High-Fidelity Simulations and Low-Order Modeling ofa Rapidly Pitching PlateJeff D. Eldredge∗and Chengjie Wang†Mechanical & Aerospace Engineering, University of California, Los AngelesLos Angeles, CA, 90095-1597, USAA thin flat plate undergoing a rapid pitch-up maneuver in a steady free stream is studiedwith both high-fidelity numerical simulations at Reynolds number 1000 and a low-orderinviscid point vortex model. The pitching rate and axis position are systematically varied,and their effect on the generated aerodynamic forces is inspected. It is found that themaximum lift and drag developed during the pitch-up both increase nearly linearly withincreasing pitch rate, though the rates of increase diminish as the pitching axis is moved aft.Furthermore, the maximum lift-to-drag ratio tends to saturate with increasing pitch rate,with the asymptotic value decreasing as the axis is moved aft. The forces predicted by thelow-order inviscid Brown–Michael model are compared with the high-fidelity results. Goodqualitative agreement is achieved, though the point vortex model tends to over-predict bothcomponents of force. The lift force obtained from the model is decomposed into inertialreaction and circulatory components, and their relative contributions are inspected.Nomenclature(˜U,˜V ) Plate centroid velocity components in ˜z frameα Plate angleα0Maximum angle of attack∆ThDuration of hold interval˙α0Nominal dimensional pitch rateΓjStrength of vortex jΩ Angular velocity of plate, ˙αθ Angular coordinate in circle planeζ Complex coordinate in circle planeζjPosition of vortex j in circle planeζ(i)jPosition of image of vortex j in circle plane˜z Plate-fixed coordinatesa Half-chord of plateasKinematic transition parameterc Chord of plateF,˜F Complex potentials in physical, circle planeFx, FyComponents of force∗Associate Professor, email: [email protected]. Member, AIAA.†Graduate student1 of 19American Institute of Aeronautics and Astronautics40th Fluid Dynamics Conference and Exhibit28 June - 1 July 2010, Chicago, IllinoisAIAA 2010-4281Copyright © 2010 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.K Dimensionless pitch rate, ˙α0c/(2U∞)P Complex fluid impulseW Complex velocity in physical planez Complex coordinate, x + iy, in physical planezcPlate centroidzj0Position of the releasing edge of vortex jzjPosition of the vortex j in physical planeI. IntroductionBiological mechanisms of flight have been the subject of much recent interest, particularly with theobjective of exploiting their remarkable capabilities in small-scale vehicles. The prevailing challenge inadapting these mechanisms for MAV use is describing, in a simplified manner, the important role of theunsteady nonlinear aerodynamics in determining the forces on the control surfaces. Reduced-order modelingof these fluid–body interactions for use in control strategies is therefore an important objective. Since theforce generation in these bio-inspired mechanics is strongly coupled to the dynamics of the vortical structures,it is paramount for models to embody these dynamics.Many of the unsteady phenomena and associated modeling challenges associated with biologically-inspiredflight mechanics can be highlighted in an ostensibly simple two-dimensional example. A rigid flat plateat initially zero angle of attack in a steady free stream is pitched steadily upward to 45 degrees, held fora short interval, then pitched downward at the same rate and held again at zero degrees. This problemhas been a recent target of study for several experimental and computational investigations, including theauthors.1–3A schematic is shown in Figure 1. This flow is especially notable for the growth and sheddingof a prominent leading-edge vortex (LEV), a feature whose influence on the force cannot be described byclassical quasi-steady aerodynamics. In particular, the growth of the vortex – on which this paper focuses –is strongly influenced by the rate of pitching and by the position of the pitching axis, and thus this problemserves as a useful test bed on which to explore reduced-order models for the unsteady aerodynamic response.Phenomenological approaches to low-order modeling of unsteady aerodynamics date from the work ofTheodorsen4and von K´arm´an & Sears.5At their heart, these models decompose the force and momenton the wing into contributions from circulatory (i.e. vortex-induced) and non-circulatory (i.e. inertial reac-tion, or added mass) effects. The models are generally based on potential flow theory, in which the circulatoryforces are accounted for by directly computing the influence of shed vorticity, whose dynamics are generallyrepresented by some simplified description: as a sheet (e.g. the flat wake of Theodorsen4and von K´arm´an& Sears,5the similarity solution for spiral evolution by Pullin & Wang,6or with full non-linear dynamics asin Nitsche & Krasny,7Jones8and Shukla & Eldredge9); a continuous sequence of point vortices;10–12or alimited number of point vortices with evolving strengths.13–15In all of these models, vorticity is generatedat salient edges via a Kutta condition.These potential flow models often do a fair job of predicting the circulatory force and moment on the wingwhen the angle of attack remains small and no LEV is generated. However, their success is less certain inflows in which the LEV plays a significant role – that is, when a Kutta condition must be applied at boththe leading and trailing edges. The physics of the interaction between the LEV with the wing are essentiallyviscous, causing the inviscid model to develop unphysical behavior. However, at higher angles, for which thedevelopment and shedding of the LEV are primarily inviscid processes, a potential flow approach seems toprovide better results.12, 16However, a method that is generally applicable to a wide range of wing kinematicsis still lacking.The objectives of this study are to explore the development of the LEV for the pitch-up problem under avariety of different pitch rates and axis positions, using both high-fidelity simulation and low-order potential2 of 19American Institute of Aeronautics and AstronauticsU∞ααXpcFigure 1. Schematic of pitching wing.flow modeling, and to assess the capability of the low-order modeling approach by comparing with thehigh-fidelity results. The high-fidelity study utilizes the viscous vortex particle