Joukowski Theorem for Multi-Vortex and Multi-Airfoil Flow (A

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Provided by Elsevier - Publisher Connector Chinese Journal of Aeronautics, (2014),27(1): 34–39

Chinese Society of Aeronautics and Astronautics & Beihang University Chinese Journal of Aeronautics

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Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil ﬂow (a lumped vortex model)

Bai Chenyuan, Wu Ziniu *

School of Aerospace, Tsinghua University, Beijing 100084, China

Received 5 January 2013; revised 20 February 2013; accepted 25 February 2013 Available online 31 July 2013

KEYWORDS Abstract For purpose of easy identification of the role of free vortices on the lift and drag and for Incompressible flow; purpose of fast or engineering evaluation of forces for each individual body, we will extend in this Induced drag; paper the Kutta–Joukowski (KJ) theorem to the case of inviscid flow with multiple free vortices and Induced lift; multiple airfoils. The major simplification used in this paper is that each airfoil is represented by a Multi-airfoils; lumped vortex, which may hold true when the distances between vortices and bodies are large Vortex enough. It is found that the Kutta–Joukowski theorem still holds provided that the local freestream velocity and the circulation of the bound vortex are modified by the induced velocity due to the outside vortices and airfoils. We will demonstrate how to use the present result to identify the role of vortices on the forces according to their position, strength and rotation direction. Moreover, we will apply the present results to a two-cylinder example of Crowdy and the Wagner example to demonstrate how to perform fast force approximation for multi-body and multi-vortex problems. The lumped vortex assumption has the advantage of giving such kinds of approximate results which are very easy to use. The lack of accuracy for such a fast evaluation will be compensated by a rig- orous extension, with the lumped vortex assumption removed and with vortex production included, in a forthcoming paper. ª 2014 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license.

1. Introduction unseparated flow. In this hypothesis, viscosity is explicitly ignored but implicitly incorporated in the Kutta condition (see 1,2 For a two-dimensional incompressible flow around a single air- Refs. ). The Kutta condition imposes a circulation or a bound foil with a sharp trailing edge at incidence, it is well known that vortex attached to the airfoil and creates a starting vortex, of an the Kutta–Joukowski (KJ) hypothesis holds at least for steady opposite sign, which moves in the downstream direction. The lift thus predicted by the Kutta–Joukowski theorem within the framework of inviscid flow theory is quite accurate * Corresponding author. Tel.: +86 10 62784116; fax: +86 10 even for real viscous flow, provided the flow is steady and 62772480. unseparated. E-mail address: [email protected] (Z. Wu). There are a number of applications where we encounter Peer review under responsibility of Editorial Committee of CJA. multiple vortices and multiple airfoils. Streitlien and Trian- tafyllou3 considered a single Joukowski airfoil surrounded with point vortices convecting freely and derived a force for- Production and hosting by Elsevier mula. In this formula the force is related to the relative velocity

1000-9361 ª 2014 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license. http://dx.doi.org/10.1016/j.cja.2013.07.022 Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil flow (a lumped vortex model) 35 4,5 of the point vortices. Smith et al. studied multi-blade flows L ¼qðV1 þ usÞCb ðGeneralized KJðaÞÞ ð2Þ with interaction. Panel methods have also been developed D ¼ qvsCb for multi-element airfoils (see Ref. 6). Crowdy7 found the complex potential to calculate the lift for a finite stack of cylinders where (us, vs) is the velocity at the (center of the) current airfoil with imposed circulation. Force decomposition for multi-body induced by an outside vortex. flows has also been considered using an integral approach (see Now, we consider M free vortices, each with a position Refs. 8,9). None of this work considered the force formula for (xj, yj) and circulation Cj for j =1,2,, M around an airfoil each airfoil in the form of the Kutta–Joukowski theorem. Wu represented by a bound vortex of circulation Cb at the center of et al.10 extended the Lagally theorem to multi-body flows with the airfoil. A point vortex j, whether it is a free or bound vortex, induces a velocity at (x, y) given by: free point vortices, but not considering the existence of any 8 bound vortex. > Cj y yj > v ðx; yÞ¼ Through finding the complex potential and using the <> x 2 2 2p ðx xjÞ þðy yjÞ Blasius theorem, Katz and Plotkin6 (see chapter 6.9, general- ð3Þ > C x x ized Kutta–Joukowski theorem) developed a generalized :> v ðx; yÞ¼ j j y 2p 2 2 Kutta–Joukowski theorem for an airfoil in interaction with ðx xjÞ þðy yjÞ another airfoil represented by a lumped vortex of opposite It is well-known that a single point vortex in relative mo- circulation. tion, at the velocity (dxj/dt,dyj/dt), to the fluid offers a force For problems with multiple airfoils and multiple free vorti- to the body given by11: ces, it is always possible to derive an exact theory or use 8 numerical computation to obtain the forces on each body. > dxj <> Lj ¼q V Cj However, for easy identification of the role of each free vortex 1 dt ðSaffmanÞð4Þ and for fast evaluation of force approximation as required by > :> dyj engineers, we require the force formulas to be explicit, captur- Dj ¼q Cj dt ing the main physics and easy to use, almost as simple as the classical Kutta–Joukowski theorem. For this purpose, we In the case of a bound vortex (bounded to a body) sur- make an extension of the generalized Kutta–Joukowski theo- rounded by M free vortices, the forces due to each vortex rem to the case of multiple airfoils with multiple free vortices, sum up to give: 6 8 using the lumped vortex assumption as by Katz and Plotkin. > XM > dxj In this assumption each airfoil is represented by a lumped vor- > L ¼qV Cb q V Cj < 1 1 dt tex at the position of the center of the airfoil. The accuracy of j¼1 ð5Þ > XM this assumption will be assessed in conclusion. The extension > dy > D ¼q j C to multiple airfoils without the lumped vortex assumption will : dt j be considered in a forthcoming paper. This paper will be orga- j¼1 nized as follows. In Section 2, we derive force formula in terms The force formulas Eq. (5) or formulas in similar forms of induced velocity for a single airfoil interacting with multiple have been frequently used to study the interaction of a cylinder free vortices. In Section 3 we extend the force formula to the with M free vortices (see Ref. 12 for more details and case of multiple body in such a way that the force formula references). holds individually for each airfoil. Supplemented by a two cyl- Now we relate the velocity (dxj/dt,dyj/dt) for each vortex to inder example and the Wagner problem, which are presented the induction by other vortices to find an explicit and new in Appendix A and B respectively, we will assess the accuracy force formula through Eq. (5). and usefulness of the present results in Section 4. In Section 4, 2 2 2 2 2 With rij ¼ðxj xiÞ þðyj yiÞ and rjb ¼ðxj xbÞ we will also discuss how to use the present results to identify 2 þðyj ybÞ , the velocity of the free vortices due to free the role of outside vortices on the lift and drag. convection induced by the bound vortex and other free vortices is: 2. Single airfoil with multiple free vortices 8 > XM > dxj Cbðyj ybÞ Ciðyj yiÞ > ¼ V < dt 1 2pr2 2pr2 Consider an incompressible two-dimensional flow around an jb i¼1;i–j ij ð6Þ > XM airfoil with a velocity field v =(u, v)atconstantdensityq,in > dy C ðx x Þ C ðx x Þ > j ¼ b j b þ i j i an unbounded domain Rf. The freestream velocity V1 is : dt 2pr2 2pr2 assumed horizontal. The circulation of the bound vortex is jb i¼1;i–j ij o defined as Cb =oA(udx + vdy) for the closed curve A along Inserting the above equations into the force formula the airfoil, with a counterclockwise path, so that a clockwise Eq. (5), we obtain: circulation has a negative sign. The classic Kutta–Joukowski 8 > XM XM XM theorem expresses the lift (L)andthedrag(D) per unit span > CbCjðyj ybÞ CjCiðyj yiÞ > L ¼qV C q q <> 1 b 2 2 as: 2prjb – 2prij j¼1 j¼1 i¼1;i j L ¼qV1Cb > XM XM XM ðKJ theoryÞð1Þ > CbCjðxj xbÞ CjCiðxj xiÞ :> D ¼q þ q D ¼ 0 2pr2 2pr2 j¼1 jb j¼1 i¼1;i–j ij Katz and Plotkin6 showed that for an airfoil in interaction with another airfoil, represented by a lumped vortex of The last terms on the right hand sides in both forces are due opposite circulation, the force for the airfoil is given by: to mutual interactions between the free vortices. For each pair 36 C. Bai, Z. Wu

2 2 i, j with i „ j, CjCiðyj yiÞ=rij þ CiCjðyi yjÞ=rji ¼ 0 holds, and vs) along this contour remains the same whether the vor- hence: tices outside of this contour are bound (fixed) or free. 8 For the problem considered here, the pressure is related to > XM y y C > qCb ð j bÞ j the velocity through the unsteady Bernoulli equation > L ¼qV1Cb < 2p x x 2 y y 2 j¼1 ð j bÞ þð j bÞ 1 2 2 p ¼/t ðu þ v ÞþC. The only difference between the > XM > qC ðx x ÞC 2 > D ¼ b j b j pressure along the contour oB for an outside FV and an : 2p 2 2 j¼1 ðxj xbÞ þðyj ybÞ outside BV, both supposed to have a circulation C which in- which can be more conveniently written as: duces a velocity potential / = Ch/(2p), comes from the term 8 /t. We have /t „ 0 for an outside FV and /t = 0 for an outside > L ¼qðV1 þ usÞCb o > BV. However, for a closedH curve B not enclosing this BV or > > XM FV, we always have /n dl ¼ 0(n is the unit outward > Cjðyj ybÞ @B!0 > us ¼ normal and l is a vector along the tangent direction) since < 2p½ðx x Þ2 þðy y Þ2 j¼1 j b j b the potential /, which is due to an outside vortex, has no sin- > ðGeneralized KJðbÞÞ > D ¼ qvsCb gularity inside oB. Therefore, there is no difference in the inte- > > grated force due to this term. Moreover, this term is linear so > XM > Cjðxj xbÞ that the conclusions still hold when there are many other out- : vs ¼q 2 2 j¼1 2p½ðxj xbÞ þðyj ybÞ side vortices (BV or FV). ð7Þ Hence the integrated forces, along the contour oB, are the same if some of the free vortices outside of oB are replaced Here us and vs are exactly the horizontal and vertical veloc- by bound ones. This means that the force formula Eq. (7) still ity components, at the location of the bound vortex, induced holds for each airfoil individually in the presence of other air- by all the free vortices. The formulas Eq. (7) are thus an exten- foils represented by bound vortices. Now we can write the sion of the generalized Kutta–Joukowski theorem Eq. (2) to above force formulas for a multi-airfoil case. For the ith the case of multiple free vortices. Remark that by using the airfoil, Blasius equation based on complex variables and using Resi- ( due theorem, Wu et al.10 obtained a similar force formula Lb;i ¼qðV1 þ us;iÞ i for the case without bound vortex. ðGeneralized KJðcÞÞ ð8Þ Db;i ¼ qvs;i i 3. Multiple airfoils with multiple free vortices where us,i = ub,i + uf,i and vs,i = vb,i + vf,i are the total induced velocity components at the ith airfoil due to other As displayed in Fig. 1, we consider N airfoils (assumed non- airfoils and free vortices. Here: rotating and fixed in this paper) interacting with M free vortices of given circulation C with m =1,2,, M. The nth air- XN XN m yn yi xn xi ðbÞ ðbÞ ub;i ¼ n; vb;i ¼ n ð9Þ foil at location ðxn ; yn Þ has an unperturbed circulation Cb,n, 2 2 – 2prin – 2prin i.e., a circulation in an oncoming uniform flow without other n¼1;n i n¼1;n i airfoils, supposed to be known from traditional theories. and For a single airfoil with a bound vortex (BV) interacting XM XM with a number of free vortices (FVs), we have proved, without y y x x u ¼ m b;i C ; v ¼ m b;i C ð10Þ working with pressure, the force formula Eq. (7). Hence using f;i 2pr2 m f;i 2pr2 m pressure integration along the bound vortex (namely, a con- m¼1 im m¼1 im tour oB of vanishing size just enclosing the bound vortex For the ith airfoil, i is the perturbed circulation of the and excluding the free vortices) also leads to the same formula. bound vortex, that is circulation of the bound vortex The instantaneous flow pattern (thus the induced velocities us subjected to influence of outside vortices and bodies. The circulation of the bound vortex without perturbation of outside vortex is determined by the Kutta condition and

Cb,i = pcAiV1Æsin(ai + /Bi), in which cAi is the chord length and /Bi is the zero lift angle of attack. When there are outside vortices, supposed to be far enough so that the lumped vortex assumption holds true, thenq theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi local freestream velocity must 2 2 be modified to be VL;i ¼ ðV1 þ us;iÞ þ vs;i and the local

angle of attack ai changed to ai + as,i (as,i is the effective angle of attack) with tan as,i = vs,i/(V1 + us,i), then we may write for each i the perturbed circulation:

VL;i sinðai þ as;i þ uBiÞ i ¼ Cb;i ð11Þ V1 sinðai þ uBiÞ Eqs. (9)–(11) now form a closed system to give the required i and us,i, vs,i for obtaining the forces through Eq. (8), and can Fig. 1 A system with multiple free vortices and multiple airfoils. be solved simply using an iterative process. Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil ﬂow (a lumped vortex model) 37

Under lumped vortex assumption we may assume us,i, vs,i to with the exact solution by Crowdy, provided the distance be small compared to V1, hence we obtain a system of N lin- between the two cylinders is large enough (so that lumped vor- earized equations for solving i. In this case: tex assumption holds). In Appendix B, we study the Wagner problem which is for an impulsively started airfoil with vortex ¼pc ðV þ u Þða þ v =V þ / Þð12Þ i Ai 1 s;i i s;i 1 Bi shedding. With the present method and with the shed vortex For vertically aligned airfoils without considering free vorti- represented by a single vortex starting at the trailing edge, ces, we have vs,i = 0 and as,i = 0, so that i ¼ Cb;ið1 þ us;i=V1Þ, we obtain quickly the force formula which compare well with and then we have from Eqs. (9) and (12): Wagner exact solution. For problems with body vortex interactions, which may XN C b;i ¼ C ð13Þ occur in many applications, the fast and easy identification i 2pV ðy y Þ n b;i n¼1;n–i 1 n i of the role of each individual vortex on the lift and drag is very important for correct interpretation of the physics of flow. A Similarly, for horizontally aligned airfoils without consider- typical example is the role of leading edge vortex. ing free vortices, we have u = 0 and ¼ C þ pc m , and s,i i b;i Ai s;i For multibody and multivortex flows, an exact but explicit then: force evaluation for each individual body is usually very complicated. Sometimes we must use numerical computation to do XN c þ Ai ¼ C ð14Þ this. However, using the present method, we may be able to i 2 x x n b;i n¼1;n–i ð n iÞ obtain force formulas without much effort and these force formulas may have an accuracy for qualitative study. This could be useful in engineering applications. 4. Applications and conclusion According to Appendix A, the evaluated force is accurate only when the distance between the bodies is large enough. Under lumped vortex assumption, for which an airfoil is rep- This is due to the lumped vortex assumption, for which the resented by a bound vortex at the center of the airfoil, we have shape of the body and image vortices are ignored. The study extended the Kutta–Joukowski theorem to the case of multiple of force formulas without lumped vortex assumption has also airfoils with multiple free vortices. The advantage of the pres- been carried out. For more details, see Appendix C. ent result is that one can first work with each airfoil individually, and then add interaction through the use of algebraic Eqs. (8)–(14). As discussed below, the usefulness of the present re- Acknowledgement sults lies in easy identification of the role of outside vortices and fast evaluation of approximate forces for relative complex This work was supported by National Basic Research Program problems. of China (2012CB720205). With the lift force formula Eqs. (8)–(10), we may identify the role of each free vortex. First consider the case when the Appendix A. Crowdy two-cylinder example free vortex is far enough from the airfoil, so that i Cb;i (which means that the circulation of the bound vortex is not perturbed by the free vortex). Then, if the bound vortex and Crowdy7 considered a potential flow for two vertically aligned the outside free vortex have opposite sign, the induced velocity circular airfoils (i = 1 for the lower airfoil and i = 2 for the by a free vortex yields a lift increase if this free vortex is above upper one) separated by a distance h with unit diameter, den- the airfoil, and a lift decrease when it is below; moreover, it in- sity, and velocity (q = cA = V1 = 1) and with fixed circula- duces a drag if it is downstream of the airfoil, and propulsion if tions Cb,1 = Cb,2 = 5. When h is large enough, the circular it is upstream of the airfoil. cylinders can be represented by lumped vortices so that the If the bound and free vortices have the same sign, then the present method applies. In the case of fixed circulations, the induced velocity by the free vortex yields a lift decrease if this use of Eqs. (8)–(10) yields: vortex is above the airfoil, and a lift increase when it is below; 8 moreover, it induces a propulsion if it is downstream of the air- > 1 25 > L ¼q 1 þ C C ¼ 5 foil, and drag if it is upstream of the airfoil. < b;1 2ph b;2 b;1 2ph The above analysis may not be true when the lumped ðA1Þ > vortex assumption is not violated. For instance, in the case :> 1 25 Lb;2 ¼q 1 Cb;1 Cb;2 ¼ 5 þ of a leading edge vortex very close to the airfoil so that the 2ph 2ph airfoil can not be simplified by a lumped vortex, the induced flow field is nonlinear and image vortex effect can If the circulations of the bound vortices are allowed to vary not be neglected, the strength of the bound vortex will be according to Eq. (13) due to satisfaction of the Kutta condi- significantly increased according to more elaborated studies Cb;1 Cb;2 tion, then 1 2 ¼ Cb;1 and 2 þ 1 ¼ Cb;2, which not presented here. 2ph 2ph For fast evaluation of forces, we consider two examples. can be solved to give: One is the two-cylinder example of Crowdy, with given bound 8 1 vortices. The other is the classical Wagner problem. For both > 5 5 5 > ¼51 1 þ examples, it is extremely complicated to obtain explicit force < 1 2ph 2ph 2ph formulas using known theories. ðA2Þ > 1 In Appendix A, we use Eqs. (8)–(10) to obtain the forces for > 5 5 5 : 2 ¼51þ 1 þ both cylinders. We remark that the lift forces compare well 2ph 2ph 2ph 38 C. Bai, Z. Wu

where C1 = pcAV1a is the circulation of the bound vortex when the starting vortex has moved infinitely downstream to the airfoil. Then, with Eq. (7), the lift and (induced) drag are found to be: 8 > 1 > 1 <> L ¼ 1 þ L1 1 þ 2V1t=cA > 1 > q 1 2 : D ¼ 1 þ C1 Fig. A1 Lift forces on each airfoil for the un-staggered biplane pcA þ 2pV1t 1 þ 2V1t=cA configuration. where L1 = qV1C1a is the final lift after a steady state is reached. Thus at t = 0, we have: According to the force formulas Eqs. (8)–(10), which are re- 8 <> 1 1 L ¼ L1 duced here to Fb;1 ¼qð1 þ Cb;2Þ 1 and Fb;2 ¼q 2 2ph :> p 2 2 1 D ¼ cAV a ð1 C Þ , we have: 4 1 2ph b;1 2 8 Thus even with the lumped vortex approximation we can > 1 > 5 5 5 5 recover the well known results of Wagner (see Refs. 11,14) that <> Fb;1 ¼ 1 51 1 þ 2ph 2ph 2ph 2ph for an impulsively starting flow of a flat plate, the initial lift is > ðA3Þ exactly one-half of the final lift (the Wagner effect) and the ini- > 5 5 5 5 1 :> p 2 2 Fb;2 ¼ 1 þ 51þ 1 þ tial drag is D ¼ c V a . Hence the lumped vortex approach, 2ph 2ph 2ph 2ph 4 A 1 though approximate but quite simpler to use than the Wagner The lift forces as a function of h are displayed in Fig. A1. approach based on the Laplace transform, gives reasonably When h > 3, the present result (see Curve b in Fig. A1) good results even for the initial behavior. We remark that determined by Eq. (A1) compares well with that of Crowdy the initial lift has also been studied for airfoils with thickness (see Curve c in Fig. A1), in the case of fixed circulations. When effect (see Refs. 14,15). h < 3, for which the cylinders can no longer be represented by lumped vortices, the present theory is no longer valid. When the circulations are allowed to depend on the distance accord- Appendix C. Extension of the results ing to Eq. (A3), the lift forces (see Curve a in Fig. A1) are high- er than those given by the fixed circulation model. By using a special momentum approach and with the help of interchange between singularity velocity and induced flow Appendix B. Wagner problem velocity, we have recently derived in a physical way the explicit force formulas for two-dimensional inviscid flow involving Consider the well-known problem of Wagner13, which is the multiple bound and free vortices, multiple airfoils and vortex 16 initial lift for a flat plate of length cA, set instantaneously into production , without use of lumped vortex assumption so that motion with a constant velocity V1 and at a small angle of at- the force formulas are exact for arbitrary distance between tack a. For this problem, an accurate and thus mathematically bodies. As in the present work, these new force formulas hold difficult solution is required to consider a vortex sheet down- individually for each airfoil thus allowing for force decompo- stream to the trailing edge. Here we consider the lumped vor- sition and the contributions to forces from singularities (such tex approximation, by assuming a bound vortex of circulation as bound and image vortices, sources and doublets) and bodies Cb(t)atx = xb, the center of the plate, and a starting vortex of out of an airfoil are related to their induced velocities at the circulation Ca(t)=Cb(t) downstream to the airfoil at x = xs. location of singularities inside this airfoil. The force contribu- The velocity induced at x = xb by the starting vortex is tion due to vortex production will related to the vortex produc- CsðtÞ tion rate and the distance between each pair of vortices in ðus; vsÞ¼ð0; Þ. Hence according to Eq. (12), 2pðxs xbÞ production. For the Crowdy example, the comparison will be good even the distance between the two cylinders is small. C ðtÞ C ¼pc V a þ s We remark that force formulas for multibody problems and b A 1 2pV ðx x Þ 1 s b in the form of induced velocities have been also studied by 10 CbðtÞ Wu et al. They followed Lagally theorem to obtain the forces ¼pcAV1 a þ 2pV1ðxs xbÞ and their results are restricted to problems without bound vortices and without vortex production. The results of Bai et al.16, If at the initial time t = 0, we assume the starting vortex is just which are obtained through a different approach, are more at the trailing edge (x (0) = c ) and the bound vortex is at the s A general since bound vortex and vortex production are 1 center of the airfoil x ¼ c , and then with dx /dt = V considered. b 2 A s 1 and thus xs = cA + V1t, the above equation can be solved to give: References 1 1 cA 1 1. Batchelor FRS. An introduction to fluid dynamics. Cam- CbðtÞ¼ 1 þ C1 ¼ 1 þ C1 bridge: Cambridge University Press; 1967. 2ðxs xbÞ 1 þ 2V1t=cA Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil flow (a lumped vortex model) 39

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