Transonic : 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, 2010

Introduction

• Despite theoretical and experimental research extending over more than 50 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 Transonic Flutter Nonlinear Characteristics

• For wings operating inside the transonic region Mcrit <

• Transonic flutter with shocks is strongly nonlinear

- Wing thickness and angle of attack affect the flutter boundary - shape becomes of importance (supercritical vs. conventional) - A mysterious transonic dip appears in the flutter boundary - Nonlinear aeroelastic mode interactions may occur

Flutter 3 Speed Linear Theory Index

U ------F - 2 bωα µ Flutter Boundary 1Transonic Dip

0.70.8 0.9 1.0 Mach No. M

O. Bendiksen UCLA Transonic Dip Effect of Thickness and Angle of Attack

300

250

200

150

2% 100 4%

50 6%

8% 0 0.7 0.8 0.9 1 1.1 1.2 Mach No. M

Dynamic pressure at flutter vs. Effect of angle of attack on experimental flutter for a swept series of wings of different airfoil boundary and transonic dip of NLR 7301 2D thickness (Dogget, et al., NASA Langley) aeroelastic model tested at DLR (Schewe, et al.)

O. Bendiksen UCLA 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, illus- trated through examples

O. Bendiksen UCLA 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?

O. Bendiksen UCLA 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· = Fx(), λ x· = A()λ xfx+ (), λ

• The validity of the Hopf linearization is crucial here:

The existence and uniform validity of the Jacobian matrix A()λ form the basis for the theory - Regular perturbation problems generally satisfy linearization assumptions - Singular perturbation problems do not permit a uniformly valid linearization - Strongly nonlinear problems often cause extra grief

O. Bendiksen UCLA Examples of Singular Problems ()

• 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

O. Bendiksen UCLA Theoretical Issues Linearized Flutter Analysis

• Nonlinear Aeroelastic Models:

x· = Fx(), λ · x = ℵ()xt , λ

xt()s = x()∞ts+ , – < s ≤ 0

• Linearization hypothesis (Hopf bifurcation)

x· = A()λ xfx+ (), λ

• For hyperbolic equilibrium points, stability near x = 0 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

O. Bendiksen UCLA 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

e A A c d A1 b A1 c

A1cr b Linear flutter Linear flutter boundary a a * UF U1 U U U1 UF U a) Supercritical b) Subcritical

O. Bendiksen UCLA Linearization Hypothesis A Closer Look

• At Mach numbers outside the transonic region, the unsteady aerodynamic prob- lem can be linearized and the velocity potential expanded in a regular asymptotic series:

Φ = U∞{}x +++ε1()ϕδ 1()εxyzt,,, 2()ϕδ 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 y˜ = λδ()y; z˜ = λδ()z; ˜t = τδ()t

Φ = U∞{}x ++εδ()ϕ1()…xy,,,,˜ z˜ ˜t χ • The distinguished limit results in the following scaling relationships: ενδ==23/ ; λδ= 13/

O. Bendiksen UCLA Transonic Limit Process Expansion

• The most general (small-disturbance) equation for the leading term ϕϕ≡ 1 is obtained by retaining certain higher-order terms:

∂ϕ γ – 1 τ ∂ϕ ∂2ϕ ∂2ϕ ∂2ϕ 2 ∂2ϕ τ ∂2ϕ χ – ------– ------+ ------+0------– ------– ------= ∂x γ + 1 U ∂˜t 2 ∂y2 ∂z2 U ∂x∂˜t 2 2 ∞ ∂x ˜ ˜ ∞ U∞ ∂˜t

where χτ and are transonic similarity parameters

1 – M2 δ23⁄ χ = ∞ ; τ = ------13⁄ 2 23⁄ 2 []()γ + 1 M∞ δ []()γ + 1 M∞

• Note: The first-order equation is nonlinear, even in the limit δ,,α w → 0

• No linearization is possible without introducing nonuniformities and destroying some of the essential physics; e.g. proper modeling of upstream wave propaga- tion, parametric excitation, etc.

• Transonic flutter problem is a singular perturbation problem of the cumulative type (using classification terminology of Cole, Hayes, and Lagerstrom)

O. Bendiksen UCLA Hopf Linearization Breakdown Strongly Nonlinear Transonic Region

• Linearization is based on assumption that we can write

x· = A()λ x + εδ()g()x, λ

where εδ()→δ0 as →δ0 and is a measure of wing amplitude(s)

• But this assumption contradicts the known facts about transonic aerodynamics

• If we attempt a perturbation solution

x()t = x0()t ++ ε1x1()t …

then x0()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 → 0 )

• The Hopf linearization breaks down because the transonic aerodynamics prob- lem cannot be linearized without introducing nonuniformities on the time scale

O. Bendiksen UCLA Crime and Punishment

• Classical linear flow theory breaks down in the transonic region, near Mach 1, 2 resulting in a singularity (11⁄ – M∞ ) 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

O. Bendiksen UCLA 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 “non- classical” or “Non-Hopfian” flutter instabilities, such as -Delayed flutter -Dirty (almost periodic?) flutter -Period-tripling flutter

O. Bendiksen UCLA Delayed Flutter Aeroelastic mode is stable but not uniformly stable...

NACA 0012 Model at Mach 0.80

0.1 0.1 θ

0.05 θ 0

0

-0.1 h/b -0.05 h/b

-0.1 -0.2 0 20406080100 0 200 400 600 800 Nondimensional time Nondimensional time Short-term behavior Long-term behavior

• Do we really have to time-march over 20-300 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 20 seconds to 3 minutes!

O. Bendiksen UCLA Delayed Flutter via LCO LCO is stable but not uniformly stable

0.4 Problem has two time scales:

0.2 θ x· = A(λ, τ)x + f(x,,λ τ) 0 τε= t, ε « 1 h/b

-0.2 Not classical Hopf

-0.4 Parametric resonances 0 200 400 600 800 become possible Nondimensional time LCO transitioning to strong (delayed) flutter

O. Bendiksen UCLA Quasiperiodic or AP(?) Flutter Non-Hopfian

0.04 4E-4 0.02 U = 1.75 0.02 h/b 3E-4 0.01 θ 0.00 2E-4 a) θ 0 c)

-0.02 1E-4 -0.01 Etot -0.04 0E+0 -0.02 0 200 400 600 800 1000 -0.04 -0.02 0 0.02 0.04 Nondimensional time h/b

0.10 8E-4 0.02 NACA 0012 BMM U = 1.77 0.05 h/b 6E-4 0.01 (Mach 0.85) θ

0.00 4E-4 b) θ 0 d)

E -0.05 tot 2E-4 -0.01

-0.10 0E+0 -0.02 0 200 400 600 800 1000 -0.04 -0.02 0 0.02 0.04 Nondimensional time h/b

Almost-periodic (?) or quasiperiodic (?) flutter of NACA 0012 model at Mach 0.85 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.

O. Bendiksen UCLA Period-Tripling Flutter Transonic Flutter at Low Mass Ratio

0.04 4E-4 µ = 5 0.02 U = 1.60 3E-4 h/b θ 0.00 2E-4

-0.02 1E-4 Etot -0.04 0E+0 0 100 200 300 400 Nondimensional time 0.10 4E-3 U = 1.65 0.05 h/b θ 3E-3

0.00 2E-3

-0.05 1E-3 Etot

-0.10 0E+0 0 100 200 300 400 Nondimensional time 0.02 0.02 µ = 6 µ = 6 0.02 0.04 U = 1.75 U = 1.77 U = 1.60 U = 1.65 0.01 0.01 0.01 0.02 θ 0 θ 0 θ 0 0

-0.01 -0.01 -0.02 -0.01

-0.02 -0.04 -0.02 -0.04 -0.02 0 0.02 0.04 -0.08 -0.04 0 0.04 0.08 -0.02 -0.04 -0.02 0 0.02 0.04 hb⁄ hb⁄ -0.04 -0.02 0 0.02 0.04 h/b hb⁄

O. Bendiksen UCLA Anomalous Mass Ratio Scaling NACA 0012 Model at Mach 0.85

8.0 250

10U ------F - 200 6.0 bω µ U α ------F- bω Period α 150 UF tripling q ------4.0 region F bω q (psf) α F 100

2.0 50

0.0 0 1 10 100 1000 10000 µ

Has practical consequences for transonic model construction...

O. Bendiksen UCLA Similarity Rules Transonic Flow

z

U∞ []wyt(), – xα()yt, Bxyt(),, ==0 z – δfu,l()xy, + ------δ wyt(), α()yt, = α0()y + θ()yt, x y (EA)

1 – M2 χ = ∞ , A˜ = []()γ + 1 M2 δ 13⁄ A ------2 23⁄ - ∞ []()γ + 1 M∞δ δ23⁄ t α = αδ⁄, w = w ⁄ δ, ˜t = ------˜ ˜ 2 13⁄ []()γ + 1 M∞

23⁄ 23⁄ δ δ C = C˜ M()χ,,A˜ α C = ------C˜ L()χ,,A˜ α˜ M ------13⁄ ˜ L 2 13⁄ []()γ + 1 M2 []()γ + 1 M∞ ∞

O. Bendiksen UCLA Similarity Rules and Scaling Laws Transonic Flutter

2 U qˆ Flutter Similarity Parameter ψ ==------2 13⁄ 2 13⁄ πµ[]() γ + 1 M∞δ []()γ + 1 M∞δ

1 2 Nondimensional ---ρ U∞ 2 2 ∞ U 1 U 2 dynamic pressure: qˆ ===------------1 2  ---mω πµ π bω µ 2 α α

U withU = ------∞- and all aerodynamic similarity parameters fixed bωα

• In inviscid case, there are 3 primary similarity parameters: χΨ, , and U

• In viscous case, the Reynolds number provides a 4th similarity parameter

O. Bendiksen UCLA Flutter Boundary as a Flutter Surface vs. Two-Parameter (2D) Flutter Boundary Plots

M 0 M 1 U 1 ------µ2 bω M2 α µ 1 0.8 b µ

a (a) µ 0.6 (c) 0 U ------F Plane bωα µ 0.4 µµ= 0 (b)

0.2

0.8 0 1.0 0 0.2 0.4 0.6 0.8 1 Mach No. M M∞ Flutter boundary as a surface in a 3D space of sim- (a) Flutter boundary for NACA 0006 model, ilarity parameters. The observed flutter boundaries corresponding to fixed mass ratio µ = 20 in wind tunnel tests are represented by curves or (b) Corresponding boundary if U∞ = a∞M∞ paths (a,b, ...) on this surface, and may differ from and µ is decreased until flutter occurs test to test if the temperature changes. (nonlinear Euler-based calculations).

O. Bendiksen UCLA Nature of Transonic Dip Appearance of “Almost Singular” Lift Curve Slope

200 1000 NACA 0012 NACA 0012

NACA 0006 NACA 0006

150 NACA 0003 NACA 0003 100 C Prandtl- Prandtl- Lα α = 0 Glauert CLα Glauert 100 α = 0

10 50

0 1 0.5 0.6 0.7 0.8 0.9 1 0.7 0.8 0.9 1 Mach No. M Mach No. M a) Linear scale b) Logarithmic scale

Almost singular behavior of the lift curve slopes of the NACA 00XX series of in transonic flow, and comparison with Prandtl-Glauert rule (Euler calculations).

O. Bendiksen UCLA Nature of Transonic Dip Almost Singular Lift Curve Slope - Experimental Data

1 Transonic scaling theory 0.9

peak Mδ 0.8 Wind Tunnel Data MTD 0.7 Ref. 18

Ref. 25

0.6 0 0.030.060.090.120.15 δ

Predicted vs. observed location of Mach Measured lift curve slope vs. Mach number for number at which C peaks, as a function symmetric airfoils of different thickness ratios, at Lα of airfoil thickness. α = 0 , and comparison with the Prandtl-Glauert rule (original wind tunnel data from Göthert).

O. Bendiksen UCLA Transonic Flutter Boundary Effect of Wing Thickness

2 43/ 23/ ()1 – M2 ⁄ M2 δ2 ------= ----- 2 43/ δ ()1 – M1 ⁄ M1 1 16/ 2 U1 ⁄ bωα µ1 M δ ------= ------2 2 2 U2 ⁄ bωα µ2 M δ 1 1

Stable point enters flutter U U F δ1 >>δ2 δ3 F ------Unstable point bω µ bω µ α δ1 α is stabilized δ2 δ 3 Boundary at t = 0; Boundary δ fixed after many flutter periods

0.8 1.0 M∞ 0.8 1.0 M∞

Qualitative sketch of the effect of wing thickness Qualitative sketch of flutter boundary “drift” on the transonic flutter boundaries. caused by nonuniformities on time scale

O. Bendiksen UCLA Correlations with Wind Tunnel Data Effect of Wing Thickness

300 4% 0.3 4% 10% 6% 250 10%p 2 8% 1U∞ ---------0.2 π bωα µ 200 ψ = ------2 13⁄ []()γ + 1 M∞δ 0.1 150

100 0 0.7 0.8 0.9 1 -2024 2 Mach No. M 1 – M χ = ------∞ Calculated vs. measured shift in flutter boundary when 2 23⁄ []()γ + 1 M∞δ wing thickness in increased from 4% (circles) to 10% (diamonds). Open symbols represent wind tunnel data NASA Langley data, after applying the from Dogget, et al. (unswept series of wings); solid transonic similarity rules for flutter. diamonds are predicted boundary based on similarity rules. Arrows connect aeroelastically similar points.

O. Bendiksen UCLA Computational Challenges

• Despite all the CFD research over the past 30 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

O. Bendiksen UCLA 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

O. Bendiksen UCLA 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 vio- lations of the conservation laws at the fluid-structure boundary

O. Bendiksen UCLA 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

0 0.2

-0.1 0.1

-0.2 0

-0.3

-0.1 -0.4

-0.2 -0.5

Mode 1 (bending) Mode 2 (torsion)-0.6 Mode 63 (ideal-0.3 structure) Mode 63 (real structure) 9.486Hz 40.767 Hz 2,444.6 Hz 2,450.7 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

O. Bendiksen UCLA Fluid-Structure Coupling Basic Questions

• What is required to obtain reliable codes that exhibit the correct aeroelastic sta- bility 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

O. Bendiksen UCLA Compatibility Requirement Inviscid Flow

w3 β z x3 3 β y3 w1 U∞ w2 β β y1 1 y η 2 2 Typical structural β h ζ x1 β finite element x2 wˆ ()ξζ,,t

x y ξ B(x,y,z,t) = 0

• Kinematic boundary condition of tangent flow is an implicit compatibility constraint between fluid and structural elements at the wing surface ∂B DB ------+ u ⋅ ∇B ≡ ------= 0 ∂t Dt • In the local element coordinates and shape functions of the FE at the wing sur- face, the compatibility constraint becomes

n n n b dqˆ i()t b ∂Ni()ξζ, b ∂Ni()ξζ, uˆ 3 = ∑ Ni()ξζ, ------+ uˆ 1 ∑ ------qˆ i()t + uˆ 2 ∑ ------qˆ i()t i = 1 dt i = 1 ∂ξ i = 1 ∂ζ

O. Bendiksen UCLA Asymptotic Compatibility Using Different Mesh Densities

• A common approach is to use a much finer mesh in the fluid domain than the cor- responding 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

n+1 n+1 Approximate n nth boundary defined structural node by fluid element Floating fluid node • On structural mesh, errors are Ol()2 ⁄ m2 . 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

O. Bendiksen UCLA Time Synchronization Bending-Torsion Model Problem

• Prototype typical section model problem with time lag ∆τ : ··  h()τ 2 h()τ 2 1 xα ------γω 0 ------U –CL()τ∆τ– + b = ------ 2 b 2 πµ x r 0 r  2CM()τ∆τ– α α α·· ()τ α ατ() where · γω = ωh ⁄τωωα; = αt; α = dα ⁄ dτ; UU= ∞ ⁄ bωα • When time delays are present, characteristic equation changes from 4th degree algebraic equation with complex conjugate roots p = p ± iω 12, 1R 1 p = p ± iω 34, 2R 2 to a transcendental equation of the form Pze(), z = 0 , where

r s m n Pzw(), = ∑ ∑ anmz w m = 0 n = 0 with zp= – ∆τ and we= z . This equation has an infinite set of complex roots.

O. Bendiksen UCLA Examples Effect of Time lag on V-g Diagram

0.005 0.005 ∆t = 0 ∆t = 0 T θ ∆t = -----α- 0 25

h/b T ∆t = ------α- -0.005 100 0 Tα T ∆t = ------∆t = -----α- 50 50 Tα -0.01 θ ∆t = ------h T 100 ∆t = -----α- 25 -0.015 -0.005 01234 0246810 Ub⁄ ω Ub⁄ ωα α (a) (b)

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 = –0.3 ; xα = 0.04 ; ωh ⁄ ωα = 0.6564 ; rα = 0.5 ; µ = 75 ); (b) destabilizing effect on typical section in supersonic flow (1st order piston theory; M∞ = 1.5; a = –0.05 ; xα = 0.05 ; ωh ⁄ ωα = 0.5 ; rα = 0.5 ; µ = 75 )

O. Bendiksen UCLA 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 sta- bility and consistency of the numerical scheme supply only necessary (but not sufficient) conditions for convergence (Lax equivalence theorem does not apply)

O. Bendiksen UCLA Time-Marching Aeroelastic Solution CFD-Based FE Model - Nonlinear Structural FE Model

z

y

x Global Eulerian coordinate system z′

y′ Undeformed element Deformed element 1 w ' β 3 y ' 3 3 u3' 3 β x ' v ' w ' 3 3 2 v ' 1 1 w2' β β x1' 1 x2' β 2 Mindlin-Reissner FE y ' v2' x′ 1 β y2' u1' u2' Local Lagrangian element coordinate system

O. Bendiksen UCLA Direct Fluid-Structure Coupling

Fluid Node Mapping FE Gaussian Point Mapping

Determine in which structural FE Determine on which fluid ele- fluid node belongs ment face GP resides

Calculate position of node Calculate position of GP in terms of FE area coords Ai in terms of fluid face area coords a) b)

Mapping of aerodynamic cell faces and structural elements at the fluid-structure boundary

O. Bendiksen UCLA NLR 7301 Unswept Wing Schewe, et al.

O. Bendiksen UCLA NLR 7301 Wing Section Validation Dilemma Transonic LCO: Theory vs. Experiment

0.02 8E-3 h/b 6E-3 0.00

θ 4E-3 Fig. 12b) -0.02 2E-3 E tot -0.04 0E+0 0 100 200 300 400 500 600 0.02 Nondimensional time 8E-3 h/b

6E-3 0.00

θ 4E-3 Fig.12d) -0.02 2E-3 Etot -0.04 0E+0 Previous calculations for MP 77 0 200 400 600 800 (from Thomas, Dowell, and Hall) Nondimensional time Direct E-L Scheme Table 1: LCO Predictions vs. experiment for MP 77

Source M∞ UF ⁄ bωα µ θLC ()hb⁄ θ LC ωω⁄φα Fig. 12b) 0.750 0.203 0.21 deg 1.37 0.781 9 deg

Fig. 12d) 0.768 0.203 0.26 deg 1.42 0.775 6 deg

Experiment 0.768 0.204 0.20 deg 1.43 0.760 4 deg

O. Bendiksen UCLA Multiple (Nested) Limit Cycles NLR 7301 Model Tested at DLR (Schewe, et al.)

0.20 4E-1

0.10 3E-1 h/b 0.00 2E-1

θ -0.10 1E-1 Etot

-0.20 0E+0 0 200 400 600 800 1000 1200 1400 Nondimensional time 0.20 4E-1

h/b 0.10 3E-1 Experiment

0.00 2E-1 0.1 0.1

-0.10 1E-1 Etot θ 0.05 0.05 -0.20 0E+0 · 300 500 700 900 1100 1300 h · --- 0 θ 0 Nondimensional time b

Euler-based simulations -0.05 -0.05

-0.1 -0.1 -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 hb⁄ θ

O. Bendiksen UCLA Swept Transport Wing G-Wing

4 G-Wing AR = 7.05 3.5 ΛLE = 31.86° 3 ΛTE = 21.02°

2.5 λ = 0.4102

2 Computational Domain 1.5 No. of nodes: 43,344 No. of faces: 430,975 1 No. of cells: 208,926

0.5 No. of structural plate elements: 96 No of structural dof’s: 168 (linear) 0 280 (nonlinear) -1 -0.5 0 0.5 1 1.5 2 2.5 3 x

O. Bendiksen UCLA Subcritical Behavior Subsonic - Nonlinear FE Code

0.80 0.40 0.4

0.60 0.30 0.2

0.40 · w w 0 TE 0.20 TE 0.20

w Wing total energy -0.2 LE 0.10 0.00 Wingdisplacements tip E tot -0.20 0.00 -0.4 0 40 80 120 160 200 240 -0.2 0 0.2 0.4 0.6 Nondimensional time wTE

0.10

0.08

0.06

W 0.04 A

0.02 Aerodynamic work Aerodynamic 0.00 E - W tot A -0.02 0 40 80 120 160 200 240 Nondimensional time

3 Stable decay of the G-Wing tip amplitudes at Mach 0.75 (ρa = 0.4177 kg/m )

O. Bendiksen UCLA Limit Cycle Flutter Onset G-Wing at Mach 0.84

0.40 0.12 0.04

w TE 0.30 0.08 0.02 · wTE 0 w 0.20 LE 0.04

E Wing total energy tot -0.02 Wing tip displacements

0.10 0.00 -0.04 300 350 400 450 500 550 600 650 700 0.22 0.24 0.26 0.28 0.3 Nondimensional time wTE 0.06 Limit cycle flutter of the G-Wing at 0.04 Mach 0.84 and an air density of W A 3 ρa = 0.4177 kg/m , corresponding to a density altitude of about 32,800 ft 0.02 (10,000 m)

E - W tot A

Aerodynamic work Aerodynamic 0.00

-0.02 300 350 400 450 500 550 600 650 700 Nondimensional time

O. Bendiksen UCLA Limit Cycle Flutter of G-Wing Nonlinear vs. Linear Structural Code

0.80 0.40 0.4 w TE wLE 0.60 0.30 0.2

0.40 Linear · ρ = 1.141 kg/m3 0.20 wTE 0 a 0.20

0.10 -0.2 0.00 Etot -0.20 0.00 -0.4 0 1020304050607080 0 0.2 0.4 0.6 0.8 Nondimensional time wTE

0.80 0.40 0.4

w 0.60 w TE LE 0.30 0.2 0.40 Nonlinear 0.20 w· 3 TE 0 ρa = 0.4177 kg/m 0.20

0.10 E -0.2 0.00 tot

-0.20 0.00 -0.4 0 1020304050607080 00.20.40.60.8 Nondimensional time wTE

O. Bendiksen UCLA Nested LCOs of Different Amplitudes Low Transonic Mach Numbers

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

-0.2

Wing tip TE velocity TE tip Wing -0.2 -0.2

-0.4 -0.4 -0.4 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Wing tip TE displacement Wint tip TE displacement Wing tip TE displacement

a) Mach 0.84 b) Mach 0.85 c) Mach 0.865

Note: All LCOs are stable (density altitude = 10 km (32,800 ft).

OOB-UCLA Nested LCOs - Continued Intermediate to High Transonic Mach Numbers

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

-0.2 -0.2 -0.2

-0.4 -0.4 -0.4 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Wing tip TE displacement Wing tip TE displacement Wing tip TE displacement

a) Mach 0.88 b) Mach 0.90 c) Mach 0.96

Note: All LCOs except the inner LCO at Mach 0.88 are weakly unstable. Density altitude = 10 km (32,800 ft).

OOB-UCLA Rapid Transition of LCO Behavior Evaporation of LCOs at Mach 0.97

0.4 0.4 0.1

0.2 0.2 0.05

0 0 0

-0.2 -0.2 -0.05 Wing tip TE velocityWing tip

-0.4 -0.4 -0.1 0 0.2 0.4 0.6 0.8 00.20.40.60.80.20.250.30.350.4 Wing tip TE displacement Wing tip TE displacement Wing tip TE displacement

a) Mach 0.965 b) Mach 0.97 (large i.c.) c) Mach 0.97 (small i.c)

Note: At Mach 0.965 both LCOs are stable. At Mach 0.97, the LCOs “evaporate.” Density altitude = 10 km (32,800 ft).

OOB-UCLA Stabilization of Unstable (Secular) LCO Mach 0.95

0.5 0.2

0.25 0.1

0 0

-0.25 -0.1 Wing tip TE velocity TE tip Wing

-050.5 -020.2 -0.25 0 0.25 0.5 0.75 0.20.30.40.50.6 Wing tip TE displacement Wing tip TE displacement

Stabilization of an unstable LCO at Mach 0.95 by increasing the air density from 0.4177 kg/m3 to 0.60 kg/m3, corresponding to an increase in the dynamic pressure of 43.6%, or a decrease in the density altitude from 10 km (32,800 ft) to about 6.86 km (22,500 ft)

OOB-UCLA 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 High-Altitude Flutter Mach 0.865

Phase plots of limit cycle flutter at high altitudes

OOB-UCLA LCO Nesting Possibilities Multiple LCOs

Other configurations are also possible

0.5

Outer LCO 0.4 10 LCO cycles (inner) 0.3 20 LCO cycles (inner) 10 LCO cycles (outer)

0.2 20 LCO cycles (outer)

LCO amplitude LCO 50 LCO cycles (outer) 0.1 Inner LCO 0 0.75 0.8 0.85 0.9 0.95 1 Mach number Multiple limit cycle flutter branches of G-Wing in transonic region

OOB-UCLA 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 lin- earized without introducing nonuniformities in time or space, even in the limit of infinitesimal amplitudes (singular problem)

O. Bendiksen UCLA Conclusions Computational

6. The fluid-structure coupling problem remains of central importance in the devel- opment 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

10.The verification and validation steps of transonic aeroelastic codes present sev- eral practical problems that have not been adequately addressed in the research literature

O. Bendiksen UCLA