Optimization of Bezier Curves for High Speed Leading Edge Geometries

Rodi, Patrick E.

doi:10.2514/6.2013-1004

Cited by 16 publications

(15 citation statements)

References 14 publications

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“…This will enable the shape to have a lower drag coefficient for the same peak heat flux, something that will be demonstrated later on. It is also worth noting that, for cold wall cases of the same wall temperature, we observe the same pattern between the characteristics of the geometries for different flow conditions, something that was also observed by Rodi [7][8] . An example of the distribution obtained for a Mach 6, 75,000ft altitude case is shown in Figure 11.…”

Section: A Cold Wall Simulationssupporting

confidence: 57%

“…The two end points would be placed at the edge of the truncated part, on the original geometry, and the intermediate one is restricted in order to satisfy first order continuity at the interfaces of the curve and the original geometry. Two flexible alternatives to this approach can be either using higher order curves with more movable control points, as is the case in Rodi's work 7 , or by using rational Bézier curves whose shape can be further adjusted with weights. The latter of the two approaches has been followed in this work and this is what we discuss next.…”

Section: Leading Edge Geometry Parameterizationmentioning

confidence: 99%

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Parametric Geometry Models for Hypersonic Aircraft Components: Blunt Leading Edges

Kontogiannis¹,

Cerminara²,

Taylor³

et al. 2015

20th AIAA International Space Planes and Hypersonic Systems and Technologies Conference

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In this paper we report the results of investigations into the efficient parameterization of blunt leading edge shapes for hypersonic aircraft geometries. The investigations mostly revolve around waverider geometries generated with inverse design techniques, such as the osculating cones waverider forebody design method. The shapes presented however, can be utilized to introduce bluntness to any wedge-like geometry with sharp leading edges. Initially, we present detailed descriptions of three different variations of the rational Bézier curve based parameterization that was developed, and the variety of shapes that can be obtained is demonstrated. Afterwards their performance is evaluated utilizing 2D CFD analysis. In our simulations, the rational Bézier curve leading edges outperform circular ones when it comes to minimizing both drag and peak heating rates or peak temperatures. Additionally, with higher order rational Bézier leading edge shapes, higher levels of geometric continuity can be achieved at the interface between the blunt part and the original wedge-like geometry, resulting in a smoother transition. Preliminary results indicate that this can potentially affect the receptivity and hence transition mechanisms. Finally, the 2D geometry formulations are extended to full 3D waverider forebody geometries. Nomenclature

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Section: A Cold Wall Simulationssupporting

confidence: 57%

Section: Leading Edge Geometry Parameterizationmentioning

confidence: 99%

Parametric Geometry Models for Hypersonic Aircraft Components: Blunt Leading Edges

Kontogiannis¹,

Cerminara²,

Taylor³

et al. 2015

20th AIAA International Space Planes and Hypersonic Systems and Technologies Conference

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“…The results will be discussed for the three specific hypersonic waveriders at design conditions, with trends identified from supplemental solutions. In the author's earlier work 17,19 , a Genetic Algorithm optimization process was used to perform the optimization by physically relocating the control points used to define the Bezier Curve. For the current study, the GA-based process has been replaced with a Particle Swarm Optimization (PSO) 20 process to perform the optimizations in Matlab using a PSO toolbox 21 from the user community.…”

Section: Introductionmentioning

confidence: 99%

“…In an effort to develop finite leading edge geometries with superior aerodynamic and/or aerothermodynamic performance, an evaluation of using Bezier Curves to create leading edge geometries for high speed vehicles was performed 17 . This earlier work demonstrated that Bezier Curve-based geometries could be optimized to offer advantages over previously employed leading edge geometry approaches such as hemi-cylindrical or power-law curve based designs.…”

Section: Introductionmentioning

confidence: 99%

Integration of Optimized Leading Edge Geometries Onto Waverider Configurations

Rodi

2015

53rd AIAA Aerospace Sciences Meeting

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Most popular waverider design approaches produce vehicles that have sharp leading edges. Manufacturing and/or thermal limits require that some degree of finite radius be employed at the leading edge. Earlier research has produced an optimized leading edge generation process suitable for high speed vehicles such as waveriders. In that effort, Bezier Curves were employed to represent the candidate leading edge cross sectional geometries which were then optimized using a number of cost functions including: minimum peak heating, minimum total heating, minimum drag, and minimum pressure gradient.In the current work, Bezier Curve leading edges have been optimized to reduce the peak leading edge laminar heating for use on waverider-based vehicles designed for three typical hypersonic applications: a Mach 5 Air-Breathing Missile, a Mach 10 Hypersonic Cruise Vehicle, and a Mach 20 Boost-Glide Vehicle. These leading edge geometries have been incorporated onto the vehicles and the resulting integrated aerodynamic performance has been quantified and compared to similar vehicles with conventional hemi-cylindrical leading edge configurations. The largest drag reduction was for the Mach 5 missile, where a 6.2% reduction was predicted when using optimized leading edges on both the inlet and fins. Nomenclature h = altitude n = power law body exponent q = local convective heating rate qbar = freestream dynamic pressure r = power law body local radius r b = power law body base radius r CL = leading edge centerline radius r swept = radius for a swept leading edge t = independent parameter in the Bezier Curve equation C p = pressure coefficient D = drag L = lift, streamwise length of waverider on a given osculating plane M = freestream Mach number X = streamwise distance from the waverider's nose, also streamwise distance on a leading edge Y = spanwise distance from the centerline of the waverider, also vertical distance on a leading edge Z = vertical distance on the waverider measured from a waterline datum θ = leading edge wedge angle θ c = cone angle of power law body on the osculating plane λ loc = local leading edge sweep angle φ = leading edge position angle, measured from the most forward point on a leading edge

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“…In the current study, the coordinates of the blunt body geometry is parameterized by using the n th degrees of Bezier-Bernstein polynomials [19][20] …”

Section: Design Optimizationmentioning

confidence: 99%

Analysis and Adjoint Design Optimization of Hypersonic Blunt Bodies

Piskin¹,

Eyi²

2014

50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference

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The main purpose of this study is to analyze hypersonic flow field around the blunt bodies and to design of that bodies in order to obtain minimum pressure drag. Modeling of non-equilibrium must be done properly. In this study, non-equilibriums of thermal and chemical modes are considered. Translational and rotational energy modes are assumed that energy exchange between these modes is so fast. Vibrational and electronic energy terms are neglected. Therefore, one temperature is used to model thermo-chemical non-equilibrium. Flow field is assumed as inviscid and continuum region. Moreover, there is not diffusion. Thus, to model chemical non-equilibrium, finite rate chemistry can be used. For the thermal-nonequilibrium, enthalpy, entropy and specific heat constants are obtained from curve fitting methods. The coupled flow field equations are solved by using Newton's methods. To solve Newton's method, Jacobian matrices evaluation is required. In terms of convergence, Jacobian matrices are obtained by using analytical methods. At the design part, sensitivities are obtained by using adjoint design methods. The aim of design part is finding a hypersonic blunt geometry with minimum pressure drag while keeping the maximum temperature smaller than the baseline value. Nomenclature w flow variable vector in computational domain F, G flux vectors H, S source vector R residual in computational domain P static pressure of mixture P e electron pressure E total energy per unit mass H total enthalpy per unit mass h s mixture enthalpy per unit mass E v mixture vibrational-electronic energy per unit mass J coordinate transformation Jacobian e v vibrational-electronic mass e sf source term of electron  , curvilinear coordinates s  species density s  mass production rate of chemical reactions , ij  viscous shear stress 1 Graduated Students, Aeronautical and Astronatical Engineering, tugba.piskin@metu.edu.tr , AIAA Member 2 Associated Professor, Aeronautical and Astronatical Engineering, seyi@metu.edu.tr, AIAA Member Downloaded by UNIVERSITY OF NEW SOUTH WALES on August 13, 2015 | http://arc.aiaa.org |

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Optimization of Bezier Curves for High Speed Leading Edge Geometries

Cited by 16 publications

References 14 publications

Parametric Geometry Models for Hypersonic Aircraft Components: Blunt Leading Edges

Parametric Geometry Models for Hypersonic Aircraft Components: Blunt Leading Edges

Integration of Optimized Leading Edge Geometries Onto Waverider Configurations

Analysis and Adjoint Design Optimization of Hypersonic Blunt Bodies

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