JOURNAL OF PROPULSION AND POWERVol. 9, No. 4, July-Aug. 1993

Verification of the Theoretical Discharge Coefficientof a Subcritical Airflow Meter

D. J. Lahti*GE Aircraft Engines, Cincinnati, Ohio 45215

andA. Hamedf

University of Cincinnati, Cincinnati, Ohio 45221

Since many modern high bypass ratio turbofan engines have flow rates that exceed the capacity of most ofthe worlds airflow calibration facilities, their airflow metering bellmouth inlets cannot be calibrated, but rathertheir discharge coefficients must be determined theoretically. The objective of this program was to verify thetheoretically derived discharged coefficient for a scale model of such a bellmouth which was small enough toallow its calibration in an existing laboratory facility. Extensive flowfield measurements were also made tofurther validate theoretical predictions. Thus, this program provides a "calibration" of the theoretical methodused, and establishes a link between a traceable airflow metering standard and large engine bellmouth inletsthat cannot be calibrated any other way.

Introduction

T HE need for highly accurate airflow measurement inaircraft engine ground testing is self-evident since engine

thrust is directly proportional to airflow rate. Historically,when high-accuracy compressible airflow measurements arerequired, flow meters designed for critical (choked flow) con-ditions are used.1 Over the period of the past 30 yr, twostandard critical flow meter designs have received wide ac-ceptance throughout industry.2 These standard flow metersare the Smith and Matz3 critical flow circular-arc throat ven-turi and the ASME critical flow nozzle. Both Smith and Matz1-3

and Stratford4 discuss many of the advantages of meteringairflow using critical flow Venturis.

Although it is desirable to utilize critical flow meters when-ever possible, because of their inherent accuracy "advan-tage," it is not possible in the case of an aircraft engine bell-mouth. The supersonic diffuser shock wave and attendantdistortion downstream of its throat could not be tolerated bythe engine fan or compressor. Therefore, engine bellmouthsrequire subcritical flow conditions. In addition, modern flowmetering bellmouths are designed to simulate as nearly aspossible the flow conditions that would exist at the engineface when it is operating with its real flight inlet. This includesthe radial velocity profile at the fan or compressor face aswell as the tip boundary layer. This is done to help assure theground test-derived engine performance will be as nearly rep-resentative as possible of the engine's performance when inflight.

Because of this, today's engine inlet bellmouths have ge-ometries that are not typical of industry standard flow metersfor which a large body of discharge coefficient calibration dataexists. Finally, some of today's engines have flow rates as highas 2500 Ib/s, and some engines planned for the near futurewill have flow rates exceeding 3000 Ib/s. Thus, the possibilityof calibrating them in any of the world's largest engine testfacilities, such as the Arnold Engineering Development Cen-ter (AEDC) in Tullahoma, Tennessee is now remote, and

Received Dec. 30,1992; revision received March 5,1993; acceptedfor publication March 9, 1993. Copyright © 1993 by the AmericanInstitute of Aeronautics and Astronautics, Inc. All rights reserved.

*Manager, Aerothermo Systems Integration. Senior Member AIAA.tProfessor, Department of Aerospace and Engineering and Ap-

plied Mechanics. Fellow AIAA.

will soon be impossible, since the highest known flow rate ofany of these facilities is about 2750 Ib/s.5

The only known way to determine the discharge coefficientsof bellmouths having subcritical flow conditions and non-standard geometries which cannot be calibrated elsewhere, isto determine them theoretically.

There are two ways of validating the theoretical methodused for this purpose. The first is to utilize the method usedfor the large bellmouth discharge coefficient predictions andapply it to a scale model bellmouth of small enough size suchthat it can be calibrated using an industry standard criticalflow venturi. The second is to directly compare the detailedflowfields measured in such a scale model bellmouth withthose predicted. This approach is similar to that used by Smithand Matz6 in their attempt to better understand the validityof the discharge coefficient of an ASME nozzle obtained fromincompressible (water) calibration for use as a compressiblesubcritical airflow meter. They did not show comparisons ofpredicted vs measured discharge coefficients, but they didshow comparisons of their flowfield predictions with mea-surements taken over a range of subcritical throat Mach num-bers. Although their predictions of the ASME nozzle flowfieldcompared reasonably well with their measurements, since pre-dicted discharge coefficients were not presented and the throatboundary layers were not measured in their tests, it is difficultto draw conclusions from their work alone as to the adequacyof theoretical methods for the prediction of the dischargecoefficients of subcritical nonstandard (i.e., large engine bell-mouth) flow meters.

It should be noted that the work of Smith and Matz6 wasaccomplished nearly 30 yr ago, and to the author's knowledge,there has been virtually no published work since then whichattempted to build on their experience. The partial successof Smith and Matz6 in predicting subcritical ASME nozzleflowfields, and the myriad successes in the use of CFD sincethen, along with the continued push for increased flow me-tering accuracy in large engine bellmouths whose dischargecoefficients could not be determined experimentally, led tothe test program and theoretical studies which are describedbelow. The main objective of this program was to experi-mentally verify the theoretically predicted discharge coeffi-cient of a scale model of a large engine bellmouth using anindustry standard ASME critical flow metering nozzle. Therationale for this was that once the prediction method had

615

616 LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT

been verified with an accepted industry standard flow meter,the method could be used on any subcritical flow meter, re-gardless of size, and in particular, very large engine bell-mouths where other means are not available for establishingtheir discharge coefficients. It was also intended that the dataobtained could be used by others to serve as a bench markfor establishing the accuracy of other theoretical methods usedfor the same purpose.

BackgroundA head-type flow meter is one that uses the measured dif-

ference between the total pressure and static pressure in themeter throat to compute the so-called velocity head (hence,the name "head meter") which is subsequently used to com-pute the flow rate. When the static pressure (and thus ve-locity) varies across the throat of the meter, this effect mustbe taken into account in order to calculate the flow rate. Thedischarge coefficient of a head-type flow meter is defined as

the ratio of the actual flow rate to the ideal one-dimensionalinviscid flow rate that would exist if all the flow were at thevelocity corresponding to the static pressure at the meter throatwall. The more uniform the velocity profile across the throatof the meter, the higher its discharge coefficient. A dischargecoefficient of unity would imply a perfectly uniform velocityprofile and no wall boundary layer. Therefore, the accuratedetermination of the discharge coefficient of a subcritical flowmeter is dependent on three primary factors. The first is theVelocity level that exists at the throat, the second is the profileof velocity (pressure) across the throat, and the third is thethickness of the throat wall boundary layer.

There are four methods for determining the discharge coef-ficient of a head-type flow meter. The first is to survey thevelocity, pressure, and temperature fields across the meterthroat to determine the pV profile which can then be inte:grated to determine the actual mass flow rate. The second isto compute these same quantities using a theoretical method

2 4 6 8 10

Fig. 1 Flow meter aerodynamic contour definition.

14

1.0r

0.9

0.8

0.7

Predicted StaticPressure Distribution

(STC)

0.6

2 4 6 8

Axial Station ~ inches

Fig. 2 Flow meter theoretical pressure distribution.

10 12

LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT 617

and then perform the same integration of the pV profile. Thethird is to physically weigh the quantity of fluid that is passedthrough the meter in a given amount of time. This is calleda primary calibration because the actual mass is measureddirectly. The fourth is to calibrate the flow meter using an-other flow meter whose discharge coefficient is already known.This type of calibration is called a secondary method, becausethe actual mass flow through the meter is not measured di-rectly, but rather, it is calculated using the known dischargecoefficient of another flow meter.

Many of the issues relating to the accuracy of determiningdischarge coefficient using these four methods are discussedin detail in Ref. 2. The test program described below wasdesigned to allow the determination of the discharge coeffi-cient of the simulated bellmouth inlet using two of the fourmethods mentioned above. The first was to make flow surveysacross the throat of the meter including the wall boundarylayer. The second was to perform a secondary calibration ofthe simulated bellmouth using an industry standard ASMEcritical flow venturi. The results of each of these tests andcomparisons with the theoretically determined discharge coef-ficients are now described.

Meter Design and Pretest Predictions

Flow Meter DesignThe flow meter aerodynamic contour definition is shown

in Fig. 1. Figure 2 shows the aerodynamic contour along withthe theoretically predicted wall pressure distribution. In Fig.1, it is seen that the meter consists of four main segments.The contracting section has an elliptical contour from the so-called "hilite" station (X = 0, R = 4.8785). The upstream6.125-in.-radius pipe is part of the facility adapter hardware.A radial line connects the two at station 0. The throat regionof the meter consists of a 1.827-in.-long cylindrical sectionfrom the end of the elliptical section to the start of the diffuser.

The cylindrical section which connects the upstream con-traction with the downstream diffuser is intended to providea region of constant physical flow area for ease of area mea-surement. Since the absolute flow rate depends on the actualthroat area, a cylindrical section is chosen to minimize thearea measurement uncertainties. Although this may not bea critical problem for small meters whose contour can beinspected with great accuracy on modern contour measur-ing machines, it is a problem on very large meters whosesize prohibits this possibility. Table 1 shows the theoret-ical coordinates compared with the inspected coordinates ofthe meter.

Pretest PredictionsThe stream tube curvature (STC) computer program7 was

selected for use in predicting the meter flowfield and potentialcomponent of discharge coefficient. It was chosen becausethe author's many years of experience with it, for predictingall subsonic internal flowfields, has shown it to give excellentresults. In addition, a key feature of the STC program is thatit is driven by global mass conservation as part of its solutionprocedure, thus making it a natural choice for such predic-tions, as opposed to other methods where numerical dissi-pation can produce global mass conservation errors.

The boundary-layer calculations were carried out using theHarris code.8 It is one of a number of validated reliable bound-ary-layer codes, and its selection was based primarily on itsbeing widely used and its thorough documentation.9

Figure 3 shows the predicted discharge coefficient vs wallstatic pressure ratio. The equation given in Fig. 3 is simplyderived from a curve fit of the calculated points for the naturaltransition case. The meter discharge coefficient depends ononly two components, an inviscid part, CDpot, that accountsfor the radial pressure variation in the throat, and a viscouspart, CDg*, that accounts for the effective boundary-layer

Table 1 Bellmouth model inspected coordinates

X0.00590.01610.02640.03670.04690.06740.08780.10840.15960.21090.26220.36470.56970.77470.97981.18481.38981.59491.79992.00492.20102.41502.62002.7225

*theo4.78874.72974.68834.65464.62544.57574.53334.49594.41634.34844.29194.19344.0393.92053.82473.74623.68153.62853.58583.55233.52743.51073.50173.5001

^inspect

4.79644.72904.68794.65454.62264.57134.52934.49254.41314.34704.28874.19014.03623.91783.82253.74393.67993.62743.58463.55143.5273.51003.50203.5010

X3.67003.67003.67004.61905.02905.43915.84916.25926.66927.07937.48937.89948.30948.71959.12959.53969.949610.359710.769711.179811.589811.692411.934313.7037

^thco

3.50003.50003.50003.50003.50493.51753.53683.56173.59153.62503.66133.69943.73843.77733.81513.85083.88343.91213.93583.95353.96433.96583.96733.9673

-^inspect

3.50013.50073.50003.49843.50353.51643.53673.56093.59143.62603.66203.70053.73953.77823.81573.85053.88373.91263.93563.95303.96353.96503.96633.9674

1.000

0.995

0.990

o 0.985

0.980

0.975

0.970

."us = .83061 + .28286(PT/PQLJ) - .1739:.03814(PT/PSW)3

T / rSW'

1.0 1.1 1.2 1.3 1.4PRESSURE RATIO - PT/PSW

1.5 1.6

Fig. 3 Discharge coefficient components.

blockage. Reference 2 describes how each of these compo-nents is determined from the STC and boundary-layer solu-tions.

Test Program DescriptionThe test facility chosen for conducting both the simulated

bellmouth flowfield measurements and discharge coefficientsecondary calibration, was the Medicine Lake AerodynamicsLaboratory of the FluiDyne Engineering Corporation locatedin Minneapolis, Minnesota. All tests were conducted in thelaboratories' channel 12 cold flow static thrust stand, normallyused for high-accuracy thrust and flow measurement exhaustnozzle testing.10 Based on a statistical analysis of many pre-vious tests, it was estimated that, for all but the lowest bell-mouth pressure ratio, measured discharge coefficients in thisfacility would have an absolute error (facility bias + randomerror) no greater than ±0.25% at a 95% confidence level.

618 LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT

i____

0.05

0.03

0.02

0.000.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

MJETFig. 4 Pressure survey probe free-jet calibration test results.

0.99 1.00 1.01 1.02 1.03 1.04 1.05M / MCL

Fig. 5 Predicted and measured throat Mach profiles.

0.710

0.705

0.695 -

0.685 -

0.67545 90 135 180 225 270 315 360

ANGULAR POSmON - DEGREES

Fig. 6 Effect of survey probe on measured throat static pressures.

[ )0.713 -

DATA POINT 38

AVG.

± 0.1J5 AIRFLOW

0°

TAP LOCATIONS270 90°

225 35*

180°

0.667

0.666

0.664

DATA POINT 47

r—||

AVG.

±. 0.1X AIRFLOW0 45 90 135 180 225 270 315 360

ANGULAR POSmON - DEGREES

Fig. 7 Measured throat static pressures without probe.

1.000

0.995

0.990

0.985

0.980

0.975

0.9701.3 1.4

PRESSURE RATIO - PT/PSW

.1 .2 .3H—T-

.4MACH NO.

.6 .7H-

.4.811.211.60 1.98 2.36 2.69 2.98REYNOLDS NO. ~ ReQ x 1CT6

Fig. 8 Discharge coefficient vs wall static pressure ratio.

The test program consisted of surveys of the radial staticpressure profiles at the meter throat, surveys of the throatwall boundary layer, and measurement of the meter overalldischarge coefficient. In addition, the pitot-static probe usedto conduct the throat radial static pressure profile surveys wascalibrated in the free jet discharge of a 6.9915-in.-diam ASMEnozzle. Details of the test facility, test apparatus, and testprocedure may be found in Ref. 2.

Test Results and DiscussionThe scale model test program was comprised of four phases.

The first phase consisted of free-jet calibrations of the tra-

LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT 619

0.7

0.6 -

Predicted StaticPressure Distribution

(STC)

4 6 8

Axial Station ~ inches

Fig. 9 Boundary-layer trip location.

10 12 14

PREDICTION. NATURALTRANSITIONPREDICTIONWITH TRIP

1.3027

DATA NATURAL \TRANSITION I POINT 28DATA WITH )TRIP

0.09 •

0.08 •

0.07 -

0.06 -

2 0.05 -I

0.04 -

0.03 -

0.02 -

0.01 -

0.00

Fig. W Predicted \s measured boundary-layer profiles.

versing pitot-static probe used for surveying the meter throatpressure profiles. Phase 2 involved surveying the meter throatusing the calibrated probe. In phase 3 the secondary calibra-tions of the meter discharge coefficients were obtained usingthe industry standard ASME low beta ratio critical flow ven-turi, and in phase 4 the meter throat boundary-layer surveyswere acquired.

Figure 4 shows the pitot-static probe calibration results fortwo axial placements in the freejet. Based on Ref. 11 data,the pressure coefficient was expected to be near zero. Thefact that it wasn't, illustrates why any throat survey devicemust be calibrated, even duplicate copies manufactured tothe same specifications. In Fig. 4, the difference in CP be-tween the two axial placements of the probe would representa flow error of 0.1% at a jet Mach number of 0.4.

0.020

0.015

£ 0.010

£ 0.005

0.000

y\\\

\V

1̂"

——————— PREDICTED NATURALTRANSITION

___ . ___ PREDICTED ALLTURBULENT

____ —— PREDICTED TRIPPED

Q MEASURED NO BL TRIPA MEASURED WITH BL TRIP

—'— IL~

J

. — ..•••••» •

3 ——

1.0 1.1 1.2 1.3 1.4 1.5PT/PSW

Fig. 11 Throat displacement thickness.

1.6

Throat static pressure profile surveys were obtained at threeflow rates which spanned the operating range of the meter.Figure 5 shows the survey results adjusted for the effects ofthe probe calibration shown in Fig. 4. Also shown in Fig. 5are the theoretically predicted profiles. All of the profilesshown are normalized by the centerline value of Mach num-ber. Although it was anticipated at the outset of the programthat the presence of the survey probe in the meter would havesome effect on the local flowfield around the probe, it wasnot expected to be as large as the measurements subsequentlyindicated. The probe produced both a blockage effect and aflowfield distortion at the wall static pressure measuring plane.Figure 6 shows the wall static pressure distribution at thethroat plane with the probe at its centerline immersion. Figure7 shows the corresponding wall static pressure distribution

620 LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT

0.95

0.90

Q POINT 6, PT/PSW = 1.0116POINT 13, PT/PSW = 1-1096

4 6 8AXIAL STATION - INCHES

0.80

0.75

0.70

0.65

0.55

0.50 4

Q POINT 38, PT/PSW = 1.4014O POINT 47, PT/PSW = 1.5035

4 6 8 10AXIAL STATION - INCHES

Fig. 12 Predicted vs measured axial wall static pressure distributions. Fig. 14 Predicted vs measured axial wall static pressure distributions.

1.00

0.90

0.80

0.75

0.70

0.65

0.60

D POINT 20, PT/PSW = i-2009

POINT 28, PT/PSW = 1.3027

4 6 8AXIAL STATION - INCHES

Fig. 13 Predicted vs measured axial wall static pressure distributions.

with no probe installed at flow rates near that of Fig. 6. Thisdistortion was caused by the probe support stem which wassuspended from only one side of the meter. Its radial blockagevaried with probe immersion and, although small, it produceda significant flowfield distortion which was large enough toproduce the disagreement between predictions and measure-ments seen in Fig. 5. This data is key evidence that showswhy static pressure survey devices should not be used in thethroat of unchoked flow meters for discharge coefficient de-termination when high accuracies are required.

Measured Discharge CoefficientsFigure 8 shows the measured discharge coefficients based

on the ASME critical flow metering nozzle calibrations com-pared with the pretest predictions for both natural transitionand an all-turbulent boundary-layer assumption. The agree-ment between the predicted discharge coefficient for naturaltransition and the measured data is within ±0.25% every-where except at the lowest pressure ratio of 1.01, where largerscatter was expected due to the known sensitivity of airflowto pressure measurement errors at low Mach number.2 In

order to investigate the influence of throat boundary-layerthickness due to transition location, a boundary-layer trip waslocated as shown in Fig. 9. Number 100 garnet sandblastinggrit, having a mean particle diameter of 0.006 in., was usedfor the trip. This grit size was chosen to initiate transitionimmediately downstream of the trip using the criteria of Ref.12. The trip location was chosen such that it was far enoughupstream of the meter throat to represent as closely as possiblean all-turbulent boundary layer, but far enough downstreamsuch that relaminarization would not occur in the strong fa-vorable pressure gradient in the contracting section. The dif-ference in discharge coefficient between the tripped anduntripped data is very close to the difference between the nat-ural transition and all-turbulent boundary-layer predictionsin Fig. 8.

Boundary-Layer SurveysOne throat boundary-layer survey was done with the trip

installed, and three were done without a trip. Figure 10 showscomparisons of the measured profiles with the pretest pre-dictions for a meter pressure ratio of 1.3027, and Fig. 11compares the predicted displacement thickness with the mea-surements for the forced and natural transition cases. In gen-eral, the agreement is very good. For the natural transitioncases, the difference between predicted and measured dis-placement thickness would represent an uncertainty in dis-charge coefficient of less than 0.03%.

Wall Static Pressure DistributionsFigures 12-14 show the axial distributions of measured wall

static pressure compared with the predictions using the STCand Harris computer codes. The agreement is seen to beexcellent over the full length of the meter for all flow rates(PTIPSW). The slight disagreement near the exit of the meteris characteristic of an uncoupled inviscid diffusing flowfieldcalculation which is adjusted for the calculated boundary-layer displacement thickness.

The excellent agreement between the measured wall pres-sure distributions and predictions, as well as between thepredicted and measured boundary layers, provides added con-fidence that the agreement between the predicted and mea-sured discharge coefficients in Fig. 8 is not just fortuitous.

ConclusionsA comprehensive subcritical flow meter design and test

program has been conducted which clearly demonstrates that

LAHTI AND HAMED: VERIFICATION OF DISCHARGE COEFFICIENT 621

current theoretical methods can predict discharge coefficientsto within 0.25% of those obtained from calibration with anindustry standard critical flow ASME nozzle. To the author'sknowledge, this is the only such program that has ever beenconducted on a subcritical flow metering device which dem-onstrates this high level of accuracy.

Discharge coefficient measurements made with a boundary-layer trip, as well as meter throat boundary-layer surveys,clearly show the importance of transition location on accuratedischarge coefficient predictions. Detailed surveys of the me-ter throat flowfield demonstrate that such surveys are notcapable of measuring the radial static pressure profiles withsufficient accuracy to determine discharge coefficient to within0.25% since the survey probe produces disturbances whichalter the flowfield being measured. Free jet calibration of thesurvey probe is incapable of correcting for these effects sincea freejet is not representative of the confined flowfield in themeter throat where probe blockage is important.

Comparisons of theoretically predicted wall static pressuredistributions with measured data suggests that as long as theflow remains subsonic, the STC and Harris boundary-layercodes provide excellent predictions of the meter flowfield.

References'Smith, R. E., and Matz, R. J., "A Theoretical Method of De-

termining the Discharge Coefficients for Venturis Operating at Crit-ical Flow Conditions," Transactions of the ASME, Series D, Journalof Basic Engineering, Vol. 84, No. 4, 1962, pp. 435-446.

2Lahti, D. J., "Theory and Experiments on Subcritical Compres-sible Gas Flow Metering," Ph.D. Dissertation, Univ. of Cincinnati,Cincinnati, OH, Aug. 1990.

3Smith, R. E., and Matz, R. J., "Verification of a TheoreticalMethod of Determining Discharge Coefficients for Venturis Oper-ating at Critical Flow Conditions," Arnold Engineering DevelopmentCenter Rept. TR-61-8, Tullahoma, TN, Sept. 1961.

4Stratford, B. S., "The Calculation of the Discharge Coefficientof Profile Choked Nozzles and the Optimum Profile for Absolute AirFlow Measurement," Journal of the Royal Aeronautical Society, Vol.68, April 1964, pp. 237-245.

5Anon., Test Facilities Handbook, 12th ed., Arnold EngineeringDevelopment Center, Arnold Air Force Station, TN, March 1984.

6Smith, R. E., and Matz, R. J., "Performance Characteristics ofan 8 Inch Diameter ASME Nozzle Operating at Compressible andIncompressible Conditions," Transactions of the ASME, Series J,Journal of Fluids Engineering, Dec. 1973, pp. 542-550.

7Keith, J. S., Ferguson, D. R., Merkle, C. L., Heck, P. H., andLahti, D. J., "Analytical Method for Predicting the Pressure Distri-bution About a Nacelle at Transonic Speeds," NASA CR-2217, July1973.

8Harris, J. E., "Numerical Solution of the Equations for Com-pressible Laminar, Transitional, and Turbulent Boundary LayerEquations and Comparisons with Experimental Data," NASA TRR-368, Aug. 1971.

9Harris, J. E., and Blanchard, D. K., "Computer Program forSolving Laminar, Transitional, or Turbulent Compressible BoundaryLayer Equations for Two-Dimensional and Axisymmetric Flow,"NASA TM 83207, 1982.

10Anon., "Model Tests for Propulsion System Installed Perfor-mance," FluiDyne Engineering Corp. Brochure, Minneapolis, MN,Sept. 1990.

HKrause, L. N., "Effects of Pressure-Rake Design Parameters onStatic-Pressure Measurement for Rakes Used in Subsonic Free Jets,"NACA TN 2520, Oct. 1951.

12Braslow, A. L., Hicks, R. M, and Harris, R. V., "Use of Grit-Type Boundary Layer Transition Trips on Wind Tunnel Models,"NASA TN-D3579, 1968.