ABSTRACT A nonlinear aeroelastic analysis system is developed using coupled numerical techniques of computational fluid dynamics (CFD) and computational structural dynamics (CSD). For efficient computation, the structural dynamic solver of the developed system is based on the modal approach. The fluid dynamic solver incorporates the modified transonic small-disturbance (TSD), Euler, and Navier-Stokes codes. In particular, the Navier-Stokes solver is employing two different turbulent models, which are the Baldwin-Lomax and Wilcox's k-ω turbulent models. Detailed aeroelastic simulation results in transonic and supersonic regimes are provided to show the aeroelastic oscillation of several wings and full aircraft models. In addition, a discussion of how to apply the proposed technology to the actual design process of high-performance aircraft is presented. INTRODUCTION Modern aircrafts are usually designed to cover a wide flight envelope, and they experience intense flow-characteristic changes while they are passing through the transonic flow region. These anomalous changes of flow can cause several types of aeroelastic problems; flutter, divergence, buffeting, limit cycle oscillations, etc. Even though all of these should be considered for aircraft design, flutter (growing oscillation of wing leading to large amplitudes and stresses) is the most dangerous problem, because it can result in total structural failure in just a few seconds. Moreover, the flutter instability is usually aggravated in the transonic and low supersonic regions and it is closely related with the unstable movement of shock waves. Hence, it is important to consider the effects of normal shock waves during an analysis of flutter.

In this study, a nonlinear aeroelastic analysis system that can be applied to a full-configuration aircraft has been developed. In this program, the unsteady aerodynamic load can be determined using the transonic small disturbance (TSD), Euler, or Navier-Stokes (NS) solvers. Selection of unsteady aerodynamic solvers depends on the purpose of a given analysis. If a precise examination is required and viscous effect is not negligible, then the Navier-Stokes code with the turbulent model should be applied to the aeroelastic analysis. Buzz, stall flutter, and buffeting problems are examples of such cases. However, because of the excessive amount of computing time and weak numerical robustness, it is nearly impossible to apply the Navier-Stokes code to a practical case involving an oscillating control-surface, a complex geometry, a whole body configuration, and other factors. In practical cases, on the other hand, the TSD has several advantages: The TSD scheme can easily reflect the aerodynamic effects of control-surface motion, because the aerodynamic grids are actually fixed and the motions are expressed by the change of slope. This peculiar feature of the TSD grids prevents a large skewness problem when the control-surface is rotated independently with other wing parts. This scheme also requires less computing time and memory capacity comparing the Euler and NS schemes. The TSD code is widely recognized as one of the most efficient and robust tools among conventional CFD-based approaches.

In this study, two separate approaches are presented. The first is a practical aeroelastic analysis for an actual aircraft model using the TSD solver; the second is a highly precise investigation of aeroelastic phenomena using the Naiver-Stokes solver with an accurate turbulent model. Detailed aeroelastic simulation results for the transonic and supersonic flow regions are also presented.

6th Korea-Japan Symposium of Frontiers in Vibration Science and TechnologyIbaraki University, July 14-15, 2005

Vibration and Aeroelastic Characteristics of Advanced Aircrafts in Transonic and Supersonic Region

In Lee, Hyuk-Jun Kwon, Jong-Yun Kim, and Jang-Hyuk Kwon

Div. of Aerospace Eng., Dept. of Mechanical Eng., Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

inlee@asdl.kaist.ac.kr

TRANSONIC AEROELASTIC SIMULATION Practical Applications of TSD Aeroelastic Solver Although the Navier-Stokes equations are the most accurate, many flow features do not depend on the viscous and turbulent effects. For example, if the thickness of wing section is small and there is no boundary layer separation, then the viscosity has little effect on the flow fields. If the viscous terms are removed from the Navier-Stokes equations, the equations become Euler equations. Furthermore, if the flow around aircraft wing is irrotational and the perturbation is small, then the TSD theory can be applied for the unsteady aerodynamic analysis. The strengths of the TSD code is that it can be applied to a realistic aircraft model with consideration of all wings and control surfaces. It would be impossible to analyze such a realistic aircraft with the Euler and Navier-Stokes aeroelastic programs, because of the excessive computing time, low robustness, and difficult mesh treatment associated with those programs.

A conservative form of the TSD equation is as follows:

0 31 2 0f ff ft x y z

∂ ∂∂ ∂+ + + =

∂ ∂ ∂ ∂ (1)

where 0 t xf A Bφ φ= − −

2 21 x x yf E F Gφ φ φ= + +

( )2 1 ,y xf Hφ φ= + 3 zf φ=

The above equations are expressed in a physical coordinate system (x, y, z, t). The term φ is the disturbance velocity potential. The definitions of the coefficients A, B, E, F, G, and H can be found in Ref. [1].

The aeroelastic oscillations of an aircraft can be determined by solving the Equation of Motion which is written in matrix form as follows:

}{Q}]{[K}q]{[C}q]{[M ggg )()()()( ttqtt =++ &&& (2)

where {q(t)}T=[q(t)1,q(t)2,…,q(t)n] is the generalized displacement vector and [Mg], [Cg], and [Kg] express the generalized mass, damping, and stiffness matrices respectively. {Q(t)} is the generalized aerodynamic forces and it is calculated by solving the Equation (1). Ordinary differential equations (ODEs), such as Equation (2) can be reduced to the state vector forms for efficient numerical calculations. For the nonlinear structural system, a typical numerical scheme such as Runge-Kutta method is used to solve the equation of motion. In this study, a fourth-order Runge-Kutta method is used and the details of the solution schemes can be found in Ref. [1].

In the present study, aeroelastic analyses are performed for the supersonic jet trainer shown in Figure 1. Figure 2 shows a full-span grid for the jet trainer. The

Figure 1: Supersonic jet trainer

Figure 2: Aeroelastic grid system for the realisticaircraft model

(a) Upper surface

(b) Lower surface

Figure 3: Steady pressure distributions of the aircraftmodel at M=0.9 and α=0o

airfoil of the main wing has the shape of NACA 64 series. Each of the horizontal and vertical tails has a biconvex airfoil.

Steady aerodynamic pressure was computed by the TSD equation. Figure 3 shows the steady pressure contours of upper and lower surfaces at M = 0.9 and the angle of attack, α = 0°. As shown here, because of the airfoil camber, the normal shock waves on the upper wing surface are stronger than those on the lower wing surface.

Aeroelastic solutions are obtained by solving the aeroelastic equation of motion and they can be represented by the superposition of mode shapes. The mode shapes and natural frequencies of the aircraft model have been obtained via free vibration analysis ,using MSC NASTRAN, and 28 elastic modes are applied to the aeroelastic analysis.

Generally, multi-disciplinary problems, including aeroelastic analysis, require a data transfer stage between mismatched points, such as the structural grid and the aerodynamic grid because those points have been subject to different engineering considerations. For example, the aerodynamic load is usually subjected to the external surface, whereas the load-carrying components are placed inside the wing or fuselage. This gives rise to the necessity of data transfer between the two different systems. In the present aeroelastic analysis, the displacements of the structural grid are interpolated to the aerodynamic grid based on the Infinite Plate Spline (IPS) method. Figure 4 shows the interpolated

mode shapes of the first symmetric and anti-symmetric bending modes. If there is a discontinuity, such as flap, then the wing structure is divided into parts: the control surface and the remaining part of the wing. Therefore, each part must be transferred independently and then superposed.

In the time-domain approach, the aeroelastic instabilities can be predicted by simulating whether the response is decaying or diverging. Figure 5 shows aeroelastic responses above and below the flutter speed. A flutter boundary exists between the two dynamic pressures corresponding to these responses. It is sometimes difficult to obtain a neutral response which indicates flutter boundary. Hence, the aeroelastic analysis should be repeated several times or else an additional signal processing technique will be required to estimate the neutral point. In this study, damping ratio and frequency are calculated using the Moving Block Method (MBM), and the flutter boundaries are predicted by an interpolation of damping ratios.

Precise Aeroelastic Analysis Using an NS Solver The most precise aerodynamic governing equations are known as Navier-Stokes equations, and they are incorporated into various schemes, such as the Direct Numerical Simulation (DNS), the Large Eddy Simulation (LES), and the Reynolds Averaged Navier-Stokes (RANS), among others. Of these schemes, the DNS yields the most exact solutions for unsteady aerodynamics; however, it is premature to apply this

(a) Symmetric bending mode

(b) Anti-symmetric bending mode Figure 4: Structural mode shapes of the aircraft model

4th

mod

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(b) Aeroelastic responses over flutter speed

Figure 5: Aeroelastic simulation results at Mach 0.9

Time (sec)

1stM

ode

0 0.05 0.1 0.15 0.2-0.03

-0.02

-0.01

0

0.01

0.02

0.03FSI = 0.283FSI = 0.298FSI = 0.308

Figure 8: Decaying and diverging responses accordingto the increase of FSI at Mach 0.96

Time (sec)

1stM

ode

0 0.05 0.1 0.15 0.2-0.03

-0.02

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0

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0.03FSI = 0.452FSI = 0.476FSI = 0.498

Figure 9: Decaying and diverging responses accordingto the increase at Mach 1.141

Time (sec)

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0 0.05 0.1 0.15 0.2-0.03

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Figure 10: Effects of the applied turbulent model influtter response at Mach 1.141 and FSI=0.498

X

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Figure 6: C-H type grid structure for AGARD 445.6wing

1.221

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Figure 7: Steady aerodynamic analysis results usingNavier-Stokes solver with k-ω turbulent model forAGARD 445.6 wing

scheme to the 3-dimensional aeroelastic problems, because the current computers are not as fast as needed.

Hence, the NS equations should be applied to the aeroelastic problem after simplifying with empirical information. A traditional way for the simplification is representing the instantaneous flow quantities by the sum of a mean value and a time dependent fluctuating value. Through this process, the NS equations become RANS equations, and the details of the RANS equations can be found in Ref. [2, 3].

In this study, a time-accurate Euler/Navier-Stokes aeroelastic solver has been developed using KFLOW [2], which embeds Baldwin-Lomax and k-ω turbulent models. The developed program adopts various numerical schemes to guarantee the accuracy and stability of the aeroelastic solution: an improved Harten-Lax-van Leer-Einfeldt (HLLE+) is applied to simultaneously satisfy the robustness and accuracy of the solution in a supersonic viscous flow, and Van Leer MUSCL provides high-resolution results near shock waves by means of higher-order extrapolation of flux at the cell boundaries. A dual-time stepping method based on the diagonalized-ADI algorithm is applied to improve the time accuracy. A second-order staggered algorithm is also employed to reduce the lagging errors between the fluid and structural solvers. The largest

Mach Number

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

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peed

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Viscous (Coarse, k-ω model)Viscous (Fine, B-L model)

Gordnier (B-L model)Lee-Rausch (B-L model)

Viscous (Fine, k-ω model)

a) Flutter speed boundary

Mach Number

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Viscous (Fine, k-ω model)

b) Flutter frequency boundary

Figure 11: Comparisons of Navier-Stokes flutterpredictions with experimental data and references

drawback of the Euler/Navier-Stokes aeroelastic program is the excessive amount of computing time required to implement it. The use of parallel processing based on the multi-grid system alleviates this problem.

Several researchers have developed aeroelastic analysis programs to address the transonic/supersonic flutter problems, and they have frequently used the experimental data of the AGARD 445.6 wing to verify their programs. Their analysis results, however, have differed from the experimental data in the low supersonic flow region. Presently, determining the cause of this difference between the experimental and computational results is an interesting issue in aeroelastic research.

In this paper, aeroelastic computations of the wing are carried out for transonic and low supersonic Mach numbers. In each case, the k-ω turbulent model is implemented to account for turbulent effects. The computed flutter points are compared with other computed results and the wind tunnel test. The flutter analysis is repeated using Baldwin-Lomax model to evaluate the influence of the applied turbulent models.

The AGARD 445.6 wing to be computed here has a sweptback angle of 45o at the 1/4 chord line, a taper ratio of 0.6576, and an aspect ratio of 1.6525. The wing section has the shape of the 65A004 airfoil. The structured aerodynamic grid for the Navier-Stokes solver is shown in Figure 6. In this study, two types of grids, coarse and fine grids, are used for the viscous/inviscid aeroelastic analysis. In the respective directions of i, j, and k, the coarse grid is 177 × 33 × 41 and the fine grid is 209 × 41 × 65.

Figure 7 shows the steady aerodynamic analysis results of the Naiver-Stokes solver with the k-ω turbulent model at Mach 1.072 and Mach 1.141. A shock wave is observed along the entire wingspan at Mach 1.072. Interestingly, at Mach 1.141, shock waves become concentrated around wing tip area. Typically, a shock wave is pushed to the trailing edge at Mach 1.141 especially with a thin airfoil. This flow pattern is caused by the sweptback shape of the wing, and it can affect aeroelastic behavior.

In this study, four structural modes are applied for the flutter analysis. The mode shapes are transferred to the aerodynamic mesh by the Thin Plate Spline (TPS) method. In this scheme, a set of discrete structural grid points is distributed within the finite plate domain and the vertical positions of the deformed plate are defined at each structural grid point only. The TPS code solves the partial differential equation to yield a continuous deformed shape for the whole surface. Once the partial differential equation is solved, the deflection at the aerodynamic points can be determined.

To obtain the aeroelastic responses, the first modal velocity is initially disturbed with an amplitude of 1.5. The level of aerodynamic load can be expressed by the flutter speed index (FSI), which is a non-dimensionalized quantity, as follows:

µωαb

UFSI f= (3)

In Equation (3), Uf is the flutter speed, b is the half root chord, ωα is the primary torsional frequency (second mode), and µ is the mass ratio.

In this study, aeroelastic computations of the wing are carried out for two Mach numbers: M∞ = 0.96 and M∞ = 1.141. In each case, the original k-ω turbulent model is implemented to account for the turbulent effects. The wing flutter analyses are repeated with various time steps to confirm whether the applied time-step is suitable, and the results show that a time step smaller than 0.01 is required for the viscous transonic flutter analysis. However, use of a time-step value smaller than 0.005 is not realistic because of the large amount of computing time. The applied Reynolds numbers are Re = 6.5×105 and Re = 8.7×105 for Mach 0.96 and Mach 1.141, respectively. Ascertaining the

exact Reynolds number is difficult, because both the air density and the air speed can change the dynamic pressure. This implies that a various Reynolds numbers can be applied without any dynamic pressure change. Fortunately, the aeroelastic responses are not sensitive to a change of the Reynolds number [4].

The variation of dynamic responses corresponding to the increase of the FSI is presented in Figures 8 and 9. Figure 8 shows that the response at FSI = 0.283 is decaying but that at FSI = 0.298 is slightly diverging. This means that the flutter boundary is located between the two FSIs corresponding to each response. The decaying and diverging flutter responses are clearer at Mach 1.141. The flutter boundary at this Mach number is between FSI = 0.452 and FSI = 0.476. The effect of the applied turbulence model is shown in Fig. 10. In this figure, the k-ω model is 2-equation model but the Baldwin-Lomax model is an algebraic 0-equation model. We expect the k-ω model provide us with a more accurate solution for the low supersonic flow region. The figure shows that the response of the k-ω model is diverging, while other two responses are converging.

The predicted flutter boundaries are described in Fig. 11. The flutter points are compared with other computed results [5-7] and experimental results [8]. Figure 11 shows that the FSI and flutter frequency boundaries of the k-ω model are closer to the experimental data than those of other researchers including our Baldwin-Lomax results. This implies that the k-ω turbulent model can provide us with more accurate solutions than the Baldwin-Lomax model because these two turbulent models use identical numerical factors, including the aerodynamic grid and the time step. CONCLUSION This study has presented aeroelastic simulation results obtained by using two different approaches: one based on the TSD equation and the other based on the Navier-Stokes equations. Our research has shown that a flutter analysis based on the TSD is well suited for practical applications, because such an analysis can consider the effects of the control surface, launcher, and tail wings. On the other hand, the Navier-Stokes solver is better suited for a precise aeroelastic analysis under the viscous effects. This study also showed how the Navier-Stokes program predicts the flutter boundary more accurately when the k-ω turbulent model is used, as compared to when the Baldwin-Lomax model is used. ACKNOWLEGEMENTS This work was supported through funding from the Korea Ministry of Science and Technology and Korea Aerospace Industry (KAI). The authors acknowledge

the support as the National Research Laboratory Program and KAI. REFERENCES 1. Kwon, H.J., Kim, D.H. and Lee, I., “Frequency and

Time Domain Flutter Computations of a Wing with Oscillating Control Surface Including Shock Interference Effects,” Aerospace Science and Technology, Vol. 8, pp. 519-532, 2004.

2. Park, S. H., “Prediction Methods of Dynamic Stability Derivatives Using the Navier-Stokes Equations,” Ph.D Thesis, Division of Aerospace Engineering, Dept. of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 2003.

3. Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries Inc., 1993.

4. Kwon, H. J., Park, S. H., Lee, J. H., Kim, Y., Lee, I., and Kwon, J. H., "Transonic Wing Flutter Simulation Using Navier-Stokes and k-ω Turbulent Model," 46th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics & Materials Conference ,Austin, USA, April, 2005.

5. Lee-Rausch, E. M., and Batina, J. T., "Wing Flutter Computations Using an Aerodynamic Model Based on the Navier-Stokes Equations," Journal of Aircraft, Vol. 33, No. 6, pp. 1139-1147, 1996.

6. Gordnier, R. E., and Melville, R. B., "Transonic Flutter Simulations Using an Implicit Aeroelastic Solver," Journal of Aircraft, Vol. 37, No. 5, pp. 872-879, 2000.

7. Farhat, C., and Lesoinne, M., "Two Efficient Staggered Algorithms for the Serial and Parallel Solution of Three-dimensional Nonlinear Transient Aeroelastic Problems," Computer Methods in Applied Mechanics and Engineering, Vol. 182, pp. 499-515, 2000.

8. Yates, E.C., Jr.,"AGARD Standard Aeroelastic Configuration for Dynamic Response, Candidate Configuration I.-Wing 445.6," NASA TM-100492, 1987.