## Abstract

In the previous part of the paper, a novel method to reconstruct the compressor nonuniform circumferential flow field using spatially undersampled data points is developed. In this part of the paper, the method is applied to two compressor research articles to further demonstrate the potential of the novel method in resolving the important flow features associated with these circumferential nonuniformities. In the first experiment, the static pressure field at the leading edge of a vaned diffuser in a high-speed centrifugal compressor is reconstructed using pressure readings from nine static pressure taps placed on the hub of the diffuser. The magnitude and phase information for the first three dominant wavelets are characterized. Additionally, the method shows significant advantages over the traditional averaging methods for calculating repeatable mean values of the static pressure. While using the multi-wavelet approximation method, the errors in the mean static pressure with one dropout measurement are 70% less than the pitchwise-averaging method. In the second experiment, the full annulus total pressure field downstream of Stator 2 in a three-stage axial compressor is reconstructed from a small segment of data representing 20% coverage of the annulus. Results show very good agreement between the reconstructed total pressure profile and the experiment at a variety of spanwise locations from near hub to near shroud. The features associated with blade row interactions accounting for passage-to-passage variations are resolved in the reconstructed total pressure profile.

## Introduction

The flow field in a compressor is circumferentially nonuniform due to the wakes from upstream stator row(s), the potential field from both upstream and downstream stator rows, and their aerodynamic interactions. Characterization of this nonuniform flow is of great importance since it can affect stage performance and blade forced response. Historically, experimental characterization of circumferential variations in the flow field is achieved by circumferential traverses, either utilizing a probe traverse mechanism or fixed instrumentation while actuating the stator rows circumferentially. This involves costly design and development of complex traverse mechanisms and introduces challenges in sealing the flow path.

Additionally, in many component or engine tests, implementation of traverse mechanisms is not possible. Rakes placed at several stations around the annulus are used to characterize the component performance. At each station, the thermodynamic properties acquired from the probes at different locations are averaged to a single value to represent the mean flow property. Therefore, depending on the location of the probes and nonuniform flow features, errors can be introduced to the calculated mean flow properties and can further propagate into calculations of the component performance. For instance, Stummann et al. [1] conducted a full annulus unsteady Reynolds-averaged Navier–Stokes (URANS) simulation of a 3.5-stage axial compressor at midspan and showed that the circumferentially nonuniform flow can cause more than a one-point error in compressor stage performance measurements. In a recent study, Chilla et al. [2] investigated the instrumentation errors caused by circumferential flow field variations in an eight-stage axial compressor representative of a small core compressor in an aero-engine. The analysis showed that a baseline probe configuration with three equally spaced probes around the annulus yields a maximum of 0.8% error in flow capacity and 2.8 points in error for compressor isentropic efficiency.

Therefore, it is of great value to resolve the compressor nonuniform circumferential flow field using spatially undersampled data. However, there is very limited research available in the open literature on this topic. To bridge this gap, a novel method to reconstruct the compressor nonuniform circumferential flow field using spatially undersampled data points from a few probes instrumented at fixed circumferential locations is proposed in Part 1 of this publication [3]. The method includes two core techniques: a particle swarm optimization algorithm for selection of optimal probe position and a multi-wavelet approximation method to reconstruct the nonuniform circumferential flow field from several dominant wave numbers. Validation of the method is performed using a representative total pressure field in a multi-stage compressor available in the open literature. The flow variations are characterized using full annulus URANS simulations, which constitute the “true” pressure field. The circumferential total pressure field is reconstructed from eight spatially distributed data points using a triple-wavelet approximation method. Results show good agreement between the reconstructed pressure field and the “true” pressure field. Following that, a sensitivity analysis of the method is conducted to study the influence of probe spacing on the error in the reconstructed signal. The method is robust and capable of reconstructing circumferentially nonuniform compressor flow field with good accuracy.

In this part of the paper, the method is applied to two compressor research articles to further demonstrate the potential of the novel method in resolving the important flow features associated with these circumferentially nonuniformities. The first experiment was performed in a high-speed centrifugal compressor for aero-engine applications to reconstruct the circumferential pressure field upstream of the diffuser leading edge. As a key component in modern centrifugal compressors, the performance of the vaned diffuser is of great importance and is typically the bottleneck component of the stage. The performance of the vaned diffuser is characterized in terms of static pressure recovery coefficient, and the mean value of the static pressure at the diffuser leading edge is one of the three key parameters to evaluate the diffuser performance. However, the pressure field is strongly nonuniform at the diffuser leading edge primarily due to the potential field of the diffuser. Therefore, proper placement of the static pressure taps and robust methods for evaluation of the mean static pressure at the diffuser leading edge is critical. In addition to the influence in calculating the diffuser performance, this circumferential nonuniform pressure acts as a temporal excitation to the rotating impeller and can cause blade vibration or failure at resonance. For example, Lusardi [4] showed that as the impeller blade sweeps through the nonuniform static pressure field of the diffuser inlet, it experiences high pressure when in close proximity to the diffuser vane and low pressure between vanes. This is one of the primary sources for the unsteady loading on impeller blades. As compressor designs advance toward higher efficiency and compact size, the gap between the impeller and the diffuser are greatly reduced in modern designs. This results in stronger diffuser potential field interactions with the impeller blades and, consequently, leads to higher forced response vibration levels of the impeller blades. As a result, it is of great value to characterize the strength of the diffuser potential field.

In the second experiment, the method is applied to reconstruct the total pressure field downstream of an embedded stator in a multi-stage axial compressor. The compressor, first reported by Fulayter [5], represents the geometrically scaled up design of the rear stages of a Rolls-Royce high-pressure compressor used for a jet engine core, matching representative Mach numbers and Reynolds numbers. With multiple stages and realistic flow conditions, the facility offers the opportunity to analyze the impact of blade row interactions for computational tool validation. Though studies of the rotor wake variability have been thoroughly carried out by Key et al. [6] and Smith et al. [7], there are limited findings on passage-to-passage flow variations for the vane rows. In this experiment, three circumferential segments of the annulus were experimentally characterized by indexing the stators, with respect to one another, past a fixed total pressure rake. The full annulus circumferential flow field is reconstructed from a small portion of data representing 20% coverage of the annulus, and comparisons between the reconstructed total pressure profiles and experimental results are made.

## Reconstruction of Diffuser Leading Edge Pressure Filed in a Single-Stage Centrifugal Compressor

The first experiment was conducted on a single-stage centrifugal compressor research facility at Purdue University, and the objective is to reconstruct the pressure field at the diffuser leading edge using measurements from nine static pressure taps. The flow path of the compressor and distribution of the steady instrumentation is shown in Fig. 1(a). The entire stage includes an inlet housing, a transonic impeller, a vaned diffuser, a bend, and deswirl vanes. The inlet housing delivers the flow to the impeller eye. The impeller is backswept and has 17 main blades plus 17 splitters. The diffuser consists of 25 aerodynamically profiled vanes. The compressor design speed is about 45,000 rpm, and the entire stage produces a total pressure ratio near 6.5 at design condition. Steady performance of the compressor stage is characterized using the total pressure and total temperature measurements at compressor inlet (station 1) and deswirl exit (station 5), and static pressure taps are located throughout the flow passage to characterize the stage and component static pressure characteristics, as shown in Fig. 1(a). The details of the research facility can be found in Ref. [8].

The diffuser leading edge static pressure measurements are selected as the focus for this study for four reasons:

The pressure field at the diffuser leading edge is primarily dominated by the potential field of the diffuser vanes, and thus provides an ideal case to examine the methodology.

The instrumentation is readily available in the original experiment setup

The performance of the vaned diffuser is evaluated in terms of static pressure recovery coefficient, and precise calculation for the mean static pressure at the diffuser leading edge is of great value in evaluating the aerodynamic performance of the vaned diffuser.

The diffuser potential field is one of the primary forcing functions for impeller forced response at resonance [9].

The distribution of the static pressure taps at the diffuser leading edge is shown in Fig. 1(b). There are a total of nine static pressure taps placed nonuniformly along the circumferential direction. Each of them is placed in a different diffuser passage at a different pitchwise location from 10% to 90% pitch. Details of the circumferential and pitchwise locations for these pressure taps are shown in Table 1.

Description | Circumferential position (deg) | Pitchwise position (%) | Passage no. |
---|---|---|---|

P1 | 52.0 | 60 | 4 |

P2 | 85.1 | 90 | 6 |

P3 | 103.9 | 20 | 8 |

P4 | 165.8 | 50 | 12 |

P5 | 198.9 | 80 | 14 |

P6 | 217.6 | 10 | 16 |

P7 | 279.5 | 40 | 20 |

P8 | 312.7 | 70 | 22 |

P9 | 350.1 | 30 | 25 |

Description | Circumferential position (deg) | Pitchwise position (%) | Passage no. |
---|---|---|---|

P1 | 52.0 | 60 | 4 |

P2 | 85.1 | 90 | 6 |

P3 | 103.9 | 20 | 8 |

P4 | 165.8 | 50 | 12 |

P5 | 198.9 | 80 | 14 |

P6 | 217.6 | 10 | 16 |

P7 | 279.5 | 40 | 20 |

P8 | 312.7 | 70 | 22 |

P9 | 350.1 | 30 | 25 |

Figure 2 shows the performance map of the compressor stage in terms of normalized total pressure ratio from 60% to 100% corrected speed from choke to high loading conditions. The operating conditions near the design loading conditions are indicated by the green circles. In the present study, the static pressure field at the diffuser leading edge was constructed near the nominal loading at design speed, indicated by a solid circle at 100% corrected speed on the compressor map.

Figure 3 shows the variation of the diffuser leading edge static pressure. In Fig. 3(a), the static pressure is shown in terms of the absolute circumferential locations of the measurements. There is an approximately 40% peak-to-peak variation in the static pressure at the diffuser leading edge. There is no apparent trend in the measurements as shown in Fig. 3(a). In contrast, the static pressure data is more informative when shown in terms of its pitchwise position, Fig. 3(b). There is a higher static pressure close to the diffuser vane and low pressure between vanes, which agrees with the findings from Lusardi in Ref. [4].

### Reconstruction of the Diffuser Leading Edge Pressure Field Using a Multi-Wavelet Approximation Method.

According to the empirical guidelines presented in Part 1 of the paper for selection of wavenumbers of most importance, a total of ten wavenumbers of interest were selected. These include the first two harmonics from the wakes at station 1 caused by the struts and rakes (Wn = 4 and 8), the first five harmonics of the diffuser counts (Wn = 25, 50, 75, 100, and 125), and the interactions between the compressor inlet struts and the vaned diffuser (Wn = 21, 17, and 34). The condition numbers of the probe set for the ten selected wavenumbers are shown in Fig. 4. The values of all the condition numbers fall in the range between 1.0 and 2.0 indicating the probe set is able to characterize all wavenumbers of interest. However, it is worth noting that this is a unique case. For instance, out of the multiple probe sets instrumented along the flow path at different stations (impeller exit, diffuser leading edge, etc.), only the probe set located at the diffuser leading edge yields a reasonable condition number.

Table 2 lists the values of Pearson’s *r*, the fitting residual, as well as the rank of individual wavenumber. The wavenumber of 25 yields the best fitting results with highest value in Pearson’s *r* and the lowest fitting residual. A single-wavelet approximation using Wn = 25 yields a value for Pearson’s *r* greater than 0.95. This indicates that the potential field of the vaned diffuser is dominant in the static pressure field at the diffuser leading edge.

Wave no. | Pearson correlation | Fitting residual | Rank |
---|---|---|---|

4 | 0.302 | 8.711 | 4 |

8 | 0.265 | 8.833 | 6 |

17 | 0.416 | 8.311 | 3 |

21 | 0.469 | 8.068 | 2 |

25 | 0.973 | 2.105 | 1 |

34 | 0.177 | 8.992 | 10 |

50 | 0.250 | 8.848 | 8 |

75 | 0.297 | 8.725 | 5 |

100 | 0.244 | 8.860 | 9 |

125 | 0.259 | 8.825 | 7 |

Wave no. | Pearson correlation | Fitting residual | Rank |
---|---|---|---|

4 | 0.302 | 8.711 | 4 |

8 | 0.265 | 8.833 | 6 |

17 | 0.416 | 8.311 | 3 |

21 | 0.469 | 8.068 | 2 |

25 | 0.973 | 2.105 | 1 |

34 | 0.177 | 8.992 | 10 |

50 | 0.250 | 8.848 | 8 |

75 | 0.297 | 8.725 | 5 |

100 | 0.244 | 8.860 | 9 |

125 | 0.259 | 8.825 | 7 |

Note: Italics font represents the best fitting results with highest value in Pearson’s *r* and the lowest fitting residual.

Furthermore, the wavenumber combinations yielding the best fitting results using single-, double-, and triple-wavelet approximations are listed in Table 3. As discussed previously, the static pressure field at the diffuser leading edge is primarily dominated by the potential of the diffuser vanes and, thus, a wavenumber of 25 yields the best fitting results for single-wavelet approximation. In addition, results indicate that the addition of the second harmonic wavenumber from the diffuser potential (Wn = 50) yields the best fitting results for double-wavelet approximation. This agrees with the findings from Sanders and Fleeter [10] showing that the variation in the static pressure field near the diffuser leading edge is dominated by the first few harmonics of the diffuser potential field. Finally, for the triple-wavelet approximation, the optimal wavenumber combination yielding the best fitting results is realized when including the effects from inlet strut-diffuser interactions (Wn = 17), and the primary and second harmonic of the wavenumber from the diffuser potential field (Wn = 25, 50).

Wave no. combination | Pearson correlation | Fitting residual, % |
---|---|---|

[25] | 0.9731 | 2.2 |

[25, 50] | 0.9927 | 1.1 |

[17, 25, 50] | 0.9996 | 0.25 |

Wave no. combination | Pearson correlation | Fitting residual, % |
---|---|---|

[25] | 0.9731 | 2.2 |

[25, 50] | 0.9927 | 1.1 |

[17, 25, 50] | 0.9996 | 0.25 |

In addition, the deviation between the measurements and fitting results at individual sensor locations are shown in Fig. 5. There is a significant improvement in data fitting from the single-wavelet to triple-wavelet approximation. For instance, there is still an approximately 4% peak-to-peak deviation between the measurements and fitting data for the base case using the single-wavelet approximation method. The deviation drops to less than 2.5% by using the double-wavelet approximation method. Finally, the reconstructed signal using wavenumbers of [17, 25, and, 50] yields the best agreement with the measurements at all the sensor locations. The peak-to-peak deviation between the measurements and fitted data is less than 0.4%. The confidence of the fit in terms of Pearson’s *r* is 99.7%, indicating no need to include extra wavelets.

Furthermore, the reconstructed pressure field at the leading edge from the triple-wavelet approximation method is shown in Fig. 6(a). The measurements at all locations are also shown, indicated by the circles, and the circumferential coverage of individual diffuser passage at the leading edge is indicated by the shaded areas. The pressure field at the leading edge of the diffuser is determined by the constructive and destructive interactions between the three wavelets, as shown in Fig. 6(b), and is the primary source for the passage-to-passage variations. In the present case, the variation in the pressure field at the diffuser leading edge is dominated by the potential field from the diffuser vanes. As shown in Fig. 6(c), the pressure variation due to the presence of diffuser vanes (Wn = 25) is approximately 14% with respect to the pitch-averaged value. The second contribution to the static pressure variation is the second harmonic of the diffuser potential (Wn = 50), yielding an approximate 4% variation in the circumferential direction at the diffuser leading edge. Finally, there is a small influence from the inlet strut-diffuser interactions (Wn = 17), causing approximately 2% of variation in the circumferential pressure field at the diffuser leading edge.

It is worth noting that there is great value in understanding the content and interactions between each component in the pressure field upstream of the diffuser leading edge. For instance, in centrifugal compressors, one of the primary causes for impeller blade failure is the effect of the potential field from the vaned diffuser. In the present case, this corresponds to wavelets with wavenumbers of 25 and 50. The potential field imposes an unsteady pressure to the impeller blades as it passes by every diffuser passage. The magnitude of the wavelet determines the magnitude of the unsteady force acting on the impeller blades, which determines the vibration amplitude of the blade if the passing frequency is close to the blade natural frequency. Although the static pressure field upstream of the diffuser leading edge in a centrifugal compressor is used to illustrate the potential of the methods in addressing some of the forced response challenges, the method can also be applied to other types of turbomachines including axial compressors and radial and axial turbines.

### Calculation of Mean Static Pressure Using the Multi-Wavelet Approximation Method.

In addition to reconstructing the detailed flow features, the method can also be used to obtain reliable mean flow properties for the characterization of engine, component, or stage efficiency. Historically, the mean flow properties have been calculated using certain averaging methods. A variety of averaging methods have emerged during the past few decades including area-average, mass-average, work-average, and momentum-average methods. However, without the detailed information of flow properties around the full annulus, the accuracy of the averaged value as a representation of the true mean flow property is limited. Additionally, one challenge occurring in almost every engine test campaign is sensor mortality. In many cases, the measurements are not recoverable and, thus, result in increased instrumentation error and even larger uncertainty in the follow-on performance evaluation. Therefore, a robust method for probe arrangement and mean value calculation is of great value.

Table 4 lists the nondimensional mean value of the diffuser leading edge static pressure obtained from circumferential-averaging, pitchwise-averaging, and triple-wavelet approximation methods. In the present case, all mean values obtained from the three methods are very close to each other. There is less than a 0.5% difference between the values from pitchwise-averaging and triple-wavelet approximation methods, and there is approximately a 3% difference between the values obtained from the circumferential-averaging and the triple-wavelet approximation methods.

Pitchwise average | Circumferential average | Multi-wavelet fitting |
---|---|---|

1.0 | 0.9695 | 1.0044 |

Pitchwise average | Circumferential average | Multi-wavelet fitting |
---|---|---|

1.0 | 0.9695 | 1.0044 |

However, for cases with sensor dropout, the method using triple-wavelet approximation yields more repeatable values in comparison with the two averaging methods. For instance, as shown in Fig. 7(a), for the case with one malfunctioning pressure measurement, the variation in the mean value of static pressure obtained using the triple-wavelet approximation method is less than 0.5%. In contrast, the variation in the mean static pressure obtained from the pitchwise-averaging method is 1.5 times larger, increasing to 2%. Additionally, the circumferential-averaging method yields the largest variation in calculation of the mean static pressure. There is an approximate 4.1% variation in the mean static pressure for cases with one sensor dropout. Furthermore, for cases with two sensor dropouts, the triple-wave approximation method still yields very repeatable mean values for the diffuser leading edge static pressure, with less than 1.2% in variation. In contrast, both averaging methods result in significantly larger variations in the mean values of the static pressure at the diffuser leading edge. For instance, the pitchwise-averaging method yields 3.7% variation while the circumferential-averaging method yields 8.1% variation in the mean value of the diffuser leading edge static pressure.

The results are shown in Fig. 7(b). In both cases, with one or two malfunctioning measurements, the method using multi-wavelet approximation method yields much smaller uncertainty in the calculated diffuser static pressure rise coefficient. The uncertainties in the diffuser static pressure coefficient caused by one or two missing measurements are 0.35% and 1.0%, respectively, using the mean value of static pressure from the triple-wavelet approximation method. In contrast, the uncertainties in the diffuser static pressure coefficient caused by one or two missing measurements are 1.7% and 3.1%, respectively, using the mean value of static pressure from the pitchwise-averaging method, and 3.2% and 6.3%, respectively, from the circumferential-averaging method.

In summary, the static pressure field at the diffuser leading edge of a vaned diffuser in a high-speed centrifugal compressor is successfully reconstructed using pressure readings from nine static pressure taps placed on the hub of the diffuser. Both the magnitude and phase information of the three most relevant wavelets are characterized using a triple-wavelet approximation method. In addition, compared with more traditional averaging methods, the multi-wavelet approximation method yields more repeatable mean values for static pressure at the diffuser leading edge.

## Reconstruction of Stator Downstream Total Pressure Field in a Multi-Stage Axial Compressor

The second experiment was conducted in the Purdue 3-Stage (P3S) Axial Compressor Research Facility using the PAX100 compressor with a reduced-vane count for stator 1 (denoted PAX101). Information associated with the layout of the facility can be found in Refs. [11,12]. The PAX100 compressor design features an inlet guide vane (IGV) followed by three stages, shown in Fig. 8. All three of the rotors are integrally bladed and each stator row is uniquely manufactured as a 180-deg segment featuring a shroud on both the inner and outer diameters. Between each stage, instrumentation ports in the casing endwall allow for various probes to be inserted into the flow field. While the compressor is in operation, each stator row also has the capability to individually circumferentially traverse an angular distance up to approximately 15 deg, or more than two stator vane passages. This enables vane clocking, allowing pitchwise measurements including wake traverses. The circumferential vane position is measured with precision string potentiometers.

In the original PAX100 configuration, the IGV, stator 1 (S1), and stator 2 (S2) all have the same vane count of 44 providing a unique environment to study the effects of vane clocking on compressor aerodynamic performance as well as rotor forced response. However, the effects of S1 and S2 on rotor 2 (R2) forced response are indistinguishable using this configuration. To address this challenge, a reduced-count S1 vane row was designed with 38 vanes (19 vanes per 180-deg segment) [13]. However, with the introduction of a different, reduced-vane count for S1 into the standard compressor configuration, the full annulus could not be captured in a single vane pass traverse. For the PAX100 configuration, with a uniform blade count of 44 for the IGV, S1, and S2, one vane passage could effectively characterize the entire annulus of the compressor (neglecting the effects of S3). For the PAX101 configuration, this characterization gets more complicated as one vane passage is no longer indicative of the entire annulus. To illustrate this complexity, a simple model was developed by Kormanik [14] to demonstrate how the blades would line up relative to one another around the annulus, Fig. 9. The circumferential positions for all blades from S1, S2/IGV (shown as a single row because of the same vane counts), and S3 are illustrated. In the PAX101 configuration, as all of the stator blade counts have a greatest common factor of 2, blades of all four stationary rows (IGV, S1, S2, and S3) only exactly line up every 180 deg (0 deg and 180 deg in Fig. 9). In addition, with decreasing the vane count of S1 by 6, the blades of IGV, S1, and S2 approximately line up every 60 deg (0 deg, 60 deg, 120 deg, 180 deg, 240 deg, and 300 deg in Fig. 9). Thus, a 6/rev pattern and a 2/rev pattern manifest around the annulus.

Since the instrumentation is stationary, S1 and S3 were clocked with respect to IGV and S2 in a particular fashion so as to imitate the location of a probe if it was able to be traversed around the annulus. These established seven clocking configurations, labeled in Fig. 9, to map out a 60 deg segment of the annulus. The offset angles of S1 and S3 with respect to S2/IGV at all seven clocking configurations are shown in Fig. 10. At each configuration, the position of IGV and S2 remains the same, while the position of S1 and S3 were adjusted with respect to IGV/S2 to match the relative blade spacing at the different annulus locations. Since the stator rows can only be traversed a finite difference, to achieve a full 60 deg sector, the clocking configuration orientation would have to be reversed just past the point when the difference between vanes exceeds 4.5 deg (configurations 5, 6, and 7 in Fig. 10).

With the clocking configurations in place, a comprehensive experimental campaign was conducted at 86% corrected speed on the 100% corrected speed peak efficiency loading line, shown in Fig. 11. This part-speed operation was selected to characterize the forcing functions for R2 forced response near the resonant crossing of the 38EO excitation of the R2 first torsion vibratory mode. In the experiment, seven-element total pressure rakes were placed behind S1, R2, and S2 at three different circumferential locations (noted as location A, B, and D). At each clocking configuration, all stators were traversed together over the length of a S1 vane passage at a resolution of 5% S1 passage, as indicated by the shaded area in Fig. 10. At the end of the test campaign, a total of 58.6 deg of effective travel around the annulus for each rake was achieved, and a total of 47.6% of the entire annulus was mapped using the three circumferentially placed rakes due to some overlap.

Figure 12 shows the total pressure measurements acquired at midspan downstream of S2. The measurements from rakes at location A, B, and D are presented. The measurements from traverses one to seven are also indicated in the figure. As expected, the stator-stator interactions result in complicated patterns of passage total pressure profile. There are evident passage–passage variations with a 6/rev feature, indicated by the dashed line.

### Reconstruction of Stator 2 Downstream Total Pressure Profile Using the Full and Reduced Datasets.

In the present study, the total pressure downstream of Stator 2 is selected for flow reconstruction. Stator 2 is an embedded stage and, thus, provides an ideal environment to examine the potential of the method in characterizing the flow features associated with blade row interaction. The total pressure field downstream of S2 is dominated by the wakes from upstream stator rows, the potential field of downstream S3, and the interactions between these stator rows. According to the empirical guidelines provided in Part 1 of the paper, a set of 20 wavenumbers were selected to reconstruct the total pressure field using the full dataset from experiments. The selected wavenumbers can be categorized into four types:

The first eight harmonics of IGV and S2 wakes (Wn = [44, 88, 132, 176, 220, 264, 308, 352]);

The first four harmonics associated with reduced-vane count of S1 (Wn = [38, 76, 114, 152]);

The first four harmonics of S3 potential field (Wn = [50, 100, 150, 200]); and

The wavelets associated with IGV-S1-S2-S3 interactions (Wn = [2, 6, 12, 24]).

This set of wavelets yields a condition number of 3.15 using the full dataset from the experiment. The reconstructed total pressure profile (black) at midspan downstream of S2 is shown in Fig. 13(a). In addition, the total pressure profile acquired from experiments is plotted on top of the reconstructed total pressure profile for comparison. The reconstructed total pressure profile agrees well with the results from experiment. The features associated with passage–passage variations are resolved in the reconstructed total pressure profile. For instance, the 6/rev features due to the S1-S2/IGV interaction are characterized in the reconstructed profile. Despite some deviations in the depth of the pressure deficit between the reconstructed and measured total pressure profiles, there is a very good overall agreement achieved between the reconstructed and measured total pressure profiles.

Furthermore, efforts were made to reconstruct the total pressure profile using a reduced wavelet set. The considerations for selection of the reduced wavelet set are twofold and need to be balanced. While a smaller number of wavelets would require fewer data points to reconstruct the flow, the reduced wavelet set should still be able to characterize the flow features of interest. After comparing the magnitudes of all the wavelets in the reconstructed total pressure profile, the first 12 dominant wavelets were selected: Wn = [6, 12, 38, 50, 44, 88, 132, 176, 220, 264, 308, 352]. The reduced wavelet set includes all eight wavelets associated with S2/IGV wakes but eliminates the higher harmonics associated with S1, S3, and the stator–stator interactions. This reduced wavelet set yields a better condition number (1.69) using the full dataset from the experiment, as shown in Table 5. The total pressure at midspan downstream of S2 was reconstructed using the first 12 dominant wavelets, and the results are shown in Fig. 13(b). There are very small differences between the reconstructed total pressure profile using 20 wavelets and the reconstructed total pressure profile using 12 wavelets. Good agreement between the reconstructed total pressure profile and the results from experiment are achieved by using a reduced number of wavelets. Therefore, 12 wavelets are sufficient to reconstruct this total pressure profile at high fidelity.

Wave no. combination | No. of wavelets | Full dataset | Reduced dataset |
---|---|---|---|

[44 88 132 176 220 264 308 352 38 76 114 152 50 100 150 200 2 6 12 24] | 20 | 2.99 | 5.81 |

[6 12 38 50 44 88 132 176 220 264 308 352] | 12 | 1.53 | 2.66 |

Wave no. combination | No. of wavelets | Full dataset | Reduced dataset |
---|---|---|---|

[44 88 132 176 220 264 308 352 38 76 114 152 50 100 150 200 2 6 12 24] | 20 | 2.99 | 5.81 |

[6 12 38 50 44 88 132 176 220 264 308 352] | 12 | 1.53 | 2.66 |

Finally, an effort was made to reconstruct the total pressure field using a reduced data size, for instance, using data from three or four traverses instead of all seven traverses. To assure a high-fidelity result, an intelligent selection of the optimal traverse combinations included in the reduced dataset was exercised to achieve a small condition number. The selected reduced dataset contains three traverses from each rake including traverse numbers two, three, and six for the rake at location A; traverse numbers two, five, and seven for the rake at location B; and traverse numbers two, four, and seven for the rake at location D. This yields a condition number of 1.97 for the case when trying to resolve 12 wavelets. The reduced dataset accounts for 43% of the full dataset and only 20% of the full annulus coverage. The reconstructed total pressure field using the reduced dataset is shown in Fig. 13(c). The segments of data used for flow reconstruction are indicated by the blue bands on the abscissa. Strong agreement in the total pressure profile between the reconstructed and experimental results is achieved in the passages where the experimental data are used for flow reconstruction. More importantly, good agreement is also achieved in the passages where the experimental data are not used for flow reconstruction. The features associated with passage–passage variations are nicely resolved in the reconstructed total pressure profile using the reduced dataset. There are very small differences between the two reconstructed total pressure profiles using the full and selected reduced datasets.

After exploring the influences of the number of wavelets and the size of the dataset, one important conclusion can be drawn: the full annulus total pressure profile downstream of S2 can be reconstructed with high-fidelity by using a small segment of the dataset and inclusion of a limited number of wavelets. Based on this finding, the total pressure profile at the near hub (12%) and near shroud (88%) are reconstructed using the reduced dataset with 12 wavelets. The results are shown in Fig. 14. The reconstructed total pressure profile at 65% span is not shown in the figure due to the physical differences in the spanwise distribution of the pressure elements for each of the three rakes used (i.e., Rake B has a pressure element at 65% spanwise location while Rakes A and D have pressure elements at 70% instead). At the other six spanwise locations, the reconstructed total pressure profiles agree well with the results from experiment including the passages where the experimental data are not used for flow reconstruction. The patterns associated with passage–passage variations are nicely resolved in the reconstructed total pressure profile. Additionally, comparing to the midspan, there is better agreement in the depth of the pressure deficit associated with wakes between the reconstructed and measured total pressure profiles achieved at the near hub and near shroud regions.

### Calculation of Stator 2 Downstream Mean Total Pressure Using the Multi-Wavelet Approximation Method.

The mean total pressure obtained from the multi-wavelet approximation method are compared with the mean total pressure obtained from experimental data using a pitchwise-averaging method, and the results from all seven spanwise locations are listed in Table 6. The column labeled F-20 contains results obtained using the full dataset with inclusion of 20 wavelets. Correspondingly, F-12 and R-12 represent the results obtained using either the full or reduced dataset with inclusion of 12 wavelets. At all the spanwise locations, the mean total pressures obtained using the multi-wavelet method are almost identical to the values calculated using the pitchwise-averaging method, with the maximum deviation less than 0.025%. In addition, the multi-wavelet method yields very repeatable values for the mean total pressure for all cases (including the full or reduced datasets with inclusion of 20 or 12 wavelets).

Span, % | Exp. | Multi-wavelet Approx. | Max dev, % | ||
---|---|---|---|---|---|

F-20 | F-10 | R-10 | |||

12 | 1.1464 | 1.1464 | 1.1464 | 1.1463 | 0.0052 |

20 | 1.1473 | 1.1473 | 1.1474 | 1.1473 | 0.0071 |

35 | 1.1480 | 1.1481 | 1.1482 | 1.1482 | 0.0184 |

50 | 1.1468 | 1.1470 | 1.1471 | 1.1471 | 0.0238 |

70 | 1.1457 | 1.1457 | 1.1459 | 1.1458 | 0.0226 |

80 | 1.1437 | 1.1436 | 1.1439 | 1.1437 | 0.0231 |

88 | 1.1407 | 1.1407 | 1.1409 | 1.1407 | 0.0212 |

Span, % | Exp. | Multi-wavelet Approx. | Max dev, % | ||
---|---|---|---|---|---|

F-20 | F-10 | R-10 | |||

12 | 1.1464 | 1.1464 | 1.1464 | 1.1463 | 0.0052 |

20 | 1.1473 | 1.1473 | 1.1474 | 1.1473 | 0.0071 |

35 | 1.1480 | 1.1481 | 1.1482 | 1.1482 | 0.0184 |

50 | 1.1468 | 1.1470 | 1.1471 | 1.1471 | 0.0238 |

70 | 1.1457 | 1.1457 | 1.1459 | 1.1458 | 0.0226 |

80 | 1.1437 | 1.1436 | 1.1439 | 1.1437 | 0.0231 |

88 | 1.1407 | 1.1407 | 1.1409 | 1.1407 | 0.0212 |

In summary, the full annulus total pressure field downstream of S2 is reconstructed from a fraction of experimental data representing 20% of full annulus coverage using a multi-wavelet approximation method. The reconstructed total pressure profile agrees well with the results from the experiment at a variety of spanwise locations from the near-hub to near-shroud regions. The patterns associated with passage-to-passage variations are resolved in the reconstructed total pressure profile. Additionally, the method proves robust in yielding a repeatable mean total pressure.

Finally, it is also worth noting that there are deviations between the reconstructed and experimental total pressure profiles. For instance, the agreement between the reconstructed and experimental total pressure profile over the traverses from Rake D is not as good as those from Rakes A and B. There are deviations in predicting the depth of the pressure deficit associated with the wake for several passages. One possible reason for these inconsistences can be the instrumentation errors associated with rake hardware variations or sensor variations contributing to measurement uncertainties, which will be investigated in future research.

## Conclusions

In this two-part publication, a novel method is introduced in Part 1 of the paper to reconstruct the compressor nonuniform circumferential flow field using spatially undersampled data. The method includes two core techniques: a particle swarm optimization algorithm for selection of optimal probe position and a multi-wavelet approximation method to reconstruct the nonuniform circumferential flow field from several dominant wavenumbers. A roadmap for implementation of the method was established including practical guidelines for selection of most important wavenumbers and evaluation of the confidence in the reconstructed circumferential flow. Validation of the method was performed using computational fluid dynamics calculations available in the open literature of the total pressure field of a multi-stage compressor. The method was shown to be robust and capable of reconstructing compressor circumferential flow with good accuracy.

In the present paper, Part 2, the method is applied to two compressor component-level experiments to further demonstrate the potential of this novel method in resolving the important flow features associated with circumferential flow field’s nonuniformity. In the first experiment, the static pressure field at the diffuser leading edge of a vaned diffuser in a high-speed centrifugal compressor is reconstructed using pressure readings from nine static pressure taps placed on the hub of the diffuser. The magnitude and phase information for the first three dominant wavelets are characterized. In the second experiment, the full annulus total pressure field downstream of Stator 2 in a three-stage axial compressor representing the rear stages of high-pressure core is reconstructed from a small segment of data representing 20% of the full annulus coverage. Results show very good agreement between the reconstructed total pressure profile and that from experiment at a variety of spanwise locations from near hub to shroud. The features associated with blade row interactions that account for passage–passage variations are nicely resolved in the reconstructed total pressure profile. Additionally, the method shows significant advantages over the traditional averaging methods to calculating suitable mean flow properties. For instance, in the first experiment, the errors in the mean static pressure with one malfunctioning measurement using multi-wavelet approximation method is 70% less in comparison with the pitchwise-averaging method. In the second experiment, the multi-wavelet method yields very repeatable values for the mean total pressure at all the selected spanwise locations for all cases of using a full or reduced dataset.

The method bridges the gap between sparsely distributed experimental data and the detailed flow field of a full annulus. Through the two experiments in different types of compressors, the method shows great potential in obtaining suitable mean flow properties for performance calculations as well as resolving the important flow features associated with circumferential nonuniformity. The method can be disruptive to the gas turbine community concerning: expectations of experimental data; how and where to place the probes; and the method to calculate suitable mean flow properties.

## 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.