1985_pneumodynamic characteristics of a circulation control rotor model (extra k on f vector)

8/3/2019 1985_Pneumodynamic Characteristics of a Circulation Control Rotor Model (Extra k on F Vector)

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P ~ ~ U I I I U U ~ I I ~ I I I I ~har~vloael

Charles B . WatkinsProfessorHoward UniversityWashington, D. C .

stics am m m

Kenneth R . ReaderSenior Aerospace EngineerDavid W . Taylor Naval Ship Researchand Development CenterBethesda, Md.

rculation Ccontrol Rotor

Subash K. D i ~ t t aGraduate AssistantHoward UniversityWashington, D .C .

Numerical and em cl.mm.sm.~.I. nulation of unsteady almo w throu eh the control valve and slotted air d uct ofa circulation control rotor are described. The numerical analysis involves the solution of the quasi-one-dimen-sional compressible fluid-dynamic equations in the blade air duct together with the coupled isentropic flowequations far flow into the blade through th e valve and o ut of the blade through th e Coanda slot. Numericalsolutions are comoared with basic emerimental results obtained for a mockup of a circulstion control rotoran d its pneumatic valving system. Thepneudynsmic phenomena th at were observed are discussed with particu laremohasis an the characteristic system time lam associated with the response of the flow variables to transientand periodic control valve inpu b

Notation T = duct static temperatureA = cross-sectional area of duct To = duct total temperatureA, = effective expansion area at valve exit T, = duct wall temperatureA, = valve area t = timecr = duct friction factor U = vector of dependent variablesC , = slot discharge coefficient V = vector of diffusion termsC, = valve discharge coefficient v, = average duct velocity in spanwise or x directionc, = specific heat at constant volume v, = approximate average duct velocity in chordwise or yDM = hydraulic diameter of duc t directione = total energy per unit volume w, = slot widthF = flux times area vector x = spanwise coordinateG = vector of nonhomogeneous terms in flow equations x, = coordinate at duct entranceh = heat transfer coefficient x,., = coordinate at duct endH, = average duct height y = chordwise coordinatek = thermal conductivity y = ratio of specific heatsL = length of duct p = densityriz = mass flow rate

= angular velocitym = mass flow rate per unit area in duct Subscriptsm. = mass flow rate oer unit area com ~u te drom isentropic I = variable computed from isentropic flow at valve exit~~, flow theorym, = mass flow rate per unit area through slutm, = mass flow rate per unit area through valvep = static pressure in ductp,, = external pressure at slot exitpo = total pressure in duct-resented at the Second Decennial Specialists' Meeting on RotorcraftDynamics, NASA Ames Research Center, Moffett Field, Calif., Nov.1984

p t = variable in pressure supply plenum or rotor hubPrefix6 = peak-to-peak value of variableIntroductionC RCULATIO N control (CC) rotor technology, applied torotary-wing aircraft or stopped-rotor vertical takeoff andlanding (VTOL) aircraft such as the X-Wing, offers distinctadvantages over conventional rotor technology. For example,CC technology provides a straightforward means of imple-

23


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C . B . WATKMS IURNALOF THE AMERICAN HELICOPTERmentlng higher harmonic control to suppress vlbrattlon and bladestresses. Both types of aircraft use a shaft-driven rotor withblades having circulation control airfoils which generate liftthrough the Coanda principle. These CC airfoils employ around edtrailing edge with a thin jet of air tangentially ejected from aspanwise slot immediately above the rounded (Coanda) surface.The iet of air suppresses boundarv laver separation and movesthe r k r sta gnatid i streamline toward ihe lower surface, thcrcbyincreasing l i f t . I.ift is increased with increasin g mass flow ratcof comp ressedair in the jet. Thc remaining control requirementsarc ohrained by cyclic m odulation o i the mass flo w ratc withvalves in the nbnrbtating system. Higher harmonic cyclic con-trol can he similarly applied for reducing blade stresses andvibration.The design of CC rotors and their pneumatic control systemsrequires anu nders tand ing and apprkciation of thc phcnonlenalnvolvcd i n thc control and distnhution of airflow to the Coendnslots in these rotors. A capability for analytical prediction ofrotor Coanda airflow is also essential. Th e term "pneumodyn-amics" has been coined to refer to the aerodynamic responsecharacteristics of CC rotor system internal ducting airflow.In a recent paper, Watkins, e t al.' describe the modelingtechniques employed in the high frequency pneumodynamicanalysis (HFPA) computer code developed for CC rotor pneu-modynamic analysis. T he present paper reviews the m odelingtechniqucs described in Ref. I and presents results obtained byapplying the code to a simple experimental mockup of a modelrotor hlade and its pneumatic v alving system. T he discussionof the results, from both the HFPA predictions and the exper-iment, focuses on th e dynamic response of the flow variablesto periodic and transient control valve inputs. These responsecharacteristics, particularly the characteristic system time lags,have implications for the performance of actual rotor systems.

TheoryThe modcling techniques cmploycd in the prcscnt researchare descrihed in Rcf. I where cm phasis is on the devails of thetheoretical formulation and numeAcal technique. The theory ispresented here in abbreviated form; a more complete discussion1s contained in Ref. 1.Basic EquationsThe HFPA code solves the quasi-one-dimensional, com-pressible flnid-dynamic equations for unsteady flow in a span-wise blowing air supply duct internal to a rotating CC rotorhlade, together with the coupled isentropic flow equations forflow into the hlade through the control valve and out of thehlade through the Coanda slot. Th e continuity, m omentum andenergy equations for flow in the duct have the conservativeform representation

where the vector of dependent variables is

and the flnx times area vector is

10

Th e gular velocer coefficievector of nonhomogeneous terms. which inctuaes tneeffects of blade rotation at an $kin rictionfactor cAx), w all heat transf i mass fluxthrou gh the slot m,(x, t) is

1For completeness, the vector of axial diffusion terms is in-cluded although it is insignificant in the present problem. It isr o 1

The fluid dynamics equations, Eqs. (2) are presented here inconservation law form since this is the form employed for thenumerical solution algorithm.The equation of state is

In the above relations, the conservative variables m and e ar edefined asm = pv,

and w,(x, t) and m,(x, I ) are the local (with respect to spanwisex location) slot width and blowing mass flux, respectively.Th e small average chordwise velocity component inside theduct can be crudelv anoroximated as the mean of the value of..chordwise velocity for flow in the duct near the slot (as deter-mined from continuitv) and the value of zero at the wall oo-posite the slot. ~e n $ ,in which H D(x) is the local effective height of the duct.The mass flux distribution through the Coanda slot is as-sumed to he well represented by one-dimensional isentropicflow theory with a discharge coefficient C,(x);

The above functional relationship is the usual expression forisentropic flow expanding through a nozzle from a plenum (inthis case the duct interior) to ambient conditions (in this casethe exterior of the rotor hlade). po(x, t) and TO(& t ) are thelocal (internal) total pressure and temperature, respectively,and p,,(x, t) is the external pressure at the exit of the Coandaslot. Dependency upon external pressure is eliminated whenthe flow is choked.


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lrounoary LOBAt the upstream bou ndary x = x,, aple num supplies blowingair to the duct through a valving system. The valve openingarea has constant and rotor-azimuth-dependent com ponents reg-ulated by collective and cyclic components, respectively, ofthe aircraft control system. The mass flow and duct pressuresare thus determined by these valve settings in concert with theductlslot configuration. Figure 1shows the idealized represen-tation adopted for the cam type valve of interest in the presentstudy. The controlled value of the valve opening is designatedas A,@), and A, is the fixed effective opening at the duct en-

trance. Like the slot flow, one-dimensional isentropic flowtheory with a discharge coefficient is used to represent flowthrough the valve. Th e valve mass flux ismu@)= Cvm,@oPt. ToptpP(x,, f) ) (5 )

where po and To are the total pressure and temperature, re-spectively, in the air supply plenum in the rotor hub and p(~,,0 is the back pressure in the duct entrance downstream of thevalve. When the flo w is choked , the dependency onp(x,, t ) iseliminated.The other boundary condition, a t x = len d, s the stagnationof the flow at the end of the duct.Th e discharge coefficients C, and C,, used as correlationparameters for a given system, relate the geometric area of asystem to the effective area. F or simple configurations such asorifice plates or venturi tubes in a well conditioned systemwhere the airflow is well behaved, the discharge coefficientvalue can be obtained from a standard engineering handbook.Finite-Difference Method

In applying the theory of th e previous section to the analysisof a rotor blade, the entire blade, including the transition duct(ducting from hub valve exit to airfoil portion of rotor blade),is discretized by dividing it into spanwise (radial) segments.The governing differential equations, Eq. ( I ) , are solved bythe implicit, "delta for m,' ' finite-difference procedure of Beamand Warm ing2 at the grid points located at segment boundaries,including the valve exit as shown in Fig. 1 and the duct end.The solution of the du ct flow is actively coupled to the flowthrough the valve by means of the upstream boundary condi-tion. With the exception of the upstream boundary conditionfor flow into the duct from the valve, the numerical analysisis rather standard and is described fully in Ref. 1.

CIRCULLIn Ref. 1

TOR MOD1

merical upstream boundary condition based on deriving forvarious valve types an auxiliary relationship, by ignoring thetime dependent term in the momentum equation between thefirst two grid points (grid point 1 and 2 in Fig. 1). The ap-proximation for the idealized valve of Fig. 1 is obtained fromthe steady momentum equation as

Approximations such as in Eq. 6) were applied because theinvestigators were unsuccessful in applying the more standardand less approximate procedures for inlet boundary conditionsof applying one-sided differencing or the method of charac-teristics. The presence of the valve poses spec ial difficulties inthe application of these standard techniques. In a recent com-munication, And erson of United Technologies Research Centerindicated that he was successful in applying the method ofcharacteristics to derive inlet boundary con ditions for the HFPAcode for a sudden expansion or gate-type valve. The extensionof his procedure to the cam valve type shown in Fig. 1 is notstraightforward due to the 90' change in flow direction. Sincethis valve is the type considered in the present investigation,Eq. (6) was retained.Numerical Solution Procedure

Th e basic approach followed in obtaining num erical solutionsto control inputs, consisting of constant (collective) settings orconstant settings with superimposed constant amplitude cyclicinputs, is to obtain the steady state or time periodic solutionsby allowing them to evo lve from a transient.Th e solution starts with the system essentially at supply pres-sure conditions. The initial valve position is assumed to befully open as specified by the sum of its Fourier componentssupplied as input. The solution is then advanced with time asthe flow in the blade adjusts itself to reach steady state flowconditions for the fully open valve. If cyclic components areto be considered, the cyclic input is imposed after fully opensteady state flow conditions are achieved. T he solution is there-after advanced further with time until steady, periodic condi-tions are attained. A harmonic analysis of pressure, temperatures,and mass flow rates is performed on the results from the finalcycle.

Experiment. The analytical formulation and the numerical methods de-scribed in the previous sections were validated by applyingthem to predict the results of a basic exp erime nt3 The exper-iment was conducted using a mockup of a rotor blade pneu-modynamic system incorporating a valve similar to the idealizedvalve of Fie. 1. The valve modulated the flow into a suhscale

AJt)Fig. 1 Idealized valve

arc ula tion control rotor hledc represented hy a stationary pluggdpipe with 3 lcn ~th wi sc lot. For sirnnlicitv. the correct rclativcmotion of an actual rotor blade pnkumddynamic system wasestablished by interchanging the normally rotating and non-rotating components. In the model, as shown in Fig. 2, thepipe was stationary and rotary motion was employed in thevalving system. The model valveconsisted of a st a ti o n q plenumsupplying air to the stationary pipe inlet nozzle, which waspartially obstructed by the su rfaces of two rotating cams mountedcontacting each other on the same shaft. T he shaft is mountedperpendicular to the nozzle axis and the effec tive nozzle areaformed by the area between the nozzle entrance and the camsurfaces varies as the cams rotate. Th e two cam surfaces each


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. , ,.", =; MLL "nmcn"=,","=AR E I N

ONE-PER-REV CA M2.00 DIA TWO-PER-REV CAM

0. kFig. 2 Experimental model

have di e re n t wavy profiles, one with a single waveform amundthe cam periphery, and the other with two distinct waveformsamund the cam periphery. Therefore, either a one-per-(cam)revolution or two-per-(cam) revolution nozzle area variation isachieved depending on which cam surface is positioned overthe nozzle entrance.The experiment was originally performed3 to establish thefeasibility of modulating the weight flow in a one-per-rev andtwo-per-rev manner and to show that the collective and cvcliccomponents of the airf low were additive. The pipeincorpo;atedseveral pressure taps distributed alone its leneth. D vnamic mea-" -surem eits of pressure were obtained for a range of camrotational speeds. In addition, quasi-dynamic results were oh-tained by positioning the cam at discrete increments of 30".Average mass flow data were collected from a venturi tubelocated upstream of the plenum.Figure 3 shows the measured valve area formed by the one-per-rev and two-per-rev cams spaced at a minimum distanceof 0.01 inch from the nozzle inlet. The points shown werecalculated from the measured gap between the cam profile andthe nozzle; the area variations are intended to approximatesinusoids as shown. T he two-per-rev cam profile is a rathercrude represen tation and contains a significant higher harmoniccontent due to the deviation. T he measured values were usedin the numerical predictions.

Results and DiscussionNumerical results were obtained for the experimental con-figuration of Fig. 2 by discretizing the pipe into 16 segmentsof approximately equal length. The pipe was assumed to beadiabatic, and the friction coefficient was represented by theformulas for fully developed flow. The integration time-stepfor the calculations simulating the quasi-dynamic experimentswas 5 x sec . For the dynamic calculations, valve cyclingwas imposed after an elapsed time of 0 .25 sec. T o ensurestability, it was found nccc


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CIRCUL, .,.,, nd slot discharge coefficients in t . ."-lution. The results obtained for the higher value C, = C, =0.95 agree more favorably with the experimental results, al-though the lower discharge coefficient is likely to be realisticphysically. That a concept as simple as a discharge coefficientcan be used to correlate the mass flow and pressure losses isvery encouraging; this is discussed in depth in Ref. 4. For theremainder of the numerical results C = C = 0.95 w ere usedwithout trying for a better correlation. As indicated in Ref. 1,strict interpretation of C, and C, is probably not desirable sincethe various deficiencies and approximations in the computermodel can, to some degree, be absorbed by adjusting them.Moreover, the concept of a constant nozzle discharge coeffi-cient is only a first approximation for the actual flow losseswhich occur ov er a range of mass flow rates in the analysis.Recent experience with the HFPA code indicates that morephysically realistic discharge coefficients can be ohtained atthe expense of decreased resolution of wave reflection phe-nomena by including second-order numerical damping in theHFPA algorithm.Figure 5. illustrates the numerical an d experimental c yclevariation of total pressure for a typical dynamic case. Thenumerical results of Fig. 5 were obtained with C = C, =0.95. The numerically computed pressure profiles have an azi-muthal phase lag from the valve area profiles of Fig. 3 ofapproximately 35". For this configuration, the phase shift isprincipally due to the finite speed of wave propagation and isreferred to as "sonic la^." A smaller portion of the delay isdue to the "capa citanc elag" effect ca"scd by the finite pipevolumc. T o facilitate com pariso nof the profile shapes the phasedifference between the vdv e m a etting and thepressuie re-sponse to it has been eliminated from both curves in Fig. 5hccause the experimental phase angle results were not ac cu ite .Table I contains data ohtained from a hdrmonic analysis ofthe dynamic numerical results of the case for which the pres-sures are plotted in Fig. 5. The numerical phase lag has beenretained. A harmonic analysis of theexperimental pressure dataobtained from the experimental curve presented in Fig. 5 isalso shown for comparison, but without phase shift since, asmentioned, an accurate determination of experimental phaseshift is not available. Magn itude and phase for the valve open-ing area, total mass flow rate, and total pressures at two lo-cations m given from the numerical results where the phasesare referred to the maximum valve opening at zero deg. (Theexperimental pressure phase shifts given in the table have anarbitrary reference). The magnitudes are all normalized withrespect to the peak-to-peak values.

5.00- - - EXPERIMENT, DYNAMIC- OMPUTATION, DYNAMIC

AZIMUTH POSITION ldeglFig. 5 Total pressure variation

Table 1 ,..,,. ...,. ..., ....,..,..,.. ". ,.e higher ha ,...,...,content of the m ass flow rate output is a rather significant 20%while the one-ner-rev valve control input contains onlv ap-proximately 10%.Thcrcforc, the mass flbw exhibits somenon -lincar amvlification. The higher harm onic content of the pressureresponse'does not exhibirs imilar amplification. ~ h digherharmonic content of all the variables for the fourth and fifth(n = 4 and 5) harmonics are in the range of m easurement ornumerical error and are included only to give some feeling forthe "noise" levels in these results.Th e phase shift phenome non is clearly illustrated by the datain the table. The first harmonic of the mass flow rate lags thevalve opening by 39". The first harmonic pressure at xl L =0.524 lags the valve opening by 36". At xlL = 0.804, this lagincreases to 42" due to the finite wave propagation speed orsonic lag. Because of the specification of a zero numericalintegration dissipation parameter in the numerical algorithmused for the cyclic calculations, computed results contain slight,stable spatial numerical oscillations which make it difficult toplace confidence in the exact amount of the lag in pressurepredicted between locations. Based on the speed of sound, thepure sonic lag between the two points shown should amountto only about one-half the lag shown. T he remaining lag maybe d ue to a physical phenomenon o r to numerical uncertainty.Although the problem pf spatial oscillations can be reduced byemploying numerical damping, as discussed previously, thisdamping suppresses multiple wave reflections and renders thecode less able to predict resonance phenomena. Fortunately,the computed m asif low rate is the result of a spatial integrationwhich has the effect of smoothing the spatial oscillations. Itshould he fairly accurate.Peak-to-Peak omparison

Figure 6 summarizes the results from the dynamic calcula-tions in the form of peak-to-peak total pressures versus fre-quency for the one-per-rev cam. In Fig. 6 the numerical solutionis able to predict fairly well the trends and magnitudes of thepressure response. One possible sour ce of error in the numericalcalculatio n;is th e a ~ s u m ~ t i ~ ~ nonstant discharge cucfficientlur the wide rangc of area variation. Somewhat more remoteare the possibilities for experimental error.Figure 7 is a plot of the dynam ic results of the peak-to-peaktotal pressure near the beginning of the slotted portion of thepipe against plenum pressure for the one-per-rev cam at twodifferent frequencies. The agreement between the measuredand numerical results in this figure, in general, is seen to befair. As shown, the peak-to-peak total pressure at the entranceof the pipe is dependent on the frequency of rotation of thecam. The data shows a droop in the peak-to-peak value ofpressure at high plenum pressure and high frequencies; thesetrends with plenum pressure and frequency are predicted bythe computer code. The peak-to-peak total pressure at the en-trance of the pipe is shown to be dependent upon the frequencyof rotation of the cam.Figure 8 displays the peak-to-peak static pressures versusrotational frequency for various plenum pressures for the one-

per-rev cam. The experimental and numerical curves generallyshow go od ag reement an d the computer code pred icts the &ends.Figures 9 and 10 are plots of the ratio of pipe end-to-entrancepressure versus cam rotational frequency for both cams. Thetwo-per-rev cam results a re included since the two-per-rev camextends the frequency range two-fold. Results are presentedfor two different plenum pressures for both one- and two-per-rev cams. These curves demonstrate the presence of a quarter-wavelength resonance phenomenon in the pipe. This phenom-enon is better illustrated inF ig. 9 for the two-per-rev cam wherea resonance peak occurs at a frequency of approximately 44 Hzfor the numerical solution at the higher pressure and around55 Hz for the corresponding experimental data. The differencesin the resonant frequencies between the numerical and exper-


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06 08 OL 09 05 OD OE OZ 01 0I 10.0

s t p a 1 me3 A~I-1x1-om1a q q!m luais!suoi s! 01 .a!du! a~n ssa rd aqs!q a q 103 ad aaueuosar pa3unouo1d aromaqL .sa!manbaq me3 A~ I-l ad -o wqt aa!mt m q sa[ leqmamosare sapuanbaq jmuosal asaqL 'arnssard ramol aql 103 ZH 08

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JL Y 1985 CIRCULPTects of Rota-. .. lency on Ma!. ".s Flow Rate. c . ..rlgures 11 and iz snow tne errect or rotat~ona~requency

I the computed mean and peak-to-peak mass flow rates, re-~ec tively, or the one-per-rev cam at several different plenum.essures. No experimental data are available for comparisonwith the peak-to-peak data. The experimental lines shown forthe mean mass flow rate represent an experimental averagesince no significant changes in average mass flow rate withfrequency were obtained in the experimental data. Likewise:ry little change was observed in the numerical results.Figure 13 shows the effect of rotational frequency on thelase of the first harmonic of the mass flow results shown ings. 11 and 12. The phase shift increases with increasingplenum pressure (and increasing mass flow rate) and increasesalmost linearly with frequency over most of the range exam -ined. The linear trend indicates that within this range there isafix ed time delay, almost independent of frequency, associatedi t h a given set of system param eters such as plenum pressure~dhlade internal geometry. At the higher pressure, capaci-nce lag is controlling, while at the lower pressure the laglould he principally sonic lag.No experimen tal phase shift data are available for comparisonbecause it is very difficult, if not impossible, to measure dy-

0.0001 ' I0 10 20 30 40 50 60 70 80 90ROTATIONAL FREQUENCY IHz1

----EXPERIMENTAL AVERAGE M OMPUTATIONCv=CS=0.95Fig. 11 Mean mass flow rate

TOR MODE-

po l= 8.84 lblinz

-151 I0 10 20 30 a 60 W 70 80 SOROTATIONhL FREOUENCV 11111

Fig. 13 Maw flow rate phase angle

$ 0.0089(L 0.0062Y5 0.00440?* 0 0025P

0.000

namic m ass flow containing higher harmonics and obtain phaseinformation.In summary, the comparison of the numerical calculationswith experiment indicates that, even with a simple constantdischarge coefficient model, the computer code satisfactorilypredicts the system performance over a wide operating range.It is able to predict the trends an d magnitudes of the total andstatic pressures as well as the mass flow rate for plenum pres-sures from 0.98 to 13.75 psig, for rotational frequencies from15 to 80 Hz, and for valve areas varying from 0 to approxi-mately 1 in2.

- .

- .'.IIv -0-p - 3.04 Iblin -PI- .. 0-0-ppe= 0.982 lb, w-.COMPUTATION -Cv=Cs=0.95

Theoretical Transient Response

0 10 20 30 40 50 60 70 80ROTATIONAL FREQUENCY IHzlFig. 12 Peak-to-peakmass flow rate

For pneuma tic control system dcsign, in addition to a knowl-edce of the pcriodic rcsponse of the system to pcriodic controlin*, t he ttans ient d yn am ics a re a~s d o fntereit. In the presentstudy an independent numerical investigation of transient dy-namics was conducted. This investigation simulated the pneu-modynamic response of the model system to step inputs incontrol valve settings. This was accomplished by suddenlyopening the control valve permitting air to flow into a hlade atatmospheric pressure, allowing steady flow to he established,and then suddenly shutting off the flow, allowing the pressureinside the pipe to return to atmospheric pressure.The results of this investigation into the system's hansientdynamics are shown in Figs. 14 to 17. Figure 14 shows thebehavior of the mass flow rate in response to the two controlactions for two plenum pressures. The rise or decay time forthe mass flow is in the approximate range 0.002 to 0.007 secand does not change significantly for the two plenum pressures.Thib suggests that in ihese c ases thc charaiteristic iimc phe-nomena are p rincipally related to thc time it takes for a pressuredisturhance.to the length of the tuhe, since this isabout 0.003 sec. This explanation is reinforced by examiningFigs. 15 and 16 which are plots of the pressure distribution atvarious elapsed times after the sudden valve opening. Figure 15shows the propagation of the incident pressure wave createdby the valve opening and Fig. 16 show s the propagation of thewave after its reflection from the end of the tuhe. The soniclag is apparently controlling the transient response to the stepinput. For a system where the capacitance lag is much greaterthan the sonic lag, the capacitance lag would tend to controlthe rise or decay. Figure 17 shows the propagation of theincident pressure wave created by sudden valve closing. Tosmooth these hansient results, second-order spatial numericaldamping has been added to the numerical algorithm' for com-puting the incremental or "delta" solution for a given timelevel from the solution for the preceding one. This was doneby computing the increment from the weighted average of thesecond-order (in time) Beam and Warming algorithm and thefirst-order (in time) algorithm of Lax5. The weighting factorsapplied to the Beam and Warming m ethod and the Lax methodwere 0.98 and 0 .02, respectively.


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0.016 SUDDEN VALVE OPENINGA" = 0.92 in 2ppO = 13.75 blIn20.014

SUDDEN VALVE CLOSINGA" = 0.001 in.?

SUDDEN VALVE OPENINGA" = 0.92 in2pL = 8.84 lblln2 I

IDDEN VALVE CLOSING\ A" = 0.001 i n 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9TIME 1x10 2aaci

Theoretical transient mass flow rate

ConclusionsThe results presented indicate that quasi-one-dimensional un-steady flow theory can he applied to predict, with reasonableaccuracy, the pneumodynamic response of an idealized cir-culation control rotor model to cyclic control valve inputs.The results also show that, for a given set of system param-eters, the phase lag in the response of the system to cycliccontrol input is a fixed time delay almost independent of fre-quency. Higher harmonic content of the mass flow rate outputexhibits some amplification due to nonlinear effects, but thehigher harmonic content of the pressure response does not.A resonance phenomenon is predicted by the numerical re-sults at approximately the frequency for quarter-wave acousticresonance in a pipe. The predicted resonant frequencies areslightly different than the experimentally observed resonantfrequencies. A possible explanation for this is the inability ofthe quasi-one-dimensional theory to predict wave reflectionfrom the expansion in the entrance section of the experimentalmodel.

SUDDEN VALVE OPEN

0 5 10 15 20 25 30 35SPANWISE LOCATION (in.)

Fig. 15 Theoretical incident transien t pressure wave

Num erical investigation of the response of the flow variablesto step inputs in the control valve area indicates that, for thecases examined, the response time is controlled by the char-acteristic time for wave propagation (sonic lag).

References'Watkins, C . B., Reader, K. R., and Duna, S. K., "Numerical andExperimental Simulation of Circulation Control Rotor Pneumodynamics,"

A IA A Paper No. 83-2551, AlAAlAHS Aircraft Design, Systems, andOperations Meeting, Fort Worth, Texas, Oct. 1983.'Beam, R. M . and Warming, R. E., "An Implicit Factored Schemeforthe Co mpressible Navier-StokesEquations," AIAA Journal, Val. 16,No. 4,pp. 393-402, May 1978.'Reader, K. R., "Evaluation of a Pneumatic Valving System for Ap-plication to a Circulation Cantm l Rotor," Naval Ship Research and De-velovment Center Rew rt 4070. M av 1973." ~ k a d e r , . R., "fhe ~ff ec ts'o f a m and Nozzle Configurationson thePerformance of a Circulation Contml Rotor Pneumatic Valving System,"David Taylor Naval Ship R esearch and Developmen t Center Repo n ASED-?91 Nnv 1977.. . ..'Lax, P. E., "Solutions of Nonlinear Hyperbolic Equations and TheirNumerical Computation," Commu,tications on Pure andAppliedMoth-emarics, Vol. 7, 1954, pp. 159-193.


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Table 1 Harmonic analysis of typical results

SUDDEN VALVE OPENING - .002125pp l = 13.75 1b/in,2 0----- 0.003000C--- 0.0037506wc 0.004375h ........ 0.0050000.006000

G- 28 I INVI

00P\

0 x 4

C Lm.

2". o n\

-

SPANWISE LOCATION (in.)Fig. 16 Theoreticnl reflected transient pressure wave

-^ ' ^ C0 - V I - -3 C - 0 0I - , = " ,- .+ - - -

c o m - VI cm o = N o oL" L" 0 0 0 0. . . . . .0 0 0 0 0 0

- - -- , - L " - =m m m m -N - I C I- - - - -

~ ) ~ . ~ ~ m c c m mu r m - m ~ - o o o~ ~ L " 0 0 0 0. . . .P x O O O O O O-- - L " -2 . : : : z'I - , - ,- - - - -

m = -5 m c -- m m - o -VI ;1 0 0 0 0. . . . . .0 0 0 0 0 0

- - -- - m m mY I L n m c cm m l - -- - - - -U ~ C u L " m - - N NP . E N N - m O 0~ m u = m o oo n - . . . . 0 9U 0 0 0 0 0 0 0 0

- -m - m- - 0 = - mN r O - mm l - l w- - - - -e D m W i l Y I Nm m m 0 0- w = o o o. 9 :0 0 0 0 0 0

-m - -w m - m. - r ( ~ ~ r n1 1 3 N I- - - - -u m C m U I =P V I V I O O O O- . .

Nr -r(-0\ W W~ m ~ oZ J U -w T WxL"- " ,- + x u= . o xw m z ~& , P IX O U "= w n 4 u. o - x

. N -z 0Z U WW Y O \ n~ r nC - 1 C - I W

.Aa Y C C 43 C x0 "0 0 4 . .

.oNz -CI ^ 0\ W WCz - 1 u -W T WTVICI" ,Y C X U= . o xW m = P, a -O X 0 ! = 0W n u uL o - E

Nz -2 - 0c-10, Wo \ w m PP X C J -4 Y* k * .A

P 40 r..

SUDDEN VALVE CLOSING t, (sec)p p k = 13.75 lbl in2 * - - - -.0000- .0005- .0015- .0020A........ 0.0040C.- . 0.0045- .0070z.5m"0.

*Pa

:?0,*flr0ZW3:::

"Y.A\0 -3 0-1 W. A n

.a - Ww " .A. m 4.eCI: u.0 "u.E I:w--Nz -- 0W= PU 0. o m W\ m .Ac .:0- 4I:C u0 5 10 15 20 25 30 35SPANWISE LOCATION (in.)

Fig. 17 Theoretical transient pressure wave

0 0 0 0 0 0 0 ~

0 - N . m , L "

S

Q;i2

1985_pneumodynamic characteristics of a circulation control rotor model (extra k on f vector)

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