Forces due to exterior singularities upon 2- and 3-dimensional bodies

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

248

2010, 22(5), supplement :248-252 DOI: 10.1016/S1001-6058(09)60202-X

Forces due to exterior singularities upon 2- and 3-dimensional bodies

Seung-joon Lee Department of Naval Arch. & Ocean Eng., Engineering College, Chungnam National University

Daejeon, Korea E-mail: [email protected]

ABSTRACT: Thrust deduction, or the resistance increase, is not fully investigated probably because of the complexity of the flow pattern in the stern region. In this work, it is assumed that the theory of potential flows may represent the most significant portion of the physics of the phenomena. Hence the propulsor and other devices in the stern region are represented by exterior singularities and their effects upon the resistance increase are estimated by taking a circular cylinder and a sphere as the 2- and 3-dimensional body, respectively. Results for a circular cylinder are summarized, and those for a sphere are given more attention. KEY WORDS: thrust deduction; potential flows; singularities; circular cylinder; sphere. 1 INTRODUCTION For predicting the power performance of a ship, it is usual to conduct both a towing test and self-propulsion test for a ship model. The difference between the resistance measured in the towing test and that of the self-propelled model is commonly termed as the thrust deduction, which is in reality the resistance increase due to the accelerated flow in the stern region. There have not been many investigations on this subject mainly because of the complexity of the flow pattern in the stern region. However, it is not irrational to assume that the most important single factor for this resistance increase is the acceleration due to the presence of the propulsor, which in turn causes the pressure upon the hull surface to decrease in the stern region[1]. In this work, a preliminary study on the thrust deduction, we will consider a body in the uniform flow of an ideal fluid and the forces upon the body due to exterior singularities behind it modeling the propulsor and other devices. First, the results for the 2-dimensional body, i.e. for a circular cylinder, will be summarized[2], and then those for the 3-dimensional body, i.e. for a sphere will be followed.

For a circular cylinder, the circle theorem is used to get the complex velocity potential and then either the Blasius theorem or the Lagally theorem[3] are cmployed to obtain the force upon the body in terms of mainly two small parameters δ and ε . δ represents the closeness of the singularity to the body, and ε the strength of the singularity. In this study, only the leading order results will be given. For a sphere, we first assume the axisymmetry of the flow field, for which the Stokes’s stream function exists. Then, the Burtler’s sphere theorem gives the stream function for the flow, and the extended Lagally theorem gives the force on the body again in terms of two small parameters as before. Since depending upon the hydrodynamic character of the device in the stern region, we may make use of various kinds of singularities, the method developed here is very simple but highly versatile and the results contain a great deal of useful information for the designer. Further studies and a short discussion are attached at the end. For completeness, a proof of the extended Lagally’s theorem is given in the appendix. 2 FORCE UPON A CIRCULAR CYLINDER As the first example we consider a circle accompanied by a sink, located at ( )1 , 1,z b aδ δ= = + where a is the radius of the circular cylinder(Fig. 1). If we let the strength of the sink Uaε , where U is the velocity of the uniform flow, we can obtain the complex velocity potential by using the circle theorem and then the force upon the cylinder by the Lagally theorem(much simpler than the Blasius theorem to apply) as follows.

2 2T X YX iYC C iCU a

εερ δ

− ⎛ ⎞= − = = +⎜ ⎟π ⎝ ⎠ (1)


249

Fig. 1 Co-ordinate system and arrangements Here, 1i = − , ρ is the density of the fluid, and

, X Y are the force applied to the body in , x y− − direction, respectively. This shows that, if

( )Oε δ= , two terms on the right-hand-side are of the

same order and ( )XC O ε= , i.e. indeed we have a resistance increase. When the sink is replaced by a point vortex, we change ε by iε and taking care of the complex conjugate part properly, we can show the following.

2T X YC C iC i εεδ

⎛ ⎞= − = +⎜ ⎟⎝ ⎠

(2)

This implies that we have a resistance increase and also the transverse force of ( )O ε . Furthermore, this can be interpreted as a vortex generator at the stern may cause a resistance increase as well as a side force. For an off-centered sink or point vortex located at

iz be μ= , by the same method it can be easily shown that

2i

i i iT X Y

eC C iC e eσ

σ μ μ εαεδ

− −⎛ ⎞= − = +⎜ ⎟

⎝ ⎠ , (3)

where ( ) ( ), 0,1 or ( /2, 1)σ α π= − for a sink and a point vortex, respectively. When 1μ , it is checked that we regain the results given in Eq. 1 and Eq. 2. When there are two symmetric sinks of the half strength, / 2Uaε , behind the circular cylinder located at iz be μ= and ibe μ− , we can obtain under the assumption 1μ

( )2 22

2 2T X YC C iC ε εδεδ μ δ

⎧ ⎫⎪ ⎪= − = + +⎨ ⎬+⎪ ⎪⎩ ⎭

, (4)

where the last term is due to the cross-effect between singularities. If ( ) ( )O Oε δ μ= = , all three terms on the right-hand-side are of the same order, and

( )XC O ε= as before.

When a dipole of strength 2Ua ε with the axis in the positive x − direction is located at z b= , the same

method is employed to yield

342T X YC C iC εεδ

⎛ ⎞= − = +⎜ ⎟⎝ ⎠ ,

(5)

for which if we assume ( )3Oε δ= two terms on the

right-hand-side are of the same order, and ( )XC O ε= as before. Eq. 5 has a similar form as the Eq. 1 and the main difference is the order relation between the two parameters δ and ε . When the axis of the dipole is arbitrary, say it is in the direction of ie σ , we can attain the following result.

342

iT X YC C iC e σ εε

δ⎛ ⎞= − = +⎜ ⎟⎝ ⎠

(6)

This enables us to have an estimate of the pressure drag increase and the side force due to an Azipod inclined at an arbitrary angle. For an off-centered dipole located at iz be μ= with its axis parallel to ie σ we can derive the following.

334

2

ii i

T X YeC C iC e

μσ μ εε

δ

−−⎛ ⎞

= − = +⎜ ⎟⎝ ⎠

(7)

The form of this equation is similar to that of Eq. 3 for a sink, and when 1μ , the resistance increase is not much different from that of the centered one. When there are two symmetric dipoles of the half strength, 2 / 2Ua ε , located at iz be μ= and ibe μ− , we can prove under the assumption 1μ

( )2

33 2 24

4 4T X YC C iC ε εδμε

δ μ δ

⎧ ⎫⎪ ⎪= − = + −⎨ ⎬

+⎪ ⎪⎩ ⎭, (8)

where the last term is again due to the cross-effect between singularities. If ( ) ( )3 3O Oε δ μ= = , all

three terms on the right-hand-side are of the same order, and ( )XC O ε= as before. Eq. 8 gives an estimate of the pressure drag increase for a twin-propulsor system, and we see that it can be less than that of the single-propulsor one because of the negative sign of the last term. As a model for the propulsor-rudder system, we may consider the combined system of a dipole and a point vortex. If ( )1 11z b aδ= = + , ( )2 21z b aδ= = + ,

2 1δ δ> , and their strengths are 21Ua ε and 2Uaiε ,

respectively, we can demonstrate

11 23

1

4 22T X YC C iC iεε εδ

⎛ ⎞= − = + +⎜ ⎟

⎝ ⎠ , (9)

for which ( ) ( ) ( )3 31 1 2 2O O Oε δ ε δ= = = is assumed.

This implies that the point vortex affects only the side force, which in turn would cause a drift of a body.


250

3 FORCE UPON A SPHERE As mentioned earlier, for an axisymmetric flow, we can make use of the Stokes stream function. Then for a sphere, we can apply the Butler’s sphere theorem to obtain the stream function, and to compute the resistance increase we may employ the extended Lagally theorem[4], which gives force upon the body. As the first example, we consider a sphere accompanied by a sink located at ( ) ( ), ,0R bθ = , and we use the spherical co-ordinate system(Fig. 2).

Fig. 2 Spherical polar co-ordinate system and arrangements

Butler’s sphere theorem gives the following stream function for the flow under discussion

32 2

2 2

1 2

1 sin2

cos 1 cos1

aU RR

R RR b R cUaR a R

ψ θ

θ θεβ

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠⎛ ⎞−− −+ + + +⎜ ⎟⎝ ⎠

(10)

We see that the strength of the sink is 2Ua ε , and c is given by 2bc a= , that is the distance of the inverse point of the sink from the center of the sphere. 2R is the distance from the inverse point to a field point, and β is defined by ( )1 /b aβ δ= + = . As well known, the image system of a sink of unit strength consists of a sink of strength 1β − located at the inverse point and

a line source of the strength 1a− distributed on the line connecting the origin and the inverse point. Force in the direction of the uniform flow upon the sphere by the sink is given by the extended version of the Lagally theorem[see Appendix] as follows

4 ( )iX m U uρ= π − + (11)

Here 2m Ua ε= − , and iu is the induced velocity in x -direction at the point of the sink by the flow except the sink itself. Since iu is given by

( )( )

322

11

iUu U εβ

β−= − −

−, (12)

substituting this into Eq. 11 we obtain the following

result. 2 2

2(4 )X U a ερ π εδ

≈ + (13)

2 2 2(2 )2 2X

XCU a

εερ π δ

= ≈ + (14)

This looks very similar to the two-dimensional result Eq. 1, and if ( )2Oε δ= , two terms on the right-hand-

side are of the same order, and have comparable contribution on the resistance increase. The first term on the right-hand-side of Eq. 14 is due to the part of the induced velocity by the dipole located at the origin. On the other hand, the second term is due to the induced velocity by the image system of the external sink, and its existence is based upon the closeness of the external sink and the internal sink of the image system. When a dipole is located behind the sphere, exactly the same method can by applied, however, the x − force on the body is given by the following extended version of the Lagally theorem[see Appendix]

4 iuX mx

ρ ∂= π∂

(15)

Here, m is the strength of the dipole, and we assume that the axis of the dipole is in the x − direction, and

iu is the same as before, that is the induced x − velocity at the point of the dipole by the flow field except by the dipole itself. x − derivative is chosen as the axis of the dipole is in that direction. Again, the Butler’s theorem gives the following stream function.

32 2 3

3 3 31 2

1 1 1 1sin2

aU R UaR R R

ψ θ εβ

⎛ ⎞⎛ ⎞= − + −⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(16)

We observe that the image of a dipole is a dipole located at the inverse point and its strength is reduced by the factor of 3 1 3β δ− ≈ − , and its axis is in the negative x − direction. From Eq. 16 we get iu as follows,

( )3 3

3 33

21ia Uau Ux x c

εβ

⎛ ⎞= − −⎜ ⎟⎜ ⎟ −⎝ ⎠

(17)

and hence its x − derivative at x b= is given by

( )4

34 43 2 2

1 23iu bUax b b a

ε

β

⎧ ⎫∂ ⎪ ⎪= +⎨ ⎬∂ ⎪ ⎪−⎩ ⎭

(18)

Substituting 3m Ua ε= and Eq. 18 into Eq. 15, we have

2 3 4412

8X U a εβρ ε β

δ−⎛ ⎞≈ π +⎜ ⎟

⎝ ⎠ (19)


251

2 2 43(6 )

2 4XXC

U aεε

ρ δ= ≈ +

π (20)

Again, this result looks alike Eq. 5, that is the result for a dipole behind a circular cylinder, and the difference is mainly due to the number of dimension of the space. If ( )4Oε δ= , two terms on the right-

hand-side of Eq. 20 are of the same order. The first term on the right-hand-side of Eq. 20 is due to the part of the induced velocity by the dipole situated at the origin, and the second by the image dipole located at the inverse point. 4 DISCUSSIONS AND CONCLUSIONS So far it has been shown how the resistance increase due to the propulsor and other devices in the stern region can be estimated in terms of two parameters, namely its strength and closeness to the body Comparing Eq. 1 and 5, as the order of singularity gets higher from a source to a dipole the order of magnitude of ε , for which two terms consisting the force upon the body are of the same order, increases by two, and this trend is confirmed also in the three-dimensional cases, i.e. from Eq. 14 and 20. Furthermore, as the number of dimension changes from two to three, the order of ε increases by one, and this is confirmed from the comparison of Eq. 1 and 14, and also from Eq. 5 and 20. If the role of a propulsor behind a ship is in essence not much different from that of a dipole behind a sphere, among the results we have shown above Eq. 20 is the closest to the reality. However, all the results for a sink and a dipole, i.e. Eq. 1, 5, 14 and 20 have the same structure, namely the force upon a body due to an external singularity is in one part proportional to its own strength and in the other given by the interplay of the strength and its closeness to the body. This latter effect is proportional to the square of the strength and inversely to some power of the closeness. In practice, the strength of the propulsor or other appendages in the stern region is determined by other design requirements, so that we may take it as given a priori. Then Eq. 20 dictates that as we put the propulsor closer to the hull, the resulting resistance increase becomes inversely proportional to the fourth power of the parameter δ . As we have shown for the two-dimensional problem, we can derive results for a wide variety of three-dimensional problems. Various combinations of singularities, symmetric arrangements and Azipod applications etc. can be carried out by essentially the same method. For three-dimensional flows which are not axisymmetric, we have Weiss’s sphere theorem[4]

for obtaining the velocity potential, and the rest is the same as before.

Dealing with problems like this, it is worth to point out that the same method we have used indeed can produce exact results, however, if it is hard to grasp the physics of the solution from those exact results, it is practically of no use to the designer. Arrangement in the stern region needs be determined in the early stage of the design, and what is needed in the initial design stage is not the exact numbers but the overall trend of relevant physical quantities as simple functions of other design parameters as possible. In this respect, it seems that the current method and results shown above are simple but still versatile enough to be justified. On the other hand, although the treatment so far offers a basic understanding of the resistance increase due to the devices in the stern region, we can still extend the method by considering more ship-like bodies, namely an ellipse in two-dimension and an ellipsoid in three-dimension. For two-dimensional problem we have a powerful tool, the conformal transform, and for three-dimensional one we can make use of ellipsoidal harmonics. Studies in this respect are undergoing. REFERENCES [1] Ship Hydrodynamics Research Committee of SNAK.

Resistance and propulsion of ships, Jisungsa, 2009: 163, in Korean.

[2] Lee S J. Characteristics of forces upon a two-dimensional circular cylinder by exterior singularities. J. SNAK, 2010, in process, in Korean.

[3] Yih C S. Fluid mechanics; A concise introduction to the theory. McGraw-Hill Book Company, 1969, Ch.4 §19, §22.

[4] Milne-Thomson L M. Theoretical Hydrodynamics. 5th Ed., Macmillan & Co. Ltd., 1968, Ch. 8 §63, Ch. 16 §42, Ch.

§13. APPENDIX In the proof of the Lagally theorem given in the literature[e.g. 4] usually it is assumed that the velocity field ( )21 /u O R= as R → ∞ . This assumption,

however, excludes the application of the theorem to the flow field generated when a body is in the uniform flow. Here, it is shown that the theorem can be proved without such assumption for the far field. First, we consider a flow consisting of the uniform flow in the x − direction and a source at a finite distance from the origin. Then we introduce a body into this flow field so that the origin is internal and the source is external to the body, and we may assume that at the far field the influence of the body is at most like that of a point dipole at the origin. Let 0S be the surface of the body, and 1S be a sphere of infinitesimal radius round the source, and 2S be a


252

Fig. 3 Schematics for the proof

sphere of very large radius, and V be the volume exterior to 0S and 1S but interior to 2S [Fig. 3]. If we

define ( )212

J nu u u n= − + i , where n is the unit

outward normal vector to the surface, and u u= , Gauss’s theorem gives

2

0d 0

jj S

J S=

=∑ ∫ (21)

Thus we have

0 1 2

d dS S S

J S J S+

= −∫ ∫ , (22)

and for the far field we may assume 2Ru iU e R−≈ + ,

where i is the unit vector in x − direction, and Re is the unit vector in the radial direction. Consequently we can get

2 2

22d d 4

S S

UJ S R i UiR

= Ω = π∫ ∫ (23)

Since 0u n =i on 0S , so that we have 212

J nu= − ,

the integral over 0S times the density of the fluid ρ gives the force F upon the body due to the hydrodynamic pressure. Therefore, we obtain

1

d 4S

F J S Uiρ

= − − π∫ . (24)

For the integral over 1S we may assume that 2

R iu e R u−≈ + , where iu is the induced velocity vector at the point of the source except the contribution by the source itself. Thus we have

1 2

22d d 4i

iS S

uJ S R uR

= − Ω = − π∫ ∫ , (25)

and finally we get ( )4 iF Ui uρ= π − + (26)

of which the x − component is ( )4 iX U uρ= π − + (27)

where iu is the x − component of iu . If we take the strength of the source into account, we get Eq. (11). When there are more than one source outside the body the extension is obvious and the term iu will be

replaced by ( )1

N

i jj

u=∑ , where ( )i ju is the induced

velocity at the point of thj source except the contribution by the thj source itself. As to the dipole, since it is necessary to assume for the far field 3( )u iU O R−≈ + the integral over 2S vanishes, and exactly by the same manner described in [4] we can show that ( )4 iF e uσρ= π ∇i (28)

where eσ is the unit vector in the direction of the dipole axis. Now, the x − component of Eq. 28 is

4 iuX ρσ

∂= π∂

, (29)

and taking the strength of the dipole into account, we get Eq. (15). In fact, the proof given above follows the way taken for the two-dimensional flows, for which the theory of analytic functions of a complex variable can be applied.

Forces due to exterior singularities upon 2- and 3-dimensional bodies

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