Theoretical investigation of aerodynamic of two dimensional porous dual sails

Theoretiqal Investigation of Aerodynamic

of Two Dimensional Porous Dual Sails

Heiu-'Jou Shaw*

Department of Naval Afchitecttire and Marine Engineering

Chih-Min Hsiun'*^*

Depiartment of Mechanical Engiiieering

NMosai-; CSeng Kung ;CJniversity •'

Tainan. Taiwaa, Republic of

China-* Associate professor ** Graduate student

Summary

A numerical method using iterating procedures has been

developed and can be used to solve extensible, porous single or

dual sails problems in inviscid flow, and the accuracy is proved

when comparing with other previous theoretical methods.

Some different cases of dual saiy problems have been

studied in this paper and the results show that for porous dual sails,

the C L of main sail is independent of chord ratio of Cj/Cm i f the

porous coefficient a is greater than 0.2, although the C L of jib and

1. Introduction

Sails have a wide application, both in transportation and sports. For sailboats, the power comes from the dragging force on the sail under a large incidence angle and from the lifting force of the curve surface of the sail as an airfoil under a small incidence angle. However, the problem of a flexible sail is more complex than a rigid airfoil. When the pressure distribution on the upper and lower surfaces of the sail change, the interaction of shape and pressure distribution should be considerd together when flexible sails are concerned.

Beside the change of shape, the dimension of the sail under the tension force which is balanced by pressure difference on the upper and lower surfaces of the saü wiU change too. The other important property is that the saü is impermeable or porous, m normal velocity on porous sail surfacej&jmpermeable or_porgug. The normal velocity on porous sail surface is not equal to zero, but to some finite value.

There are two methods to get solution concerning a two-dimensional sail problem in inviscid flow. The saü equation obtained by Thwaites[l] used the thin airfoil theory and Nielsen[2] used a similar method to solve this problem. Tnis method linearizes the problem and therefore was only applied to a small incidence angle. Van den-Broecks useènonlinear method to solve a more complicated sail equation [3] which can be apphed to a large incidence angle. The other method is an iterating procedure. The pressure distribution is first obtained from'^panel method when the shape of a saü is fixed, and then the pressure difference on the saü surface^is used to s o l v ^ h e shape and tension force on the sail. The above procedures repeat beföfèra convergent solution is obtained. Murai[4], Jackson[5], Matteis and Socio[6] used a similar concepts and numerical method to solve the problem. A n d Greenhalgh[7] used a sunilar numerical method to solve the pressure difference and sail length with a given tension force.

For extensible and porous saüs, Jackson[8] presented a simple model to solve an extensible ƒ aü and ref. [6] used the same model. Barakat Richard[9] used the same method like in r e f . [ l ] to solve the porous dual saüs problem assummg that the normal velocity on the saü was proportional to the pressure difference between upper and lower surfaces of the sail. Murata and Tanaka[10] used the same assumption and appHed a series of Jacobi polynomials to obtain the pressure

distribution of a single porous sail. In ref.[6], the same assumption was also used to solve the porous sail problem.

In practical application, large sailboats have two or more sails. Sailboats of 15~25m length usually have two sails as shown m F i g . l . The one on the post is called main sail and the other j i b sail.

For the problems of dual sails, the mteraction between the two should be considered. Myall and B e r g e r [ l l ] used the method similar to r e f . [ l ] to solve the interaction between the two inextensible sails. Wiersma[12] used the vortex sheet method to work out the integral equation of the two rigid sails. Ref.[5] used iterating procedures to solve the problems of impermeable dual sails.

This paper uses theoretical and numerical methods similar to those used in ref. [5] with the simple model in ref. [8] for extensible sails, and the assumption of normal velocity on sails like in ref. [9] is taken to develop a numerical method to solve the extensible porous dual sails problem.

2. Theoretical Analysis

2.1 Pressure distribution of sails

The continuity equation for a two-dimensional incompressible steady flow is

V . V = 0 (1)

I f the flow velocity is uniform in infinite distance, for inviscid flow, we can assume reasonably that this flow is also irrotational.

that means ;

V x V = 0 (2)

and there exists a stream function satisfied

3 T a T u=

v=-dX dY

(3)

where u and v are the velocity components in the direction of the X and the Y axis in Cartesian coordinates.

To substitute equation (3) into (1), we have

V2^=0 (4)

which is the governing equation of the velocity and pressure distribution of a two-dimensional sail problem in an inviscid, irrotational, incompressible steady flow.

The boundary conditions of this problem are as follows:

(1) The flow field is not influenced at infinite distance away from the sails, so

-V ( X , Y ) = ï L when X 2 + Y 2 - ^ o o (5)

(2) I f the sail is made by impermeable material, the boundary condition on the surface of the sail is

V n = 0 ₍₆₎

where n is the normal unit velocity vector on the surface,

(3) When the sail is porous, the boundary condition on the surface of the sail is

where VQ(S) is the permeant velocity along the surface of the sail with the

same direction of n .

2.2 The shape of sails

Assume that the curvature of the sail shown m Fig.2, is very small. For inviscid flow, we can assume reasonably that the tension force T along the surface of the sail is constant, then

where dp is the pressure difference between upper and lower surface of the saü.

The boundary condition of equation (8) reveals that the two end points of the saü are fixed.

Today, most of the saüs are made of l ^ o n and other sünüar polymer. Smce the stress-stram relationship of those materials is very comphcated, this paper assumes that the length deformation of the saü keeps within 3% under tension force T. Within this range, the Young's moulus E is ahnost constant and the deformation is elastic.

Let At and B be the thickness and breadth of the saü respectively. Under the tension force T, the deformation As of a unit length is

V . n = V n ( s ) ₍₇₎

de.

T0'éefines-exteïïsiöïH:o^ = — ^ , equation (6) becomes

to-'-i-^ EAtB

This simple model is similar to ref. [8] and [6].

In the problem of the two-dimensional sail, i f the distance between the two end points is C and the initial length is lo, the initial excess length of sail slack 8 is

, lo-c lo ^

^ = " c " = c - i (11)

I f the deformation of an extensible sail under the tension force T is AE, the final length of the sail becomes

1 = C(l+e X l + A E ) = C(l+e )(1+p T) (12)

which is the governing equation of the length of an extensible sail. Equation

(12) is also apphed to inextensible sails. In that case, E becomes mfinite and "p =0; therefore, equation (12) can be simplified as

1 = C(l+£ ) = lo (13)

Overall, the governing equations of an impermeable or porous sail in an inviscid, irrotational, and incompressible steady flow are equations (4), (8) and (12). The boundary conditions are equations (5), (6), (7) or(13) and the two end points of the sail are fixed.

3. Numerical Method

3.1 Pressure distribution on the sail surface

In this paper, the velocity field and pressure distribution on the surface of the sail is solved by using Vortex Panel Method.

The geometric diagram of vortex panel is shown in Fig. 3. Let the original point of the local coordinate system be the first edge of the j t h panel. The component of velocity at an arbitrary point P induced by the unit strength vortex at the quarter length of the jth panel can be presented as follows :

where (x',y') and (x,y) are the coordmate values of the vortex point K of the jth panel and point P,and r is the distance from P to K.

The velocity V induced by the jth vortex panel basing on global coordinate system is'.

V=(ux cos Gj - Uy sin Gj) Cx + (ux sin Oj +Uy cos Gj )ey (16)

where e^x , e^y are the unit vector of the global coordinate system and ej is the angle between the j t h vortex panel and X axis shown in Fig. 3.

To take the three-quarter chord on the ith panel as the control point, the total induced velocity of the ith control point by aU vortex on the saü can be written as -in r 2 (14) 1 x-x' (15) N Vi

=Jsn

4

where K^j and K j j are the influenced coefficient as equation (16),

= Ux cos Gj - Uy sm 9j Y

Ky = Ux sin Oj + Uy cos Gj (18)

The total velocity of the ith control pomt is the sum of vectors of uniform

velocity in infinite distance Uoo and V j , so /

ÏÏi= \L +Vi (19)

or

^ N N

ï ^ = ( Uoocosa

+XTj

j = l j = l

where a is the angle of attack.

Let the normal velocity at the ith control point be Vni, and then the boundary condition (6) or (7) becomes;

Ui • ni =Vni ^ (21)

where ni is the normal unit vector of the ith vortex panel on the sail.

I f the sail is impermeable, Vni becomes zero.

From Kutta-Joukowski Theorem,

po, U«7i=Api Asi , (22)

or A p i = p « UooTi/Asi (23)

where Api is the pressure difference between the upper and lower surface of the ith panel on the sail, pco the density of flow, Asi, Ti the length, vortex strength of the ith panel respectively.

Assume Vni =k A p i , where k is a constant, the Vni can be written as

VQ i = k poo UooYi / Asi = a / Asi (24)

The a shown above is the porous coefficient and is assumed to be a constant in this paper. This model is similar to those in ref.[9], [6] and [101.

From equations (20), (21) and (24), we can get the equation system shown as foUows:

^ , , Y i - Y i ^ l Xi-Xi-Hi

r Yj-Yi+l. X i - X j ^ i .

=-[ ( )cosa + ( ) sma ] Uoo (25)

In order to make sure that the flow passes through the trailing edge of the sail smoothly, the Kutta condition TN =0 must be satisfied at tiie last panel on the sail. So we have ji, i=l,2 ,...,N-1, that is N-1 unknown. And we take the middle point of the ith panel, where i=2, 3,..., N , as control points so that equation (25) is satisfied at tiiese control points.

When the strength of each vortex is known, the hfting coefficient C L and the dimensionless pressure difference coefficient ACpi can be calculated from :

N

C L = 2 X r . (26)

i = l

and ACpi= 2 r , / A s i ^ (27)

i f we take Uoo as unit.

3.2 The shape of the sail

If the pressure difference APi of all panels are known, the shape of the sail can be found from equation (8) by using the method of Jackson [5].

. First, from dimensionless equation (8), we get •

de

. dCp ds =2 C C T s i n y / (28) where C T is the tension force coefficient and C is the chord length.

The above equation can be presented as following pattern when we divide the sail into N panels,

6i+i Qi

ACpi Asi =2 C C T sin ( — - j ) i = l , 2, , N (29) Next, we define the dimensionless extension coefficient as :

|3=P" | p o o U « 2 c

PooU«2c

(30)

2AtE

and then the length of the ith panel of the sail at C T is •

Asi = Asoi (1+p C T ) 7 (31) where Asoi is the initial length of the ith panel when the tension force is not

loaded.

3.3 Procedures of Numerical Calculation

The procedures of numerical calculation are as follows :

(1) Set up the angle of attack a, coordinates of the end points of the sail, initial excess length e, extension coefficient {3 and porous coefficient a. I f e is not equal to zero, the initial shape of the sail is assumed to be an arc, Qlse-an(plate

(2) Divide the sail mto N panels as shown in Fig.4, and there are N+1 points totally.

(3) Use equations ( 2 5 ) , (26) and (27) to solve , C t a n d ACpi.

(4) Choose an initial value of Cj. Substitute ACpi into equation (31) and use an iterating method to solve a final value of C T and find the shape of the sail satisfied the equation (29).

(5) After a new shape is found, repeat (3) and (4) before C T of the sail converges to an assigned range.

There are two convergent values assigned in the above calculation. One is in step (4) where the range of convergence should be mdicated for the modification of C T to be stopped. The other is in step (5) where the range should be indicated for A C T between two values to converge.

This problem has been discussed m ref. [5]. The conclusion is

A C A C T

C " - ^ 2 ^ (32)

A C T

The value of AC is constrained when is assigned. In the example of

A C T

this paper, it is convergent if - Q J T ^ 0.001 and, therefor, the convergent domain of

AC is 0.0005C£ .

In calculation, i f we have two sails, there are totally 2(N-1) unknowns of Xi-The original ( N - l ) x ( N - l ) matrix derived fi-om equation (20) becomes a (2N-2)x(2N-2) matrix .

4. Numerical Results and Discussion

4.1 Influence of the number of panel

Table 1 shows the comparison between the C L values of a rigid thin arc airfoil obtained from the analysis solution and the present numerical method using different number of panel N . The maximum height of the camber and the chord length of the airfoü are h and C, and h/C=0.06133. The numbers of panel N on the sail are 20, 40 and 60 respectively. When N increases, the numerical result of C L is close to analysis solution slowly. However, with a large N , the computing time increases greatly proportional to ( N - l ) x ( N - l ) . Therefore, the calculation in this paper set N=40.

4.2 Comparison of results with those of other references

The first comparison between the results obtained fi-om the present method and ref.[5] is shown in Fig. 5. It is in agreement with the work done by Jackson who used different numerical scheme to obtain the pressure distribution of the sail.

The other results used to compare come from ref.[8]. Fig. 6(a) shows that the C L value derived from present method is in agreement with those from Murai and Matteis, but the C T value is a httle greater in Fig. 6(b).

For extensible sail, Fig. 7 shows the comparison between results using present method and those done by Jackson and Matteis, and the difference is small.

For the porous sail, we make some comparison between ref. [10] and the present numerical method when a is 0, 0.2, 0.4 and 0.8 respectively. The result is shown in Fig. 8 and it has good agreement between those two method.

4.3 Interaction influence of porous two sails

In ref. [5] Jackson made a study about the mteraction influence of impermeable inextensible dual saüs. From equation (12), the extension length of Üie saü panel is proportional to CT, SO that the behavior of extensible two saüs is reasonably simüar to that of inextensible saüs. The other hnportant parameter of

sails is porous coefficient, which is also the main parameter when the interaction influence of porous two sails is studied m following section.

There are many variables m the geometry of dual sails. The notation used in two sails is defined m Fig. 9. The most hnportant parameters those can be changed when (fij, (j)m and Sm are fixed will be a', Cj and Cm, i f the Sj and Em are also be fixed.

Fig. 10 and 11 show the relation between C L and a in some different values

C j

of ^ and a'. There are some sunilar results when a increases and a' is fixed at different special values. Fnst, the C L values of jib and mam sails decrease when c increases and is the same as a single sail. Next, the C L values of dual sails also

Ci

decrease when the chord length ratio pT" increases. The reason is the interaction

effect between the jib and main sails that had been discussed m ref. [ 5 ] . But when a'=5°, firom Fig. 10, we can find that i f a is greater than 0.2, the C L value of main

C j

sail is independent on p ^ . And Fig. 11 shov/s that the variation of C L of main sail

C j

in different ^ conditions is also very small i f a is greater than 0.2.

Fig. 12, 13, and 1 4 are the shapes, pressure distribution and velocity vector

diagrams at some different values of impermeable two sails. It can be found

that the maximum pressure difference ACp^ajc between the upper and lower

surfaces of the jib sail decreases greatly when value increases and then makes

*^m

the C L of the j i b sail decrease. One the other hand, the ACp^ax of main sail also

C j

decreases when R v a l u e increases and the location of ACpmax is more close to the

leading edge. Fig. 1 5 is the comparison of different two sails shapes when is

Cm

0.8 and 1.4. It shows cleariy that the maximum height of the camber of the main

saü is more close to the leading edge when = 1 . 4 .

Fig. 1 6 shows the shapes, pressure distribution and velocity vector diagram of two sails with same parameter except a, and Fig. 1 7 shows the comparison of

ihe shapes of porous and hnpermeable sails. The pressure difference ACp of j i b and main sails with porousness is much very different from that of impermeable sails. For the j i b sail, ACp at front part near to the leadmg edge is very small when compared to Fig. 14, and it explains the reason why the C L of j i b sail decreases largely (about 120%) when a iucreases from 0 to 0.3. However, the difference of C L between porous and impermeable main sails decreased about 60% just because the distribution of ACp has more similar type than jib sails. But i f we

check the shapes of sails shown i n Fig. 17, it can be found that the maximum height of the camber of porous maü sail is more close to the trailing edge.

5. Conclusions and Suggestion

A numerical method using iterating procedures has been developed and it can be used to solve extensible, porous single or dual sail^ problems in inviscid flow^- aad^the solution is more accurate wheg-comparing with other previous theoretical methods. t < = - ^ a x ^

Some differen«2c cases of the dual sails problem have been studied using numerical method presented ui this paper. The results show that for independent

on ^ i f the porous coefficient a is greater than 0.2, althougth the C L of j i b and I

main sail w i l l decrease when a increases. ^

Smce the problem^ is restricted to 2-D sails and mviscid flow, it has Httle practicability in real 3-D sail^ design. But we can more or less obtain some ideas of the behavior of porous dual saüs.

ó.References

[1] Thwaites, B., The aerodynamic theory of sails. Part I - Two dimensional sails'. Proceedings of the Royal Society, Series A, Vol. 261 (1961).

[2] Nielsen, J.N., "Theory of flexible aerodynamic surfaces'. Trans. ASME, Journal of Applied Mechanics, Vol.30 (1963).

[3] Vanden-Broeck, J. M,,'Nonlinear two-dimensional sail theory,' Phys. Fluids, Vol.25 (1982).

[4] Murai, H and Maruyama, S., Theoretical mvestigation of aerodynamic of double membrane sail wing aüfoil sections,' Journal of Aircraft, Vol. 17(1980).

[5] Jackson, P.S., Two-dimensional sails in inviscid flow,' Journal of Ship Research , Vol.28 , No.l(1984).

[6] Matteis, G. de and Socio, L . de, 'Nonlinear aerodynamics of a two-dimensional membrane airfoil with separation,' Journal of Aircraft ,Vol.23, No.ll(1989).

[7] Greenhalgh, S., Curtiss Jr., H . C. and Smith, B., 'Aerodynamic properties of a two-dimensional mextensible flexible airfoil,' A I A A Journal, Vol. 22 (1984).

[8] Jackson, P.S., 'A shnple model for elastic two-dimensional sails,' A I A A Journal, Vol. 21 (1983).

[9] Barkat Richard, 'Incompressible flow around porous two-dimensional sails and wings,' J. Math. Phys., Vol. 47 (1968).

[10] Murata, S. and Tanaka, S., 'Aerodynamic characteristics of a two-dimensional porous saü,' J. Fluid Mech., Vol. 206 (1989).

[11] MyaU, J.O. and Berger, S. A., 'Interaction between a part of two-dimensional saüs for the case of smoothly attached flow,' Proceedmgs of the Royal Society, Series A, Vol. 310 (1969).

[12] Wiersma, A. K., 'Note on the mteraction of two overiappmg rigid sails, Part I : Two-dimensional sails,' Intemational Shipbuilding Progress, Vol. 26 (1979).

Nomenclature

B width of sail

C chord length of sail AC chord length error of sail C L lifting coefficient

Cp pressure coefficient

Cp+ pressure coefficient on upper surface of sail pressure coefficient on lower surface of sail ACp Cp+ -

Cp-C T tension coefficient

A C T error of tension coefficient E modulus of elasticity

ê unit vector

F L total Hfting force

X

K|j the X component of induced velocity of the ith control point by the unit strength vector at the jth panel

K j j the Y component of induced velocity of the ith control

point by the unit strength vector at the jth panel 1 the finial length of extensible sail

lo the mitial length of extensible sail N number of panel

n unit vector normal to surface of sail Poo pressure of uniform flow

dp pressure different of sail surface Ap pressme different of sail panel As the finial length of saü panel Aso the initial length of saü panel T tension force

t thickness of sail

Uoo uniform flow velocity

Greek Symbols

a angle of attack

a' effective angle of attack

j3 extension coefficient

7 vortex strength e excess length \|/ stream fimction

^xn rotation angle of main sail a porous coefficient o f saü Poo density of uniform flow

Subscripts

i , j index of panel j j i b saü

Table 1. The comparison of values of a rigid thin arc airfoil with h/c 0.06133 by different nmnber of panel N .

a N=20 N=40 N=60 _solution Analysis

0° 0.7105 0.7459 0.7554 0.7688

4° 1.144 1.1817 1.1915 1.2019

8° 1.5727 1.6117 1.6218 1.6292

j i b sail

Figure 1 The sailboat with jib and main sail.

dp

Figure 2 The force balanced diagram of a small

C L _25.0 20.0 15.0 10.0 • Nielsen O Jackson A Present 5.0 0.0 0.5 1.0 1.5 2.0 2.5

(a) C L values of present method and ref. [5].

3.0 a C T 5.0 4.0 3.0 2.0 1.0 • Nielsen O Jackson A Present 0.0 0.5 1.0 1.5 2.0 2.5

(b) C T values of present method and ref. [5].

3.0

C L 2.0

6.0 8.0

a(degree)

(a) C L values of present method and ref.[8].

C T 3.5

8.0

a(degree)

(b) C T values of present method and ref. [8].

C L 3.0 2.5 • Jackson O Matteis A Present 6.0 8.0 a(degree)

(a) C L values of present method and ref. [5] and ref.[8].

C T 3.0 2.5 2.0 1.1 1.0 0.0 • Jackson O Matteis A Present 2.0 4.0 6.0 _8.0 a(degree)

15.0

a

obtain from present method and ref. [10] . The a in present method is fixed as 5 degree.

Y

a

jib sail

main sail

C L 4.0. O . O L 0.0 • Cj/Cm=0.8; Main sail O Cj/Cm=1.0;Main sail A C j / C m = 1.4; Main sail Figure 10 0.1 O C /Cm=0.8: Jib sail + Cj/Cm=1.0;Jib sail X Cj/Cm=1.4:Jib sail 0.2 0.3 Ö The C L -Ö diagram of numerical results at different Cj/Cm and « ' = 5 ° ,(t)j =4)m=30°. Sm=l, £j=£m=0.05.

C L 4.0.

0.0

• Cj/Cm=0.8;Main sail <> Cj/Cm=0.8;Jib sail

(a) The shapes and pressure distribution of dual sail.

^ unit velocity vector

y V y

y / y

y y y

y y y

y y y

y y y

y y y

y y y

/ y y

/ / y

/ / y

/ / /

/ / /

/ / /

III

y y y y^y^yyy-^^.^^

y^ y^ y^ y^ y^y y .^^^^^

y, y^ y^ y, y, y^ y^ ^ ^ ^

y y y y yy^-^--^ -^

y y yyyy'

'y/y/y/^y^y^

/ 7 7 7

yyyyy

-/, 7 / /

'yyyyyy.

^

yyy-^ yyy-^ y yyyyy-^

(b) The shapes and velocity vector diagram of dual sail.

— unit velocity vector

/ y y y y y y yy^^^-~

/ y y y y y y y ^

/ y y y y y yy

9/y9/y7/y/yZ

/ y

y y

y y

y /

y /

/ /

/ /

/ /

/ /

/ /

/ /

(b) The shapes and velocity vector diagram of dual sail.

- C p 8 h

2 \

-- 2

(a) The shapes and pressure distribution of dual saü.

unit velocity vector

y /

/ z

y, y^ y y

y, y, y, y, y, yyy

y/yy/y/y

/zy/y

^/ y ^/ ^/ y yyy

y y y y y y y^ y

y/yzy/yi

/ / / / / 7

/,

7 / / /

(b) The shapes and velocity vector diagram of dual saü. Figure 14 The shapes , pressure distribution and velocity vector

(a) The shapes and pressure distribution of dual sails.

unit velocity vector

y y y y y y y y y / _{y y y y} / y y y y / • / • y y y y / / • _{y y y y y} /

y

(b) The shapes and velocity vector diagram of dual sails.

Figure 16 The shapes , pressure distribution and velocity vector. diagram of dual sails which Sm=l, (J)j =4)m=30''.

Figure 17 The comparison of different dual sails shaps when