estimating the low-speed sidewash.pdf

7/25/2019 Estimating the Low-Speed Sidewash.pdf

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1088 J. AIRCRAFT, VOL. 39, NO. 6: ENGINEERING NOTES

5Fremaux, C. M., Spin-Tunnel Investigation of a 1/28-Scale Model ofthe NASA F-18 High Alpha Research Vehicle (HARV) With and WithoutTails, NASA TR-201687, April 1997.

6Croom, M., Kenney, H., Murri, D., and Lawson, K., Research on theF/A-18E/F Using a 22%-Dynamically Scaled Drop Model, AIAA Paper2000-3913, Aug. 2000.

7Phillips, W. F., Hailey, C. E., and Gebert, G. A., Review of AttitudeRepresentations Used for Aircraft Kinematics,Journal of Aircraft, Vol. 38,No. 4, 2001, pp. 718737.

8Etkin, B., and Reid, L. D., Eulers Equations of Motion,Dynamics of

Flight: Stabilityand Control, 3rd ed., Wiley, New York, 1996, pp. 100101.9Phillips, W. F., and Anderson, E. A., Predicting the Contribution of

Running Propellers to Aircraft Stability Derivatives, AIAA Paper 2002-0390, Jan. 2002.

Estimating the Low-Speed SidewashGradient on a Vertical Stabilizer

W. F. Phillips

Utah State University, Logan, Utah 84322-4130

Nomenclature

b = wingspanb0 = wingtip vortex spacingCLw = wing lift coefcientRAw = wing aspect ratioV

z = zcomponent of induced velocityV1 = magnitude of the freestream velocity

x = axial coordinateNx = dimensionlessaxial coordinate, 2x=b

y = normal coordinateNy = dimensionlessnormal coordinate,2y=bz = spanwise coordinate,measured from the aircraft

plane of symmetryNz = dimensionlessspanwise coordinate,2z=bz

0 = spanwise coordinate,measured from the centerlinemidway between the two wingtip vortices

Nz0 = dimensionlessspanwise coordinate,2z0=b = sideslip angle0wt = wingtip vortex strength"s = sidewash angleb = spacing between the wingtip vortices divided by the

wingspanv = ratio of wingtip vortex strength to that of an elliptic

wing having the same lift coefcient and aspect ratio = sidewash factor3 = quarter-chordwing sweep angle

Introduction

T HE sidewash induced on a vertical stabilizer by the wingtipvortices shed from the main wing can have a signicant effecton the static yaw stability of an airplane. For a vertical stabilizermounted above the main wing, the sidewash gradient is negativeand has a stabilizing effect on the airplane. The sidewash gradientproduced by the wingtip vortices can be estimated using a vortexmodel that was recently presentedfor predictingthe downwash pro-duced by a wing of arbitrary planform.1 This vortex model, which

Received 30 May 2002; revision received 2 August 2002; accepted forpublication 8 August 2002. Copyright c 2002 by W. F. Phillips. Published

by the American Institute of Aeronautics and Astronautics, Inc., with per-mission. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 0021-8669/02 $10.00 in correspondence with the CCC.

Professor,Mechanical and AerospaceE ngineering Department. MemberAIAA.

is shown in Fig. 1, was originally proposed by McCormick2 forestimating the downwash produced by an elliptic wing.

Analytical Model

Relativeto thecoordinatesystemshownin Fig.1, the z componentof velocity induced by this model of the wingtip vortices, at thearbitrary point in space (x , y, z), is readily found from the BiotSavart law to be

Vz

D0wt

4

( y

y2 Cz C 1

2b02

241 C x 12 b0 tan 3q

x 12

b0 tan 32

C y2 Cz C 1

2b0235

y

y2 Cz 1

2b02

241 C x 12 b0 tan 3qx 1

2b0 tan 3

2C y2 C

z 1

2b02359=; (1)

The wingtip vortex strength is proportional to the product of thewing lift coefcient and airspeed. The wingtip vortex strength andspacing can be evaluatedfrom lifting-line theory.1 When the tradi-tional sign convention that sidewash is positive from left to rightis used, Eq. (1) is combined with lifting-line theory, and the smallangle approximation is applied, the sidewash angle can be writtenas

"s D V

z

V1D

CLwv

2RAw

Ny

Ny2 C .Nz b/2

"

1 C Nx btan 3p

. Nx btan 3/2 C Ny2 C .Nz b/2

#

Ny

Ny2 C .Nz C b/2

"1 C

Nx btan 3p. Nx btan 3/2 C Ny2 C .Nz C b/2

#) (2)

where the parametersv and b can be evaluated analytically fromthe known planform shape of the wing using the results presentedby Phillips et al.1

Equation (2) can be directly applied to determine the sidewashangle only when the sideslip angle is zero and the aircraft plane of

Fig. 1 Vortex model used to estimate the sidewash gradient for the

vertical stabilizer.


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J. AIRCRAFT, VOL. 39, NO. 6: ENGINEERING NOTES 1089

Fig. 2 Effect of sideslip on the position of the wingtip vortices relativeto the vertical stabilizer.

symmetry is midway between the two trailing vortices. When theairplane has some component of sideslip, the wingtip vortices aredisplaced relative to the position of the vertical tail, as shown inFig. 2. For small sideslip angles, Eq. (2) gives a very close approx-imation for the sidewash angle, if the zcoordinate, measured from

the aircraftplane of symmetry, is replaced with az

0

coordinate,mea-suredfrom the centerlinemidway between the two wingtip vortices,as shown in Fig. 2. Whenthe small angleapproximationis used, thez

0 coordinate is related to the zcoordinateaccording to

z0./ D z cos

x 1

2b0 tan 3

sin D z

x 1

2b0 tan 3

or

Nz0./ D Nz . Nx btan 3/ (3)

Within this small angle approximation, the sidewash gradient canbe written as

@"s

@ D

@ Nz0

@

@"s

@ Nz0

Nz0 D Nz(4)

where, from Eq. (2)

@"s

@ Nz0

Nz0 D Nz

DCLwv

2RAw

( 2 Ny.Nz b/Ny2 C .Nz b/

22

"1 C

Nx btan 3p. Nx btan 3/2 C Ny2 C .Nz b/2

#

C 2 Ny.Nz C b/Ny

2

C .Nz C b/22

"1 C

Nx btan 3p. Nx btan 3/2 C Ny2 C .Nz C b/2

#

Ny

Ny2 C .Nz b/2

24 . Nx btan 3/.Nz b/

. Nx btan 3/2 C Ny2 C .Nz b/2 3

2

35

CNy

Ny2 C .Nz C b/2

2

4. Nx btan 3/.Nz C b/

. Nx btan 3/2 C Ny2 C .Nz C b/232

3

5

9=;(5)

and from Eq. (3)

@ Nz0

@D . Nx btan 3/ (6)

Fig. 3 Effect of vertical stabilizer positionon the sidewash gradient inthe plane of symmetry.

The sidewash gradient induced at an arbitrary point in space canbe estimated by using Eqs. (5) and (6) in Eq. (4). In the plane ofsymmetry this reduces to

@"s

@. Nx; Ny; 0/ D

v

b

CLw

RAw(7)

where

. Nx; Ny; 0/ D4 Ny. Nx btan 3/

2b

2 Ny2 C 2

b2

"1 C

Nx btan 3p. Nx btan 3/2 C Ny2 C

2b

#

C 2 Nyb

2

Ny2 C 2b

24 . Nx btan 3/2b

. Nx btan 3/2 C Ny2 C 2b

32

35 (8)

The sidewash factor depends on the planform shape of thewing and the position of the tail relative to the wing. However, theplanform shape of the wing affects the value of only through3and b . Thus, is a unique function of Nx=b t an3and Ny=b , asshown in Fig. 3.

It is important to remember the orientation of the coordinatesys-tem usedin the developmentof Eqs. (5) and (8).As shown in Figs. 1and 2, the x z plane of this coordinate system coincides with theplane of the trailing wingtip vortices, and thex axis is aligned withthe equilibrium relative wind. Thus, the y coordinate is the verticaldistance measured above the plane of the trailing wingtip vortices.When the downwash from this vortex modelwas computed accord-ing to the relations presented by Phillips et al.,1 the y coordinatewas found to have only a second-order effect on downwash. How-ever, as can be seen from Eqs. (5) and (8), they coordinate has arst-ordereffecton the sidewash. Thus, the orientationof thex axiswithin the aircraft plane of symmetry is critical for predicting thesidewash gradient. This orientation changeswith angle of attack as

shown in Fig. 4. The y

coordinate of any point on the aft verticalstabilizer decreases with increasing angle of attack. From Eq. (5),it is seen that the sidewash on the vertical stabilizer is proportionalto the product of the wing lift coefcient and the dimensionless ycoordinate Ny. As angleof attackis increased,thewingliftcoefcientincreases but Ny decreases.Thus, for a given wing conguration, themagnitude of the sidewash gradient does not increase linearly with


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1090 J. AIRCRAFT, VOL. 39, NO. 6: ENGINEERING NOTES

Fig. 4 Effect ofaircraft angleof attackon theyposition of the verticalstabilizer.

lift coefcient. In fact,as theangleof attackincreasesfrom zerolift,the magnitudeof thesidewashgradientwill, at somepoint,begin todecrease with increasing angle of attack. Ultimately, the sidewash

gradientbecomespositiveand destabilizingwhen theangle of attackis large enough to put the vertical stabilizer below the wingtip vor-tices. Normally the wing would stall well beforethis angle of attackwas reached.However, in poststall maneuvers, this can produce animportant destabilizing effect.

Conclusions

The sidewash gradient predicted from the present analyticalmodel increases rapidly with decreasing wing span. Thus, aircrafthaving wings of low aspect ratio produce large sidewash gradients.In addition, because the sidewash gradient is proportional to the

wing lift coefcient, slow ight with high-lift devices deployed canalso resultin a signicant increase in the sidewash gradient inducedon the vertical stabilizer.

The present model also points out that the sidewash gradient onthe vertical stabilizer increases rapidly with its distance aft of thewingtips. Because the yaw stability derivative increases with theproduct of the area of the vertical stabilizer and its distance aft ofthe center of gravity, an important consideration in tail design isthe tradeoff between the tail area and length required to attain the

desired level of yaw stability. As the tail is moved aft, the requiredsize of the verticalstabilizeris decreased,which reduces tail weightand drag. However, moving the tail aft increases the weight anddrag associated with the structure supportingthe tail. Thus, there isan optimum in the tradeoff between tail area and length. Becausethe sidewash gradient on a high vertical tail becomes more stabi-lizing with increasing tail length, sidewash moves this optimumin the direction of smaller tails placed farther aft of the center ofgravity. As the wingspan is decreased, this effect becomes moreimportant.However, moving the vertical stabilizer farther aft of thewingtips will aggravatethe destabilizingeffect of sidewash in post-stall maneuvers.

The proposed model does not account for the effects of the fuse-lage on the sidewash gradient. The fuselage can signicantly alter

the oweld induced by the wingtip vortices. Depending on fuse-lage geometry, the sidewashgradientinducedon a verticalstabilizerby the wingtip vortices canbe alteredby as much as 30% as a resultof ow around the fuselage.

References1 Phillips, W. F., Anderson, E. A., Jenkins, J. C., and Sunouchi, S.,

Estimating the Low-Speed Downwash Angle on an Aft Tail, Journal ofAircraft, Vol. 39, No. 4, 2002, pp. 600608.

2 McCormick, B. W., Downwash Angle, Aerodynamics, Aeronau-tics, and Flight Mechanics, 2nd ed., Wiley, New York, 1995, pp. 479482.

estimating the low-speed sidewash.pdf

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