fig. 5(b)), <uzuz′> is the primary contributor in the shear layer, but not near the TLV center. Furthermore, <uzuz′> has another peak at z/cA = 0.1, i.e., upstream of the TLV, in the region where the backward tip leakage flow meets the main passage flow near the endwall.

Fig. 5
Fig. 5
Close modal

The values of <urur′> near the TLV center for the ACG case (Fig. 5(c)) are much lower compared with those for the smooth endwall (Fig. 5(d)), but they are high in the corner vortex inside the groove. As for the circumferential normal stress, <uθuθ′> near the TLV center for the smooth endwall is higher than that of the ACG case. In a previous study, the high <uθuθ′> near the TLV center of the smooth endwall has been attributed to circumferential contraction and associated early phases of TLV breakup [12,15]. Within the groove, <uθuθ′> has the highest peak in the region (z/cA = 1.8 and r* = 1.08) where the corner vortex interacts with the groove, indicating the flow structure inside the groove is far from being two-dimensional [25]. As discussed later, this high stress is generated by circumferential contraction when the flow decelerates circumferentially with the groove. One final note is the source for high TKE at the PS for the smooth endwall case. As Figs. 5(b), 5(d), and 5(f) show, all the normal stress components are elevated, but they have very different spatial distributions. The main contributors are <urur′> and <uθuθ′> along the PS surface, but <urur′> is the main peak in the tip gap and near the endwall. As discussed later, most of the elevated turbulence along the PS is not generated locally. It is originated from the previous TLV and is advected by the mean flow to the vicinity of the blade.

Figure 6 compares the normal Reynolds stress components for the high flowrate. Here again, the dominant term for the grooved endwall is still <uzuz′> (Fig. 6(a)), and its magnitude is much higher than the corresponding values for the smooth endwall (Fig. 6(b)). However, the confined <urur′> peak (Fig. 6(c)) is higher than that of <uθuθ′> (Fig. 6(e)) and is comparable to the corresponding peak for the smooth endwall (Fig. 6(d)). For the latter, <urur′> is still the dominant contributor to the TKE. In both cases, the values of <uθuθ′> are lower than the other components, in contrast to the trends observed at φ = 0.25 (Figs. 5(e) and 5(f)). Finally, within the groove, both <urur′> and <uθuθ′> are much lower than the axial component in the corner vortex, also in contrast to the trends observed at the lower flowrate.

Fig. 6
Fig. 6
Close modal
To explain the spatial inhomogeneity and anisotropy of the turbulence, one has to start by examining the local turbulence production rate. The TKE production rate is P = 0.5(Pzz + Prr + Pθθ), where the individual terms corresponding to each of the normal stresses in the laboratory reference frame are as follows:
$Pzz=−2(∂Uz∂z+∂Uz∂r+1r∂Uz∂θ)$
(1)
$Prr=−2[∂Ur∂z+∂Ur∂r+(1r∂Ur∂θ−Uθr)−Uθr]$
(2)
$Pθθ=−2[∂Uθ∂z+∂Uθ∂r+(1r∂Uθ∂θ+Urr)+Uθr]$
(3)

Currently, most of the terms in Eqs. (1)(3) can be calculated from the SPIV data, except for the terms in Pzz and Prr, which involves ∂Uz/∂θ or ∂Ur/∂θ. The values of ∂Uθ/∂θ can be obtained from the in-plane gradients using the continuity equation. In a previous study involving a smooth enwall, the out-of-plane derivatives have been measured by recording data in a series of closely spaced meridional planes [15]. In the present analysis involving the ACGs, the discussion is restricted to the impact of available components. Some of the dominant terms in the TKE production rates are shown in Fig. 7 for φ = 0.25 and in Fig. 8 for φ = 0.35. The same terms for the smooth endwall are included as insets as a basis for the comparison of trends.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

Figure 7(a) shows the axial contraction/stretching term, namely, −2<u′z2>∂zUz/(ΩUT2). Around the TLV center, this term has a diagonal four-quadrant pattern, with the sign changing from negative values where the flow is stretched axially, to positive values in regions of axial contraction. Qualitatively, this pattern is similar to that for the smooth endwall (Fig. 7(a) inset), indicating that the flow mechanisms are similar. The particularly high production peak between the TLV and the corner vortex is the result of a strong axial contraction when the tip leakage flow is entrained into the groove, corresponding to the high <uzuz′> there (Fig. 5(a)). However, the region occupied by the corner vortex has negative values, indicating that the flow there is stretched axially. In Fig. 7(b), the distribution of axial shear production, −2<u′zu′r>∂rUz/(ΩUT2), shows that this term is high, as expected, in the shear layer connecting the TLV to the SS tip corner, and the groove corner to the corner vortex center. The effect of the latter is partially canceled by the negative production rate due to axial stretching (Fig. 7(a)). Accordingly, the TKE peak associated within the corner vortex is significantly lower than that involving the TLV. The shear production rates in the shear layer connecting the SS corner to the TLV are high for both the smooth and grooved endwalls (Fig. 7(b) inset).

As for the distribution of <urur′>, the trend can be explained by looking at the radial contraction/stretching production term, −2<u′ru′r>∂rUr/(ΩUT2), shown in Fig. 7(c). Entrainment of the tip leakage flow into the groove involves significant axial contraction (discussed above) and radial extension (∂rUr > 0) between the two vortices. The radial extension generates a broad area of negative production for <urur′> staring from the TLV center to the lower part of the corner vortex. Conversely, in the corner vortex, the flow is stretched axially and contracted radially (∂rUr < 0), resulting in a high contribution to the production of <u′ru′r> there. These trends explain why <u′ru′r> is small near the TLV center but high in the corner vortex (Fig. 5(c)). Note that the spatial distributions of −2<u′ru′r>∂rUr/(ΩUT2) around the TLV for the grooved and smooth (Fig. 7(c) inset) endwalls share the inclined four-quadrant pattern, but the details differ. For example, the negative regions are connected together for the grooved endwall rather than the positive ones that are connected for the smooth endwall. Finally, the differences in the circumferential contraction/stretching production term, −2<u′θu′θ>∂θUθ/rUT2), are striking (Fig. 7(d)). For the grooved endwall, the circumferential production is high around the TLV and the corner vortex centers, while it is only slightly elevated for the smooth endwall. The large positive values are associated with circumferential contraction (∂θUθ < 0), as the tip flow with high Uθ is slowed down at the entrance to the groove. In addition, the positive production region upstream of the TLV is associated with the circumferential velocity gradient between the flow with Uθ < 0 reinjected from the groove and the tip leakage flow with Uθ > 0. As a direct result of the elevated production due to circumferential contraction, <uθuθ′> is high around the TLV and the corner vortices (Fig. 5(e)). This mechanism does not exist for the smooth endwall case.

The comparisons for φ = 0.35 (Fig. 8) show a few trends that are similar to those observed in Fig. 7, including the four-quadrant pattern around the TLV for the axial (Fig. 8(a)) and radial (Fig. 8(c)) productions terms, the high shear production (−2<u′zu′r>∂rUz/(ΩUT2)) in the shear layer connecting the TLV to the SS corner (Fig. 8(b)), and the low circumferential contraction for the untreated enwall case (Fig. 8(d) inset). However, there are several differences as well. First, the positive −2<u′z2>∂zUz/(ΩUT2) region upstream of the TLV now has a higher peak value (Fig. 8(a)). As is evident from Fig. 2(c), this strong axial contraction is associated with rapid deceleration of the axial flow upstream of the TLV. Second, the axial shear production (−2<u′zu′r>∂rUz/(ΩUT2)) in the shear layer connected to the corner vortex is low (Fig. 8(b)), in contrast to that in Fig. 7(b). Third, the high circumferential contraction term (−2r−1<u′θu′θ>∂θUθ/(ΩUT2)) around the TLV and the corner vortex now disappears, with only a small peak left under the TLV. This phenomenon indicates that the circumferential deceleration of the flow exiting from the groove at a high flowrate is much milder than that occurring at φ = 0.25. However, this term becomes the dominant source of turbulence production again at a later phase (showing later).

### Turbulence at Higher Chord Fractions and Downstream of the Groove.

Figure 9 examines the evolution of flow and turbulence at φ = 0.35, when the blade section is located downstream of the groove. The ensemble-averaged circumferential vorticity distribution and in-plane velocity vectors are shown in the top row, followed by the TKE in the second row, and the normal Reynolds stress components in the third to fifth rows. At s/c = 0.76 (Fig. 9(a)), the TLV is much larger and has lower peak vorticity than that at s/c = 0.44 (Fig. 2(c)). A long shear layer with high vorticity still connects the now-fragmented TLV to the blade SS tip corner. This shear layer defines the boundary between the forward passage flow and the backward tip leakage flow. The corner vortex at the groove corner is now in the process of migrating into the passage. Due to the interactions with this corner vortex, the upstream end of the TLV is latched to the groove downstream corner instead of moving downstream with the blade. These trends persist at s/c = 0.87 (Fig. 9(b)), where the peak vorticity for both vortices decreases further.

Fig. 9
Fig. 9
Close modal

Along with the expansion of the TLV, at s/c = 0.76 (Fig. 9(c)), the area occupied by elevated TKE also expands significantly compared to that at s/c = 0.55 (Fig. 4(e)). The TKE is also broadly elevated in the corner vortex and the region of vortex interactions. In addition, The TKE is high on the PS where the remnants of a previous TLV approach the current blade. An unexpected increase in turbulence intensity occurs s/c = 0.87 (Fig. 9(d)), mostly owing to an increase in <uzuz′> (compare Figs. 9(e) and 9(f)), which is still the dominant term over a large fraction of the sample area. For both phases, the radial velocity fluctuations increase in the shear layer and under the TLV (Figs. 9(g) and 9(h) compared to those measured at s/c = 0.44 (Fig. 6(c)). The circumferential fluctuations increase significantly (Figs. 9(i) and 9(j)) in the region between the TLV and the corner vortex, making it the new dominant contributor to the TKE there. To explain some of these trends, two important production rate terms are compared in Fig. 10. As is evident, the higher <uzuz′> at s/c = 0.87 is caused by an increase of axial shear production (−2<u′zu′r>∂rUz/(ΩUT2)) shown in Figs. 10(a) and 10(b). This trend is associated with an increase in the Reynold shear stress <u′zu′r> (not shown) in the shear layer. The increase in <u′θu′θ> is caused by a particularly high production rate owing to circumferential contraction near the downstream corner of the groove. As discussed earlier, at this flowrate, the TLV, which has high circumferential velocity, entrains fluid with low circumferential velocity out from the groove, causing rapid deceleration of the flow in the region between the TLV and the corner vortex, near the corner of the groove. These phenomena demonstrate how interactions of the passage flow with the groove contribute directly to the expansion of the TLV and generation of high turbulence. These trends are likely contributors to the performance degradation of machine performance near the best efficiency point.

Fig. 10
Fig. 10
Close modal

### A Brief Discussion on Eddy-Viscosity Model and Anisotropy.

Eddy-viscosity-based Reynolds stress models are still widely used as closures in applications of RANS equations. However, the above discussion shows strong anisotropy in Reynolds stresses, which challenges the fundamental assumption of these models. Our past studies [15,16] have shown large spatial variations of eddy viscosity calculated from the measured overall production rate and/or from individual Reynolds stress terms, which is caused by the non-equilibrium condition as the turbulence cannot respond to the fast-changing strain fields. Figure 11 provides such a sample distribution of eddy viscosity at φ = 0.35, calculated from the axial normal stress, namely νT,zz = (<u′z2> − 2/3k)/Szz, and normalized by the kinematic viscosity of the fluid (ν). The deviatoric part of the normal stress (Fig. 11(a)) and Szz (Fig. 11(b)) are included in the figure for comparison. The four quadrants sharped Szz distribution around the tip leakage vortex is a common feature observed in our past studies [16], which affects the four-quadrant pattern in turbulence production rate (Figs. 7(a) and 8(a)). However, a comparison between Figs. 11(a) and 11(b) clearly shows that the Reynolds shear stress and the strain rate are spatially misaligned. As a result, the calculated νT,zz/ν (Fig. 11(c)) varies from negative to large positive numbers, showing no functional correlations between the stress and the strain rate. This trend applies not only to the vicinity of the TLV but also to a broad area within and below the groove, in part owing to the low stresses there. The large variations persist if the same calculations are repeated using the other stress and strain-rate components, but the values and spatial distributions vary, in agreement with previous results [15,16]. Interestingly, as discussed in Ref. [16] and mentioned in the Introduction, when trends for the same stress in different machines and operating conditions are compared, the results show spatial similarity.

Fig. 11
Fig. 11
Close modal

Lastly, Fig. 12 shows several anisotropy invariant maps (AIM), which is a convenient means to demonstrate the spatial distribution of the anisotropy of the deviatoric part of the Reynolds stress tensor. The analysis is based on the Lumley Triangle [30,31] and involves calculations of the invariant of the anisotropy tensor (bij = τij/τkk − δij/3, where τij is the Reynolds stress tensor) at every point. The results then color-coded using red-green-blue (RGB) colors, as suggested by Emory and Iaccarino in Ref. [32]. The reference color code is provided as an equilateral triangle (shown on the top right-hand corner in Fig. 12), with its three vertices representing the number of dominant components of the Reynolds stress tensor eigenvalues. The numbers one (x1c, red), two (x2c, green), and three (x3c, blue) indicate that the local turbulence is dominated by one, two, or three stress components, respectively, implying, e.g., that blue regions correspond to nearly isotropic turbulence. The results show substantial spatial variations in color, but also some persistent trends among the two flowrates and/or the grooved and smooth endwalls. For example, the area occupied by the shear layer connecting the TLV to the blade SS corner is predominantly red, consistent with the previously discussed prominence of <uzuz′> due to the local high shear production. A large area surrounding the TLV is green for all the cases, consistent with the previously observed presence of two dominant terms, namely, the radial and circumferential stress components. In the shear layer extending from the upstream corner of the groove and separating between the outflow from the groove and the main passage flow, the color is mainly red, consistent with the dominance of <uzuz′>. Only inside of the groove, one can observe blue regions, indicating nearly isotropic turbulence. Clearly, these AIM maps are instrumental for demonstrating and characterizing the spatial distribution of Reynolds stress anisotropy. Note, however, that they do not provide the principal directions of the eigenvectors involved. The partial similarities among different cases suggest that such maps could be used as guidance for the development of locally tuned, data-driven turbulence models that account for the turbulence anisotropy.

Fig. 12
Fig. 12
Close modal

## Conclusion

The present study utilizes a large database obtained in the JHU refractive index-matched facility for evaluating the effect of axial casing grooves on the evolution of turbulence in the tip region of an axial turbomachine. The distributions of turbulent kinetic energy, Reynolds normal stresses, dominant TKE and Reynolds stress production terms, as well as the anisotropy invariant maps and the eddy viscosity calculated from the measured data, are discussed and compared with those obtained for a smooth endwall in the same machine. The comparisons are performed at two flowrates, the lower one corresponding to prestall conditions in the untreated machine and the higher one slightly below the best efficiency point. Several trends can be identified as general conclusions: (a) interactions of the tip flow and the tip leakage vortex with the groove introduce several new phenomena with elevated turbulence. The most prominent ones are the highly turbulent corner vortex and its interaction with the TLV as well as the shear layer extending from the upstream corner of the groove at the interface between the flow injected from the groove and the main passage flow. (b) Many of the observed changes to the distribution of turbulence can be explained by examining the corresponding production rate terms, in particular interactions of normal stresses with the TLV, corner vortex-induced axial contraction, and the Reynolds shear stress with regions of high shear strain rates. (c) Irrespective of flowrate and presence of grooves, the turbulence in the rotor passage and the groove is highly inhomogeneous and anisotropic, with the anisotropy invariant maps revealing spatially varying structures ranging from 1D to 2D and 3D turbulence. While the levels of anisotropy levels vary, some common features persist, such as 1D turbulence in the shear layers, 2D turbulence in the area surrounding the TLV, and nearly isotropic turbulence within parts of the groove. (d) While presenting only one sample in the present paper for a grooved endwall, consistent with prior publications, the magnitudes, alignment, and even signs of the Reynolds stresses are generally uncorrelated with the local strain-rate field. Consequently, eddy-viscosity-based Reynolds stress models are unsuitable for characterizing the flow in the tip region of turbomachines. As shown in Ref. [15] for a smooth endwall, in some areas, e.g., in the shear layer connecting the TLV to the SS corner, the contribution of the turbulent stresses to the flow dynamics is comparable to that of the mean flow advection term, indicating that using the wrong turbulence model is expected to have a substantial impact on the predicted flow.

Other phenomena are flowrate or blade phase dependent. First, with the introduction of grooves, the dominant contributor to the TKE shifts from radial velocity fluctuations for a smooth wall to axial fluctuation for the grooved wall. This shift is predominantly (but not only) associated with increase axial contraction as the inflow or outflow from the groove interacts with the passage flow. The circumferential contraction associated with the outflux from the groove also increases the production of the circumferential velocity fluctuations in several places, including the TLV and the corner vortex at low flowrates, and around the downstream corner of the groove at high flowrates. Second, at low flowrates, the turbulence level near the groove increases substantially when the blade tip is located nearby. The TLV is partially entrained into the groove, and a corner vortex is generated owing to flow separation at the downstream corner of the groove. In contrast, at high flowrates, as the blade section shifts away from the groove, the corner vortex and flow within the groove are entrained by the TLV into the main passage, and the TLV becomes fragmented, expanding to a broad area. These processes cause an increase in shear production of axial turbulence and the above-mentioned very high circumferential contraction that increases circumferential velocity fluctuations. The latter phenomenon does not happen for a smooth endwall and might be a contributor to the degradation of performance and the decrease in efficiency of the grooved machine at high flowrates.

## Acknowledgment

This project along with the facilities and instrumentation involved has been funded in part by NASA and in part by ONR. The authors would like to thank Chunill Hah and Michael Hathaway from NASA Glenn for their guidance and 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 data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper. Data provided by a third party are listed in Acknowledgment.

## Funding

(1) Office of Naval Research (ONR) and United Technologies Research Center (UTRC).

(2) NASA Glenn Research Center.

## Nomenclature

c =

h =

width of the rotor blade tip gap

k =

turbulent kinetic energy

s =

A =

through flow area

H =

L =

nominal distance from the hub to the inner casing endwall

P =

production rate of TKE, production rate of Reynolds stress

T =

motor input torque

r* =

uʹ =

velocity fluctuation

pexit =

static pressure at stator outlet

pin =

static pressure at IGV inlet

Sij =

mean strain-rate tensor

UT =

Vz =

average axial velocity in the rotor passage

r, z, θ =

ur, uz, uθ =

ρ =

NaI solution density

φ =

flow coefficient

ψSS =

static-to-static pressure rise coefficient

ω =

vorticity

Ω =

rotor angular velocity

< > =

ensemble-averaged quantity

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