A Study on the Deep-Surge Frequencies in Various ...

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International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2017.10.4.363

Vol. 10, No. 4, October-December 2017 ISSN (Online): 1882-9554

Original Paper

A Study on the Deep-Surge Frequencies in Various Conditions of

Axial Flow Compressors and Flow-paths

Nobuyuki Yamaguchi1

1Department of Mechanical Engineering, Meisei University (retired)

2-1-25, Akanedai, Aoba-ku, Yokohama City, 227-0066, Japan, [email protected]

Abstract

Frequencies of deep surges and their behaviors in axial flow compressors were surveyed numerically. Relative surge

frequencies, normalized by the basic acoustical resonance frequencies, are seen to tend to lower together with increases

in the stalling pressure ratios, i.e. increases in the number of stages and the compressor tip speed, and also together with

increases in the flow-path sectional area ratios. However, it appears difficult to express simply the general behaviors of

the relative frequencies affected by the various factors. In order to know the essential behaviors, a modified reduced

surge frequency is proposed, which is a dimensionless number comparing the mass flow filling and emptying the

plenum volume in surge and the mass flow provided by the compressor. The modified reduced surge frequencies are

found to have or approach a definite and nearly constant value in conditions of deep surges. The parameter suggests the

fundamental mechanism of deep surges and could be used to determine approximate frequencies of deep-surges in

various conditions of compressors and flow-paths.

Keywords: Fluid Machine, Axial Flow Compressor, Surge, Analytical Simulation, Frequency, Fluid Dynamics

1. Introduction

Information on the oscillation frequencies in surge phenomena in compressors and fans are very important, although the horribly

large amplitudes of the related flows and pressures are likely to attract attentions. The frequencies are ones of essential keys to get an

insight into the vibrational nature of the phenomena and also ones of aspects to take preventive measures into consideration.

Nevertheless, the information on the surge frequencies does not appear to have reached the stage of generalized one.

Greitzer [1, 2] has proposed a basic parameter named “B”, which is a kind of reduced frequency paying attention to the flow-path

resonance frequency approximated by a Helmholtz resonator. Although it has been shown by him that various phase of surge

phenomena ranging from deep surges to stall stagnations could occur in dependence on the magnitude of the B parameter, the detailed

situation of the surge frequencies have not been studied. In the subsequent many studies with respect to surge phenomena, details of the

behaviours of the surge frequencies have curiously not been paid attention to. Maybe, it could have been more important at the stage to

manage to reproduce analytically the surge phenomena as truly as possible (for example, Boyer and O’Brien [3]), with the frequencies

regarded as some specific and individual data belonging to the particular cases. In the circumstances, the general and comprehensive

nature of the surge frequencies has not been made clear.

In some recent analytical experiences of the author in deep surges in multi-stage axial compressors (Yamaguchi [4, 5]), the deep-

surge frequencies tended to change much for changes in the compressor speeds, where the consistent nature of the behaviour was

difficult to grasp. Large deviations from the near-resonant conditions appear in high-pressure-ratio compressors and multi-stage

machines, although the frequencies appear to be very near the flow-path resonance frequencies for low-pressure fans and compressors.

Yamaguchi [4] suggested a dimensionless parameter of volume-modified reduced surge frequency in relation with stall stagnation

boundaries, but no further detailed investigations into the frequencies in deep surges have been conducted subsequently.

Characteristic scenes with respect to the surge frequencies could be summed into the following three groups according to the

author’s present stage of analytical experiences.

The first is the ordinary deep-surge situation where the deep surge means the one whose every surge loop encircles or passes

by the initial steady-state stalling point in the flow-vs-pressure domain. They are common ones both in site and in analyses, but

the information on the frequencies are rather by bits and pieces, having been not yet put to order. The frequencies are often seen to

be much lower than the flow-path resonance frequencies.

The second is the situation of very low surge frequency in high-speed multi-stage compressors. In analytical results on multi-

stage compressors, very low deep-surge frequencies often occur where the surge period is elongated by intervention of small

pressure peaks between neighboring ordinary major pressure peaks (Yamaguchi [6]). The small peaks were found to occur as

Received February 14 2017;accepted for publication June 12 2017: Review conducted by Hideaki Tamaki. (Paper number O17006J)

Corresponding author: Nobuyuki Yamaguchi, [email protected]

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incomplete recoveries resulting from temporary unstalling of rear-stages and its failure in the environment of whole stages in stall.

The situation is somewhat different from the ordinary deep-surge one described above.

The last is the situation of surge frequencies abruptly changeable in the immediate neighborhood of the stall stagnation

boundaries (Yamaguchi [8]). In the deep-surge zone very near the stagnation boundary, subharmonic deep surges often appear,

which could be considered to be a precursor of stall stagnation events. Stall stagnations here include not only decaying oscillations

converged onto some stalled point, but also mild surge loops outside of the steady-state stalling point. In the sense, in addition to

the zero frequency of the decayed condition, some frequencies other than zero could be present, particularly on the way toward the

converged conditions.

The above three groups are the main characteristic situations of surge frequencies distinguished in the author’s mind.

Particularly, the first two groups on the deep surges remain unclear, which the author considers necessary to make clearer. If a

reasonable formula for estimation of the deep-surge frequencies become available, it would contribute much in studying further

the essential dynamics of the deep surge phenomena, and also in planning the countermeasures to the flow oscillations in the

system. So, the present study surveyed deep-surge frequencies numerical-experimentally and devised a rule of thumb.

The last item with respect to the stagnation stall concerns has been already examined (Yamaguchi [8]) and the information in

the neighborhood of the stall stagnation boundaries will be only briefly described here.

The present study is based only on numerical results using a code for surge transient analysis, “SRGTRAN” developed by the

author (Yamaguchi [9]). The conclusions might be rather qualitative in the sense. The author wishes that the conclusions would be

substantiated experimentally in the future.

2. Outline of the Analyses

The contents described here are based on the results by use of a code SRGTRAN for the surge transient analysis and simulation,

developed by the author (Yamaguchi [9]).

Figure 1 is a schematic presentation of the flow-path system, consisted of a suction flow-path, a compressor, a delivery plenum,

and an exit valve path. The system has an open inlet and an exit terminated by a flow-adjusting valve.

The geometrical variables are as follows: Lc** and Lp: lengths of

the suction flow-path from the inlet to the compressor exit, and the

delivery plenum, respectively. Ac2 and Ap: sectional areas of the exit

of the last stage of the compressor for analysis, and delivery plenum,

respectively. At the same time, Ac2 and Lc** are the representative

sectional area and the length for normalization, respectively. The

plenum inlet is connected to the exit of the compressor immediately

downstream, thus the distance between both locations is assumed to

be very small.

The following dimensionless parameters are defined for features

of the flow-paths.

Sectional area ratio of the flow-path: AR=Ap/Ac2 (1)

Length ratio of the flow-path: LR=Lp/Lc** (2)

The suction flow-path and the delivery flow-path in the present

analyses are configured by fourteen control volumes (CVs), and

thirty-four CVs, respectively. The stages of the compressor are

represented by corresponding respective CVs. The suction flow-path

is given the same geometrical dimensions throughout the analyses.

Table 1 gives the main numerical figures and dimensions of the

compressors and the flow-paths for the analyses. The compressors are

designed for 11300rpm, having number of stages 1, 3, 5, and 9.

The stage characteristics of each stage is assumed to be the same,

and is shown in Fig. 2. Here,

Flow coefficient:

tst

stt

uA

Q (3)

Pressure coefficient: 2

1

1

21

)1(

t

stTp

Ptu

Tc

(4)

Temperature coefficient: 2

12

21

)(

t

TTp

Ttu

TTc (5)

Here, ut : representative tip speed of the compressor, Ast : average annulus area of the stage, πst : Stage total pressure ratio,

Qst : average volumetric flow in the stage, TT1 and TT2: total temperature at the stage inlet and the stage exit, respectively, CP:

specific heat at constant pressure, κ: ratio of heats. Subscript st stands for the concerned stage.

Fig. 2 Stage characteristics for each stage

Fig. 1 Schematic view of the flow-path system

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The stage characteristics requires to cover the wide range of flow from the turbine action zone to the reversed flow zone. Figure

2 is made up on the basis of the author’s experience and literature survey results. It is the same as the one that the author has

employed in the previous studies (Yamaguchi [4-9]).

The following situations are analyzed by use of code SRGTRAN, whose results are surveyed paying attention to deep surges

and the surge frequencies fs0.

Compressor rpms: 12000, 10000, 7000, and 5000.

Flow-path geometries

Length ratios LR: 10, 1, and 0.3.

Area ratios AR: 1-10.

Table 1 includes also steady-state stalling pressure ratios PR, evaluated by use of a stage-stacking procedure for the compressor

performances in the initial stage of the SRGTRAN analyses. They are required for computations of concerned parameters, such as

given by Eqs. (7), (9) and (11), and are convenient also to see the overall tendency as in Fig. 4.

Table 1 Main dimensions and numerical figures for analyses

Name Comp19 Comp15 Comp13 Comp11

Stages 9 5 3 1

Inlet Annulus Area of Stage 1:

Ac1 (m2)

0.10314 0.10314 0.10314 0.10314

Exit Annulus Area of

the Last Stage: Ac2 (m2)

0.0975 0.07688 0.06057 0.04125

Reference Radius: r t1 (m) 0.254 0.254 0.254 0.254

Design rpm (rpm0) 11300 11300 11300 11300

Design ut1 (m/s) 300.6 300.6 300.6 300.6

Suction Temperature (K) 288.2 288.2 288.2 288.2

Suction Pressure (Pa) 101300 101300 101300 101300

Stalling PR

12000rpm 5.38 2.87 1.97 1.28

10000rpm 3.52 2.16 1.62 1.19

7000rpm 1.48 1.47 1.28 1.09

5000rpm 1.2 1.14 1.04

Suction Length: Lc** (m) 4.631 4.323 4.17 4.093

3. Global Tendency of Deep Surge Frequencies

In the study, length ratios LR and sectional area ratios AR for the delivery flow-paths were varied with the suction flow-path

geometry kept the same. In the specified flow-path configurations, the deep-surge frequencies were surveyed by use of

SRGTRAN for variation of number in stages and rotational speeds of the compressor.

First of all, a relative surge frequency is evaluated from normalization of a resulting deep-surge frequency fs0 relative to a

basic resonance frequency f1 in the flow-path system.

Relative deep-surge frequency fs0/f1 (6)

Here, the basic resonance frequency f1 is calculated for the acoustic resonance condition of a still air column in the flow-path

identical with the surge analysis with the boundary conditions of the open inlet and the closed exit. The pressure is supposed

uniform to be the atmospheric one throughout the path. The temperatures upstream and downstream of the compressor are

assumed to be equal respectively to the suction temperature and to the exit temperature for the compressor stalling condition. The

resonance frequency thus evaluated could naturally be a rather approximate one, since, in the actual surge situations, the pressure

and the temperature could significantly change locally and temporarily. But as a measure of the related frequencies it would be

difficult to replace by some other simple parameters.

Figure 3 shows some examples of the tendencies of the relative deep-surge frequencies against the relative compressor rpms

normalized by the design rpm (rpm0 of 11300 rpm, here). The data conditions are given in the explanatory notes in terms of the

stage numbers, flow-path area ratios AR and length ratios LR. For compressors of smaller numbers of stages and for the lower

speeds, the relative surge frequencies are seen to be near unity; namely, the deep-surge frequencies are near the system resonance

frequencies.

On the other hand, for compressors having more stages, the relative deep-surge frequencies are seen to tend to lower, and the

tendency becomes stronger for higher speeds.

Figure 4 shows the same data plotted against the compressor stalling pressure ratios, PR. For the values of PR near unity, the

relative deep-surge frequencies are near unity, but lower for increasing values of PR. For the value of PR greater than 3.5, the

relative surge frequencies are seen to fall more abruptly, approaching toward 0.1. The behavior of the relative deep-surge

frequencies could be considered more consistently in terms of the stalling pressure ratios in Fig.4 than in terms of the compressor

speed ratios in Fig. 3.

In Figs. 3 and 4, it is generally understood that, the relative deep-surge frequencies fs0/f1 tend to fall from unity toward lower

levels for higher stalling pressure ratios PR. The behavior appears to change in rather complicated manners affected both by the

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stalling pressure ratios influenced by the compressor stage numbers and the speeds, and by the flow-path geometries, typified by

the sectional area ratios AR and the length ratios LR.

Fig. 5 Contours of relative deep-surge frequencies in relations of the sectional area ratios of flow-paths and the stalling pressure

ratios. (a) Nine-stage compressor, Comp19, (b) Five-stage compressor, Comp15, and (c) Single-stage compressor, Comp11.

(1) flow-path length ratio LR of 0.3, (2) LR of 1.0, and (3) LR of 10.0.

Figure 5 shows contour maps of the relative deep-surge frequencies fs0/f1 with the abscissa for the flow-path sectional area ratio

AR and the ordinate for the stalling pressure ratios PR. Figure 5(a), (b), and (c) are for compressors having stages 9, 5, and 1,

respectively. Figure 5(1), (2), and (3) are for flow-path sectional area ratios LR of 0.3, 1, and 10, respectively. They are contour

Fig. 3 Tendency of relative deep-surge frequencies

against variations in the compressor speeds

relative to the design one.

Fig. 4 Tendency of relative deep-surge frequencies

against the compressor stalling pressure ratios

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maps corresponding to Fig. 4, in which only a few selected typical geometrical conditions are shown. In the contour maps in Fig.

5, the area ratios are given basically by about ten data points horizontally, except some failed-calculation conditions.

As a whole, the relative deep-surge frequencies tend to lower together with increasing flow-path area ratios AR and increasing

stalling pressure ratios PR where both ratios appear to affect the tendency in some manner of combination. Effects of the length

ratios are not clear in these figures.

The contour lines of the relative surge frequencies often show some local irregular changes in the behavior, particularly near

the borders. The causes are not understood at present. The situations around the boundaries are put aside for future study. The

present study pays attention mainly to the deep-surge regions.

4. Behaviors of Volume-modified Reduced Surge Frequencies

It is observed in the previous section that the global behaviors of the relative deep-surge frequencies fs0/f1 are affected in a

combined manner by the flow-path sectional area ratio AR and the stalling pressure ratio PR. The nature of the behaviors have not

yet been understood.

Here, let us pay attention to the behaviors of the volume-modified reduced surge frequency fRPVVs (Yamaguchi [4]). It is a

dimensionless parameter resulted from a ratio of the mass flow in the filling and emptying action during the surge in the delivery

volume to the mass flow supplied by the compressor. It is expressed as follows;

)1

1(1

0

2PRA

A

u

Lff

C

P

t

PsRPVVS (7)

As can be understood, it is constructed of a reduced frequency with respect to the surge frequency fs0, the delivery length Lp, and

the rotor tip speed ut, modified by the sectional area ratios AR, and a pressure-ratio-effect term.

Although it could have a physical validity as it is and shows relatively well-correlated behaviors of data (Yamaguchi [6]), some

further trials have yielded the following improved formula having a better correlation.

LR

LRff RPVVsRPVVsm

1 (8)

More concretely,

)1

1()(

1

**

0

2PRA

A

u

LLLff

C

P

t

PCPsRPVVsm

(9)

The right-hand side of the above formula suggests indirectly that the representative length should reflect not only the effect of the

delivery plenum length LP, but also that of the whole length of the flow-path (LC**+LP). It could be reasonable in consideration of

the nature of the surge phenomena as the flow dynamics in the whole flow-path system.

Thus the effective representative length of the concerned flow-path will be defined as follows;

)(**

PCPeff LLLL (10)

Hereafter, the modified reduced surge frequency fRPVVsm is paid attention to.

As another parameter, the following area-pressure ratio parameter, or modified area ratio, is employed.

)1

1(1

*

2PRA

AAPR

C

P (11)

It reflects the global effects of the levels of the stalling pressure ratios on the area ratios.

Figures 6-9 show behaviors of the relative deep-surge frequencies fs0/f1 and the modified reduced deep-surge frequencies

fRPVVsm against the modified sectional area ratios APR* for the compressors having single stage through nine stages, respectively.

The relative deep-surge frequencies fs0/f1 change much from near unity to roughly 0.2, affected by changes in the modified flow-

path area ratios and the length ratios. The behaviors of the relative surge frequencies appear difficult to describe simply.

On the other hand, the modified reduced surge frequencies fRPVVsm show a relatively well-correlated manner of behaviors,

tending to the value of nearly 0.1. The green-colored chain-line approximates the behavior.

For the nine-stage compressor, Comp19, however, scatters in the data are seen to increase in Fig. 9, where upper data points

and lower ones having the same marks show fs0/f1, and fRPVVsm, respectively. The differences in the behavior tendencies suggest

some other features of surge phenomena.

Thus, although the relative surge frequencies fs0/f1 are convenient to estimate the levels of the frequencies directly in

comparison of the system resonance frequencies, the modified reduced surge frequencies fRPVVsm are better to reflect the essential

nature of the deep surge phenomena. It could be said that the deep-surges occur so that the values of the modified reduced surge

frequency fRPVVsm tend to have or approach some definite value.

All the data points of fRPVVsm vs. APR* are gathered together in Fig. 10. The approximate tendency could be given

approximately by the following two straight lines shown as dotted lines.

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Fig. 6 Behaviors of relative deep-surge frequencies fs0/f1

and modified reduced surge frequencies fRPVVsm against

modified sectional area ratios APR* for the single-stage

compressor, Comp11



modified sectional area ratios APR* for the three-stage

compressor, Comp13



modified sectional area ratios APR* for the five-stage

compressor, Comp15

Fig. 9 Behaviors of relative deep-surge frequencies f

s0/f1 and modified reduced surge frequencies fRPVVsm

against modified sectional area ratios APR* for the

nine-stage compressor, Comp19

Fig. 10 Tendency of the modified reduced deep-surge frequencies fRPVVsm against

the modified area ratios APR* for all compressors including Comp11 through to

Comp19 (superposition of all points in Figs. 6-9)

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1.0RPVVsmf for APR* > 0.7 (12)

*APRfRPVVsm for APR* < 0.7 (13)

For sufficiently large values of APR*, the modified reduced frequencies fRPVVsm tend to have a definite value, and for smaller

values of APR*, the frequencies fRPVVsm tend to change proportionally to the square root of the APR*.

The effects of the flow-path area ratios and the stalling pressure ratios are put together into APR*, and the effect of the flow-

path length ratio LR is included in the effective representative length √{Lp(Lc**+Lp)}.

In addition to the above, the data points encircled by a dotted ellipse line in Fig. 10 indicate occurrence of deep surges having

further lower frequencies for the nine-stage compressor Comp19 at 12000rpm. As shown later, the situation suggests ultra-low

frequencies caused by interventions of incomplete surge recoveries in high-speed operations of high-pressure ratio compressors.

5. Tendency of Surge Frequencies

The behavior of the following reduced surge frequency is surveyed.

t

effs

RPsmu

Lff

0 (14)

The parameter fRPsm has an ordinary simple form of reduced frequency easier to understand. At the same time, the following

relation holds from the definition of fRPVVsm, Eq. (9);

*

0

APR

f

u

Lff RPVVsm

t

effs

RPsm (15)

Figures 11-14 show the behaviors of the reduced surge frequencies fRPsm against the modified area ratios APR*. They have

nearly the same tendencies, all of which are superposed in Fig. 15, where the data points form a relatively narrow band of

distribution. In correlation with Eqs. (12) and (13), the tendencies are approximately expressed as follows;

#2-1 Improved

Fig. 11 Behavior of reduced surge frequency fRPsm

against the modified area ratio APR* for the

single-stage compressor Comp11


against the modified area ratio APR* for the three-

stage compressor Comp13


against the modified area ratio APR* for the five-



against the modified area ratio APR* for the nine-


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*1

APRfRPsm for sufficiently large values of APR* (16)

*1

APRfRPsm for sufficiently small values of APR* (17)

Figures 10 and 15 could be the most compact representation of the macroscopic behaviors of the deep-surge frequencies fs0, in

terms of the major parameters, such as the compressor stalling pressure ratios PR, and the flow-path geometries given by AR and

LR.

However, some deviations from the average behavior are seen for the data points in the shaded zone in Fig. 14 and encircled

by a dotted ellipse line in Fig. 15, which are for the nine-stage compressor Comp19 at 12000 rpm, indicating occurrences of

further lower surge frequencies. The situation suggests ultra-low frequencies caused by interventions of incomplete surge

recoveries in high-speed operations of high-pressure ratio compressors, as shown later.

6. Tendency of Surge Frequencies for Changing Compressor Speeds

In order to see more plainly the effects of the compressor speeds and the flow-path geometries, behaviors of the modified

reduced deep-surge frequencies fRPVVsm against the flow-path volume ratios VR are shown in Figures 16-19. The groups for the

same length ratio LR are seen to gather together. Here

VR=AR*LR (18)

A point of intersection of a vertical line for the specified value of VR on the abscissa with a curve for a wanted compressor rpm in

the data group for the specified LR value will give the value of fRPVVsm for the situation. For the specified values of VR and LR, the

value of AR is determined by Eq. (18). Thus the effect of compressor tip speeds or rpms on fRPVVsm is known. The design rpm for

the present study is 11300 rpm.

The tendency of the behaviors of fRPVVsm affected by compressor rpms thus obtained are seen to change relatively little for

conditions of a specified compressor and a specified geometry of the flow-path, in many situations. It could be said that

frequencies of deep surges are determined so as to give some definite value of the modified reduced surge frequencies fRPVVsm

during the speed changes in most situations.

However, for the single-stage compressor Comp11, the situation is unsettled yet where the level of the fRPVVsm values increases

with increases in AR, and at the same time, with increases in the compressor rpms. As a whole, it is in the range of 0.02-0.07, less

than 0.1of the ordinary level for other compressors.

For the nine-stage compressor Comp19, the level of the fRPVVsm values keeps nearly 0.1 for compressor speeds lower than the

design speed, but it drops to further lower level for a higher speed, 12000rpm, and for the flow-path length ratio LR of 0.3 and 1. It

corresponds to the situation in Fig. 9 where the relative surge frequencies fs0/f1 approach 0.1. The situation will be explained later

in relation with Figs. 21 and 22.

Fig. 15 Tendency of the reduced deep-surge frequencies fRPsm against the modified

area ratios APR* for the compressors having a single stage through to nine stages

(superposition of all points in Figs. 11-14)

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7. Surge Loops

Some examples of surge loops will be shown here in relation with the variations in the frequencies.

7.1 Three-stage compressor, Comp13, LR=0.3 and AR=2

Figure 20 shows surge conditions for the three-stage compressor, Comp13, and the flow-path condition of LR of 0.3 and AR of

2, (1) the pressure oscillogram, (2) the mass flow oscillogram, and (3) surge loops in the pressure-mass flow domain at the exit of

the compressor last stage. The compressor speeds are (a) 12000 rpm, (b) 10000 rpm, (c) 7000 rpm, and (d) 5000 rpm. Numerical

figures 2, 15, and 16 in the explanatory notes in Fig. 20(a) and (b) stand for near the inlet of the suction flow-path, the inlet of the

first stage of the compressor and the exit of the last stage, respectively. The surge behavior at 5000 rpm in Fig. 20(d) shows loops

completely within the stalled zone, meaning the stall-stagnation condition. The surge behavior at 7000 rpm in Fig. 20(c) shows

loops passing just near the stalling point, meaning the conditions near the stall-stagnation boundary. The surge behaviors at

10000rpm in Fig. 20 (b) and 12000 rpm in Fig. 20 (a) are deep surges, respectively. The surge frequencies are seen to change little

in these environments.

Detailed situations with respect to the stall stagnations can be found in Yamaguchi [8]. Very near the stall stagnation

boundaries, subharmonic deep surges can often be observed. In the stalled zone after the stagnation, small loops of mild surges

can be kept as seen in Fig. 20(b) or can decay to a point in the zone.

7.2 Nine-stage compressor, Comp19, LR=1 and AR=3.5

Figure 21 shows surge conditions for the nine-stage compressor, Comp19, and the flow-path condition of LR of 1.0 and AR of

3.5, (1) the pressure oscillogram, (2) the mass flow oscillogram, and (3) surge loops in the pressure-mass flow domain at the exit

of the last stage of the compressor. The compressor speeds are (a) 12000 rpm, (b) 10000 rpm, and (c) 7000 rpm. All the cases are

in the deep-surge situations.

The distinctive feature of the situation is that, as seen in Fig. 21(a-1), small peak(s) exist between neighboring high pressure

peaks of ordinary pressure recovery and re-stalling, encircled by a small red dotted-line circle. The situation is observed at 12000

Fig. 16 Behaviors of the modified reduced surge

frequency fRPVVsm against the flow-path volume ratio

VR for the single-stage compressor Comp11



VR for the three-stage compressor Comp13



VR for the five-stage compressor Comp15


frequency fRPVVsm against the flow-path volume ra

tio VR for the nine-stage compressor Comp19

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rpm, which is higher than the design rpm of the compressor. Also in the behaviors of the mass flows, corresponding small

recovery is observed, as can be seen in Fig. 21(a-2) and (a-3). It immediately returns to near-zero flow, having failed in achieving

complete recovery from the stalled conditions. The phenomenon might be named incomplete recovery.

The detailed situation of the incomplete recovery is shown in Fig. 22 for a complete surge period. Figure 22(a) and (b) show the

oscillograms of the pressure and the mass flow, respectively, where the respective tag numbers indicate the inlet (15) and the exit

(16) of the compressor, the plenum (29 and 39), and the flow-path exit (49). A small peak of pressure recovery is located around

time k of 110000, halfway between the neighboring major peaks of pressure while the mass flow shows an abrupt increase from a

near-zero negative level to a rather high positive level, followed by an abrupt return to the near-zero negative level previously

observed. Figure 22(c) shows the pressure-vs-mass flow loops at the compressor exit and in the plenum. The nearly-horizontal

movements of both trajectories in the bottom zone are corresponding to the small recovery. The movement appears to be induced

by relatively free oscillation of the flow in the plenum. The mass flow in the plenum is seen to be oscillatory also in Fig. 22(b).

Figure 22(d) shows the behaviors of the coefficients of flow and pressure for stages 1, 5 and 9. Before the incomplete recovery all

the flow coefficients are negative or in the reversed flow condition. In the incomplete recovery, the stage flow coefficients recover

abruptly from the negative or reversed condition to a positive large one. In the condition, the stage flow coefficient increases

stage-by-stage from a near-stall one of the first stage toward the very large one of the last stage reaching as far as the zone of

negative pressure rise or turbine action, as can be seen in the levels of the ninth-stage coefficients of flow and pressureφ9 andψ9.

The recovery fails abruptly because of the abrupt decrease in the flow coefficient down to negative or reversed one and resulting

failure in increasing the pressure ratios.

Similar phenomena have been observed analytically in multi-stage compressors for other conditions of compressor speeds

and flow-path geometries. In the previous study (Yamaguchi [6]), it is found that the small peaks are corresponding to

instantaneous recoveries and re-stalling of only some rear stages during the low-pressure phase in the deep surge cycles. The

intervention of the incomplete recoveries elongates the surge period time, which, in turn, lower the surge frequency. Resulting

values of fRPVVsm and fRPsm tend also to lower. More than one small peaks of the incomplete surge recoveries could appear also in

the same surge period, depending on the situation.

Fig. 20 Variations of surge behaviors for the three-stage compressor, Comp13 for changes in the compressor

speeds. LR=0.3 and AR=2. (a) 12000 rpm, (b)10000 rpm, (c) 7000 rpm, and (d) 5000 rpm. And, (1) pressure

oscillogram, (b) mass flow oscillogram, and (c) surge loop p vs. W at the compressor exit.

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Fig. 22 Local conditions of Comp19 at 12000 rpm for the flo

w-path geometry of LR of 1 and AR of 3.5. Conditions near th

e incomplete recovery are expanded in time.

Mass

flow

Fig. 21 Variations of surge behaviors for the nine-stage compressor, Comp19 for changes in the compressor speeds.

LR=1 and AR=3.5. (a) 12000 rpm, (b)10000 rpm, and (c) 7000 rpm. And, (1) pressure oscillogram, (b) mass flow

oscillogram, and (c) surge loop p vs. W at the compressor exit. Red dotted-line circles in Fig. 20(a) indicate

intervening incomplete surge recovery phenomena.

Fig. 22 Local conditions of Comp19 at 12000 rpm for the flow-path geometry of LR of 1 and AR of 3.5.

Behaviors around the incomplete recovery are expanded in time.

Mass flow

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Concerning the situation, it is supposed as a possibility that, although the deep-surge frequency should have been given

fundamentally by the fRPVVsm value of approximately 0.1, the deep-surge recoveries in the situations might have been degenerated

into alternate recoveries of complete one and incomplete one, one after another, which could apparently have brought about the

apparently lowered deep-surge frequency, thus reducing the apparent fRPVVsm value to roughly a half of 0.1, or 0.03-0.06. It might

be a result of delicate interactions among the free oscillations in the plenum, the compressor stage working conditions changing

stage-by-stage, and other origins. At the present stage, the degeneration argument is only a proposition, and the author would like

to continue to investigate the mechanisms of the lowered deep-surge frequencies further.

It is to be careful here also that, in the present study, the stage characteristics of the rear stages are assumed not to change in the

environment of stalled front stages. If the stalling of the front stages could have significant effect on the characteristics of the rear

stages, the situation might change. The incomplete recovery phenomena have been observed only analytically as far as the author

knows. The author would like to mention the phenomena only as an analytical possibility.

8. Summary on the Deep-Surge Frequencies From the above analytical observations, it could be said that the deep-surge frequencies fs0 are determined so as to give the

modified reduced surge frequency fRPVVsm basically a definite value, for example 0.1 or some near-by value. The surge frequencies

fs0 are generally lower significantly than the acoustical resonance frequency in the system. The work required for the surge action

of filling and emptying the delivery volume could have reduced the frequency. The condition could be named a deep surge with a

lowered frequency caused by the filling and emptying action of surge.

On the other hand, for low-pressure-ratio compressors having a small number of stages, the relative surge frequency fs0/f1 is

nearly unity, which situation could be named a resonant or near-resonant surge. In the situation, the values of fPVVsm tend to be

lower somewhat than 0.1.

The resonance frequency f1 is principally the fundamental one for surges, which could be suppressed gradually in the presence

of the convection effects in the surge actions of large amplitude for larger pressure ratios.

For high-speed operation of high-pressure multi-stage compressors, both fs0/f1 and fRPVVsm tend to drop to levels lower than

those expected for ordinary deep surges, resulting in the lowered levels of fs0/f1 of roughly 0.1 and fRPVVsm of roughly 0.03-0.06. It

could be named a deep surge with a very low frequency caused by both of the surge action and the intervention of incomplete

recoveries. As a possibility, the situation could have been brought about by degeneration of the deep surges for fRPVVsm of roughly

0.1 into surges of alternate deep one and incomplete one where the apparent fRPVVsm values are reduced to about a half of 0.1. The

phenomenon requires further study, since it is uncertain

at the present stage whether it occurs actually or it is

only an analytical possibility, and at the same time, the

cause has not been made clear yet.

9. Conditions at Stall-Stagnation

Boundaries

Here, frequency conditions at the stall stagnation

boundaries are examined along the line of thinking

described above. In the neighborhood of stall stagnation

boundaries, a variety of surge behaviors have been

observed. For example, in the stalled zone, continued

mild surges and/or convergence or decaying onto a point

on the stalled branch are observed, depending on the

situations. In the deep-surge zone, deep surges with

subharmonic frequencies could often occur very near

the stagnation boundaries (Yamaguchi [8]). The

subharmonic surges are events containing both a deep

surge loop and one or more mild-surge loops, both

having nearly the same surge period, thus resulting in a

global surge period of some integer multiple of the basic

surge period, i.e., a deep surge having the subharmonic

frequency. The occurrence of subharmonic surges could

be regarded to be a precursor of the stall stagnation.

On the basis of examinations and considerations on

such many numerical results, the possibility of stall

stagnations is considered to be deeply related with the

condition whether the initial infinitesimal oscillation

having an acoustical resonance frequency could be

amplified or damped in the particular situation. In

general, surge oscillations could develop through

amplification of the acoustical disturbances to large-

amplitude disturbances having another frequency under

influence of the non-linear nature of the phenomenon. In

Fig. 23 Behaviors of the modified reduced resonance

frequency fR1mstg and the flow-path sectional area-pressure

ratio APRstg for the stall stagnation boundaries against the

flow-path length ratio LR

375

the sense, the concerned frequency to be paid attention to in the stall stagnation event is not the surge frequency fs0 but the

acoustical resonance frequency f1.

Thus a parameter of volume-modified reduced resonance frequency with particular respect to the resonance frequency has

been proposed and it proves to give a relatively well-correlated manner of behavior of the stagnation boundaries (Yamaguchi[6,

8]). Figure 23 shows the behaviors of the flow-path sectional area-pressure ratio APRstg and the reduced resonance frequencies

fR1mstg, given by Eq. (20) below, against the flow-path length ratios LR for the abscissa. Here, the subscript stg stands for the

conditions at the stagnation boundaries. The reduced resonance frequency given below by Eq. (20) has been improved from that

proposed previously by Yamaguchi [6], i.e., Eq. (19). The data are for compressors having 1-9 stages, at 10000 rpm, and a single-

stage compressor at 15000-6000 rpms.

The previous paper (Yamaguchi [6]) has shown that the modified reduced resonance frequency fRPVV1 given below by Eq. (19)

can reduce the many numerical-experimental data at the stall stagnation boundaries into a rather well-correlated curve. The term

√(Mt*PR) is added to adjust the tendency of the numerical-experiment data.

PRMPRA

A

u

Lff

t

P

t

PRPVV

C

1)

11(

1

11

2

(19)

Furthermore, if the effective representative length of the flow-path Leff is employed in place of Lp in Eq. (19) similarly to the

derivation of Eq. (9), the correlation might become better. The following formula is proposed;

PRMPRA

A

u

LLLff

t

P

t

pcp

mstgR

C

1)

11(

)(

1

**

11

2

(20)

The stagnation boundary conditions given by the above reduced resonance frequency fR1mstg are shown in the lower part of Fig. 23.

The improvement has proved effective in reducing the dependence of the parameter on the flow-path length ratios.

The values of the improved reduced resonance frequency fR1mstg at the stall stagnation boundaries are given in average by the

following level of magnitude, though some scatters are present for variations in the length ratios and compressors.

fR1mstg ~ 0.07 (21)

With respect to occurrence of surges, when the value of fR1mstg given by the right-hand side of Eq. (20) is smaller than the

threshold value given approximately by Eq. (21), the stall will decay or stagnates. On the other hand, when the value of fR1mstg is

larger than the threshold value given by Eq. (21), the infinitesimal disturbance having the acoustic frequency f1 will be amplified

and transformed into a deep-surge oscillation having a frequency fs0 evaluated from Eqs. (12) and (13).

The area-pressure ratio has the following definition at stall stagnation boundary;

APRstg=AR*PR (22)

With respect to the flow-path geometrical conditions, minimum values of APRstg range roughly from 1.5 to 2.5 for LR of 1 to 10

in large, as shown in the upper part of Fig. 23.

The threshold condition given above by Eqs. (20) and (21) is similar in form to the following one in Greitzer’s B parameter

(Greitzer [1 and 2]).

B = 0.8 (23)

Here, the B parameter is given by the following formula.

mcc

m

uLfLf

uB

11

1

4

1

4 (24)

When the resonance frequency f1 is approximated by the Helmholtz’ resonator formula, then Eq. (24) is given as follows;

cc

ppm

cc

pm

LA

LA

a

u

LA

V

a

uB

22 (25)

Here, um: average peripheral speed of the rotor blades, a: speed of sound, and Lc: length of the suction flow-path.

Yamaguchi [6] evaluated the threshold values of the Greitzer’s B parameter for cases of analytical stagnation conditions over a

wide variety of compressors and flow-path geometries. The results has indicated that the threshold values of the B parameter

distribute over a wide range roughly from 0.2 for small LR values to 10 for large LR values, in which the B value of 0.8 stands for

some restricted area near the average conditions. In comparison of these results, Eqs. (20) and (21) could be said to predict the

stall stagnation condition more reasonably.

10. Conclusions

Behaviors of deep-surge frequencies over a relatively wide range of conditions of compressors and flow-path geometries were

surveyed. The following conclusions are obtained.

(1) When the stalling pressure ratios are near unity and the pressure-modified sectional area ratio APR* are relatively small, the

relative surge frequencies fs0/f1 tend to be near unity, namely the deep-surge frequencies are near the resonance one. The

surge could be named resonant surge or near-resonant surge.

(2) With increases in the stalling pressure-ratios PR and in area ratios AR, the relative deep-surge frequencies fs0/f1 decrease,

falling to as low as 0.2-0.1.

(3) A modified reduced surge frequency fRPVVsm is important in describing the deep-surge situations of higher stalling pressure

ratios. It is a dimensionless parameter evaluating the filling and emptying actions in surges. It contains the surge frequency

376

fs0, the flow-path sectional area ratio AR, the compressor tip speed ut, the representative flow-path length including the

delivery length Lp and the total length of the flow-path (Lp+Lc**), and a pressure-ratio-affected term. Although the relative

surge frequency indicates only the frequency level, the modified reduced surge frequency is a more essential parameter

related with the surge actions.

(4) Deep surges tend to have the parameter fRPVVsm of roughly 0.1 or tend to approach the level. It could be the general condition

determining the deep surge frequencies. The condition could be named a deep surge with a lowered frequency caused by the

filling and emptying action of surge.

(5) For situations of compressors having sufficiently many stages, the relative surge frequencies fs0/f1 tend to fall as low as 0.1,

and the modified reduced surge frequencies fRPVVsm tend to lower below 0.1. In the situation, small pressure peaks appear

between neighboring pressure recovery peaks of pressure oscillations, caused by incomplete surge recoveries of rear stages.

It could be named a deep surge with a very low frequency caused by both of the surge action and the intervention of

incomplete recoveries. As its possible cause, the situation could have been brought about by degeneration of the deep surges

into a sequence of alternately occurring deep one and incomplete one where the apparent fRPVVsm values tend to drop to about

a half of 0.1.

(6) When the stall stagnation boundary conditions are satisfied, the deep surges disappear and a stall stagnation occurs. The

improved condition for the stall stagnation is the modified reduced resonance frequency fR1mstg of nearly 0.07. Near the

boundary, surge frequencies are changeable. For example, subharmonic deep-surges could often occur above the boundary.

Below the boundary occur various surge phenomena, such as mild surges within the stalled region, decaying surge

oscillations onto some point in the stalled area, etc.

11. Postword

The study could have made clear to some extent the brief behaviors of the deep-surge frequencies, which have been overlooked

so far. The possibility of estimation of the rough order of magnitude of the surge frequencies will contribute much not only to

understanding of the surge phenomena itself, but also to consideration of preventive measures against the oscillation problems in

the actual sites.

However, the results have been based solely on the numerical-experimental procedures. Confirmations with practical data will

be required, since situations that have not been confirmed practically or experimentally could possibly exist. For more details,

further analyses and experiments will be wanted.

Nomenclature

a Sound of speed (m/s) Lp Length of the delivery flow-path (m)

Ac1 Annulus sectional area of the first stage inlet LR Length ratio of the flow-path

(m2) PR Stalling pressure ratio

Ac Sectional area of the suction-compressor flow- rpm Compressor speed (rpm)

path (m2) rpm0 Compressor design speed (rpm)

Ac2 Sectional area of the exit of the last stage f the Leff Effective representative length of the flow-

compressor (m2) path (m)

Ap Sectional area of the delivery plenum (m2) rt1 Tip radius of the first rotor of the compressor

APR Sectional area-pressure ratio of the flow-path (m)

APRstg APR at the stall stagnation boundary um Mean peripheral speed of rotor of the

AR Sectional area ratio of the flow-path compressor (m)

B Greitzer’s B parameter ut1 Tip peripheral speed of the first rotor of the

f1 Basic resonance frequency of the whole flow- compressor (m)

path (Hz) VR Volume ratio of the flow-path

fR Reduced frequency κ Ratio of specific heats

fs0 Deep-surge frequency (Hz) φt Stage flow coefficient

fs0/f1 Relative surge frequency ψPt Stage pressure coefficient

fR1mstg Modified reduced resonance frequency for the ψTt Stage temperature coefficient

stagnation boundary

fRPVV1 Reduced resonance frequency for the subscript

stagnation boundary stg Stall stagnation boundary

fRPVVs Volume-modified reduced deep-surge s Deep-surge condition, or surge,

frequency 1 Inlet to the compressor, or basic frequency

fRPVVsm Modified reduced deep-surge frequency 2 Exit from the compressor

Lc Length of the suction duct and the compressor c Compressor

(m) p Delivery plenum

Lc** Length of the suction duct and the compressor t Rotor tip

(m)

377

References

[1] Greitzer, E. M., 1976, “Surge and Rotating Stall in Axial Flow Compressors Part I-Theoretical Compression System Model,”

ASME, J. Engineering for Power, Vol.98, pp.190-198

[2] Greitzer, E. M., 1976, “Surge and Rotating Stall in Axial Flow Compressors Part II-Experimental Results and Comparison

with Theory,” ASME, J. Engineering for Power, Vol.98, pp.199-217

[3] Boyer, K. M., and O’Brien, W. F., 1989, “Model Predictions for Improved Recoverability of a Multistage Axial-Flow

Compressor,” AIAA-89-2687

[4] Yamaguchi, N., 2014, “Surge Phenomena Analytically Predicted in a Multi-stage Axial Flow Compressor in the Reduced-

speed Zone,” International Journal of Fluid Machinery and Systems, Vol. 7, No. 3, pp. 110-124

[5] Yamaguchi, N., 2014, “A Study on the Fundamental Surge Frequencies in Multi-Stage Axial Flow Compressor Systems,”

International Journal of Fluid Machinery and Systems, Vol. 7, No. 4, pp. 160-172

[6] Yamaguchi, N., 2016, “A Comparison of Surge Behaviors in Multi-Stage and Single-Stage Axial Flow Compressors,”


[7] Yamaguchi, N., 2013, “Analytical Study on Stall Stagnation Boundaries in Axial-Flow Compressor and Duct Systems,”


[8] Yamaguchi, N., 2016, “Analytical Surge Behaviors in Systems of a Single-stage Axial Flow Compressor and Flow-paths,”


[9] Yamaguchi, N., 2013, ”Development of a Simulation Method of Surge Transient Flow Phenomena in a Multistage Axial Flow

Compressor and Duct Systems,” International Journal of Fluid Machinery and Systems, Vol. 6, No. 4 , pp. 189-199

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