Research Article External Aerodynamics Simulations in a

Research ArticleExternal Aerodynamics Simulations ina Rotating Frame of Reference

Filomena Cariglino1 Nicola Ceresola2 and Renzo Arina1

1 Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy2 Alenia Aermacchi SpA Corso Francia 426 10100 Torino Italy

Correspondence should be addressed to Filomena Cariglino filomenacariglinopolitoit

Received 13 November 2013 Accepted 19 March 2014 Published 14 July 2014

Academic Editor James J McGuirk

Copyright copy 2014 Filomena Cariglino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents the development of a tool integrated in the UNS3D code proprietary of Alenia Aermacchi for the simulationof external aerodynamic flow in a rotating reference frame with the main objective of predicting propeller-aircraft integrationeffects The equations in a rotating frame of reference have been formulated in terms of the absolute velocity components in thisway the artificial dissipation needed for convergence is lessened as the Coriolis source term is only introduced in the momentumequation An Explicit Algebraic Reynolds Stress turbulence model is used The first assessment of effectiveness of this method ismade computing stability derivatives of a NACA 0012 airfoil Finally steady Navier-Stokes and Euler simulations of a four-bladesingle-rotating propeller are presented demonstrating the efficiency of the chosen approach in terms of computational cost

1 Introduction

Themain objective of the present work is to calculate the flowfield around a propeller Unsteady Reynolds averagedNavier-Stokes equations (RANS) represent the state of the art for thenumerical prediction of the viscous flow around propellers[1 2] However unsteady Navier-Stokes simulations for suchcomplex flows are computationally expensive Thereforethese simulations are generally carried out only using reducedordermethodologies [3] In our case to simulate the unsteadyviscous flow around a propeller we use the Navier-Stokes orEuler equations in a noninertial (rotating) reference framein which the propeller is at rest In this way the simulationscan be performed in steady-state mode The advantageof this method is the relatively lower computational costneeded to obtain a CFD solution in comparison to a fullyunsteady simulation Moreover the present method permitsthe insertion of the propeller as a building block into the fullcomputational model of an aircraft to investigate propeller-aircraft integration issues from an aerodynamics point ofview

When the Navier-Stokes equations for a steady viscouscompressible flow are written in a rotating frame of referencethere are two choices regarding the velocity vector compo-nents Either they can be the components with respect tothe absolute (inertial) frame hereafter called the absolutevelocity components [4] or they can be the components withrespect to the rotating (noninertial) frame hereafter calledthe relative velocity components [5] In this paper the firstformulation has been chosen because it has the advantage ofenabling a steady-state formulation as the flow field can beviewed as a steady state in the rotating frame In order to fullyuse the UNS3D code which is written in the absolute velocitycomponents source terms are added to take the coordinate-system rotation into account in the governing equations

To assess this method two applications are presentedthe first application is the determination of the stabilityderivatives for a NACA 0012 airfoil and the second is thesimulation of the flow field around a rotating propeller Thefirst application allows us to assess the method by comparingthe results to a reference test case for which several authors

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2014 Article ID 654037 14 pageshttpdxdoiorg1011552014654037

2 International Journal of Aerospace Engineering

have obtained a solution [6ndash9] In this case the flow field iscomputed using Euler equations The second application is amore complex test case which has the purpose to evaluate theaccuracy efficiency and robustness of the current method topredict the complex flow field of a rotating propeller For thistest case a comparison with experimental results for cruiseconditions in terms of thrust coefficient is also made

2 Governing Equations

The governing equations which are numerically resolvedusing the UNS3D code are summarized below The Navier-Stokes equations in a noninertial frame of reference areexpressed as

120597997888rarr119883

120597119905+ nabla

997888rarr119865119868+ nabla

997888rarr119865119881=

997888rarr119876119883 (1)

where 997888rarr119883 is defined as

997888rarr119883 = [

[

120588

120588997888rarrV

120588119864

]

]

(2)

997888rarr119865119868and 997888rarr

119865119881are the respective flux vectors

119865119868119894=

[[[[[

[

120588V119894

120588V1V119894+ 1199011205751119894

120588V2V119894+ 1199011205752119894

120588V3V119894+ 1199011205753119894

(120588119864 + 119901) V119894

]]]]]

]

(3)

minus119865119881119894

=

[[[[[

[

0

1205911198941

1205911198942

1205911198943

119902119894+ V119895120591119894119895

]]]]]

]

(4)

and the source terms are contained within 997888rarr119876119883

997888rarr119876119883

=[[

[

0

120588997888rarr119891119890

119882119891

]]

]

(5)

119882119891is equal to the following relation

119882119891= 120588

997888rarr119891119890sdot997888rarrV (6)

21 Formulation in Rotating Frame for the Absolute VelocityNormally the governing equations for rotating systems aresolved for the relative velocity components in the relativeframe of reference However in an external aerodynamiccontext such as the propeller-wing configuration two mainadvantages can be derived through taking instead the abso-lute velocity components in the relative reference frameThe first and most important one is that the grid blockcontaining the propeller can be interfaced more easily withinthe full aircraftmodelThe second is that a smaller numerical

effect of source terms is present as will be shown below Toexpress (1) in terms of a relative reference frame the followingrelations for substantial and local derivatives are used

119863119887

119863119905=

1198631015840119887

1198631199051015840

119863119861

119863119905=

1198631015840119861

1198631199051015840+997888rarr120596 times

997888rarr119861

120597119887

120597119905=

1205971015840119887

1205971199051015840minus (

997888rarr120596 times

997888rarr119903 ) sdot nabla119887

120597119861

120597119905=

1205971015840119861

1205971199051015840minus (


997888rarr119903 ) sdot nabla

997888rarr119861 +


997888rarr119861

(7)

where the prime 1015840 denotes the operation with respect to therelative reference frame By using relations (7) the right-handside of (1) becomes

997888rarr119876119883

= [

[

0

minus120588 (997888rarr120596 times

997888rarrV )

0

]

]

(8)


997888rarr120596 = (

120596119909

120596119910

120596119911

) (9)

With this formulation the source term vector (see (8))contains only the contribution of the Coriolis force and thecontribution of the centrifugal force is omitted In this waythe magnitude of the source term is greatly reduced and asmaller amount of artificial dissipation is required to ensureconvergence

To take into account the rotation of the coordinate system(3) is modified in the following way

119865119868119894=

[[[[[[

[

120588V119894

120588V1(V119894minus V119887119894) + 119901120575

1119894

120588V2(V119894minus V119887119894) + 119901120575

2119894

120588V3(V119894minus V119887119894) + 119901120575

3119894

(120588119864 + 119901) V119894

]]]]]]

]

(10)

where 997888rarrV119887is defined as

997888rarrV119887=


997888rarr119903 (11)

It is noteworthy to mention that the above expressionfor the Navier-Stokes equations is valid only for a rotatingreference frame Linear acceleration components should beadded in case one wants to express the equations of motionin a general noninertial frame

In addition it should be pointed out that it is not possiblein the present context to impose any radial equilibriumcondition to the pressure as it is normally done in turboma-chinery

An important thing to be noted here is that except forthe source term 997888rarr

119876119883 the functional form of the noninertial

International Journal of Aerospace Engineering 3

Navier-Stokes equations is the same as the functional formof the standard conservative equations defined for inertialreference frames and including the Algebraic LagrangianEulerian (ALE) approach for generalized motion of thegrid Therefore it is possible to implement a conservativeformulation in terms of the conservative variables 997888rarr119883 definedin (2) and the introduction of the ALE approach permits alocal application of the noninertial frame of reference as abuilding block in a more complex configuration frameworkwithout any interface between the noninertial and inertialpart of the same mesh because this formulation guaranteesthe flux conservation In applications it is sufficient to specifyas input the bounds of the rotational region and in this way arotational speed is imposed at the nodes inside this region

22 Enforcement of Compatibility Conditions The numericalsource error due to the noninertial reference frame can beexamined analytically by imposing the conservation of thefreestream In this case all the flow derivatives are zero andthe velocity vector is

997888rarr119881 =

997888rarr119881infin

+ (997888rarr120596 times

997888rarr119903 ) (12)

where 997888rarr119881infin

is equal to


= (

119906infin

Vinfin

119908infin

) (13)

The continuity momentum and energy equations (see(1)) can then be reduced to the following expressions

120588 [997888rarr120596 sdot (nabla times

997888rarr119903 )] = 0 (14)

120588997888rarr119881 [

997888rarr120596 sdot (nabla times

997888rarr119903 )] + 120588


sdot nabla997888rarr119881 = minus120588 (


997888rarr119881) (15)


997888rarr119903 )] = 0 (16)

For the first application case where we have a steadyrotation parallel to the 119910-direction (14) can be reduced to thefollowing expression

0 = 120596119910(120597119911

120597119909minus

120597119909

120597119911) (17)

that is identically zero for any nonzero angular velocity 120596119910

whereas for the second case with a rotation parallel to 119909-direction (14) becomes

0 = 120596119909(minus

120597119911

120597119910+

120597119910

120597119911) (18)

In both cases for the numerical formulation the right-hand side is not exactly zero however producing a freestreamerror

Using the results of (17) and (18) and denoting the right-hand side as 119878

Ω a simple and straightforward source term

correction can be applied in (1) In particular an additional

source term 997888rarr119870 can be included to exactly cancel the free-

stream error

120597997888rarr119883

120597119905+ nabla

997888rarr119865119868+ nabla

997888rarr119865119881=

997888rarr119876119883+997888rarr119870 (19)

where 997888rarr119870 for the NACA 0012 is

997888rarr119870 = 119878

Ω

[[[[[

[

120588

0

120588V2

0

120588119864

]]]]]

]

(20)

whereas for the propeller (20) becomes

997888rarr119870 = 119878

Ω

[[[[[

[

120588

120588V1

0

0

120588119864

]]]]]

]

(21)

3 Numerical Method

The computations have been performed using the codeUNS3D The solution algorithm is based on a finite volumenode centred approach operating on a hybrid unstructuredgridThe artificial dissipationmodel is derived from the non-linear scheme of Jameson [10] with no eigenvalue blendingScalar or matrix dissipation can be chosen

The Navier-Stokes equations are integrated in time witha second order backward difference scheme and dual timestepping A five-stage Runge-Kutta scheme is used to drivetoward zero the residual at each time step With the useof residual averaging a local CFL number of 49 could beemployed in the multistage subiteration process The Alge-braic Lagrangian Eulerian approach for generalized motionof the grid is included [11]

The Weiss and Smith version of low Mach numberpreconditioning is implemented in the code [12] A sensordepending on cell Reynolds number was also introduced toavoid applying the preconditioning inside boundary layersFor the computation of the present test case its applicationwas found to be beneficial in order to reduce numericaldissipation and enhance convergence

Matrix dissipation was also found to be beneficial allow-ing a strong reduction of the dissipation associated withconvective eigenvalues hence enabling a better resolution ofvortices

31 RANS Turbulence Model The k-120596 turbulence modelproposed by Hellsten [13] has been employed The modelconstants have been calibrated requiring consistent behaviornear boundaries between turbulent and laminar flow insideshear flows and for zero pressure gradient wall flows Inparticular the calibration has been considered taking intoaccount a variable 119888

120583 as it is the case if an algebraic stress

model (EARSM) is includedThe Wallin-Johansson Explicit Algebraic Stress Model

(WJ-EARSM) [14] is implemented using Hellstenrsquos k-120596 as


20

0

minus20

minus20 0 20 40

X

Y

(a)

05

0

minus05

minus05 0 05

X

Y

1

(b)

Figure 1 Mesh (a) wide-view and (b) close-view

the basic RANS model The model is an exact solution ofthe corresponding ARSM in two-dimensional mean flow Inthree dimensions there is still a complete while approximatesolution

The full anisotropic version of the model is used that isthe anisotropic part of the Reynolds stress tensor is directlyintroduced in the momentum equations while the isotropicpart is taken into account in the form of an effective variable119888120583

4 Boundary Conditions

The boundary conditions along solid walls for Navier-Stokes(viscous) flows are different from those for Euler flows In thecase of viscous flows the velocity of the flow must vanish atthe walls while in the case of Euler flows it is only requiredthat the flow does not go through the wall

As a consequence of the foregoing statement at the airfoilthe condition of nonpenetration has been imposed whereason the blade surface no-slip and no-penetration conditionsare used by setting the absolute velocity equal to the absolutelocal blade velocity and the adiabatic wall condition and zero-normal pressure gradient condition at the wall are imposed atthe blade surface

In general the boundary conditions applied at the far-field boundary are the same for Navier-Stokes and Eulerflows therefore the far-field boundaries are treated by usingcharacteristic boundary conditions

5 Numerical Results

51 Model Validation Steady Rotary NACA 0012 Airfoil Tovalidate the numerical model the stability derivatives fora NACA 0012 were computed using finite differences and

compared with the results obtained by Limache and Cliff [6]In the experiment an airfoil is submitted to a steady rotationperformed at constant incidence 120572 for a given pitch rate 119902generating a steady flow field in a reference frame attached tothe airfoil The radius of the loop is inversely proportional to119902 Thus as 119902 reduces to zero the radius approaches infinityand steady level flight is recovered

The results presented below are all computed for an angleof attack equal to zero so we use the wind-axis referenceframe for the computation of the derivatives

52 Numerical Results In Figure 1 is shown the 2D unstruc-tured grid The outer boundary is at a distance 30 times thelength of the airfoilrsquos chord with respect to the grid centercoincident with the leading edge of the airfoil The grid ismade by 12334 nodes and 12096 elements

To verify the implementation of 3D Navier-Stokes equa-tions in terms of absolute velocities we compare results forthe NACA 0012 airfoil rotating at a finite 119902 to those producedby Limache [7] simulating inviscid flow around aNACA 0012airfoil at Mach equal to 02 for nondimensional pitch rate119902 equal to 0 001 003 and 005 In fact at the present testconditions (low Mach number and low incidence) we do notexpect that the integral quantities computed using viscousand inviscid methods respectively differ significantly

The 119862119901distributions around the airfoil are shown in

Figure 2 where the computed 119862119901distributions and stream-

lines of relative velocity are compared with those presentedin [7] for 119902 = 001 In Figure 3 the pressure coefficientcontours and streamlines of relative velocity in the wholecomputational domain are shown

Finally in Table 1 we compare the values of 119862119897and 119862

119898

from our implementation to the references results The twoimplementations match quite well over a range of 119902 values


1

05

0

minus05

minus05 0 05 1 15

X

Y

minus1

(a)

X

Y

1

05

0

minus05

minus05 0 05 1 15

minus1

(b)

Figure 2 119862119901contours and streamlines comparison for rotating NACA 0012 airfoil at Mach = 02 and 120572 = 0∘ (a) present work 119902 = 001 (b)

Limache [7] 119902 = 001

30

20

0

minus10

minus20

minus30minus20 0 20

X

10

Y

(a)

minus30 minus20 minus10 0 10 20 30

30

20

0

minus10

minus20

Y

X

10

(b)


Limache [7] 119902 = 001

Table 1 Comparison of lift and moment coefficients for the NACA0012 at 120572 = 0∘ and Mach = 02 at various values of 119902 (results fromLimache [7] are in parentheses)

119902 119862119897

119862119898

001 minus0051 (minus0053) minus002 (minus0018)003 minus0153 (minus0157) minus006 (minus0053)005 minus026 (minus0262) minus01 (minus0088)

The stability derivatives are calculated using finite differ-ences

119862(◼)119902

asymp119862(◼)

(119872 120572 119902 + Δ119902) minus 119862(◼)

(119872 120572 119902 )

Δ119902 (22)

In Table 2 the stability derivatives are compared with theresults obtained by Limache


1200∘ di

am

Spinners Cylindrical

section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL

Figure 4 Geometrical and experimental model used by Biermann and Haetman [15]

Table 2 Comparison of stability derivatives for the NACA 0012 at120572 = 0∘ and Mach = 02

Derivatives UNS3D Limache and Cliff [6]119862119897119902

minus5225 minus5250119862119898119902

minus1932 minus1766

It is possible to conclude that the results obtained byUNS3D are in good agreement with the numerical resultsobtained by Limache and Cliff [6]

53 Geometrical Model In Figure 4 it is possible to seethe geometry of the experimental model used by Biermannand Haetman [15] The experimental results were performedfor four- and six-blade single-rotating and dual-rotatingpropellers with and without the symmetrical wing in placeThe maximum propeller speed was 550 rpm The resultsfor four-blade single-rotating propeller were made up withtwo two-way hubs mounted in tandem and the spacingbetween front and rear blades is not equal and therefore thefront blade led the rear by 854 deg In this paper only thefour-blade single-rotating propellers with and without wingare considered for the comparison with the experimentalresults in terms of thrust coefficient The propeller namelya Hamilton Standard 3155-6 consists of four blades installedon a single way hub in front of a streamline body or nacellehousing the engine needed to spin it The four blades arestreamlined using Clark Y profiles and the angle between twoblades is 90 deg

In the report of Biermann and Haetman [15] severalblade pitch angles defined as the angle between the rotationplane and the airfoil chord at 75 of the radius of thepropeller in the range of 20 deg to 65 deg were studiedIn our case it was decided to investigate a propeller witha blade pitch angle of 45 deg The propeller diameter is308m Starting from the geometrical details reported inthe mentioned report a mathematical model describingthe propeller has been created with CATIA V5 (Figure 5)

Table 3 Operating conditions investigated

Advance ratio Velocity ms Rotational speed rps143 491744 111415 491744 10618 491744 8920 491744 8024 491744 66

The wing shaped using NACA 0012 airfoils is located in amidposition of the spinner and set at an angle of attack of0 deg Wing chord is 18288mm and wing span is 42418mm

54 Results for the Four-Blade Single-Rotating Propeller +Spinner (Viscous) A 3D unstructured grid of the propeller +spinner has been generated with ICEM-CFD (Figure 6) Thisgrid is strongly refined in the region around the blades andon the blade surfaces It is made by

(i) 5672824 nodes(ii) 16265544 elements(iii) 21 prismatic layers on solid surfaces to correctlymatch

boundary layer behavior

In the code UNS3D it is possible to specify an arbitraryvelocity for a specific group of nodes within the mesh(ALE formulation) The resulting fluxes are automaticallyinterfaced in order to ensure conservation at the boundarybetween rotating and nonrotating zonesThe solution is thencomputed specifying a rotational velocity as described in(11) only for the nodes inside the rotating block (Figure 7)and taking into account the source terms (see (8)) andthe correction terms as in (23) It is worth noting that theformulation described above is steady The solution is indeedan instantaneous snapshot of the flow field which can be seenas frozen in correspondence with a fixed phase angle

Five different operating conditions shown in Table 3were investigatedThe axial undisturbed velocity has been set


XY

Z

(a)

XY

Z

(b)

Figure 5 CATIA model (a) propeller with wing and (b) propeller without wing

Figure 6 Grid generated with CFD-ICEM

equal to 491744ms corresponding to the maximum windtunnel test speed of 110mph [15]

In Figure 8 it is possible to note a general increasing levelof the relative Mach number from the nacelle surface towardthe tip which is the result of the increasing rotational speedwith increasing radius

As the propeller rotates it induces swirls in the slip-stream and the blade tip vortices pass by periodicallyThis phenomenon is more evident when the advance ratiodecreases In fact for lower advance ratios we see a strongvortex shedding which starts from each blade and travelsdownstream with the perturbation velocity creating strongspiral type regions in the rear wake for each blade (Figure 9)Furthermore strong hub and tip vortices (Figure 10) arecontinuously shed from the respective blade regions andldquoabsorbrdquo the weaker vorticity regions at inner blade radiiproducing also spiral type patterns

In Figures 11 and 12 the azimuthal velocity profilesdownstream of the blades are shown at different position of 119909

Table 4 Experimental and computational results for the thrustcoefficient

Advance ratio 119869 Experimental 119862119879

Computational 119862119879

143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005

and it is possible to note how the swirl induced by the rotationof the propeller vanishes with increasing distance from theblades In particular the fast decay of the slipstream swirlin the region far from the spinner is due to the coarse meshand consequently to the high levels of numerical dissipationin that region

With the aim of comparing the results with thoseobtained by Biermann and Haetman [15] the thrust coeffi-cient defined as

119862119879=

119879

12058811989921198634 (23)

has been calculated Following the experimental procedureadopted by Biermann and Haetman [15] the thrust force 119879

has been obtained by integrating the forces along 119909-directionon all the blade surfaces and subtracting the drag force due tothe blades alone in case of zero thrust coefficient

In Figure 13 the obtained thrust coefficients for fivedifferent advance ratios are plotted and compared with thoseobtained by Biermann and Haetman [15]

The computed thrust coefficients are in good agreementwith the experimental values as listed in Table 4


YX

Z

(a) (b)

Figure 7 Rotating block within the mesh (a) propeller and (b) mesh inside the block

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)

Figure 8 Relative Mach number contours in the plane (119910 119911) (a) 119869 = 18 and (b) 119869 = 143

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)

Figure 9 X-velocity in the plane 119909 = 750mm (a) 119869 = 18 and (b) 119869 = 15

As the experimental errors are unknown it is not possibleto determine whether the computed results are within therange of the experimental uncertainty or to give a preciseassessment of the quality of the results

55 Results for the Four-Blade Single-Rotating Propeller + Spin-ner +Wing (Inviscid) Steady Euler results for the propeller +

spinner + wing are presented in this section The grid isgenerated with ICEM-CFD (Figure 14) and it is made by

(i) 2173935 nodes(ii) 10718702 elementsA rotational velocity has been imposed for the nodes

inside the block around the blades as indicated in Figure 14


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z

Figure 10 X-vorticity in the plane (119909 119911) for 119869 = 143

Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm

Figure 11 Sections for the analysis of the azimuthal velocity profiles

The investigated operating conditions are identical to theprevious case

The Mach distributions in Figure 15 clearly show theeffect of the propeller slipstream that washes the wing Inparticular the impact of the swirl velocity component is verypronounced It should be noted that theMach number distri-butions much like the pressure distributions as presented inFigure 16 are affected by both the local propeller inducedflowangles and the dynamic pressure increases in the slipstream

Another phenomenon that is clearly visible due to theinterference between the propeller and the wing is therise of vortices around the juncture of spinner and wing(Figure 17) A shedding of these vortices can be evidentwhich is indicative of the high gradient of spanwise load onthe wing

In fact the streamwise and spanwise locations of bladevortices are staggered on the upper and lower surfaces of wingand interacted vortices are induced near spinner

In order to make a further verification of the method thenumerical calculations have been compared to experimentaldata (Figure 18)




143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054

Again the computational results are in good agreementwith the experiment and the maximum error is around5 (Table 5) Also in this case the experimental errors areunknown

6 Conclusions

The Alenia Aermacchi UNS3D code was modified intro-ducing the capability of flow simulations in a noninertialframe of reference The modified code was at first applied


4000

2000

0

minus2000

minus4000

minus04 minus02 0 02 minus04 minus02 0 02


minus04 minus02 0 02



4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

Figure 12 Azimuthal velocity profiles downstream the blades at positions of Figure 11 for 119869 = 18


025

020

015

010

005

000000 050 100 150 200 250 300

Experimental resultsComputational results

CT

J

Figure 13 Computed versus experimental thrust coefficient

X

Y

Z

(a) (b)


to the computation of damping derivatives of a rotatingairfoil and then to the prediction of the performance of apropeller following the experimental test case described byBiermann and Haetman [15] for different rotational speedsIn the first case good agreement has been obtained with thenumerical results of [7] In the second the results are in goodagreement with the experimental data within the propelleroperating range The computational results clearly showedthe effect of the swirl velocity and the increased total pressureon the spinner and the wing Therefore this approachfacilitates the identification of typical flow phenomena likethe deformation of the slipstream when passing the wingbeing able to model aerodynamic phenomena linked to thepropeller-airframe integration

Nomenclature

119887 Generic scalar997888rarr119861 Generic vector

119888 Airfoil chord119862(◼) Generic aerodynamic coefficient

119862119897 Lift coefficient

119862119898 Pitching moment coefficient

119862119901 Pressure coefficient

119862(◼)119902

Generic stability derivative119862119879 Thrust coefficient

119863 Propeller diameter [m]119864 Total energy per unit of mass [Jkg]997888rarr119891119890 Vector of external forces [N]

997888rarr119865119868 Inviscid flux vector

997888rarr119865119881 Viscous flux vector

119869 Advance ratio119872 Mach number119899 Propeller rotational speed [rps]119901 Pressure [Pa]119902 Pitch rate


XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08

CpY = 1000mmCpY = minus1000mm

(c)

Figure 15 Mach number (a) upper side for 119869 = 18 (b) lower side for 119869 = 18 and (c) 119862119901on surface for 119910 = 1000mm and 119910 = minus1000mm

2000

0

minus2000

minus2000 0 2000 4000

Y

Z

Figure 16 Total pressure loss in the plane (119910 119911) for 119869 = 18

3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z

Figure 17 Vorticity magnitude in the plane (119910 119911) for 119869 = 18


030

025

020

015

010

005

000000 050 100 150 200 250 300

Experimental resultsNumerical results

CT

J


119902 Normalized pitch rate (1199021198882119881)119902119894 Heat flux [Wm2]

997888rarr119876119883 Source vector

997888rarr119903 Position vector relative to the rotation

center [m]119905 Time in physical space [s]119879 Effective thrust [N]119906infin Vinfin 119908infin Freestream velocity components [ms]

997888rarrV 997888rarr119881 Velocity vector [ms]997888rarrV119887 Rotational speed vector of the coordinate

system [ms]V119894 Generic component of the velocity vector

[ms]997888rarr119881infin Freestream velocity vector [ms]

V1 V2 V3 Components of velocity vector [ms]

119882119891 Work of the external forces [J]

997888rarr119883 Vector of conservative variables119909 119910 119911 Position vector components relative to the

rotation center [m]120572 Angle of attack120575119894119895 Kronecker symbol

120588 Density [kgm3]120591119894119895 Generic component of the shear tensor

[Pa]997888rarr120596 Rotational speed vector [radsec]120596119909 120596119910 120596119911 Components of rotational speed vector

[rads]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work has been performed through a closecooperation with Alenia Aermacchi SpA and through theuse of Confidential Information and Data property of AleniaAermacchi SpA which remains the sole owner of all suchrelevant IP rights The results of the present work shall betherefore property of Alenia Aermacchi SpA

References

[1] B Chen and F Stern ldquoComputational fluid dynamics of four-quadrant marine-propulsor flowrdquo Journal of Ship Research vol43 no 4 pp 218ndash228 1999

[2] K SMajety Solutions to theNavier-Stokes equations in a non-in-ertial reference frame [MS thesis] Mississippi State University2003

[3] J Quilan J Eaton and M OFlaherty ldquoComputational inves-tigation of propulsion integration effects on wing propellerinstallationrdquo in Proceedings of the 34th Aerospace SciencesMeeting and Exhibit AIAA Paper 96-0673 Reno Nev USAJanuary 1996

[4] R K Agarwal and J E Deese ldquoEuler calculations for flowfield ofa helicopter rotor in Hoverrdquo in AIM 4th Applied AerodynamicsConference 1987

[5] M D Hathway R M Chriss J R Wood and A J StrazisarldquoExperimental and computational investigation of the NASAlow-speed centrifugal compressor flow fieldrdquo in Proceedings ofthe 37th ASME International Gas Turbine Conference 1993

[6] A C Limache and E M Cliff ldquoAerodynamic sensitivity theoryfor rotary stability derivativesrdquo Journal of Aircraft vol 37 no 4pp 676ndash683 2000

[7] A C Limache Aerodynamic modeling using computationalfluid dynamics and sensitivity equations [PhD thesis] VirginiaPolytechnic Institute and State University Blacksburg Va USA2000

[8] M A Park L L Green R C Montgomery and D L RaneyldquoDetermination of stability and control derivatives using com-putational fluid dynamics and automatic Differentiationrdquo inProceedings of the 17th AIAA Applied Aerodynamics ConferenceAIAA Paper pp 99ndash3136 Norfolk Va USA June 1999

[9] M A Park and L L Green ldquoSteady-state computation ofconstant rotational rate dynamic stability derivativesrdquo in Pro-ceedings of the 18th Applied Aerodynamics Conference AIAAPaper 20004321 Denver Colo USA 2000

[10] V Selmin and L Formaggia ldquoUnified construction of finite ele-ment and finite volume discretizations for compressible flowsrdquoInternational Journal for Numerical Methods in Engineering vol39 no 1 pp 1ndash32 1996

[11] L Djayapertapa N Ceresola et al ldquoTime accurate methodsrdquoTech Rep AG38TP154 GARTEUR AD 2006

[12] J M Weiss and W A Smith ldquoPreconditioning applied tovariable and constant density flowsrdquo AIAA Journal vol 33 no11 pp 2050ndash2057 1995

[13] A Hellsten ldquoNew advanced k-120596 turbulence mode for high-lift aerodynamicrdquo in Proceedings of the 42nd AIAA AerospaceSciences Meeting and Exhibit Reno Nev USA January 2004


[14] S Wallin and A V Johansson ldquoAn explicit algebraic Reynoldsstress model for incompressible and compressible turbulentflowsrdquo Journal of Fluid Mechanics vol 403 pp 89ndash132 2000

[15] D Biermann and E P Haetman ldquoWind tunnel tests of four-and six-blade single- and dual-rotating tractor propellersrdquo TechRep 747 NACA 1942 httpnixnasagovsearchjspR=199-30091825ampqs=N3D42941292432B172B4294448544

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



have obtained a solution [6ndash9] In this case the flow field iscomputed using Euler equations The second application is amore complex test case which has the purpose to evaluate theaccuracy efficiency and robustness of the current method topredict the complex flow field of a rotating propeller For thistest case a comparison with experimental results for cruiseconditions in terms of thrust coefficient is also made

2 Governing Equations

The governing equations which are numerically resolvedusing the UNS3D code are summarized below The Navier-Stokes equations in a noninertial frame of reference areexpressed as

120597997888rarr119883

120597119905+ nabla

997888rarr119865119868+ nabla

997888rarr119865119881=

997888rarr119876119883 (1)


997888rarr119883 = [

[

120588

120588997888rarrV

120588119864

]

]

(2)

997888rarr119865119868and 997888rarr

119865119881are the respective flux vectors

119865119868119894=

[[[[[

[

120588V119894

120588V1V119894+ 1199011205751119894

120588V2V119894+ 1199011205752119894

120588V3V119894+ 1199011205753119894

(120588119864 + 119901) V119894

]]]]]

]

(3)

minus119865119881119894

=

[[[[[

[

0

1205911198941

1205911198942

1205911198943

119902119894+ V119895120591119894119895

]]]]]

]

(4)

and the source terms are contained within 997888rarr119876119883

997888rarr119876119883

=[[

[

0

120588997888rarr119891119890

119882119891

]]

]

(5)

119882119891is equal to the following relation

119882119891= 120588

997888rarr119891119890sdot997888rarrV (6)

21 Formulation in Rotating Frame for the Absolute VelocityNormally the governing equations for rotating systems aresolved for the relative velocity components in the relativeframe of reference However in an external aerodynamiccontext such as the propeller-wing configuration two mainadvantages can be derived through taking instead the abso-lute velocity components in the relative reference frameThe first and most important one is that the grid blockcontaining the propeller can be interfaced more easily withinthe full aircraftmodelThe second is that a smaller numerical

effect of source terms is present as will be shown below Toexpress (1) in terms of a relative reference frame the followingrelations for substantial and local derivatives are used

119863119887

119863119905=

1198631015840119887

1198631199051015840

119863119861

119863119905=

1198631015840119861

1198631199051015840+997888rarr120596 times

997888rarr119861

120597119887

120597119905=

1205971015840119887

1205971199051015840minus (


997888rarr119903 ) sdot nabla119887

120597119861

120597119905=

1205971015840119861

1205971199051015840minus (


997888rarr119903 ) sdot nabla

997888rarr119861 +


997888rarr119861

(7)

where the prime 1015840 denotes the operation with respect to therelative reference frame By using relations (7) the right-handside of (1) becomes

997888rarr119876119883

= [

[

0

minus120588 (997888rarr120596 times

997888rarrV )

0

]

]

(8)


997888rarr120596 = (

120596119909

120596119910

120596119911

) (9)

With this formulation the source term vector (see (8))contains only the contribution of the Coriolis force and thecontribution of the centrifugal force is omitted In this waythe magnitude of the source term is greatly reduced and asmaller amount of artificial dissipation is required to ensureconvergence

To take into account the rotation of the coordinate system(3) is modified in the following way

119865119868119894=

[[[[[[

[

120588V119894

120588V1(V119894minus V119887119894) + 119901120575

1119894

120588V2(V119894minus V119887119894) + 119901120575

2119894

120588V3(V119894minus V119887119894) + 119901120575

3119894

(120588119864 + 119901) V119894

]]]]]]

]

(10)

where 997888rarrV119887is defined as

997888rarrV119887=


997888rarr119903 (11)

It is noteworthy to mention that the above expressionfor the Navier-Stokes equations is valid only for a rotatingreference frame Linear acceleration components should beadded in case one wants to express the equations of motionin a general noninertial frame

In addition it should be pointed out that it is not possiblein the present context to impose any radial equilibriumcondition to the pressure as it is normally done in turboma-chinery

An important thing to be noted here is that except forthe source term 997888rarr

119876119883 the functional form of the noninertial




997888rarr119881 =


+ (997888rarr120596 times

997888rarr119903 ) (12)


is equal to


= (

119906infin

Vinfin

119908infin

) (13)



997888rarr119903 )] = 0 (14)

120588997888rarr119881 [


997888rarr119903 )] + 120588




997888rarr119881) (15)


997888rarr119903 )] = 0 (16)


0 = 120596119910(120597119911

120597119909minus

120597119909

120597119911) (17)



0 = 120596119909(minus

120597119911

120597119910+

120597119910

120597119911) (18)






stream error

120597997888rarr119883

120597119905+ nabla

997888rarr119865119868+ nabla

997888rarr119865119881=

997888rarr119876119883+997888rarr119870 (19)


997888rarr119870 = 119878

Ω

[[[[[

[

120588

0

120588V2

0

120588119864

]]]]]

]

(20)


997888rarr119870 = 119878

Ω

[[[[[

[

120588

120588V1

0

0

120588119864

]]]]]

]

(21)

3 Numerical Method










20

0

minus20

minus20 0 20 40

X

Y

(a)

05

0

minus05

minus05 0 05

X

Y

1

(b)








5 Numerical Results










119898



1

05

0

minus05

minus05 0 05 1 15

X

Y

minus1

(a)

X

Y

1

05

0

minus05

minus05 0 05 1 15

minus1

(b)


Limache [7] 119902 = 001

30

20

0

minus10

minus20

minus30minus20 0 20

X

10

Y

(a)


30

20

0

minus10

minus20

Y

X

10

(b)


Limache [7] 119902 = 001


119902 119862119897

119862119898



119862(◼)119902

asymp119862(◼)

(119872 120572 119902 + Δ119902) minus 119862(◼)

(119872 120572 119902 )

Δ119902 (22)



1200∘ di

am


section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL




minus5225 minus5250119862119898119902

minus1932 minus1766













XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












997888rarr119881 =


+ (997888rarr120596 times

997888rarr119903 ) (12)


is equal to


= (

119906infin

Vinfin

119908infin

) (13)



997888rarr119903 )] = 0 (14)

120588997888rarr119881 [


997888rarr119903 )] + 120588




997888rarr119881) (15)


997888rarr119903 )] = 0 (16)


0 = 120596119910(120597119911

120597119909minus

120597119909

120597119911) (17)



0 = 120596119909(minus

120597119911

120597119910+

120597119910

120597119911) (18)






stream error

120597997888rarr119883

120597119905+ nabla

997888rarr119865119868+ nabla

997888rarr119865119881=

997888rarr119876119883+997888rarr119870 (19)


997888rarr119870 = 119878

Ω

[[[[[

[

120588

0

120588V2

0

120588119864

]]]]]

]

(20)


997888rarr119870 = 119878

Ω

[[[[[

[

120588

120588V1

0

0

120588119864

]]]]]

]

(21)

3 Numerical Method










20

0

minus20

minus20 0 20 40

X

Y

(a)

05

0

minus05

minus05 0 05

X

Y

1

(b)








5 Numerical Results










119898



1

05

0

minus05

minus05 0 05 1 15

X

Y

minus1

(a)

X

Y

1

05

0

minus05

minus05 0 05 1 15

minus1

(b)


Limache [7] 119902 = 001

30

20

0

minus10

minus20

minus30minus20 0 20

X

10

Y

(a)


30

20

0

minus10

minus20

Y

X

10

(b)


Limache [7] 119902 = 001


119902 119862119897

119862119898



119862(◼)119902

asymp119862(◼)

(119872 120572 119902 + Δ119902) minus 119862(◼)

(119872 120572 119902 )

Δ119902 (22)



1200∘ di

am


section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL




minus5225 minus5250119862119898119902

minus1932 minus1766













XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20

0

minus20

minus20 0 20 40

X

Y

(a)

05

0

minus05

minus05 0 05

X

Y

1

(b)








5 Numerical Results










119898



1

05

0

minus05

minus05 0 05 1 15

X

Y

minus1

(a)

X

Y

1

05

0

minus05

minus05 0 05 1 15

minus1

(b)


Limache [7] 119902 = 001

30

20

0

minus10

minus20

minus30minus20 0 20

X

10

Y

(a)


30

20

0

minus10

minus20

Y

X

10

(b)


Limache [7] 119902 = 001


119902 119862119897

119862119898



119862(◼)119902

asymp119862(◼)

(119872 120572 119902 + Δ119902) minus 119862(◼)

(119872 120572 119902 )

Δ119902 (22)



1200∘ di

am


section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL




minus5225 minus5250119862119898119902

minus1932 minus1766













XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

05

0

minus05

minus05 0 05 1 15

X

Y

minus1

(a)

X

Y

1

05

0

minus05

minus05 0 05 1 15

minus1

(b)


Limache [7] 119902 = 001

30

20

0

minus10

minus20

minus30minus20 0 20

X

10

Y

(a)


30

20

0

minus10

minus20

Y

X

10

(b)


Limache [7] 119902 = 001


119902 119862119897

119862119898



119862(◼)119902

asymp119862(◼)

(119872 120572 119902 + Δ119902) minus 119862(◼)

(119872 120572 119902 )

Δ119902 (22)



1200∘ di

am


section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL




minus5225 minus5250119862119898119902

minus1932 minus1766













XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1200∘ di

am


section

section

600∘

600∘

720∘

1800∘608∘

338∘

120572

1670∘

120∘

120∘

120∘

120∘

120∘

320∘

diam320∘

diam uer2060∘

3123∘

2753∘

NACA 0012

CL




minus5225 minus5250119862119898119902

minus1932 minus1766













XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










XY

Z

(a)

XY

Z

(b)










143 02192 021715 02175 021518 01782 017420 01374 0137624 00487 005



119862119879=

119879

12058811989921198634 (23)






YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










YX

Z

(a) (b)


2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

0 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(a)

1500

1000

500

0

minus500

minus1000

minus1500

minus1000 0 1000 2000

02802661540252308023846202246150210769019692301830770169231015538501415380127692011384601

Y

Rela

tive M

ach

num

ber

Z

(b)


3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500


Y

12112510509750908250750675060525045037503022501500750

minus

Z U

(a)

2000

0

0 2000

Y

12112510509750908250750675060525045037503022501500750

Z U

minus2000

minus2000

(b)








3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2500

0 2000 4000

005

004

003

002

001

0

minus001

minus002

minus003

minus004

minus005

X

X-v

ortic

ity

Z


Y X

Z

x=215

mm

x=1671

mm

x=3633

mm

x=7350

mm

x=835

mm

x=2669

mm

x=4341

mm










143 0206 0215 02052 019618 0175 01820 014 013524 0049 0054


6 Conclusions



4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4000

2000

0

minus2000

minus4000






4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

4000

2000

0

minus2000

minus4000

Z Z

Z

Z

Z

Z

Z

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf

VV inf



025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










025

020

015

010

005

000000 050 100 150 200 250 300


CT

J


X

Y

Z

(a) (b)



Nomenclature






119862(◼)119902







XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

000000 050 100 150 200 250 300


CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










XY

Z

Mac

h

03

0285

027

0255

024

0225

021

0195

018

0165

015

(a)

Y X

Z

Mac

h

030292502850277502702625025502475024023250225021750210202501950187501801725016501575015

Z = 1000mm

Z = minus1000mm

(b)

Cp

X

minus1

minus05

0

05

1

150 02 04 06 08


(c)


2000

0

minus2000

minus2000 0 2000 4000

Y

Z


3500

3000

2500

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

minus2000 0 2000

Y

Z



030

025

020

015

010

005

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Acknowledgments


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RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










030

025

020

015

010

005

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CT

J















[rads]



Acknowledgments


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

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