Three dimensional droplet trajectory code for propellers of …icing.ae.illinois.edu/papers/98/AIAA-1998-197.pdf · 2003-06-23 · of motion utilizing the two-dimensional flow field

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Three Dimensional Droplet Trajectory Codefor Propellers of Arbitrary Geometry

K.A. Farag* and M. B. Bragg*University of Illinois at Urbana-Champaign

ABSTRACT

A three-dimensional droplet impingement codefor aircraft propellers has been developed. The codeallows the modeling of propellers of arbitrary geom-etry in three dimensions, but is limited to model-ing axi-symmetric spinner / nacelles combinations.The code was used to analyze several 3-D wind tun-nel propeller-nacelle models which were experimen-tally studied. The code was also used to validatea theoretical relationship between two-dimensionaland three-dimensional impingement efficiencies forrotating propellers. The effects of the spinner, en-gine nacelle geometry, contraction of the slipstream,and blade-to-blade interference on the impingementefficiency and limits of impingement were studied.Parametric studies of the effects of the advance ra-tio J and droplet size on the efficiencies and limitswere performed.

NOMENCLATUREAd Droplet's cross-sectional Area.B Buoyancy force vector.CD Droplet drag coefficient.Ci Local lift coefficient.d droplet diameter.D Drag force vector.Fr Froude number of the flow.

* PH.D. Candidate The Ohio State University, Colum-bus, Ohio, Visiting Scholar, University of Illinois at Urbana-Champaign, Member AIAA.

+ Professor, Department of Aeronautical and Astronauti-cal Engineering, University of Illinois at Urbana-Champaign,Associate Fellow AIAA.

Copyright ©1998 by K. A. Farag, and M. B. Bragg. Pub-lished by the American Institute of Aeronautics and Astro-nautics, Inc. with permission.

G Gravitational force vector.g Gravitational acceleration.J Advance ratio.K Droplet's inertia parameter.A'o Droplet's modified inertia parameter.L Characteristic length.Rs Slip velocity Reynolds NumberjRoo Freestream Reynolds Number.R Propeller radius.T Position vector.T Non-dimensional time.UQO Freestream velocity.U Flow velocity vector.V Droplet velocity vector.Vsiip Droplet slip velocity.£ Non-dimensional x coordinate.T] Non-dimensional y coordinate.C Non-dimensional z coordinate.u Rotation rate.Pd Droplet's mass density.p Freestream mass density.a Freestream to droplet mass density ratio.

INTRODUCTION

The increased demands for fuel efficiency has inten-sified the search for efficient propellers for commer-cial and general aviation aircraft. One aspect of thedesign of such propellers is the ability to predicttheir susceptibility to accrete ice when operating inicing conditions. Icing occurs on propeller bladeswhen supercooled water droplets suspended in theatmosphere impinge on the rotating blades. Themass flux of liquid water striking the blades as wellas the most aft locations of the impinging droplettrajectories, limits of impingement, are importantparameters used in the design of propeller ice pro-tection.


Far Field

CO

Near Field

Fig. 1 The two-stage flow-field solutions used in the BETAPPROP code.

Virtually all previous efforts to computationallypredict the impingement efficiency and limits of im-pingement of supercooled water droplets on pro-pellers and rotor blades are based on the classicalpropeller strip analysis such as the work done byKorkan et al.1.2>3. * Guffond et al.s.6 calculated the3-D catch efficiency on a non-rotating helicopter ro-tor.

In such an approach, the propeller blade is di-vided into a finite number of strips; each of which ismodeled as a two-dimensional airfoil. The flow fieldabout each blade strip is computed separately andan iteration is performed on the total integrated in-flow which is adjusted (by varying the induced angleof attack distribution) until agreement with the in-flow value obtained from the classical momentumtheory is achieved.

Once the distribution of the induced flow anglealong the span of the propeller blade is found, theeffective angle of attack of each blade section is de-termined. The flow field around each blade stripis then modeled as a two-dimensional airfoil oper-ating at the effective angle of attack obtained fromthe above iteration. Droplet trajectories are thencomputed by integration of the droplet equationsof motion utilizing the two-dimensional flow fieldfor the propeller strips. The impingement limitsand the impingement efficiency are calculated in theusual way. Two-dimensional strip analysis does ne-glect three-dimensional flow-field effects which maybe important in the study of the icing characteristicsof propeller blades.

In order to thoroughly understand the effectsand significance of any three-dimensional effectson the icing characteristics of propeller blades, athree-dimensional droplet trajectory and impinge-ment code was developed. The code has a versatile3-D surface geometry generator capable of modelingarbitrary 3-D multiple-blade propellers with spin-ner. The code, however, is limited to modeling onlyaxi-symmetric engine-nacelle combinations.

COMPUTATIONAL APPROACH

The code utilizes two separate three-dimensional po-tential flow solutions of the propeller-spinner andengine-nacelle assembly as shown in Fig. 1. Thefirst is an optional far-field solution using vortexlattice methods (VLM). The use of this simplifiedmodel was intended to reduce the CPU time inthe analysis of highly complex propeller geometries.Further, the VLM solution can be used to verify theconvergence of the PM solution.

In the vortex lattice solution, the propeller bladesare modeled as highly twisted thin surfaces lyingon the mean-chamber lines of the airfoil sectionsalong the span of the blade. A system of three-dimensional horseshoe vortices are used which arebound to the mean chamber surfaces with their trail-ing filaments convected downstream along a helix ofconstant pitch. The magnitude of the circulationassociated with every horseshoe is determined byimposing flow tangency boundary conditions at a fi-nite number of control points on the mean chambersurface.


The second flow-field method is based on a first-order panel method (PM). This solution can be usedin the near-field in combination with the VLM inthe far-field, or can be used alone for the entire flowfield. In the PM solution, the three-dimensional ge-ometry is modeled by a discrete array of 3-D rectan-gular panels tangent to the surface of the blade. Itis further assumed that on every panel there existsa point source and a doublet of constant strength.The strength of the sources and doublets is obtaineddirectly by requiring the resulting flow field to betangent to the discretized surface at the geomet-ric centeroids of the surface panels (Control Points).Figure 2 shows the geometry of the panels used inthe VLM and PM solutions.

Tnilmi Vatiees

Fig. 2 The approximating geometry of the near andfar-field surfaces

Panel methods fail to give accurate velocities atpoints within one-panel characteristic length mea-sured from the sides of the panels. For that reason,a three-dimensional right rectangular grid is con-structed off of the surface of the blade. The grid inFig. 3 is constructed to cover only the panel cor-ners that are later targeted and hit by calculateddroplet trajectories. Accurate velocity values areassigned to the outer and inner edges of the surfacegrid. Linear interpolation is used to find the velocityanywhere inside the grid.

The three dimensional geometry of one of the pro-pellers studied is shown in Fig. 4. The main bladewas approximated by 16 radial panel columns. Thepropeller airfoil sections were approximated by 30chordwise panels. Every radial panel column onwhich /? calculations are desired, is further subdi-vided into five subpanel columns for the construc-

tion of the surface interpolation grid.The equations of motion for a supercooled droplet

suspended in the atmosphere were developed in arotating reference frame to simplify the calculation.Buoyancy, gravitational and pressure gradient forceswere neglected. The equations of motion are inte-grated in time assuming no initial relative motionwith respect to the freestream. The resulting tra-jectories are examined for surface penetration. Aniteration is performed on the droplets initial loca-tion until a trajectory piercing the surface is found.Four separate trajectories striking the four cornersof the surface panels are found. The four impingingtrajectories are assumed to bound the stream-tubeof liquid striking the surface. The impingement effi-ciency is calculated as the ratio of mass striking thesurface non-dimensionalized by the freestream massflux at a given radial location.

Fig. 3 The surface interpolation grid.

Aerodynamic ModelThe BETAPROP code used in this analysis uti-

lizes two separate flow field models to calculate thepotential flow. A vortex lattice potential flow solverin the far-field region (one propeller diameter up-stream from the plane of rotation). Closer to theplane of rotation and near the surface of the blade,the code utilizes a first order panel method poten-tial flow solution. The panel solution is obtainedfrom a version of PMARC7 after undergoing sub-stantial alteration and adaptation to rotating ref-erence frames. PMARC is a first order panel codethat is capable of modeling external and internalincompressible potential flow.

The PM part of the code uses a constant sourceand doublet distributions on flat rectangular pan-


els that are used to approximate the propeller andnacelle surface in question. The PM module in thecode has an additional option where panels distantfrom the point where the local velocity is desired;in the far field, are approximated as point sourceand point doublets. The use of this option causessignificant computational time savings with reason-able accuracy.

Fig. 4 The three-dimensional model of the propellerand wake system modeled by BETAPROP.

Trajectory EquationsIcing occurs when supercooled water droplets sus-

pended in the atmosphere impinge and freezw on thesurface of the propeller blades and the spinner / na-celle assembly. The water droplet diameters of 10 to50 [im were considered here. These droplets expe-rience a relatively low Reynolds numbers (less than103). Therefore, it can be assumed that the dropletsremain spherical in shape. For the low cloud liquidwater content (LWC) assumed in the code, it canalso be assumed that the droplets do not effect theflow field. A propeller fixed non-inertial referenceframe will be used to eliminate the time dependencefrom the flow field solution. In the rotating referenceframe of the code, the propeller's plane of rotation isassumed to be in the x-z plane. As depicted in Fig.1, the propeller is assumed to be rotating about they axis with angular velocity vector u = [0,cj, 0]T. Ina non-inertial reference frame, Newton's second lawapplied to the droplet motion takes the form:

, = m (r 23 x x fj (1)

where the vector f= [x,y,z]T is the position vectorof the droplet with respect to the rotating frame at-tached to the propeller. The droplet mass m in Eq.(1) is multiplied by the absolute acceleration terms.Performing the time differentiation and the crossproducts, the absolute acceleration components inEq. (1) become

2u>z —y

z — — o>2z(2)

The forces acting on the droplet are the grav-ity force G, buoyancy force B and the drag forceD. The droplet's drag is in the direction of thedroplet's slip velocity Vsnp = [VSl, V3y, V3z]T whichis the vector difference between the droplet's veloc-ity U = [Ux,Uy, UZ}T and the fluid velocity V =[vx,vy,Vz]T at droplet position f. The aerodynamicdrag force on the droplet calculated in the rotatingreference frame is given by

= ^P\\VMp\\VMpAdCD (3)

where A& and CD are the droplet's cross-sectionalarea and drag coefficient, respectively. It is assumedthat the droplet is a sphere of mass density pd andof diameter d, with mass m = pd1^- and cross-sectional area Ad = ?r^-. With the magnitude of theslip velocity defined as Vs = \\\V,np\\\ = \\V - U\\\,the drag force can be expressed as

= K-Z-PCD

V V*s ' sxV V* sv syVV' s 'sz

(4)

Since the gravity and buoyancy forces are station-ary with respect to an inertia! reference frame, eachwill appear to be time dependent to an observer ro-tating with the coordinate system (Z,T/ ,Z) . Giventhe rotation is about the y axis at a constant an-gular velocity o>, both forces will have componentsin the x and z direction only. In vector componentform, the buoyancy and gravitational forces are

— cos(ort)0

s n

= 9P-cos(art)

0— sin(urt)

(5)

(6)


Summing all the forces and substituting in Eq.(1) we get the following set of differential equationswhich describe the motion of the droplet as seenfrom the rotating reference frame:

3CDaVsVsx

4d g(a — l)cos(a>i) =

3CDaVsVs

(7)sy

3CDaVsVsl

4d

4

— cr)sin(u;/)

z — 2u>x — u

Introducing the characteristic length L of the pro-peller and the characteristic velocity U^, we definethe non-dimensional coordinates (£,T/,£), the non-dimensional angular velocity u = ^-, dimension-less time T = t^- and dimensionless slip velocityVs = (y"S jr115 £/"•)• We define, respectively, theinertia parameter, the freestream Reynolds numberbased on the droplet diameter, and the Froude num-ber of the flow

r, _ Pdd2^18/ii

Uo

-Roo =

Fr =

Substitution yields the non-dimensional form ofthe droplet equation of motion in a rotating refer-ence frame:

(8)

24K sy_- *

1 g

The spherical water droplets experience Reynoldsnumbers in the range of

0 > Rs < 1000 (9)

For Reynolds number within this range, Langmuirand Blodgett,8 determined the following expressionfor the fraction D^°°, which is valid for low-speedatmospheric droplets where compressibility effectsare negligible

where the slip velocity Reynolds number is definedby

R.= EM = RooVs (11)

Equations (8) contain two similarity parametersK and R^ that characterize the particle's motion inthe flow field around a given body of characteristicdimension L. It is possible to combine the inertiaparameter K and the freestream Reynolds number.Roo into a single similarity parameter K0, called themodified inertia parameter by defining K0 as

The integral in the equation above was first evalu-ated by Bragg9 and can be expressed in the followingrelation

= 18K Roo3 arctan76

(13)

In aircraft icing research, forces experienced bythe droplets due to gravity and buoyancy are or-ders of magnitude smaller than the drag force onthe droplets. It is therefore customary to neglectthe effects of gravity and buoyancy in the equationsof motion of the water droplets. This is a practicalassumption for the size and mass of droplets oftenencountered in atmospheric icing clouds. Defining(p(Rs) = C^fs i and introducing the state variables(Xi, X2,..., X6) = (£, £, 77,77, C, C) Tne equations ofmotion of the droplet can be written as the six first-order coupled ordinary differential equations:

•"•Y \- A2 I -

2

KU (14)

Xs =

K


Ao

Fig. 5 Local impingement efficiency on a 3D surface.

The state space equations above are numericallyintegrated with respect to time. Only initial condi-tions are required to start the numerical integration.It is assumed that the droplet starts at rest with re-spect to the freestream velocity vector as seen fromthe rotating reference frame. Therefore, a dropletfar upstream will appear to an observer on the ro-tating propeller to have velocity components thatdepend on the z and a; coordinates of the dropletinitial position in space. The axial velocity compo-nent, however, will be the freestream velocity in thenegative y direction.

A point Q far upstream from the propeller withthe non-dimensional coordinates coordinates RQ =(£> T?> C) at time T = 0 will have the initial veloc-ity VQ = —UQ X RQ. Performing the vector crossproduct and combining the result with the initialaxial velocity, the initial velocity at point Q can beexpressed as

(-Of(15)

Local Emingement Efficiency /3The local collection efficiency of the surface is de-

fined in three dimensions as the ratio of the cross-sectional area of the mass tube far upstream of thebody to the surface area of the body bounded bythe impinging stream tube, i.e., Fig. 5.

Assuming that the droplets are at rest with re-spect to freestream, the four trajectories imping-ing on the corner points of a surface panel coveringthe area where the /3 value is desired are found by

iteration on the initial conditions. The upstreamcross-sectional area of the bounding stream tube A0and the panel surface area AS are respectively cal-culated. Note that AQ is normal to the freestream.The average local collection efficiency is then com-puted by:

(16)

Total Collection Efficiency EThe total collection efficiency on a given radial

segment of the propeller that has been approxi-mated by a set of N flat surface panels is dennedby

(17)= /3d(A)

where Ap is the projected area of the segment ofthe blade under consideration. In this 3D analysis,Ap will be taken as the projected frontal area of thepropeller segment as seen from the axial direction.The total collection efficiency will then be noted asE2D- In two-dimensional icing analysis, the expres-sion for E takes a similar form. The surface integralin Eq. (17) is replaced by a line integral evaluatedover the surface of the airfoil and Ap is replaced bythe projected height of the airfoil section taken per-pendicular to the local relative velocity vector. Theresulting two-dimensional total collection shall bereferred to as

CODE VALIDATION

The code was used to model a 36 inch diam-eter propeller blade rotating at 500 RPM. Thefreestream velocity was 40.0^. The correspondingadvance ration was J = 1.6. The 4.5-inch chord pro-peller blade was assumed to have a constant Clark-Yairfoil section. The blade cross-section is tapered into a 1.25 inch diameter cylinder at the root. Thetapering starts from 18.05% station and continuesto the base of the blade. The blade was linearlytwisted with /375% = 39.35°.

To validate the code, the propeller blade abovewas modeled as a single blade without a spinneror nacelle. This was done in order to compare theperformance results of each 3D solution to the two-dimensional results uncorrected for the effects ofmultiple blades and center body / nacelle blockage.


0.

0.

^0.

lQ.

2D Strip TheoryVortex Lattice Method3D Panel Method

0.1 0. 3 0. 5 0. 7 0. 9r/R

Fig. 6 Radial thrust distribution.

The flow fields were determined from the VLMand PM modules of the code. In both cases, theresulting lift force was resolved in the thrust direc-tion and the radial distribution of thrust was de-termined. In Fig. 6 the radial thrust distributionas calculated by the VLM , PM and the 2D striptheory results calculated by the Lan and Roskam10

method or that outlined by McCormick11. With theexception of the tip region, both solutions predictsimilar thrust characteristics. The 3D panel solu-tion predict a 10% lower thrust than the 2D striptheory and the VLM. This is attributed to the abil-ity of the PM to model the tip losses more accuratelythan the other two methods.

Comparison to 2-D DataBy writing a conservation of mass expression for

the liquid water impinging on the surface of theblade to that crossing a stream tube perpendicu-lar to the effective velocity vector, and further non-dimensionalizing by the freestream mass flux, it canreadily be shown that fl^D and ?iDCOTrfl,tfd. are re-lated by the expression:

(18)

where f is a function defined by:

1 + '*(*)'I 1/2

J(19)

corrcctcd >Equation (18) implies that PiDcThe factor /"(J, £) defined by Eq. (19) accountsfor the increase in liquid mass impingement due tothe rotation of the propeller.

0.5 -

-0.2

Fig. 7 BETAPROP compared to the /32£>correct(:d fora two-bladed propeller with J=2.0, d = 20/im,^ =0.50

In Fig.7, the BETAPROP code results for £ =50% radial location on a two-blade propeller arecompared to the /?2£> and ^2£)corre<:ted- It is im-portant to note that values for (S^o appearing inthe right hand side of Eq. (18) are calculated byBragg's AIRDROP12 code, fan was computed forthe Clark-Y airfoil operating at the same lift coeffi-cient. The droplet inertia parameter and Reynoldsnumber used to calculate the fop curve were foundusing the airfoil chord and effective velocity as thecharacteristic length and velocity, respectively.

Consistently, the BETAPROP curves closely re-semble the 02Dcorrectcd curves which to a certain de-gree validates the relationship in Eq. (18) and theBETAPROP code. Although both methods predictthe same upper limits of impingement, the three-dimensional approach predicts the lower limits ofimpingement closer to the trailing edge. /?3Dmoi con-sistently was predicted on the upper surface corre-sponding to ^ = +.015.

Comparison to 3-D Experimental DataIn order to experimentally validate the BE-

TAPROP code, the droplet impingement on a four-bladed wind tunnel propeller model was experimen-tally measured by Riechhold.13 The propeller modeland the tunnel settings were selected to be repre-sentative of a commuter turbo-prop at cruise con-ditions. The 36-inch diameter four-bladed propeller


was tested in the UIUC wind tunnel which was fittedwith five NASA standard spray nozzles. Thin stripsof chromatography paper were placed on the pro-peller surface at the 25%, 35%, 50%, 70% and 90%radial locations. Dye laden water was injected intothe tunnel via the spray system for a given spraytime. The new dye-tracer technique developed byBragg et al.14 was then used to determine the col-lection efficiency on the propeller blade.

-0.2 -0.1

Fig. 8 Impingement efficiency for the baseline con-figuration, ji = 0.50, d = 15/xm, KQ = .006, J = 1.6

The baseline propeller test configuration was afour-bladed propeller with (375% = 40° together withan axi-symmetric engine nacelle 10 inches in diam-eter. The propeller impingement experiments wereperformed in 15/im, 20/zm, and 30/nn MVD sprayclouds. Additional experiments were conducted us-ing a 16-inch axi-symmetric nacelle in two-and four-blade configurations.

The experimental and computational ft^D resultsat £ = 50% for the MVD values of 15, 20 and 30/im( the baseline configuration) are shown in Fig. 8,9, and 10, respectively. Figure 11 displays resultsat the same radial location corresponding to thelarge nacelle case. The experimental data gener-ally were in agreement with the trends predictedby the BETAPROP code. The 3D code predictedthe experimental upper limits of impingement well.Consistently however, the experimental /3max valueswere larger than the predicted 3D values. The dif-ference in fimax was as high as 35% as in Fig. 9. Forthe large nacelle cases however, the differences in

the /3max values did not exceed 15% at every radialstation. The impingement efficiencies calculated byBETAPROP for the 16-inch nacelle case were in bet-ter agreement with the experimental data than forthe calculations for the 10-inch nacelle case.

1.6

1.2

0.8

0.4

I I I I I I I I I | I I I I | I M I | I I I I | I I I I | I I I I | I I I I

'. -•—— BETAPROP H ———*——— EXP.

-0.2 -0.1

Fig. 9 Impingement efficiency for the baseline con-figuration, £ = 0.50, d = 20/zm, K0 = .01, J = 1.6

BETAPROP j | ———*——— EXP.

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Fig. 10 Impingement efficiency for the baseline con-figuration, £ = 0.50, d - 30/zm, KQ = .02, J = 1.6

The code generally predicted the lower impinge-ment limits to be closer to the trailing edge. For thelarge nacelle configuration, the computational forjcurves in Figs. 11, 12 and 13 indicated that moremass was collected on the lower blade surface. The


experimental results do not show such trends.It is important to note that the calculated un-

certainties for this experimental technique of Bragget al.14 is 15.44%. The uncertainties are primarilydue to the uncertainty in the spray cloud MVD andvariations in the ambient conditions of the differ-ent clouds tested in the tunnel. The uncertaintyin the experimental impingement efficiency valueswere calculated by Reichhold13 using in part themeasured standard deviation in the mass of the dyecollected on the freestream collector, the standarddeviation in the propeller data and that of the cor-responding collector at each radial location. Theuncertainty in the experimental impingement effi-ciencies were reported to be less than or equal 16%at /3max. The uncertainty increase as the impinge-ment limits were approached to 30% - 50% of thelocal 0.

1.5

0.5

BETAPROP EXP. -

0 |i u-i-r+Ti . i lj[ ^ 'if.

-0.4 -0.2 0.2S/C

-0.2

Fig. 11 Impingement efficiency for the large nacelleconfiguration, ^ = 0.50, d = 20/um, KQ = .01, J =1.6

In an attempt to remove some of the uncertaintyin the experimental data for the purpose of compari-son, the experimental impingement efficiency curveswere normalized by /3moz and compared to theircomputational counterparts in Figs. 14-17. Com-parison of the normalized beta curves exhibit dif-ferences between the experimental data and thosefrom the BETAPROP code. The difference couldbe due in part to the uncertainty in the experimen-tal data. The tendency for codes to over predict theexperimental ft value near the limits of impingement

Fig. 12 Impingement efficiency for the large nacelleconfiguration, ^ = 0.50, d = 30/^m, K0 = .02, J =1.6

-0.4 s/c 0.2

Fig. 13 Impingement efficiency for the large nacelleconfiguration, ^ = 0.70, d = 30/zm, KQ = .02, J —1.6


1.2

0.8

Eea.

0.4

i i 1 1 1 1 1 i 1 1 1 1 i 1 1 1 i i 1 1 i 1 1 1 1 1 1 1 i | 1 1 1 1 1 1 1 1 1......... EXP. ———A——— BETAPROP .

0 I •' i*1 '* i v 1*1-0.2 -0.1

Fig. 14 /3/pmax for the baseline configuration,0.50, MVI> = 15/im, #0 = -006, J = 1.6

-0.2

is very similar to that seen in the two-dimensionalcomparisons to the experimental work done by Pa-padakis et al.15

Based on the agreement of the BETAPROP coderesults 02DcorrectCd and the general agreement withthe trends of the experimental data available atthis time, the BETAPROP code can be used withconfidence to analyze other three-dimensional pro-peller/nacelle combinations.

RESULTS AND DISCUSSION

The code has been used in a study to determinethe significance of three-dimensional effects on thedroplet trajectories in the vicinity of 3D propellers.The radial variation of the impingement efficiencyfor the baseline configuration with MVD of 15/tm,20/im and 30/zm are shown in Figs. 18, 19 and 20respectively.

At very radial location, increasing the droplet sizeincreases /3max and causes the lower and upper limitsof impingement to move closer to the trailing edge.In Fig. 18, the 70% station of the baseline case wasexamined. Changing the droplet size from 15/z to30/i caused a 13% increase in J3max. The lower andupper impingement limits also increased by 80% and55%, respectively.

With the propeller rotation rate held constant at500 RPM, the number of blades was reduced fromthe baseline four-blade configuration to two blades.

Fig. 15 /3/(3max for the baseline configuration,0.50, MVD = 20/xm, KQ = .01, J = 1.6

0.2 -0.1 0 0.1

Fig. 16 P/Pmax f°r the baseline configuration,0.50, MVD = 30/rni, KQ = .02, J = 1.6

10


1.2

0.8XC3E

CO.

ea.0.4

i i i i I i i i i I i i i i I i i i i | i i i i I i i i i | i i i i | i i i i------- EXP. ———*——— BETAPROP

-0.2 -0.1

Fig. 17 /3/(3max for the large nacelle configuration,ft = 0.50, MVD = 20/zm, K0 = .01, J = 1.6

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Fig. 19 Impingement efficiency for the baseline con-figuration d — 2Qfj,m, KQ — .01, J = 1.6

1.5

0.5

1 i ' ' ' ' i ' ' ' ' i ' r T^fT ' r '--• - - R=.25

• - R=.35I -•-•— R=.50.

jn\ ....A.... R= 90

•'/ i;

l •//4 \V/'\* '/)// i» ;^/' ^ •ai, v \f V./. v_\

» ' 1 ' V,

-0.1 0S/C

0.1

Fig. 18 Impingement efficiency for the baseline con- Fig. 20 Impingement efficiency for the baseline con-figuration d = 15/im, A'0 = .006, J = 1.6 figuration d = 30/^m, A'0 = .02, J = 1.6

11


4 Blades--------- 2 Blades

-0.3 -0.2 0.2 0.3

Fig. 21 Impingement efficiency of two propellers,r/R = 0.35, d = 30//m, K0 = .0210, J = 1.6, 10"diameter nacelle.

i i i i I i i i i I i i i i I i i i i I i i i i

4 Blades ——•—— 2 Blades

-0.3

Fig. 22 Impingement efficiency of two propellers,r/R = 0.50, d = 30/im, K0 = .0210, J = 1.6, 10"diameter nacelle.

It was observed that such reduction of the numberof blades had essentially no effect on the icing char-acteristics near the root and tip of the blade. Atthe 35% radial location, Fig. 21, the lower limit ofimpingement was slightly displaced further back to-wards the trailing edge. The rest of the /3 curve andthe upper limit of impingement remained essentiallyunchanged.

Only at the mid-span of the blade, Fig. 22, didthe lower and upper limits of impingement move to-wards the stagnation point. The maximum effectsof the reduction of the number of blades occurredat the 50% radial location. The movement of theimpingement limits decreased in magnitude at the70% radial location to completely vanish at the 90%radial location. For the same power settings, vary-ing the number of blades had no influence on /3max

nor (S/C)0mat.To achieve optimum propulsive efficiency, typical

turboprop must operate over a wide range of ad-vance ratio J. In take off and landing configura-tions, characterized by low flight velocities, constantspeed propellers are set at high rotational rates;2200 - 2500 RPM; and low pitch, hence low J set-tings. When higher flight velocities are reached atcruise configuration, propeller RPM is usually setin the range of 1800 - 2200 RPM and the propellerpitch is varied ( Via the governer ) to maintain con-stant rotational speed.

The icing characteristics of propellers greatly de-pend on the propeller power settings. In Fig. 23, thelocal impingement efficiency at the 50% radial loca-tion of the propeller is plotted for different advanceratio J. In each case, the blade pitch was variedto match the same local lift coefficient C\ = 0.717,and hence a fixed local effective angle of attack,aejf — 2.34°. For the graphs in Fig. 23, the valuesof the 2D and 3D modified inertia parameters andthe rotation correction factor, Eq. (19), are listedin table 1.

It is evident that at low J values resulting fromlow Uoo, the propeller experiences substantial in-

J

0.81.21.62.02.4

tfco

2030405075

urel

44.0649.4156.0563.5789.63

A'o

.0059

.0082

.0103

.0122

.0165

Ko2D

.0889

.0970

.1066

.1170

.1496

T

2.2301.6471.4011.2721.195

Table 1 The variation of icing parameters with ad-vance ratio J, jj = 0.5, d = 20/^m.

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I ' ' ' ' I ' ' ' ' I

J = 0.8 U=20 ft/s

— J = 1.2 U=30 ft/s

= 1.6 U=40ft/s

--*- J = 2.0 U=50ft/S

—-*.„ j = 2.4 U=75ft/s

-0.4 -0.3 -0.2

Fig. 23 Icing characteristics of two bladed propellerswith a variable J, r/R = 0.50, d = 20/^m, C\ = .717

crease in j3max along the leading edge of the blade.This occurred despite the fact that KQ and Ko2D in-creased with increasing J. This can be explained bynoting that with the rotation rate held constant, theblade with a low J setting have experienced more ro-tations within a given translation distance forward,than a blade at a higher J setting. Therefore, thepropeller blade with the low J setting would sweepa larger volume.

As the axial flow velocity increases, the propellercompletes fewer rotations as it advances forward anequal distance, hence a stream tube of less volumewill be swept by the blade segment. Although theupper limit of impingement remains relatively unaf-fected by variations in J, the lower limit of impinge-ment moves farther from the stagnation point as Jincreases. The movement of the lower limit towardsthe trailing is the result of the slight increase in themagnitude of the effective velocity vector. Dropletsapproaching the blade with the larger effective ve-locity have higher KQ values. Given that C\ is con-stant, the lower limits of impingement move furtherback on the lower surface.

The graphs in Fig. 23 show the total collection tobe dependent on the advance ratio. Integrating allthe distributions according to Eq. (17), the result-ing E3D values are plotted in Fig. 24 as a functionof J. It is noted that such dependence is nonlinear.

0.16

0.1430

0.12

0.10.5 1.5

J

2.5

Fig. 24 Total collection for a two bladed propellerconfiguration at r/R = 0.50, d = 20^m, C\ - .717

Figure 24 shows the collection reaches a minimum atJ = 1.75. With the RPM held constant, operatingthe propeller at J < 1.75 increases the total collec-tion due to the rotational effects described earlier.If J > 1.75, the effects of high KQ values dominateand cause £3^ to increase.

The blockage effects of the nacelle on the impinge-ment characteristics of the propeller blade have beenconsidered. Figures 25 and 26 show the /3 distribu-tion at the 50% and 70% radial locations of the base-line configuration, respectively. For comparison, theimpingement efficiencies for the 16-inch and 22-inchnacelle cases are also depicted on the same graph.In addition, the (3 distribution for the No Nacelle( spinner only) geometry as well as the ft^D distri-bution corrected for rotational effects via Eq. (18)have been plotted. It is noted that the 10" diam-eter nacelle corresponds to 7.71% area blockage ofthe flow through the 36" diameter propeller disc.The 16-and 22-inch diameter nacelles correspond to19.75% and 37.35% area blockage, respectively.

Increasing the nacelle diameter causes substan-tial movement of the lower limit of impingement to-wards the trailing edge. A trend of lesser magnitudewas also present on the upper limit of impingement.The upper impingement limit shifted towards thestagnation point with increasing nacelle diameter.Pmax decreased by less than 10% as the nacelle di-ameter was varied over the range of 10 - 22 inches.The movement of the lower limit of impingement

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10" Nacelle

• 16" Nacelle

• 22" Nacelle

• No Nacelle

20 Effective

-0.8

Fig. 25 Propeller with varying nacelle configura-tions. r/R = 0.50, d = SQfim, J = 1.6

1.5

P 1

0.5

-•—— 16" Nacelle .

- - - - 22" Nacelle

-» - No Nacelle -

2D Effective

-0.4 -0.2 0S/C

0.2 0.4

Fig. 26 Propeller with varying nacelle configura-tions. r/R = 0.70, d = 30/xm, J = 1.6

towards the trailing edge correspond to an increasein the angle of attack. This is caused by the re-duction in axial velocity caused by the nacelle. Itis important to realize that significant changes inthe impingement characteristics of the blade haveemerged due to the three-dimensional aerodynamicblockage caused by the nacelle. Note that the 2Dimpingement data corrected for rotational effects,shown in Figs. 25 and 26, underestimate the limitsof impingement and the total collection.

For the 20/zm and 30/wn droplet sizes, the effectsof the nacelle on the impingement efficiency pro-files were insignificant near the root of the bladeat the 25% radial location. The effects of the na-celle blockage for these two droplet sizes begin toappear starting at the 35% radial location. The na-celle effects on the 20 and 30 pm droplet sizes reachmaximum at the 50% radial location. The nacelleblockage effects on the 20//m and 15/xm particlesstart to decrease from the 50% radial location un-til they virtually vanish at the 90% radial location.Droplets with moderate inertia experience signifi-cant radial acceleration due to the axi-symmetricflow about the spinner and nacelle. The outward ra-dial acceleration of the large droplets reduces theirimpingement at the root of the blade and enhancesit at the midspan of the blade. The axi-symmetricspinner and nacelle cause a radial droplet sorting

effect along the blade.The small 15/zm droplets, however, were affected

by the nacelle blockage at every radial location onthe blade. This greater sensitivity to flow distur-bance, caused by the nacelle blockage, is due to thedroplets low inertia. The aerodynamic drag of lowmass droplets, hence low inertia, tend to make thedroplets follow the stream lines of the flow field.

CONCLUSIONS

A droplet trajectory code for propellers, BE-TAPROP, was described. A three-dimensional vor-tex lattice method and first-order panel methodwere used to calculate the, invicid incompressibleflow. This method allowed the modeling of com-plex 3D propeller geometries with axi-symmetric na-celles. The droplet equations of motion in a non-inertial reference frame were integrated in the 3Dflow. The impinging trajectories were used to cal-culate the local impingement efficiency, limits ofimpingement and total collection. The 3D resultsobtained from the BETAPROP code were com-pared to 3D experimental data and 2D computa-tional data corrected for rotational effects. Compar-ison of the 3D code results with the 3D experimen-tal data shows reasonable agreement. When singleblades without a center body or nacelle are mod-eled, the 3D results exhibit very good agreement

14


with the two-dimensional AIRDROP code resultscorrected for rotational effects. For some moder-ately loaded propellers with slender spinner/nacellegeometry, the two-dimensional droplet impingementcodes, corrected for rotational effects, reasonablypredicted the impingement efficiencies and the lim-its of impingement. The BETAPROP code resultsindicate that the nacelle/spinner geometry is themost important three-dimensional flow-field factoraffecting the impingement efficiency and the limitsof impingement on propeller blades.

ACKNOWLEDGMENT

This research was supported in part by BF-Goodrich De-Icing systems. The BETAPROP codedescribed in this paper is the property of BF-Goodrich. The authors would like to thank Mr.Dave Sweet for his significant contributions to thiswork.

REFERENCES1 Korkan, K. D. , Dadone, L. and Shaw, R. J. Per-formance Degradation of Propeller/Rotor Systemddue to Rime Ice Accretion. AIAA Paper Num-ber 82-282, The 20th Aerospace Sciences Meeting,1982.

2Korkan, K. D., Dadone, L. , and Shaw, R. J. Per-formance Degradation of Propeller Systems due toIce Accretion. Jornal of Aircraft., Vol. 21 No. 1January, 1984.

"Millere, T. L. , Korkan, K. D. , Shaw, R. J. An-alytical Determination of Propeller PerformanceDegradation due to Ice Accretion. AIAA paperNumber 85-0339, The 23rd AIAA Aerospace Sci-ences Meeting, 1985.

4 Korkan, K. D., Dadone, and Shaw, R. J. Per-formance Degradation of Helicopters due to Icing- A review. Paper Presented at the 41st AnnualAmerican Helicopter Society Forum and Technol-ogy Disply, Fort Worth, Texas, May 15-17, 1985.

sHedde, T. and Guffond, D. Development of Three-Dimensional Icing Code, Comparison with Experi-mental Shapes. AIAA paper Number 92-0041, The30th Aerospace Sciences Meeting, Reno NV, Jan-uary, 1992.

6 Guffond, D. Prediction of Ice Accretion: Compar-ison Between 2D and 3D Codes. Presentation toSAE Subcommittee AC-9C. Meeting No. 21, Air-craft Icing Technology. Toulouse France, Septem-ber 27-30, 1994.

?Ashby, D. L., Dudley, M. R., and Iguchi S. K.Development and Validation of an Advanced Low-Order Panel Method. NASA TM 101024, October,1988.

8Langmuir, I. and Blodgett, K. A MathematicalInvestigation of Droplet Trajectories. In collectedworks of Irving Langmuir. New York: PergamonPress, pp 348-393, 1946.

9Bragg, M.B. A similarity Analysis of the DropletTrajectory Equation. The AIAA Journal, Volume20, Number 12, Page 1681, December, 1982.

10Lan, C. E. and Roskam, Jan. Airplane Aerody-namics and Performance. Roskam Aviation andEngineering Corporation, Ottawa, Kansas, 2ndedition, chapter 7, 1988.

11 McCormick, B. W. Aerodynamics, aeronauticsand flight Mechanics. John Wiely & Sons, Inc.,2nd edition, Chapter 6, 1995.

12Bragg, M. B. Rime Ice Accretion and Its Effectson Airfoil performance. Ph.D. Dissertation, TheOhio State University, Columbus, Ohio, 1981.

13Reichhold, J. D. Experimental Determination ofthe Droplet Impingement on a Propeller Using aModified Dye-Tracer Technique. Master Thesis,Aeronautical And Astronautical Engineering, TheUniversity of Illinois at Urbana-Champaign, Jan-uary, 1997.

14Bragg, M. B., Cebrzynski, M. Reichhold, J.D., Sweet, D., Waples, T., and Shick, R.An Experimental Method for Water Droplet Im-pingement Measurement. American HelicopterSociety/Society of Automotive Engineers Inter-national Icing Symposium, Montreal, Canada,September 18-21, 1995.

"Papadakis. M., Elangovan, R., Freund Jr., G.A., Breer, M., Zumwalt, G. W. and Whitmer,L. An Experimental Method for Measuring Wa-ter Droplet Impingement Efficiency on Two- andThree-Dimensional Bodies. NASA Contractor Re-port 4257, November, 1989.

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Three dimensional droplet trajectory code for propellers of …icing.ae.illinois.edu/papers/98/AIAA-1998-197.pdf · 2003-06-23 · of motion utilizing the two-dimensional flow field