numerical analysis of an wing sail ... analysis of an wing sail aerodynamic characteristics using...
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NUMERICAL ANALYSIS OF AN WING SAIL
AERODYNAMIC CHARACTERISTICS USING
COMPUTATIONAL FLUID DYNAMICS - PART I
Fernando Joel Lopes Gamboa
IST
2010
Abstract
The present work applies Computational Fluid Dy-namics (CFD) to calculate the ow around a wingsail airfoil and a methodology is presented to de-termine the best turbulent and discretization mod-els for this particular ow, which includes laminar-turbulent transition.
This study compares the numerical results withexperimental data for the drag and lift coe-cients (CD and CL), using four dierent turbu-lence models and its variants: Spalart - Allmaras(Standard), k − ε (Standard and Low Reynoldsvariants),k − ω (SST and Gamma-Re-Theta vari-ants) and Reynolds Stress Turbulence (QuadraticStrain).
A representative ow of the wing sail is chosen,which is based on an airfoil NACA0015, with aReynolds number of 2.0 × 106. Due to the lowReynolds number and the possibility of having asignicant laminar zone in the boundary layer, thewall functions eect in each turbulent model wasinvestigated.
The numerical results were obtained with a seto grids with a reasonable size and the numericaluncertainty estimated using a methodology basedin the Grid Convergence Index.
One can conclude that the agreement betweennumerical and experimental results for the CL isachieved in all turbulence models. However the CDagreement is not achieved for any model except forthe k − ω, Gamma-Re-Theta. For this model theagreement is achieved for both aerodynamic coe-cients, making it the appropriated model for a wingsail aerodynamic analysis.
The Gamma-Re-Theta does not employ walllaws, therefore the boundary layer is numericallysolved. Consequently the mesh above the foils sur-face must be specially rened to obtain consisted
and accurate numerical results.
1 Introduction
The wings sail's rise has been a notorious fact innowadays. The remarkable BWM Oracle [13] vic-tory , rigged with a wing sail, in the America'sCup against Alinghi [7], (rigged with conventionalcloth's sails) showed the potential of this type ofsails. The wing sails have a thick airfoil, as the air-foils used in the general aviation. This type of air-foils can oer more lift than the ne airfoils whichcompose the cloth sails.The yacht industry is a very competitive one,
where the regattas itself increase the demand fornew products and innovations either for the shapeof hulls or sails.The use of the Computational Fluid Dynamics
(CFD's) increased as a direct result from this de-mand, in fact since the 32th America's Cup in 2007that CFD's are now the state of the art in yacht de-sign, used as tool for ow analysis and optimizationfor hulls and sails.The used of CFD's allows a quick prediction on
forces and pressures acting on a surface, conse-quently the need for expensive wind tunnel testsis minor and a shape's optimization can be simpli-ed.Despite of the potential oered by the CFD's,
there are some problems with them. First of allthe discretization of the continuum introduces nu-merical errors. These errors can be minimized,but for that is required an extra computationaleort, which consequentially consumes time andmoney. The CFD's also have some problems withthe numerical turbulence models, which try to solvethe Reynolds Average Navier-Stoques Equations(RANSE). Furthermore, these turbulence modelsstill require some experimental data in order to val-
1
idate the numerical results obtained.There are 4 turbulent models which are often
used by the industry: Spalart - Allmaras [11] , k -epsilon (k−ε) [5], k-omega (k−ω) [6] and ReynoldsStress Turbulence [9]. The rst three are Eddy vis-cosity models. In all the models, the most commonprocedure is to used these models using the walllaws to calculate the boundary layer characteris-tics. These semi-empiric wall laws were developedassuming that the ow is fully turbulent. Howeversome ows, specially the ows with low Reynoldsnumbers, such as the ow around a sail, there isa signicant part of the boundary layer that is inthe laminar regime. So, the numerical predictionof this kind of ows can fail, and the agreementwith the experimental data is not achieved. Thisis being one of the biggest problems in CFD's: thecorrect prediction of the boundary layer and thetransitional ow.That diculty is discussed by Rumsey [8] who
studied the lift and drag of a NACA0012 airfoilat low Reynolds numbers. To predict those forcesthe author use the Spalartas- Allmaras [11] andthe k − ω [6], in the variant Menter Shear StressTransport (SST). The achieved conclusions are thatthese models are intended for fully turbulent highReynolds number computations, and the use ofthem for transitional ows, such as low Reynoldsows, is not appropriate.Eça [3] presents a study on the numerical cal-
culation of the friction resistance coecient for aninnitely thin plate as a function of the Reynoldsnumber. Seven eddy-viscosity models have beenused, four of which are the mentioned above andused in the present study. The study compares thenumerical results with the ITTC line and anotherthree estimation lines. The results obtained leadthe author to conclude that none of the turbulencemodels selected is able to model the transition fromlaminar to turbulent ow. Despite of that, the(k − ω) model is the model whom results show abetter agreement with the expected results.Firooz [4] makes a numerical study using the
Spalartas-Allmaras and the k − ε turbulence mod-els to solve the ow around a NACA4412 with alow Reynolds number (2× 106). This paper is use-ful to the present work in two ways: First of all,this study presents the discretization domain,i.e.,the dimensions of the numerical tunnel used. Lastlyit's concluded that both the turbulence models havegood agreement with the experimental data if theboundary layer is numerical solved without walllaws, but if wall laws are applied the agreement isnot achieved. This demonstrate the negative con-sequences of using models fully turbulent in lowReynolds number's ows.
Sorensen [12] oers an alternative to the conven-tional fully turbulent models, the γRe − θ modelwhich is variant of the k − ω. This correlationbased transition model has lately shown promis-ing results, and it's used to solve the ow andpredict Lift, Drag ant the transition point in twothick airfoils. The numerical results have a out-standing agreement with the experimental data anda substantial improvement in predicting the aero-dynamic forces acting on a airfoil on a ow withlow Reynolds number. This paper and its resultsencouraged the use this correlation model in thepresent paper to compare with the others models.
The present study's aim is to determine the mostsuitable turbulence and discretization models to beused in CFD's calculations for ows around wingsails and have the best possible agreement with thereality.
In order to do that this study will compare thenumerical results with experimental data for theCD (Drag Coecient) and CL (Lift Coecient),using four dierent turbulence models and its vari-ants: Spalart - Allmaras (Standard), k − ε (Stan-dard and Low Reynolds variants),k − ω (SST andGamma-Re-Theta variants) and nally ReynoldsStress Turbulence (Quadratic Strain).
To obtain the numerical results it's used the com-mercial CFD software Star-CCM+. This softwarehas an algorithm where the user can dene if thewall laws are used or if the boundary layer is nu-merical solved. This function is apllied and for eachturbulence model the ow is solved with and with-out wall laws.
The ow to be solved is the ow around a NACA0015 with a Reynolds number or 2 × 106. It waschosen this thick airfoil due to it geometrical sim-ilarity to the airfoils currently used in wing sails.The Reynolds number of 2 × 106 corresponds tonumber of a ow around a airfoil of 4.8m chord, anaverage boom's chord in a conventional sail, and awind speed of approximately 12.7 knots.
For a bigger sample were obtained numerical re-sults for four angles of attack (AOA), below theangle of separation. For each angle of atack andturbulence model (with and without wall laws), theow is computed using several geometrically simi-lar Cartesian grids to enable a reliable estimationof the numerical uncertainty.
Finally the numerical results are compared withexperimental data and conclusions were takenabout the inuence of wall laws, turbulence and dis-cretization models, and which combination of theseparameters oers the best numerical prediction ofa low Reynolds number's ow around a wing sailwith thick airfoils along its span.
2
2 Turbulence Models
As said previously, this study will make use fourturbulence models and its variants in the numericalcalculations:
• SpalartAllmaras (Standard), one-equationmodel
• k − ε(Standard, Low Reynolds), two-equationmodel
• k − ω (SST, Gamma-Re-Theta), two-equationmodel
• Reynolds Stress Transport models
The rst three turbulence models are eddy vis-cosity models that are often used in CFD's stud-ies. The last turbulence model are also known assecond-moment closure models, and are the mostcomplex turbulence models in STAR-CCM+.
2.1 SpalartAllmaras, One-Equation Model
This model [11] only solves one transport equationthat calculates the turbulent viscosity. This is incontrast to many of the early one-equation modelsthat solve an equation for the transport of turbulentkinetic energy and required an algebraic prescrip-tion of a length scale.The original model was developed primarily for
the aerospace industry, so it's expected to computeviable numerical results for the wing sail's ow.This study will use in his calculations the Standardvariant of this model.The Spalart Allmaras [11] (SA) solves the follow-ing transport equation:
ux∂v
∂x+ uy
∂v
∂y= cb1v
+1
σx[∇× (v + v)∇v + cb2(∇v × v)]
−cw1fw1v
d(1)
Where ux and uy are the Cartesian velocity com-ponents on the xx axis and yy axes, respectively.v = (ux, uy, uz) is the velocity vector and nally vis the time-averaged velocity.The subsequent parameters are dened in this
way:
S = Ω +v
k2d2fv2; Ω =
∣∣∣∣∂uy∂x − ∂ux∂y
∣∣∣∣
fw = g
[1 + c6w3
g6 + c6w3
]1
6; fv2 = 1− χ
1 + χfv1
g = r + cw2
(r6 − r
(; r =
v
Sk2d2
χ =v
v; fv1 =
χ3
χ3 − c3v1(2)
The model constants are: k = 0.41, cb1 = 0.1355,cb2 = 0.622, cw1 = 3.2391, cw2 = 0.3, cw3 = 2,cv1 = 7.1 and nally σx = 2/3.
2.2 The k − ε, two-equation model
The k-epsilon [5] turbulence model is a two-equation model in which transport equations aresolved for the turbulent kinetic energy k, and itsdissipation rate, ε. Various forms of the K-Epsilonmodel have been in use for several decades, and ithas become the most widely used model for indus-trial applications.Since the inception of the k-epsilon model, there
have been countless attempts to improve it. Themost signicant of these improvements have beenincorporated into STAR-CCM+. In its originalform, the k-epsilon turbulence model was appliedwith wall functions, but was later modied touse the following approaches for resolving the vis-cous sublayer: Low-Reynolds number and Two-layer. The present work numerical results, obtainedwith this model, use the Low-Reynolds number ap-proach, in the case study where the viscous sublayeris numerically solved.
2.3 The k − ω, two-equation model
The k-omega [6] model is a two-equation model thatis an alternative to the k-epsilon model.The transport equations solved are for the turbu-
lent kinetic energy k, as the k-epsilon model, anda quantity called ω, which is dened as the spe-cic dissipation rate, that is, the dissipation rateper unit turbulent kinetic energy.One reported advantage of the k-omega model
over the k-epsilon model is its improved perfor-mance for boundary layers under adverse pressuregradients. Perhaps the most signicant advantage,however, is that it may be applied throughout theboundary layer, including the viscous-dominatedregion, without further modication. Furthermore,the standard k-omega model can be used in thismode without requiring the computation of walldistance.There are four model variants in the Star-CCM+:
Standard k-omega, SST (Shear Stress Turbulence)
3
k-omega, SST k-omega detached eddy model andnally the Gamma-Re-Theta [12].This study will use the variant SST k-omega with
and without wall laws and the variant Gamma-Re-Theta, which solves numerically the viscous sub-layer.For all the model variants, the two transport
equations are the same:
ux∂k
∂x+ uy
∂k
∂y= vtS
2
+∇×[(v +
vtσ k
)∇k]
−β∗ωk (3)
ux∂ω
∂x+ uy
∂ω
∂y= αS2
+∇×[(v +
vtσ ω
)∇ω]
−β∗ω2 + Fω1
ω∇k ×∇ω (4)
Despite the fact that the transportation equationsremain the same for all the k−ω variants, the eddy-viscosity term computation for the SST is calcu-lated in a dierent way. This term it's obtainedwith the following expressions:
vt =a1k
max(a1ω, F2Ω); a1 = 0.31
F2 = tanh arg22
arg22 = max
(2√k
0.09ωd2,
500v
ωd2
)(5)
With this constants: β∗ = 0.09, β1 = 0.075,σk1 = 1/0.85, σw1 = 2, α2 = 0.4404, β2 = 0.0828and nally, σk2 = 1.17.
2.4 The γRe−θ (Gamma-Re-Theta)variant of the k−ω, two-equationmodel
The Gamma-Re-Theta transition model is acorrelation-based transition model that has beenspecically formulated for unstructured CFD'scodes.The evaluation of momentum thickness Reynolds
number is avoided by relating this quantity tovorticity-based Reynolds number. In addition, acorrelation for transition onset momentum thick-ness Reynolds number dened in the free stream ispropagated into the boundary layer by a transportequation. An intermittency transport equation is
further used in such a way that the source termsattempt to mimic the behaviour of algebraic engi-neering correlations.The Gamma-Re-Theta transition model, as orig-
inally published, is incomplete, since two criticalcorrelations were claimed to be proprietary andhence omitted. One justication for such an omis-sion is that the model provides a "framework" forusers to implement their own correlations. All thenumerical results obtained with this model use theFlenght correlation. This correlation uses the fol-lowing constants: ca1 = 1, ca2 = 0.03, ce2 = 50,cθt = 0.03, and nally, σθt = 2.These constants are the same used by Sorensen
[12], this was intended, due to the good agreementbetween the numerical results and the experimentaldata in that study.
2.5 Reynolds Stress Transport mod-els
Unlike the models mentioned before, these modelsaren't eddy-viscosity models.These are the most complex models in the ow
solver Star-CCM+. Instead of solving only oneor two transport equations as the previous mod-els, the Reynold Stress Transport computes all theReynolds Stress. In order to do that, these mod-els solve seven more equations, which naturally re-quires more computational time and eort.
3 Flow Solver, Computational
Domain, Boundary Condi-
tions and Grid Sets
All the numerical computation on this work weredone with the commercial software Star-CCM+. Inthis CFD software the ow is solved numericallyusing the nite element method, more accuratelythe nite volume method.In the nite volume method, the solution domain
is subdivided into a nite number of small controlvolumes, corresponding to the cells of a computa-tional grid. Discrete versions of the integral formof the continuum transport equations are appliedto each control volume. It's obtained in this waya set of linear of algebraic equations, with the to-tal number of unknowns in each equation systemcorresponding to the number of cells in the grid.However, if the equations are non-linear, which is
the RANSE case, iterative techniques are employedin order to linearise all the equations to be solved.This set of equations is solved using an algebraicmultigrid solver.
4
This software not only is used to solve numeri-cally the ow, but also used as the mesh generator.The incorporated mesh generator has some meshcontrol parameters, which revealed to be very im-portant in getting a good ow discretization. Thechoice of these parameters is strongly inuenced bythe kind of wall treatment employed, that is, if thewall laws are used or not in the boundary layer so-lution.The parameters modied and studied in detail
were the following ones:
Base Size
This parameter sets all mesh size parameters for thesurface mesh. Each cell size in the computationaldomain is a function, in percentage, of the base size.Exception made for the NACA0015 suface cells,
where for a better dimensional control of the cells,the size was imposed with an absolute value.So if the base size value decreases the number
of cells on the grid increases and the grid is ner,which leads to a better ow discretization and con-sequently minor numerical errors.
Near Wall Prism Thickness
This parameter is one of the most important pa-rameters studied, it controls the thickness of therst layer of cells above the airfoil's surface. Be-cause of the possibility of having signicant laminarregime in the airfoil's boundary layer due the lowReynolds number, the fully turbulent models em-ploying wall laws can lead to signicant errors, sothe boundary layer probably need to be completelysolved without wall laws.In order to do that the cells near the airfoils sur-
face must have the minimum size possible. In thatway each dierent boundary layer zone has severalmesh layers, leading to a better mesh renementand discretization.This near wall prism thickness specially concerns
to the rst boundary layer zone, the viscous sub-layer. This zone has a huge inuence in the shearstress component of drag, and if the boundary layeris going to be solved without wall laws it's requiredthat the distance to the wall made dimensionless,y+ should have a value between 0 and 1 in the air-foils surface. This y+ guaranties that the boundarylayer and specially the viscous sublayer has enoughcells to be resolved in a proper way, also means thatthe rst cell layer thickness has a minor value thanthe viscous sublayer thickness.If wall laws are going to be employed in the tur-
bulence models the mesh requirements aren't so re-stricted. The cells size can be bigger and the y+
can be higher than 30.
The viscous sublayer thickness is directly re-lated to the Reynolds number and the characteris-tic length, which is the airfoil chord in this case. Inorder to estimate the minimum required thicknessof the rst layer of cells that guaranties a maxi-mum y+ with the value 1, was used an algorithmdeveloped by NASA [2].According to the output of this algorithm, with
the input of 2× 106 for the ow Reynolds numberand 1m as the airfoil chord length, the thickness ofrst layer of cells above the airfoils surface shouldhave the minimum value of 0.0123mm. Consideringthis output value all the grids used for the calcula-tions without wall laws have the Near Wall Prism
Thickness value under 0.0123mm.For the calculations employing the wall laws, was
mentioned before that the y+ can be higher than30. Regarding this fact the NASA's algorithm wasonce again used, but now having as a target y+ thevalue of 30, instead of the value 1 without wall laws.The output value was a minimum 0.0.3719mm forthe rst layer thickness. This value was used as areference value for the grids used with wall laws.
Number of Prism Layers
The Number of Prism Layers parameter controlsthe number of layers generated for each boundarysurface that is allowed to have prism layers. If thisvalue increases the mesh renement increases aswell. In the other hand the computational eortalso increases, meaning that more computationaltime and resources are required.A little increase in the number of layers automat-
ically implies a huge dierence in the number of thecells in the computational grid.Concerning about the computational eort and
the mesh renement, the maximum number of lay-ers for the grids used with wall functions is 30. Forthe grid sets used in numerical calculations withoutwall laws this number is bigger, 90. This increasein the number of layers it's explained by the needto have several layers in each dierent zone of theboundary layer. In this way each boundary layerzone is discretise independently, and the boundarylayer can be solved numerically without the directuse of wall laws.
Prism Layer Thickness
The Prism Layer Thickness controls the totaloverall thickness of all the prism layers. This prismlayer thickness ideally should have at least the valueof the boundary layer thickness, however the lastthickness is an unknown.In order to estimate the magnitude of the bound-
ary layer thickness was used the Von Kárman Equa-tion:
5
δ = 0.37L
(1
Re
) 15
(6)
This equation introduces the boundary layerthickness δ, as function of the Reynolds number(2 × 106) and as a function of the characteristiclength, which is the NACA0015 airfoil's chord of1m.
This equation is deduced for fully turbulent ows,so it can be only used as an approach to estimatethe require prism layer thickness.
Substituting the values in the equation, the esti-mated boundary layer thickness is 0.02m. Regard-ing this result the prism layer thickness is alwayssuperior to 0.02m for all grid sets.
Control Volumes
The mesh type used in this study regardless theuse of wall functions was a mixture between the Otype with the H type. The O type was obtained bythe use of the prism layer generator and its param-eters.
The H type was obtained using control volumes.These control volumes dene volumes with severalshapes where the mesh is locally rened. The meshwas rene in the next critical areas:
• The trailing and leading edge of theNACA0015 airfoil
• The surrounding NACA0015 area
• The NACA0015 wake
• A narrow band from the the inlet to theNACA0015
• Narrow bands perpendiculars to theNACA0015 chord
The resulting mesh can be observed in Figure 1.
After the mesh parameters choice, regarding foreach parameter if wall laws were going to be used ornot, was established a set of ve dierent grids withan increasing degree of renement for the numer-ical calculations using turbulent models with wallfunctions.
For the numerical results obtained without walllaws were generated seven grids, again with an in-creasing degree of renement,i.e., with several num-ber of cells. Both the grid sets are in the Table 1and Table 2.
Figure 1: Resulting mesh with the control volumes.
Table 1: Grid sets for numerical calculations with walllaws.
Char.Nr.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
1.18 32249 0.35 1 15 0.081.16 33100 0.35 1 18 0.081.13 35064 0.35 0.6 25 0.081.11 36218 0.35 0.3 30 0.0851 44828 0.3 1 22 0.08
3.1 Computational Domain
The dimensions choice for the numerical wind tun-nel, is an important key process in CFD's owanalysis. In one hand, as much larger the tunnelgets, less blockage eects exists. In the other hand,the existence of a big tunnel automatically leadsto more cells in the grid. This means more equa-tions to be solved, which will require a additionalcomputational time an eort.
In sum the numerical wind tunnel dimensions area compromise between reducing the blockage andhave a sustainable grid to be solved with the avail-able computational resources.
Wolfe [14] in a numerical study on a thick airfoiluses an quadrangular numerical wind tunnel werethe airfoil was in the center and the distance fromthe airfoil to the walls was 10C where C is the air-foils chord length.
Firooz [4] studing the ground efect with aNACA4412 airfoil used a rectangular numericalwind tunnel with the inlet at a distance of 3C fromthe airfoil, the outlet at 5C and nally the lateral
6
Table 2: Grid sets for numerical calculations withoutwall laws.
Char.Nr.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
1.99 30299 0.48 0.003 35 0.041.96 31149 0.48 0.002 40 0.041.70 41395 0.42 0.0018 50 0.081.54 50552 0.40 0.0016 65 0.081.52 51635 0.35 0.0014 75 0.081.36 64576 0.3 0.001 80 0.081 120071 0.2 0.001 90 0.08
wall (not including with the ground) was at a dis-tance of 4C from the airfoil.Considering this two studies and some prelimi-
nary tests, the numerical wind tunnel dimensionsare in the Figure 2.
Figure 2: Numerical wind tunnel dimensions used in
the present work.
For setting the angles of attack in this study, wasthe airfoil that changed his position inside the tun-nel, instead of changing the ow direction in theinlet. This technique is applied for reducing block-age eects either in numerical wind tunnels or realwind tunnels.Another tunnel characteristic to reduce the
blockage is the lateral wall distance of 5C from theairfoil. This superior distance accounts for the wakethat will develop behind the airfoil trailing edge.
3.2 Boundary Conditions
Inlet
This boundary is dene as Velocity Inlet in theStar-CCM+, is from this boundary that the uid
is emanated to the wind tunnel interior. The mostimportant parameter to be dened is the velocity,it direction and magnitude.As said before, the ow analysed will have a
Reynolds Number of 2 × 106 and the NACA0015will have 1m of chord. The correspondent velocityfor this characteristic length and Reynolds Numberis 31.7m/s.The direction of the vector velocity is normal to
the inlet.
Outlet
This boundary is dene as Pressure Outlet in theStar-CCM+. In this boundary the velocity is ex-trapolated from the uid domain, i.e., from thewind tunnel's interior using reconstruction gradi-ents.The parameters used in this boundary were the
default ones in the Star-CCM+.
Lateral Walls
The lateral walls are a required boundary thatencloses a nite uid volume, however their pres-ence can interfere with the ow and originate badresults.To minimize as much as possible the lateral walls
interference, they were dened as Slip Walls. In thisway both lateral walls will not have boundary layer,that would be a negative fact if existed, interferingwith the ow inside the wind tunnel.
4 NACA0015 Characteristics
and experimental data
The airfoil studied was the NACA0015. This airfoilis one airfoil from the four-digit NACA series. TheNACA 0015 airfoil is symmetrical, with no camberand it has a 15% thickness to chord length ratio.The choice of this airfoil has two main reasons:
• It's a thick airfoil, consequently is geometri-cally identical to the airfoils used in the wingsail.
• The NACA0015 is a well know airfoil with abig quantity of experimental data available.
The chosen chord was 1m having in considerationalso two main reasons: First of all a reduced chordwill lead to a better descritization, for the samenumber of cells the cell characteristic size is smallerin comparison with a airfoil with a bigger chord.Lastly the chord of 1m is closer to the NACA0015chord used in the experimental trials in the wind
7
tunnel and which data will be compared with thenumerical results.The experimental data was obtained from a re-
port of Sandia Laboratories [10]. In this reportthere are presented the Drag Coecients (CD) andLift Coecients (CL) for a range of angles of attackand for several Reynolds numbers, were is includedthe present study Reynolds number of 2× 106.Regarding the angles of attack and as it was said
earlier, this study will analyse four angles of at-tack in a range were the separation doesn't occur.The separation is well identied in the airfoil po-lar curve, and it occurs in the angle of attack werethere is a sudden loss of lift, this phenomena is alsoknown as stall.The experimental data suggests that the
NACA0015 stall occurs in the angle range between10o and 15o. Considering this information the nu-merical results are obtained for the following anglesof attack:3o, 5o, 7o and 10o.The experimental results for the CD and CL
which will be compared with the numerical results,are presented for each angle of attack studied inTable 3.
Table 3: NACA0015 experimental results in Re = 2 ×106
.
AOA 3o 5o 7o 10o
CD 0.0075 0.0083 0.0098 0.0133CL 0.33 0.55 0.77 1.0433
5 Numerical Data Acquired
For each angle of attack (3o, 5o, 7o and 10o), itwill be obtained the numerical result for the DragCoecient (CD) and Lift Coecient (CL), usingfour turbulence models and its variants with andwithout wall laws.The wall treatment is dened by all y+ in the
Star- CCM+ if wall laws are used, and low y+
if wall laws aren't used and the boundary layer isnumerically solved.The Table 4 and 5 resume the numerical data
acquired for each turbulent model, variant and walltreatment.
6 Uncertainty in CFD Predic-
tions
The quantication of the uncertainty of a numericalprediction is commonly known as verications of
Table 4: Numerical data obtained with wall laws
TurbulentModel
Variant Wall Treat-ment
Spalart-Allmaras
Standard all y+
K-Epsilon Standard all y+
K-Omega SST all y+
ReynoldsStress Turbu-lence
QuadraticPressureStrain
all y+
Table 5: Numerical data obtained without wall laws
TurbulentModel
Variant Wall Treat-ment
Spalart-Allmaras
Standard low y+
K-Epsilon Standard/LowReynoldsNumber
low y+
K-Omega SST low y+
K-Omega SST/Gamma-Re-Theta
low y+
ReynoldsStress Turbu-lence
QuadraticPressureStrain
low y+
calculations.
There are many procedures proposed in the exis-tent literature which are based on grid renementstudies and Richardson extrapolation (RE), like thewell known Grid Convergence Index (GCI) intro-duced by Roache [1].
Alternatives are based on various techniques ap-plied to single grid calculations. In this paper thestrategy used in determining the numerical uncer-tainty was the grid renement method based in theGCI.
This model stands that the numerical uncer-tainty U , associated to the use of certain a gridi, is dened by:
U = FS |δRE | (7)
Where FS is the safety factor, assumed as 1.25and δRE is the error estimation obtained by ex-trapolation. The error is assumed as being only thediscretization error, meaning that it's assumed thatthe round-of errors and iterative errors are negligi-ble. The error estimation is given Eq.8.
δRE = φi − φ0 (8)
8
Where φi is the numerical solution of any localor integral scalar quantity on a given grid i, in thispaper is the CD or CL. φ0 is the estimated exactsolution which is unknown.To determine the φ0 was used the following
methodology: Firstly each grids i was related withthe most rened grid using the cell number. Thisrelation between grids is called grid characteristicnumber ri, and it's dened by Eq.9.
ri =
√h1hi
(9)
Where h1 is the number of cells in the grid mostrened and hi is the number of cells in the grid i.After the calculus of the ri for each grid i, was
made an linear interpolation between the severalnumerical results obtained with the dierent gridsi and the correspondent grid characteristic number.This interpolation origins an straight line which lin-ear equation is y = mri + b.The parameter b, known as y-intercept corre-
sponds to the estimated numerical result if theri = 0. The ri is equal to zero when grid i hasin innity number of cells, limhi→∞
√h1/hi = 0,
therefore the discretization errors are also equal tozero. Consequently the parameter b, is the esti-mated exact solution φ0, that is the estimated so-lution without numerical errors.With the knowledge of the φ0 value , the nu-
merical uncertain for each grid can be calculatedand it can be dene the range of values where thenumerical solution can variate, i.e., the uncertaintyinterval. The uncertainty interval is given by Eq.10.
grid i numerical solution ∈ [φi − U, φi + U ]](10)
The comparison between the numerical resultsand between the experimental data is done in twodierent ways. Firstly is veried if for each gridthe experimental result was inside the range of thenumerical uncertainty dened in Eq.10. Finally iscalculated the relative error between the estimatedexact solution φ0. and the experimental result. Therelative error equation can be found in Eq.11.
relative error =|φ0 − φexp|
φexp(11)
Where φexp is the quantity experimental result,in this paper can be either CD or CL.Regarding the linear interpolation used to obtain
the φ0, it's needed at least 3 numerical results form3 dierent grids. Were obtained 5 dierent numer-ical results, for each AOA, using wall laws and 7numerical results without wall laws . However in
the numerical analyse were only used the 3 resultswith better agreement between then, in order toreduce the numerical noise eect.
7 Numerical Results
Despite the fact that it were obtained the numeri-cal results for all the analyse angles of attack, thepresented results in this section are only regardingthe angle of attack of 5o, the conclusions obtainedwith these results are the same for the other angles,so present them is redundant.
7.1 Numerical Results with walllaws for the AOA of 5o
Figure 3: Numerical Results for the CD using turbulent
models employing wall laws.
9
Figure 4: Numerical Results for the CL using turbulent
models employing wall laws.
The graphics showed in the Figure 3 and 4 havethe same layout: In the xx axes are representedthe dierent grid characteristics numbers. In theyy axes are represented the numerical values eitherfor CD or CL.
The vertical bar in each numerical result rep-resents the uncertainty interval, calculated withEq.10.
For each set of results using the same turbuletmodel, can be seen a straight line and it linear equa-tion. This straight line is the result of the linear in-terpolation between the 3 chosen numerical resultsand, as said before, the y-intercept in each linearequation is the the estimated exact solution φ0.
Finally the thick horizontal line in each graphicand identied as Exp(Sandia) represents the re-spectively experimental result in the Sandia Re-port [10].
In this section all the gures have the same lay-out.
7.1.1 Numerical Results with wall laws Dis-
cussion
In this subsection the numerical results for the CDand CL for each turbulent model employing walllaws will be discussed.
Spalart-Allmaras, Standard - all y+
The numerical results using the Spalart - All-maras model with wall laws show a big uncertaintyinterval in both aerodynamic coecients. This isdue to the big dierence of values between each nu-merical result obtained for each grid i and the esti-mated exact solution φ0. Consequently in order tohave grid independence the cell number in the gridshould be substantially increased. Another conse-quence of the big uncertainty interval is that all theexperimental are inside the interval.Despite that, the agreement between the nu-
merical results and the experimental data is notachieved for CD. This is can be easily seen consid-ering that the maximum relative error between theestimated exact solution and the experimental CDis 90% at the AOA of 7o.Regarding the CL the agreement is improved, as
example the relative error between the φ0 for theCL and the experimental result at the AOA of 3o isabout 10% and at the AOA of 3o the error is only7%.It's concluded that this model fails in the CD pre-
diction, and the agreement for the CL is satisfac-tory. However the big uncertainty interval showsthat this model requires a highly rened grid inorder to have grid independence in the numericalresults.
K-Epsilon, Standard - all y+
The numerical results using the k-epsilon modelwith wall laws show a big uncertainty intervals forboth aerodynamic coecients. As a consequencepractically all the experimental results are insidethe intervals.Comparing directly the estimated exact solutions
and the experimental results for the CL, the agree-ment between these two values is notorious in allAOA analysed, where the maximum relative erroris around 9.4% for the angle of 3o and the minimumonly 2.8% at 10o.The opposite happens to the CD results. The
agreement is not achieved: Not only some experi-mental results aren't inside the uncertainty intervalas also the direct comparison between φ0 and theexperimental result shows that the maximum rel-ative error is 30.8% for the angle of 10o being theothers errors the following ones: 18.4% at 7o, 1.2%at 5o and nally 2.7% at 3o.
10
Considering these relative errors, it's concludedthat this model fails in the CD prediction, and theagreement for the CL is satisfactory. However thebig uncertainty interval shows that this model re-quires a highly rened grid in order to have gridindependence in the numerical results.
K-Omega, Menter Shear Stress Transport -
all y+
As the previously models analysed so far, the k-omega numerical results show a big uncertainty in-tervals for both aerodynamic coecients. Almostevery experimental results are inside these uncer-tainty intervals, exception made for two CL resultsat 5o.Comparing now the estimated exact solutions
and the experimental results for the CL, the agree-ment between these two values is quite notorious inall AOA analysed.The maximum relative error isaround 16.4% for the angle of 3o and the minimumvalue is only 4% at 5o.Regarding the CD results there is an substan-
tial improvement of the agreement between the thenumerical results and the experimental data: Themaximum relative error is only 24% at 3o and theminimum error is around 14.3% at 10o.This improvement in the Drag Coecient makes
this models the most suitable model to analyse theow with Re = 2 × 106 around a thick airfoil sofar. As the previous models the numerical resultsfor the CL show an regular agreement with the ex-perimental data.
Reynolds Stress Turbulence, Quadratic
Pressure Strain - all y+
As the previously models analysed, the ReynoldsStress Turbulence results show a big uncertaintyinterval in both aerodynamic coecients, meaningthat a more rene grid was needed in order to guar-antee the results grid independence.This model requires a higher iteration time, so
it's not suitable to use if there are time limitations.All experimental results are inside the uncer-
tainty intervals.Comparing now the estimated exact solutions
and the experimental results for the CL, the agree-ment between them is the same seen in the previ-ous models .The maximum relative error is around16.5% for the angle of 7o and the minimum valueis only 3.2% at 10o.For the CD the agreement is not achieved, as
can be seen by the relative errors obtained: Themaximum relative error is around 37.5% at 7o andthe minimum error is around 13.5% at 10o.
Considering these relative errors, it's concludedthat this model fails in the CD prediction, and theagreement for the CL is satisfactory. However thetime factor should also be a concern, due to theextra time that this model requires.
7.2 Numerical Results without walllaws for the AOA of 5o
Figure 5: Numerical Results for the CD using turbulent
models without wall laws
7.2.1 Numerical Results without wall laws
Discussion
In this subsection the numerical results for the CDand CL for each turbulent model without wall lawswill be discussed. The Figure 5 and 6 show thenumerical results obtained for the AOA of 5o forboth aerodynamic coecients.
Spalart-Allmaras, Standard - low y+
Contrary to what happened in the Spalart - All-maras model with wall laws, the numerical results
11
Figure 6: Numerical Results for the CL using turbulent
models without wall laws
without wall laws show a small uncertainty intervalsfor both aerodynamic coecients. This is due tothe similar values of each numerical result obtainedfor each grid i and the estimated exact solution φ0.The direct consequence of this fact is that the nu-merical results aren't so dependent in the grid cellnumber. This means that the convergence to a nalnumerical result happens faster.
Due to the smaller uncertainty intervals, practi-cally all the experimental results are outside of theintervals in both aerodynamic coecients studied.Despite of that fact, the numerical results are quitesimilar to the experimental data.
This is can be easily seen considering that themaximum relative error between the estimated ex-act solution and the experimental CL is 6.2% in theAOA of 10o.
Regarding the CD, the agreement is improvedcomparing with the results with wall laws: as ex-ample the maximum relative error between the φ0and the experimental results, occurs at the AOAof 7o and it's about 59.2% and the minimum erroroccours at the AOA of 3o and is 41.3%.
It's concluded that this model fails in the CDprediction, and the agreement for the CL is satis-factory.
K-Epsilon, Standard Low Reynolds - low y+
The numerical results using the k-epsilon modelwithout wall laws show again a small uncertaintyinterval in both aerodynamic coecients. As a con-sequence practically all the experimental results areoutside the intervals.Comparing directly the estimated exact solutions
and the experimental results for the CL, the agree-ment between these two values is notorious in allAOA analysed, where the maximum relative erroris around 8.7% for the angle of 5o and only 3% at3o. This is a slightly improve comparing to the re-sults for this model using wall laws.The opposite happens to the CD results. The
agreement is not achieved: Not only some experi-mental results aren't inside the uncertainty intervalas also the direct comparison between φ0 and theexperimental result shows that the maximum rel-ative error is 64.7% for the angle of 10o being theothers errors the following ones: 57.1% at 7o, 48.2%at 5o and nally 45.3% at 3o.Considering these relative errors, it's concluded
that this model fails in the CD prediction, and theagreement for the CL is satisfactory.
K-Omega, Menter Shear Stress Transport -
low y+
As the models analysed so far, the k-omega nu-merical results not only show a small uncertaintyintervals in both aerodynamic coecients but alsothese intervals are slightly smaller than the othermodels intervals.All experimental results are outside the uncer-
tainty intervals.Comparing now the estimated exact solutions
and the experimental results for the CL, the agree-ment between these two values is extremely satis-factory in all AOA analysed. The maximum rela-tive error is only 6.6% for the angle of 3o and theminimum value is only 3.8% at 10o.Regarding the CD results, there are irregularities
between the numerical results and the experimentaldata, as can be seen in the maximum relative errorthat is 49% at 7o and the minimum error is around42.6% at 10o.The drag coecient results make this model not
suitable to predict the aerodynamic forces in anthick airfoil in a ow with Re = 2× 106.
Reynolds Stress Turbulence, Quadratic
Pressure Strain - low y+
12
As the previously models analysed, the ReynoldsStress Turbulence results show again small uncer-tainty intervals in both aerodynamic coecients,meaning that the grid independence is close. Thismodel requires an higher iteration time, so it's notsuitable to use if there are time limitations. Allexperimental results are outside the uncertainty in-tervals.Comparing now the estimated exact solutions
and the experimental results for the CL, the agree-ment between them is the same seen in the previ-ous models .The maximum relative error is around11.4% for the angle of 10o and the minimum valueis only 3.7% at 7o.For the CD, the agreement is not achieved as
can be seen by the relative errors obtained: Themaximum relative error is around 108% at 3o andthe minimum error is around 57.1% at 7o.Considering these relative errors, it's concluded
that this model fails in the CD prediction, and theagreement for the CL is satisfactory. However thetime factor should also be a concern, due to theextra time that this model requires.
K-Omega, Menter Shear Stress Transport,
Gamma-Re-Theta - low y+
As the previously models without wall laws anal-ysed , the Gamma-Re-Theta results show a smalluncertainty intervals in both aerodynamic coe-cients. Almost every experimental results are out-side the uncertainty intervals.However there is not only an outstanding agree-
ment between the estimated exact solutions and ex-perimental results, but the agreement is also veri-able for each numerical result in each dierent gridi for both aerodynamic coecients.Comparing now the estimated exact solution and
the experimental result for the CL, the agreementbetween these two values is quite notorious in allAOA analysed. The maximum relative error isaround 6.6% for the angle of 3o and the minimumvalue is only 4% at 10o.Regarding the CD results, the agreement be-
tween numerical results and experimental resultsis by far the best one among all the models anal-ysed, with and without wall laws. The maximumrelative error is just only 18.7% at 3o and the min-imum error is around 2% at 7o. The grids and thediscretization model used have success in guaranteethat the y+ parameter was always under the valueof 1 at the airfoil surface, as required.With these results, it's concluded that the
Gamma-Re-Theta model can predict the uid be-haviour around a NACA0015 with Re = 2 × 106,and it originate numerical results with a excellent
agreement with the experimental data. Making thismodel the most suitable to be used in the aerody-namic analysis of a wing sail.
8 Conclusions
The present paper proposes and applies a method-ology to study the most suitable turbulent and dis-cretization model to perform a numerical aerody-namic analysis on a wing sail.Were applied four turbulent models and its vari-
ants: Spalart - Allmaras (Standard), k − ε (Stan-dard and Low Reynolds variant),k − ω (SST andGamma-Re-Theta variants) and nally ReynoldsStress Turbulence (Quadratic Strain).The NACA0015 with Re = 2× 106 at four AOA
(3o, 5o, 7o and 10o) was selected as a case studyfor validation. Furthermore, this airfoil and Re re-sembles a realist on a typical yacht, namely the Rewhich corresponds to ow around an airfoil with a4.8m chord(an average boom's length), in a wind-speed of approximately 12.7knts.Due to the low Reynolds number and the pos-
sibility of having a signicant laminar zone in theboundary layer, the wall functions eect was stud-ied in each turbulent model.After the numerical uncertainty estimation us-
ing a method based in the Grid Convergence Index(GCI), the numerical results of CD and CL werecompared with experimental data. From this com-parison some conclusions and observations regard-ing the turbulent models and wall treatment areobtained:
• For all turbulent models employing wall func-tions, the numerical results for the CL havea remarkable good agreement with the experi-mental data, as expected.
However, the contrary happens for the CD,since the agreement is not achieved and theprediction of this aerodynamic coecient fails.This is due to the fact that the boundarylayer is modelled as fully turbulent neglectingthe large laminar part, characteristic in lowerReynolds numbers.
Another fact of the models using wall functionsis the big uncertainty intervals, so the grid in-dependence is not guarantee.
• Regarding the models without wall functions,the agreement between numerical and experi-mental results is slightly improved.
Despite of that, the CD agreement is still notachieved for any model except for the Gamma-Re-Theta model. For this model the agree-
13
ment is achieved for both aerodynamic coe-cients making this particular model suitable tobe used in a aerodynamic analysis on a wingsail.
The turbulent models without wall functionsalso show an improvement in the uncertaintyintervals, they are smaller than the ones ob-tained with wall functions. This means thatgrid independence is achieved quickly and withless number or cells, reducing the computa-tional eort.
Regarding the discretization model some conclu-sions can be draw as well. In such low Reynoldsnumber as Re = 2×106 and with the existence of asignicant laminar regime on the airfoil boundarylayer, the wall functions should not be used and theboundary layer should be numerically solved.Therefore some grid aspects must be concerned:
• It's advisable to use the smaller near wall prismthickness possible, this not only guaranteesthat the y+ is under 1 in all the airfol surfaceas required, but also guarantees that the vis-cous sublayer is computed independently fromthe other boundary layer zones.
• The use of a large number of prism layers is alsoadvisable. This guarantees that each boundarylayer zone has several layers to solve the ow.
• At last, the overall prism layer thickness shouldhave at least the boundary layer thickness toassure a good discretization.
References
[1] Verication and Validation in Computational
Science in Engineering. Hermosa Publishers,1998.
[2] NASA Viscous Grid Spacing Calculator.http://geolab.larc.nasa.gov/apps/yplus/.1997.
[3] L. Eça and M. Hoekstra. The numerical fric-tion line. J Mar Sci Technol, 13:328345, 2008.
[4] A. Firooz and M. Gadami. Turbulence owfor naca 4412 in unbounded ow and groundeect. Int. Conference on Boundary and Inte-
rior Layers, 2006.
[5] W.P. Jones and B.E. Lander. The predictionof laminarization with a two-equation modelof turbulence. Int. J. Heat and Mass Transfer,15:301314, 1972.
[6] F.R. Menter. Two-equation eddy-viscosity tur-bulence models for engineering applications.AIAA, 32:15981605, 1994.
[7] Alinghi Defender of the 33rd America's Cup.http://www.alinghi.com. 2010.
[8] Christopher L. Rumsey and Philippe R.Spalart. Turbulence model behavior in lowreynolds number regions of aerodynamic ow-elds. AIAA, 44:982993, 2008.
[9] S. Sarkar and L. Balakrishnan. Applicationof a reynolds-stress turbulence model to thecompressible shear layer. NASA, (CR 182002),1972.
[10] Robert E. Sheldahi and Paul C. Klimas. Aero-dynamic characteristics of seven symmetricalairfoil sections trough 180-degrees for use inaerodynamic analysis of vertical axes in windtrubines. Technical report, National TechnicalInformation Service and U. S. Department ofCommerce and Sandia Laboratories, 1981.
[11] PR Spalart and S.R. Allmaras. A one-equation turbulence model for aerodynamicows. AIAA, (92-0439), 1992.
[12] Niels N. Sørensen. Cfd modelling of laminar-turbulent transition for airfoils and rotors us-ing the gamma-re-theta model. Wind Energy,12:715733, 2009.
[13] BMW ORACLE RACING TEAM.http://bmworacleracing.com. 2010.
[14] Walter P. Wolfe and Stuart S. Ochs. Ccfd cal-culations of s809 aerodynamic characteristics.AIAA, (97-0973), 1997.
14
NUMERICAL ANALYSIS OF AN WING SAIL
AERODYNAMIC CHARACTERISTICS USING
COMPUTATIONAL FLUID DYNAMICS - PART II
Fernando Joel Lopes Gamboa
IST
2010
Abstract
The aim of the present work is to developed anecient a wing sail using a Reynolds AveragedNavierStokes (RANS)-based Computational FluidDynamics (CFD) method. A Reynolds number ofRe = 2 × 106 is considered, which corresponds toa ow around an aerofoil with 4.8m of chord, anaverage boom's chord in a conventional sail, and aapparent wind speed of approximately 12.7 knots.A parametric numerical aerodynamic analysis is
carry out to select the most ecient wing sail aero-foil. This analysis is done for several aerofoils vary-ing the maximum camber, position of the maximumcamber in the chord direction and the existence ofa inection point in the suction side. The eciencyof the aerofoil is assessment in terms of the aerody-namic propulsive and lateral forces (FX and FY ).It's concluded that the best aerofoil analysed has
a maximum camber position at 20% of the chord,a maximum camber of 22% of the chord and a in-ection point in the suction side near the trailingedge.After the two-dimensional aerofoil is selected, the
development proceeds to the 3D wing sail analysis.The results demonstrate that the best wing sail
has a small gap between the wing sail lower tip andthe deck, the aspect ratio is high and nally the dis-tribution of chords is elliptical. These conclusionsare supported by the analytical lifting line theory.
1 Introduction
The wings sail's rise has been a notorious fact innowadays. The remarkable BWM Oracle [17] vic-tory , rigged with a wing sail, in the America'sCup against Alinghi [12], (rigged with conventionalcloth's sails) showed the potential of this type ofsails and make the interest in this type of rigging
grow substantially.There are several applications for wing sails: In
land yachts to increase the performance as demon-strated by Khayyat [9]. Used as the main propul-sion system in autonomous sailing boat as the 30ftcatamaran studied by Elkaim [6]. Wing sails arealso used in extremely fast sail boast either foraccomplish velocity records, as the Endeavour, orfor regatta purposes such as the C-Class Catama-rans [3] and the America's Class BWM Oracle [17].The wing sails utilization aiming a bigger perfor-
mance is easily explained by the presence of thickaerofoils along its span. This type of aerofoils canoer more lift than the ne aerofoils which composethe cloth sails, consequently the wing sail has thepotential to provide a bigger thrust than a clothsail.The yacht industry is a very competitive one,
where the regattas itself increase the demand fornew products and innovations either for the shapeof hulls or sails. To keep up for this demand,new optimization and design techniques are ap-plied. One of state of the art optimization tech-nique are the CFD's (Computational Fluid Dynam-ics), which utilization has grow substantially in thepast years.In fact since the 32th America's Cup in 2007 that
CFD's are now often used in yacht design as tool forow analysis and optimization for hulls and sails.They allow a quick prediction on forces and eld
pressure acting on a surface, consequently the needfor expensive wind tunnel tests is minor and ashape's optimization can be simplied.Despite of the potential oered by the CFD's,
there are some problems to be solved. First ofall the discretization of the continuum introducesnumerical errors. Those errors can be minimized,but for that is required an extra computationaleort, which consequentially consumes time andmoney. The CFD's also have some problems with
1
the numerical turbulence models, which try to solvethe Reynolds Average Navier-Stoques Equations(RANSE). Furthermore, these turbulence modelsstill require some experimental data in order to val-idate the numerical results obtained.There are 4 turbulence models which are often
used by the industry: Spalart - Allmaras [15] , k -epsilon (k−ε) [8], k-omega (k−ω) [11] and ReynoldsStress Turbulence [14]. The rst three are Eddyviscosity models.In all the models, the most common procedure is
to used these models using the wall laws to calculatethe boundary layer characteristics.These semi-empiric wall laws were developed as-
suming that the ow is fully turbulent. Howeversome ows, specially the ows with low Reynoldsnumbers, such as the ow around a sail, there area signicant part of the boundary layer that is inthe laminar regime. So, the numerical prediction ofthis kind of ows can fail, and the agreement withthe experimental data is not achieved.This is presently one of the biggest problems in
CFD's: the correct prediction of the boundary layerand the transitional ow.That diculty is discussed by Rumsey [13] who
studied the lift and drag of a NACA0012 aero-foil at low Reynolds numbers. To predict thoseforces the author use the Spalartas- Allmaras [15]and the k − ω [11], in the variant Menter ShearStress Transport (SST). The achieved conclusionsare that these models are intended for fully tur-bulent high Reynolds number computations, andthe use of them for transitional ows, such as lowReynolds ows, is not appropriate.Sorensen [16] oers an alternative to the conven-
tional fully turbulent models, the γRe − θ modelwhich is variant of the k − ω. This correlationbased transition model has lately shown promis-ing results, and it's used to solve the ow and pre-dict Lift, Drag ant the transition point in two thickaerofoils. The numerical results have a outstand-ing agreement with the experimental data and asubstantial improvement in predicting the aerody-namic forces acting on a aerofoil on a ow with lowReynolds number.Despite of those diculties, the use of CFD's to
predict the ow around a wing and cloth sails canbe observed in several studies:Masuyama [10] made a study numerically calcu-
lates the forces originated by a rigging composedby cloth sails of a 10.3m length overall sail yacht.Were employed two numerical methods, the rstwas the Vortex Lattice Method (VLM), a per-fect uid method. Lastly a Reynolds AveragedNavierStokes (RANS)-based computational uiddynamics method. These results were compared
with experimental results from a real boat and rig-ging.
Couser [5] studied the Mirror, a open class 3.3mlength over all sail boat. This study was performedto making small adjustments, within the tolerancesallowed by the class rule, to the sails and underwa-ter foils. It was used the Vortex Lattice Method asnumerical method.
Ciortan [4] has done a numerical study in two rigcongurations, both with cloth sails,reproducingexperimental tests carried in a wind tunnel. Therst conguration was mast-main sail and theother, mast-jib-main sail. The numerical methodused to perform this study was based in a ReynoldsAveraged NavierStokes turbulence model, theShear Stress Turbulence, a variant of the k − ω.The present study's aim is to develop and op-
timize a wing sail using a Reynolds AveragedNavierStokes (RANS)-based Computational FluidDynamics (CFD) method.
In order to perform this study was rstly useda parametric aerodynamic analysis of the two-dimensional aerofoil to be applied in the nal three-dimensional wing sail.
The two-dimensional numerical aerodynamicanalysis was done for several aerofoils varying themaximum camber, position of the maximum cam-ber in the chord direction and the existence of ainection point in the suction side.
Were obtained numerical results for CD (DragCoecient) and CL (Lift Coecient), and fromthese results the Surge an Sway forces adimension-alized, CX and CY respectively, are calculated foran interval of true wind angles. Finally is chosenthe aerofoil to be applied in the nal 3D wing sail.
The chosen two-dimensional aerofoil is applied inthree dierent wing sails along their span in orderto study the aspect ratio and elliptical chords dis-tribution eect in the Lift(L), Drag(D) and conse-quently in the Surge (FX) and Sway (FY ) results.After the comparison between the dierent numericresults obtained and some theoretical results, thenal wing sail shape is determined.
2 Theory, Numerical Solution,
Flow Solver and Boundary
Conditions
The CFD's methodology used in all numeric com-putations in the present work is based in theRANSE (Reynolds Averaged Navier Stokes Equa-tions. The turbulent and discretization modelchoice to perform such calculus is critical.
2
So it's required to conduct a validation of the nu-merical results and a quantication of the numericaluncertainty associated, conrming that such mod-els are able to predict correctly the ow. This iscommonly know as verication of calculations.The verications of calculations and the choice of
a turbulent and discretization models was done ina previous numerical work, Gamboa [7].In this previous numerical study was compared
the numerical results with experimental data forthe CD and CL , using four dierent turbu-lence models and its variants: Spalart - Allmaras(Standard), k − ε (Standard and Low Reynoldsvariants),k − ω (SST and Gamma-Re-Theta vari-ants) and Reynolds Stress Turbulence (QuadraticStrain).It was chosen a representative ow of the wing
sail, based on an aerofoil NACA0015, with aReynolds number of 2.0 × 106. Due to the lowReynolds number and the possibility of having asignicant laminar zone in the boundary layer, wasstudied the wall functions eect in each turbulentmodel.The numerical results were obtained with a set
o grids with a reasonable size and the numericaluncertainty estimated using a methodology basedin the Grid Convergence Index.It was concluded in that study that the agree-
ment between numerical and experimental resultsfor the CL is achieved in all turbulence models.However the CD agreement is not achieved for anymodel except for the k−ω, Gamma-Re-Theta. Forthis model the agreement is achieved for both aero-dynamic coecients, making it suitable to be usedin a wing sail aerodynamic analysis.Regarding this conclusion about the variant
Gamma-Re-Theta of the k − ω turbulence model,this was the model chosen to perform all numeri-cal results in the present paper. A brief explana-tion about the main model, k − ω and its variantGamma-Re-Theta will be made now:
2.1 The k − ω, two-equation model
The k-omega [11] model is a two-equation modelthat is an alternative to the k-epsilon model.The transport equations solved are for the turbu-
lent kinetic energy k, as the k-epsilon model, anda quantity called ω, which is dened as the spe-cic dissipation rate, that is, the dissipation rateper unit turbulent kinetic energy.One reported advantage of the k-omega model
over the k-epsilon model is its improved perfor-mance for boundary layers under adverse pressuregradients. Perhaps the most signicant advantage,however, is that it may be applied throughout the
boundary layer, including the viscous-dominatedregion, without further modication. Furthermore,the standard k-omega model can be used in thismode without requiring the computation of walldistance.There are four model variants in the literature:
Standard k-omega, SST (Shear Stress Turbulence)k-omega, SST k-omega detached eddy model andnally the Gamma-Re-Theta [16].For all the model variants, the two transport
equations are the same:
ux∂k
∂x+ uy
∂k
∂y= vtS
2
+∇×[(v +
vtσ k
)∇k]
−β∗ωk (1)
ux∂ω
∂x+ uy
∂ω
∂y= αS2
+∇×[(v +
vtσ ω
)∇ω]
−β∗ω2 + Fω1
ω∇k ×∇ω (2)
Despite the fact that the transportation equationsremain the same for all the k−ω variants, the eddy-viscosity term computation for the SST is calcu-lated in a dierent way. This term it's obtainedwith the following expressions:
vt =a1k
max(a1ω, F2Ω); a1 = 0.31
F2 = tanh arg22
arg22 = max
(2√k
0.09ωd2,
500v
ωd2
)(3)
With this constants: β∗ = 0.09, β1 = 0.075,σk1 = 1/0.85, σw1 = 2, α2 = 0.4404, β2 = 0.0828and nally, σk2 = 1.17.
2.1.1 The γRe−θ (Gamma-Re-Theta) vari-ant of the k − ω, two-equation model
The Gamma-Re-Theta transition model is acorrelation-based transition model that has beenspecically formulated for unstructured CFD'scodes. This variant is applied in the present studybased not only in the k − ω but also on the ShearStress Turbulence.The evaluation of momentum thickness Reynolds
number is avoided by relating this quantity tovorticity-based Reynolds number. In addition, a
3
correlation for transition onset momentum thick-ness Reynolds number dened in the free stream ispropagated into the boundary layer by a transportequation. An intermittency transport equation isfurther used in such a way that the source termsattempt to mimic the behaviour of algebraic engi-neering correlations.The Gamma-Re-Theta transition model, as orig-
inally published, is incomplete, since two criticalcorrelations were claimed to be proprietary andhence omitted. One justication for such an omis-sion is that the model provides a "framework" forusers to implement their own correlations. All thenumerical results obtained with this model use theFlenght correlation. This correlation uses the fol-lowing constants: ca1 = 1, ca2 = 0.03, ce2 = 50,cθt = 0.03, and nally, σθt = 2.These constants are the same used by Sorensen
[16], this was intended, due to the good agreementbetween the numerical results and the experimentaldata in that study.
2.2 Flow Solver
All the numerical computation on this work weredone with the commercial software Star-CCM+. Inthis CFD software the ow is solved numericallyusing the nite element method, more accuratelythe nite volume method.In the nite volume method, the solution domain
is subdivided into a nite number of small controlvolumes, corresponding to the cells of a computa-tional grid. Discrete versions of the integral formof the continuum transport equations are appliedto each control volume. It's obtained in this waya set of linear of algebraic equations, with the to-tal number of unknowns in each equation systemcorresponding to the number of cells in the grid.However, if the equations are non-linear, which is
the RANSE case, iterative techniques are employedin order to linearise all the equations to be solved.This set of equations is solved using an algebraicmultigrid solver.This software not only is used to solve numeri-
cally the ow, but also used as the mesh generator.The incorporated mesh generator has some mesh
control parameters, which are quite in getting agood ow discretization. The choice of these pa-rameters is strongly inuenced by the kind of walltreatment employed, that is, if the wall laws areused or not in the boundary layer solution.The parameters used on the mesh, either for the
two-dimensional mesh or three-dimensional mesh,were the following ones:
Base Size
This parameter sets all mesh size parameters for thesurface mesh. Each cell size in the computationaldomain is a function, in percentage, of the base size.So if the base size value decreases the number
of cells on the grid increases and the grid is ner,which leads to a better ow discretization and con-sequently minor numerical errors.
Near Wall Prism Thickness
This parameter controls the thickness of the rstlayer of cells above the aerofoil's surface.This near wall prism thickness specially con-
cerns about the rst boundary layer zone, the vis-cous sublayer. This zone has a huge inuence inthe shear stress component of drag, and since theboundary layer is going to be solved without walllaws it's required that the distance to the wall madedimensionless, y+ should have a value between 0and 1 in the aerofoils/wing surface. This y+ guar-anties that the boundary layer and specially theviscous sublayer has enough cells to be resolved ina proper way, also means that the rst cell layerthickness has a minor value than the viscous sub-layer thickness.The viscous sublayer thickness is directly re-
lated to the Reynolds number and the characteristiclength, which is the aerofoil chord in this case. Inorder to estimate the minimum required thicknessof the rst layer of cells that guaranties a maxi-mum y+ with the value 1, was used an algorithmdeveloped by NASA [2].
Number of Prism Layers
The Number of Prism Layers parameter controlsthe number of layers generated for each boundarysurface that is allowed to have prism layers. If thisvalue increases the mesh renement increases aswell. In the other hand the computational eortalso increases, meaning that more computationaltime and resources are required.
Prism Layer Thickness
The Prism Layer Thickness controls the totaloverall thickness of all the prism layers. This prismlayer thickness ideally should have at least the valueof the boundary layer thickness, however the lastthickness is an unknown.In order to estimate the magnitude of the bound-
ary layer thickness was used the Von Kárman Equa-tion Eq.4, deduced for fully turbulent ows.
δ = 0.37L
(1
Re
) 15
(4)
4
This equation introduces the boundary layerthickness δ, as function of the Reynolds number(2 × 106) and as a function of the characteristiclength L, which is chord's length.
Control Volumes
The mesh type used in this study was a mixturebetween the O type with the H type. The O typewas obtained by the use of the prism layer generatorand its parameters.The H type was obtained using control volumes.
These control volumes dene volumes with severalshapes where the mesh is locally rened. The meshwas rene in the next critical areas:
• The trailing and leading edge of aerofoil/wingsail
• The surrounding the aerofoil/wing sail
• The aerofoil/wing sail wake
• A narrow band from the the inlet to the aero-foil/wing sail
• Narrow bands perpendiculars to the aero-foil/wing sail
2.3 Boundary Conditions
Inlet
This boundary is dene as Velocity Inlet in theStar-CCM+. The most important parameter to bedened is the velocity, it direction and magnitude.All ows studied will have a Reynolds Number
of 2 × 106 which corresponds to number of a owaround a aerofoil of 4.8m chord, an average boom'schord in a conventional sail, and a wind speed ofapproximately 12.7 knots. Therefore the velocitymagnitude is the one that guarantees that ReynoldsNumber for the ow.The direction of the vector velocity is normal to
the inlet.
Outlet
This boundary is dene as Pressure Outlet in theStar-CCM+. In this boundary the velocity is ex-trapolated from the uid domain, i.e., from thewind tunnel's interior using reconstruction gradi-ents.The parameters used in this boundary were the
default ones in the Star-CCM+.
Lateral Walls
The lateral walls are a required boundary thatencloses a nite uid volume, however their pres-ence can interfere with the ow and originate badresults.To minimize as much as possible the lateral walls
interference, they were dened as Slip Walls. In thisway both lateral walls will not have boundary layerand then not interfering with the ow inside thewind tunnel.
Tunnel Base (3D Wing Sail Flow)
The wind tunnel base will be treated as a lateralwall, due to the possibility of interference from aexistent boundary layer on this surface, the tun-nel base is dened as a Slip Wall. In this way theboundary layer don't exist.
Tunnel Top (3D Wing Sail Flow)
The wind tunnel top design has to main consid-erations: First the boundary condition of this sur-face. It was chosen to be dened as a Pressure
Outlet, in this way the ow with a velocity normalto this surface will pass it avoiding reections ef-fects to the tunnel interior. The another concernis regarding the distance between the wing sail tipand the tunnel top. As bigger the distance, minorare the blockage eects, however the cell number inthe grid increases as well the computation eort.It was chosen a distance of 10m, which is a com-
promise between reducing blockage eect and beable to solve the ow with the existent computa-tional hardware.
3 Method of Performance As-
sessment
Either wing sails or cloth sails are the main propul-sion device in a sail boat, so in the sail optimiza-tion and design process it's necessary to quantifythe force that a sail can provide to propel the boatforward, that is the Surge Force, FX and the forcethat drives the sailboat sideways, the Sway ForceFY . The rst one must be maximized and the sec-ond one minimized.The numerical aerodynamic analysis on the two-
dimensional ow around the airfols provides thegenerated forces, Lift and Drag. These forces aredecomposed in a referential which orientation is de-ned by the Angle of Attack (AOA) as show in Fig-ure 1.Where V is the wind velocity vector, L is the
Lift Force, D is the Drag Force and nally α is theAngle of Attack.
5
Figure 1: Resulting mesh with the control volumes.
To derive the Surge and Sway forces from the Liftand Drag is necessary to introduce a new variable,the True Wind Angle (TWA) dened as β, whichis the angle between the wind direction and thelongitudinal boat axes.The trigonometric relations between α and β can
be seen in Figure 2. From these relations the Forceson the boat referential can be derived from Lift andDrag and seen in Eq. (5).
FX = L sinβ −D cosβ
FY = L cosβ −D sinβ(5)
However for simplicity reasons in the two-dimensional ow around the aerofoils, the Surge
and Sway forces are adimensionalized by1
1
2ρV 2A
,
where V is the velocity vector, ρ is the uid densityand nally A is the lateral sail area. In this way andwithout any information lost, the Surge and Swayforces are directly related to the CD and CL, whichare the numerical results obtained from the CFDcode in the two-dimensional aerofoils study.The adimensional relations between Lift, Drag,
Surge and Sway Forces can be observed in Eq.(6)and (7).
FX1
2ρV 2A
=L
1
2ρV 2A
sinβ − D1
2ρV 2A
cosβ
FY1
2ρV 2A
= − L1
2ρV 2A
cosβ − D1
2ρV 2A
sinβ
(6)
⇐⇒
CX = CL sinβ − CD cosβ
CY = −CL cosβ − CD sinβ(7)
Figure 2: Resulting mesh with the control volumes.
The adimensional Surge force CX , is nominatedSurge Coecient and the adimensional Sway forceCY , is nominated Sway Coecient.
The comparison between the dierent aerofoilsshapes will be based in the CX , whose value shouldbe the biggest possible and the CY , related to thenegative sideways sail boat motion, must be min-imized.Therefore, the aerofoils with the minimumvalues of CY and the maximum values of CX areconsidered the best ones.
This force coecients are calculated for a rangeof True Wind Angles from the optimum angle ofattack of each aerofoil and 180o, covering all thepossible wind directions possible at sailing.
For each aerofoil are obtained the numerical re-sults for the CD and CL at several angles of attack,starting from 0o with increases of 5o.
Due to the fact that the CL value is much big-ger than the CD, the inuence of this aerodynamiccoecient in the calculus either for CX or CY isstronger as can be seen in Eq.(7).
So in the numerical analysis for each aerofoil the
6
angle of attack with the biggest CL is considered theangle of attack optimum. It's the CL and CD at theangle of attack optimum that are used in Eq.(7) tocalculate the CX and CY for each aerofoil.The mesh used is a mesh generated in the previ-
ous study Gamboa [7], was chosen this mesh beinga compromise between a good discretization and areduce computational eort. The characteristics ofthe chosen mesh and computational domain can beseen in Figure 3 and Table 1.
Figure 3: Numerical wind tunnel dimensions used in
the present work.
Table 1: Grid sets for numerical calculations withoutwall laws.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
31149 0.48 0.002 40 0.04
All aerofoils analysed have a chord 1m and theow has a Reynolds number of Re = 2 × 106, cor-respondent to a ow's Re around an aerofoil witha 4.8m chord (a typical boom's length), in a windspeed of approximately 12.7 knots.The two-dimensional numerical aerodynamic
analysis was done for several aerofoils, varying themaximum camber, position of the maximum cam-ber in the chord direction and the existence of ainection point in the suction side.Were studied ve aerofoils with dierent shapes,
with well dened objectives. The parametric studybegan with the study of the position of the maxi-mum camber in the chord direction. Therefore weredesigned two aerofoils: Aerofoil 1 and Aerofoil 2.These aerofoils have the same maximum camberand suction side, however the Aerofoil 1 position
of the maximum camber at is at 43% of the chord'slength, for Aerofoil 2 this position is at 22%.It's chosen the position of the maximum camber
from the aerofoil that generate more lift.After that were created two aerofoils with the
chosen position of the maximum camber and suc-tion side, but with dierent maximum camber, todetermine which value of maximum camber gener-ates more lift.Finally is added a inection point and concluded
if that change has some eect The results obtainedand the others aerofoils characteristics are show inthe following section.
4 Comparison of two-
dimensional aerofoils
At each subsection are presented the pair of aero-foils for each geometrical characteristic analysed.It's also presented each aerofoil characteristics andthe numerical results for the CD and CL at the dif-ferent angles of attack analysed.Finally for each subsection conclusions are draw
regarding the analysed geometrical characteristic.
4.1 Aerofoils 1 and 2, position of the
maximum camber in the chord
direction
Is this subsection it will be discussed the eect ofthe position of the maximum camber in the chord.direction
Table 2: Aerofoil 1 geometrical characteristics and nu-merical results obtained.
Chord [m] Max.Camber[m]
Max.CamberPosition[x/c]
1 0.21 0.43
AOA 5o 10o 15o
CD 0.00897 0.0157 0.0189CL 0.693 1.282 1.222
7
Table 3: Aerofoil 2 geometrical characteristics and nu-merical results obtained.
Chord [m] Max.Camber[m]
Max.CamberPosition[x/c]
1 0.21 0.2
AOA 5o 10o 15o
CD 0.0141 0.0225 0.0353CL 0.751 1.27 1.673
With the numerical results presented in Table 2and 3, one can conclude that the proximity of theposition of the maximum camber to the leadingedge is benecial. Taking into account the maxi-mum CL, due to its stronger inuence in the Surgeand Sway coecients, its value increases substan-tially from 1.282, obtained with Aerofoil 1, to 1.673with Aerofoil 2.
Regarding this results the following aerofoils willhave the position of the maximum camber near theleading edge.
4.2 Aerofoils 3 and 4, Maximum
Camber
Is this subsection it will be discussed the eect ofmaximum camber.
In order to do that were created two dierentaerofoils: Aerofoil 3 and Aerofoil 4. The rst onewith a minor maximum camber than the secondone. Furthermore, both aerofoils have dierentmaximum cambers than the others aerofoils anal-ysed before.
In order to restrict the analysis only to the eectof the maximum camber, both aerofoils have thesame maximum camber position.
As it can be observed in Table 4 and 5, a de-crease in the maximum camber also leads to a CLdecrease, comparing to the results obtained withAerofoil 2.
The increase of the maximum camber in Aerofoil4 generates a maximum lift with the value of 1.660at 15o, which is similar to the value of 1.67 at 15o
obtained with Aerofoil 2. However the Drag gen-
Table 4: Aerofoil 3 geometrical characteristics and nu-merical results obtained.
Chord [m] Max.Camber[m]
Max.CamberPosition[x/c]
1 0.19 0.2
AOA 5o 10o 15o
CD 0.0118 0.0194 0.0316CL 0.662 1.174 1.586
Table 5: Aerofoil 4 geometrical characteristics and nu-merical results obtained.
Chord [m] Max.Camber[m]
Max.CamberPosition[x/c]
1 0.26 0.2
AOA 10o 15o 20o
CD 0.0306 0.0517 0.242CL 1.310 1.660 0.871
erated also increased substantially, making this foilnot suitable to be applied in the wing sail.
It's concluded that a change of the maximumcamber value is not benecial. So the followingstudy about the eect of a inection point in thesuction side will be applied in a aerofoil based inthe Aerofoil 2, which is the aerofoil with the bestCL so far.
8
4.3 Aerofoil 5, Existence of an inec-
tion point in the suction side
Is this subsection it will be discussed the eect ofthe existence of an inection point in the suctionside. In the literature it's well know the benetsof having an inection point in the suction side,specially at low Reynolds ows, as the wing sailcase.
To study this geometrical parameter, it was takenthe Aerofoil 2 shape and modied the suction sideinserting an inection point near the trailing edge.It was chosen the Aerofoil 2 as the base Aerofoil dueto fact that is this aerofoil with the best results sofar either for the CL or the CD coecients.
The resulting aerofoil is the Aerofoil 5.
Table 6: Aerofoil 5 geometrical characteristics and nu-merical results obtained
Chord [m] Max.Camber[m]
Max.CamberPosition[x/c]
1 0.22 20
AOA 10o 15o
CD 0.0254 0.0397CL 1.174 1.944
As it can be seen in Table 6, the existence of a in-ection point increased substantially the maximumCL obtain so far, from the value of 1.67 (with Aero-foil 2 ), to the value of 1.944, without compromisinga big increase in the CD.
This fact is easily explain by the existence of ahigh pressure zone in the inection point due to thebarrier created by the double curvature that slowsthe ow and therefore increases locally the pressure.
With this results, and considering the maximumCL obtained from all analysed aerofoils, the chosenaerofoil to be applied in the wing sail is the Aerofoil5, and the wing sail will be analysed in a ow witha angle of attack of 15o to match the angle of attackwere is obtained the maximum lift.
The choice of the Aerofoil 5 to the wing sail isagain supported by the results obtain for the CXand the CY , which can be observed in Figures 4
and 5.
Figure 4: CX results for the analysed aerofoils
Figure 5: CY results for the analysed aerofoils
In both graphics in the vertical axe is representedeither CX or CY , in the horizontal axe are repre-sented the True Wind Angles, β.
For each aerofoil are represented the CX and CYcalculated with Eq. (7) considering the CL and CDof the optimum angle of attack.
The range of TWA analysed are for each aerofoilfrom the optimum angle of attack until 180o
The Figure 4 shows the remarkable superiorityof the aerofoil 5 in providing Surge force. For allthe TWA, the surge coecient is superior compar-ing to the others aerofoils, for instance at 90o therelative dierence between Aerofoil 5 and Aerofoil
1 is about 34%.
However the Sway coecient for the Aerofoil 5
is also superior in absolute value for each angle oftrue angle analysed.
9
In other to compensate the additional Swayforces that can be generated by the wing sail, atthe sail boat design process the appendices musthave special concern about.It was assumed that the superior eciency in
generating a bigger Surge force compensates thefact that there is also an additional Sway force.
5 3D Flow calculations for a
rigid wing sail
In this section will be presented the 3D analysis ofthe wing sail that lead to a proposal of the nalwing sail shape.
5.1 Computational Domain and
Grid Sets
First of all is presented the numerical domain usedin all 3D computations. Due to the fact that nowthe ow is a three-dimensional ow, new distancesto numerical wind tunnel top and base must be de-ned. The distances for the outlet, inlet and lateralwalls are the same seen in the Figure 3.It was dened that the wing sail superior tip will
be at a distance of 10m from the tunnel top, inorder to reduce the blockage eects. The bottom ofthe sail will be at 0.56m (an average distance froma sail to the deck), from the tunnel base. Thesedistances can be seen in Figure 6.
Figure 6: CX results for the analysed aerofoils
Regarding the grid sets, each of the three anal-ysed wing sail shape has a set of grids. The useof several grids with dierent degrees of renementhas the purpose to estimate the numerical uncer-tainty based on the Grid Convergence Index (GCI)introduced by Roache [1].This model stands that the numerical uncer-
tainty U , associated to the use of certain a gridi, is dened by:
U = FS |δRE | (8)
Where FS is the safety factor, assumed as 1.25and δRE is the error estimation obtained by extrap-olation.The error is assumed as being only the discretiza-
tion error, meaning that it's assumed that theround-of errors and iterative errors are negligible.The error estimation is given Eq.8.
δRE = φi − φ0 (9)
Where φi is the numerical solution of any localor integral scalar quantity on a given grid i, in thispaper is the L or D. φ0 is the estimated exactsolution which is unknown.To determine φ0, it's used the several numerical
results obtained from the dierent grids and donea linear interpolation between this results and thegrid characteristic number dened in Eq.(10).
ri =h1hi
(10)
Where h1 is the number of cells in the grid mostrened and hi is the number of cells in the grid i.The y-intercept in the resultant linear equation
corresponds to the estimated numerical result if theri = 0, i.e., the estimated exact solution φ0. The riis equal to zero when grid i has in innity number ofcells, limhi→∞ h1/hi = 0, therefore the discretiza-tion errors are also equal to zero.Finally the uncertainty interval is given by
Eq.(11).
grid i numerical solution ∈ [φi − U, φi + U ]](11)
The grid sets for each wing sail are in the Tables7, 8 and 9.
Table 7: Grid sets used in Wing Sail 1
Char.Nr.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
1 2560781 0.60 0.1 35 0.11.148 2231027 0.65 0.1 35 0.11.315 1947753 0.70 0.1 35 0.11.940 1320147 0.85 0.1 35 0.12.308 1109385 0.9 0.1 35 0.1
5.2 Analysed Wing Sail Shapes
Were analysed three wing sail shapes in order tostudy the aspect ratio and elliptical distribution ofchords.
10
Table 8: Grid sets used in Wing Sail 2
Char.Nr.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
1 1453251 1.8 0.1 35 0.11.077 1349733 1.9 0.1 35 0.11.157 1256509 2.2 0.1 35 0.11.293 1123789 4 0.1 35 0.12.308 713318 5 0.1 35 0.1
Table 9: Grid sets used in Wing Sail 3
Char.Nr.
CellNr.
BaseSize[m]
NearWallThick.[mm]
Nr.Lay-ers
PrismLayerThick.[m]
1 1340924 2.38 0.1 35 0.11.083 1237866 2.4 0.1 35 0.11.492 898551 2.8 0.1 35 0.1
All analysed wing sails are composed by Aerofoil5 along their span, no torsion is applied. As saidbefore, all wing sails are going to be analysed witha angle of attack of 15o.The Wing Sail 1 and Wing Sail 2 have the same
constant chord length (4.8m) along their span ,being the only dierence between them the spanlength, and consequently the aspect ratio.The Wing Sail 3 has an trapezoidal approxima-
tion of a elliptical distribution of chords. A briefdiscretion of each wing sail is done e the followingparagraphs.
Wing Sail 1
Table 10: Wing Sail 1 characteristics.
SailArea[m2]
Chord[m] SpanLength[m]
AspectRatio
72 4.8 15 3.125
The fact that this sail has a sail area of 72m2
isn't random, this is the combined sail area of amain sail and a genoa(108%) of a Beneteau First35, a sail boat representative of the yachts worldeet either for its size or sail area. This wing sailcan be seen in Figure 7.
Figure 7: Wing Sail 1
Wing Sail 2
Table 11: Wing Sail 2 characteristics
.
SailArea[m2]
Chord[m] SpanLength[m]
AspectRatio
92.16 4.8 19.2 4
This wing sail objective is to be directly com-pared with Wing Sail 1 and conclude about theaspect ratio eect on the results. Wing Sail 2 canbe seen in Figure 8.
Wing Sail 3
Table 12: Wing Sail 3 characteristics
.
SailArea[m2]
Chord[m] SpanLength[m]
AspectRatio
72 Variable 17.49 4.25
The sail area of this wing sail is the same thanwing sail 1, in this way it's possible to comparedirectly both sails and conclude about the ellipticaldistribution eect on the results.Wing Sail 3 canbe seen in Figure 9.
11
Figure 8: Wing Sail 2
Figure 9: Wing Sail 2
5.3 Numerical Results
The numerical results are all obtained for a windspeed of 12.7 knots, correspondent to a ow withRe = 2 × 106. The numerical data obtained foreach grid and wing sail is the Lift (L) and Drag(D) forces. Afterwards both forces are decomposedwith Eq.(5) in Surge and Sway forces. Furthermorethe numerical Lift will be compared with the theo-retical Lift from the potential theory.The numerical results obtained for the Wing Sail
1 are presented in Figure 10, in Figure 11 for Wing
Sail 2 and nally in Figure 12 for Wing Sail 3.All those gures the same layout: In the xx axes
are represented the dierent mesh characteristicsnumbers. In the yy axes are represented the nu-
Figure 10: L and D numerical results for Wing Sail 1
merical values either for D or L.
The vertical bar in each numerical result rep-resents the uncertainty interval, calculated withEq.(11).
For each set of results, can be seen a straightline and it linear equation. This straight line isthe result of the linear interpolation between the 3chosen numerical results and the y-intercept in eachlinear equation is the the estimated exact solutionφ0. In the present work is considered that the valueof L and D from each sail to be compared is thevalue of φ0, the estimated exact solution.
It's possible to compare the numerical resultswith the theoretical expected CL derived from thepotential theory and the lifting line theory. Thistheory stands that due to the tip and wake vorticesthere is a induce drag in three-dimensional wingswith a nite span length, in contrary to innitewings which can be modelled as two-dimensionalaerofoils, as the Aerofoi l two-dimensional 5 study.This induce drag will decrease the expected Lift,and as bigger the Aspect Ratio minor is the induceddrag. The relation between the Lift coecient fora innite wing and for a nite wing is deduced bythis theory and can be seen in Eq.(5.3).
12
Figure 11: L and D numerical results for Wing Sail 2
CL =AR
AR+ 2Cl∞ (12)
Where CL is the three-dimensional wing lift co-ecient dened by Eq.(13), AR is the aspect ratioand Cl∞ is the bidimensional lift coecient of theaerofoil used in the wing, in this case is the CL ofthe Aerofoil 5 at 15o of AOA.
CL =L
1
2ρV 2A
(13)
Where V is the velocity vector, ρ is the uid den-sity and nally A is the lateral sail areaThe Table13 shows the theoretical CL calculated
with Eq. and the numerical CL for each wing sailanalysed.Considering for each wing sail its L and D the
estimated exact solution φ0, the Surge Force andSway force can be calculated using Eq.(5) The re-sults obtained are presented in Figure 13.
5.4 Results Analysis
Firstly will be compared the numerical and the the-oretical CL between each wing sail.
Figure 12: L and D numerical results for Wing Sail 3
Table 13: Wing Sail 2 characteristics.
WingSail 1
WingSail 2
WingSail 3
AR 3.125 4 4.25
CL Theory 1.185 1.296 1.322
CL Num. 1.334 1.462 1.401
The results presented in Table 13, show that forall the wing sails the numerical CL is always biggerthan the theoretical CL.
This fact can be easily explained by the groundeect. Due to the proximity between the lower tipof the wing sails to the wind tunnel base, the tipvortex is substantially reduced, therefore the liftingis increased.
The theoretical CL is calculated considering anisolated wind and didn't account for this eect. Inpractice the wing sails have a eective aspect ratiohigher than the aspect ratio without proximity tothe ground.
This eective aspect ratio is accounted by thelifting theory and can be explained by the mirrorimage in the potential theory. The sails special case
13
Figure 13: FX and FX results for all Wing Sail analysed
and this increase of lift by reducing the gap betweenthe sail and the deck is discussed in the literatureRegarding now the aspect ratio eect, the com-
parison between the numerical CL for Wing Sail
1 and Wing Sail 2, show the benets of a biggeraspect ratio. The increase of the CL from 1.334 ob-tained with wing sail 1 to 1.462 withWing Sail 2 isnotorious and it's obtained 9.6% more lift (regard-ing to Wing Sail 1 ) only by changing the aspectratio from 3.125 to 4.The benets of an higher aspect ratio are also
seen in the CD where this aerodynamic coecientis minor in the wing sail 2 than in wing sail 1.Facing now the elliptical distribution of chords,
the Wing Sail 3 can be compared directly withWing Sail 1 due to the same sail area. Wing Sail 3
has a bigger CL than the one calculated for Wing
Sail 1. This corresponds to a increase of 5% in lift.It can be either for the aspect ratio which is the
biggest one of all ails or for the elliptical distribu-tion of chords. Despite of that there is also a minorincrease in the CD, which is a negative fact.
So no direct conclusion can be taken about theelliptical distribution of chords.
Regarding now the Fiqure13, concerning aboutthe Surge and Sway forces , the Wing Sail 2 hasthe biggest Surge and Sway force at all angles ofTWA. This is due to the fact that is the sail withthe biggest area, therefore it generates more aero-dynamic forces
The results obtain forWing Sail 1 andWing Sail
e seem to be similar, however Wing Sail 3 has aslight advantage in generate Surge force, this ad-vantage is constant for all TWA angles. In the otherhand the Sway force, which should be as minor aspossible,is also a little higher in absolute value atWing Sail 3.
5.5 Final Wing Sail Shape
Regarding the previous the nal wing shape shouldhave a high aspect ratio. About the elliptical dis-tribution of chords, it was not possible to concludeabout the positive aspect of having a elliptical dis-tribution in the wing sail. Despite of that the po-tential theory and the lifting line theory demon-strate that this distribution is the best one possi-ble to reduce the induced drag and regarding thosefacts it's assumed that the nal wing sail shape isthe Wing Sail 3.
5.6 Conclusions
A wing sail was successfully developed using aReynolds Averaged NavierStokes (RANS)-basedComputational Fluid Dynamics (CFD) method
This development started with a parametric nu-merical aerodynamic analysis for several aerofoilsvarying the maximum camber, position of the max-imum camber in the chord direction and the exis-tence of an inection point in the suction side.
After the most ecient 2D aerofoil is selected,this is used to design several 3D wing sails and in-vestigate the eects of aspect ratio and ellipticaldistribution of chords.
The conclusions obtained can be subdivided intwo parts: The aerofoil choice and the wing sailshape choice.
Regarding the parametric aerofoil study, one canconclude the following:
• The airfoil generates substantially more liftwhen the maximum camber position is movedfrom the mid chord to 20% of the chord.
14
• The best result in terms of propulsive forceis obtained for a aerofoil with 22% of maxi-mum camber. An increase in maximum cam-ber leads to a increase in the drag coecientand a lower maximum camber value generatesalso a minor lift.
• The inclusion of a inection point in the suc-tion side was the most benecial geometricalchange in the results. This inection point in-creases substantially the lift obtained withoutcompromising the drag.
The chosen aerofoil was has a maximum camberposition at 20% of the chord, a maximum camberof 22% of the chord and a inection point in thesuction side near the trailing edge.
Regarding the wing sail 3D shape, one can con-clude that:
• All the 3 wing sails Lift are superior to the onescalculated by using the analytical lifting linetheory. This happens because of the groundeect. Due to this eect is concluded that asmall gap between the wing sail lower tip andthe deck should be adopted.
• The analytical lifting line theory indicates thata larger aspect ratio increases the Lift in a -nite wing. This is conrmed by the numericalresults.
• The analytical lifting line theory lead to theconclusion that a elliptical chord distributionis optimum in terms of induced Drag of a nitewing. No clear conclusions were obtained fromthe numerical study in this regard. Despite ofthat, the propulsive force is slightly higher atall wind angles for the sail with elliptical dis-tribution. Considering this fact and the liftingline theory conclusion, it's concluded that thenal wing sail should have this type of chorddistribution.
The nal wing sail shape is the shape of Wing
Sail 3, composed by Aerofoil 5 along its span. Ithas a small gap between the wing sail lower tip andthe deck, the aspect ratio is high and nally thedistribution of chords is elliptical.
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