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Transonic Aeroelasticity: Theoretical and Computational Challenges 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, 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 <<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 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. Mach number 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 (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 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 Aerodynamics 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
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