radial turbine rotor response to pulsating inlet flows

Teng Caoe-mail: [email protected]

Liping XuWhittle Laboratory,

Department of Engineering,

University of Cambridge,

Cambridge CB3 0DY, UK

Mingyang Yang

Ricardo F. Martinez-Botas

Dept. of Mechanical Engineering,

Imperial College London,

London SW72AZ, UK

Radial Turbine Rotor Responseto Pulsating Inlet FlowsThe performance of automotive turbocharger turbines has long been realized to be quitedifferent under pulsating flow conditions compared to that under the equivalent steadyand quasi-steady conditions on which the conventional design concept is based. How-ever, the mechanisms of this phenomenon are still intensively investigated nowadays.This paper presents an investigation of the response of a stand-alone rotor to inlet pulsat-ing flow conditions by using a validated unsteady Reynolds-averaged Navier–Stokessolver (URANS). The effects of the frequency, the amplitude, and the temporal gradientof pulse waves on the instantaneous and cycle integrated performance of a radial turbinerotor in isolation were studied, decoupled from the upstream turbine volute. A numericalmethod was used to help gain the physical understanding of these effects. A validation ofthe numerical method against the experiments on a full configuration of the turbine wasperformed prior to the numerical tool being used in the investigation. The rotor was thentaken out to be studied in isolation. The results show that the turbine rotor alone can betreated as a quasi-steady device only in terms of cycle integrated performance; however,instantaneously, the rotor behaves unsteadily, which increasingly deviates from thequasi-steady performance as the local reduced frequency of the pulsating wave isincreased. This deviation is dominated by the effect of quasi-steady time lag; at higherlocal reduced frequency, the transient effects also become significant. Based on thisstudy, an interpretation and a model of estimating the quasi-steady time lag have beenproposed; a criterion for unsteadiness based on the temporal local reduced frequencyconcept is developed, which reduces to the K criterion proposed in the published litera-ture when cycle averaged. This in turn emphasizes the importance of the pressure wavegradient in time. [DOI: 10.1115/1.4025948]

Introduction

Turbochargers are widely used in the automobile industry fortheir significant benefits for fuel saving, emission reduction, andpower enhancement of engines. The turbine part of turbochargersis directly mounted downstream of the exhaust manifold ofengines in order to transfer the wasted energy from the exhaustgas into mechanical work. High performance of the turbine is cru-cial for a good overall and transient performance of the wholeengine system.

However, the working environment of the turbocharger, espe-cially in the turbine part, is not as ideal as would be preferred.Due to reciprocating nature of internal combustion (IC) engines, apulsating flow is fed into the turbine, which causes an inherentlyunsteady operating environment. This causes highly complex anddisturbed flow inside the turbine [1] and severely challenges theconventional design philosophy based on the steady flowconditions.

During the past few decades, in order to assess the effects ofthe pulsating flow on turbocharger turbines, a significant amountof research has been undertaken using many kinds of methods,including experimental measurements, one-dimensional model-ing, and more recently three-dimensional numerical simulations.

The turbine performance has long been realized to be quite dif-ferent under pulsating flow conditions compared to that under theequivalent steady and quasi-steady conditions. This includes twoaspects: One is the comparison of turbine overall or cycle inte-grated performance; the other is the comparison of turbine instan-taneous characteristics. Wallace and Blair [2], Benson andScrimshaw [3], Kosuge et al. [4], Capobianco et al. [5] and Capo-bianco and Gambarotta [6] are among some of the earliestresearchers that reported deviations of cycle integrated turbinepower output, efficiency, mass flow capacity under nonsteady, and

quasi-steady flow conditions, using experimental data. With thedevelopments in experimental measurements and computationalfluid dynamics (CFD), instantaneous turbine performance can becaptured. One of the most important characteristics of the turbineperformance under pulse flow conditions is the “hysteresis loop,”the loop seen in power output against pressure ratio, as well as inmass flow rate against pressure ratio. Such hysteresis loops indi-cate instantaneous deviations of the performance from quasi-steady conditions. Dale and Watson [7] observed the hysteresisloop by experiment in the mid-1980s. Yeo and Baines [8], Winter-bone et al. [9], Abidat et al. [10], Karamanis et al. [11], Szymkoet al. [12], Rajoo and Martinez-Botas [13], and Copeland et al.[14] also reported this phenomenon experimentally later. Suchhysteresis loops were also captured by one-dimensional models[10,12,15–17]. One of the common features in these models isthat the volute and the turbine rotor are treated as unsteady andquasi-steady devices, respectively. Recently, more complex three-dimensional CFD models were used to help obtain better under-standings of the phenomenon. Lam et al. [18] are among the firstresearchers to study the radial nozzled turbine by employing CFDmethods. Palfreyman and Martinez-Botas [1] investigated theflow field in a mixed flow turbocharger turbine by using a three-dimensional numerical method validated with experimental data.Following this work, recently, Padzillah et al. [19] also tried toassess the unsteady flow effects on a nozzled turbocharger turbineusing a validated numerical method. These validated CFD resultsgreatly strengthened people’s confidence in the numerical methodto help understand the phenomenon. The large eddy simulation(LES) method was used by Hellstrom and Fuchs [20] to investi-gate effects of secondary perturbation under pulsating flow.Nevertheless, it is not known how this CFD result compares withexperimental data.

In order to understand unsteady phenomenon of the turbineunder pulsating flow conditions, the pulse frequency effect wasintensively investigated in the past [1–26]; a direct link betweenthe turbine performance and the pulse frequency was found. Manyresearchers have also introduced criteria based on the pulse

Contributed by the International Gas Turbine Institute (IGTI) of ASME forpublication in the JOURNAL OF TURBOMACHINERY. Manuscript received July 30, 2013;final manuscript received August 6, 2013; published online December 27, 2013.Editor: Ronald Bunker.

Journal of Turbomachinery JULY 2014, Vol. 136 / 071003-1Copyright VC 2014 by ASME

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frequency to judge the unsteadiness of the turbine system respondto the inlet pulsating flow. A Strouhal number was employed byChen and Winterbone [21] to respectively treat the volute and therotor as unsteady and quasi-steady devices. However, the wavelength is usually longer than the pulse length. To take this intoconsideration, Szymko et al. [12] introduced a modified Strouhalnumber which contains pulse length fraction factor. Costall andMartinez-Botas [26] afterwards developed a Fourier seriesStrouhal number by using a Fourier decomposition method todecompose pulse waves into many components and then inte-grated them up with energy weighting factors. Apart from thepulse frequency effects, Benson [22] and Kosuge et al. [4]reported the importance of wave form shapes on the turbine per-formance. Their results show that the wave form has large impacton the turbine performance. However, the physics behind theseeffects are still not very clear. Kosuge et al. [4], Capobianco et al.[5] and Capobianco and Gambarotta [6] showed experimentalresults indicating that the pulse amplitude has impact on the tur-bine performance. Copeland et al. [25] introduced a K criterionthat took the amplitude effect into considerations. Kosuge et al.[4] and Copeland et al. [25] also mentioned the importance of thepressure wave gradient to the turbine performance, but detailedrotor performance needs to be further investigated.

It can be learned from previous research that these effects (fre-quency, amplitude, and temporal pressure gradient) had beenintroduced to be important to the whole turbine stage. However,how does each component (the volute, the rotor, and sometimesthe nozzle) play its part in the process? Or, first, whether turbinerotor can be treated as a quasi-steady device? This is importantbecause not only it is crucial for the understanding of the wholeproblem but also how useful simple turbine models based onsteady flow data can be, which serves as the foundation for practi-cal design systems currently used in industry [23].

Many researchers have already tried to answer the questionraised above. Most of the arguments on the quasi-steady behaviorof turbine rotors are based on the Strouhal number (Chen et al.[16], Chen and Winterbone [21], Baines et al. [15], Abidat et al.[10], Szymko et al. [12], and Costall et al. [17]). However, thearguments are not directly connected to the rotor performance.The experimental evidence to support the quasi-steady manner ofthe rotor was given by Yeo and Baines [8]. They concluded thatthe turbine was operating in a quasi-steady manner during pulseflow by showing similar rotor inlet conditions (absolute velocity,flow angle, and incidence angle) under nonsteady and steady con-ditions. However, in contrast, Winterbone et al. [9] carried out aseries of measurements on turbine performances; based on themeasurements, they concluded that the rotor does not behavequasi-steadily by showing that the rotor caused most of the timelag between pressure and torque trace [24]. Copeland et al. [25]employed CFD methods and used a particle tracking technique tohelp calculate the Strouhal number of each turbine component.Based on the Strouhal number, they reported that for the pulse fre-quency of 84 Hz, the rotor could behave unsteadily because aStrouhal number of 0.1 was calculated, but detailed rotor stand-alone performance has not been shown to further validate thecriteria.

It is felt that there has not been sufficient evidence to supportthe answers to the question either way and contradictory opinionsabout the question exist. Crucially, in the past, most of the studieswere carried out with the whole turbine system, lumping both thevolute and the rotor together. It would be very difficult to distin-guish the effects contributed by different components due to thestrong interactions between the two: the volute tongue effect[1,19], the wave reflection effect [2,18], the nozzle effect [13],and the circumferential nonuniformity effect [19] by the volute.

However, if a validated unsteady CFD tool is used, it is possibleto isolate the rotor from the influence of the upstream volute. Thiscan be used to study various effects of the pulsating flow in gen-eral and to give a clear evidence to support findings on the per-formance of a hypothetical turbine rotor alone setup.

In this paper, a single radial turbine rotor response to inlet pulseflows is investigated systematically. A validation of the numericalmethod against the experiments on a full configuration of the tur-bine is first performed. Then, in order to eliminate the interactionbetween the volute and the rotor, a rotor is studied in isolationunder different pulse flow conditions. Finally, an interpretationand a model of estimating the quasi-steady time lag are proposed;a criterion for unsteadiness based on the temporal local reducedfrequency concept is developed, which reduces to the K criterionof Copeland et al. [25] when cycle averaged.

Numerical Methods and Validation

Numerical Methods. Both steady and unsteady state calcula-tions of the turbocharger turbine were performed using the CFDcode TBLOCK (Denton [27]). TBLOCK is a Reynolds-averagedNavier–Stokes equation solver that is based on finite volumemethod and uses simple explicit “SCREE” scheme [28] and multi-block structured meshing strategy. The turbulence model used inthis paper is a simple mixing length model. Wall functions areused to obtain the surface shear stress.

The turbine used in this study is a Mitsubishi Heavy Industryradial vaneless turbocharger turbine, as shown in Fig. 1. The tur-bine includes an inlet transition duct (which connects the voluteand the upstream pulse generator), the volute, the rotor, and theexhaust pipe. The back clearance of the turbine rotor was alsomodeled, as can be found in Fig. 1(c).

Mesh independency studies were carried out prior to the study;the criteria of the mesh independency study were that the gradientof the turbine efficiency over the mesh size was lower than 0.1%per million mesh points. This resulted in a reasonably fine meshwith about 12� 106 nodes which kept yþ lower than 5. The distri-bution of the mesh was about 2� 106 nodes within the inlet transi-tion duct and the volute and about 10� 106 nodes within the rotorpassage and the exhaust pipe.

The boundary conditions used in the CFD validations weregiven by the experiments. Total pressure and total temperature inabsolute frame of reference were imposed on the inlet boundary

Fig. 1 Computational model of the turbine: (a) the transitionduct, the turbine volute, and the exit pipe; (b) the turbine rotor;(c) the back clearance of the turbine rotor

071003-2 / Vol. 136, JULY 2014 Transactions of the ASME


of the computational domain. Due to very small exit static pres-sure variation (within 1%), a cycle averaged exit static pressurevalue measured from experiment was fixed on the exit casing andthe spanwise variation was obtained by simple radial equilibrium.The variation of measured rotating speed was less than 1%; there-fore, a constant rotating speed was set for the turbine rotor. Theentire wall boundaries are assumed to be adiabatic and a fullydeveloped turbulence assumption was used; the turbulence inten-sity was set to a medium level of 5%.

The interface treatment in the steady state calculations was“frozen rotor” method. In unsteady calculations, a “sliding plane”approach was used.

A dual time-stepping approach was used for the unsteady calcu-lations in this study. In order to decide the scale of physical timestep, a time step independency study had been carried out. Thecriteria of the time step independency study was that the shape ofthe hysteresis loop of turbine power against pressure ratio did notchange when further increasing the number of physical time stepsof the unsteady calculation. This study showed that in terms of theturbine’s rotation, about 12 steps per turbine passage passing issufficient for the temporal resolution required for the presentcases.

Validations of the Numerical Method. In order to build upconfidence on the CFD solver, validations of the computationalmethod against the experiments on a full configuration of the tur-bine described above were performed under both steady and pul-sating flow conditions.

The turbine was tested at the Imperial College London. The fa-cility used is able to simulate an engine exhaust pulsation underlow temperature and also extended loading capability with aneddy current dynamometer [12]. The test rig is driven by com-pressed air supplied through three screw-type compressors, capa-ble to deliver up to 1 kg/s of mass flow rate at maximum pressureat 5 bar (absolute). A rotary air pulse generator with two rotatingchopper plates is used to simulate the engine exhaust pulse flow atdifferent frequency.

The test was carried out under both steady and pulsating flowconditions at turbine reduced speed (x=

ffiffiffiffiffiffiffiffiffiTo;in

p) about 45

rps � K�0:5. For the pulsating flow conditions, the turbine wastested under inlet pulse flow with frequency of 60 Hz. The instan-taneous pressure and temperature at turbine inlet, the pressure atthe turbine’s exit, and the turbine’s rotational speed were meas-ured to provide boundary conditions for the numerical calcula-tions. The torque of the turbine and the inlet mass flow rate werealso instantaneously recorded by using a permanent magnet eddycurrent dynamometer and a constant temperature type hot wiresensor, respectively.

A group of nondimensionalized parameters was defined inorder to compare the results of the CFD and the experiment. Theinlet mass flow function is defined as

m� ffiffiffiffiffiffiffiffiffi

T0;in

pp0;in

(1)

the velocity ratio

xrinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2cpT0 1� ðpout

p0;inÞc�1c

� �s (2)

the pressure ratio

p0;in

pout

(3)

and the total to static isentropic efficiency (EFF)

1� T0;out

T0;in

� �� ð1� pout

p0;in

� �c�1c Þ

(4)

For the steady state validations, as shown in Figs. 2(a) and 2(b),it can be seen that the results of inlet mass flow function given byCFD solver agree with the experimental data quite well, within2% difference of the measurements; the CFD solver also showsvery good capability of predicting the total-to-static efficiency ofthe turbine, although there is a slightly bigger deviation underhigh velocity ratio conditions.

Figure 3 shows the comparisons between results of the CFDand the experiment under pulse flow conditions. The instantane-ous mass flow function comparisons indicate that the overall inletmass flow level and the general shape of the hysteresis loop arecaptured by CFD reasonably well. However, the CFD seems tocapture more wave dynamics in the peak mass flow region. Figure3(b) shows the turbine power output comparisons between theresults given by the CFD and the measurements; the CFD poweroutput is processed using the blade torque and the turbine rotationspeed, the same as for the experimental results. As can be seen,the shape of the instantaneous power trace shows good agreementwith the experimental data. It is interesting to note that the inletmass flow differences between the CFD and the test data do notinfluence much the downstream turbine power prediction.

The validations of the numerical method show that the CFDsolver used in this study is able to capture both steady andunsteady behaviors of the radial turbine reasonably well and it canbe used to study the rotor alone configuration with confidence.

Fig. 2 Steady state turbine performance: (a) inlet mass flowfunction versus pressure ratio; (b) efficiency versus velocityratio

Journal of Turbomachinery JULY 2014, Vol. 136 / 071003-3


Isolated Rotor Response

The turbine rotor was then studied in isolation using the numer-ical method under a series of pulsating inflow conditions; theeffects of the pulse frequency, amplitude, and temporal gradienton the turbine rotor performances were examined, respectively.Due to the absence of the volute, the uncertainty in the inlet flowangle needs to be confirmed. The rotor inlet flow angle at abovevalidation condition was extracted out as shown in Fig. 4. As canbe seen, the rotor inlet absolute flow angle is fairly constant duringthe pulse period while the relative flow angle varies from �5 to�60 deg. This is mainly due to the change of the absolute velocitymagnitude rather than the flow angle. Therefore, to model the tur-bine rotor, a fixed inlet absolute flow angle was imposed. Thiswas obtained by averaging the value from the validation case withvolute mounted on. The other operating conditions were kept thesame with the validation case; the inlet pulse wave shape wasredesigned for the investigation purpose at similar levels as thevalidation case.

The Effect of Pulse Wave Frequency. Pulse wave frequencyis believed to have significant impact on the performance of thewhole turbine system based on many previous researches[1–4,7–26]. However, it is still interesting to see how the rotorresponds to different pulsating frequencies. To investigate theeffect of pulse wave frequency, three pulse waves with differentfrequency varying from 60 Hz to 200 Hz were designed, as shownin Fig. 5. The range of the pulse frequency covers the range of

current pulse turbocharging applications and extends to higher fre-quency regions.

All of the waves have the same wave shape with a sinusoidpulse part and a constant tail part. Each part takes up 3/4 and 1/4of the whole pulse period, respectively. The amplitude is kept thesame for all the three waves, which is about 18% of the meanvalue.

The unsteady calculations of the turbine rotor are carried outunder these three inlet pulsating flow conditions. The instantane-ous responses of the turbine rotor to the pulse waves with differentfrequencies are shown in Fig. 6. The hysteresis loops of the inletmass flow against pressure ratio and the power output againstpressure ratio can be observed, which shows an instantaneousdeviation from the quasi-steady behaviors. As can be seen, thehysteresis loops essentially encapsulate and are symmetrical aboutthe quasi-steady data, and the higher the pulse frequencies are, thebigger the loops becomes.

One of the possible causes for the hysteresis loop of the workoutput against pressure ratio is the quasi-steady time lag[11,12,15,18,24]. To confirm this point, the work output traceswere shifted back by a certain amount of time (The exact time, theway of calculating it and the physical interpretation of it are pre-sented later in the Quasi-Steady Time Lag section). It can be seenfrom the results after phase shifting shown in Fig. 7 that the hys-teresis loops of the turbine power disappear. Moreover, the instan-taneous performances of the rotor follow its quasi-steadybehaviors. This applies to all the cases studied with different

Fig. 3 Turbine performances under pulsating flow conditionscomparing with quasi-steady data: (a) inlet mass flow functionversus pressure ratio; (b) power output versus pressure ratio

Fig. 4 Circumferential averaged flow angle at the inlet of theturbine rotor

Fig. 5 Designed pulse wave forms with different frequencies



frequencies. Therefore, the quasi-steady time lag assumption isvindicated.

To assess the pulse frequency effect on the cycle integrated per-formances of the rotor, the unsteady performance is compared tothe quasi-steady and equivalent steady mean flow performance.The unsteady performance can be directly obtained from theunsteady CFD results; the quasi-steady performance is defined asthat at each time instance, the rotor is assumed to steadily respond

to the inlet boundary condition. In practice, a series of steady stateresults that cover the operating range of the rotor under pulsatingflow condition are first calculated, then according to the inlet pulseflow condition at each moment, the quasi-steady performance ofthe rotor is interpolated from curve fitting based on the steadydata. The steady mean flow performance is the steady state solu-tion under the condition of time mean value of the inlet pulsatingflow. The quasi-steady comparison factors were defined by Ben-son and Scrimshaw [3] and Benson [22] as

Cm;qs ¼

ðT

0

m�

in;unsðtÞdtðT

0

m�

in;qsðtÞdt

(5)

CP;qs ¼

ðT

0

PunsðtÞdtðT

0

PqsðtÞdt

(6)

Following the similar way, a steady mean comparison factors arealso defined as

Cm;mean ¼

ðT

0

m�

in;unsðtÞdt

m� in;meanT(7)

CP;mean ¼

ðT

0

PunsðtÞdt

PmeanT(8)

Figure 8 shows that the quasi-steady comparison factors Cm;qs

and CP;qs are all about 1 and they are almost independent to thepulse frequency. The steady mean comparison factors Cm;mean andCP;mean are slightly lower than 1 and also almost independent tothe pulse frequency.

In conclusion, under these three pulse flow conditions, the rotorperformance instantaneously deviates from the quasi-steadybehavior that is mainly due to the quasi-steady time lag, and thehigher the frequency, the larger the deviation. However, from thecycle integrated performance point of view, the turbine rotorbehaves as a quasi-steady device and the predictions using themean steady value slightly overpredict the turbine performance.All of the cycle integrated comparisons are almost independent ofthe inlet pulse flow frequency.

The Effect of Pulse Wave Amplitude. As introduced before,Kosuge et al. [4], Capobianco et al. [5], Capobianco and

Fig. 6 Turbine rotor responses to different inlet pulse frequen-cies: (a) inlet mass flow function versus pressure ratio, (b)power output versus pressure ratio

Fig. 7 Phase shifted turbine power output against pressureratio

Fig. 8 Turbine rotor cycle integrated performances under dif-ferent inlet pulse frequency conditions



Gambarotta [6], and Copeland et al. [25] had indicated the impor-tance of pulse amplitude effect on the turbine performance. In thispaper, in order to study the effects of pulse amplitude on the rotorperformance, three pulse waves are designed to have the same fre-quency of 60 Hz but different amplitudes, which are 17.8%,35.6%, and 53.4% of the mean value, (amplitude 1, 2, 3, respec-tively), as shown in Fig. 9; all of the wave forms have the samemean values.

Figures 10(a) and 10(b) show the instantaneous inlet mass flowfunction and power output against the pressure ratio. It is interest-

ing to note that the hysteresis loops can also be observed from theplots of power output against pressure ratio and the inlet massflow against pressure ratio; the hysteresis loop becomes biggerwith increase of the pulse amplitude.

The phase shifting is also applied to these cases, and it can beseen from Fig. 11 that the hysteresis loops of the power output dis-appear and the turbine performance follows the quasi-steadybehaviors. This indicates that the mechanisms of quasi-steadytime lag for the hysteresis loop remain valid in these cases of dif-ferent wave amplitudes studied.

The comparisons of cycle integrated unsteady turbine perform-ance against the quasi-steady and the steady mean performanceare shown in Fig. 12. The comparison factors of Cm;qs and CP;qs

are still about 1 and almost independent of the wave amplitude.However, significant variations of Cm;mean and CP;mean can beobserved. The Cm;mean drops from 0.996 to 0.939 and the CP;mean

drops from 0.995 to 0.965 when the amplitude increases from17.8% to 53.4% of the mean value; it also indicates that the rotorhas lower mass flow capacity and efficiency under higher pulsewave amplitude conditions.

The phenomenon actually can be explained using steady char-acteristics. If the turbine is designed based on the mean value ofthe pulse flow, when the pulse amplitude is high, there will be abigger portion of energy that is away from the mean value and isdistributed to the region at off design or low efficiency of the tur-bine; in this case, the mean value will be an overestimate for theturbine performance, especially when the rotor performs badly atoff-design conditions.

Fig. 9 Designed pulse wave forms with different amplitudes

Fig. 10 Turbine rotor responses to different inlet pulse ampli-tudes: (a) inlet mass flow function versus pressure ratio, (b)power output versus pressure ratio

Fig. 11 Phase shifted turbine power output versus pressureratio

Fig. 12 Turbine rotor cycle integrated performances under dif-ferent inlet pulse amplitude conditions



The Effect of Temporal Gradient. The results have shownthat both pulse wave frequency and amplitude have impacts onthe turbine rotor instantaneous behaviors, and the higher fre-quency and the larger amplitude the wave, the bigger the hystere-sis loop of the power output. From further analyses of thecontributions of the pulse frequency and the pulse amplitude, it isfound that both of them change the local temporal gradient of thepulse wave. Kosuge et al. [4] and Copeland et al. [25] also men-tioned the importance of the pressure wave gradient to the turbinestage performance, but little information of detailed rotor per-formance was given in the references. In order to further investi-gate the effect of the temporal gradient of pressure wave, fourdifferent wave shapes were designed based on sinusoidal wavewith different wave front gradient, as shown in Fig. 13. The wavefront gradient was defined as Dp=Dt (Dp is the pressure ratio am-plitude of the wave front, Dt is the time period of the wave front).All of the waves have the same fundamental frequency and theamplitude.

Figures 14(a) and 14(b) show the instantaneous turbine per-formance of inlet mass flow function and power output. It can beseen that significant hysteresis loops of the turbine performanceare found for the “steep” raising wave. From the plot of poweroutput against pressure ratio, it is clear to see the trend of the sizeof the loop varying with the wave front sharpness; the sharper thewave front, the bigger the loop. The size of the loop is only deter-mined by the gradient of the wave front (or the raising period); theturbine behaves the same during the falling period because thegradient of the falling period is the same for all four waves.

By applying the phase shifting on the power output trace andplotting against the pressure ratio, it can be seen from Fig. 15 thatthe rotor performances under pulse wave temporal gradientDp=Dt ¼44.16 and 176.64 follows the quasi-steady performancequite well; however, increasing deviations are found in the casesunder Dp=Dt ¼353.28 and infinite wave conditions. This indicatesthat when the local temporal gradient of the pressure wave is high,instantaneously, the turbine rotor no longer behaves quasi-steadily. Therefore, apart from the quasi-steady time lag, theremust be other mechanisms that influence the turbine instantaneousperformance. One of them could be due to the transient effect ofwave actions inside the turbine passage. A further study on flowdetails to confirm this assumption is currently underway.

However, no matter how far the rotor instantaneous behaviorsdeviate from its quasi-steady manner, its cycle integrated perform-ances seem not to be influenced very much by the sharpness of thewave gradient (within 0.2% variations), as shown in Fig. 16. Thereason for that is largely because when temporal local reduced fre-quency is high, the time scale of the corresponding portion of thewave is small comparing to that of the overall wave. Therefore,the influence on overall performance shall also be small.

Quasi-Steady Time Lag

In the early research, a time phase lag had been found betweenthe inlet pressure trace and the rotor torque. However, differentresearchers have reported quite different results. Winterbone et al. [24]

Fig. 13 Designed pulse wave forms with different wave frontgradients

Fig. 14 Turbine rotor responses to different inlet pulse wavefront gradients: (a) inlet mass flow function versus pressure ra-tio, (b) power output versus pressure ratio

Fig. 15 Phase shifted turbine power output versus pressureratio



reported a phase lag that was much larger than the wave traveltime through the casing. Baines et al. [15] suggested that the timelag should be estimated by the bulk flow travel time. Karamaniset al. [11] found that the sonic travel time should be used to shiftthe torque trace in order to avoid an unrealistic unsteady effi-ciency. Szymko et al. [12] proposed method of using local sonicvelocity plus local bulk flow velocity for the phase shifting. How-ever, Lam et al. [18] showed CFD results to indicate that the timelag is a kind of combined effect, both acoustic and convectivephenomena are important to the unsteady performance of the tur-bine by respectively considering the pressure and temperaturetraveling speed. The phase shifting used in this paper is found tobe about 0:125trev, which is somewhere between the one calcu-lated using convection velocity and that using pressure wave trav-eling speed. This seems to echo the suggestions given by Lamet al. [18]. One factor to bear in mind is that the model studied inthe present paper is only the rotor itself. In other literature, the fullconfiguration of the turbine including both volute (some with noz-zle vane) and the turbine rotor was considered, which has consid-erably longer characteristic length.

The physics of the cause of the quasi-steady time lag can beattributed to two effects: One is that pressure wave, which willchange the local flow velocity and, therefore, the blade circula-tion, needs time to travel though the turbine passage; the other isthat due to the inertia of the flow inside the turbine passage, theflow needs time to respond to the traveling pressure wave. Thetime lag due to the inertia of the flow can be observed from thephase difference between the mass flow and the pressure ratioshown in each case. Therefore, the overall time lag tl can be esti-mated based on the pressure traveling speed (tp, suggested bySzymko et al. [12]) and the inertia of the flow (ti), shown asfollows:

tl ¼ tp þ ti (9)

tp ¼L

uþ c(10)

ti ¼Dm�

inV

DpS2in

(11)

In these equations, L is the length of the turbine passage, c is theaveraged speed of sound of the turbine inlet and outlet, u is theaveraged fluid velocity of the turbine inlet and outlet, V is volumeof the turbine passage, Dm

�in is the pulse amplitude of the mass

flow rate at the turbine inlet, Dp is the pulse amplitude of the pres-sure at the turbine inlet, and Sin is the area of the turbine inlet.This definition of the time scale was used to calculate the time lagfor the shifting in the previous sections. For the cases studied inthis paper, the ratios of tp/ti are all at an order of 1.

A Criterion for Unsteadiness

It has been introduced above that many researchers have al-ready developed criteria to judge the unsteadiness of the turbinesystem response to the inlet pulsating flow. One of the commonal-ities in the previous criteria is that turbines were treated to respondto the overall pulse wave. Szymko et al. [12] pushed it forward byconsidering the real pulse length fraction to the overall wavelength. However, for the real pulse part, the conventional Strouhalnumber is still used. In the present study, it has been found thatthe rotor behaves very locally in time, e.g., for the sharp wavefront case, and its instantaneous behaviors highly rely on the localtemporal gradient of pressure waves. If the gradient is small theeffect of quasi-steady time lag dominates and if it is large, thenthe transient effects have to be taken into the account. Therefore,to consider the importance the pressure gradient mentioned byKosuge et al. [4] and Copeland et al. [25] and the wave amplitudeeffects [4–6,25], in the present study, a criterion was developed tohelp judge when the turbine rotor can be treated as a quasi-steadydevice in terms of instantaneous performance.

The temporal local gradient of the pressure wave can beexpressed as

@pðtÞ@t

�� (12)

The time mean value of the pressure wave p and the average timefor fluid particle traveling through the turbine passage tf are usedto normalize the pressure gradient, as shown in Eq. (13):

@pðtÞ=p

@t=tf

�� (13)

For the convenience of practical applications, it can be rewritten as

DpðtÞj jp

tf

Dt) eðtÞj jblocalðtÞ

where

eðtÞj j ¼ DpðtÞj jp

; blocalðtÞ ¼tfDt

(14)

This leads to the definition of a temporal local pressure correctedreduced frequency. Figure 17 shows a sketch to illustrate this con-cept. The temporal local reduced frequency actually is the ratio offluid particle travel time tf and the temporal local disturbance pe-riod Dt. The selection of Dt depends on the accuracy one wants. Ifit is very small, the local reduced frequency may come huge; how-ever, this will be corrected by the Dp, and even if the Dt tends tobe zero, it will end up with a finite number that is the pressure gra-dient. Figure 17 also shows a variation of |e(t)|blocal(t) of the abovepressure wave. It can be observed that the number is shooting upduring the period of high pressure gradient, and for the case stud-ied, it shows that when |e(t)|blocal(t) is higher than about 0.07, thetransient effects become significant for the turbine instantaneousperformance and vice versa. The importance of this “temporallocal” concept is that, first, it indicates that the rotor responds tothe pulse wave locally in time, in terms of the instantaneous per-formance, not to the whole pulse wave; second, it can pin downthe temporal location of the unsteadiness.

Interestingly, when the |e(t)|blocal(t) is time averaged over thewhole pulse cycle and divided by the ratio of specific heat capaci-ties c,

ðT

0

eðtÞj jblocalðtÞdt

cT¼

Xn

k¼1

Dpkj jp

tf

cT¼ 2A

cp

tf

T) PSt ¼ K

Fig. 16 Turbine rotor cycle integrated performances under dif-ferent inlet pulse wave front gradient conditions



where

P ¼ 2A

cp; St ¼

tfT

(15)

This reduces to the K a criterion that has been developed by Cope-land et al. [25] from a one-dimensional mass storage point ofview. This agreement in turn emphasizes the importance of thetemporal pressure gradient. The K criterion can be used here toindicate the unsteadiness of the rotor in terms of the cycle inte-grated performance. In the present study, the largest K value cal-culated is about 0.02. The results also show that for all the cases,the rotor behaves quasi-steadily in terms of cycle integrated per-formance. These results support the conclusion drawn from the Kcriterion from a cycle integrated performance point of view; how-ever, at the same time the existence of instantaneous unsteadinesshighlights the need for having a temporal local pressure correctedreduced frequency that can take into account and pin down thelocation of the unsteadiness in time.

Concluding Remarks

The response of a standalone rotor to inlet pulsating flow condi-tions has been investigated by using validated URANS. The vali-dations of the numerical method show that the CFD solver used inthis study is able to capture both steady and unsteady behaviors ofthe radial turbine reasonably well. The results of the investigationshow that the turbine rotor itself can be treated as a quasi-steadydevice only in terms of cycle integrated performance. This findingsupports the general understanding of most previous researchers.However, instantaneously, the results indicate that the rotorbehaves unsteadily when the temporal local pressure correctedreduced frequency is high. This also provides supporting evidenceto the statements given by the other researchers [24,25]. The

unsteady effects cannot be neglected if the transient performanceis to be taking into consideration.

The instantaneous performance of the rotor, or the hysteresisloop, is found to be dominated by the quasi-steady time delay.This delay is in turn determined by two factors: First, the time thepressure wave takes to travel though the passage to change localfluid velocity and, therefore, the blade loading, and second, thefluid also needs time to response to the pressure wave due to theinertia. Both have impact on the instantaneous turbine workoutput.

In terms of design, the deviation of cycle integrated perform-ance in reality from the nominal design conditions will be signifi-cant only when the amplitude of incoming pulse is large. Thereason for that is because if the turbine is designed based on timeaveraged value of incoming pulse flow, when the pulse amplitudeis high, there will be a bigger portion of energy that is away fromthe mean value or the design point and is distributed to the regionof far off design and low efficiency of the turbine. In such case,the mean value will be an overestimate of the true turbine per-formance, especially when the turbine performs badly at off-design conditions.

A criterion for unsteadiness based on the temporal localreduced frequency concept is developed, which reduces to the Kcriterion [25] when cycle averaged. For the rotor instantaneousperformance, simplified sharp wave front wave forms have beendesigned and studied. It is found when the value of |e(t)|blocal(t) islarger than about 0.07, the transient effect becomes significant.Interestingly, when |e(t)|blocal(t) is time averaged over the wholepulse cycle, it becomes the K criterion that has been developed byCopeland et al. [25] from a one-dimensional mass storage point ofview. This in turn emphasizes the importance of the temporalpressure gradient. Furthermore, the results shown in this paperalso support the conclusion of quasi-steady rotor behavior drawnfrom the K criterion in terms of cycle integrated performance;however, at the same time the existence of instantaneous unsteadi-ness highlights the need of having a temporal local pressure cor-rected reduced frequency that can take into account and pin downthe location of the unsteadiness in time.

Acknowledgment

The authors would like to thank Mitsubishi Heavy Industry fortheir support and permission to publish this wok. The authorswould also like to thank Professor Nick Cumpsty for his discus-sion and comments on the research.

Nomenclature

A ¼ pulse wave amplitudec ¼ speed of sound

Cm ¼ mass flow rate comparison factorCP ¼ power output comparison factor

EFF ¼ total-static isentropic efficiencyn ¼ discretization intervalL ¼ characteristic lengthm� ¼ mass flow ratep ¼ pressureP ¼ power outputr ¼ radiusS ¼ surface areat ¼ time

T ¼ temperature or one pulse period timeu ¼ flow velocityV ¼ volume

Greek Symbols

b ¼ reduced frequencyc ¼ the ratio of specific heat capacitiese ¼ pressure correction factor

K ¼ lamda criterion [25]

Fig. 17 Sketch to show temporal local concept



p ¼ pressure ratioP ¼ amplitude correction factorq ¼ densityx ¼ turbine rotational speed

Subscripts

f ¼ fluid particlei ¼ fluid inertia

in ¼ turbine inletk ¼ series indexl ¼ quasi-steady time lag

local ¼ temporal localmean ¼ mean value

out ¼ turbine outletp ¼ pressure

qs ¼ quasi-steady stateref ¼ referencerev ¼ turbine revolutionuns ¼ unsteady state

0 ¼ total value

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http://dx.doi.org/10.1115/1.1812322

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http://dx.doi.org/10.1115/GT2007-28317

http://dx.doi.org/10.1115/1.2927983

radial turbine rotor response to pulsating inlet flows

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