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Aerodynamics Glossary Wing Design Tip A most difficult aspect of wing design can be choosing the correct airfoil cross sectional shape. Although most airfoil shapes can support flight, only the right one will save thousands of dollars in operational costs over the life of the aircraft. MultiSurface Aerodynamics is a digital wind tunnel that can compare the performance of many airfoil shapes to make the airfoil selection process easy. Aerodynamic Center The aerodynamic center is a point along the airfoil or wing about which the moment coefficient does not vary with an angle of attack change. Airfoil An airfoil is the cross section of a wing. The airfoil shape and variations in angle of attack are primarily responsible for the lift and profile drag of the wing. Angle of Attack The angle of attack is defined as the angle between the plane of the wing (airfoil chord) and the direction of motion (free stream velocity). The angle of attack can be varied to increase or decrease the lift acting on the wing. An increase in lift often results in an increase in drag. Center of Pressure A point along the airfoil about which the moment due to the lift is zero, i.e., it is the point of action of the lift. The center of pressure will change its position when the angle of attack changes. Chord The chord is the dimension of the airfoil from its leading edge to trailing edge. Circulation Circulation is a measure of the vorticity in the flow field. For an inviscid flow field, the lift is equal to the product of the circulation about the airfoil, the density and the velocity. Computational Fluid Dynamics (CFD) Computational fluid dynamics is the term given to a variety of numerical mathematical techniques applied to solving the equations that govern fluid flows and aerodynamics. Modern CFD results can rival the accuracy of wind tunnels in testing airfoils, wings and entire airplanes for certain test configurations. Density The mass of a substance contained in a given volume divided by the volume. For a incompressible fluid, the density is considered to be constant throughout the flow field. However, for a compressible fluid, the density can vary from one location to the next in the flow field. The speed of sound in a fluid depends on the ratio of pressure changes to density changes in the fluid. Drag Drag is an aerodynamic force opposing the direction of motion. Drag can be due to surface viscosity (friction drag), pressure differences due to the shape of an object (form drag), lift acting on an finite wing (induced drag) and other energy loss mechanisms in

the flow such as wave drag due to shock waves and inefficiencies in engines. Drag Coefficient The drag coefficient is defined as the drag/(dynamic pressure * reference area). The reference area is usually the plan-form or flat projection (the wing's shadow at noon) area of the wing. Dynamic Pressure The dynamic pressure is defied as the product of the density and the square of the velocity divided by two. The dynamic pressure has units of pressure, i.e. Force/Area. The dynamic pressure is used to non-dimensionalize forces and pressures in aerodynamics. Flap Deflection Angle The flap deflection angle is the angle between the deflected flap and the chord line. The angle is positive for a downwards deflection of the flap. Deflect the flap downwards to increase the airfoil's lift. Lift The lift is a force acting perpendicular to the direction of flight. The lift is equal to the fluid density multiplied by the circulation about the airfoil and the free stream velocity. In level flight, the lift developed by an airplane's must be equal to the weight of the entire airplane. Lift Coefficient The lift coefficient is defined as the lift/(dynamic pressure * reference area). The reference area is usually the plan-form area of a wing or horizontal projection of the wing. Mean aerodynamic chord This chord is located along the wing and has the aerodynamic property of the twodimensional wing. NACA Airfoils NACA airfoils are wing cross section designs invented by the NACA organization. NACA eventually became NASA (National Aeronautics and Space Administration). Here are a few popular airplanes that have NACA airfoil wings:

Airplane

Root Airfoil

Tip Airfoil

Beech 50 Twin Bonanza

NACA 23014.1

NACA 23012

B-17 Flying Fortress

NACA 0012

NACA 0010

Cessna 152

NACA 2412

NACA 0012

Cessna 172 1973-later

NACA 2412

NACA 2412 mod

Cessna 550 Citation II

NACA 23014

NACA 23012

Douglas DC-3

NACA 2215

NACA 2206

Fairchild A-10 Thunderbolt II

NACA 6716

NACA 6713

Sikorsky S-61 SH-3 Sea King

NACA 0012

NACA 0012

Panel Method This numerical method places singularities along the airfoil. In the case of VisualFoil or 3DFoil , the singularities are vortices. The vorticity is distributed linearly along the panel.

Plain Flap A plain flap is a hinge attachment near the trailing edge of an airfoil. The length of the flap is measured as a percentage of the chord and the deflection is measured in degrees. Pressure Coefficient The pressure coefficient is a non-dimensional form of the pressure. It is defined as the difference of the free stream and local static pressures all divided by the dynamic pressure. Reynolds Number The Reynolds number is a non-dimensional parameter that compares the inertia to viscous forces. If the Reynolds number is low, then viscosity plays an importatant part in the simulations. Stall At low angles of attack, the lift developed by an airfoil or wing will increase with an increase in angle of attack. However, there is a maximum angle of attack after which the lift will decrease instead of increase with increasing angle of attack. This is know as stall. Knowing the stall angle of attack is extremely important for predicting the minimum landing and takeoff speeds of an airplane. Streamlines Contours in the flow field that are tangent to the velocity vector. Wing Loading The total weight of the airplane divided by the plan form area of the wing. Wing Span The span is the total length of the wing.

About Dr. Hanley Dr. Patrick E. Hanley, is the owner and founder of Hanley Innovations, a small business specializing in the development of aerodynamics and fluid dynamics simulation software for education and industry. Dr. Hanley earned his B.S. degree (summa cum laude) in aerospace engineering from Polytechnic Institute of New York and his S.M. and Ph.D. degrees from the department of Aeronautics and Astronautics of Massachusetts Institute of Technology (MIT). He also completed a minor in the area of management of innovation and technology at MIT's Sloan School of Management. After graduating from MIT, Dr. Hanley joined the Mechanical Engineering faculty at the University of Connecticut where he formulated and taught courses in aerodynamics, compressible fluids, introductory fluid mechanics and heat transfer. As a faculty member, he won the highly competitive National Science Foundation Research Initiation Award, the NASA-ASEE Summer Faculty Fellowship and three consecutive research awards from NASA Lewis Research center to study compressible viscous flows in turbomachinery using pseudospectral methods. This research led to the successful education of four (4) Ph.D students and four (4) Masters degree students. In addition Dr. Hanley can be credited with a number of publications including the pioneering work in multi-domain pseudospectral methods for compressible viscous flows entitled "A Strategy for the Efficient Simulation of Viscous Compressible Flows using a Multidomain Pseudospectral Method" which can be found in Journal of Computational Physics, Vol 108, No. 1, pp. 153-158, September 1993. MultiSurface AerodynamicsTM is the leading CAE (computer aided engineering) product for multiple wing design & analysis on the PC. MultiSurface turns your PC into a powerful wind tunnel & diagnostic tool for creating & testing airfoils & 3D wing systems.The userinterface is self-contained and has tools for design, analysis, presentations and prototyping. MultiSurface computes the lift, total drag, moments and stability derivatives of your custom wing system in just seconds on a Windows PC. Use different airfoils and wing shapes to optimize your design on the fly. No advanced aerodynamics or computer programming skills are required to use the software.

Design & Test Wings On Your PC!

MultiSurface Aerodynamics features include: stand-alone application (built-in pre & post processing tools) easily enter and manipulate complex wing system; instantly compute lift, drag (profile and vortex), moments and stability derivatives; calculate trim, neutral point and ground effect. ability to read-in, modify and analyze custom airfoil shapes to test the effectiveness of original designs. users do not need to have advanced aerodynamics or computer programming skills accurate solvers for realistic simulations computer aided design interface to model and edit 3D wing systems (joined wing for example) interactive visualizations graphs & tables for comparative analysis

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ability to export 2D & 3D CAD files for rapid prototyping. More Examples Sailplane Analysis Joined Wings Endplates (.pdf) Sprint Car Wing (.pdf) MultiSurface Validation Hydrofoil Analysis Elliptical Wing Other Wing Shapes Applications MultiSurface Aerodynamics is the ultimate desktop wind tunnel software. Use it to design and test complex wing systems directly on your Windows PC. It was developed by Patrick Hanley, Ph.D. for accuracy, efficiency and ease of use and is by far the quickest method on the market for developing wings for aircraft, water craft and automobiles. MultiSurface is used in the following areas:       

Light aircraft wing design & analysis. This includes kites, powered parachutes, parafoils and other sporting equipment. UAV design & analysis Multi-plane wing configurations including joined wings Sailboat/yacht keel and rudder design & analysis Up-wind sail design & analysis Ship/sailboat hydrofoil design & analysis Human powered hydrofoil design & analysis

   

Water sport equipment design. This includes sailboard fins and sail configurations. Equipment for marine industry such as foils for stabilizing vessels & trawl doors. Automobile wing, spoilers and surfaces designed for safety and fuel efficiency. Lectures and presentations in the area of aerodynamics design and analysis.

Online Tutorials & Demos Tutorial - How to enter and analyze a wing. View tutorial now. Demo - How to improve a sailboat keel. View demo now Demo - How to compare the performance of two keels. View demo now. Demo - Using the mirror imaging feature for wing design. View demo now. Examples Design and analyze joined wings for UAV applications. More information ... Unmanned aerial vehicles design and analysis. More information ... Sailboat keel & rudder analysis and design. More information ... Analyze hydrofoil with front and rear foils. More information Analyze hydrofoils in shallow water. More information ...

...

Analyze an elliptical wing. More information ... Analyze a full-sized sailplane. More information ... Design & analyze full-sized race car wings and spoilers. More information ... Analyze a full-sized sprint car wing with endplates (pdf file). More information ... Educational Use Introduce engaging experiments, design content & capstone projects into your lectures & labs . More information ... Who Uses MultiSurface Aerodynamics MultiSurface Aerodynamics is best suited for users who possess a good understanding of aerodynamics, hydrodynamics and related engineering principles. Our customers leverage their design & engineering knowledge with MultiSurface Aerodynamics to develop outstanding products and services. The following companies, consultants and universities are among the users of MultiSurface Aerodynamics:           

AeroScience CDE Danish Marine Design ApS Clyde Booth (race car wing engineer) Hugh Welbourn (British yacht designer) Island Engineering, Inc Kimokeo Engineering Korea Ocean Research & Development Institute Mike Mageria (race car wing designer) MultiHull Power Paradis Nautica AS Sea State

   

United States Air Force Universitat Politècnica de Catalunya University of California, San Diego University of Edinburgh

Computer Requirements The perpetual licence of MultiSurface Aerodynamics requires a PC running Windows 95 or Later. The leased (online) versions requires a PC running Windows 95 or Later and a connection to the internet. The leased software accesses the internet for login purposes only

Overview VisualFoil Plus is used by engineers and designers to compute the lift, drag and moment coefficients for airfoils in subsonic, transonic and supersonic flows. It generates accurate aerodynamics data for experimental or existing airfoils where the information might be incomplete or unknown. The software can analyze and select airfoils for aircraft, water craft, industrial machinery and other products. VisualFoilPlus produces graphs of lift, drag and moment coefficient versus angle of attack; surface pressure coefficient, velocity, Mach number and temperature. The software produces contour plots of pressure, Mach number, temperature and total pressure. VisualFoil Plus has a large library of built-in airfoils which includes NACA 4, 5 & 6-digit airfoils. In addition, the user can enter custom airfoils for analysis and modify existing airfoil shapes. Please click here to purchase VisualFoil Plus. Benefits      

Eliminate steep learning curve for new engineers & reduce training costs. Share airfoil data & results across your entire department Understand and make critical design decision in transonic & supersonic flow regime . Understand the behavior of airfoils used in the design of wings, helicopter blades, hydrofoils, fans, propellers and other equipment. Easy-to-use tool for designing more efficient wing & machine blade cross-sections. Windows based interfaces requires little or no training.

Incompressible Flow Solver  

Linear strength vortex panel method Coupled boundary layer solver

      

Lift, drag & moment coefficient calculations Prediction of transition & separation points Stall & maximum lift prediction Surface graphs of pressure & velocity ratio Contour graphs of Cp Graphs of Cl, Cd & Cm versus angle of attack Graphs of Cl vs Cd.

Compressible Flow Solver       

2-dimensional compressible Euler equations using a finite-volume solver 2nd order flux vector splitting 2nd order flux-difference splitting 4-stage Runge-Kutta time marching scheme using local time steps Automatic o-grid generation Contour plots of pressure, Mach number, temperature & total pressure Surface plots of Cp, Mach number, velocity ratio & temperature

Other Features of VisualFoil Plus           

Incompressible flow solver using vortex panel method coupled with boundary layer solver Compressible flow solver based on Euler Equations Built-in library of NACA 4, 5 & 6-digit airfoils Custom airfoil analysis Computes Cl, Cd & Cm Stall Model Built-in graphs Built-in tables Export tables to .csv files Export airfoil to .dxf files Airfoil plotting

Articles  

High Speed Blade Screening (.pdf file) VisualFoil Plus for Propeller Analysis

Computer System Requirements VisualFoil Plus is a stand-alone software package that requires Windows 95 or later. We recommend a pentium based PC running Windows XP. The leased version of VisualFoi Plus also requires an internet connection. Purchase How to Purchase VisualFoil Plus Note: VisualFoil Plus is available exclusively on hanleyinnovations.com.

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Price

Purchase

Lease for Three Months

$695.00

Buy CD Now

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$1,695.00

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$2,995.00

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Perpetual (3 Users) $6,995.00 Buy CD Now Telephone Orders: Please call us at (352) 687-4466 to place your credit card order by telephone. We can provide you with a download URL & password after the transaction is processed. Fax Your Orders: Please click here to fax your order. (.pdf order form) Order by Mail: Click here to order by mail using a check, money order or credit card. ** The leased versions of VisualFoil Plus are identical to the full perpetual version. Unlike the perpetual version, however, they require an internet connection to enter a username & password.. Note: All prices subject to change without notice.

Other Payment Methods We also accept payment by Bank Transfer (SWIFT for example). Please email us for details. Request More Information: For more information, please email us at [email protected]. We can also be reached by telephone at (352) 687-4466. Our mailing address is Hanley Innovations, Attn. VisualFoil Plus, PO Box 831514, Ocala, Fl 34483-1314. See Also: MultiElement Airfoils, MultiSurface Aerodynamics

About Dr. Hanley Dr. Patrick E. Hanley, is the owner and founder of Hanley Innovations, a small business specializing in the development of aerodynamics and fluid dynamics simulation software for education and industry. Dr. Hanley earned his B.S. degree (summa cum laude) in aerospace engineering from Polytechnic Institute of New York and his S.M. and Ph.D. degrees from the department of Aeronautics and Astronautics of Massachusetts Institute of Technology (MIT). He also completed a minor in the area of management of innovation and technology at MIT's Sloan School of Management. After graduating from MIT, Dr. Hanley joined the Mechanical Engineering faculty at the University of Connecticut where he formulated and taught courses in aerodynamics, compressible fluids, introductory fluid mechanics and heat transfer. As a faculty member, he won the highly competitive National Science Foundation Research Initiation Award, the NASA-ASEE Summer Faculty Fellowship and three consecutive research awards from NASA Lewis Research center to study compressible viscous flows in turbomachinery using pseudospectral methods. This research led to the successful education of four (4) Ph.D students and four (4) Masters degree students. In addition Dr. Hanley can be credited with a number of publications including the pioneering work in multi-domain pseudospectral methods for compressible viscous flows entitled "A Strategy for the Efficient Simulation of Viscous Compressible Flows using a Multi-domain Pseudospectral Method" which can be found in Journal of Computational Physics, Vol 108, No. 1, pp. 153-158, September 1993.

As owner and chief software author of Hanley Innovations, Dr. Hanley has written a number of software packages including AirfoilBrowser, Airfoil Organizer, Science Graphs, VisualFoil, ModelFoil, Aerodynamics in Plain English, Center of Gravity Calculator, WingAnalysis, SmockSoft, PerpeturalPaper amongst other

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Page 1

A Study of Shape Parameterisation Methods for Airfoil Optimisation Wenbin Song ∗

and Andrew J. Keane †

University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom This paper presents a study on parameterisation methods for airfoil shape optimisation within a CAD­based design optimisation framework. The objective of the paper is to study the effect of different methods on airfoil shape optimisation when using computational fluid dynamics (CFD). Parameterisation of geometry is one of the essential requirements in shape optimisation, and it presents further challenges when carrying out multidisciplinary design optimisation, as it is critically important to maintain shape consistency between analysis domains, while providing different analysis models from the same CAD definition. It is usually the case that there are numerous possibilities in defining the parametric model, and it will prescribe to a large extent the scope of the search space and landscape of the objective function. This paper adopts design of experiments and optimisation approaches to study several representative parameterisation methods in terms of flexibility and accuracy of the methods for aerodynamic shape optimisation.

I. Introduction

A

ircraft design is a complex decision making process and according to Raymer, 1

can usually be broken down into three phases, conceptual design, preliminary design, and detailed design. Aerodynamic design occurs throughout these steps. Two different approaches are often employed in the aerodynamic design: 1) inverse design and 2) direct numerical optimisation. The first method tries to solve for a geometry that produces a prescribed pressure distribution. On the other hand, direct numerical optimisation methods couple a geometry definition and aerodynamic analysis code in an iterative process to produce optimum designs subject to various constraints. Depending on whether the goal is to improve on an existing design or to create a completely new design, different parameterisation methods are often required. If the new design only requires small changes to the initial geometry, a localized parameterisation approach is often used. But when conducting a study of a radically new concept, the parameterisation method needs to accommodate a wider range of new shapes.

Airfoils have been represented in a number of different ways in the past. For example, coordinates have been directly used to fit airfoil shapes using B­splines and Bzier curves 2

via interpolation methods. Analytical functions have also been derived to represent families of airfoils, for example, in the work reported by Hicks and Henne. 3

In a more recent work, 4

Non­uniform rational B­splines (NURBS) were used first to approximate existing airfoils, then adopted as a general parameterisation method to be used in optimisation. The concept of using relatively few orthogonal functions to represent a large number of functions has also been exploited, for example in a work reported by Robinson and Keane, 5

where a set of orthogonal functions was developed using numerical methods. These functions were then used to represent a family of airfoils in a wing design study. However, the basis functions derived by Robinson and Keane 5

were believed to be dependent on the particular familiar of airfoils. Although these numerically derived basis functions can be used in the design of a particular set of airfoils, other airfoils may not be adequately represented using them. The choice of parameterisation method, when coupled with optimisation techniques to find desirable shapes in terms of user­defined objective functions and constraints, has a major effect on the final results, efficiency and effectiveness of particular search strategies. Giving the same CFD models, the parameterisation ∗

Research Fellow, School of Engineering Sciences, University of Southampton, Highfield, SO17 1BJ, United Kingdom, and AIAA Member. †

Professor, School of Engineering Sciences, University of Southampton, Highfield, SO17 1BJ, United Kingdom. 1 of 8 American Institute of Aeronautics and Astronautics

Page 2 effectively defines the optimisation problem formulation, the topology of the design space, and the landscape of the objective functions. Although it is vitally important, it is also very difficult to come up with a set of effective criteria that can be readily used to evaluate the pros and cons of different parameterisation methods. Wu 6

compared three geometric representations in three case studies of cascade blade design using adjoint methods. After carrying out the optimisation, it has concluded that one of the methods using geometry parameters is not suitable for two occasions. There are a number of key issues that need to be addressed in the choise of parameterisation methods. The first issue is the flexibility of any parameterisation method. Flexibility is interpreted here as the ability to represent a wide range of different shapes. Some parameterisation methods, for example, coordinate­ based methods, can accurately represent a variety of dramatically different shapes and can also reflect subtle changes in local areas, however it would be very difficult to use such an approach for optimisation problems using high fidelity codes due to the large number of design variables and complexity of the design space. On the other hand, methods using fewer variables may not be capable of generating shapes with high accuracy, especially when used in inverse design problems where a target pressure distribution is sought. The second issue when considering parameterisation methods is the accuracy or the optimum objective functions that the final shape can achieve either in an inverse design study or direct optimisation work, respectively. The accuracy should be measured in both geometric and aerodynamic senses. However, the optimal objective function cannot be obtained without actually carrying out the optimisation, therefore, here an inverse design approach is adopted to compare different parameterisations. In this work, three datum airfoils, two from the NACA supercritical airfoils family (NACA0406 and NACA0610) and the third being the RAE2822, are used as reference shapes to compare different param­ eterisation methods for airfoil design. The paper is organised as follows. Section two describes different parameterisation methods for airfoil design. The geometry modelling and flow analysis of the airfoil prob­ lems are described in section three. Results and discussions are presented in section four, with conclusions given in section five.

II. Parameterisation of Airfoil Geometry

Figure 1. The first three numerically derived or­ thogonal basis functions.

Geometry parameterisation methods have attracted renewed interests in recent years, especially in the con­ text of multidisciplinary design optimisation (MDO). Samareh 7

identified three categories of parameterisation methods in the context of MDO. These include the dis­ crete approach, CAD­based approaches, and free­form de­ formation methods. Indeed, all these different approaches could be implemented in most modern CAD systems. Several parameterisation methods have been proposed in previous papers for airfoil geometry, for example, the NACA supercritical airfoils are defined as a series of y­ coordinates at prescribed chord wise locations. 8

The sec­ ond approach models the geometry as the linear combina­ tion of a basis airfoil and a set of perturbation functions, defined either analytically 4

or numerically, 5

as shown in Eq. (1). The coefficients of the perturbation functions in­ volved are then considered as the design variables. A set of such orthogonal basis functions derived from a group of base airfoils was developed by Robinson and Keane 5

to provide an efficient means to define the airfoil for optimi­ sation study in preliminary design, for example. y(x) = α 0

y 0

(x) + w i

f i

(x) (1) A third, and more geometrically intuitive method, is to use geometric parameters such as leading edge radius, thickness­to­chord ratio or maximum thickness to define the airfoil shape. An airfoil parameterisation using 11 geometry parameters was presented by Sobieczky 9

and used by Oyama etc. 10

A fourth method uses the control points of Non­uniform Rational B­splines (NURBS) curves to define the airfoils. 4

This method is 2 of 8 American Institute of Aeronautics and Astronautics

Page 3 also used by Li 11, 12

in which a B­pline interpolation through 35 points is used to define the airfoil geometry. The advantage of this approach is that free­form geometrical shapes can be accommodated with fewer design variables compared to the direct use of coordinates. However, due to difficulties in controlling the relative

positions of the control points, free­form parameterisations are usually used in the inverse design approach, where only a subset of control points are allowed to change in a relatively small range to meet a target pressure distribution. One of the key issues in deciding the parameterisation method is the balance the requirements of ro­ bustness and flexibility, and these decisions are also strongly dependent on the goal of the design activity. Although free­form parameterisations may well be able to generate radical new shapes, this is not suitable for designs where the aim is to meet a specific pressure distribution, due to the poor efficiency caused by the large search space that arises in the optimisation process. Another disadvantage of free form parameterisa­ tion is the inherent difficulties encountered when trying to generate airfoil­like shapes: Usually additional geometrical constraints need to be imposed. Two different parameterisation methods are implemented in the current work to compare their effectiveness. The first approach is to use a set of numerically derived basis function to define the airfoil, 5

in this case, only a small number of design variables are involved. The basis functions used are illustrated in Figure 1. The second approach uses a B­spline interpolation based on 34 points as shown in Figure 2. The y coordinates of the points are used as design variables while the chord wise coordinates of the points are fixed. Both methods are implemented in the CAD system ProEngineer. 13

Airfoil shapes from the family of supercritical airfoils 8

and RAE2822 are chosen in the current work as ref­ erence airfoils in the comparison. The number of parameters involved in the two parameterisation methods are summarized in Table 1. Table 1. Summary of different parameter methods for airfoils

Method Number of parameters Description of the parameters Numerical Basis Functions 5 Weights for the basis airfoil functions B­spline interpolation 34 Point coordinates

III. Geometry Modelling and Flow Analysis X/C Y / C

0 0.25 0.5 0.75 1 ­0.03 ­0.025 ­0.02 ­0.015 ­0.01 ­0.005 0 0.005 0.01 0.015 0.02 0.025 0.03

Figure 2. Airfoil Modelling using B­spline inter­ polation.

To compare the flexibilities of different parameterisa­ tion methods in representing different airfoil shapes, three existing airfoils NACA0406, NACA0610, and RAE4822 are chosen as the modelling targets for these parameter­ isations. The difference between the target airfoil and approximated airfoil is defined as diff = abs(f target

(x i

) − f(x i

))

(2) where x i

(i = 1, ..., n) are the chordwise coordinates used in the definition of the target airfoils. This quantity is minimised using a global optimisation algorithm for dif­ ferent parameterisation methods and then used as an in­ dication of the flexibility of the methods. To remove the effect of the optimisation techniques that may not pro­ duce the optimum results, a large number of iterations has been carried out for the minimisation problem. The airfoil models are all implemented using ProEngi­ neer and exported in the form of a STEP file, which is then imported into Gambit. A boundary layer is attached to the lower and upper surface of the airfoils, and size functions are also used to give better control of the mesh and to reduce the computational time for the problem. An unstructured mesh generated using Gambit 14

for solving the N­S equations is shown in Figure 3. The mesh contains 11356 cells (compared with 87305 cells without using the size function). In both cases, 3 of 8 American Institute of Aeronautics and Astronautics

Page 4 the node spacing on the airfoil surfaces and farfield circle are the same, with size functions giving better control of the transition of size of the cells in between. The computation time is reduced from around 40 minutes to less than 20 minutes for most geometries on a Xeon 2.4Ghz compute node with 1Gb memory. The flow model used in the current work is based on the Navier­Stoke model from Fluent. 14

The pressure distribution of the upper and lower surfaces are used in the comparison. Here, the cruise condition (M ∞

= 0.73) is used when calculating the lift and drag values. The Spalart­Allmaras viscosity model is used.

IV. Results and Analysis Figure 3. Unstructured mesh used for solving the N­S equations by Fluent

It is not straightforward to compare alternative parameterisation methods. There are two impor­ tant considerations when a parameterisation model is built around an existing geometry: the first is the flexibility of the model, i.e., how many different shapes can this model represent. The second is be the robustness of the model, i.e., can the model gen­ erates the desired shape for large number of different designs. In general models with more design vari­ ables will be able to represent more complex shapes, and will be more likely to produce novel designs us­ ing optimisation. However, that will be more expen­ sive in the search as the design space will have higher dimensions, and chances of failure or not generating desired shapes will be higher. The best results for approximations of the tar­ get airfoils are shown in Table 2. The results are produced by minimising the objective function computed using (2). A genetic algorithm (GA) from OP­ TIONS 15

is used in the current work, however, the first population is not generated randomly, rather, it is generated using a Design of Experiment (DoE) method plus one point describing a user specified base

design to obtain more uniform coverage of the design space as well as to provide the best possible guess. The DoE method used here is a Latin Hyper Cube method. The base design specified by the user can play an important role in accelerating the search process as the GA used in this work always maintains the best solution in the population. A single base design is used in the orthogonal basis function approach for all three target airfoils and in the B­spline approach for RAE2822 approximation. The two base designs used in B­spline interpolation for NACA0406 and NACA0610 are NACA0403 and NACA0606, respectively. It can be seen that the approximations using orthogonal basis function can produce better results for the NACA supercritical airfoils NACA0406 and NACA0610 than for the RAE2822, this is not surprising, as the set of basis functions used were derived from a family of airfoils containing these two. The errors are believed to be caused by the smoothing process in the derivation of these basis functions. 5

For NACA0406 and NACA0610, the orthogonal basis functions also produce better results than the B­spline interpolation approach, this is because more variables are used in the B­spline interpolation and so it would be much more expensive to obtain the optimum if a comparable number of iterations were used for both cases. However, a different story arises for RAE2822. Since it was not included in the process of deriving the basis functions, this leads to greater error when compared with the B­spline interpolation approach. This indicates the wider applicability of the B­spline approach, at the higher cost of reaching the optimum. Table 2. Best Approximation of target airfoils for different parameterisations

Method RAE2822 NACA0406 NACA0610 Numerical Basis Functions 0.2217 0.0582 0.1595 B­spline interpolation 0.1552 0.0758 0.1993 However, similarity in geometry does not always guarantee similar pressure distribution, especially for 4 of 8 American Institute of Aeronautics and Astronautics

Page 5 transonic flows where small perturbations in shape will lead to large variations in pressure. Therefore, the pressure distributions of the approximated shape are also compared to that of the original airfoils, as shown in Figure 4. It can be seen from Figure 4 that the orthogonal basis function approach always produces smooth pressure distributions; also errors are generally bigger in the leading and trailing edge areas than in the middle section of the airfoils. Figure 5 shows the results of approximation using the B­spline interpolation approach. It can be seen that close agreement can be achieved using B­spline interpolation through 34 points apart from the leading edge area, which indicates that more points need to be placed within this area to achieve better results. Moreover the chordwise coordinates can also be varied, but this would involve higher computational cost while not necessarily increasing the accuracy of the approximation. Another advantage of the B­spline approach is its ability to carry out local shape tunning by varying a subset of the coordinates, while it would be difficult to perform this with the basis function approach, in which, any changes in the coefficients will change the shape globally. The approach adopted in the current work is essentially an inverse design method. However, it is not used to seek a prescribed pressure distribution, as the definition of the pressure distribution itself is a design problem and accurately re­generating the prescribed pressure distribution often leads to degradation of performance in other conditions. This method can be used to evaluate different parameterisations before carrying out optimisations using the high fidelity codes.

V. Conclusions

Two airfoil parameterisation approaches are studied in this paper to analyse their flexibility and robust­ ness in producing optimal shapes when used in optimisation studies. Global optimisation methods are used to analysis the accuracy these two parameterisations can achieve when used to model three target airfoils. The B­spline approach produces better results in terms of accuracy at a higher computational cost while the basis function approach is more efficient while producing less accurate results. Further work will in­

volve the combination of the basis function approach in the initial stages of design combined with B­spline interpolation for the final tuning of the shapes.

Acknowledgement

The work described here is supported by the UK e­Science Pilot project: Grid­Enabled Optimisation and Design Search for Engineering (Geodise) (UK EPSRC GR/R67705/01). Financial support from EPSRC is greatly acknowledged.

References 1

Raymer, D.P., Aircraft Design: A Conceptual Approach, AIAA Educational Series 1999. 2

Cosentino, G. B. and Holst, T. L., Numerical Optimisation Design of Advanced Transonic Wing Configurations, May, 1984. 3

Hicks, R. M. and Henne, P. A., Wing Design by Numerical Optimisation, Journal of Aircraft, Vol. 15, No. 7, 1978, pp. 407­413. 4

Lepine, J., Guibault, F., Trepanier, J.­Y., and Pepin, F., Optimized Nonuniform Rational B­spline Geometrical Repre­ sentation for Aerodynamic Design of Wings, AIAA Journal, Vol. 39, No. 11, 2001. 5

Robinson, G. M. and Keane, A. J., Concise Orthogonal Representation of Supercritical Airfoils, Journal of Aircraft, Vol. 38, No. 3, 2001, pp. 580­583. 6

Wu, H.Y., Yang, S.C., Liu, F. and Tsai, H.M., Comparison of Three Geometric Representations of Airfoils for Aero­ dynamic Optimization, AIAA 2003­4095, 16th AIAA Computational Fluid Dynamics Conference, June 23­26, Orlando, FL, 2003 7

Samareh, J. A., Survey of shape parameterization techniques for high­fidelity multidisciplinary shape optimisation, AIAA Journal, Vol. 39, No. 5, 2001, pp. 877­884. 8

Charles D.H., NASA Supercritical Airfoils, A Matrix of Family­Related Airfoils, NASA Technical Paper 2969, Mar. 1990. 9

Sobieczky, H., Parametric Airfoils and Wings, Notes on Numerical Fluid Mechanics, edited by K. Fujii and G. S. Du­ likravich, Vol. 68, Vieweg Verlag, 1998, pp. 71­88. 10

Oyama, A., Obayashi, S., and Nakamura, T., Real­coded Adaptive Range Genetic Algorithm Applied to Transonic Wing Optimisation, Springer, Paris, France, 2000, pp. 712­721. 11

Li, W., Huyse, L., and Padula, S., Robust airfoil optimisation to achieve drag reduction over a range of Mach numbers, Structural and Multidisciplinary Optimisaiton, Vol. 24, 2002, pp. 38­50. 5 of 8 American Institute of Aeronautics and Astronautics

Page 6 X/C Y / C 0 0.25 0.5 0.75 1 ­0.06 ­0.05 ­0.04 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 RAE2822 Orthfoil­Approximation

(a) Approximation of geometry to RAE2822 by basis functions approach X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1

­0.5 0 0.5 1 RAE2822 Orthfoil­approximation

(b) Comparison of pressure distributions for RAE2822 using basis functions X/C Y / C

0 0.25 0.5 0.75 1 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 NACA0406 Orthfoil­Approximation

(c) Approximation of geometry to NACA0406 by basis func­ tions approach X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1 ­0.5 0 0.5 1 NAC0406 Orthfoil­approximation

(d) Comparison of pressure distributions for NACA0406 using basis functions X/C Y / C

0 0.25 0.5 0.75 1 ­0.05 ­0.04 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 0.04 0.05 NACA0610 Orthfoil­Approximation

(e) Approximation of geometry to NACA0610 by basis func­ tions approach X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1 ­0.5 0 0.5 1 NACA0610 Orthfoil­approximation

(f) Comparison of pressure distributions for NACA0610 using basis functions

Figure 4. Comparisons of geometry and pressure distributions for approximations using orthogonal basis functions 6 of 8 American Institute of Aeronautics and Astronautics

Page 7 X/C Y / C 0 0.25 0.5 0.75 1 ­0.06 ­0.05 ­0.04 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 RAE2822 B­Spline interpolation

(a) Approximation of geometry to RAE2822 by B­spline in­ terpolation X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1 ­0.5 0 0.5 1 RAE2822 B­Spline Interpolation

(b) Comparison of pressure distributions for RAE2822 using B­spline interpolation X/C Y / C

0 0.25 0.5 0.75 1 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 NACA0406 B­Spline Interpolation

(c) Approximation of geometry to NACA0406 by B­spline in­ terpolation X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1 ­0.5

0 0.5 1 NAC0406 B­Spline Interpolation

(d) Comparison of pressure distributions for NACA0406 using B­spline interpolation X/C Y / C

0 0.25 0.5 0.75 1 ­0.05 ­0.04 ­0.03 ­0.02 ­0.01 0 0.01 0.02 0.03 0.04 0.05 NACA0610 B­Spline Interpolation

(e) Approximation of geometry to NACA0610 by B­spline in­ terpolation X/C P r e s s u r e C o e f f i c i e n t s 0 0.25 0.5 0.75 1 ­1 ­0.5 0 0.5 1 NACA0610 B­Spline Interpolation

(f) Comparison of pressure distributions for NACA0610 using B­spline interpolation Figure 5. Comparisons of geometry and pressure distributions for approximations using B­spline interpolations 7 of 8 American Institute of Aeronautics and Astronautics

Page 8 12

Li, W., Profile Optimisation Method for Robust Airfoil Optimisation in Viscous Flow, NASA /TM­2003­212408, 2003. 13

ProEngineer, http://www.ptc.com, 2004. 14

Fluent, http://www.fluent.com, 2004. 15

Keane, A.J., OPTIONS Design Exploration System, http://www.soton.ac.uk/˜ajk, 2004 8 of 8 American Institute of Aeronautics and Astronautics

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A Two­Dimensional Multigrid­Driven Navier­Stokes Solver for Multiprocessor Architectures Juan J. Alonso, Todd J. Mitty, Luigi Martinelli, and Antony Jameson Department of Mechanical and Aerospace Engineering Princeton University Princeton, New Jersey 08544 U.S.A.

Abstract A two­dimensional unsteady Navier­Stokes solver has been parallelized using a domain decomposition approach and the PVM message passing library. Several options for the treatment of multigrid and implicit residual smoothing are examined. Results for the unsteady flow over a pitching NACA 64A010 airfoil are presented.

1 INTRODUCTION In recent years, computational fluid dynamics (CFD) has been gaining acceptance as a design tool in industry. Advancements in algorithm development and computational hardware have led to more complex modeling of fluid flows. Although current inviscid models can accurately predict the coefficient of lift for an airfoil in transonic flow, viscous effects such as shock wave/boundary layer interaction can significantly modify important aspects ofa flow. Furthermore, unsteady phenomena ofpatent viscous character, such as buffeting, can not be predicted with the help ofinviscid models. Such viscous phenomena directly impact the design ofengineering configurations, and therefore, it is necessary to enhance the viscous prediction capability of CFD tools. Increasingly complex fluid flow models require high performance computing facilities. A cost effective solution for problems of this type requiring fast CPUs and large internal memory is the use of a parallel computing paradigm. For computational efficiency, one typically incorporates convergence acceleration tech­ niques such as multigrid and implicit residual smoothing. Message passing becomes necessary in this new environment, and severely limits the performance of processes that are inherently communication intensive. In this paper we present a parallelized version ofa well established Navier­Stokes solver, FLO103 [1]. This computer program has recently been enhanced to include Jameson’s implicit multigrid approach [2] for the efficient calculation ofunsteady viscous flows. Calculations are performed on an IBM SP1 multiproces­ sor computer with a domain decomposition approach, and message passing is handled by PVM (Parallel Virtual Machine) software [3]. Several methods for implementing convergence acceleration techniques such as multigrid and implicit residual smoothing are studied.

2 NAVIER­STOKES EQUATIONS DISCRETIZATION The two­dimensional, unsteady, compressible Navier­Stokes equations may be written in divergence form for a Cartesian coordinate system (x, y) as ∂ w ∂t + ∂ f ∂x + ∂ g ∂y = ∂

R ∂x + ∂ S ∂y , (1) where w is the vector offlow variables, f and g are the convective fluxes, and R and S are the viscous fluxes in each ofthe coordinate directions. With Reynolds averaging, turbulence effects can be taken into account with a turbulence model. In this work, a Baldwin­Lomax model was used. In integral form, Equation 1 can 1

Page 2 be applied to each finite volume ofa computational domain to yield a set ofcoupled first­order differential equations ofthe form d dt (w ij

V ij

) + E(w ij

) + NS(w ij

) + D(w ij

) = 0 , (2) where E(w ij

) are the convective Euler fluxes, NS(w ij

) are the Navier­Stokes viscous fluxes, and D(w ij

) are the artificial dissipation fluxes added for numerical stability. For unsteady problems, Equation 2 is modified by introducing a pseudo­time formulation to improve computational performance [2].

3 PARALLELIZATION STRATEGY

FLO103P is parallelized using a domain decomposition model, a SIMD (Single Input Multiple Data) strategy, and the PVM Library for message passing. Flows were computed on a C­mesh of size n i

×n j

= 1024×64. This domain was decomposed into subdomains containing n i

N p

×n j

points, where N p

is the number ofsubdomains used. Communication between subdomains was performed through halo cells surrounding each subdomain boundary. A two­level halo was sufficient to calculate the convective, viscous, and dissipative fluxes for all cells contained in each processor. In the coarser levels ofthe multigrid sequence, a single level halo suffices since a simplified model ofthe artificial dissipation terms is used. For problems with a low task granularity (ratio ofthe number ofbytes received by a processor to the number of floating point operations it performs), large parallel efficiencies can be obtained. Unfortunately,

convergence acceleration techniques developed in the 1980s base their success on global communication in the computational domain. Thus, current multigrid and implicit residual smoothing techniques [4] are bound to hinder parallel performance in traditional mesh sizes. In order to effectively deal with the parallelization ofthese two techniques, we propose several ideas. In this paper, static load balancing is performed at the beginning of the calculation. Since the number ofcells remains constant throughout the calculations, the domain can be partitioned into subdomains with an equal number ofcells. Except for some domains where an additional message across the wake is required, this provides for perfect load balancing.

4 PARALLEL MULTIGRID The full approximation multigrid technique enhances the convergence rate of a scheme by performing com­ putations on a series ofincreasingly coarser meshes. The calculations performed in each ofthese meshes are driven by the residuals at the previous finer level, and the results obtained are interpolated to the corre­ sponding finer mesh. The explicit time­stepping acts as a smoothing operator for the high frequency errors in each level, thus damping the dominant error mode at the finest level and accelerating convergence. A more detailed discussion of this procedure can be found in [4]. If domain decomposition is used for parallel calculations, the size ofthe meshes contained in each processor at the coarser levels ofthe multigrid cycle is quite small (typically 8 or 16 cells). Most ofthe CPU time is then wasted sending information back and forth between processors. This situation worsens for multigrid W­cycles, where equilibrium in the coarser meshes is established repeatedly before traversing the series back to the finest level. Previous authors [5] have attempted to deal with this problem in the Euler equations by limiting message passing to only some stages in the Runge­Kutta time­stepping scheme, or passing the boundary data exclusively at the finest level in the series. This usually led to a decrease in the convergence rate ofthe numerical algorithm, presenting a clear tradeoff between the improved parallel performance and the increasing number of cycles required for a similar level ofconvergence. In this work, we examine three different approaches to the implementation ofthe multigrid algorithm. First, the full multigrid algorithm is implemented with message passing at all required points such that the parallel program exactly recovers the results ofthe serial code. Second, multigrid is applied independently within each subdomain to completely avoid inter­processor communication. Third, and last, at coarse multigrid levels where communication overhead dominates CPU time, a single processor will be used to gather, compute, and scatter information. Parallel speed­up curves for these three cases are also presented. 2

Page 3 "DUMB" PROCESSOR W­CYCLE Parallel T T T T T T T

Fine Coarse Single Proc.

"LAZY" PROCESSOR W­CYCLE Parallel T T T T T T T Fine Coarse No work

Figure 1: “Lazy”­“Dumb” Multigrid Approach.

4.1 Full multigrid approximation

This first approach was an attempt to implement a parallel program that exactly reproduced the output of the single processor code in which message passing is not necessary. In order to achieve this goal, boundary information was passed among processors on all multigrid levels at the beginning of all five stages of the Runge­Kutta time­stepping. Additional messages were required in order to process convergence information, calculate force coefficients, and compute the eddy viscosity in the turbulence model. This procedure recovers the serial version convergence rate, at the expense ofpoor parallel performance.

4.2 Implicit multigrid within subdomains

The second approach used to deal with the multigrid algorithm was to completely decouple the subdomains on the coarser levels ofthe series in order to minimize the number ofmessages passed when very little computation is being performed. As expected, this approach restores parallel efficiency to higher levels, but it suffers from a decreased convergence rate. Clearly, the success of the multigrid technique is due to the

increased communication between different parts ofthe computational mesh at all levels, and the transfer of this information is limited by the isolation of the coarser levels in each subdomain. The degradation of the convergence rate is especially bad when interdomain boundaries lie close to regions ofhigh gradients (such as shock waves and stagnation points). In some ofthese cases, a converged solution cannot be obtained.

4.3 “Lazy”­“Dumb” multigrid approach

In order to preserve the convergence rate ofthe original multigrid algorithm, and somehow improve the parallel efficiency ofthe first approach we propose the following procedure: calculations on the finer levels are performed as in 4.1; when the algorithm shifts the solution to a user­specified coarse mesh, all the processors in the calculation pass their flow variables and grid locations to a single processor which computes the coarser levels of multigrid without the need for message passing. This processor is termed “dumb” since it performs everyone else’s work. The rest ofthe processors in the computation wait until the calculation needs to be performed on the finer levels, at which point they receive the information from the “dumb” processor and proceed once more in parallel. These processors are called “lazy” since they avoid carrying out part ofthe work that was, in theory, assigned to them. In our program, the level at which the transfer is done can be specified as an input, allowing for investigations of the optimal location of this transition point. Figure 1 presents this procedure graphically. Note that this construction can be extended to a hierarchy of“dumb” processors for larger calculations involving more processors. 3

Page 4

5 IMPLICIT RESIDUAL SMOOTHING Implicit residual smoothing (or averaging) is a technique that couples the residuals at any given point with those ofall the other cells in the domain, increasing the support ofthe scheme, and allowing a larger time step than that permitted by the Courant­Friedrichs­Lewy (CFL) restriction. Once more, this is a communication intensive procedure that negatively impacts parallel performance in a multiprocessor environment.

5.1 Fully implicit residual smoothing

Serial implementations ofthis technique usually split the problem into the implicit coupling along each of the two coordinate directions. The direction which is normal to the airfoil surface presents no difficulties since all the required data resides in the appropriate processors (in this decomposition), allowing calculations to proceed in parallel. In the coordinate direction that is parallel to the airfoil surface, the residuals that are to be coupled reside in different processors. Moreover, the solution procedure (Thomas algorithm for a tridiagonal system) is inherently serial. A brute force method allows only one processor to be active at a time in the forward elimination and back substitution phases, while the remaining processors are idle. Since the residual smoothing procedure consumes about 30% ofthe time ofthe total calculation, this idle time causes parallel performance to drop­off considerably.

5.2 Implicit residual smoothing within subdomains

In a very similar fashion to the multigrid performed implicitly within blocks, and in an attempt to reduce the amount ofmessages passed, residual smoothing was performed implicitly within blocks, with no global coupling ofthe residuals. Once more, as expected, although parallel performance improves, the convergence rate degrades to unacceptable levels. As in the corresponding multigrid procedure, this lack ofcoupling between domains often led to instability in the calculations.

5.3 Iterative implicit residual smoothing

An alternative to the previous approach is to perform an iterative solution of the smoothing problem. Messages containing the boundary residuals are passed at the beginning ofeach iteration, and the relaxation process proceeds in parallel. It has been observed in practice that in order to obtain the increase in CFL number provided by the fully implicit version (from 3 to about 6), at least two iterations are required. In these calculations, three iterations were performed and a CFL number of 6 was used without stability problems. The matrix problem is setup in each subdomain and a Jacobi relaxation procedure is performed on all subdomains concurrently.

6 RESULTS A summary of the results for the different multigrid approaches is presented in Figure 2. The full multigrid approach produces results with an optimum convergence rate, but with a parallel performance that degrades for a large number of processors. One must take into account, that for these meshes (n i

× n j

= 1024 × 64) granularity becomes too large as the number ofprocessors increases. This makes efficient parallel calculations not viable for a number of processors larger than 8. The multigrid performed implicitly within blocks (see Section 4.2) exhibits better parallel performance, but at a very high cost. When the total cost to achieve a solution converged to the same level through these two procedures is computed, the second technique requires about twice the number ofcycles making this approach less than desirable. Finally, the “lazy”­ “dumb” version ofthe algorithm performs slightly better when the “dumb” processor computes the two coarsest levels ofthe sequence. When the three coarsest levels are assigned to the “dumb” processor, parallel performance degrades since the amount of time required by the “dumb” processor exceeds that needed for all processors (with poor parallel performance). Notice that wall time was used to compute the parallel efficiencies in all cases. Ifthe unused CPU time ofthe “lazy” processors were to be factored in, the gain would be larger. It must also be mentioned that the implementation ofthis approach introduces a higher level ofcomplexity to the multigrid algorithm, since the memory management on both types ofnodes differs 4

Page 5 considerably from the traditional one. As a payoff for the additional work, this approach exhibits the exact same rate ofconvergence as the full multigrid algorithm. Evaluating the performance of multigrid in meshes that are relatively small places a strong restriction on the parallel efficiencies that can be achieved. Calculations using very large meshes (three­dimensional calculations, two­dimensional LES calculations, etc) will benefit from the full multigrid algorithm and will achieve parallel performances on the order of 90% since most of the time will be spent on the finer levels of the sequence. Nevertheless, for engineering calculations, reasonable speed­ups can be obtained through this method. The performance results of the different methods used to implement residual smoothing are presented in Figure 3. The fully implicit approach is clearly unacceptable even for a small number of processors. While the approach that uses implicit residual smoothing within blocks exhibits a parallel performance of 96% for 8 processors, the degradation ofthe convergence rate disqualifies it as a possible candidate. Finally, the iterated residual smoothing preserves both the convergence rate and a reasonable parallel speed­up, and is chosen as the candidate for engineering calculations. Figure 4 presents a detail of the domain decomposition for a calculation involving four processors for a typical airfoil section. Figure 5 shows the pressure contours at different points on the oscillation period of a NACA 64A010 airfoil at a Reynolds number of 10 6

, M ∞

= 0.796, and a reduced frequency, k c

= 0.202. These results were obtained with the O­mesh version ofthe program. One can clearly see how the shock waves strengthen and weaken as the airfoil pitches up and down. The lines that intersect the pressure contour lines are the interprocessor boundaries ofthe O­mesh. Perfect continuity ofthese contour lines validates the accuracy ofthe code even for the cases where strong shocks are close to the interprocessor boundaries. For more details, please refer to [6].

7 CONCLUSIONS A parallelized, two­dimensional, unsteady Navier­Stokes flow solver has been developed. Parallelization was realized using a domain decomposition approach to achieve proper load balancing and computational efficiency. PVM communication software was used for message passing between processors. Strategies for dealing with two convergence acceleration techniques, namely implicit residual smoothing and multigrid, and the performance ofeach ofthese techniques have been evaluated. It is observed that for meshes of sizes typically used in engineering calculations, acceptable parallel performances can be achieved with up to 8 processors. A larger number of processors is not suitable for this type ofcalculation. For larger meshes, the multigrid technique is still quite favorable even in multiprocessor architectures. Implicit residual smoothing can be performed in an iterative fashion without an observable impact in the convergence rate while retaining good parallel performance. All calculations used the public distribution ofthe PVM software developed at Oak Ridge National Labs. Preliminary results with the IBM optimized version ofthis message passing standard (PVMe), on both the SP1 and SP2 platforms, confirm the expected trends in which parallel performances improve considerably. Finally, results for the unsteady

pressure field around a pitching NACA 64A010 at M ∞

= 0.796 were computed on the IBM SP1 system.

References [1] Martinelli, L., Jameson, A., Validation ofa Multigrid Method for the Reynolds Averaged Equations, AIAA Paper 88­0414, AIAA 26th Aerospace Sciences Meeting, Reno, January 1988. [2] Jameson, A., Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows Past Airfoils and Wings, AIAA Paper 91­1546, AIAA 10th Computational Fluid Dynamics Conference, Honolulu, June 1991. [3] Geist, A., Beguelin A., Dongarra J., Jiang W., Manchek, R., Sunderam V., PVM 3 User’s Guide and Reference Manual, Oak Ridge National Laboratory, May 1993. 5

Page 6 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Number of Processors Parallel Speedup Comparison of Parallel Speedups of Multigrid Methods solid Full multigrid −−−−− Implicit within blocks −.−.− Lazy−Dumb − 2 levels in dumb processor ..... Lazy−Dumb − 3 levels in dumb processor

Figure 2: Summary ofParallel Speed­ups for Differ­ ent Approaches to the Multigrid Technique. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Number of Processors Parallel Speedup Comparison of Parallel Speedups of Residual Smoothing Methods solid fully implicit −.−.− implicit within blocks −−−−− explicit iteration

Figure 3: Summary ofParallel Speed­ups for Differ­ ent Implicit Residual Smoothing Approaches. 129x6 5 129x6 5 129x6 5 129x6 5

Figure 4: Domain Decomposition for a NACA 0012 Airfoil at a 10 degree Angle of Attack. [4] Jameson, A., Transonic Flow Calculations, Princeton University Report 1651, March 1984, in Numerical Methods in Fluid Dynamics, edited by F. Brezzi, Lecture Notes in Mathematics, Vol. 1127, Springer­ Verlag, 1985, pp. 156­242. [5] Yadlin, Y. and Caughey, D. A., Block Multigrid Implicit Solution ofthe Euler Equations ofCompressible Fluid Flow, AIAA Journal, 29(5):712­719. [6] Alonso, J. J., Martinelli, L., and Jameson A., Multigrid Unsteady Navier­Stokes Calculations with Aeroelastic Applications, 33rd AIAA Aerospace Sciences Meeting, AIAA Paper 95­0048, Reno, NV, January, 1995. 6

Page 7

1 2 3 4

5 6 Figure 5: Mach Number Contours. Pitching Airfoil Case. Re = 1.0 × 10 6

, M ∞

= 0.796, K c

= 0.202. Read figures by lines. 7

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Page 1 W.H. Mason 4/5/06

8. High­Lift Aerodynamics 8.1 Introduction: Why high lift? For transonic transports, the high­lift system design is a critical part of the configuration  design. To achieve “reasonable” field performance while also obtaining efficient transonic  cruise the design will require a fairly sophisticated high lift system. From a paper by Boeing aerodynamicists 1

: (presumably referring to the B­777) • “A 0.10 increase in lift coefficient at constant angle of attack is equivalent to reducing  the approach attitude by one degree. For a given aft body­to­ground clearance angle, the landing gear may be shortened for a savings of airplane empty weight of 1400 lb. • “A 1.5% increase in maximum lift coefficient is equivalent to a 6600 lb increase in  payload at a fixed approach speed” • “A 1% increase in take­off L/D is equivalent to a 2800 lb increase in payload or a 150  nm increase in range.” For fighters, “devices” are also scheduled allow efficient maneuver.

High­lift systems are also critical for STOVL and V/STOL aircraft. They also use the  propulsion system to help generate the lift. It always seemed to me peculiar to design fighter aircraft, or virtually any military aircraft, to operate from traditional runways. The one thing the  adversary is going to know is the exact location of your runways. So a STOVL capability seems to be  critical in a serious confrontation. Current status: Typical values of C Lmax

are shown in Table 1. They come from papers by Brune and McMasters, 2

Roskam and Lan, 3

and Sanders. 4

Table 1. Values of C Lmax

for some airplanes.* * Note that there is a significant variation of values from different sources.

Clearly the 727 emphasized short fields, and thus required a higher C Lmax

Anyone who ever looked out the window while landing in a 727 noticed the elaborate high lift system  employed. Model C Lmax

B­47/B­52 1.8 367­80/KC­135 1.78 707­320/E­3A 2.2 727 2.79 DC­9 3.0 737­200 3.2 747/E­4A 2.45 767 2.45 777 2.5

Page 2 8­2 W. H. Mason, Configuration Aerodynamics Some Key Aspects: • Compressibility can be important early. • Reynolds number scaling from WT to flight may be problematic. • Today simple high lift systems are critical, the high manufacturing cost for high lift  systems is important. Classes of problems • High lift for a single element airfoil • Multi­element airfoils • Use of “blowing” in some form: Powered Lift Computing: • Requires consideration of viscous immediately. (unlike typical cruise airfoil analysis  and design, where some insight can usually be gained ignoring viscous effects). • Predicting high­lift is done almost entirely with Navier­Stokes (RANS) codes  (exception, Prof. Mark Drela’s MSES 5

code). A recent summary of the computational capability is by Rumsey and Ying. 6

Single element airfoils • The key example of how to obtain high lift on a single element airfoil is the story of Liebeck’s high lift airfoil 7

and the Stratford “pressure recovery” shape of the pressure distribution. This introduces the classic paper by A.M.O. Smith. 8

See section 8.5 below. Multi­element airfoils: • Understanding the physics: Section 6.3 of A.M.O. Smith’s paper 8

is critical to understanding the physics. Know what is meant by: 1. The slat effect, 2. The circulation effect, 3., The dumping effect, 4. Off­the­surface pressure recovery, and 5. The fresh boundary layer  effect. Note: some people combine the circulation and dumping effects and call it the “vane  effect”. See section 8.5 below for details. • Design: See the recent survey by C.P. van Dam, 9

• Existing systems on commercial transports: Rudolph has surveyed the high­lift systems  on current subsonic transport aircraft. 10

8.2 Types of Trailing Edge Devices To begin we include a number of examples of high­lift devices as drawn by Dick Kita of Grumman. 11

These are fairly realistic drawings, as opposed to many of the drawings in textbooks, which are cartoonish. Kita worked in high lift for many years. I’m aware of his work on the Gulfstream II and F­14, but he worked on many other Grumman aircraft high­lift  systems as well. Of the large number of papers addressing high lift, several deserve mention. Pepper, et al 12

provides a more current look at high lift, while the Boeing 777 high lift system  development, as well as the overall design process, is available in the paper by Nield. 13

A valuable description of high lift on transports in contained in Gratzer. 14

The use of powered lift is covered in the survey by Korbacher. 15

Somewhat dated but valuable resources are the book by Hoerner and Borst, 16

and the book by McCormick 17

(recently reissued unchanged from the 1967 edition as a Dover paperback). Perhaps the best chapter on high lift in a basic text is the chapter in Shevell. 18

Page 3 High­Lift Aerodynamics 8­3 a) basic devices Figure 8­1. Examples of typical trailing edge devices. Note that the Fowler flap also adds  area, so that part of the C Lmax

increase is simply due to the use of the original reference area in computing the C L

. (from Dick Kita’s Grumman talk, Feb. 1985) Page 4 8­4 W. H. Mason, Configuration Aerodynamics b) other trailing edge devices Figure 8­1. Examples of typical trailing edge devices. (from Dick Kita’s Grumman talk, Feb. 1985) Page 5

High­Lift Aerodynamics 8­5 8.3 Types of Leading edge devices Figure 8­2. Typical leading edge device concepts. (from Dick Kita’s Grumman talk, Feb. 1985) Page 6 8­6 W. H. Mason, Configuration Aerodynamics Figure 8­3. The actual high­lift system employed on the F­14. (from Dick Kita’s Grumman talk, Feb. 1985) Notice that the F­14 has a fairly elaborate scheme, where the cove region is smoothed  with a moving flap, and the upper surface of the slat also has a movable piece to fair the flap.  Also note the spoilers. Many fighter airplanes use spoilers instead of ailerons for roll control,  although their use varies from company to company. Page 7 High­Lift Aerodynamics 8­7 8.4 Aerodynamics of Leading and Trailing Edge Devices Figure 8­4. Typical effect of flaps on lift (from Dick Kita’s Grumman talk, Feb. 1985) This figure illustrates the effects of flap deflection on lift. Note that the angle of attack for C Lmax

actually decreases as the flap is deflected. Page 8 8­8 W. H. Mason, Configuration Aerodynamics Figure 8­5. Effect of flap extension on lift. (from Dick Kita’s Grumman talk, Feb. 1985) Flaps that extend in a Fowler motion also benefit from additional wing area. Because we continue to use the same reference area, the C Lmax

value increases. Page 9 High­Lift Aerodynamics 8­9 Figure 8­6. Typical variation of C Lmax

with Mach and Reynolds Number. (from Dick Kita’s Grumman talk, Feb. 1985) In general, we expect C Lmax

to increase with Reynolds number as shown here for a clean airfoil. However, sometimes the projection is not so straightforward, and C Lmax

may even decrease. 19

Considering the adverse effect of Mach number on C Lmax

, notice the low Mach numbers at which the effects take place. It’s rather surprising to the uninitiated. Page 10 8­10 W. H. Mason, Configuration Aerodynamics Figure 8­7. Effect of leading edge slats. (from Dick Kita’s Grumman talk, Feb. 1985) Leading edge slats work to protect the leading edge from separation. Therefore, they  don’t really “do anything” until you reach the angle of attack where the leading edge flow would “let  go” without the slats for protection. Thus the slats allow the lift to continue to rise to higher  angles of attack. Page 11 High­Lift Aerodynamics 8­11 Figure 8­8. Effects of various types of leading edge devices on C Lmax

. (from Dick Kita’s Grumman talk, Feb. 1985) Different leading edge devices differ in their effectiveness. This is Kita’s estimate of  how each type of device affects the performance of the wing. Page 12 8­12 W. H. Mason, Configuration Aerodynamics Figure 8­9. Estimated performance of various types of high lift systems. (from Dick Kita’s Grumman talk, Feb. 1985) This is Kita’s estimate of the typical “best” performance you can get from various types  of high­ lift systems. You can see that sweeping the trailing reduces the effectiveness of all the  systems. The curve labeled “advanced” is typical of projections made in advanced departments,  where the assumption is that an advanced technology development effort can improve the  performance of any system. This may or may not be true. Page 13 High­Lift Aerodynamics 8­13

Figure 8­10. Effects of flap deflection on drag. (from Dick Kita’s Grumman talk, Feb. 1985) In addition to lift, flaps make a large change in drag. Clearly, you don’t want the flaps  deployed at low lift, and thus flaps aren’t deflected in cruise. As lift increases, there may be an  optimum flap deflection schedule, and this is done, for example, on the F­18, where the flaps are scheduled with angle of attack and Mach number. This was also done on the Grumman  X­29. Page 14 8­14 W. H. Mason, Configuration Aerodynamics Figure 8­11. Flap effects on pitching moment. (from Dick Kita’s Grumman talk, Feb. 1985) Flap deflection also produces a large change in pitching moment. This is an important consideration, since you need to be able to trim this pitching moment. In the case of the  Beech Starship, the canards actually changed sweep to be able to generate the required force. So this effect cannot be ignored in developing the high lift system. Page 15 High­Lift Aerodynamics 8­15 Figure 8­12. Definition of gap and overlap. (from Dick Kita’s Grumman talk, Feb. 1985) In developing the high lift system, the selection of the “gap” between the slot and main  element, and the “overlap” have been found to be two of the key parameters. A lot of wind tunnel  time, and more recently computer resources, are spent trying to identify the values of these  parameters that produce the highest lift. Figure 8­12 provides Kita’s definition of these parameters. 8.5 Physics of high lift: A.M.O. Smith’s analysis of the high lift aerodynamics Now that we’ve surveyed the characteristics of high lift systems, we need to examine the physical basis for the operation and limits of high lift systems. A.M.O. Smith “wrote the  book” on the physics of high­lift systems 8

and is required reading. His message: you need to carry as much lift (load) as you can on the airfoil’s upper surface without separating the boundary  layer. His classic paper describes the physics associated with the high­lift characteristics we  described

above, and how to achieve the available high lift performance. We summarize his  description here. To obtain insight into the characteristics of pressure distributions as they affect boundary layer separation, Smith introduced the use of a canonical pressure distribution. He felt  strongly that this was necessary to understand and compare possible separation on different  airfoils. It is Page 16 8­16 W. H. Mason, Configuration Aerodynamics essentially another type of dimensionless or “scaled” pressure distribution. For boundary  layer investigations it is found that the best scaling factor is the velocity just before the  deceleration begins. In Smith’s canonical system, C p

= 0 represents the start of the pressure rise and C p

= +1 the maximum possible value, that is, u e

= 0.The velocity at the start of the pressure rise is u 0

. Thus, the canonical pressure distribution is defined as C p

= 1− u e

u 0

⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2

The next step is to examine the best way to specify the pressure distribution to allow the pressure to recover to as close as possible to C p

= +1. Smith made a parametric study of various possibilities to gain insight into the “best” way to prescribe a pressure distribution to  delay separation. However, here we look at his limiting case. It makes use of an analysis by  Stratford

that was done to estimate separation before the days when boundary layer computer  programs were available and their use routine (1959). Using Stratford’s analysis, it is possible to  define a pressure distribution where the boundary layer is everywhere just on the verge of  separation. To do this, we manipulate the Stratford criteria: C p

x dC p

dx ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 10 −6

R

( ) 1 10 1 2

= S Note that originally, Stratford used this relation to say that separation occurred when the  quantity on the LHS of the equation reached the value of S (typically 0.35). However, we can  define a Cp distribution using this relation that is everywhere equal to S, just on the verge of  separation, and this pressure distribution is the best way to achieve a very large pressure recovery without separation. Figure 8­13 shows the resulting pressure distribution. Examining this pressure distribution, several key observations can be made. The initial slope is infinite, and then decreases. Thus, when the boundary layer is thin, it can withstand a very large pressure  gradient.

As the boundary layer thickens (either when it starts to recover, or as it recovers) it  cannot sustain the large pressure gradient, and the pressure gradient to maintain attached flow  decreases. This illustrates the idea that thick boundary layers are more likely to separate than thin  boundary layers. Note also that the Reynolds number effect is relatively weak. Finally, the  boundary layer could recover all the way to C p

= +1, but to attain this, x would need to go to infinity. These shapes are the best possible pressure distributions to use to recover the pressure without separating the boundary layer. as Smith notes, the only way to do better is to use some  sort of active boundary layer control (suction or blowing). Page 17 High­Lift Aerodynamics 8­17 ­0.20 0.0 0.20 0.40 0.60 0.80 1.0 0.0 0.50 1.0 1.5 2.0 Canonical Cp

x ­ feet solid line U 0

/v = 10^6 dashed line U 0

/v = 10^7

Figure 8­13. Stratford Limiting flows for two different Reynolds numbers. Single element airfoils: The key is how to obtain high lift on a single element airfoil, and  is essentially the story of Liebeck’s high lift airfoil 7

and the Stratford “pressure recovery” shape of the pressure distribution described above, as told by Smith. 8

The question of how much lift you

can obtain on a single element airfoil involves how low the pressure can be on the upper  surface, and how the pressure can recover to a positive pressure coefficient at the trailing edge  and keep the boundary layer attached. Smith describes two aspects of the problem. In the first case, he explains the limit of the pressure coefficient in terms of the vacuum C p

when a zero pressure is specified on the airfoil upper surface. Thus, using the definition of C p

, C p

= p − p ∞

1 2 ρ ∞

U ∞ 2

we can obtain an alternate form using q = 1 2 ρ ∞

U ∞ 2

= γ 2 P ∞

M ∞ 2

, that is C p

= p − p ∞

γ 2 p ∞

M ∞ 2

Page 18 8­18 W. H. Mason, Configuration Aerodynamics and vacuum C p

occurs when the pressure is zero: C p vac

= −2 γ M ∞ 2

Then, he points out that only a value of 70% of the vacuum C p

has been achieved in practice. This results in a C p

limit of M ∞ 2

C p

= −1 for γ of 1.4. Next, Liebeck and Smith used the analytical analysis by Stratford illustrated above to  specify a pressure distribution that allows the most lift to be obtained. Given this pressure  distribution, an inverse method is used to obtain the associated airfoil shape. The result is the Liebeck  family of high lift airfoils. Multi­element airfoils: Understanding the physics: read section 6.3 of A.M.O. Smith’s  paper. The five ideas are: 1. The slat effect. The slat “protect”s the leading edge of the main element. That’s why its effect is only observed near C Lmax

of the single element. Thought of as a point vortex, the slat velocity acts to reduce the velocity around the leading edge of the main element. 2. The circulation effect. The downstream element causes the upstream element to be in a high velocity region, inclined to its mean line. To meet the Kutta condition, the circulation has  to be increased. Instead of the airfoil deflecting as a plain flap, the trailing edge is placed in an  inclined flow, something else (the downstream element) turns the flow. 3. The dumping effect. The trailing edge of the forward element is in a region of velocity appreciably higher than the freestream velocity. Thus, the boundary layer can come off  the

forward element at a higher velocity. You don’t have to recover back to Cp = +0.2 for  attached flow, relieving the pressure rise on the boundary layer, alleviating separation problems  and permitting increased lift. The suction lift can be increased in proportion to U TE 2

for the same margin against separation. 4. Off­the­surface pressure recovery. The boundary layer leaves the trailing edge faster  than the freestream, and now becomes a wake (a viscous phenomena). The recovery back to  freestream velocity can be more efficient way from contact with the wall. Wakes withstand more  than boundary layers. Note that the wake can actually “separate” out in the flowfield. Note: for a well designed high lift system the local boundary layers and wakes remain separate. If they  merge, everything is more complicated. 5. The fresh boundary layer effect. Thin boundary layers can sustain a greater pressure  gradient than a boundary layer. Thus, three thin boundary layers (on three airfoil elements) are  more effective than one thick boundary layer (single element). Note: some people combine the circulation and dumping effects and call it the “vane  effect”. 8.6 Computational methods for high lift Significant effort has been devoted to improving prediction capabilities for high lift  systems. As stated in the introduction, the best recent survey is by Runsey and Ying. 6

In the meantime, for low speed predictions of the maximum lift for a single element airfoil, XFOIL can be  used. experience shows that it’s predictions are slightly higher than experimental results.  Nevertheless, this is rea remarkable capability of a code than be run on a laptop PC. Page 19 High­Lift Aerodynamics 8­19 8.7 Passive and active boundary layer control Passive Boundary Layer Control: The boundary layer can be prevented from separating  by the use of vortex generators, snags and fences. 20

Active boundary layer control: If suction or blowing is used to suppress boundary layer separation, the blowing (which is generally preferred to suction) is known as boundary  layer control (BLC), if the amount of blowing exceeds the value required for BLC, then the  blowing is know as “powered lift”. The key parameter used to describe the amount of blowing is the blowing coefficient, defined as: C µ

= m j

V j

qc where the subscript j refers to the jet, and q is the dynamic pressure. Blowing for BLC  was used often in early fighters, but is not used nearly as much today. The F­4 Phantom originally  had blowing on both the leading edges and over the trailing edge flap. However, to improve  transonic maneuver characteristics and to improve resistance to departure, the leading edge  blowing was replaced by leading edge slats. 21

8.8 Powered Lift A huge class of concepts have been tried to increase maximum lift using high pressure air from the engine. Some examples of powered lift concepts are: • propeller slipstream deflection (Brequet 941/McDonnell Model 188) • externally blown flaps (McDonnell DouglasYC­15/C­17) • internally blown flaps • upper surface blowing (Boeing YC­14, NASA QSRA, Ball­Bartoe JetWing) • vectored thrust (AV­8 Harrier) • jet flaps (Hunting H.126) • jet augmentor wings (NASA­deHavilland Augmentor Wing Aircraft) • circulation control (advocated by Hokie Bob Englar 22

, A­6 CCW) 8­9 Configuration Integration issues • The best airplane C Lmax

you can achieve with a mechanical high lift system is about 3 – 3.5 • The military and civil air regulations require a margin between C Lmax

and the operating C L

of the airplane. This must be accounted for during design. For example, the approach speed must be 1.3 times the stall speed. This would suggest that the maximum approach C L

is only 59% of the C Lmax

. However, some relief is available because the measured stall speed, V smin

is usually about 0.94 times the stall speed in 1g steady flight (the wind tunnel case). This means that you can use a C L

of about 67% of the C Lmax

.

9,23

• Increased span can be used to reduce the induced drag, so a bigger flap angle can be  used before a climb limit is encountered, but this is a heavy solution. • Sweep decreases max lift Page 20 8­20 W. H. Mason, Configuration Aerodynamics • 2D to 3D: lots of losses – don’t be mislead by 2D C Lmax

values, the real 3D value will be much less. • The maximum C L

available for takeoff and landing for many swept­wing airplanes is actually the limit on angle of attack to avoid tailscrape. • The best high­lift configuration integration description also involves powered lift. The YC­14 AIAA Case Study 24

is highly recommended. Finally, issues not covered but worth mentioning: the concept of the Gurney flap 7

and the need for accuracy around the leading edge. ∗

8­10 Exercises 1. Examine the predictive capability of XFOIL for C Lmax

. Use the data from Abbott and von Doenhoff supplied previously for the NACA 0012 and 4412 airfoils at a Reynolds  number of 6 million and free transition. Comment on your results. 2. Read the high lift paper by A.M.O. Smith. Summarize what you learned in one page.  Pay

special attention to the details of single and multielement airfoils described in Sections 3­ 6. 8­11 References 1

Garner, P.L., Meredith, P.T., and Stoner, R.C., “Areas for Future CFD Development as Illustrated by Transport Aircraft Applications,” AIAA Paper 91­1527, 1991. 2

G.W. Brune and J.H. McMasters, “Computational Aerodynamics Applied to High Lift Systems,” Chapter 10 of Applied Computational Aerodynamics, P. Henne, Ed. Progress  in Astronautics and Aeronautics, Vol. 125, AIAA, Washington, 1990. 3

Jan Roskam and C­T Edward Lan, Airplane Aerodynamics and Performance,  DARcorporation, Kansas, 1997, pp. 343. 4

Karl L. Sanders, “High­Lift Devices, A Weight and Performance Trade­Off  Methodology,” Society of Allied Weight Engineers (SAWE) Technical Paper 761, May 1969. 5

Mark Drela, “Design and Optimization Method for Multi­Element Airfoils,” AIAA Paper 93­ 0969, Feb. 1993. 6

Christopher L. Rumsey and Susan X. Ying, “Prediction of high lift: review of present  CFD capability,” Progress in Aerospace Sciences, Vol. 38, pp. 145­180, 2002. (note that  articles in Progress in Aerospace Sciences are available for download through the Virginia Tech  University Library if you search Addison and have a vt.edu address) 7

Robert H. Liebeck, “Design of Subsonic Airfoils for High Lift,” Journal of Aircraft, Vol. 15, No. 9, Sept. 1978, pp. 547­561. 8

A.M.O. Smith, “High­Lift Aerodynamics,” 37 th

Wright Bothers Lecture, Journal of Aircraft, Vol. 12, No. 6, June 1975 9

C.P. van Dam, “The aerodynamic design of multi­element high­lift systems for transport airplanes,” Progress in Aerospace Sciences, Vol. 38, pp. 101­144, 2002. (note that  articles in Progress in Aerospace Sciences are available for download through the Virginia Tech  University Library if you search Addison and have a vt.edu address) 10

Peter K.C. Rudolph, “High­Lift Systems on Commercial Subsonic Airliners,” NASA CR

4746, Sept. 1996. ∗

Maintaining an accurate leading edge contour is critical at high lift conditions. Once after a navy depot had repainted an F­14 wing, a small ridge was left on the leading edge where the upper and lower surface paint overlapped. This was enough to cause early stall. Ed Heinemann reported a similar experience on a  Douglas airplane during World War II in his autobiography.

Page 21 High­Lift Aerodynamics 8­21 11

Dick Kita, “Mechanical High Lift Systems”, Grumman Aerodynamics Lecture Series, Grumman Aerospace Corporation, Feb. 1985. 12

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