Transonic Aeroelasticity: Theoretical and Computational Challenges

Transcription

1 Transonic Aeroelasticity: Theoretical and Computational Challenges Oddvar O. Bendiksen Department of Mechanical and Aerospace Engineering University of California, Los Angeles, CA Aeroelasticity Workshop DCTA - Brazil July 1-2, 21

2 Introduction Despite theoretical and experimental research extending over more than 5 years, we still do not have a good understanding of transonic flutter Transonic flutter prediction remains among the most challenging problems in aeroelasticity, both from a theoretical and a computational standpoint Problem is also of considerable practical importance, because - Large transport aircraft cruise at transonic Mach numbers - Supersonic fighters must be capable of sustained operation near Mach 1, where the flutter margin often is at a minimum O. Bendiksen UCLA

3 Transonic Flutter Nonlinear Characteristics For wings operating inside the transonic region M crit < M < 1, strong oscillating shocks appear on upper and/or lower wing surfaces during flutter Transonic flutter with shocks is strongly nonlinear - Wing thickness and angle of attack affect the flutter boundary - Airfoil shape becomes of importance (supercritical vs. conventional) - A mysterious transonic dip appears in the flutter boundary - Nonlinear aeroelastic mode interactions may occur Flutter Speed Index Linear Theory 3 U F bω α µ Flutter Boundary 2 1Transonic Dip Mach No. M

4 Transonic Dip Effect of Thickness and Angle of Attack % 4% 6% 8% Mach No. M Dynamic pressure at flutter vs. Mach number for a swept series of wings of different airfoil thickness (Dogget, et al., NASA Langley) Effect of angle of attack on experimental flutter boundary and transonic dip of NLR 731 2D aeroelastic model tested at DLR (Schewe, et al.)

5 Objectives of Lecture Attempt a theoretical explanation of the key characteristics of transonic flutter and the transonic dip Explain the mechanism responsible for the observed sensitivity to wing thickness and angle of attack Provide an overview of the main theoretical and computational challenges, illustrated through examples

6 Research Challenges Motivating Questions Theoretical: Understanding Transonic Flutter - Is there a rational explanation for the location and shape of the transonic dip? - Why is the dip markedly different from wing to wing? - Can all transonic flutter instabilities be predicted by formulating and solving a linearized aeroelastic eigenvalue problem? - How do we distinguish between flutter and aeroelastic (forced) response? Computational: Predicting Transonic Flutter - What level of modeling is necessary? - Why do different codes based on similar CFD yield different predictions? - Why are LCO amplitudes so difficult to predict? - When do we need to use nonlinear structural models?

7 Observations Linear methods do a reasonable job of capturing the correct flutter behavior of wings in subsonic and supersonic flows, outside the transonic region The theoretical basis for this success is the Hopf bifurcation model for classical flutter, based on a valid linearization of the aeroelastic equations: x = Fxλ (, ) x = A( λ)x+ f( x, λ) The validity of the Hopf linearization is crucial here: The existence and uniform validity of the Jacobian matrix form the basis for the theory A( λ) - Regular perturbation problems generally satisfy linearization assumptions - Singular perturbation problems do not permit a uniformly valid linearization - Strongly nonlinear problems often cause extra grief

8 Examples of Singular Problems (Fluid Dynamics) Flows at low Reynolds numbers (cumulative/secular type singularity) - Stokes linearized solution is not uniformly valid (far field singularity) - Stokes Paradox was identified and corrected by Oseen Flows at high Reynolds numbers (layer type singularity) - Limit of zero viscosity yields Euler equations (different boundary conditions) - Small viscosity requires a boundary layer near body to satisfy no slip b.c. Supersonic flow wave structure (cumulative type singularity) - Linarized Ackeret solution is correct near body, but fails in the far field - In linear solution, weak shocks never coalesce (straight characteristics) - In actual flow, weak shocks always coalesce (cumulative nonuniformity) Flows at transonic Mach numbers (cumulative type singularity) - Linearized solution is not uniformly valid, even near the body - Nonuniformities occur both in space and time - Nonuniformities affect validity of Hopf linearization in flutter calculations

9 Theoretical Issues Linearized Flutter Analysis Nonlinear Aeroelastic Models: x = Fxλ (, ) x = ℵ( x t, λ) x t ( s) = x( t+ s), < s Linearization hypothesis (Hopf bifurcation) x = A( λ)x+ f( x, λ) For hyperbolic equilibrium points, stability near x = is determined by the eigenvalues of A (Hartman-Grobman) Recent results suggest that the Hopf linearization breaks down in the strongly nonlinear transonic region, in and near the transonic dip

10 The Hopf Bifurcation Classical Viewpoint Comes in two flavors -Supercritical (soft flutter) -Subcritical (hard flutter) Implies that small limit cycles do exist near linear flutter boundary, but assumes that -linearization is possible -no nonuniformities occur on the time scale A A 1 Linear flutter boundary c b a A A 1 A 1cr e d c b a U F U 1 U U * U 1 U F a) Supercritical b) Subcritical U

11 Linearization Hypothesis A Closer Look At Mach numbers outside the transonic region, the unsteady aerodynamic problem can be linearized and the velocity potential expanded in a regular asymptotic series: Φ = U { x + ε 1 ( δ)ϕ 1 ( xyzt,,, ) + ε 2 ( δ)ϕ 2 ( xyzt,,, ) + } where δ is a thickness parameter of the wing In the transonic region this regular expansion procedure fails, because the magnitudes of certain neglected terms have grown to order one To fix the problem one can use strained coordinates and a nonlinear perturbation expansion of the form ỹ = λδ ( )y; z = λδ ( )z; t = τδ ( )t Φ = U { x + εδ ( )ϕ 1 ( xỹz,,, t, χ) + } The distinguished limit results in the following scaling relationships: ε = ν = δ 2/ 3 ; λ = δ 1/ 3

12 Transonic Aerodynamics Limit Process Expansion The most general (small-disturbance) equation for the leading term ϕ ϕ 1 is obtained by retaining certain higher-order terms: χ ϕ x γ τ γ ϕ U t 2 ϕ x ϕ ỹ 2 2 ϕ ϕ τ ϕ = z 2 U x t U2 t 2 where χ and τ are transonic similarity parameters χ 1 M 2 = ; τ [( γ + 1)M 2 δ] 2 3 = δ [( γ + 1)M 2 ] 1 3 Note: The first-order equation is nonlinear, even in the limit δ, α, w No linearization is possible without introducing nonuniformities and destroying some of the essential physics; e.g. proper modeling of upstream wave propagation, parametric excitation, etc. Transonic flutter problem is a singular perturbation problem of the cumulative type (using classification terminology of Cole, Hayes, and Lagerstrom)

13 Hopf Linearization Breakdown Strongly Nonlinear Transonic Region Linearization is based on assumption that we can write where εδ ( ) as δ and δ is a measure of wing amplitude(s) But this assumption contradicts the known facts about transonic aerodynamics If we attempt a perturbation solution x = A( λ)x + εδ ( )g( x, λ) x() t = x () t + ε 1 x 1 () t + then x () t cannot be expected to satisfy the linearized equation x = A( λ)x because the aerodynamic forces are governed by nonlinear equations, even in the limit of small disturbances ( δ ) The Hopf linearization breaks down because the transonic aerodynamics problem cannot be linearized without introducing nonuniformities on the time scale

14 Crime and Punishment Classical linear flow theory breaks down in the transonic region, near Mach 1, resulting in a singularity ( 1 1 M 2 ) at Mach 1 The breakdown is of a mathematical rather than a physical nature (magnitudes of certain terms have been estimated incorrectly) Blowup at Mach 1 is related to the use of a linear governing equation, which only permits upstream disturbances to travel at a constant speed (linear acoustic speed of sound) Linearized solution is not uniformly valid in space and time To remove the infinities and nonuniformities we must allow the local speed of sound to vary as a nonlinear function of local disturbances, and this requires a nonlinear field equation for the fluid

15 Limitations of the Hopfian Viewpoint Recent computational results suggest that the Hopf bifurcation has its limitations as a theoretical model for explaining all types of transonic flutter Nonlinear mode interactions and temporal nonuniformities can give rise to nonclassical or Non-Hopfian flutter instabilities, such as -Delayed flutter -Dirty (almost periodic?) flutter -Period-tripling flutter

16 Delayed Flutter Aeroelastic mode is stable but not uniformly stable... NACA 12 Model at Mach θ.5 θ -.5 h/b -.1 h/b Nondimensional time Nondimensional time Short-term behavior Long-term behavior Do we really have to time-march over 2-3 flutter oscillations to determine stability? If flutter mode frequency is of the order of 1 Hz or less, delay before flutter onset could be as long as 2 seconds to 3 minutes!

17 Delayed Flutter via LCO LCO is stable but not uniformly stable.4 Problem has two time scales: h/b θ x = A( λ, τ)x + f( x, λ, τ) τ = εt, ε «1 Not classical Hopf Nondimensional time Parametric resonances become possible LCO transitioning to strong (delayed) flutter

18 Quasiperiodic or AP(?) Flutter Non-Hopfian.4.2. h/b U = 1.75 θ 4E-4 3E-4 2E-4 a).2.1 θ c) -.2 1E E tot -.4 E Nondimensional time h/b.1.5 h/b U = 1.77 θ 8E-4 6E NACA 12 BMM (Mach.85). 4E-4 b) θ d) -.5 E tot 2E E Nondimensional time h/b Almost-periodic (?) or quasiperiodic (?) flutter of NACA 12 model at Mach.85 and µ = 6 : a) at U = 1.75 (slightly below flutter boundary); b) at U = 1.77 (slightly above flutter boundary); c-d) corresponding phase plots.

19 Period-Tripling Flutter Transonic Flutter at Low Mass Ratio.4.2 µ = 5 U = 1.6 h/b θ 4E-4 3E-4. 2E E-4 E tot Nondimensional time U = 1.65 h/b θ E+ 4E-3 3E-3. 2E E tot 1E Nondimensional time U = E+ U = µ = 6 U = µ = 6 U = θ θ θ h b h b h b h/b

20 Anomalous Mass Ratio Scaling NACA 12 Model at Mach U F bω α Period tripling region q F 1U F bω α µ U F bω α q F (psf) µ Has practical consequences for transonic wind tunnel model construction...

21 Similarity Rules Transonic Flow z U Bxyt (,, ) = = z δ f u,l ( x, y) wyt (, ) α( y, t) = α ( y) + θ( y, t) [ wyt (, ) xα( y, t) ] δ x y (EA) χ 2 1 M = , Ã = [( γ + 1)M 2 [( γ + 1)M 2 δ] 2 3 δ] 1 3 A α = α δ, w = w δ, t = δ 2 3 t [( γ + 1)M ] 1 3 C δ 2 3 δ L = C L ( χ, Ã, α ) C 2 3 M = C M ( χ, Ã, α ) 2 [( γ + 1)M ] [( γ + 1)M ] 1 3

22 Similarity Rules and Scaling Laws Transonic Flutter Flutter Similarity Parameter ψ U 2 = = πµ [( γ + 1)M 2 δ] 1 3 qˆ [( γ + 1)M 2 δ] 1 3 Nondimensional dynamic pressure: qˆ ρ U mω 2 α U 2 = = = πµ 1 -- U π bω α µ with U = U bω α and all aerodynamic similarity parameters fixed In inviscid case, there are 3 primary similarity parameters: χψ,, and U In viscous case, the Reynolds number provides a 4th similarity parameter

23 Flutter Boundary as a Flutter Surface vs. Two-Parameter (2D) Flutter Boundary Plots U bω α µ 1 µ 2 b M M1 M 2 µ 1.8 µ a Plane µ = µ U F bω α µ.6.4 (b) (a) (c) M Flutter boundary as a surface in a 3D space of similarity parameters. The observed flutter boundaries in wind tunnel tests are represented by curves or paths (a,b,...) on this surface, and may differ from test to test if the temperature changes Mach No. M (a) Flutter boundary for NACA 6 model, corresponding to fixed mass ratio µ = 2 U (b) Corresponding boundary if = a M and µ is decreased until flutter occurs (nonlinear Euler-based calculations).

24 Nature of Transonic Dip Appearance of Almost Singular Lift Curve Slope 2 NACA 12 1 NACA 12 NACA 6 NACA 3 NACA 6 15 NACA 3 1 C Lα Prandtlα = Glauert C Lα 1 α = Prandtl- Glauert Mach No. M Mach No. M a) Linear scale b) Logarithmic scale Almost singular behavior of the lift curve slopes of the NACA XX series of airfoils in transonic flow, and comparison with Prandtl-Glauert rule (Euler calculations).

25 Nature of Transonic Dip Almost Singular Lift Curve Slope - Experimental Data 1.9 Transonic scaling theory M δ peak M TD.8.7 Wind Tunnel Data Ref. 18 Ref δ Measured lift curve slope vs. Mach number for symmetric airfoils of different thickness ratios, at α =, and comparison with the Prandtl-Glauert rule (original wind tunnel data from Göthert). Predicted vs. observed location of Mach number at which C Lα peaks, as a function of airfoil thickness.

26 Transonic Flutter Boundary Effect of Wing Thickness 2 ( 1 M 2 ) M4/ ( 1 M 1 ) M4/ 3 1 = δ 2 2/ δ 1 U 1 bω α µ U 2 bω α µ 2 = 2 1/ 6 M 2δ M 1δ1 U F bω α µ δ 1 > δ 2 > δ 3 δ 1 δ2 U F bω α µ Stable point enters flutter Unstable point is stabilized δ 3 Boundary after many flutter periods Boundary at t = ; fixed δ.8 1. M Qualitative sketch of the effect of wing thickness on the transonic flutter boundaries M Qualitative sketch of flutter boundary drift caused by nonuniformities on time scale

27 Correlations with Wind Tunnel Data Effect of Wing Thickness % 1% 1%p ψ = 1 U π bω α µ [ ( γ + 1)M δ].3.2 4% 6% 8% Mach No. M Calculated vs. measured shift in flutter boundary when wing thickness in increased from 4% (circles) to 1% (diamonds). Open symbols represent wind tunnel data from Dogget, et al. (unswept series of wings); solid diamonds are predicted boundary based on similarity rules. Arrows connect aeroelastically similar points χ 2 1 M = [ ( γ + 1)M δ] NASA Langley data, after applying the transonic similarity rules for flutter.

28 Computational Challenges Despite all the CFD research over the past 3 years, we still do not have a good understanding of the essential requirements for a good nonlinear flutter code The chief challenges are associated with the problem of coupling CFD codes to FE structural codes: 1) Fluid-structure coupling 2) Time synchronization 3) Code validation Time synchronization is most important, because a loosely coupled aeroelastic code using classical modal coupling can be made to converge to an incorrect aeroelastic solution as the mesh is refined and the time-step is reduced

29 Fluid-Structure Coupling Correct implementation of fluid-structure boundary conditions is of fundamental importance in aeroelastic stability calculations Both spatial compatibility and time synchronization requirements must be met, to assure that the time-marching simulations exhibit the physically correct stability behavior and LCO amplitudes Inconsistent or inaccurate implementations can result in a set of discretized eqs. that are not dynamically equivalent to the exact aeroelastic model: -May be missing some orbits (LCO/flutter) or fixed points (divergence) -May contain extraneous flutter orbits or divergences

30 Current Practice Most Codes In classical approach to time-marching flutter calculations, the equations for the fluid and structure are discretized and time-marched separately Most aeroelastic codes are implemented in this manner, using separate software modules for the fluid and structural domains - Codes are then coupled by imposing the kinematic boundary conditions - In loosely coupled schemes, strict synchronization is relaxed and the coupling is accomplished in an approximate manner This approach is simpler to implement, but is known to lead to local spurious violations of the conservation laws at the fluid-structure boundary

31 Modal vs. Direct Approach For strongly nonlinear problems such as transonic flutter, the coupling problem becomes progressively more difficult if a modal approach is used - Higher-order modes tend to be highly oscillatory, and transonic CFD codes are sensitive to rapid changes in surface slopes and curvatures - What modal amplitudes should be used in calculating the aero forces? - Superposition principle breaks down, and convergence becomes an issue These problems do not arise in the direct method of fluid-structure coupling Mode 1 (bending) Mode 2 (torsion) Mode 63 (ideal structure) Mode 63 (real structure) 9.486Hz Hz 2,444.6 Hz 2,45.7 Hz Oscillatory nature of higher-order natural modes of a low-aspect-ratio wing, and the sensitivity of higher mode shapes to small structural imperfections within normal manufacturing tolerances -.3

32 Fluid-Structure Coupling Basic Questions What is required to obtain reliable codes that exhibit the correct aeroelastic stability behaviors for a diverse class of problems? How can (should) coupling schemes be validated? How should fluid-structure coupling schemes be formulated and implemented to obtain - High accuracy - Required consistency - High scalability

33 Compatibility Requirement Inviscid Flow z w 3 β x3 3 U w 1 Typical structural finite element η ζ β y1 ŵ( ξ, ζ, t) β x1 1 h 2 β x2 w 2 β y3 β y 2 x B(x,y,z,t) = y ξ Kinematic boundary condition of tangent flow is an implicit compatibility constraint between fluid and structural elements at the wing surface B u B DB = t Dt In the local element coordinates and shape functions of the FE at the wing surface, the compatibility constraint becomes û 3 b = n i = 1 N i ( ξ, ζ) dqˆ i() t b n N û i ( ξζ, ) b n N dt qˆ ξ i () t û i ( ξ, ζ) qˆ ζ i () t i = 1 i = 1

34 Asymptotic Compatibility Using Different Mesh Densities A common approach is to use a much finer mesh in the fluid domain than the corresponding structural mesh at the boundary In this case compatibility errors can be shown to be of second order on the fluid mesh, provided that all fluid nodes are properly coupled to structure (see figure) Fluid-structure boundary defined by structural FE solution Structural boundary nth structural node n+1 Approximate boundary defined by fluid element n n+1 Floating fluid node On structural mesh, errors are Ol ( 2 m 2 ). As m these errors can be made arbitrarily small (compatibility in an asymptotic sense) Convergence may still become an issue, because of Morawetz s Theorem

35 Time Synchronization Bending-Torsion Model Problem Prototype typical section model problem with time lag τ: where 1 x α x α r α 2 h ( τ) b + α ( τ) γ ω 2 r α 2 h( τ) b ατ ( ) = C L ( τ τ) πµ 2C M ( τ τ) U 2 γ ω = ω h ω α ; τ = ω α t; α = dα dτ; U = U bω α When time delays are present, characteristic equation changes from 4th degree algebraic equation with complex conjugate roots p 1 2 = ± p 3, 4 = ±, p 1R iω 1 p 2R iω 2 to a transcendental equation of the form r Pze (, z ) =, where Pzw (, ) = a nm z m w n m = n = with z = p τ and w = e z. This equation has an infinite set of complex roots. s

36 Examples Effect of Time lag on V-g Diagram.5 t = θ.5 t t = T = α h t t t = = = T α T α T α θ h/b U bω α U bω α (a) (b) t t = = T α T α Effect of time lag on aeroelastic stability: (a) stabilizing effect on typical section in incompressible flow, with quasistatic aerodynamics evaluated based on retarded structural state ( a =.3; x α =.4; ω h ω α =.6564; r α =.5; µ = 75); (b) destabilizing effect on typical section in supersonic flow (1st order piston theory; M = 1.5; a =.5; x α =.5; ω h ω α =.5; r α =.5; µ = 75)

37 Code Verification and Validation A Significant Challenge in Transonic Aeroelasticity Verification: Solving the equations right Validation: Solving the right equations Separation of the verification and validation steps may be difficult in a real-world setting -What do we mean by solving the equations right? (Need to define right ) -Numerical accuracy is always finite and bugs may be present -Aeroelastic solution still depends on the numerical scheme used and how the fluid-structure coupling is implemented If a strict interpretation of solving the equations right is adopted, then the widely used loosely coupled codes can never be verified, far less validated For nonlinear transonic problems convergence becomes an issue, because stability and consistency of the numerical scheme supply only necessary (but not sufficient) conditions for convergence (Lax equivalence theorem does not apply)

38 Time-Marching Aeroelastic Solution CFD-Based FE Model - Nonlinear Structural FE Model z y x z Global Eulerian coordinate system 2 1 v 1 ' y 3 Undeformed element Deformed element w 1 ' w 3 ' 3 β y3 ' u 3 ' w 2 ' β x3 ' v 3 ' x β x1 ' β 1 x2 ' β y1 2 Mindlin-Reissner FE ' v 2 ' β y2 ' u 1 ' u 2 ' Local Lagrangian element coordinate system

39 Direct Fluid-Structure Coupling Fluid Node Mapping FE Gaussian Point Mapping Determine in which structural FE fluid node belongs Determine on which fluid element face GP resides Calculate position of node in terms of FE area coords A i Calculate position of GP in terms of fluid face area coords a) b) Mapping of aerodynamic cell faces and structural elements at the fluid-structure boundary

40 NLR 731 Unswept Wing Schewe, et al.

41 NLR 731 Wing Section Validation Dilemma Transonic LCO: Theory vs. Experiment.2 h/b 8E-3. θ 6E-3 4E-3 Fig. 12b) -.2 E tot 2E E Nondimensional time 8E-3 h/b. 6E-3 θ 4E-3 Fig.12d) -.2 E tot 2E-3 Previous calculations for MP 77 (from Thomas, Dowell, and Hall) -.4 Table 1: LCO Predictions vs. experiment for MP Nondimensional time Direct E-L Scheme E+ Source M U F bω α µ θ LC ( h bθ) LC ω ω α φ Fig. 12b) deg deg Fig. 12d) deg deg Experiment deg deg

42 Multiple (Nested) Limit Cycles NLR 731 Model Tested at DLR (Schewe, et al.).2 4E-1.1. h/b 3E-1 2E E tot θ 1E E Nondimensional time 4E-1.1 h/b 3E-1 Experiment. 2E E tot 1E E Nondimensional time Euler-based simulations θ.5 h -- b θ h b θ

43 Swept Transport Wing G-Wing G-Wing AR = 7.5 Λ LE = Λ TE = 21.2 λ = x Computational Domain No. of nodes: 43,344 No. of faces: 43,975 No. of cells: 28,926 No. of structural plate elements: 96 No of structural dof s: 168 (linear) 28 (nonlinear)

44 Subcritical Behavior Subsonic - Nonlinear FE Code Wing tip displacements w TE w LE Wing total energy.2 w TE Nondimensional time E tot w TE.1.8 Aerodynamic work W A. E tot - W A Nondimensional time Stable decay of the G-Wing tip amplitudes at Mach.75 ( ρ a =.4177 kg/m 3 )

45 Limit Cycle Flutter Onset G-Wing at Mach Wing tip displacements.3.2 E tot w TE w LE.8.4 Wing total energy.2 w TE Nondimensional time.6 Aerodynamic work.4.2. W A E tot - W A w TE Limit cycle flutter of the G-Wing at Mach.84 and an air density of ρ a =.4177 kg/m 3, corresponding to a density altitude of about 32,8 ft (1, m) Nondimensional time

46 Limit Cycle Flutter of G-Wing Nonlinear vs. Linear Structural Code w TE w LE.3.2 w TE.2 Linear ρ a = kg/m 3. E tot Nondimensional time w TE w LE w TE.3.2 w TE.2 Nonlinear ρ a =.4177 kg/m 3. E tot Nondimensional time w TE

47 Nested LCOs of Different Amplitudes Low Transonic Mach Numbers Wing tip TE velocity Wing tip TE displacement Wint tip TE displacement Wing tip TE displacement a) Mach.84 b) Mach.85 c) Mach.865 Note: All LCOs are stable (density altitude = 1 km (32,8 ft). OOB-UCLA

48 Nested LCOs - Continued Intermediate to High Transonic Mach Numbers Wing tip TE displacement Wing tip TE displacement Wing tip TE displacement a) Mach.88 b) Mach.9 c) Mach.96 Note: All LCOs except the inner LCO at Mach.88 are weakly unstable. Density altitude = 1 km (32,8 ft). OOB-UCLA

49 Rapid Transition of LCO Behavior Evaporation of LCOs at Mach Wing tip TE velocity Wing tip TE displacement Wing tip TE displacement Wing tip TE displacement a) Mach.965 b) Mach.97 (large i.c.) c) Mach.97 (small i.c) Note: At Mach.965 both LCOs are stable. At Mach.97, the LCOs evaporate. Density altitude = 1 km (32,8 ft). OOB-UCLA

50 Stabilization of Unstable (Secular) LCO Mach Wing tip TE velocity Wing tip TE displacement Wing tip TE displacement Stabilization of an unstable LCO at Mach.95 by increasing the air density from.4177 kg/m 3 to.6 kg/m 3, corresponding to an increase in the dynamic pressure of 43.6%, or a decrease in the density altitude from 1 km (32,8 ft) to about 6.86 km (22,5 ft) OOB-UCLA

51 High-Altitude Flutter A Real Possibility? Limit cycle flutter amplitude vs. altitude at a constant Mach number (inner LCO, reached through small perturbations from steady equilibrium state) OOB-UCLA

52 High-Altitude Flutter Mach.865 Phase plots of limit cycle flutter at high altitudes OOB-UCLA

53 LCO Nesting Possibilities Multiple LCOs Other configurations are also possible LCO amplitude.5 Outer LCO Inner LCO Mach number 1 LCO cycles (inner) 2 LCO cycles (inner) 1 LCO cycles (outer) 2 LCO cycles (outer) 5 LCO cycles (outer) Multiple limit cycle flutter branches of G-Wing in transonic region OOB-UCLA

54 Conclusions Theoretical 1. Transonic flutter should not be considered classical bending-torsion flutter, because flutter near the transonic dip is often triggered by nonlinear interactions between modes (not coalescence flutter) 2. Near the transonic dip, no meaningful uniformly valid linearization of the aeroelastic problem is possible 3. The transonic dip is primarily associated with the occurrence of highly nonlinear lift and moment curve slopes (almost singular behavior). The part-chord shocks on the wing surface play a fundamental role 4. Certain nonlinear transonic flutter instabilities near the transonic dip cannot be described using the classical Hopf theory 5. The breakdown occurs because the unsteady aerodynamic forces cannot be linearized without introducing nonuniformities in time or space, even in the limit of infinitesimal amplitudes (singular problem)

55 Conclusions Computational 6. The fluid-structure coupling problem remains of central importance in the development of accurate and reliable flutter codes 7. Both spatial compatibility and time synchronization requirements must be met, to assure that the time-marching simulations exhibit the correct aeroelastic stability behavior 8. Codes capable of predicting correct transonic LCO amplitudes must satisfy strict energy balance requirements at the fluid-structure boundary 9. In the strongly nonlinear transonic region near the dip, time-invariance is lost and it may be necessary to calculate several dozen flutter cycles before the correct stability behavior can be assessed 1.The verification and validation steps of transonic aeroelastic codes present several practical problems that have not been adequately addressed in the research literature