Upwind Sail Performance Prediction for a VPP including “Flying Shape” Analysis

Brian Maskew, Flow Simulation Consultant, Winthrop, WA

Frank DeBord, Chesapeake Marine Technology LLC, Easton, MD

ABSTRACT

A coupled aerodynamic/structures approach is

presented for predicting the flying shape and performance

of yacht sails in upwind conditions. The method is

incorporated in a flow simulation computer program, and is

part of an ultimate objective for a simultaneous aero-

elastic/hydro analysis in a Dynamic Velocity Prediction

Program (DVPP), that will include a 6DOF motion solver,

and at some point could include calculations in waves. The

time-stepping aerodynamic module uses an advanced vortex lattice method for the sails and a panel method with

special base separation treatment to represent the above-

water part of the hull and mast. A coupled inverse

boundary layer analysis is applied on all surfaces including

both sides of each sail membrane; this computes the skin-

friction drag and the source displacement effects of the

boundary layers and wakes, including bubble and leeside

“trailing-edge” type separations.

At each step, the computed aerodynamic pressure and

skin-friction loads are transferred to a coupled structures

module that uses a network grid of tension “cords” in each sail membrane, each cord representing a collection of fiber

“strings”. The solution of a structural equilibrium matrix

provides the displacements needed to achieve balance

between the aerodynamic and tension loads at each grid

point as the shape iterations proceed.

Details of the methodology used are presented and

comparisons of predicted aerodynamic forces to wind

tunnel results and an existing VPP sail model are provided.

In addition, predictions are compared to some simple

experiments to demonstrate the aerodynamic/structural coupling necessary to predict flying shape. Finally, an

outline is given for incorporation of this methodology into

the planned Dynamic Velocity Prediction Program.

NOTATION

B/L

S/Line

F&M

GFF

POV

6DOF

FS

SR VR

VT

VW

VA

VN

VBL

VW,L

ρ

ζ

μ

μW

n P

q

CF

Cp

ΔCp δFP δFF δF

δM

γ E

PR

A

Ӕ

ε

δl

l

T

THE 19th

CHESAPEAKE SAILING YACHT SYMPOSIUM

ANNAPOLIS, MARYLAND, MARCH 2009

Boundary Layer

Stream Line

Force and Moment Vectors

Ground-Fixed-Frame of reference

Position, Orientation, Velocity of Configuration

Six Degrees Of Freedom

Free Surface

Reference area (m2) Reference speed (m/sec)

True wind velocity at height HW (m/sec)

Wind speed at height z (m/sec)

Apparent wind velocity (m/sec)

Velocity component normal to Surface (m/sec)

VN due to B/L Displacement (m/sec)

Velocity on Windward and Leeward surfaces

of sail membrane (m/sec)

Mass Density of air (kg/m3)

Source value (m/sec)

Doublet value (m2/sec) Doublet value on a Wake point (m2/sec)

Unit normal vector to Surface

Local pressure (Pa)

Reference dynamic pressure, ½ ρVR2 (Pa)

Skin Friction Coefficient

Pressure Coefficient, P/q

Jump in Cp across the sail membrane

Pressure Force increment (N)

Skin Friction Force increment (N)

Total Force increment (N)

Moment increment (N-m) Vorticity vector on sail = (VW - VL) (m/sec)

Modulus of Elasticity (Pa)

Poisson Ratio

Cross-sectional area of a fiber (m2)

Strength of each fiber, A * E (N)

Tensile strain in a Cord = δl/l

The extension in length of a Cord (m)

Original unstrained length of a Cord (m)

Tension in a Cord (N)

81

INTRODUCTION

Over the years the VPP approach has developed into a

well accepted tool for predicting the performance of sailing

yachts, and is applied not only as a design aid but also as a

tactical tool for improving racing strategies. The VPP background has been well covered by earlier papers (e.g.,

Jacquin et al (2005), Korpus (2007)), and clearly, there is a

wide range of methods. Although most VPP’s are for

steady-state conditions, there are some that cover the

unsteady conditions due to motions in waves and

maneuvering; for example, Day (2002), Krebber and

Hochkirch (2006), Harris (2002, 2005) and Binns et al

(2008). Largely, the unsteady force and moment data are

generated in separate modules and are based on linearized

boundary conditions.

The power of a VPP depends heavily on the aero- and

hydro-dynamic force-and-moment database assembled for a configuration, and to be effective the database should be

extensive, covering a wide range of conditions. Ideally, this

database should be from full scale measurements, but such

experiments are difficult to setup, time consuming and

costly and also prone to measurement inaccuracies. Model-

scale experiments are easier to setup but are still relatively

expensive, and are subject to scale effects. In particular,

scaling model sail performance data is a major obstacle,

due mainly to issues related to accurately representing full-

scale flying shapes and trim and also wind gradient.

Consequently, much reliance is placed on CFD predictions being more practical when populating a large data base.

Predicting a yacht’s force and moment data using

CFD methods faces problems associated with modeling

complex interaction of waves, viscosity, flow separation,

turbulence and vortex wakes. Predicting sail performance is

especially difficult due to the fact that sail material is both

extensible and flexible, so the sail shape varies in

accordance with the local forces and vice-versa. “Off-the-

shelf” methods for this so-called “flying shape” prediction

are typically beyond reach as a routine tool since they are

time consuming, need a high level of user expertise and demand costly computational resources (Ranzenbach and

Kleene, (2002)). In assembling a practical CFD

performance prediction method, therefore, compromises

are generally made and not all “real-word” aspects are

covered in one package. Potential flow predictions are

rapid but without viscous treatment they lose some of the

real-world effects of boundary layers and separation unless

supplemented by empirical data. In principle, RANS

methods capture most “real-world” effects but can be very

time consuming when populating a high-detail data base

for a VPP operation, and this may not be practical for a time-critical design operation.

One of the most advanced approaches for predicting

steady VPP data is the RANS treatment presented by Rich

Korpus (2007), but even Rich admits this is not for

everyone. First a certain amount of expertise is needed to

run such a program, but above all a significant computer

power is required to deal with an adequate density of grid

points (in excess of 5M points) in a “reasonable” time

scale. High grid density is essential to capture flow details

such as bubble separations and vortex/surface interactions.

Solutions with low grid densities inherently have poor resolution of boundary layer effects and this may lead to

late prediction of separation and hence over-estimation of

lift. Also, dissipation in a sparse grid can cause rapid decay

in the downwind vorticity and hence under-prediction of

vortex-wake influences on sails that are downwind.

This paper presents an approach to predicting yacht

sail upwind performance data based on an advanced

Boundary Element Method, and is aimed between the

RANS and Potential Flow approaches, hopefully leaning

towards the “real world” accuracy of the RANS, but with

the speed advantage of the potential flow approach--- at least for the steady VPP. But in addition, the sail flying

shape analysis is included here using a coupled structural

analysis developed internally so that the procedure is self-

contained. The aero- and structural models for this

capability are described in the following Sections together

with a discussion of some verification and validation

results. The coupled program, FloSim, is fully operational

and a proprietary version with certain specialized features

for sail design, has been delivered as program SpiderSim II

to the Fiorenzi Marine Group, Srl. in Italy as part of their

SpiderTech sail design/ manufacture operation. A follow-on paper on this aspect is planned for the future.

In a planned next phase, the present aeroelastic

method will be combined with the existing hydrodynamic

version of FloSim, to provide a simultaneous aeroelastic-

hydrodynamic solution for a complete yacht in an unsteady

non-linear, time-stepping formulation. A 6DOF motion

solver and a wave generator will be added incrementally to

provide a Dynamic VPP, or DVPP, an outline of which is

presented herein. The feasibility of this approach was

demonstrated earlier in Maskew (1993).

AERODYNAMIC ANALYSIS

The Flow Problem and Assumptions

Consider the aerodynamic flow problem of a heeled

multi-sail yacht configuration, including mast and hull,

moving with uniform speed over a calm free surface in the

presence of a wind with gradient. For the present objective

the free surface is regarded as a horizontal plane of

reflection. A Ground-Fixed-Frame (GFF) coordinate

system is established with the X and Y axes in the horizontal plane, X being directed “downwind” and Y to

the starboard, with Z vertically upward. The true course is

directed along the negative X axis and the true wind angle,

gamma (γ), is positive for a port tack, see Figure 1.

82

Figure 1– Onset Flow Diagram

The true wind velocity, VT, is defined at a certain

height, HW, in the wind gradient and so the apparent wind

speed, VA, and angle, beta (β), vary with height up the mast with the yacht moving at velocity, VB.

The vertical gradient, or shear, in the wind represents

a rotational flow field that strictly presents a problem for a

Boundary Element Method. Although the wind gradient

vorticity effects could be treated using field distributions of

doublets this would significantly increase the complexity of

the model with possibly very little “payoff” in improved

results. In the present model, therefore, the wind gradient is

represented simply with a vertical variation of horizontal

wind speed and the total head is assumed to be a function

of height only. This simplification assumes the vertical perturbation of the onset flow streamlines is small.

Following the above simplification, we further

assume that regions of the flow field that are dominated by

viscous and rotational effects are confined to thin boundary

layers and thin wake shear layers associated with the

vorticity in the flow around the yacht configuration. The

rest of the flow field is assumed to be inviscid, irrotational

and incompressible and therefore governed by Laplace’s

equation. These assumptions do not preclude a simple

representation of extensive separations such as bluff-body

base flows (e.g., in the lee of the hull) or “bubble” flows such as behind the mast, see the discussion below under

“Flow Separation Model”.

Based on the above assumptions, and using Green’s

theorem, the “near-field” boundaries of the flow region,

i.e., the configuration surfaces and vortex wakes, are

represented by doublet and source singularity distributions.

The latter include source terms representing the

displacement effect of each viscous region. The flow region

is also bounded by an infinite horizontal reflection plane at

the calm water surface.

Boundary Conditions

In order to evaluate the singularity distributions the

following Boundary Conditions are applied on the

configuration surfaces, keeping in mind that the magnitude

and direction of the apparent onset flow varies with height

when yacht velocity is combined with the wind gradient:

(a) On the sail surfaces:

(i) A Neuman boundary condition of prescribed

normal velocity is applied, leading to an integral

equation for the unknown doublet distribution.

(ii) Source terms are evaluated based on the

boundary layer displacement thickness properties

computed along local streamlines on both surfaces

of each sail membrane; the windward and leeward

source terms are combined to produce a resultant

displacement term and also a resultant normal

velocity term (due to the difference across the membrane) which is included in (i).

(b) On the mast and hull surfaces:

(i) On the external (i.e., wetted) surfaces a

Neuman boundary condition of prescribed

resultant normal velocity is applied. This is a

combination of the normal component of the onset

flow and a term representing the displacement

effect of the boundary layer. This establishes the

source distribution.

(ii) On the internal surface the Dirichlet boundary condition of zero perturbation potential

is applied. This leads to an integral equation in

terms of the unknown doublet distribution.

(c) On the wake surfaces:

(i) An unsteady Kutta condition is applied at

points along the trailing edge of each lifting

component in the configuration to establish

circulation values, and hence doublet values shed

onto the wake surfaces.

(ii) Source terms on the wake surfaces are

evaluated based on the boundary layer characteristics computed along local streamlines.

Windward and leeward sources are combined.

(iii) Since a wake surface cannot support a load, a

force-free condition is applied.

Hence, the problem has mixed Neumann and Dirichlet

boundary conditions. Also, it is a non-linear problem due

to the mutual interaction between the configuration surface

singularities, the boundary layer displacement effect, and

the location and singularity strengths of wake surfaces.

Numerical Aerodynamic Model

In the numerical model, the non-linear problem is

dealt with using a time-stepping procedure. The flow is

started impulsively from rest and the wake elements are

convected with the local flow at each time step. This forms

Wind Gradient Wind Gradient

0.0

1.0

h =

z/H

W

VW/VT

VW/VT

VB ; Boat Velocity

Leeway Angle, λ

GFF X axis

γ True Course

VT ; True Wind at height HW

β

VA

83

the shape of each wake surface over time and also satisfies

the force-free condition. The singularity distributions are

discretized into piecewise uniform doublet and source

distributions on quadrilateral panels. The source value, ζ,

on each surface panel is evaluated directly from the current

normal velocity at the panel center:

ζ = (VN + VBL - n•VA)/4π ………………….(1)

Where:

VN is the resultant normal velocity at the panel center;

VBL is the computed boundary layer displacement

term from the previous step , and

n•VA is the normal component of the local apparent

wind, n being the panel outward normal vector.

On a solid boundary, VN is generally zero but where there

is extensive flow separation, VN is used to provide an

equivalent “displacement source” distribution (see later under “Flow Separation Model”).

The surface boundary conditions are applied at each

panel center; the integral equations in Green’s theorem are

thereby converted to a set of simultaneous equations in

terms of the panel doublet values, and hence a matrix of

influence coefficients is formed, see, for example, Maskew

(1987). The doublet values are obtained at each time step

using a GMRES iterative matrix solver.

On the sail surfaces, where the Neuman normal velocity boundary condition is applied, the uniform doublet

panels are treated as quadrilateral vortex “rings” as in an

earlier method, (Maskew (1970)--- see also Maskew (1976)

Paper#10). This treatment is similar to that of a vortex

lattice method except each panel has a closed vortex

system; this is more convenient when dealing with

cambered surfaces. In the earlier work, as in current

“standard” vortex lattice practice, a ¼ panel chordwise shift

was applied to the lattice; this makes the location of the

center of pressure independent of the panel chordwise

density but does not affect the calculated lift.

Contrary to the above “standard” practice, the present

model does not apply the ¼ panel shift and so the sail

edges coincide with panel edges and this is more

convenient when the mast and boom are included in the

model. This decision was made because relatively high

panel chordwise densities (30-40) are used: (a) to provide

good resolution for boundary layer calculations and

separation bubbles; (b) to resolve flow details in the close

interaction with overlapping sails and passing vortex wakes

and (c) to provide a reasonable transition in panel size

across the mast/mainsail junction. With such high grid densities the lattice shift for the center-of-pressure issue

would be very small (~ ½ % of the chord).

To further justify the decision not to use the ¼ panel

shift, Figure 2 shows a computed 2-D potential flow

velocity distribution compared with the analytic Joukowski

transformation solution for a circular-arc camberline with

camber, f/c, = 0.1 and incidence α = 5.0 degrees. The

comparison is very close even though only 20 equally

spaced vortices were used, yet typically, as stated above, 30

or 40 vortices are generally used across the chord. The

computed lift, drag and moment coefficients for this case (using the Kutta-Joukowski theorem; see later) are

1.79956, 0.0, -0.76705 compared with the Joukowski

transformation values of 1.79948, 0.0, -0.76777,

respectively; the center-of-pressure error is about 0.1%.

Figure 2–Comparison of Analytic and Calculated Velocity distributions on a 0.1 f/c Circular Arc at α = 5.0

Some vortex lattice methods use a “cosine” or “Lan”

(Lan (1974)) panel distribution across the chord to produce

small intervals at the leading and trailing edges. This

provides a slightly better accuracy than with uniform

spacing when using a small number of panels (say less than

10 across the chord), but the differences are negligible

when using the current higher panel densities, moreover,

cosine spacing here would give unreasonably small panels

at the beginning and end. The high density equal grid is therefore preferred here, at least in the chordwise direction;

a “cosine” distribution is generally used in the vertical

direction (where the panel density is typically lower), since

this provides more “spanwise” grid detail for the vortex

wake roll-up at the sail foot and head.

Inviscid Aerodynamic Analysis

At each time step the perturbation velocity and

pressure distributions are evaluated based on the current

doublet solution and source values. Analysis of velocities

and pressures on the mast and hull (i.e., closed) surfaces is relatively straightforward; the gradient of the doublet

distribution provides the tangential perturbation velocity

and the local source value gives the perturbation normal

velocity. These are combined with the non-uniform onset

flow to evaluate the local flow relative to the surface. The

pressure value is adjusted for the change in total head in the

wind gradient at the local height above the reflection plane.

Joukowski Transformation

Upper

Lower 2

1

0

V

/VR

0.0 0.2 0.4 0.6 0.8 1.0

X/C

Calculation

84

Analysis of velocities and pressures on the (open) sail

surfaces is slightly more complicated than the above; the

gradient of the doublet distribution now provides the local

vorticity vector, γ; this is the difference between the

windward and leeward tangential velocity vectors. These

individual vectors are evaluated using γ and the mean velocity. The latter is obtained by adding to the local onset

flow the singularity-induced contributions from all the

panels in the configuration (surface and wake) but

excluding the local panel contribution to the tangential

component. The normal velocity component on each face

is half the local source value combined with the residual

value in the mean normal velocity; the latter will be non

zero due to the differential in the windward and leeward

boundary layer displacement values applied in the

boundary conditions, (see (a) (ii) above).

The singularity values, velocities, etc. at the panel

vertices are obtained by interpolating in the neighboring panel center-point values. Off-body velocity calculations

for wake convection and for flow survey purposes, then use

distributed singularities for the panel influence coefficients;

this allows velocity calculation points to approach very

closely to the singularity sheets whereas for the basic

lattice the off-body velocities deteriorate rapidly closer

than about one panel width (Maskew (1976) Paper#20).

Boundary layer Analysis

With the surface velocity distribution established, streamlines can be traced on the sail windward and leeward

surfaces and on the mast and hull. 2-D Boundary layer

calculations are then performed along each surface

streamline using Cebeci’s (1999) inverse code. The

boundary layer properties, including displacement source

term and skin friction coefficient, are transferred onto the

local panels for use in the next time step; the displacement

source term is included in the surface and wake panel

influences when forming the next influence coefficient

matrix, and the panel skin friction coefficient is used in the

force and moment integration.

The boundary layer calculation on each streamline

starts at an upstream stagnation point on the mast or near

the sail luff, or near the hull bow. The calculations start

with laminar flow and proceed either to laminar separation

or to natural transition to turbulent flow. If transition is

established the turbulent calculation proceeds until either

the leech is reached or conditions for turbulent separation

are reached. If laminar separation occurs before transition

the program looks for transition to turbulence in the bubble

with possible turbulent reattachment, otherwise there is

total separation.

Flow Separation Model

Bluff body separations present a fundamental problem

to a potential flow method; the basic flow simply goes

around the sharp corner and recovers to stagnation pressure

in the base. Consequently, the potential flow form drag is

essentially zero. Fortunately, singularity sheet modeling

provides two basic ways to represent separated flow in a

simple way: a vortex sheet model and a source outflow

model. Both models provide a reasonable base pressure

value for the purpose of drag prediction. At the same time they provide a good account of the displacement effect of

the separated wake so that neighboring parts of the

configuration no longer see the abrupt “potential flow”

contraction in the streamlines.

As far as the flow outside the separated region is

concerned, the source and vortex sheet methods are

equivalent, but they differ in the representation of the flow

inside the wake. While the source model “fills” the base

flow with fluid travelling at essentially the same speed as

the local onset flow, the vortex sheet model provides a

simple representation of the free shear layer separating the external flow from the “base” flow, see Figure 3. Hence,

the vortex sheet model provides a reasonable, albeit simple,

representation of the essentially stationary flow inside the

wake (Maskew and Dvorak (1977). The vortex sheet

strength is part of the unsteady potential flow solution and

this provides the jump in total pressure across the separated

shear layer through the unsteady pressure term in

Bernoulli’s equation. Hence, the base pressure is also part

of the solution. In FloSim the vortex sheet is actually

represented by a doublet distribution; the gradient of the

doublet being the local vorticity value.

Figure 3 – Modeling Separated Base Flow

Where the details of the internal wake flow are not of

direct concern, e.g., on the lee side of the hull, the source

outflow model is adequate and provides the displacement effect for the onset flow for the sails.

When the boundary layer calculations predict a

bubble separation, e.g., on the lee side of the mast or on the

lee side of the Genoa luff, this too can be represented either

by a vortex sheet or a source model, see Figure 4. Based on

Vortex Sheet Model

Onset Flow

Simple Potential Flow Model

has stagnation Cp in the Base

V~ 0; Base Cp is part of solution

External Streamlines

Source Outflow Model; Base Cp~ 0

85

the observations of Wilkinson (1989) and others, the static

pressure at separation remains constant over most of the

extent of the bubble, then a rapid recovery occurs near the

bubble end. The recovery corresponds to the spread of the

turbulent flow following transition of the separated laminar

shear layer. The source model for the bubble comprises a source/sink distribution, the net strength being zero for a

closed bubble. The peak source value depends on the

height of the bubble contour from the boundary. For the

vortex sheet model the vorticity strength is determined by

the velocity at separation and remains constant over the

extent of the bubble. For short bubbles it is reasonable to

place the constant vorticity sheet directly on the body

surface. Both models have been formulated in FloSim but

the source model is currently favored as being more

“robust”, especially for the mast/main situation.

Figure 4 – Modeling Bubble Separation

Vortex Wake Model

Wakes are singularity surfaces that emanate from

separation lines such as the leech of each sail. They are

defined by a set of wake doublet points. New doublet

points are created at the trailing edge at each time step and

existing wake doublet points convect downstream at the

local velocity and maintain a constant doublet value over

time. This value is established as the point leaves the separation line, being determined by the unsteady trailing

edge Kutta condition:

∂μW/∂t + Vm ∂μW/∂s = 0 …………….(2)

Where μW is the doublet value at the trailing edge and

Vm is the mean convection speed leaving the trailing edge.

The derivative of μW with distance, s, is along the local

mean streamline. For the present case we are interested in

the final steady solution and the derivatives quickly go to

zero, however, for the more general DVPP case these terms will reflect the changing conditions due to yacht motions.

The wake points also carry a source term that

represents the displacement effect of the wake downstream

from the shedding line. This distribution is established by

the inverse boundary layer calculation along the streamline

crossing the local trailing edge. The wake “source” term is

actually a sink representing entrainment and this typically

decays to zero strength quite quickly with streamwise

distance and so the wake source panels are constructed on a

much finer grid than that of the wake doublets; actually, the wake source points are interpolated along the wake

streamwise lines established by the doublet points.

On a multi-sail rig, as the points on a vortex wake

from one sail move with the flow they may impinge on a

downstream sail; for example, the wake from the head of a

Genoa is likely to impinge on the mast or the luff region of

the Main sail. Such an impingement may cause a numerical

problem in the evaluation of local perturbation velocities,

so procedures are installed to keep the wake points above

the configuration surfaces, otherwise extremely fine time-

stepping would be needed and that could significantly increase run time.

Force and Moment Evaluation

In the FloSim program the potential flow force is

modified due to the computed boundary layer displacement

effect acting on the local boundary conditions, for example,

the differential in boundary layer displacement source term

between the windward and lee sides of a sail membrane

leaves a residual normal velocity term that effectively

reduces the sail camber. In addition, the summation for total force and moment includes the tangential force on

each panel due to the local boundary layer skin friction.

The details are as follows:

The force acting at a point on a surface has two

components; one due to pressure and the other skin friction.

The elemental pressure force, δFP , acting on an element of

surface, dS, with outward local normal vector, n, is:

δFP = -P n dS ………………………….(3)

where:

P = q CP is the local pressure; q = ½ ρ VR

2 is the reference dynamic pressure,

ρ being the air mass density and

VR the reference velocity magnitude

CP = 1- (V/VR)2 - H(z); H is the normalized

total head adjustment in the wind

gradient to maintain a uniform static

CP distribution with height.

The elemental friction force δFF , acting on an

element of surface, dS, is:

δFF = q CF V |V | dS / VR

2 ……………….(4)

where:

CF is the local skin friction coefficient value

V is the flow velocity vector relative to the surface

point.

Source Model

Source Sink

γ = constant

Vortex Sheet Model

Bubble Boundary

86

The elemental force and moment at the point are then:

δF = δFP + δFF …..………….….…..(5)

δM = R ᴧ δF …..………….…………..(6)

Where R is the local position vector.

The integration of Equations 5 and 6 over all the

surfaces in the configuration gives the total aerodynamic force and moment. For the numerical model, the integral is

discretized into a summation of panel contributions based

on uniform pressure and skin friction over each panel.

For a sail the integral needs to cover both sides of the

membrane, however, it is more efficient to consider each

panel as having an “upper” and a “lower” side and to

integrate the combination, i.e., the difference in the case of

pressure but the sum in the case of the skin friction term.

There is another aspect of the sail model that needs a

modified approach for the pressure force; when a sail has a

free luff (i.e., no mast), the pressure force summation described above has an error due to a mathematical

singularity in the vortex lattice at the sharp leading edge.

This error can be illustrated in the case of a flat plate at

small incidence, α, in 2-D conditions; the potential flow

analytical drag force is zero but the pressure integration can

only produces a normal force, Cn, say, hence the numerical

drag component would be Cn sin(α) rather than zero. An

artificial “leading edge suction force” is generally

introduced to overcome this, based on the leading edge

“singularity”. In real conditions, the singularity collapses

because the flow separates forming a leading edge bubble. Whereas an analytical value for the leading edge suction

force can be obtained for the case of a flat plate under 2-D

conditions, the situation is less clear for the general 3-D

case involving camber, interference effects from other sails

and vortex wakes, and especially when including a

treatment for the separation bubble.

An alternative way of evaluating the pressure force

contribution was used in the earlier quadrilateral vortex

lattice model (Maskew (1970) and is the preferred method

for sail surfaces in the present program. It is derived as

follows: First, substitute for P in Equation 3 and form the

pressure difference across the membrane, using:

ΔP = ρ (VW2 – VL

2)/2

Where subscripts W and L are for the

windward and leeward sides, respectively.

Then for a sail, with n directed to the windward side:

δFP = ρ (VW2 – VL

2)/2 n dS …………. (7)

Using the triple vector product expansion on the nV2 terms,

we can convert Equation 7 to the vector product of the

local mean velocity, Vm, and the surface vorticity vector, γ:

δFP = ρ Vm ᴧ γ dS ………………………(8)

Where: Vm = (VW + VL)/2 and γ = (VW - VL)

Equation 8 is the general form of the Kutta -

Joukowski theorem and when applied to the 2-D cambered

plate case it correctly produces the potential flow zero drag

(see Figure 2 discussion). In general the mean velocity

comprises onset flow, surface and wake contributions,

hence the force includes the induced drag. For the present

method, the sail pressure force and moment integration is

discretized into a summation of the panel edge vortex

segment contributions.

STRUCTURAL ANALYSIS

Sail Membrane Structural Model

Although rigorous formulations exist for the non-

linear analysis of an elastic sail membrane using Finite

Elements, (e.g. Coiro, et al, (2002), Jackson and Christie,

(1987)), the simple treatment used here is an extension of

that used in Maskew (1993) as this proved to be an

effective and “robust” approach in earlier work. The model uses a network of “cords”. These are straight lines joining

the vertices, or structures node points, in a rectangular grid.

Figure 5 shows the cords in one panel interval. The model

is essentially a 3-D extension of the 2-D approach by de

Matteis and de Socio (1986). This is a physically intuitive

representation of a woven fabric, the cords representing

bundles of fibers in the weft and warp directions. In the

current model diagonal cords have been added to support

shear (or bias) loads and this provides even better

representation for anistropic materials such as those with

modern fiber “strings” between Mylar membranes. Cords

along free edges can be “beefed up” where needed to represent reinforcing ropes.

Figure 5 – Cords on a Structures Panel

The cords are simply-connected at the grid vertices

and can carry tension only. Each cord represents a strip of

the material, or more correctly, the bundle of fibers or

“strings” whose orientation most closely matches the

direction of the cord across the interval. Essentially, the

internal loads in the material fibers are transferred into the

cords according to the relative orientation. Thus for a

v

u (Chordwise)

Structures Grid Vertices

87

woven material, depending on the cut, the u- v- and

diagonal cords represent the weft (or fill) and warp fibers,

and bias strength. For Mylar sheets, the u-v- and diagonal

cords represent the emd, etd and eshear properties.

The sail structural model uses a grid of equally spaced

rows (u-wise across the sail chord) and columns (v-wise up the mast). The structures grid is now constructed on the sail

surface independently of the aero panel grid after initial

investigations demonstrated that a common grid for the

aerodynamic and structural analyses involved two many

compromises; this was especially so when batten treatment

was added. The program generates the structures grid

automatically in a number of regions vertically; the region

edges coincide with battens (if these are present) to ensure

the batten lines coincide with certain grid lines.

COUPLED PROGRAM

Figure 6 shows the flowchart for the coupled program. The

overall flow is an anticlockwise time- stepping loop with

the basic Boundary Element Program on the left. On

returning for another time step the streamline/boundary

layer calculations evaluate the boundary layer displacement

source term, VBL, and the skin friction coefficient, CF; these

are passed over to the Influence Coefficient Matrix

formation and Force and Moment Integration routines,

respectively, for the next time step. The Wake routine

convects the vortex wake points with the local flow to form

the new wake geometry for the next step and the Sail Flying Shape calculation is performed using the latest

pressure jump and friction distributions, ΔCP, CF,

respectively. The new geometry is then assembled in the

GFF at the start of the next time step.

Figure 6–Flow Chart for the Aero-Structures Program

Because separate structures and aero grids are used,

data transfer between them requires interpolation; for the

ΔCP and CF transfer, each structures vertex point is

projected normally onto the aero grid and a distance2

weighted scheme is used to obtain the local value for the

grid point from the values at the set of aeropoints neighboring the projected location. When the new

geometry has been obtained (see below), the new aero grid

is constructed by interpolation in the structures grid.

The non-linear aero-elastic problem for the sail

membrane is solved iteratively. The given geometry is

assumed to be the mould shape at zero strain and a small

normal displacement is applied to establish an initial strain

so that the equilibrium calculations can proceed.

The local aerodynamic force (pressure plus friction) is

evaluated on the area represented by each structures grid point. This must be balanced by tension forces in its

attached cords. Figure 7 shows a cut through the surface

and the area of the membrane associated with a grid point.

Figure 7– Force Equilibrium at a Grid Point

The aerodynamic force at the point depends on the

normal projection of the local area. Thus, referring to Figure 7, for normal and tangential equilibrium (per unit

length normal to the cut):

T1 = R (P + F Tanθ) ---------------------- (9)

T2 = R (P - F Tanθ)

Where P = q ΔCP and F = q CF V2/VR

2 for flow in this

plane; R is the local radius of curvature.

The tension load in each cord is:

T =Ӕ(1 - PR ε)ε ------------------- (10)

Where; Ӕ is the strength of the cord, being the accumulated

strength of the bundle of fibers the cord represents.

PR is the Poisson ratio for the fiber material; this is

normally between 0.3 and 0.4.

ε is the tensile strain, δl/l, where δl is the current

absolute extension and l is the original cord length in

the mould geometry.

CF

VBL

S/Line, B/L calc.

Initial Conditions &

Geometry Definition

Assemble Model in GFF

Form I.C. Matrix; Solve for μ

Analysis for Surface Cp;

Integrate for F&M

Sail Flying Shape

Wake Convection

Converged ?

Add Results to DataBase

OutPut File; FloViz File

New

del

ta,

twis

t, V

B,

VT

, γ

, h

eel

New Geometry

Yes

No

ΔCp

Time Loop

CF

OutPut

More Cases ?

Yes

No

Stop

T2

T1 R θ

2θ 2F Rsinθ

Rsinθ θ

Cord Model

FabricTension

Area represented by the Grid Point

Local Radius

of Curvature

Pressure, P,

& Friction, F,

per unit area

2P Rsinθ

88

The Ӕ strength of each fiber is the product of its

cross-sectional area, A and its modulus of elasticity, E.

Where a fiber crosses each grid interval its strength is

transferred to the nearest cords in accordance with their

orientation relative to the fiber.

On each sail, there are 3N force components to be

restored to equilibrium, where N is the number of “free”

grid points. Certain grid points are treated as “fixed”, e.g.,

those along a mast or boom. A displacement of a grid point

to strain the attached cords so that the tension loads balance

the local aerodynamic load, will also affect the loads at

neighboring grid points. Hence a set of 3N simultaneous

equilibrium equations in 3N unknown displacements must

be solved. A force/displacement coefficient matrix is

constructed by considering incremental x,y,z displacements

at each grid point relative to the current geometry. The

matrix solution provides the displacement vector for each grid point, however, since this is a non-linear problem (the

aerodynamic load will now change), damping is applied

before forming the new shape. A new aerodynamic

solution is then obtained for the next step.

When battens are present, each of these is treated as a

separate beam calculation (with end-load) based on the

displacement from its initial shape. The batten loads are

then assembled at the structures grid nodes and included in

the force equilibrium equation at those nodes.

Aero-Structures Verification Exercises

Two exercises were carried out to test the basic

numerical operation of the coupled procedure. The first

examines the sensitivity of the Cord model to grid density

under extreme conditions, while the second test examines

the convergence properties in a more regular situation.

For the first test an originally flat, high aspect ratio

rectangular membrane of unit chord, is constrained along

its long edges and a uniform pressure applied. Based on Equation 9 with no friction, the center span of the

membrane should take up a circular arc shape. Assuming

the material has extreme stretch properties, we now

increase the pressure until the arc becomes a semi-circle.

The radius is 0.5 so the strain is now π/2-1.0, i.e., 0.5708.

Let the structural strength, Ӕ=103 N with Poisson Ratio

0.3, then Equation 10 gives T=473.06 N. Equation 9, then

gives P = 946.12 Pa.

A case was run with this uniform pressure applied to

three such membranes with grid densities 5, 15, and 30 intervals, respectively, across the chord. The section cut in

Figure 8 shows a convincing convergence to the final shape

after 20 iterations. (Note: a blue semicircle is also drawn on

the figure). All three membranes were actually converged

within 12 steps with final radius values of 0.4920, 0.4990

and 0.4994 for 5, 15, and 30 intervals, respectively, the

corresponding tension values being 450.9, 471.9, and 473.0

compared with the theoretical value of 473.06 N. These

tension differences would be significantly smaller in the

case of more reasonable curvature.

Figure 8– Grid Density Study with Uniform Pressure

The second test of the coupled procedure is for a

simple sail case. This is a vertical trapezoidal sail raised 1m

above the horizontal “free surface” reflection plane. The

foot and head chord lengths are 5m and 2m, respectively

and the luff is 10m.The initial (strain-free) camber line is a NACA 6-series mean line with a = 0.8 and maximum

camber ratio of 0.125; this produces an ideal sectional Cl of

1.841 with uniform loading back to 0.8c. The aerodynamic

panel grid has 30 panels equally-spaced across the chord

and 10 panels vertically in a full cosine spacing (small at

head and foot and large in the middle). The structural grid

has 10 equally-spaced panels in both the chordwise and

vertical directions. The results below are for a structural

strength of Ӕ =105 N in the u-wise cords; since these cords

represent 1m width of material, this strength requires fibers

providing 1000 N per cm. The v-wise cords have Ӕ 104 N.

Figure 9 shows a general view of the configuration at

the end of a run; the contours represent ΔCP across the

membrane. The apparent-wind “flag” on the right indicates

the beta angle of 25 deg and the sheeting angle on the sail

foot is 13.5 giving a local angle of attack of 11.5 degrees at

the chordline. The true course is essentially along the –ve

X axis, so negative Fx force will be in the thrust direction.

Figure 9–Test 2; General View showing ΔCP contours

P=946.12 Pa

Final Shapes;

Grid densities: 5, 15, 30

Initial Shape

89

Figure 10a shows a section cut through the pressure

distribution at 50% height before the structural relaxation

starts. Since the 11.5 degrees angle of attack is slightly

higher than the “ideal” angle of attack for this section, a

small lee side suction at the luff is superimposed on the

otherwise uniform distribution for this camberline. In the adverse pressure gradient in the lee-side recovery region

from .8c to the leech, the boundary layer analysis is just

showing a small separation near the leech (not shown here)

and this would expand if we tried to go higher.

Figure 10– Test 2; Section cut through chordwise pressure

distribution at 50% height before and after relaxation.

Figure 10(b) shows the section cut through the pressure distribution after the flying shape iteration has

converged. Compared with the initial conditions, the

pressure loading has increased in the mid-chord region with

a forward shift in center of pressure. The maximum camber

has increased and moved forward but the curvature of the

camberline has decreased in the forward and aft regions

and the forward stagnation is now captured closer to the

luff since the small suction peak has been reduced. The lee-

side adverse pressure gradient towards the leech has been

reduced and has alleviated the boundary layer separation.

The convergence of the aero/structural iteration is extremely well behaved; Figure 11 shows the history of Fx

(thrust) and Fy (side force) in Newtons. The structural

relaxation starts at step 40; although 40 steps of aero

calculation were clearly not needed here, the forces being

converged in the first 10 steps, such lengths are needed in a

multi-sail case to ensure the wake development from the

forward sail has gone beyond the aft sail’s leech. Also, the

aero calculation is relatively quick. After the flying shape

has settled down the thrust has increased by about 3% but the side force has gone up by about 10%.

Figure 11–Test 2; History of Thrust and Side-Force

Figure 12 shows the time history for the deflection,

ΔY and ΔCP for a point near mid chord. These histories are very well behaved, being essentially converged in less than

10 steps after the structural relaxation started.

Figure 12–Test 2; History of Deflection, ΔY

and pressure differential, ΔCP, for a typical point

The time histories of the u-wise and v-wise strain

components for the same point as for Figure 12 are shown

in Figure 13. The relatively high strains after the first step

indicate that the applied initial displacement may have been

too low, the procedure then reacted with an overshoot,

however, recovery from that was very rapid and well

behaved, and again, the histories are essentially converged

before 10 steps. The strains in the two directions are

comparable at this mid-chord position, however, the v-wise

Fy

Fx

N

ΔYm

ΔCP

ΔY

ΔCP

(b) Final Step

(a) Before Relaxation

Cp windward and leeward

Section shape

Cp

90

cord strengths are an order of magnitude smaller than the u-

wise values, so the tensions in the vertical direction are

significantly smaller than in the u-wise direction.

Overall, the convergence characteristics appear very

well behaved, and although not as extreme as the Test 1

case, the extensions here are still larger than would be the case in practice, so these results demonstrate a good “safety

margin” for more regular applications.

Figure 13–Test 2: History of U-wise and V-Wise Strain for a typical point

Aero-Structures Validation Case

For the purpose of validation a test apparatus was

constructed comprising a rigid rectangular base panel, 1.2m

by 1.5 m, to which a sample of sail material was attached

and sealed along the edges. Figure 14 shows the FloSim

model together with a display of the 240 fibers in the

membrane. These are 1600 denier Pen fibers sandwiched

between two 1mil Mylar membranes. The fibers tend to be concentrated to the right of center on the membrane and

this is reflected in the distribution of assembled cord

strength, shown in the color contours.

Figure 14–Structures Validation Test Membrane

A pressure of 42 mm of water was applied in the

cavity and the membrane displacement was measured using

a laser scan with 0.01mm precision. The laser data was

obtained in a file with 100 scan lines each with 102 points.

This data was entered as a “dummy” surface patch in the

FloSim run to facilitate section comparisons in the FloViz graphics display.

Unfortunately, the “rigid” base panel also deflected

under the pressure, so the measured edge deflections were

applied in FloSim using its “bent mast” option for edge

points that are declared “rigid” initially. Calculations were

carried out with structures grids ranging from 12x10 to

40x32 but no significant difference was observed in the

computed membrane shape. This was consistent with the

earlier test shown in Figure 8. Shape convergence took

less than15 steps. Figure 15 shows a general view of the

deformed geometry and the membrane peak deflection is to the left of center, consistent with the fiber strength

distribution shown in Figure 14. The initially rigid base

geometry is also shown (blue) and so its edge distortion

(applied on the membrane edge) is shown in the figure.

Figure 15–Computed Deflection of the Membrane

Section cuts through the calculated membrane

geometry at 25%, 50 % and 75% across the panel are

compared with the measured data in Figure 16. The vertical

coordinates have been doubled to visualize the small differences in shape. The largest error is under 0.2% of

chord and occurs on the right for the upper section.

Figure 16–Comparison of Measured and Calculated Shapes

StrainV

StrainU

0

z/h:

0.75

0.50

0.25

Note: Section vertical offset has been doubled

Calculated shape

Pressure: 42 mm H2O

Base Panel

edge deflection

91

VPP TREATMENT

Background

The purpose of steady Velocity Prediction

Programs (VPP’s) for sailing yachts is to calculate the equilibrium state of the vessel when forces due to the sails

are balanced with hydrostatic and hydrodynamic forces.

Most of these programs can be traced back to the Irving

Pratt Project (Kerwin 1976) completed by MIT and the

Delft University in the mid nineteen seventies. This project

used parametric model tests to create semi-empirical

predictors of hydrodynamic forces as functions of vessel

characteristics, and full-scale performance analyses

combined with vessel-specific tank data to create semi-

empirical sail force models as functions of sail plan

characteristics. This approach has been successfully used

to handicap racing yachts and assist designers with performance optimization since commercial versions of the

programs were available.

For the steady cases, the basic equilibrium solvers

have been improved, but the most significant advances in

VPP technology have been in the areas of predicting

hydrodynamic and aerodynamic forces for specific

configurations more accurately. Since the mid nineteen

eighties, numerical methods for both hydrodynamic and

aerodynamic force predictions have evolved to the point

where potential flow panel methods, panel methods with

viscous boundary layers, and Reynolds Averaged Navier Stokes methods are routinely used to calculate forces for

input to various VPP’s. However, these computational

efforts still rely on experimental methods such as tank and

wind tunnel testing for validation, and computational

results are typically input to the VPP as tabular data that

can be interpolated to find forces for a given set of

operating conditions.

From the perspective of sail forces, competitive

pressures on sail makers and yacht designers have led to

force predictions that are much more sophisticated than the

original approach of using generic semi-empirical sail force

coefficients. Currently, the state-of-the-art is to calculate sail-plan-specific sail forces for a range of velocities and

apparent wind angles, including effects of sail trim changes

and flying rather than static shape. These tabulated forces

are then used by the VPP in conjunction with a

hydrodynamic force model (typically from a tank test or

CFD) to solve for the operating conditions where the sail

forces are optimized for a given set of hydrodynamic

characteristics and wind conditions. This level of detail has

given the sail designers capabilities to optimize sail designs

for a specific boat, and it has also given the yacht designers

capabilities to optimize boat design assuming that each candidate will have optimum sails.

For a dynamic simulation of events such as

tacking, pre-start maneuvering (Binns 2008), or operation

in waves, aerodynamic and hydrodynamic forces must be

determined at each time step including added mass and

damping effects, in addition to average steady forces. For

these cases, the current approach of using force look-up

tables based on calculations and/or experiments may not be

adequate to include all effects. This is the motivator

behind development of the DVPP as described below.

Sail Force Example

As a check on the ability of the aerodynamic

model discussed above to accurately calculate steady sail

force, a case study was run for a relatively complex sail

plan for which wind tunnel data was available. The panel

model shown in Figure 17 represents a large ketch

designed by Gerard Dykstra and Partners and Reichel-Pugh

Yacht Design, with Project Manager Jens Cornelsen

GmbH. As part of the design effort for this project, wind

tunnel tests were completed at the Wolfson Unit for Marine

Technology and Industrial Aerodynamics at the University

of Southampton. In addition, the Wolfson VPP, WinDesign, was used to do performance analyses. A series

of FloSim runs was completed for one of the upwind sail

configurations tested to provide a comparison of predicted

forces with the wind tunnel results and the empirical sail

force model incorporated in WinDesign.

Figure 17 – Panel Model for Panamax Ketch

Figures 18 and 19 compare the lift and drag

coefficients for the complete sail plan versus apparent wind

angle. The solid lines in the figures are fits to the Wolfson

wind tunnel data that were used in the VPP, and the dashed lines represent the original empirical sail force model built

into the VPP.

Figure 18 – Comparison of Sail Lift Coefficients

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 10 20 30 40 50 60

AWA (deg)

Cl

Wind Tunnel Fit

VPP Empirical

Wind Tunnel Raw

FloSim Raw

92

The relatively large differences between these two

predictions were expected for this configuration, and are

one of the reasons that the wind tunnel tests were

completed. The fit to the wind tunnel data represents a

smoothed fit to those test results that exhibited the

maximum driving force for each apparent wind angle. These data points are shown in the figures as triangles.

Figure 19 – Comparison of Sail Drag Coefficients

The calculations were completed in a similar

manner in that sail angle of attack, camber and twist were

varied systematically to find the trim where the drive force

was maximized. The data points at maximum driving force

are shown as diamonds in the figures. Since flying shape

data was not available from the wind tunnel tests, the

calculations were completed with rigid sails. The specific

test series for comparison was selected from those tested in the wind tunnel because it included both upwind apparent

wind angles and reaching apparent wind angles where

separation is likely on portions of the sails. As shown in

the figures, the calculations slightly over-predict lift and

under-predict drag as compared to the wind tunnel tests.

Calculated heeling moments were somewhat higher than

those measured in the wind tunnel, and this is believed to

be due to rigid modeling of the sails in the calculations,

instead of letting the sails deform as the sheets were eased.

Figure 20 shows the computational model with wakes for

an apparent wind angle of 20 degrees.

.

Figure 20 – Heeling Force Distribution for Ketch

IMPLEMENTING THE DVPP

For the planned Dynamic VPP the Aeroelastic

module described above will be combined with the existing

hydrodynamic module in FloSim. The flow chart for the

overall procedure is shown in Figure 21.

Figure 21–Flow Chart for Aero/Struct/Hydro Program

Compared with the aeroelastic procedure flow chart

shown earlier in Figure 6, the basic Boundary Element

routines on the left of the new chart have an extra step to

regrid the hull and free surface to the changing waterline.

Because the yacht is now moving in up to six degrees of

freedom and because the free surface is now deforming

under wave action (see below), we need to recompute the intersection line between the hull and free surfaces at each

step. Both surfaces are then regridded to the new waterline.

On the right side of the chart there are two new steps.

At the top we have the 6DOF solver that predicts the

motion of the yacht in response to waves, control

deflections, sail trim, gusts, etc; this is driven by the total

force and moment computed for the configuration in the

previous analysis step. The moment of inertia properties of

the configuration are now required input for a case. The

routine integrates the equations of motion over the time

step interval and provides the new Position, Orientation and Velocity (translational and rotational) of the yacht.

These are applied at the start of the next step when

assembling the yacht in the GFF for the updated panel

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60

AWA (deg)

Cd

Wind Tunnel Fit

VPP Empirical

Wind Tunnel Raw

FloSim Raw

Treat the Free

Surface Waves

F&M

No

Sail Flying Shape

OutPut

Yes

Regrid Hull & FS to

Current Waterline

Yes More Cases ?

No

Stop

CF

VBL

S/Line, B/L calc.

Initial Conditions &

Geometry Definition

Assemble Model in GFF

Form I.C. Matrix;

Solve for μ and σ

Analysis for Surface Cp;

Integrate for F&M

6 DOF Motion

Solver

Wake Convection

Converged ?

Add Results to DataBase

OutPut File; FloViz File

New

del

ta,

twis

t, V

B,

VT

, γ

, h

eel

New Geometry

& POV

ΔCp

Time Loop

CF

93

model. The unsteady translational and rotational velocities

require additional terms in the expression for the surface

source in Equation 1:

ζ = (VN + VBL – n•VA + n•VF + n•Ω ᴧ R)/4π ….(11)

Where:

VF is the instantaneous translational velocity of the

yacht Frame of reference, and

Ω is its instantaneous rotational velocity;

R is the offset of the local panel center from the

yacht reference frame origin.

Also, the pressure coefficient term in Equation 3 for

solid boundaries must be reformed for a point moving with

velocity VS relative to the Ground-Fixed-Frame, and must

now include ∂φ/∂t, the unsteady pressure term;

in the air: CP = (VS

2 - V

2)/VR

2 + 2∂φ/∂t - H(z);

and in the water:

CP = (VS2 - V2)/VR

2 – Z/Fr2 + 2∂φ/∂t

Where:

V is the fluid velocity relative to the point;

VR is the reference velocity magnitude

Fr is the Froude Number and Z is the height above the

mean free surface; t is time in seconds

Note that the onset flow in the air and water regions are treated separately--- we define the true wind above the free

surface and a water current below.

The other additional step on the right side of the flow

chart is the free surface treatment for wave development.

This is already operational in the hydrodynamic version of

FloSim. Basically, the configuration, comprising the hull,

keel, bulb, etc., is placed in an initially flat free surface and

the flow characteristics, wave deformation and

hydrodynamic load and moment are calculated in a non-linear , time-stepping treatment that starts impulsively from

rest and proceeds to steady conditions. The “transient

wave” problem that is generally associated with an

impulsive start is alleviated using a “sieve” treatment in

which the hull disturbance is applied in a very gradual

manner using a “porosity” function. The doublet

distribution on the free surface is obtained from Bernoulli’s

equation; this is applied on the (wavy) free surface:

∂φ/∂t = (P - PREF)/ρW + V2/2 + gZ ----------(12)

Where: P is the local static pressure, and PREF is the

reference ambient atmospheric pressure;

ρW is the mass density of water; V is the water

velocity; the convention here is that V is the

negative gradient of phi, i.e., V = - φ ; g is the gravitational constant.

Following a particle in the free surface, the total derivative

of the velocity potential, φ is;

dφ/dt = ∂φ/∂t - V2 = (P - PREF)/ρW - V2/2 + gZ

from Equation 12; this is integrated over the next time step

using 2nd order Adams Bashforth scheme to get φ and hence doublet:

μ = φ/4 π …………………………….. (13)

Note that the local static pressure, P, in Equation 12,

will now be obtained from the “air” solution on the wavy

surface in the presence of the sails. (Previously it was

declared constant and equal to PREF). Before transferring P

to the underside of the free surface, a surface-tension term

can be applied, based on the local surface curvature.

The Dirichlet boundary condition of zero perturbation

potential is applied on the free surface. Here the doublet

value is known (Equation 13) and the integral equation is

formed in terms of the unknown source distribution (i.e.,

the local normal component of the perturbation velocity).

When the source values are obtained, the perturbation

velocity, V, can be evaluated at each point. The kinematic

boundary condition on the free-surface point is then

satisfied by moving the points with the local flow:

dr/dt = V

Again, this is integrated over the time step using 2nd

order Adams Bashforth scheme to get the new free surface geometry.

A wave-generator can be included in this formulation

by introducing a cyclical pressure function at the upstream

“edge” of the free surface model. The waves will then

propagate downstream.

In the sail flying shape treatment on the right side of

the flow chart, an extra term is required in the structural

equilibrium equation; the changing motion of the yacht

results in an acceleration at each grid point and so an

inertia force must be included:

FI = ρm(Ȑ – gk)ΔA

Where:

ρm is the sail membrane mass density per unit area;

Ȑ is the local acceleration of the point due to yacht

motion; (acceleration provided by the 6DOF solver)

k is the vertical unit vector for the GFF;

ΔA is the sail surface area represented by the grid point

HydroDynamic Case Example

Results are presented below for a hydrodynamic run

on a generic 1/5th scale model yacht comprising a hull, fin,

bulb and rudder. This covers five speed cases in one run

from 6ft/sec to 8 ft/sec in 0.5 ft/sec increments. Each case

has a transition period between speeds followed by a

“settling down” period for convergence at constant speed.

94

Figure 22 shows the calculated dynamic pressure

coefficient contours at the end of the run; the flow

streamlines are shown on the starboard half. The time

histories of calculated resistance coefficient, Cfx, and trim

moment coefficient, Cmy, are shown in Figure 23; a speed

schedule of sail moment contribution was included in the Sink and Trim equilibrium calculation.

Figure 22 – Calculated Dynamic Pressure Coefficient

Contours for a Generic Yacht at Fr 0.403

Figure 23–Time History of Cfx, Cmy for a Generic Yacht

with 5 Speeds in a Run

Figure 24 shows the close comparison of the

calculated total resistance of the generic yacht with that

measured in a tank test. The skin friction contribution is

provided automatically from the boundary layer analysis

and is included in the figure. Finally, Figure 25, shows the

calculated wave contours at the first speed, Fr =0.302.

Computer Requirements

FloSim and its graphics program, FloViz, have been written specifically to run on a PC under MicroSoft

Windows XP. A 3Ghz or faster chip with 2GB memory is

recommended; 2 GB will treat about 16,000 panels.

Typically, a complete yacht hydrodynamic configuration at

moderate to high speed (above Fr 0.3) needs about 12000

panels in the asymmetric case with heel and yaw. These

numbers include the panels for the free surface. The ketch

aero model discussed in the VPP Section above, used about 4000 panels, hence the 2GB memory capacity should

cover the combined capability.

Figure 24 – Generic Yacht Calculated Resistance versus

Tank Data

Figure 25 – Calculated Wave Contours for Generic Yacht

CONCLUSIONS

A time-stepping aero-elastic method has been

presented for predicting the flying shape and performance

of upwind sails. The procedure combines an advanced

Boundary Element Method, including coupled boundary

layer and vortex wake convection calculations, with a

simple structural treatment of a sail membrane. The results

presented here demonstrate excellent convergence

characteristics with robust behavior and good accuracy.

The outline of a planned dynamic VPP, or DVPP, has

been discussed in which the current aero-elastic analysis

will be coupled with an existing non-linear hydrodynamic

program to provide simultaneous aero/hydro solutions in a

time-stepping procedure. A 6DOF motion solver and

wavemaker will be added incrementally. The ultimate aim

is to provide a capability to analyze transient events such as

wave encounter, gust encounter and maneuver response to

rudder and sail trim changes.

Calculated Resistance compared with Tank Data

Speed ft/sec (Model Scale)

1 2 3 4 5 6 7 8 9R

esis

tan

ce

Lb

(M

od

el S

ca

le)

0

2

4

6

8

10

12

14

16

18

Tank Data

Calculated; FloSim

Calculated Friction Component

Fr= 0.302

Cfx

Cmy

Cfx

, C

my

95

ACKNOWLEDGEMENTS

Special thanks go to Alessandro Gherardi of Fiorenzi

Marine Group, Srl., Italy, who constructed and carried out

the membrane test for validation of the aero-structures

model. Alberto Fiorenzi of Fiorenzi Marine Group, Srl., kindly gave permission for the data to be used for this

paper and this is greatly appreciated.

Wind tunnel and VPP sail force data for the Panamax

Ketch were used with the permission of Jens Cornelsen

GmbH Yacht Consultant (Project Manager), Gerard

Dykstra and Partners, and Reichel-Pugh Yacht Design. It

was a privilege to work with this project team, and we

greatly appreciate the opportunity to use this unique data

set for comparison to calculations.

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