CORE Metadata, citation and similar papers at core.ac.uk 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 [email protected] www.sciencedirect.com 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 out- side 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 demon- strate 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 ig- nored 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 com- plex 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,ÁÁÁ, 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 vor- tex, induces a velocity at (x, y) given by: free point vortices, but not considering the existence of any 8 bound vortex. > Cj y À yj > v ðx; yÞ¼À Á Through finding the complex potential and using the <> x 2 2 2p ðx À xjÞ þðy À yjÞ Blasius theorem, Katz and Plotkin6 (see chapter 6.9, general- ð3Þ > C x À x ized Kutta–Joukowski theorem) developed a generalized :> v ðx; yÞ¼ j Á j y 2p 2 2 Kutta–Joukowski theorem for an airfoil in interaction with ðx À xjÞ þðy À 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 À 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 À q V À 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 À xiÞ þðyj À yiÞ and rjb ¼ðxj À xbÞ we will also discuss how to use the present results to identify 2 þðyj À 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 À ybÞ Ciðyj À yiÞ > ¼ V À À < 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 À x Þ C ðx À 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 =òoA(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 À ybÞ CjCiðyj À yiÞ > L ¼qV C À q À q <> 1 b 2 2 as: 2prjb – 2prij j¼1 j¼1 i¼1;i j L ¼qV1Cb > XM XM XM ðKJ theoryÞð1Þ > CbCjðxj À xbÞ CjCiðxj À 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.