Remarks on the Model

Thissign inversioninthe indued veloities omes from thewaytheregularization

by nite part integrals is applied. By Biot-Savart law, the veloity indued by a

pointinthevortiitywakeonapointinthelifting-linedependsontheinverseofthe

distanebetweenthosetwopoints. Aspointsgetloser,theirinueneinreaseand

indued veloitiestend to innity when thedistanetends to zero. Guermond and

Sellier

[Guermond1991℄presentedamethodusing Hadamard'snite partintegrals

inordertooveromethesingularityproblem. Intheirapproah,thewinggeometry

and thewake vortiityweredesribed usinganalytial funtions,andtherefore this

enables toalulateanalytiallytheinduedveloitiesofallthewakeonapoint

M ₀

on thewing:

where

L

^{represents the urved} lifting-line. It is important to notie that the reg-ularization by nite part integrals is performed not only in the diretion of the

freestream, i.e. along the

x

^{axis, but also spanwise, i.e. along the}

y

^{axis. On the}

ontrary, Muller proposes only a regularization along the

x

^{axis, and thus it leads}

to unphysial induedveloityontributions.

The present analysishas shown how nite part integrals both inthe inner and

theouterdomainsinversethesignofthe singularpanelontribution onindued

ve-wakeomponentintheinnerdomainmayompensate itandstabilizethewhole

sys-tem.

However,inthegeneralase, theonvergeneannotbeensuredduetothesingular

panel omponent, as it has been notied when implementing this methodology in

HOST.Thus,nitepartintegralsseemnot tobeadaptedinthepresentformtothe

alulationof awake desribed withdisretepanels.

The Appendix Cpresentsanalternative formfor theintegrationof theindued

veloities inthe singular panel. This new method exploits the approah proposed

by Muller but nite part integrals are performed in both hordwise and spanwise

diretions. Thisenablesthenewapproahtoprovidemoreregularinduedveloities

regardlessboth the hordwise and the spanwisesize of thepanel.

Developing and Implementing an

UCLL Method in HOST without

Finite Part Integrals

Contents

4.1 The CompleteUnsteadyCurved Theory . . . 121

4.2 Implementationin HOST . . . 124

4.3 Conludingremarks . . . 125

Asexplained inhapter3,theunsteadymodel implementation inAILEode is

notadaptedtotheonsideredaseinHOST,wherebladespresentsmallaspetratios

andanimportantnumberofpanelsdueto thenitepartintegrals. Thus,anew

ap-proahtoimplementtheunsteadyurved lifting-linetheoryproposedbyGuermond

and Sellier [Guermond1991℄ has been oneived in order to remove Hadamard's

nite partintegrals. Thishapter presents indetail this newapproah, exposes its

hypothesisand limitsand presents itsimplementation inHOST.

4.1 The Complete Unsteady Curved Theory

The asymptoti approah presented byMuller hasbeen modiedto remove two of

theterms alulated innite part integrals, in theinner and intheouter domains.

Henetheimpliitequationoftheirulation

Γ(M )

^{inEq.3.1hasbeenmodiedas}

followsEq. (4.1).

Γ = πc cos Λ U _∞ α + w _{I ∪O wi} + w _{O\O wi} + w _c

(4.1)

where:

• w _{I∪O wi}

^{: represents the veloity indued by the domain}

I ∪ O _wi

^{, whih has}

a pure two-dimensional origin. Based on the linearized unsteady thin-airfoil

theory,we obtain Eq. (4.2).

w _{I ∪O wi} (y, t) = ˙ h(y, t) + c 2 α(y, t) ˙

+ 1

πcU ∞ cos Λ Z +∞

c/2

s ν + c/2 ν − c/2 − 1

!

γ _w (ν, t)dν

(4.2)

• w _{O\O wi}

^{: represents theveloityinduedbytheouter domain,thewholewake}

exeptthe longitudinal vortiity

γ y

^{inthe wake panels behind theonsidered}

setion. For this domain, the harateristi length is the wingspan, and so

the wingis supposedto be alifting-line. The wake is modeledbya lattie of

vortexpanelsand vortexlaments. Theexpressionoftheveloityinduedby

thewakeis showninEq. (4.3).

The term

w _{O\O wi}

^{orrespondsto the}

V ~

^{ind omponent ontained inthe airfoil}

plane andnormalto theloalhord.

• w _c

^{: representsthe} omplementary induedveloity, whih takes into aount theloalsweepandurvature,inordertoaddtheeetofspanwiseirulation

variations onthe lifting line. Equation(4.4) an be deduedfrom thestudies

donebyGuermond[Guermond 1990℄fortheaseofasteadyurvedlifting-line

theorybyfatorizing by

Γ

^and

∂Γ/∂y

^.

w c (y) = a 1 Γ(y) + a 2

Again, the mathing between outer and inner domains is done by linking the

vortexintensities ofthese two domains:

γ _w (ξ, y) = γ _y (ξ − c/2, y)

^{. Thisis whatthe}

urrentunsteadyorretionimplementedinHOSTatuallylaks: theairfoilmotion

eet is onsidered, but the two-dimensional unsteady wake eet on the airfoil is

not.

Following the linearizedunsteady thin-airfoil theory, thetwo-dimensional wake

inuene should be onsidered from the trailing edge (

ξ = c/2

^{) to innity}

down-stream. However, intheHOSTode,astheairfoil isreduedto alifting-line, wake

panels shed by eah setionstart immediately after thealulation point. Inorder

to avoidalulating twie theveloities indued bythe two-dimensional wake (see

O _wi

^{domain inFig.20),aseriesof MINTwakepanels mustbe treateddierentlyin}

thealulationof the total induedveloity. Indeed,these panelsonly onsiderthe

other omponent of thevortiity:

γ _x

^{,i.e. theradial variation oftheirulation.}

Theoretially,weshouldonsideraninnitetwo-dimensionalwakeandonsideronly

γ _x

^{inalltheMINTwakepanelsbehindtheonsideredsetion. However,asshownin}

b

M

b

γ _y ¹

b b b

γ _y ⁿ

b

/2 /2

/4

X

(a)Unsteadyairfoilmodel. Innerdomain

y

x

b M

MINTwake

γ _x γ _y

γ _x ¹ γ _x ⁿ

Only

γ x

(b)MINTwakemodel. Outerdomain

Figure 4.1: The unsteadyairfoilmodelasimplementedinHOST-MINTode

thetwo-dimensional wakeintheairfoilformulation aneleahotherafteraertain

distane. Ifwe introduethe parameter

X

representingthelimitbetweeninnerand outer domains (

X ≫ c/2

^{), the outer terms of theintegrals beome equivalent and}

sothey ompensate eah other. Thus, we an onsider theairfoil formulation from

c/2

^to

X

^{in our wake integrals. In the urrent HOST} implementation,

X

param-eter has been hosen

X = 3c

^{. Eah panel length an be estimated by onsidering}

that they are onveted at the relative veloity of thesetion, without adding any

induedveloity. Thisestimatedpanellengthisthenusedtodetermine thenumber

of panels to be onsidered in thetwo-dimensional unsteady wake, and hene to be

removed fromthe MINTwake(as showninFig.4.1).

Again, as the urrent airfoil formulation is derived from a small perturbation

theory,itassumes aat two-dimensional wake,whih isnot stritlytheaseinour

free-wake model. Thus, an additional assumption must be done: we will onsider

that wake deformation of the rst panels, i.e. panels at distane

ξ ≤ X

^{, has a}

negligible impat on indued veloities over the blade. This hypothesis has been

Finally,Eq. (4.1) an be expressedasshowninEq. (4.5).

where we an distinguishseveral termsofindued veloities:

•

^{The inner domain} (thin-airfoil theory): omposed by thequasi-steady airfoil motion term and the near wake term in airfoil formulation. The terms in

therst andseond linesof Eq.(4.5) orrespondto theinner domain indued

veloities.

•

^{The outerdomain}

w _O

^{: wherethewakeisdividedinpanels andintegratedby}

the Gauss-Legendre quadrature rule. A speial treatment is needed for the

alulation ofwake panels behind theonsidered setion, asonlyradial

varia-tions of irulation

γ _x

^{areonsidered. The third line of Eq.(4.5) orresponds}

to the veloitiesindued bythe outerdomain indoublet formulation.

•

^The omplementary terms

w c

^{(from Guermond and Sellier): due to the}

lo-al sweep and urvature. The fourth line of Eq.(4.5) orresponds to these

omplementary terms.

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