# Harmonic Analysis of Diffuser Inlet Static Pressure Distortion for Centrifugal Stages

D. Japikse, Sharon E Wight

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D. Japikse, Sharon E Wight. Harmonic Analysis of Diffuser Inlet Static Pressure Distortion for Centrifugal Stages. 17th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC2017), Dec 2017, Maui, United States. hal-03004932

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# Harmonic Analysis of Diffuser Inlet Static Pressure Distortion for Centrifugal Stages

D. Japikse<fnref n="1" />

Sharon E. Wight<fnref n="2" />

<figure><img src="image_2.png" /></figure>

ISROMAC 2017

International
Symposium on
Transport

Phenomena and
Dynamics of

Rotating Machinery

Hawaii, Maui

December 16-21
2017

## Abstract

Preliminary studies of the impeller exit/vaned diffuser inlet static pressure field distortion have been reported by Japikse and Krivitzky (2016). The present work extends the study to a set of nine different impellers with a wide variety of vaned diffusers from four different test rigs and a three-stage industrial compressor. Pertinent work of similar cases from others is reviewed. In all cases where a high number of uniformly spaced pressure taps ($p_2$) were used to measure the time-averaged local static pressure, a clear set of harmonics in a Fourier representation of the time-averaged diffuser inlet pressure field was evident, which effectively establishes the signature or fingerprint of the specific stage. In all such cases, the strongest harmonic amplitude for the resulting pressure trace analysis was the diffuser vane count which exhibits various amplitude and phase shifting with change in flow rate for a given speed operation. The next largest amplitude harmonic was usually (but not always) the first, second, or third spatial harmonic, which can never be neglected for pressure trace representation. Significant study of pressure field phase angle shifts and amplitude shifts occur for these lowest spatial frequencies, representing part of the static pressure field when moving from flow point to flow point along an operating speed line. In certain cases, for example, with a certain 14-vane diffuser, spatial subharmonics of 6, 8, and 12 may also play a role in fully representing the steady, circumferential, inlet static pressure field. The cause of this unexpected pressure field distortion is not completely understood at this point. Hypotheses are suggested, and the next steps for research are outlined. The historical hypothesis of equal flow per diffuser passage is carefully reviewed and rejected, and hence periodicity is not established for these radial discharge centrifugal stages. Various tests conducted to test invariance to rig and environmental factors are reviewed. The role that CFD can play in understanding this phenomenon is discussed. Early CFD team study observations are shared. First order modeling is suggested.

## Keywords

Radial compressor, radial pump, distortion, non-axisymmetric flow, periodicity, harmonic analysis

## Nomenclature

$C\theta$ Absolute frame velocity, tangential component

$C_m$ Absolute frame velocity, meridional component

$r_2$ Radius to impeller exit

$m, M$ Meridional distance from impeller leading edge; mass flow rate

$p, p_0$ Static pressure, total pressure; subscripts 2,5 refer to impeller and diffuser exit respectively

$K$ Diffuser loss coefficient

$Zd$ Diffuser vane number

$\alpha$ Absolute flow angle from meridional reference

$\delta$ Phase angle shift or difference or increment

$\theta$ Tangential coordinate

<footnote n="1">¹Concepts NREC, LLC, Wilder, VT, USA, dj@conceptsnrec.com</footnote>

<footnote n="2">²Concepts NREC, LLC, Wilder, VT, USA, sew@conceptsnrec.com</footnote>
# INTRODUCTION

It is commonly assumed that the static pressure at the exit of a centrifugal compressor, blower or pump impeller ought to be circumferentially uniform, or, in fact, is uniform, if 1) the stage inlet and outlet are uniform (axisymmetric), 2) that all parts are precisely manufactured, and 3) that the impeller and other components are perfectly centered and lie in a plane orthogonal to the centerline in an axisymmetric construction. This is the common concept for a stage with a vaneless exit element and no volute; it is recognized that a vaned element, located downstream of an impeller, should reflect pressure variations from the flow entering the vaned element onto the impeller exit flow at the diffuser vane number. The common concept holds that the diffuser vane flow would be periodic. Historically, it has been largely unknown how strongly the interaction (coupling) between an impeller and a diffuser might be manifested. It is almost always assumed that periodically repeating flow conditions (periodic flow) exist for modern modelling of such stages. Recent detailed investigations provide strong information that challenges such beliefs.

In 1977, the critical work of Eckardt on centrifugal compressors with vaneless diffusers raised pertinent questions. His work showed very little evidence of circumferential distortion, but the steps required to get this flow are fascinating. First, he used a 50% linear pinched vaneless diffuser with a 360° axisymmetric exit comprised of a tight 90° bend from radial to axial with an axisymmetric throttling ring exactly at stage exit. This tight exit configuration should have worked very well; but it didn't. To get uniform flow, it was also required to impose additional pinch before the bend by using a supplemental hub-side pinch ring of considerable tightness. While this study was extremely important to the field of research in general, it also showed that an additional field needed study: the root cause of non-axisymmetric flow in centrifugal stages.

The authors' findings in this area cover several decades and multiple experiences. In 1982, a program was pursued for flow in various axisymmetric stages with vaneless diffusers and various return channel vane systems. In a baseline test with all return vanes removed, the $p_2$ static pressure naturally deviated from circular (pressure versus angular position or theta) and took an oblong or pseudo-elliptical form. When various return vane cascades were employed, the pressure field became nearly circular in circumferential pattern, essentially due to the resistance of the vanes. No specific studies were made at that point, but it was noted that the 180° symmetric bend at the top of the system may be able to impose radial components of force that could initiate a non-axisymmetric field.

Two interesting references for related studies are S. Berten et al, (2011) and P. Harley 2015) as discussed previously by Japikse and Krivitzky (2016). The first authors studied a diffuser pump and made the comment: "The difference, which is observed here, could be explained by non-uniformities in the diffuser flow in different diffuser channels, creating a non-uniform, circumferential pressure distribution around the impeller." No justification for this opinion is given in the paper, but it is a logical option. The second author studied a turbocharger with a volute, and in a personal communication, Harley noted (while referring to his different impellers—C4, C5, and C6): "C5 in particular has 2 clear maxima circumferentially, C4 has less pronounced maxima, but they are there...I have a C-6 not presented in the paper which is much more pronounced, and I didn't publish it due to self-doubt." It is probably this last statement that points to the reason why this phenomenon is so long in being resolved; likely, all the authors just cited, including the present ones, have felt this self-doubt. In the early 1990's, diffuser consortium research showed that there were serious issues with uniform $p_2$ measurements, even in seemly well-constructed test facilities, and that periodicity was difficult to establish, but a wide team of involved contributors was not ready to challenge traditional assumptions.

Current research on diffuser stages has changed the picture. This current work used a set of eight different impellers with a wide variety of diffusers (more than 25) from three different test rigs plus data from a three-stage industrial compressor. After a very thorough examination of all related construction and operational issues, it is firmly asserted (see Japikse and Krivitzky, 2016) that any geometry/hardware non-symmetries are very small (e.g. a measured operating impeller clearance of typically 0.009 +- 0.0005 inches (0.23 mm +- 0.013mm) with component assembly variations of about 0.002 +- 0.001 inches or 0.051mm +- 0.025mm). All components are CNC machined with surface accuracy also of 0.002 +- 0.001 inches. The exit flow is discharged into an axisymmetric plenum with four exit ports; pressure measurements in the plenum or collector confirm only a very slight pressure variation, at a level well below the variations discussed herein. These are very small variations (see Figure 2, Japikse and Krivitzky, op. cit., and related discussion) and there are no known mechanisms that might allow such variances to manifest the actual $p_2$ variations, and especially their specific wave forms and other flow characteristics. It is recognized that the slight collector pressure variations might play some role in the actual circumferential fixation or location of the phenomenon discussed herein, but having a causal effect is unlikely. Such variations at a larger scale can surely be found in nearly all other research studies.
# 1. REVIEW OF NEW DATA

A variety of data has been acquired with various compressor impellers and different vaned diffusers and always tested with a clean inlet and an axisymmetric collector with four exit pipes (measured pressures in the collector confirm that it is essentially constant pressure). Variations in $p_2$ for such cases are presented in this section. Typically, the quantity of measurements is about 3 times the diffuser vane count and usually a multiple that permits easy recognition of the total vane count. The taps are 0.015inch (0.38 mm) diameter and are drilled precisely with a CNC five-axis mill. The experimental setup has been tested for sensitivity to environmental effects. Several different inlets,

several different downstream orifice-runs, and several perturbations to the collector have been checked. Only when one of the four exit ports was completely blanked off to force a strong collector distortion could any effect be found on the measured diffuser inlet pressure pattern. In this latter case, the $p_2$ distortion was slightly reduced!

In the subsequent sections, sample data from six different diffuser sets are evaluated. Only the data from the highest speeds are shown in this report (135,000 rpm for the small stage and ½ of this for the 2X scaled stage), but very comparable results were found for the lower speed line data sets.

## 1a. Double divergent diffuser (15 vane) tested with the small (68.6 mm, 2.70 inch, diameter) impeller

Pressure data for this case are shown in Figure 1, with the raw data (measured pressure data are shown as small brown tick marks) in the top set and normalized data in the bottom set. The trace lines result from a proper Fourier analysis of the complete pressure field, which gives the full basis of this study. Figure 2 gives the corresponding amplitude and phase angle shifts for the series expansions.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Fourier expansion on $p_2$ pressures around the full data set; DD diffuser and small impeller; top data set is unscaled and bottom set is scaled about the average value (60 mm DD diffuser, 135,000 rpm).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data. (60 mm DD diffuser, 135,000 rpm)</figcaption></figure>
1b. Airfoil diffuser (16 vanes) tested with the small (68.6 mm, 2.70 inch, diameter) impeller

Pressure data for this case are shown in Figure 3 with the raw data in the top set and normalized data in the bottom set. The trace lines result from a proper Fourier analysis of the complete pressure field. Figure 4 gives the corresponding amplitude and phase angle shifts for the series expansion.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Fourier expansion on $p_2$ pressures with the full data set; airfoil diffuser and small impeller; top data set is unscaled and bottom set is scaled about the average value (60 mm LSA nom diffuser, 135,000 rpm)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data (60 mm LSA nom diffuser, 135,000 rpm).</figcaption></figure>
1c. Flat plate 14-vane diffuser (tested with the small impeller [68.6 mm, 2.70 inch, diameter])

Pressure data for this case are shown in Figure 5 with the raw data in the top set and normalized data in the bottom set.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Fourier expansion on $p_2$ pressures around the full data set; flat-plate diffuser and small impeller; top data set is unscaled and bottom set is scaled about the average value (60 mm FP M4 diffuser, 135,000 rpm).</figcaption></figure>

The trace lines above result from a proper Fourier analysis of the complete pressure field. Figure 6 gives the corresponding amplitude and phase angle shifts for the series expansions.

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data (60 mm FP M4 diffuser, 135,000 rpm).</figcaption></figure>
1d. Close coupled (2%) flat-plate diffuser (tested with the small [68.6 mm, 2.70 inch, diameter] impeller)

Pressure data for this case are shown in Figure 7 with the raw data in the top set and normalized data in the bottom set. The trace lines result from a proper Fourier analysis of the complete pressure field. Figure 8 gives the corresponding amplitude and phase angle shifts for the series expansions.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Fourier expansion on $p_2$ pressures around the full data set; close coupled (2%) flat-plate diffuser and small impeller; top data set is unscaled and bottom set is scaled about the average value.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data (60 mm 2% FP nom diffuser, 135,000 rpm)</figcaption></figure>
1e. Airfoil diffuser (15 vanes) tested with the large (2X) impeller

Pressure data for this case are shown in Figure 9 with the raw data in the top set and normalized data in the bottom set. The trace lines result from a proper Fourier analysis of the complete pressure field, which gives the full basis of this study. Figure 10 gives the corresponding amplitude and phase angle shifts for the series expansions.

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Fourier expansion on $p_2$ pressures around the full data set; airfoil diffuser and large (2X) impeller; top data set is unscaled and bottom set is scaled about the average value (120mm LSA nom, 67500 rpm).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data (120mm LSA nom, 67500 rpm).</figcaption></figure>
1f. Flat-plate diffuser (15 vanes) tested with the large (2X) impeller

Pressure data for this case are shown in Figure 11 with the raw data in the top set and normalized data in the bottom set. The trace lines result from a proper Fourier analysis of the complete pressure field, which gives the full basis of this study. Figure 12 gives the corresponding amplitude and phase angle shifts for the series expansions.

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Fourier expansion on $p_2$ pressures; flat-plate diffuser and large (2X) impeller; top data set is unscaled and bottom set is scaled about the average value (120mm FP nom, 67500 rpm).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Modal amplitudes (top traces, above) and phase angle shifts (bottom traces, above) for the Fourier series expansion of $p_2$ data (120mm FP nom, 67500 rpm).</figcaption></figure>
## 2. DATA EVALUATION

### 2a. General Discussion

Several basic effects can be observed in the trends including 1) the characteristic waveform (spatially fixed) with various amplitudes being shown, 2) mostly monatomic phase shifting, 3) some exceptions where phase-shifted lines cross each other, 4) strong phase shifting at low harmonic indices, and 5) certain possible common prime numbers that might be central to the harmonic analysis.

All wave forms, such as Figures 1, 3, 5, 7, 9, and 11 show a sinusoidal characteristic with a strong one per diffuser vane number relationship, but frequently other harmonics are evident as well such as the first harmonic which distorts the pressure field to one side (but not always), and even the 6–9 harmonics split, which is easily seen in Figure 1 (count the humps to see the effect).

Phase shifting occurs across each data speedline and most of the shifting tracks monotonically with incidence or flow. There are examples where the phase shifting gets complex: Figures 2, 4, 6, 8, 10, and 12 generally have most phase angle shifting occurring on the first 3 or 4 spatial harmonics as the flow distribution between the impeller and the diffuser establishes its preferred orientation amongst the vanes for each operating point.

It is not yet established as to which harmonics are most critical to this coupling problem. For the Figure 1 & 2 cases, 15 diffuser vanes are employed. This count of 15 has integer constituents of 1, 3, 5, and 15. Including simple multiples of these would give 1, 2, 3, 5, 6, 9, 10, and 15. In the cited figures, one finds principally 1, 6, 9, and 15 for amplitude effects and 1, 2, and 3 for phase angle effects.

For the Figures 3 & 4 case using 16 vanes, the integer numbers are 1, 2, 4, 8, and 16 with a fuller set using low multiples as well of 1, 2, 3, 4, 6, 8, 9, 12, and 16. In the cited figures, one finds principally 1, 4, and 16 for the amplitudes and 1 & 2 for the phase angle shifts.

For the Figures 5 & 6 case using 14 vanes, the integer numbers are 1, 2, 7, and 14 with a fuller set using low multiples as well of 1, 2, 3, 4, 6, 8, 9, 12, and 16. In the cited figures, one finds principally 1, 8, and 14 for the amplitudes and 1 & 2 for the phase angle shifts.

For the Figures 7 & 8 case using 14 vanes with 2% coupling, the integer numbers are 1, 2, 4, 7, and 14 with a fuller set using low multiples as well of 1, 2, 3, 4, 6, 8, 9, 12, and 16. In the cited figures, one

finds principally 1, 6, 8, and 14 for the amplitudes and 1 & 2 for the phase angle shifts.

For the Figures 9 & 10 case using 15 vanes in the larger rig, the integer numbers are 1, 3, 5, and 15 with a fuller set using low multiples as well of 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 16. In the cited figures, one finds principally 1, 2, and 15 for the amplitudes and 1 & 2 for the phase angle shifts.

For the Figures 11 & 12 case also using 15 vanes in the larger test rig, the integer numbers are 1, 3, 5, and 15 with a fuller set using low multiples as well giving 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, and 16. In the cited figures, one finds principally 1, 9, and 15 for the amplitudes and 1, 2, 4, 5, 7, 8, 10, and 12 for the phase angle shifts.

Despite using a collector with four exit ports, the index 4 shows up only once in the amplitude numbers and once in the phase angle shift numbers and these two are not for the same case. There is no credible evidence in these p₂ wave forms of any impact of the collector and its exits.

These harmonic number combinations change when vanes are more closely coupled to the impeller. Other harmonics are also involved, but to much lower effect, as a careful review of the data above shows. Although many harmonics are involved, certain prime numbers often are naturally excluded from each case and, in fact, have only rarely been found in the data sets.

A test with two impellers, but one with 2 additional vanes, gave some interesting results (not shown above). While the basic harmonics are nearly the same, differences were found. The higher vane impeller had a loss of choke margin of 1.8% and a surge flow penalty of 2.15% but the higher impeller vane count gave a higher p₂ pressure of about 5%. In general, the higher impeller vane number gave rise to a more complex and distorted flow field than the impeller with the 2 fewer vanes.

### 2b. Specific Trends

The data shown above, by way of example, are a small subset of all data acquired over the past several years by the authors. From the full data collection, it can be firmly asserted that the impeller exit static pressures are not uniform with or without consideration of the diffuser vane number and the harmonics. There is no evidence in this study to support the time-honored notion of an essentially uniform impeller exit pressure. Hence there is no true periodicity in the flow field.

There exists, based on a careful Fourier analysis of an adequate field of measurements, a definite,
spatially fixed, harmonic balancing process for the pressure field at the impeller exit/diffuser inlet that establishes the actual $p_2$ pressure field distribution. Essential characteristics, as currently researched, include:

1. The actual pressure field shifts slightly for every point on a stage characteristic and most of the (phase angle) shifting gives a variation that might be related to the diffuser inlet incidence;

2. The harmonic with the highest amplitude is always the vane number harmonic;

3. The second highest harmonic amplitude is the first harmonic for about half of the cases, but the other half show a higher order harmonic as the second strongest;

4. The exhibited harmonics follow basic integer relationships with the diffuser number and simple multiples thereof;

5. The active harmonic numbers for the pressure amplitudes do not change for an entire stage but the magnitudes associated with each harmonic can shift up or down as the local circumferential pressures vary;

6. Different diffuser designs reveal different harmonics;

7. The actual amplitudes for the vane number harmonic can be nearly constant or highly variable at different flow rates;

8. Phase angle shifting involves mostly the first two harmonics but occasionally the third and fourth harmonics are engaged; small variations exist at higher harmonics;

9. The effects are strong for a small stage (68.7 mm diameter impeller) and are somewhat weaker for a 2X stage of exact scaling (137.2 mm or 5.400 inches);

10. A case with close coupling has caused an increase in amplitudes and the number of harmonics–this 2% coupling case (see above) used the same harmonics (compared to the 10% coupling) for the phase angle changes and kept the same harmonics for the amplitudes but added an additional harmonic as well.

11. In most cases, there is amplitude modulation occurring at each critical harmonic number, but little or none at the Figure 1 case for the vane number harmonic (the corresponding tests on the 2X rig (not included here) showed the wider range of amplitude modulation).

These observations are made using several

different rigs with uniform inlet flows and an axisymmetric exit collector with negligible, measured, collector circumferential pressure variations. It is rather inconceivable that these variations in $p_2$ are substantial measurement error or an accidental creation of chance. In addition, $p_2$ values for each of three stages in an industrial gear driven compressor tested at the author's facility have been examined. Each stage had four taps apiece which is far too few for a harmonic analysis. But a study of their values across the operating characteristic clearly showed significant variance and with different trends for each stage; such data is not conclusive by itself but is consistent with the basic thesis of the present work.

## 2c. Possible Hypotheses and Modeling

The data and trends presented above are based on details and general observations from four different test rigs available to the authors and other existing studies. No support has come from these rigs for a true circumferentially uniform pressure at impeller exit.

Part of the problem with the historical understanding may come from a slight misunderstanding of just what the stage exit condition actually is and how a stage overall characteristic might best be understood. Figures 13 and 14 illustrate some of these issues. Figure 13 is a common speedline characteristic for a compressor or a pump with a system (or rig) resistance line added. The ordinate axis plots pressure difference (rise) but can be pressure ratio as well (either can be used for either pumps or compressors). The classical approach is to use the exit total pressure to reckon the thermodynamic state for pumps (almost always) and (frequently) for compressors; when a static state is used for compressors, a cycle implication is introduced that the leaving dynamic head is dealt with elsewhere by convention, but it is not a full thermodynamic record. Additionally shown is the impeller exit pressure, $p_{02}$, which is very, very difficult to measure precisely, but is important for understanding the physics of this problem. It is also very difficult to directly and precisely measure the exact $p_{05}$, which also does not yield a simple average or single number. The resistance line defines the actual pressure rise and the operating flow of the pump or compressor. When the resistance changes, the pump operating point changes. The resistance line is just what the name says: resistance or losses in the system elements against which the pump operates. By common convention and thermodynamic principles, losses are based on the drop of total pressure. So, the resistance line is setting a balance between the system and the compressor/pump based on the
<figure><img src="image_15.png" /><figcaption>Figure 13. Common stage characteristic</figcaption></figure>

<caption>Figure 14. Stage characteristic with uniform collector pressures, p<sub>5</sub>, and variable diffuser passage flow rates</caption>

common total pressure (still true even when a compressor designer chooses to use a static exit condition). By historical practice, the stage exit total pressure is rarely actually measured but rather a static pressure and an estimated exit dynamic head based on the flow rate are used. In fact, to be precise, one would have to make detailed total and static pressure and flow angle surveys of the compete exit profiles at any exit port to deduce the exit total pressure. This is very expensive and not to be expected in any commercial work. While conventional practice has indeed worked very well in commercial application, it approximates the actual p<sub>05</sub> value. The actual total pressure from the impeller, p<sub>02</sub>, is higher than the stage exit value due to losses in the downstream elements. It is just as hard to evaluate.

The second illustration, Figure 14, shows the conventional total pressure balance at p<sub>05</sub> which is taken to be the point which sets the stage mass flow rate. When attention is now focused back to the impeller exit question, there is no way to assert that either the impeller exit static pressure or the total pressure ought to be uniform at some fixed value for a given operating point. For this research, the true downstream known condition is a uniformly fixed circumferential static pressure at the stage discharge, i.e., the collector impact, and there is no requirement that the total pressure would be uniform at this state or that each passage should have equal mass flow. So, the conventional norm of using a fixed value of total pressure at stage exit is just that: a conventional norm which may, in fact, be an average of some varying values, especially for this case of a fixed exit static pressure.

A better appreciation of the operating point on the Figure 14 diagram may be obtained by looking at the p<sub>5</sub> options, which are shown as a different characteristic for the full regime of flows, but also illustrating the 'p<sub>5</sub> shelf' of operation which can be made up of many different increments of flow which represent discreet flow rates per individual diffuser passage. Whereas point '1' on this chart gives the system match on total pressure by convention, point '2' and the associated 'shelf' shows the possible variations in incremental flow, which then give the individual total pressure values from the p<sub>05</sub> characteristic which should average out to give the proscribed exit total pressure value, p<sub>05</sub>. This observation is fully consistent with the physics of the problem and only states that the exit static pressure is uniform, as we have indeed created in the laboratory work. This is the closest to uniform flow that one can approach, but only in the partial sense presented here. It should also be noted that the local passage flow rate cannot go to higher values than implied by the total pressure limit as shown on the sketch (point 3). The distribution of passage flow rates, δm<sub>i</sub>, must balance the implied overall system curve flow rate, hence illustrating the complex process of adjusting individual diffuser passage flow rates to meet this overall condition. It seems that this may be just what is revealed in the Figs. 1-12, i.e., the rebalancing of flow between individual diffuser passages to meet the overall stage pressure on the resistance line.

One pertinent question has been resolved and that is: what is the average pressure at r<sub>2</sub>? A calculation of the integrated average pressure from the Fourier series representation is in excellent agreement with the a<sub>0</sub> term of the series itself. The user must recall, however, that this is only the average on the front surface and that the distribution on the rear surface has its own set of issues (see Japikse and Krivitzky, op cit.).
### 3. MODELING AND CFD EXAMINATION

#### 3a. Modeling

Modeling is complex. Current thinking is to look at individual flow per diffuser passage as an iterative process where certain conservation principles must be met, including: 1) Overall mass flow rate must equal the sum of all passage flows; 2) Variations in diffuser inlet angular velocity components must balance over 360°; 3) the mass averaged $p_{05}$ must sum to the values given on the stage characteristic, 4) either $p_2$ data is used directly or a simplified Harmonic equation is used for $p_2$ representation (3 or 4 terms) and possibly 2 (or more) phase angle shift variables with these 5 or 6 variables found by iterative search methods to meet the conditions above.

The representation for the Fourier expansion is: $$p_2(\theta) = a_0 + \sum (a_n \cos(n(\theta + \delta))) \quad \text{for } n = 1, 2, 3, \dots, \infty$$

This equation represents the content of Figures 1-12. The plotted harmonic amplitudes are the $a_n$ terms and the phase angle shifts are the $\delta_n$ terms. For modeling purposes, it appears that the $a_0$, $a_1$, and $a_{2d}$ must, at least, be included, plus usually one or two more terms. For phase angle, the first two terms and sometimes up to the 5th term must be included; the effects of the small shifting of phase angles across the modal range is not clear and will require analytical testing. Finding the unknown values for a design case is a challenge. Some boundary conditions and basic principles are imposed, written for the 1st passage such as:

Mass flow rate: $m = \sum \delta m_i$

Stage total pressure: $p_{05} = \sum (p_{05i} * \delta m_i) / \sum \delta m_i$

Diffuser passage loss: $p_{05i} = p_{02i} + K*(p_{02i} - p_{2i})$

Pressure limits: $p_5 < or = p_{05}$

Impeller exit total pressure: $p_{02i} = f$ (impeller model per $p_{2i}$ with $p_5$ fixed); (resulting $p_{05i}$ gives the channel location and $\delta m_i$ per Figure 14)

Stage Power: $P = \sum W_i * \delta m_i$

Typical modeling unknowns might include 5 $a_i$ and 5 $\theta_i$ values, sometimes less. The $\delta m_i$ for each passage will result from the modeling process. Since developing analytical tools is contemplated, it is useful to know as much as possible about the $a_i$ and $\theta_i$ characteristics as possible. Figures 15 and 16 (on the next page), based on Figures 1 and 2, help to build this understanding.

<figure><img src="image_16.png" /><figcaption>Figure 15. Active amplitude coefficients for the Figures 1, 2 configuration plotted vs. the overall mass flow rate (60 mm-Bd8, Double Divergence Diffuser, 135,000 rpm)</figcaption></figure>
Figure 15 shows the key amplitude coefficients for the first example above. Only four terms are moderately strong contributors to the overall Fourier series data match. The variations with mass flow are modest, but noticeable. Figure 16, below, gives the phase angle coefficients as a function of mass flow rate; again, not all terms are active and some of the terms plotted might be negligible as well, such as a3, a8, and a10. The spike value in Figure 16 corresponds to deep choke and might safely be overlooked. It will be necessary to assess a replica of this variation through the modeling process.

<figure><img src="image_17.png" /><figcaption>**Figure 16.** Active phase angle coefficients for the Figures 1, 2 configuration plotted vs. the overall mass flow rate (60 mm-Bd8, Double Divergence Diffuser, 135,000 rpm)</figcaption></figure>

To solve this system of equations, an optimization search method will be required to vary the (10?) unknowns until the best $p_2$ distribution is found that preserves the overall mass flow rate and the anticipated $p_{05}$. These calculations must include a full set of flows along a speedline. N.B.: Instead of a constant $p_5$ (see 'shelf' in Figure 14), a $p_5$ distribution with theta is certainly an alternative to model volutes and other downstream devices.

### 3b. CFD Modeling

CFD discussions have been conducted with three different commercial CFD code groups. This data was fully unexpected by all groups and no suggestions for CFD evaluation has been available from any of the three at this time. Current thinking of the authors is that empirical wave forms may have to be developed and then CFD codes trained to match said forms, but even this concept is presently under strong review. Clearly, periodicity does not exist in these flows and hence nearly all common commercial codes cannot presently handle this class of flow physics by current practice.

To approach a good CFD modeling of these flows, it appears that one will need a full 360° model (no periodicity assumption), unsteady computational

algorithms, uniform inlet conditions and, for the data sets presented herein, an imposed constant static pressure at diffuser exit and the total mass flow for the entire flowfield, but not for each passage. It may still be necessary to train the system to a preferred $p_2$ pattern, but this last point needs more research.

Additionally, the flow analysis is complicated by a lesson learned for studies with a vaneless diffuser. Non-circumferential pressure variations were also found in a purely vaneless diffuser build and these had the largest amplitudes yet seen, except for the 2% coupled case. This is part of a separate but related investigation.

## 4. CONCLUSIONS

It has been demonstrated that the common state for impeller exit static pressure for radial compressor and pump stages is not uniform but rather a complex balance of various harmonics with modulated amplitudes and lower harmonic plus vane number phase shifts. A variety of higher order, but lower scale, effects are also seen in the data. Increased impeller vane number or reduced vaneless gap increased coupling has been shown to increase distortion and flow field complexity (more phase shifting and more amplitude variation for the different harmonic terms). Precise periodicity does not exist,
and while this assumption may be quite helpful to guide some designs, many problems will not be precisely modeled with the periodicity assumption. It is hypothesized that each diffuser passage can have a different flow rate to reach the intended exit total pressure. This study of the pressure field asymmetries has long been anticipated in the reviewed works of Eckardt, Japikse, Berten, and Harley, covering a span of 40+ years; hopefully these predecessors will find some satisfaction in the present data.

## 5. FUTURE WORK

Some of the future work will attack the more obvious issues. We hope to clock or index the test covers so that, in two subsequent tests, we can double the number of $p_2$ taps and hence look at finer scale details and correctly resolve higher number diffuser vane sets. More work must be conducted to establish the related distribution on the rear side of the flow passage. Further tests with the fully instrumented cover and blanked off collector ports should also be made while carefully checking waveforms. For a select case, probably the channel diffuser first, we wish to install $p_{05}$ probes in the center of each passage and locate $p_5$ taps in each individual passage as an indicator of the local flow per passage and then plot these values similar to the $p_2$ data. Additionally, we might test some of the hypothesis by creating a 1D branched circuit model of the entire stage.

At a higher level, we continue to look for very high fidelity CFD and continue our discussions around the industry to find them. The use of our L2F velocimeter to study details in each individual passage, at blade passing frequency, is an additional option.

## 6. ACKNOWLEDGMENTS

The authors wish to acknowledge the excellent work of the test technicians, Tom Dansereau, Rob Burrell and their Director, Andrew Provo and Advisor David M. Karon. Likewise, many machinists in the Concepts NREC shop, under the direction of Sean McDermott, are to be thanked for superb parts. Finally, Eric Krivitzky is thanked for his continuous support of this work and for providing the coding necessary to process the harmonic analysis.

## 7. REFERENCES

[1] Eckardt, D., and Trültzsch, K. J., 1977, "Radial Compressor Report No. 182", Research Report on Combustion Machinery, Research Institution DFVLR, Köln 90, Germany, June 30, 1977.

[2] Japikse, D., and Osborne, C., 1982, "Vaneless Diffuser, Return Bend and Return Channel Performance Investigation", TN-346, Creare R&D, Inc., and Technical Memorandum No. 7, Concepts ETI, Inc.

[3] Hinch, D. V., Japikse, D., and Yoshinaka, T., 1997, "Final Report for the Advanced Diffuser Consortium", Advanced Diffuser Performance for Centrifugal Compressors and Pumps, Phase V, Technical Memorandum No. 565, Concepts ETI, Inc.

[4] Berten, S., Hentschel, S., Kieselbach, K., and DuPont, P., 2011, "Experimental and Numerical Analysis of Pressure Pulsations and Mechanical Deformations in a Centrifugal Pump Impeller", ASME-JSME-KSME 2011 Joint Fluids Engineering Conference: Volume 1, Symposia – Parts A, B, C, and D, Hamamatsu, Japan, July 24-29, 2011.

[6] Harley, P., Spence, S., Filsinger, D., Dietrich, M., and Early, J., "Experimental and Numerical Benchmarking of an Improved Meanline Modelling Method for Automotive Turbocharger Centrifugal Compressors", 2015, ASME Turbo Expo 2015: Turbine Technical Conference and Exposition, Montréal, Canada, June 15-19, 2015.

[7] Japikse, D., and Wight, S. E., 2015, "High-Performance Diffusers with Strong Impeller-Diffuser Coupling", Advanced Centrifugal Pump and Compressor Consortium for Diffuser & Volute Design, Phase VI, Technical Memorandum No. 1817, Rev. 4, Concepts ETI, Inc.