AIAA 2003Œ3531 - Stanford Universityaero-comlab.stanford.edu/Papers/shankaran.aiaa.03-3501.pdf · AIAA 2003Œ3531 Numerical Analysis and Design of Upwind Sails Sriram and Antony

AIAA 2003–3531Numerical Analysis and Design ofUpwind Sails

Sriram and Antony JamesonDept of Aeronautics and AstronauticsStanford University, Stanford, CA

Margot G GerritsenDepartment of Petroleum EngineeringStanford University, Stanford, CA

21st AIAA Applied Aerodynamics ConferenceJune 23–26, 2003/Orlando, FL

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Numerical Analysis and Design of Upwind Sails

Sriram and Antony Jameson ∗

Dept of Aeronautics and Astronautics

Stanford University, Stanford, CA

Margot G Gerritsen †

Department of Petroleum Engineering

Stanford University, Stanford, CA

The aim of this study is to develop and validate numerical methods that solve the in-viscid field equations (Euler) to simulate and design upwind sails. The three dimensionalEuler equations for compressible flow are modified using the idea of artificial compress-ibility and discretized on unstructured tetrahedral grids to provide estimates of lift anddrag for upwind sail configurations. Convergence acceleration techniques like multigridand residual averaging are used along with parallel computing platforms to enable thesesimulations to be performed in a few minutes. To account for the elastic nature of thesail cloth, this flow solver is coupled to NASTRAN to provide estimates of the deflectionscaused by the pressure loading. The results from the aeroelastic simulations show thatthe major effect of the sail elasticity was in altering the pressure distribution aroundthe leading edge of the head and the main sail. Adjoint based design methods have alsobeen developed, and used to induce changes to the camber distribution of the main sail.The design process resulted in a camber distribution that allows smooth entry of theflow through the leading edge of the main sail thereby, reducing the leading edge suctionpeaks that would be detrimental to the growth of the boundary layer.

1. Introduction

Races like the Americas Cup have seen significantimprovements in the design of both the hull and thesails over the last two decades. Competing syndicatesare constantly pushing the aerodynamic and structurallimits of the designs as improvements of less than 0.5 %in the speed of the boat results in savings of about 25-35 seconds which is typically greater than the marginof victory for these races. For the windward leg of therace, a good measure of the performance of a design isthe distance that the boat travels directly to windwardin a given time. This performance index (called thespeed-made-good by the boat) is dependent on boththe speed of the boat and her true sailing course whichin turn are dependent on the aerodynamic and hydro-dynamic forces produced by the sails and the hull. Ingeneral, the performance of the boat can be improvedby reducing the resistance of the hull and the drag ofthe sails. However changes in the aerodynamic andhydrodynamic forces alter the equilibrium of the sailboat which then has to adjust its speed and sailingcourse to maintain equilibrium. Hence, the design ofsails boats has to be carried out in an environmentwhere the analysis and design procedures for the sailsand the hull are tightly coupled to realize a meaningfuloverall design.

Traditionally, designers have used Velocity Predic-tion Programs (VPP)to design a boat with good over-all performance. These programs solve for the motionof the boat using force and moment balance relations

∗Professor, Stanford University†Professor, Stanford University

Copyright c© 2003 by authors. Published by the American In-stitute of Aeronautics and Astronautics, Inc. with permission.

while ensuring that the resulting motion is stable.These programs typically compute the aerodynamicand hydrodynamic forces using potential flow solvers,which are computationally inexpensive and easy to im-plement, enabling the designer to evaluate a wide arrayof designs. However, there exists large regions of ro-tational flow and significant viscous interaction, wherethe assumptions of potential flow are not valid.

Alternate descriptions of the flow field that accountfor the rotational nature of the flow can provide moreaccurate estimates of the induced drag (which ac-counts for 60 % of the total drag) for the upwind legof the race. Modeling the viscous effects during theupwind and downwind leg of the race will help in thedesign of sails and hulls with reduced viscous drag.Simulations of the evolution of the free-surface of thesea and its interaction with the hull can help providea virtual tool that can supplement tow tank testing.The development of robust and accurate tools to pre-dict the above mentioned physical phenomena couldlead to improved designs that can benefit the racingcommunity in particular and the sailing world in gen-eral.

Computational techniques that solve the Euler orthe Navier-Stokes equations are increasingly beingused by competing syndicates in races like the Ameri-cas Cup. For sail configurations, this desire stems froma need to understand the influence of the mast on theboundary layer and pressure distribution on the mainsail, the effect of camber and planform variations ofthe sails on the driving and heeling force produced bythem and the interaction of the boundary layer profileof the air over the surface of the water and the gapbetween the boom and the deck on the performance

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American Institute of Aeronautics and Astronautics Paper 2003–3531

of the sail. Traditionally, experimental methods alongwith potential flow solvers have been widely used toquantify these effects. While these approaches are in-valuable either for validation purposes or during theearly stages of design, the potential advantages of highfidelity computational methods makes them attractivecandidates during the later stages of the design pro-cess.

The aim of this study is to develop and validatenumerical methods that solve the inviscid Euler fieldequations to simulate and design upwind sails. Thethree dimensional Euler equations for compressibleflow are modified using the idea of artificial com-pressibility, and discretized on unstructured tetrahe-dral grids to provide estimates of lift and drag forupwind sail configurations. Convergence accelerationtechniques like multigrid and residual averaging areused along with parallel computing platforms to enablethese simulations to be performed in a few minutes. Toaccount for the elastic nature of the sail cloth, this flowsolver is coupled to NASTRAN to provide estimates ofthe deflections caused by the pressure loading. The re-sults of this aeroelastic simulation, illustrate that themajor effect of the sail elasticity, is in altering the pres-sure distribution around the leading edge of the headand the main sail.

We also address the issue of how to calculate im-proved sail shapes using an optimization procedurebased on the adjoint method. This is used to in-duce changes to the camber distribution of the mainsail with the goal of reducing the leading edge suctionpeaks that could be detrimental to the growth of theboundary layer. The design process results in an cam-ber distribution that allows smooth entry of the flowthrough the leading edge of the main sail, thereby re-ducing the leading edge suction peaks.

2. Analysis with CFD

2.1 Finite Volume Discretization of the Flow

Equations

A vast repertoire of computational codes have beendeveloped by Antony Jameson to analyze aerodynamicconfigurations in transonic flight.1, 2 These codesmodel the fluid as a compressible fluid and a varietyof numerical techniques have been developed to effi-ciently solve the governing equations of a compressiblefluid with embedded supersonic regions.

In the limit of truly incompressible flow, or zeroMach number, alternate methods must be used tocompute the flow field to preserve the accuracy, robust-ness and convergence properties of the flow solution.The fundamental problem in transitioning from a com-pressible fluid model to an incompressible one is theloss of the evolution equation for the density. Sincethe density is constant, a time dependent constraintmust be imposed on the continuity equations to en-sure a divergence-free velocity field. In addition, the

eigenvalues resulting from the system of conventionalhyperbolic Euler equations for compressible flows be-come infinite in the limit of incompressible flow. Thisis due to the fact that incompressible flows exhibit in-finite sound speeds. Hence, the use of compressibleflow solvers in the incompressible flow limit introduceswidely varying eigen-speeds, resulting in extremelystiff equations.

To overcome this difficulty, the present work usesthe artificial compressibility method, an approach firstproposed by Chorin in 19673 as a method to solveviscous flows. Artificial compressibility methods in-troduce a psuedotemporal equation for the pressurethrough the continuity equation. This approach re-moves the troublesome sound waves associated withcompressible flow formulations as the Mach numberapproaches zero. The eigenvalues of the original sys-tem are now replaced with an artificial set that rendersthe new set of equations well-conditioned for numericalcomputation. When combined with multigrid acceler-ation procedures, artificial compressibility proves tobe particularly effective4. Converged solutions of in-compressible flows over a main sail can be obtained inabout 75-100 multigrid cycles.

Using the idea of artificial compressibility, the equa-tions of motion of an incompressible, inviscid fluid canbe cast in the following form.

∂w

∂t+ P

{

∂F

∂x+∂G

∂y+∂H

dz

}

= 0. (1)

Here, the dependent variables w, the inviscid flux vec-tors f , g and h and the pre-conditioning matrix P aredescribed by

w =

p

u

v

w

, F =

u

u2 + p

uv

uw

, G =

v

vu

v2 + p

vw

,

(2)

H =

w

wu

wv

w2 + p

, P =

Γ2 0 0 00 1 0 00 0 1 00 0 0 1

(3)

This system of equations has no physical meaninguntil a steady state is reached. At steady state, thetime dependent pressure term drops from the continu-ity equation resulting in the true steady state equa-tions for an incompressible flow. Further, Γ can beselected to accelerate the time decay to steady state.Using the finite volume approach, the governing equa-tions can be cast in the integral form for each compu-tational volume in the domain as follows,

Conservation of Mass

d

dt

∫

V

pdV +

∫

S

Γ2 (u · n) dS = 0 (4)

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Conservation of Momentum

d

dt

∫

V

udV +

∫

S

u(u · n)dS = −

∫

S

pndS (5)

Spatial discretization of equation (4) and (5) leadsto the following equation for each point of interest inthe computational mesh.

d

dtViwi +

∑

k

Fk.nkSk = 0 (6)

where p is the pressure, u is the velocity vector, n isthe unit normal at the surface of the control volume,V and S are the volume and surface area of the controlvolume respectively, F is the flux through the controlvolume and the summation of the fluxes is over thecontrol volume that surrounds each node of the mesh.

In the present study, a cell-vertex scheme is used forthe implementation of the finite volume scheme on un-structured tetrahedra. Dual meshes constructed fromplanes bisecting each edge of the mesh are used toaccumulate the fluxes at each node. Boundary condi-tions are then enforced along the triangular faces thatlie on the boundary to account for the one-sided con-trol volumes for the nodes on the boundary. The restof the discussion in this section outlines the details ofthe implementation of the spatial discretization opera-tors when used with artificial compressibility methods,the evaluation of the numerical diffusion terms and themultigrid algorithm.

In Equation (3), Γ is called the artificial compress-ibility parameter in the light of the analogy that maybe drawn between the above equations and the equa-tions of motion for a compressible fluid whose equationof state is given by p = Γ2ρ. Thus, ρ is an artificialdensity and Γ may be referred to as an artificial speedof sound. When the temporal derivatives tend to zero,the set of equations satisfy precisely the incompressibleEuler equations, with the consequence that the correctpressure may be established using the artificial com-pressibility formulation. The pre-conditioning matrix,P , may be viewed as a device to create a well posed sys-tem of hyperbolic equations that are to be integratedto steady state along lines similar to well establishedcompressible flow Finite Volume formulations. In ad-dition, the artificial compressibility parameter may beviewed as a relaxation parameter for the pressure iter-ation.

The eigenvalues of the system of equations in equa-tion (1), are given by

λ1 = U, λ2 = U, λ3 = U + a, λ4 = U − a

where,a2 = U2 + Γ2(ψ2 + η2 + ξ2)

andU = uψ + vη + wξ

The terms ψ, η, ξ represent the slopes of the character-istic system of waves, are arbitrary and defined.

The choice of Γ is crucial in determining the conver-gence and stability properties of the numerical scheme.Typically, the convergence and stability rate of thescheme is dictated by the slowest waves and the sta-bility of the scheme by the fastest. In the limit of largeΓ, the difference in wave speeds can be large. Althoughthis situation would presumably lead to a more accu-rate solution through the penalty effect in the pressureequation, very small time steps would be required toensure stability. Conversely, for small Γ, the differencein the maximum and minimum wave speeds may besignificantly reduced, but at the expense of accuracy.Thus a compromise between the extremes is achievedby choosing Γ to be

Γ2 = C(u2 + v2 + w2)

where C is a constant of the order of unity. In regionsof high velocity and low pressure where suction occurs,Γ is large to improve accuracy, and in regions of lowvelocity, Γ is correspondingly reduced.

Under these assumptions on the choice of the pre-conditioner, P , and the application of the Finite Vol-ume method for a cell-vertex scheme results in a setof ordinary differential equation for each node of thecomputational mesh,

d

dt(Viw) + PQi = 0. (7)

where Vi is the volume around each node and Qi repre-sents the flux through the faces of the control volume.To prevent, odd-even decoupling at adjacent nodeswhich may lead to oscillatory solutions, a dissipationterm is added to the flux calculation to modify theabove equation

d

dt(Viw) + P [Qi −Di] = 0. (8)

ord

dt(Viw) +Ri = 0. (9)

where Ri is the residual at each node in the computa-tional mesh.

The resulting system of equations are integrated intime using an explicit multistage scheme with coeffi-cients that maximize the stability region of the time-stepping scheme. To further accelerate convergenceto steady state, local time-stepping and residual aver-aging techniques are introduced. Detailed numericalanalysis of the spatial discretization6, 7 and the timestepping scheme8 can be obtained from previous stud-ies listed in the citations.

2.2 Dissipation

Numerous research efforts during the eighties led toto the development of a mathematical frame-work to

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add numerical diffusion to the discretized equations.While the focus of these studies were on deriving stableand accurate techniques to resolve shocks in the flow-field, the mathematical frame-work can be inheritedfor incompressible flows that use the artificial com-pressibility method with some modifications that limitthe amount of numerical diffusion.

Local Extremum Diminishing (LED) schemes thatguarantee that new extrema are not generated duringthe evolution of the solution have proven to be robustand efficient. These schemes limit the reconstructedsolution and fluxes at cell interfaces by using limitersthat can be constructed from gradient informationfrom a stencil of points around each computationalpoint. A variety of choices exist for the form of thelimiters, and the JST scheme 8 can be reformulated inthe class of SLIP schemes for a particular choice of thelimiter. Over the years the JST scheme has proven tobe robust and accurate for a variety of aerodynamicflows. It can be represented as

dj+ 1

2

= ε2j+ 1

2

∆wj+ 1

2

−ε4j+ 1

2

(

∆wj+ 3

2

− 2∆wj+ 1

2

+ ∆wj− 1

2

)

When used for incompressible flows, the first termon the R.H.S. can be dropped and the remainder usedas the numerical diffusion term that prevents odd-evencoupling. All the computations in this study has beenperformed using the JST scheme as the basis for nu-merical dissipation.

2.3 Multigrid

Multigrid techniques are widely used to acceleratethe convergence of a system of equations to steadystate. A general framework for the development offull-approximation multigrid methods for non-linearequations can be outlined as follows.

Consider,

Lu = F

discretized on a mesh with spacing h as

Lhvh = Fh

This can be rewritten as

Lh(vh + δvh) = Fh

where δv represents correction to the present estimatevh or

Lhδvh +Rh = 0 (10)

where Rh is the residual. on the coarse grid, the aboveequation can be replaced by

L2hδv2h + Ih2hRh = 0 (11)

where Ih2h represents the aggregation or restriction op-

erator. The correction to the present fine grid solutioncan be represented as

vnewh = vh + I2h

h δvh

where I2hh represents an interpolation operator. We

can add and subtract the following from equation (11)

L2hv2h −F2h = R2h

to get

L2h(v2h + δv2h) −F2h + Ih2hRh = 0

This leads to the full approximation scheme (FAS)

L2h(v+2h) −F2h + Ih

2hRh −R2h = 0

Thenv+

h = vh + Ih2h(v+

2h − v2h)

In the present work a series of non-nested meshes areused for the multigrid cycle and the solution and resid-ual from each mesh are aggregated to the coarser meshwhile interpolating the correction from the coarser tothe fine mesh. Detailed descriptions of the multigridscheme can be obtained from9.

For unstructured grids, the nature of the grids tobe used in the multigrid cycle is a question of ongoingdebate. In the present study a series of non-nestedmeshes was generated by a grid generator (MESH-PLANE). The initial solution from a particular meshis advanced in pseudo-time to obtain new estimatesof the flow variables. On transfer to the next level ofthe multigrid, the solution for the coarse grid meshpoints is interpolated from the four nodes of the finemesh tetrahedron that contains this node. Further,the accumulated residual at each fine mesh point is dis-tributed to each node of the tetrahedron in the coarsemesh that that encloses the fine mesh node. The in-terpolating factors for each node are computed fromweights which are based on the volume included by agiven node and opposite face of the tetrahedron (fig-ure 1). This reduces to a second order interpolationscheme on equilateral tetrahedra and has been foundto be sufficient for the present calculations. The es-timate of the residuals from the fine mesh is used toadvance the solution on the coarse mesh where thelarger scale errors can be more efficiently accomplishedon the coarse mesh. Further levels in the multigridcycle involve the same operations are before, therebyusing grids that are coarser and coarser to convect theerror terms out of the computational domain faster.The ascending side of the multigrid cycle estimates acorrection from each grid which is then interpolatedto the next finer mesh in the sequence (figure 2). Thecorrections from the coarser mesh are transferred using

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n1

n2

n3

a1

a2

a3

p

Fig. 1 Interpolation coefficients for use in themultigrid cycle

fine grid nodes

coarse grid nodes

��

��

Fig. 2 Transfer of solution, residuals and correc-tions between the fine and coarse mesh

similar interpolating factors as for the aggregation op-erations. Multigrid cycles which progress in the shapeof a W have generally been found to provide fasterconvergence to steady state than the simple V cycle.

3. Parallel Implementation of the

Unstructured Multigrid Flow Solver

To exploit the availability of modern parallel com-puting platforms, a combination of the AIRPLANE1

and FLO7710 computational programs, was paral-lelized. Due to the unstructured nature of the compu-tational grid, a wide variety of possible data structuresto implement the underlying numerical algorithms ex-ist. The following sections outline the choice of datastructures and algorithms that were made to paral-lelize the flow solver.

3.1 Domain Decomposition, Load Balancing

A modified coordinate bisection method was usedto recursively divide a given computational mesh intosub-domains (figure 3). The sub-domains were cre-ated such that they contained approximately the samenumber of computational nodes. No effort was madeto optimize the domain decomposition process so as tominimize the number of edges that are shared between

9

8

7

6

54

3

10y

x

16

15

11

12

1314

21

Fig. 3 Domain decomposition of a rectangular re-gion using a modified bisection method

inter-processor boundary

edges that are shared between processors

Fig. 4 Halo nodes and the distribution of edgesalong processor boundaries

the sub-domains. The sub-domains are distributedamong the available processors in a manner that mini-mizes a combination of the computational cost associ-ated with the domains in each processor and the costof communication among the processors for a givendistribution. This methodology was found to result inwell balanced load distributions.

As the flow solver uses an edge-based data struc-ture to accumulate the fluxes at each vertex, the edgessurrounding the nodes that lie within a partition areaccumulated. If an edge connects a pair of nodes thatlie across processor boundaries, this edge is duplicatedin the two processors and ‘halo’ nodes are constructedfor both processors. This idea is illustrated further infigure 4.

3.2 Parallel Implementation of the Multigrid

Algorithm

The use of multigrid techniques for flow analysis ne-cessitates the need to exchange information betweenfine and coarse grid points and vice-versa. While the

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2 4 6 8 10 12 14 162

4

6

8

10

12

14

16

Number of Processors

Spe

edU

pActual SpeedUpIdeal SpeedUp

Fig. 5 Speedup from the parallel implementation

residuals from the fine grid points need to be accumu-lated at the coarse grid points, the corrections fromthe coarse grid points need to be transferred back tothe fine grid points. Hence, efficient point locationmethods are used to determine the interpolation co-efficients associated with each computational node inthe multigrid cycle. The required search methodologyhas been implemented using an octree-based searchroutine. Each grid in the multigrid cycle is recursivelydivided into octants that contain a certain number ofpoints. Using this data structure, a given point is iden-tified within an octant and the closest node closest inthis octant is determined. The tetrahedra connected tothis node are checked to see if they contain the searchpoint. Once such a tetrahedron is identified, inter-polation coefficients to aggregate the residuals and tointerpolate the corrections are constructed. Further,to reduce the communication cost among processorsduring the transfer of information between the fineand coarse grids, sub-domains on the coarser gridsare constructed and distributed so as to conform tothe division and distribution of the fine mesh. Typ-ical computational times to build the octree and tobuild the interpolation tables are in the range of a fewminutes for a sequence of meshes containing a millionnodes. The efficiency of the parallel implementation isshown in figure 5.

4. Aeroelastic Computations to Predict the

Flying Shape of Upwind Sails

The incompressible flow solver described in the pre-vious sections has been used to obtain the character-istics of sail configurations of high performance yachtsfor races like the Americas Cup. Modifications tothe inlet boundary condition were incorporated so asto simulate the boundary layer profile over the sea.Figure 7 shows a representative head and main sailcombination used in the Americas Cup. Due to thefast turn-around times of the flow solver it was possible

to use this computational package to obtain aerody-namic characteristics of the sail configurations for avariety of wind conditions, and also study the effect ofvarying twist, camber and sail trim. Further, in orderto predict the flying shape of the sail geometries, thisflow solver was coupled to NASTRAN. The aeroelas-tic package uses an iterative algorithm that transfersthe pressure loading obtained from the flow solver tothe structural model, and uses the deflections from thestructural analysis to modify the computational meshfor the fluid. This process is iteratively carried outuntil the deflections are below a particular threshold.

4.1 Structural Model

In the structural model of the sail the sail clothis discretized into quadrilateral membrane finite ele-ments with four nodes (after neglecting the presenceof batten pockets). These elements withstand all ex-ternal forces through tension and are incapable ofresisting bending moments. The translational and ro-tational degrees of freedom along the foot of the mainsail are suppressed. Along the mast, the translationaldegrees of freedom are inhibited while allowing forrotational motion. For the head sail, the point of at-tachment of the foot to the rig is constrained. Theleech of the main and head sail are allowed to movefreely to induce a geometric twist due to the aerody-namic loading. The mast is assumed to be rigid duringthe structural and aeroelastic calculations. The pres-ence of battens and tension cables and other structuralelements of the sail rig is neglected from this analysis.

The linear system of equations relating the displace-ments to the force field is advanced to a steady state byan iterative process that incrementally adds the loadwhile obtaining a converged displacement field for eachstep. This procedure was introduced in order to allowfor large deflections of the sail geometry. Wrinklingof the structure, which is an important considerationespecially around the leading edge (luff) and at thesail tip, is not anticipated by this model, but the useof a numerical algorithm to track large deformationsallows for wrinkling models to be included at a laterstage. Figures 13 and 14 pictorially depict the bound-ary conditions used for the structural model.

4.2 Aeroelastic Coupling Procedure

The pressure loading from the flow solver is fed tothe structural analysis to estimate the deflected shapeof the sail. To enable the transfer of loads and displace-ments to be conservative, the fluid mesh on the surfaceand the structural mesh are identical, eliminating theneed for interpolation. The deflected shape of the sailis used to deform the computational mesh. The pop-ular ‘spring-analogy’ method has been used to trackthe mesh deformations. While this method restrictsthe allowable deflections and may impair the qualityof the deformed mesh, it provides a simple tool to track

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mesh deformations. The deformed mesh is then usedto compute a new pressure loading for the sail. Thisiterative process offers no formal guarantee of conver-gence, but it usually predicts the deflected shape toreasonable accuracy in a few steps (typically 5 for sailgeometries).

4.3 Results of Aeroelastic Simulations

The deflected shapes of the head and the main sailare shown in figures 19 and 20. It can be seen fromthese plots that the lower sections of the head andthe main sail do not undergo appreciable deformation.The largest deflections occur in the mid-sections of themain sail. As the point of attachment of the main sailto the mast and the leading edge of the head sail werenot allowed to move, the aerodynamic loading changesthe twist of the sail geometry. This has a favorableinfluence on the pressure distribution, especially onthe head sail (figures 17 and 18). The pressure dis-tribution over the head and sail after the aeroelasticsimulation highlights the need to perform aeroelasticanalysis to obtain the flying shape of these sail geome-tries. While the lift and the drag distribution of thedeformed shape is not significantly different from theundeformed shape, the pressure distribution over thesail sections show that the twist and the camber distri-bution of the head and the main sail have been alteredto provide a smooth entry for the flow over the leadingedge of both components.

5. Aerodynamic Shape Optimization

With the availability of high performance computingplatforms and robust numerical methods to simulatefluid flows, it is possible to shift attention to auto-mated design procedures which combine CFD withgradient-based optimization techniques. Typically,in gradient-based optimization techniques, a controlfunction to be optimized (the sail shape, for example)is parameterized with a set of design variables and asuitable cost function to be minimized is defined. Foraerodynamic problems, the cost function is typicallylift, drag or a specified target pressure distribution.Then, a constraint, the governing equations can be in-troduced in order to express the dependence betweenthe cost function and the control function. The sensi-tivity derivatives of the cost function with respect tothe design variables are calculated in order to get adirection of improvement. Finally, a step is taken inthis direction and the procedure is repeated until con-vergence is achieved. Finding a fast and accurate wayof calculating the necessary gradient information is es-sential to developing an effective design method sincethis can be the most time consuming portion of thedesign process. This is particularly true in problemswhich involve a very large number of design variablesas is the case in typical three dimensional shape opti-mization.

The control theory approach11–13 has dramatic com-putational cost advantages over the finite-differencemethod of calculating gradients. With this approachthe necessary gradients are obtained through the solu-tion of an adjoint system of equations of the governingequations of interest. The adjoint method is extremelyefficient since the computational expense incurred inthe calculation of the complete gradient is effectivelyindependent of the number of design variables.

In this study, a continuous adjoint formulation hasbeen used to derive the adjoint system of equations.Hence, the adjoint equations are derived directly fromthe governing equations and then discretized. Hence,this approach has the advantage over the discrete ad-joint formulation in that the resulting adjoint equa-tions are independent of the form of discretized flowequations. The adjoint system of equations have a sim-ilar form to the governing equations of the flow andhence the numerical methods developed for the flowequations1, 8, 14 can be reused for the adjoint equations.The gradient is derived directly from the adjoint solu-tion and the surface motion independent of the meshmodification. This is critical for this design methodol-ogy to work on unstructured meshes. If the gradientdepends on the form of the mesh modification, thenthe field integral in the gradient calculation has to berecomputed for mesh modifications corresponding toeach design variable. To reduce the computational costwith this approach,15–17 the number of design variableswould have to reduced by parameterizing the geom-etry. However, this reduced set of design variableswould be incapable of recovering all possible shapevariations. Using the gradients computed with thisnew formulation, a steepest descent method is used toimprove an existing design.

5.1 The General Formulation of the Adjoint

Approach to Optimal Design

For flow about an airfoil, or wing, the aerodynamicproperties which define the cost function are functionsof the flow-field variables, w, and the physical loca-tion of the boundary, which may be represented bythe function, F , say. Then

I = I(w,F),

and a change in F results in a change

δI =∂IT

∂wδw +

∂IT

∂FδF , (12)

in the cost function. Using control theory, the gov-erning equations of the flow field are introduced as aconstraint in such a way that the final expression forthe gradient does not require re-evaluation of the flow-field. In order to achieve this, δw must be eliminatedfrom equation (12). Suppose that the governing equa-tion R which expresses the dependence of w and Fwithin the flow field domain D can be written as

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R(w,F) = 0 (13)

Then δw is determined from the equation

δR =

[

∂R

∂w

]

δw +

[

∂R

∂F

]

δF = 0 (14)

Next, introducing a Lagrange Multiplier ψ, we have

δI =∂IT

∂wδw+

∂IT

∂FδF −ψT

([

∂R

∂w

]

δw +

[

∂R

∂F

]

δF

)

δI =

(

∂IT

∂w− ψT

[

∂R

∂w

]

∂w

)

δw+

(

∂IT

∂F− ψT

[

∂R

∂F

]

δF

)

Choosing ψ to satisfy the adjoint equation

[

∂R

∂w

]T

ψ =∂I

∂w(15)

the first term is eliminated and we find that

δI = GδF (16)

where

G =∂IT

∂F− ψT

[

∂R

∂F

]

(17)

This process allows for elimination of the terms thatdepend on the flow solution with the result that thegradient with respect with an arbitrary number of de-sign variables can be determined without the need foradditional flow field evaluations.

After taking a step in the negative gradient direc-tion, the gradient is recalculated and the process re-peated to follow the path of steepest descent until aminimum is reached. In order to avoid violating con-straints, such as the minimum acceptable wing thick-ness, the gradient can be projected into an allowablesubspace within which the constraints are satisfied. Inthis way one can devise procedures which must neces-sarily converge at least to a local minimum and whichcan be accelerated by the use of more sophisticateddescent methods such as conjugate gradient or quasi-Newton algorithms. There is a possibility of more thanone local minimum, but in any case this method willlead to an improvement over the original design.

5.2 Design Using the Euler Equations

The application of control theory to aerodynamicdesign problems is illustrated in this section for thecase of three-dimensional wing design using the com-pressible Euler equations as the mathematical model.It proves convenient to denote the Cartesian coordi-nates and velocity components by x1, x2, x3 and u1,u2, u3, and to use the convention that summation over

i = 1 to 3 is implied by a repeated index i. Then, thethree-dimensional Euler equations may be written as

∂w

∂t+∂fi

∂xi

= 0 in D, (18)

where

w =

ρ

ρu1

ρu2

ρu3

ρE

, fi =

ρui

ρuiu1 + pδi1ρuiu2 + pδi2ρuiu3 + pδi3

ρuiH

(19)

and δij is the Kronecker delta function. Also,

p = (γ − 1) ρ

{

E −1

2

(

u2i

)

}

, (20)

and

ρH = ρE + p (21)

where γ is the ratio of the specific heats.Consider a transformation to coordinates ξ1, ξ2, ξ3

where

Kij =

[

∂xi

∂ξj

]

, J = det (K) , K−1ij =

[

∂ξi

∂xj

]

,

and

S = JK−1.

The elements of S are the cofactors of K, and in afinite volume discretization they are just the face areasof the computational cells projected in the x1, x2, andx3 directions. Using the permutation tensor εijk wecan express the elements of S as

Sij =1

2εjpqεirs

∂xp

∂ξr

∂xq

∂ξs. (22)

Then

∂

∂ξiSij =

1

2εjpqεirs

(

∂2xp

∂ξr∂ξi

∂xq

∂ξs+∂xp

∂ξr

∂2xq

∂ξs∂ξi

)

= 0. (23)

Now, multiplying equation (18) by J and applyingthe chain rule,

J∂w

∂t+R (w) = 0 (24)

where

R (w) = Sij

∂fj

∂ξi=

∂

∂ξi(Sijfj) , (25)

using (23). We can write the transformed fluxes interms of the scaled contravariant velocity components

Ui = Sijuj

8 of 21


as

Fi = Sijfj =

ρUi

ρUiu1 + Si1p

ρUiu2 + Si2p

ρUiu3 + Si3p

ρUiH

.

Assume now that the new computational coordinatesystem conforms to the wing in such a way that thewing surface BW is represented by ξ2 = 0. Then theflow is determined as the steady state solution of equa-tion (24) subject to the flow tangency condition

U2 = 0 on BW . (26)

At the far field boundary BF , conditions are specifiedfor incoming waves, as in the two-dimensional case,while outgoing waves are determined by the solution.

The weak form of the Euler equations for steady flowcan be written as

∫

D

∂φT

∂ξiFidD =

∫

B

niφTFidB, (27)

where the test vector φ is an arbitrary differentiablefunction and ni is the outward normal at the bound-ary. If a differentiable solution w is obtained to thisequation, it can be integrated by parts to give

∫

D

φT ∂Fi

∂ξidD = 0

and since this is true for any φ, the differential formcan be recovered. If the solution is discontinuous (27)may be integrated by parts separately on either side ofthe discontinuity to recover the shock jump conditions.

Suppose now that it is desired to control the surfacepressure by varying the wing shape. For this pur-pose, it is convenient to retain a fixed computationaldomain. Then variations in the shape result in corre-sponding variations in the mapping derivatives definedby K. As an example, consider the case of an inverseproblem, where we introduce the cost function

I =1

2

∫ ∫

BW

(p− pd)2dξ1dξ3,

where pd is the desired pressure. The design problemis now treated as a control problem where the con-trol function is the wing shape, which is to be chosento minimize I subject to the constraints defined bythe flow equations (24). A variation in the shape willcause a variation δp in the pressure and consequentlya variation in the cost function

δI =

∫ ∫

BW

(p− pd) δp dξ1dξ3. (28)

Since p depends on w through the equation of state(20–21), the variation δp can be determined from thevariation δw. Define the Jacobian matrices

Ai =∂fi

∂w, Ci = SijAj . (29)

The weak form of the equation for δw in the steadystate becomes

∫

D

∂φT

∂ξiδFidD =

∫

B

(niφT δFi)dB,

whereδFi = Ciδw + δSijfj ,

which should hold for any differential test function φ.This equation may be added to the variation in thecost function, which may now be written as

δI =

∫ ∫

BW

(p− pd) δp dξ1dξ3

−

∫

D

(

∂φT

∂ξiδFi

)

dD

+

∫

B

(

niφT δFi

)

dB. (30)

On the wing surfaceBW , n1 = n3 = 0. Thus, it followsfrom equation (26) that

δF2 =

0

S21δp

S22δp

S23δp

0

+

0

δS21p

δS22p

δS23p

0

. (31)

Since the weak equation for δw should hold for anarbitrary choice of the test vector φ, we are free tochoose φ to simplify the resulting expressions. There-fore we set φ = ψ, where the co-state vector ψ is thesolution of the adjoint equation

∂ψ

∂t− CT

i

∂ψ

∂ξi= 0 in D. (32)

At the outer boundary incoming characteristics for ψcorrespond to outgoing characteristics for δw. Con-sequently one can choose boundary conditions for ψsuch that

niψTCiδw = 0.

Then, if the coordinate transformation is such thatδS is negligible in the far field, the only remainingboundary term is

−

∫ ∫

BW

ψT δF2 dξ1dξ3.

Thus, by letting ψ satisfy the boundary condition,

S21ψ2 + S22ψ3 + S23ψ4 = (p− pd) on BW , (33)

we find finally that

δI = −

∫

D

∂ψT

∂ξi

δSijfjdD

−

∫ ∫

BW

(δS21ψ2 + δS22ψ3 + δS23ψ4) p dξ1dξ3. (34)

9 of 21


Here the expression for the cost variation dependson the mesh variations throughout the domain whichappear in the field integral. However, the true gradientfor a shape variation should not depend on the wayin which the mesh is deformed, but only on the trueflow solution. In the next section, we show how thefield integral can be eliminated to produce a reducedgradient formula which depends only on the boundarymovement.

5.3 Adjoint Equations for the Euler Equations

Modified by the Artificial Compressibility

Method

Although the adjoint equation represents a linearset of partial differential equations for the adjoint vari-ables, they are of the same form of the flow equations.The numerical solution procedures developed for theflow equations are applied to the adjoint system withthe appropriate boundary conditions. The adjointco-state flux terms are modified to account for the in-troduction of the artificial compressibility terms in thegoverning flow equations. The methodology followedhere is derived from the work of Luigi Martinelli andG. Cowles.20 The adjoint field equations can be ex-pressed as a time dependent system of the form,

∂ψ

∂t− [Ai]

T ∂ψ

∂xi

= 0 (35)

where

ψ =

p

φ1

φ2

φ3

(36)

Hence, this system can be integrated to steady stateusing a pre-conditioner similar to that used in themethod of artificial compressibility. The adjoint ‘con-tinuity’ equation is augmented by a time derivative ofthe adjoint pressure p.

∂p

∂t− Γ2 ∂φi

∂xi

= 0 (37)

The form of Γ is identical to that used for the flowequations since the magnitude of the eigenvalues ofthe flux Jacobians for the two systems are identical.Together with equation (37), the adjoint system is dis-cretized and solved in a manner that is consistent withthat used for the flow equation.

5.4 Reduced Gradient Formulations

Continuous adjoint formulations have generally useda form of the gradient that depends on the manner inwhich the mesh is modified for perturbations in eachdesign variable. To represent all possible shapes thecontrol surface should be regarded as a free surface. Ifthe surface mesh points are used to define the surface,this leaves the designer with a thousands of designvariables. On an unstructured mesh evaluating the

gradient by perturbing each design variable in turn,would be prohibitively expensive because of the needto determine corresponding perturbations of the entiremesh. This would inhibit the use of this design tool inany meaningful design process.

In order to avoid this difficulty an alternate for-mulation to the gradient calculation is followed inthis study. This idea was developed by Jameson andSangho Kim18 and was validated for two and three di-mensional problems with structured grids. However,as it is possible to devise mesh modification routinesthat are computationally cheap on structured grids,the major benefit of this alternate gradient formu-lation is for general three dimensional unstructuredgrids. To complete the formulation of the controltheory approach to shape optimization, the gradientformulations are outlined next. The formulation forthe reduced gradients in the continuous limit is pre-sented in the context of transformation between thephysical domain and the computational domain, andis easily extended to unstructured grid methods wherethese transformations are not explicitly used.

The evaluation of the field integral in equation (34)requires the evaluation of the metric variations δSij

throughout the domain. However, the true gradientshould not depend on the way the mesh is modified.

Consider the case of a mesh variation with a fixedboundary. Then,

δI = 0

but there is a variation in the transformed flux,

δFi = Ciδw + δSijfj .

Here the true solution is unchanged. Thus, the vari-ation δw is due to the mesh movement δx at fixedboundary configuration. Therefore

δw = ∇w · δx =∂w

∂xj

δxj (= δw∗)

and since∂

∂ξiδFi = 0,

it follows that

∂

∂ξi(δSijfj) = −

∂

∂ξi(Ciδw

∗) . (38)

It is verified by Jameson and Sangho Kim18 that thisrelation holds in the general case with boundary move-ment. Now

∫

D

ψT δRdD =

∫

D

ψT ∂

∂ξiCi (δw − δw∗) dD

=

∫

B

ψTCi (δw − δw∗) dB

−

∫

D

∂ψT

∂ξiCi (δw − δw∗) dD. (39)

10 of 21


Here on the wall boundary

C2δw = δF2 − δS2jfj . (40)

Thus, by choosing ψ to satisfy the adjoint equationand the adjoint boundary condition, we have finallythe reduced gradient formulation that

δI =

∫

BW

ψT (δS2jfj + C2δw∗) dξ1dξ3

−

∫ ∫

BW

(δS21ψ2 + δS22ψ3 + δS23ψ4) p dξ1dξ3.(41)

5.5 The need for a Sobolev inner product in

the definition of the gradient

Another key issue for successful implementation ofthe continuous adjoint method is the choice of anappropriate inner product for the definition of the gra-dient. It turns out that there is an enormous benefitfrom the use of a modified Sobolev gradient, which en-ables the generation of a sequence of smooth shapes.

When the metric perturbations, δSij , are related tothe surface motion, equation (41) finally yields the costvariation in the form of an inner product defined overthe surface

δI = (G, δF) =

∫

B

GδFdξB

where δF denotes the surface displacement. Then theupdate

Fn+1 = Fn − λGn

would result in an improvement

δI = −λ (G,G)) ≤ 0

It turns out, however, that the gradient is generally afunction in a lower smoothness class than the initialshape, with the result that there is a progressive lossof smoothness with each iteration.

This can be corrected by introducing a modified gra-dient which corresponds to a weighted Sobolev innerproduct, of the form

〈u, v〉 =

∫

(uv + εu′

v′

)dξ

This is equivalent to replacing G by G where in onedimension

G −∂

∂ξε∂G

∂ξ= G,

with G = 0 at the end points and making a shapechange

δF = −λG

This both preserves the smoothness of the re-designed shape, and acts as a preconditioner whichreduces the number of design steps needed to reach anoptimum solution.

6. Mesh Deformation

The modifications to the shape of the boundaryare transferred to the volume mesh using the springmethod. This approach has been found to be adequatefor the computations performed in this study.

The spring method can be mathematically concep-tualized as solving the following equation

∂∆xi

∂t+

N∑

j=1

Kij(∆xi − ∆xj) = 0

where the Kij is the stiffness of the edge connectingnode i to node j and its value is inversely proportionalto the length of this edge, ∆xi is the displacementof node i and ∆xj is the displacement of node j, theopposite end of the edge. The position of static equilib-rium of the mesh is computed using a Jacobi iterationwith known initial values for the surface displacements.

7. Results

The overall design process is illustrated in figure 6.

Mesh deformation

Pre−processor

Generate sequence ofmeshes

Repeat until design

Gradient evaluation,camber line changes

Adjoint Solver (Parallel)

Flow Solver (Parallel)

criterion is satisfiedFig. 6 Flow chart of the overall design process

11 of 21


7.1 Shape Optimization of Airfoils in

Incompressible Flow

In order to validate the design procedure, two di-mensional problems were studied first. An inversedesign problem that recovers the pressure distribu-tion over the Onera M6 airfoil was used to validatethe gradient formulations and the adjoint solution. Athree level multigrid cycle was used for obtain steadystate solutions for the flow and the adjoint equations.The grids were generated using a conformal mappingtechnique. The initial airfoil shape had a NACA 0012profile and the initial pressure distribution is shown infigure 21. Around 40 design cycles were required torecover the target pressure and shape by the designprocess (figure 21).

7.2 Inverse Design of Wings in Incompressible

Flow

To validate the design process for three dimensionalflows, a test problem similar to the two dimensionalcase was used. The initial wing had the planform ofthe Onera M6 but had NACA 0012 airfoil sections.The target pressure distribution corresponded to thesteady state pressure distribution over the Onera M6wing. Three levels of multigrid were used to obtainsteady state flow and adjoint solutions. The mesheswere generated using an automated grid generator andinterpolation coefficients were accumulated in a pre-processing step. The parallel implementation of theflow and adjoint solvers were used to reduce the com-putational time of the design process. Modificationsto the shape of the wing were transmitted to the inte-rior mesh using the spring deformation method whichworked well for this problem.

Figures 22,23,24,25 show that the target pressuredistribution has been recovered in about 50 designcycles. These computations took under 30 minutesrequiring 8 processors of an SGI Origin 300.

7.3 Inverse Design for Sail Geometries

The results of the flow and aeroelastic simulations,show that the interaction of the head sail with the mainreduces the development of sharp pressure gradientsaround the leading edge. This interaction is crucialto the performance of the main sail as it allows themain sail to be set at a higher angle to the center-lineof the boat. These results also show that the regionabove the head sail has large suction peaks which isa cause of concern. The aerodynamic shape optimiza-tion procedure validated in the previous sections wasused to redesign the main sail, with an aim of reducingthe pressure gradient around the luff of the main sail.An inverse design procedure was employed and thetarget pressure distribution was obtained by smooth-ing the pressure distribution on the main sail obtainedfrom the aeroelastic analysis. Figures 26,27,28 and 29show that a significant portion of the leading edge of

the main sail has been redesigned to allow for smoothentry of the flow. The associated reduction in sharpsuction peaks should have a favorable affect on thegrowth of the boundary layer over the upper surface.The change to the sections are shown in figure

8. Conclusions

Rotational inviscid flow solutions to sail configura-tions can now obtained in a cheap and efficient manner.These can be used to substitute potential flow modelsto obtain better estimates of the induced drag. Ad-joint based design methods can be used to redesignsail shapes to smooth out suction peaks and improvethe sail performance. The use of gradient formula-tions that depend only of the surface mesh allowsadjoint based methods to be used for unstructuredgrids in a computationally efficient manner. Hence, itis now possible to devise completely automated shapeoptimization procedures for sail configurations. Fur-ther exploiting the flexibility of unstructured grids,it is possible to devise a hierarchy of optimizationprocedures that can be used to tackle planform op-timization, sail settings, hull/keel shapes and othercomponents of the overall design process. This holdsgreat promise for the design of improved sails.

9. Ongoing and future directions of Research

Our current efforts are focussed on developing com-putational methods for viscous simulations. Prelim-inary results from a two equation turbulence modelhave provided good estimates of the friction drag forairplane configurations. Boundary layer methods cou-pled to inviscid flow solutions are also being investi-gated to provide a computationally cheap method toestimate the viscous effects. Redesign of hull and keelshapes are within sight, and it is hoped that in thenear future, a combined analysis and design tool forhigh performance boats will be realized.

References1A. Jameson and T.J. Baker, Improvements to the Aircraft

Euler Method, AIAA Paper 87-0353, 25th AIAA Aerospace

Sciences Meeting, Reno, January, 1987.2J. Reuther, A. Jameson, J. Farmer, L. Martinelli and

D. Saunders, Aerodynamic Shape optimization of complex

aircraft configurations via an adjoint method, AIAA Paper 96-

0094, 34th AIAA Aerospace Sciences Meeting, Reno, January,

1996.3A. Chorin, A Numerical Method for Solving the Incompress-

ible Viscous Flow problem, Journal of Computational Physics

Vol. 2, pp 12-26, 1967.4J. Farmer, L. Martinelli and A. Jameson, A Fast Multigrid

Method for Solving Incompressible Hydrodynamic Problems

with Free Surfaces, AIAA Paper 93-0767, 31st AIAA Aerospace

Sciences Meeting, Reno, January, 1993.5A. Rizzi and L. Eriksson, Computation of inviscid incom-

rpessible flow with rotation, Journal of Fluid Mechanics Vol.

153, pp 275-312, 1985.6A. Jameson, Analysis and Design of numerical schemes for

gas dynamics II, Artificial diffusion and discrete shock structure,

International Journal of Computational Fluid Dynamics, Vol.

5, pp. 1-38, 1995.

12 of 21


7A. Jameson, Multigrid algorithms for compressible flow cal-

culations, In W. Hackbusch and U. Trottenberg, editors, Lecture

Notes in Mathematics, Vol. 1228, pages 166–201. Proceedings of

the 2nd European Conference on Multigrid Methods, Cologne,

1985, Springer-Verlag, 1986.8A. Jameson, W. Schmidt and E. Turkel, Numerical Solution

of the Euler equations by finite volume methods using Runge-

Kutta time stepping schemes, AIAA Paper 81-1259, June, 1981.9A. Jameson, Multigrid algorithms for compressible flow

calculations, Proceedings of the 2nd European Conference on

Multigrid Methods, Cologne, 1985.10A. Jameson, Unpublished notes and computational tech-

niques for the solution of the unsteady Euler equations on un-

structured grids using multigrid methods.11J.L. Lions, Optimal Control of Systems Governed by Par-

tial Differential Equations, Springer-Verlag, New York, 1971.

Translated by S.K. Mitter.12O. Pironneau, Optimal Shape Design for Elliptic Systems,

Springer-Verlag, New York, 1984.13A. Jameson, Optimum Aerodynamic Design Using Control

Theory, Computational Fluid Dynamics Review 1995 Wiley,

1995.14T.J. Barth, Apects of unstructured grids and finite vol-

ume solvers for the Euler and Navier-Stokes equations, AIAA

Paper 91-0237, 29th AIAA Aerospace Sciences Meeting, Reno,

January, 1994.15K. Anderson and V. Venkatakrishnan, Aerodynamic De-

sign Optimization on Unstructured grids using a continuous ad-

joint formulation, AIAA Paper 97-0643, 34th AIAA Aerospace

Sciences Meeting, Reno, January, 1997.16J. Elliot and J. Peraire, Aerodynamic design using un-

structured meshes, AIAA Paper 96-1941, 33rd AIAA Aerospace

Sciences Meeting, Reno January, 1996.17S. E. Cliff, S.D. Thomas, T. J. Baker, A. Jameson and

R. M. Hicks, Aerodynamic Shape optimization using unstruc-

tured grid method, AIAA Paper 02-5550, 9th AIAA Sympo-

sium on Multidisciplinary Analysis and Optimization, Atlanta,

September, 2002.18A. Jameson and Sangho Kim, Reduction of the Adjoint

Gradient Formula in the Continuous Limit, AIAA Paper, 41st

AIAA Aerospace Sciences Meeting, Reno January, 2003.19A. Jameson, L. Martinelli and J. Vassberg, Using CFD

for Aerodynamics - A critical Assesment, Proceedings of ICAS

2002, September 8-13, 2002, Toronto, Canada.20G. Cowles and Luigi Martinelli, A Control-Theory Based

Method for Shape Design in Incompressible Viscous Flow using

RANS, AIAA Fluids 2000-2544, June 19-22, Denver, CO.

13 of 21


32 m

24 m

Twisted inflowwith boundary layerprofile

Main Sail

10 m

10 m

2.3 m

Jib

Fig. 7 Sail geometry

0 50 100 150 200 250 300 350 400 450 50010−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102Convergence History

Number of iterations

Log(

Err

or)

1 grid2 grid3 grid

Fig. 8 Convergence history for sail geometrieswith artificial compressibility

AIRPLANE CP from -1.0000 to -0.5000

Fig. 9 Pressure contours on the leeward side

AIRPLANE CP from -0.6000 to -0.1000

Fig. 10 Pressure contours on the windward side

14 of 21


0 5 10 15 20 25−0.2

0

0.2

0.4

0.6

0.8

1

1.2Lift and Drag distribution along the height of the head sail

Height along the span

ClCd

Fig. 11 Spanwise force distributions on the headsail

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

1.4Lift and Drag distribution along the height of the head sail

Height along the span

ClCd

Fig. 12 Spanwise force distributions on the mainsail

Leech is allowed

to move freely

Translational degrees of freedom

suppressed at the mast

Mast is assumed to be rigid

Translational and

rotational degrees of freedom suppressed at the boom

Fig. 13 Boundary conditions for the main sail

the stayfreedom supressed alongTranslational degrees of

All degrees of freedom suppressed

mast

clew foot

leech

luff

head

stay

Fig. 14 Boundary conditions for the head sail

15 of 21


M2F TNZ ALPHA 19.000 Z 2.587

CL 0.8872 CD 0.2378 CM -0.5607

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

+++++++++++++++++++++++++++++++

+++

++++++ + + ++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+

+

M2F TNZ ALPHA 19.000 Z 9.509

CL 0.8018 CD 0.1745 CM -0.5067

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++

++++++++++++++++++++++++++++++

++

++++

++

+ + ++

++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + +

+

+

M2F TNZ ALPHA 19.000 Z 16.409

CL 0.5918 CD 0.1136 CM -0.4305

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++

++++++++++++++++++++++++++++

+++

++++++

++

+ + + ++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + + +

+

Fig. 15 Pressure distributions along sections at 1,25 and 85 percent of the height of head sail

M2F TNZ ALPHA 19.000 Z 3.885

CL 0.8119 CD 0.3222 CM -0.5526

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

++++++++++++++++++++++++++++++++++++++++ + + + + + + + +

++

++

+ ++ + + + + + + + + + + + + + + + + + + + + + +

+

+

M2F TNZ ALPHA 19.000 Z 10.532

CL 0.7847 CD 0.2737 CM -0.5064

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

++++++++++++++++++++++++++++++++++++++++ + + + +

++

++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + ++

+

M2F TNZ ALPHA 19.000 Z 18.929

CL 0.9173 CD 0.2608 CM -0.5521

NCYC 500 RES0.180E-07

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

+++++++++++++++++++++++++++++++++++++++

+ + + ++ +

++ +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + ++

+

Fig. 16 Pressure distributions along sections at 1,25 and 85 percent of the height of the main sail

16 of 21


M2F TNZ ALPHA 19.000 Z 2.587

CL 0.8897 CD 0.1600 CM -0.4765

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp ++

++++++++++++++++++++++++++++++++++++++

+

++ + +

+ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

M2F TNZ ALPHA 19.000 Z 9.509

CL 1.0856 CD 0.0757 CM -0.5412

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++

++++++++++++++++++++++++++++++++++++

++

+

++

++

++

++

++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

M2F TNZ ALPHA 19.000 Z 16.409

CL 1.0096 CD -0.0163 CM -0.5429

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++++++

+++++++++++++++++++++++++++++++

+++++ + +

++

++

++

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + ++

Fig. 17 Pressure distributions along sections at 1,25 and 85 percent of the height of head sail afteraeroelastic analysis

M2F TNZ ALPHA 19.000 Z 3.885

CL 0.7112 CD 0.2590 CM -0.4177

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

++++++++++++++++++++++++++++++++++++++

+++

+ + + + + + + + + + ++ + + + + + + + + + + + + + + + + + + + + + + + + + +

++

M2F TNZ ALPHA 19.000 Z 10.532

CL 0.9060 CD 0.2505 CM -0.4739

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

+++++++++++++++++++++++++++++++++++++++

++

+

+ + + + + + + ++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + ++

+

M2F TNZ ALPHA 19.000 Z 18.929

CL 1.1717 CD 0.2199 CM -0.5570

NCYC 500 RES0.538E-02

1.

00

0.50

0.

00 -

0.50

-1.

00 -

1.50

-2.

00 -

2.50

-3.

00

Cp

+++++++++++++++++++++++++++++++++++++++

+

+

+

++

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ++

Fig. 18 Pressure distributions along sections at 1,25 and 85 percent of the height of main sail afteraeroelastic analysis

17 of 21


2 4 6 8 10 12 14−0.5

0

0.5

1

1.5

2

Z = 25

Deformed and original section geometry along the height of the head sail

Z = 18Z = 14Z = 8Z = 3.6

OriginalDeformed

Fig. 19 Original and deformed sail sections for thehead sail

10 11 12 13 14 15 16 17 18 19 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Z = 5

Z = 8.5

Z = 14

Z = 19

Z = 24

Z = 31

Deformed and original section geometry along the height of the main sail

OriginalDeformed

Fig. 20 Original and deformed sail sections for themain sail

NACA 0012 TO ONERA MACH 0.000 ALPHA 1.796

CL 0.2117 CD 0.0041 CM -0.0029

GRID 160X32 NDES 0 RES0.469E-03 GMAX 0.100E-05

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++

+++++

+

+

+

+++

+

+

+

+

+

+

+++++++++++++++++++++++++++ + + + + + + + + + + + + + + + +

+ +

NACA 0012 TO ONERA MACH 0.000 ALPHA 2.015

CL 0.2116 CD 0.0059 CM -0.0053

GRID 160X32 NDES 90 RES0.195E-04 GMAX 0.161E-04

0.1E

+01

0.8E

+00

0.4E

+00

0.0E

+00

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++

++++++++++++++++++++++++++++++++++++++++++

+

+

+

+++

+

+

+

+

++++

+

+

+++++++++++++++++++++++ + + + + + + + + + + + + + + + +

++o

oooooooooooooooooooooooooooooooooooooooo

ooo

o

o

o

o

o

o

o

o

o

oooo

o

o

oooooooooooooooooooo o o o o o o o o o o o o o o o o o o

oo

o

Fig. 21 Initial and final pressure distribution, o isthe target pressure distribution, x is the computedpressure distribution for the redesigned airfoil

18 of 21


NACA 0012 TO ONERA M6 ALPHA 3.060 Z 0.000

CL 0.1890 CD 0.0186 CM -0.0594

NCYC 10 RES0.194E-02 DESIGN CYCLE 50

0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o

+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o+o

+o

Fig. 22 Attained(+,x) and target(o) pressure dis-tributions at 0 % of the wing span


CL 0.2089 CD 0.0063 CM -0.0543


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o



CL 0.2299 CD 0.0032 CM -0.0573


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o



CL 0.3340 CD 0.0219 CM -0.1151


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o+o+o+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o +o +o +o +o +o +o +o +o +o

+o


19 of 21


TNZ : INVERSE DESIGN ALPHA 19.000 Z 4.975

CL 0.7850 CD 0.2799 CM -0.4489


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o

+

o

+o

+

o+

o

+

o

+o+o

+o+o

+o+o+o

+o

+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o+o+

o+o+o+o

Fig. 26 Initial (o) and final pressure distributionat 15% height on the main sail


CL 0.9992 CD 0.2965 CM -0.5381


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+

o+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o+o+o+

o+

o

+

o

+

o

+

o

+

o

+

o+o+o

+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+

o+o+

o+o+o+o



CL 1.4461 CD 0.2735 CM -0.6545


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o+o

+o+o+o+o+o+o+o+o+o+o+o+o+o+

o+o

+o

+o

+o

+o

+o

+o

+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+

o+o

+o

+o+

o+

o

+

o

+o

+

o

+

o

+

o

+

o

+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+

o+o+o+

o+o+o+o+o+o+o+o+o+o+o+o+o+o+o+o



CL 1.5637 CD 0.2928 CM -0.6885


0.1E

+01

0.5E

+00

0.0E

+00

-.5E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+o

+o+o+o

+o+o+o+o+o+o+o+o+o+o+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+o

+o

+o+

o+

o

+

o

+

o

+

o

+

o+

o

+

o

+o+o+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o

+

o+o+o+o+

o+o+o+o+o+o+o+o+o+o+o+o+o

+o+o


20 of 21


11 12 13 14 15 16 17 18 19−3

−2

−1

0

1

2

3

Initial and deformed sections at 15 percent height

x

y

InitialRedesign

Fig. 30 Initial and redesigned camber line at 15%of height

11 12 13 14 15 16 17 18

−2

−1

0

1

2

3


x

y

InitialRedesign


11 12 13 14 15 16 17

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3


x

y

InitialRedesign

Fig. 32 Initial and redesigned camber line at 75% of height

11.5 12 12.5 13 13.5 14 14.5 15 15.5 16 16.5

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5


x

y

InitialRedesign


21 of 21


Documents

AIAA 2003Œ3531 - Stanford Universityaero-comlab.stanford.edu/Papers/shankaran.aiaa.03-3501.pdf · AIAA 2003Œ3531 Numerical Analysis and Design of Upwind Sails Sriram and Antony