VORTICAL FLOW COMPUTATIONS O N S W E P T FLEXIBLE W I N G S U S I N G NAVIER-STOKES EQUATIONS

Guru P. Guruswamy' Applied Computational Fluids Branch

NASA Ames Research Center, hloflett Field, California 94035

Abstract

A procedure to conple the Navier-Stokes sol~~t ions with modal structural equations of motion is presented for com-

The Lavier-Stokes How equations are solved by a finite-blfference scheme with dynamic grids. The coupled aeroelastic equa- tions of motion are solved using the linear-acceleration method. The configuration-adaptive dynamic grids are time-accurately generated using the acroelastically deformed shape of the wing. The calculations are compared with the experiment when available. Eflects of flexibility, sweep an- gle, and pitch rate are demonstratedfor !lows with vortices.

Introduction

uting aeroelaatie responses of swept flexible win 3

Flows with vortices play an important role in the de- sign and development of aircraft. In general, strong vor- tices form on a wing at large angles of attack. For wings with large sweep angles, strong vortices can even form at moderate angles of attack. Formation of vortices changes the aerodynamic load distribution on a wing. Vortices formed on aircraft have been known to cause several types of instabilities such as wing rock for rigid delta wings' and aeroelastic oscillations for highly swept flexible wings.' Such instabilities can severely impair the performance of an aircraft. On the other hand, vortical flows can also play a positive role in the design of an aircraft. Vortical flows associated with rapid, unsteady motions can increase the unsteady lift, which can be used for maneuvering the aircraft.'

To date, most of the calcelations for wings with vor- tical flows have been restricted to steady and unsteady computations on rigid wings. It is necessary to account for the wing flexibility in order to accurately compute flows. The aeroelastic deformation resulting from the flexibility of a wing can considerably change the nature of the flow. Strong interactions between the vortical flows and the struc- tures can lead to sustained aeroelastic oscillations for highly swept wings.' Also it is necessary to include the flexibility for proper correlations of cornpitted data with experiments, particularly with those obtained from flight tests. In order to account for the flexibility of B wing, it is necessary to solve the aerodynamic and aeroelxstic equations of motion simultaneously. I n this IT 'i. the flow is modeled using the Navicr-Stokes equations i , .d is coupled with the aeroelastic equations of motion.

Computational methods for analyzing wing problems involving steady flow3 liave advanced to the level of us- ing Navier-Stokes equations, and calculations on full air- craft configiirations are in progress.' In comparison, for unsteady flows associated with such moving components as oscillating wings, the most advanced codes in terms ofgeo- metric complexity use metbods based on the poten'tial flow

* Research Scientist, AIAA Associate Fellow

Copyrighi 0 1989 by the American Iniliiulc of Acroniulin and Aflronaulici. he. NO copyright ii arrencd in the

United Stain under Title 17. U.S. Code. The U.S. Covcm- men1 ha it royally-frrr license 10 exercse i l l rights under rhe EOPYrighl claimed herein for Governmenial Q W Q O S ~ .

Ail other rights are r ~ ~ m v e d by ihc copyright oumcr.

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theory.' Somc reasons fm the la i n tlie development of un- steady methods when compareckto steady methods are: 1 ) the complexitv of the physics associated with the move- ment of flexible components, 2) the complexity in model- ing the flow because of the moving grids, and 3) the lack of development of fad time-accurate mrthods. One of the main reasons lor .developing efficient unsteady methods is to understand the iriteraction of complex Rows, including vortices, with such moving strxctural components as wings of an aircraft.

The most advanced codes used for seroelastic annly- sea, such as X'l'RAN3ST.'(the Air Force/NASA code for transonic aeroelastic analysis of aircraft), use the tran- sonic small perturbation ?quation. The NASA Ames Re- search Center's version of XTRAN3S (ATRANS) , is be- ing used for generic research in unsteady aerodynamics and aeroelasticity of almost full aircraft eonfigurntione.a Though codes based on potential flow theory give practical results, they cannot he used lor flows involving vortices and separation primarily because the potential flow theory sa- sumes tha t the flow is irrotational and inviscid. Given the availabilty of new efficient numerical techniques and faster computer^,^ it is appropriate to consider Euler/Navier- Stokes equations for aeroelastic applications.

Codes based on the Euler/Navier-Stokes equations have already been applied for practical problems involving steady flows. .Generic flow-solvers such as ARC3D," a three- dimensional Etrler/Navier-Stokes flow-solver written for gen- eral coordinates, have been used for some scientific inves- tigations. Such generic solvers have resulted in such usefiil codes as the TNS Transonic Navier-Stokes) code' which

been successfully used to compute complex separated flows on full aircraft configurations.

At Amcs a code, ENSAERO," is being devrloped for computing the unsteady aerodynamics and aeroelastic- ity of aircrkft by using the Euler/Navier-Stokes eqtiations. The capnbility of ENSAEKO to compute aeroelastic re- sponses by simultaneously integratin tlie Euler equations and the modal structural equations ofmotion, using aeroe- lastically adaptive dynamic grids, has been demonstrated." The Row is solved by time-accurate, finite-difference schemes baaed on the Benm-Warming algorithm. Recenlly the ca- pnbility of ENSAERO was extended to couple solutions from the Navier-Stokes equations with the modal struc- tural equations. The capability of ENSAERO to aumeri- cally simulate vortical flows ?n flexible rectangular wjngs is demonstrated in Ref. 12.

In this work, the capability of ENSAERO is extended to model the Eukr/Navier-Stokes e hations for swept wings. Computaiions are made for vorticaq-fldw conditions about flexible swept wings and the results are compared with available experiments. The focmation of vortices and their dynamics on flexible wings is demonstrated, as are the ef- fects of flexibility, sweep angle, and pitch rate for wings in unsteady ramp motion with vortical flows.

uses a zonal-grid sc 6 eme for complex geometries. TNS has

Aerodynamics Equations and Approximations

The strong, cnnsrrvation-law form of the Navier-Stokes equations is used for shock-capturing purposes. The equa- tions in Carterian coordinates in nondimensional form can be written as13

a Q a E a F ac a E , aF" BG, at a. ay % = K + ~ + F - + - + - +

s = J-'

where

Q =

~ 0 r (C + c: +c:)u< + (P/3)(C.U( + cvvc + c.woc.

r(C: + c: + C : ) q + ( P / 3 ) ( c a q + C U T + C.wc)Cv

r(S: + e: + ZZ)WC + (@/3)(C.U< + C V Y + c.woc. {(e: + c: + c:)[o.S@(u' + uz + WZ)<

+ pPr-'(-r - 1)- ' (a2)(1 + ( ~ / 3 ) ( c . u + Cvu + (.w)

- x K.U< + C,V( + C Z W O }

with

rzz = A(ua + uy + w.) + 2pu,

rvl = A(% + v, +w') + Zpv,

r,, = A(u= + v1 + w') t 2pw.

rrv = ry* = P(U, + v.)

7.2 = 7.. = r(u. + w.)

ryz = Tzv = r(v. + w,)

0, = 7d'r- 'a=er +UT.. + vre9 + wr.. By = yuPr-'B,er + urrr + urvy + wrri p. = 7.Pr-'azcr + UT.. + vr., + wr**

e, = ep-' - O.S(U' +I? + wz)

(3)

The Cartesian velocity coniponents, u, u, and w are nondimensionalized hy 4- (the free-stream s p u d of sound); density p is nondimensionalized by pe and the total en- ergy per unit volume, e, is nondimensionalized by p-aL. Pressure can he found from the ideal gm law M

(4) p = (7 - 1)[e - O.SP(U' + vz t wZ)/

and 7 is the ratio of the specific heats. Also, IC is the corf- ficient of thermal conductivity; p is the dynamic viscosity; and A , the second coefficient of viscosity from Stokes' hy-

othesis is, - 2 / 3 p . The Reynolds number is Re and the brandtl number is Pr.

To enhance nummcal accuracy ani1 dficimcy and to handle boundary conditions more eMily, the governing equa- tions are transformed from Cartesian coordinates to gen- eral curvilinear coordinates where

r = t

The resulting transformd equations *re not much more complicated than the original Cartesian set and can be written in nondimensional lorrn as

where'indicates the transformed quantities. In order to s o h Eq. (R for the full flow-field, a wry

Reynolds-number Rows, the viscous effects are confined to a thin layer near rigid boundaries. Because of computer storage and speed limitations, in most practical cases there are only enough rid points to resolve the gradients normal to the body by cfuustering the grid in the normal direction. Resolution along the body is similar to that used in invis- cid flow. As a result, even though the full derivatives are retained in the equations, the gradients along the body are not resolved unless the streamwise and eireumferen- tial grid spacings are sufficiently small. lience, fur many Navier-Stokes computations, the viscous derivatives along the body are dropped. This leads to the thin-layer Navier- Stokes equations." In this paper, the thin-layer Navier- Stokes form of Eq. (6) is used lor modeling the flow.

The thin-layer model requires a boundary-layer-type coordinate system. In our case, the ( and 7 directions are along the streamwise and spanwise directions of the wing, respectively, and the viscous derivatives msociated with these directions are dropped, whereaa the terms in {, normal to the body, are retained and the body surface is mappedonto a constant surface. Thus, Eq. ( 6 ) simplifies to

fine'grid is required throug b out the Row-field. III high-

a,Q +ac& + a,P +a& = Re-'a(S (7)

where

Aeroelastic Equat ions of Mot ion

The governing aeroeiastic equations of motion of a flexible win are solved using the Rayleigh-Ritz method.18 In this metfod, the resulting aerodaatic displacements at any time are expmaed as a lunction of a finite set of as- sumed modes. The contribution of each msumcd mode to the total motion is derived by Lyrange's equation. Fur- thermore, it is assumed that the delormation of the con- tinuous wing structure can be represented by deflections at a set of discrete points. This assumption facilitates the use of discrete structural data, such as the modal vector, the modal stiffnesa matrix, and the modal mms matrix. These can be generated from a finite-element analysis or from experimental influence-coefficient measurements. In this study, the finite-element method is used to obtain the modal data.

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It is assumed that the deformed shape of the wing can he represented by a set of discrete displacements at selected nodes. From the modal analysis, the displacement vector { d } can be expressed aa

v Id) = 141tq) ( 8 )

where [$I is the modal matrix and { q } is the generalized displacement vector. The final matrix form of the aerorlastic equations of mo- tion is

lAfl{4} + ICltd} + IKIIqk = IF} (9)

where [MI, [GI, and [K] are the modal mass, damping, and stiffness matrices, respectively; (F} is the aerody- namic force vector defined as ($)pU&[+JT[d][AC,}; and [A\ is the diagonal area matrix of the aerodynamic control points.

The aeroelastic equation of motion, Eq. (9), IS solved hy a numerical integratioir technique based on the linear- acceleration method."

Aeroelastic-Conflguration-Adaptive Grids

One of the major problrms in computational aero- dynamics using the Euler equations lies in the area of grid generation. For the case of steady flows, advanced techniques srich aa zonal grids' are bring wed. Grid- generation techniques for aeroelastic calculations that in- volve nloving components are in an early stage of develop- ment. 'The effects of the aeroelastie-configuration-adaptive dynamic grids on the stability and accuracy of the numer- ical schemes are yet to he studied in detail.

nique for aeroelastic applications. The scheme satisfies the general requirements of a grid required for implicit finite- difference schemes used in the Some of the requirements are: 1) the p! IinFs in-t normal to the wing surface in thr chor wise directron, 2 ) the grid cells are smoothly stretched away from the wing surface, 3)the outer boundaries are located far from the wing to mini- mize the effect of boundary rrflections, and 4) the grids adapt to the rleformed wing position at each time step. The type of grid used is a C-H grid. The method by which the grid is generated at each time step, based on the aeroe- lastic position of the wing computed by using Eq. (9), IS described.

At the end of each time step, the deformrd shape of the a ing is computed usin Eq (9) The ( and 7 rid dis tributions on the grid surface 'corr;sponding to t t e wing surface ({ grid index = 1) are obtained from previotisly assumed distributions. These distributions are selected to satisfy the general requirements of a grid for accurate com- putations. In this work, the grid in the t dirtction is se- ected so that the grid spacing is close on the wing and stretches exponentially to outer boundaries. The grid near the nose is made finer than the rent of the wing to aecu- rately model the nose geometry. In the spanwise direction, a uniformly distributed grid spacing is used on the wing. To model the wingtip a finer grid spacing is used. Away from the wingtip, the 7 grid spacing stretches exponen- tially. The { grid spacing is computed at each time step using the deformed shape of the wing computed by using Eq. (9). The { grid lines start normal to the surface in the chordwise directions and their spacing stretches expo- nentially to a fixed outer boundary. To prevent the outer boundaries from moving, the grid is sheared in the ( direc- tion. The mrtrics required in the computational domain are computed using the following relations

This work developed an analytical grid-generation tech-

E t = -z& - YrFv - .&

c, = -z& - Y?CU ~ 4 7, = -Xy7= - y.qu - :7vz (10)

The grid velocities zr, y,, and L~ required in Eq. (IO) are computed using the grids at new and old time levels. This adapl.ive grid generation scheme is incorporated in ENSAERO. The abi1it.v of this rcbeme to compute accurate aeroelastic responses has been deniunstrated i n Ref. 1 I .

Results

Steady Pressures

The aerodynamic and aeroelilstic solution procedims presented iii the earlier sections are incorporated in EN- SAERO. The current version of ENSAERO can compute flowsuver flexible wings using either the Euleror the Navier- Stokes equations. The same diagonal form of the Bearn- Warming scheme is used to solve both equation sets. The resulting computer code is validated with experimental data for steady and unsteady Rows over rectangular wings by using both the Euler and Navier-Stokes Equations. De- tailed resnlts are presented in Refs. ll and 12. In this pa- per, computations are made on swept rectangular wings. All computations are made using the Navier-Stokes equa- tions for laminar flows wilh a 91 x 20 x 40 grid.

Steady-state compotat.ions are made on a swrpt rect- angular wing of aspect ratio 3.0 with NACA 0012 airfoil sections to validate the code. The leading-edge sweep angle is 20". For this wing, steady experiinental data are avail- able in Ref. 18. Figure 1 shows the computed and the measured steady pressures at A l , = 0.695, a = Z O O , and Re, (Reynolds number based on the root chord) = 5.95 x lo4. The computed steady pressiires compare well with the experimental data for all span s l d o n s . Slight discrep- ancies near the leading edge of the 25% semispan station can,be attributed to the wind tunnel wall effects. The cur- rent version of ENSAERO does not model the wind tunnel walls.

In Fig. 2, the computed steady pressures at AI, = 0.836, a = Z O O , snd Re, = 8.0 x IO' ate shown. As reported in Ref. 19, the wind tunnel wall effects are pro- nounced for this case. As a result, computations are not compared with the experiment, however, the results are compared with computations perfmnied using the TNS code.19 In Fig. 2, the steady pressures from TNS are shown for the 50% semispan station. The comparison between the present results and those from TNS is favorable. He- cause both computations use the Beam-Warming scheme, the slight differences in the results can he attributed to the difference in the numher of grid points. The number of grid points used in the cornputations and in Ref. 19 are 72,800 and 165,321, respectively. Because of the coarseness of the present rid, the shock is slightly smeared.

To femonstrate the limitations of potential codes, com- putations are made using ATRAN3S, which uses the classi- cal transonic small perturbakion (TSP) equation, and EN- SAERO using the Euler option. Figure 3 shows steady pressures from both ENSAERO (Euler) and ATRAN3S (inviscid) at A4, = 0.836 and a = 2.0°. The results from the transonic small pertwhation theory show significantly stronger shock waves. Attempts have been made elsewhere to apply corrections for vorticity and entropy, to improve the TSP solutions. Given the present computer capabili- ties, it is efficient to use more exact equations as warranteil by the situation.

For any givbn flow conditions it is difficult to esti- mate the magnitude of viscous effects. To accurately com- pute viscous effects the Navier-Stokes equations need to he solved. However, because the Navier-Stokes equations are computationally more expensive, they should be used only

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increased by increasing the pitch rates. Such increases i n the dynamic lift can be used in manerivering an aircraft.' Because the dynamic lift is an unsteady phenomenon and it is associated with the presenceof vortices, it is important to model it accurately. The present computational tool can he used to accurately compute the dpnaniir lift o v w wings, including the wing flexibility. To illustrate this further, computations are made for the swept flrxible wing for a pitch rate A = U.05, and the resnlts are compared witit those obtained earlier for A = 0.10. The unsteady lifts are plotted against the time in degrees in Fig. 14. The increase in the dynamic lift at higher pitch rate A = 0.10 can be seen in Fig. 14.

The above computations are made on both CRAY- XMP48 and CRAY-2 cnmpnters avaliahle at Ames. The current version of ENSAERO runs a t 60 MFLOPS(mil1ion floating point operations per second). The central process- ing unit time per time step per grid point is 31 x IO-' see and the memory required per grid point is 66 words.

' i d

as individual cases warrant. In order to accomplish this efficiently, an important option of switching between the Euler and Navier-Stokes equations is provided in the input da ta of ENSAERO. Figure 4 shows the steady pressures computed using both the Eider and Nsvier-Stokes options of ENSAERO at iw, = 0.836, (I = 2.0°, and Re, = 8.0 x 10'. Even for these moderate flow conditions there are significant viscous effects. As expected, the viscous effects have moved the shock backward and weakened it.

Unsteady Computations on Wings in Ramp Motions

The unsteady and aeroelastic computations made us- ing ENSAERO are validated with experiments for rect- angular wings as reported iii Refs. I1 and 12. In this section, the unsteady computations for rigid and flexible swept wings are made when wings are undergoing ramp- type motion from 0 to 20'. Both wings are of an aspect ratio of 4.0 with NACA 0015 airfoil sections, and their leading-edge sweep angles are 30'. The structurnl proper- ties of the fiexible wing arc modeled using four modes. The mode shapes and frequencies of the first four modes com- puted using the finitedement method are shown in Fig. 5.

Figure 6 shows unsteady pressures a t fonr span sta- tions for the rigid wing at M, = 0.50, Re, = 2.0 x IOs, and a pitch rate A = 0.10. The pitch rate A is defined as ac/ lJ , . The unsteady computations are started from the converged steady state solution a t a = 0.0". The results in Fig. 6 show the dynamics of vortex development. As the wing is pitched toward the maximum angle of attack, a rapid pressure decrease (increasing negative pressure) oc- curs over the leading edge. The leading-edge vortex starts moving downstream rapidly afler the wing has reached an angle of at1ac.k of ahout 18.0". A similar phenomenon h w been observed in an experimental study at a low Mach number."

The same computations made on the rigid wing are also made on the flexible wing. Figure 7 shows results for the flexible wing, which has B pattern siuiilar to the one shown in Fig. 6 for the rigid wing. However, the leading- edge vortex moves at a slightly faster rate than it does on the rigid wing. This is shown in Fig. 8 by plotting the unsteady pressures for both rigid and flexible wings at the nondimensional time of 7.0. The vortex core on the flexible wing is farther downstream then the vortex core on the rigid win Figure 9 shows the unsteady sectional- lift coefficients &ted against the time for both the rigid wing and the flexible wing. At a given time, the lift over the flexible wing is higher than the lift over the ri id wing. This is due to the incrcaqe in the an le of attacf caused by the wing flexibility. The deforme! shape of the wing and the velocity contours i n the free-stream direction are shown in Fig. 10. The presence of the leading-edge vortex can also be seen in Fig. 10.

Computations are made for the unswept flexible wing a t the same flow conditions as the swept flexible wing to study the effect of the sweep angle. The unsteady pressures for the unswept wing are shown in Fig. 11. The flow pattern for the unswept wing is highly two-dimensional when compared to the swept wing shown in Fig. 8. The sweep effects on the unsteady pressures and lift coefficients are illustrated in Fig. 12 and 13, respectively. In both figures, the plots for the unswept wing do not differ much from section to section when compared to the plot for the swept wing. The unsteady pressures in Fig. 12, for T = 7.0, show that the vortex core for the unswept wing has convected downstream with almost the same speed for all four span sections; for the swept wing the vortex core has convected faster toward.the tip.

One area where vdrtieal flows play an importnnt role is in increasing the dynamic lift. From earlier studies3 on airfoils, it has been observed that the dynamic lift can be

Conclusions

The following conclusions can be made based on the present work. 1. A time-accurate numerical procedure has been devel- oped for computing the unsteady flows on swept flexible wings using the Navier-Stokes equations with configuration- adaptive dynamic grids. 2. The compnted results c o n are well with the avajlable experimental data for steady t?ows. 3. Coupling solutions from the Navier-Stokes equations with the modal structural equations of motion have been demonstrated for swept rectangular wings. 4. Unsteady computations lor flows involving vortices are madeon a flexible wing, and the effects offlcxihility, sweep angle, and pitch rate on vortical flows have been demon- strated. 5. Additional work needs to be completed for turbulent flows.

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References

Nquyen, L. E., Yip, L. P., and Chambers, J. R., "Self- Induced Wing Rock Oscillations of Slender Delta Wings," AIAA Paper 81-1883, hug. 1981. ' Dobbs, S. K. and Miller, G. D., "Self-Induced Oseil- lation Wind Tunnel Test of a Variable Sweep Wing," AIAA Paper 85.0739, Apr. 1985. Mnbey, D. G., "On theProspects for Increasing Dy- namic Lift," Aeronaut. J . Roy. Aeronaut. SOC. Mar. 1988. DD. 95.105. .. ' Guruswamy, G. P., Goorjian, P. M., Ide, H. and Miller, G. I). "Transonic Aeroelastic Analysis of the B-1 Wing," J . of Aircraft Vol. 23, No. 7, July 1986, pp. 547-553. Flores, J . , Chaderjian, N., and Sorenson, R. "The Numerical Simulation of Transonic Separated Flow About the Complete F-16A," AIAA paper 88-2506, Jiine 1988. ' Guruswamy, G. P. and Tu, E. L,"Effects of Symmet- ric and Asymmetric Modes on Transonic Aeroelas- tic Characteristics of Full-Span Wing-Body Configu- ration," AIAA Paper 88-2308, Apr. 1988. ' Borland, C. J. and Rizretta, D. P., ''Transonic Un- steady Aerodynamics for Aeroelastic Applications. Vol- ume I - Technical Development Summary," AFWAL- TR-80-3107, June 1982.

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Huttsell , L.J. and Cooley, D.E. “The Background and Status of the Joint Air Force/NASA Transonic Un- steady Aerodynamic Program(XTRAN3S),” AFWAL- Th.I-86-154-FIBRC, Jan. 1986.

e Peterson, V. L. and Ballhaus, W. F., “History of the Numerical Aerodynamic Simulation Program,” Coit. ference on Super Computing in Aerospace, NASA CP- 2454, 1987. Pulliam, T. H., and Steger. J. L.“Impli~it Finite- DiKerence Simiilations of Three Dimensional Compress. ible Flow,” AIAA 3. , Vol. 18, No. 1, Jan. 1980. Guruawamy, G . P., “Time-Accurate Unsteady Aero- dynamic and Aeroelastic Calculations of Wings Using Euler Equations,” AIAA Paper 88-2281, Apr. 1988.

I’ Guruswamy, G . P., “Numerical Simulation of Vortical Flows on Flexible Wings,” AIAA Paper 89-0537, Jan. 1989. ’’ Peyret, R. and Viviand, H., “Computation of Viscous Compressible Flows Based on Navier-Stokes Equations,” AGARD AG-212, 1975.

l 4 Pulliam, T. H., and Chanssre, D. S., “A Diagonal Form of an Implicit Approximate Factorization Algo- rithm,” J. of Comp. Phys., Vol. 39, No. 2, Feb. 1981, pp. 347-363.

Beam, R. and Warming, R. F.. “An Implicit Finite- Difference Algorithm for Hyperbolic Systems in Con- servation Law Form,”J. of Comp. Phys., Vol. 22, No. 9, Sept. 1976, pp. 87-110. Bisplinghoff, R. L., Ashley, H. and €Inifman, R. L., Aeroelasticity, Addison Wesley, Nov. 1957.

I’ Guruswamy, P. and Yang, T. Y., “Aeroelastic Time Response Analysis of Thin Airfoils by Transonic Code LTRAN2,” Computers and Fluids, Vol. 9, No. 4, Dec. 1980, pp. 409-425. Lockman, W. K., and Seegmiller, H. L., “An Experi- mental Investigation of the Subcritical and Supercriti- cal Flow About a Swept Semispan Wing,” NASA ThI- 84367, 1983.

le Kaynak, U., Hold, T . L., and Cantwell, H. J . , “Com- putations of Transonic Separated Wing Flows Using an l?iiler/Navier-Stokes Zonal Appronch,” NASA Tbl- 88311, 1986. Robinson, M . C. and Wissler, J. B., ‘‘ IJnsteadp Stir- face Pressure Measurements on a Pitching Rectangu- lar Wing,” A I A A Paper 88.0328, Jan. 1988.

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SWEPT WING, NACA 0012, AR = 3.0. TR - 1.0 SWEEP ANGLE = 20" ,

SWEPT WING, NACA 0012 AR = 3.0. TR = 1.0

SWEEP ANGLE = 20" 78% SEMISPAN

78% SEMISPAN M, = 0.826 M, ~0.695. o! = 2.0" Rec - 5.95 x lo6

oi - 2.0"

- ENSAERO -1.0, INAVIER-STOKES)

Fig. 1 Comparison between computed and measured steady pressures for 20' swept wing.

1 4

ENSAERO INAVIER-STOKES1

-1.0, I - A TNS INAVIERSTOKESI

1.0 - 0 .2 .4 .6 .8 1.0

x/c

Fig. 2 Comparison of computed steady pressures between ENSAERO and TNS codes for 20° swept wing.

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SWEPT WING, NACA 0012 AR - 3.0, TR - 1.0

SWEEP ANGLE - 20" M, - 0.826 - -+., OL - 2.0"

I . 5 ~

.. ........

\ ENSAERO (EULER)

UPPER LOWER

ROOT ATRANS (TSP) ............... UPPER -1.0, ... J ---_ ............. LOWER

..-

W'

1.0 J 1 0 2 .4 .6 .8 1.0

XIC

Fig. 3 Comparison of steady pressures between Euler and tran.wnic small perturbation calculations.

SWEPT WING AR - 3.0, TR - 1.0

SWEEP ANGLE - 20"

M, - 0.826 a = 2.0"

ENSAERO

~ NAVIERSTOKES

EULER

-1 - 4 -.5 - > 0 0 -

v1 .5 - z l 0 .z .4 .6 .E 1:o

XIC

Fig. 4 Comparison of steady pressures between Navier- Stokes and Euler calculations.

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1 .o MODE 1 127 HI)

1 .o MODE 2 184 H d

1 .o MODE 3 (150 Hzl

1 .o MODE 4 1260 Hz) i L,

Fig. 5 Mode shapes of the rectangular wing.

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RIGID SWEPT WING IN RAMP MOTION NACA 0015. AR = 4.0, TR - 1.0, SWEEP ANGLE - 30"

M, - 0.5, A - 0.1 = 2 x 106

RAMP MOTION

75% SEMISPAN

FLEXIBLE SWEPT WING, NACA 0015. AR = 4.0, TR = 1.0. SWEEP ANGLE M, - 0.5. A - 0.1

Rec - 2 x lo8

30"

75% SEMISPAN

RAMP MOTION

1 - h- 14 5.00 0----- 18 8.25

> 0--- 20 8.25

0-.- 20 9.50

J

a. -3 * -2 @ .....,....... 20 7 .w

v 2 -1

g o 2 1 = 2-

Fig. 6 Unsteady prasures on the swept rigid wing in ramp motion.

= 2- 0 .z .4 .e .6 1.0

x/c

Fig. 7 Unsteady pressures on the swept flexible wing in ramp motion.

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SWEPT WING IN RAMP MOTION

2 -

1 - -1

NACA 0015. AR = 4.0, TR - 1.0. SWEEP ANGLE - 30”

... M, = O S , A - 0.1

R - 2.0 x lo6 0C

. . . . . . . . . .

RIGID

FLEXIBLE ._._ -____

I = 7.0 1

-RIGID ----- FLEXIBLE

. . . . . . . . .. ..._.._

& - 3 b 0 -2

. . ... _ _ > ..__ ....__ 2 -1

e o 3 ’ v)

2 0 .2 .4 .B .8 1.0 x/c

Fig. 8 Comparison of unsteady pressuns between rigid and flexible swept wings.

SWEPT WING IN RAMP MOTION NACA 0015, AR = 4.0. TR - 1.0, SWEEP ANGLE - 30’ 1 75% SEMISPAN

1

Ld

0 2.5 5.0 7.5 10.0 12.5 TIME

Fig. 9 Comparison of unsteady lift coefficients between rigid and flexible swept wings.

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FLEXIBLE SWEPT WING NACA W15. AR - 4.0. TR = 1.0 SWEEP ANGLE = 30"

......

r = 7.0, RAMP ANGLE - 20"

Fig. 10 Velocity contours on the flexible swept wing in ramp motion. . . .

UNSWEPT WING IN RAMP MOTION NACA 0015, AR - 4.0, TR - 1.0

M, - 0.5, A - 0.1

Rec - 2 x 10' 75% SEMISPAN

V

a r J

@-- 14 5.00 .-. m----- 18 6.25

3- 0 .2 .4 ..6 .8 1.0

x/c

c/ Fig. 11 Unsteady pressures on the w e p t flexible wing in ramp motion.

FLEXIELE WINGS IN RAMP MOTION NACA 0015, AR = 4.0. TR = 1.0

M, - O S , A = 0.1

Ret- 2 X IO6

T - 7.0

75% SEMISPAN

SWEEP ANGLE - 0.0" ....... 30.0'

> 0 -1 a u o

I

0 .2 . .4 .6 .8 1.0 x/c

= 2 J

Fig. 12 Comparison of unsteady pressures between swept and unswept flexible wings.

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FLEXleLE WINGS IN RAMP MOTION NACA 0015. AR - 4.0, TR - 1.0

75% SEMISPAN M, - 0.5, A - 0.1

Rec - 2 x 106

..

1

SWEEP ANGLE - 0.0" --- 30.0'

\/'' ..' ROO?

O 6 30 TIME

Fig. 13 Comparison of unsteady lifts between swept and unswept flexible wings.

SWEPT FLEXIELE WINGS IN RAMP MOTION NACA M15. AR - 4.0, TR - 1.0

WEEP ANGLE M. I 0.6 R.

1 _.-.. /" VZ5%, ...

2 1

I fi

-0.10 ----0.06

0 10 20 30 TIME, d.g

Fig. 14 Effects of pitch rate on unsteady lift coefficients for the swept wing.

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