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1 Electronic Transactions on umerical Analysis. Volume 3, pp , 8. Copyright 8,. SS TERACTO OF COMPRESSBLE FLOW AD A MOVG ARFOL MART RŮŽČKA, MLOSLAV FESTAUER, JAROMÍR HORÁČEK, AD PETR SVÁČEK Astract. The suject of this paper is the numerical simulation of the interaction of two-dimensional incompressile viscous flow and a virating airfoil. A solid airfoil with two degrees of freedom can rotate around an elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element solution of the avier-stokes equations coupled with a system of ordinary differential equations descriing the airfoil motion. The time-dependent computational domain and a moving grid are taken into account with the aid of the Aritrary Lagrangian-Eulerian formulation of the avier-stokes equations. High Reynolds numers require the application of a suitale stailization of the finite element discretization. umerical tests prove that the developed method is sufficiently accurate and roust. The results are compared with experiments. Key words. aeroelasticity, avier-stokes equations, aritrary Lagrangian-Eulerian formulation, finite element method, stailization for high Reynolds numers AMS suject classifications. 5M, 7M1, 7D5 1. ntroduction. The interaction of fluid flow with virating odies plays a significant role in many areas of engineering. We can mention, for example, development of airplanes or turines and some prolems from civil engineering see, e.g., [. n the design of airplanes the point of interest is the analysis of deformations and virations of wings induced y flowing air. n this paper we are concerned with numerical solution of an aeroelastic prolem of two dimensional viscous incompressile flow over an airfoil with two degrees of freedom in a wind tunnel. The airfoil is represented y a solid ody, which can perform vertical and torsional virations. A mathematical model of the flow is formed y the system of twodimensional non-stationary avier-stokes equations and the continuity equation, equipped with initial and mixed oundary conditions. Due to the moving airfoil, the computational domain is time-dependent. This requires the use of a suitale technique for the simulation on a moving computational grid. Here we apply the Aritrary Lagrangian-Eulerian (ALE method [11. The flow prolem is discretized y the stailized finite element method, which leads to a large system of nonlinear algeraic equations. We overcome the nonlinearity y the use of the Oseen linearization, resulting in a sequence of linear systems of saddle-point type. They are solved y the direct solver UMFPACK [3. The numerical results are compared with wind tunnel measurements.. Mathematical model. We consider two-dimensional non-stationary viscous incompressile flow past a virating airfoil inserted into a channel (wind tunnel in a time interval, where. The symol denotes the computational domain occupied y the fluid at time. The oundary of the domain consists of mutually disjoint sets and on which different types of oundary conditions are prescried. By we denote impermeale walls and the inlet, through which the fluid flows into the domain, denotes the outlet, where the fluid flows out and is the oundary of the profile at the time. n Received ovemer 7, 7. Accepted for pulication June 3, 8. Pulished online on Feruary, 9. Recommended y O. Steinach. Charles University Prague, Faculty of Mathematics and Physics, Sokolovská 83, Praha 8, Czech Repulic (mart.in.ruza@seznam.cz, feist@karlin.mff.cuni.cz. nstitute of Thermomechanics of the Academy of Sciences of the Czech Repulic, Dolejškova 5, 18 Praha 8, Czech Repulic (jaromirh@it.cas.cz. Czech Technical University Prague, Faculty of Mechanical Engineering, Karlovo n. 13, Praha, Czech Repulic (svacek@marian.fsik.cvut.cz. 13

2 C M C 14 M. RŮŽČKA, M. FESTAUER, J. HORÁČEK, AD P. SVÁČEK FG..1. Model scheme. contrast to!, we assume that and are independent of time. The flow is characterized y the velocity field $%(', and the kinematic pressure *+*,', for '.-/ 5 and -. The kinematic pressure is defined as 143, where 1 is the pressure and 3 is the constant fluid density. The motion of the profile is descried y functions 78, representing the rotation around the elastic axis TR, and 9:,, denoting the vertical displacement see Figure ALE formulation of the avier-stokes equations. The time dependent computational domain can e treated with the aid of a smooth, one-to-one ALE mapping [11 (.1 DC =< 7>@?AB CF?AE' G H J The C coordinates of points 'K-. are called spatial coordinates, the coordinates of points -L7> are called ALE coordinates or reference coordinates. The ALE mapping reflects the deformation of the computational domain see Figure.. We define the domain velocity in the following way M CF G C$ (. O ' This velocity can e expressed in spatial coordinates as (.3 Let us consider a function UVWU!(' real numers. Let Z y (.4 7- (' G QP LRS,'G T J C$ '/-V X- U!,' 7-LY, where Y is the set of U! [U! \. We define the ALE derivative of the function U _^ U!,' = M U ` CF DC RS,'G

3 o o q p r ~ p p t p s ~ TERACTO OF COMPRESSBLE FLUD AD A MOVG ARFOL 15 FG... The comparison of Lagrange(left and ALE(right transformations. The application of the chain rule gives _^ (.5 _^ Ua U `c edgf U By using this relation we can otain the avier-stokes equations in the form (. 8:h dif div/ f *_hkjlm/ in in The symol j denotes the kinematic viscosity of the fluid. We assume that j n is constant... Equations for the moving airfoil. The equations of airfoil motion are derived from the Lagrange equations for the generalized coordinates 9 and. These equations have the form [ (.7 where the stiffness matrix wv$xzy7y x} {y x{y x} { Q~ op, :q 8 :s, the viscous damping q v y7y zy The force vector u and the generalized coordinates u8 Gv hˆ, Š 8 8 =.u8 and the mass matrix s y7 } ~ read, Gv 9:, 78 vƒ have the form Q~ The symol ˆ stands for Š the component of the aerodynamic force acting on the profile in the vertical direction, is the torsional aerodynamic moment with respect to the elastic

4 Š j 1 M. RŮŽČKA, M. FESTAUER, J. HORÁČEK, AD P. SVÁČEK axis,,,, are the coefficients of structural { damping, and denote the static moment around the elastic axis TR, the moment of inertia around TR, the mass of the profile and the stiffnesses of the profile elastic support. acting on the airfoil are given y the relations y7y y {y x yxy x } x y x {y The force ˆ Š and moment ê$h Œ}}Ž ` O (.8 % H S Œ }Ž ` 8 H šhœg P( S ž Ÿ 8 h O S ž Ÿ, T O (.9 S Here is the airfoil depth in the direction orthogonal to the plane S, representing the length of a wing segment in consideration. Further, is the unit outer normal to on, is the Kronecker symol, i.e., $ for! and for ª : S are the coordinates of points on,, =«, are the coordinates of the elastic axis ', and (.1 53 h* j v `±.3. nitial and oundary conditions. The avier-stokes equations are completed y the initial condition ³ W > ' - > and the following oundary conditions. On we prescrie the Dirichlet oundary condition { (.1 µ n7 `± On the outlet we consider the so-called do-nothing oundary condition hn *_h *O ¹šº}» ` (.13 on ` where *` ¹šº is a given reference pressure. On we consider the condition Ž (.14 µ Moreover, we equip system (.7 with the initial conditions ( : =¼> X r =W S r GW9½> 9: GW9 S where > 9½> 9 S are input parameters of the model. The initial value prolem (.7, (.15 is transformed to a prolem for a first-order system and then discretized y the fourthorder Runge-Kutta method. The interaction of a fluid and an airfoil consists in the solution of the flow prolem (., (.11 (.14 coupled with the structural model (.7 and (.15. n what follows, we shall e concerned with the discretization of the flow prolem and descrie the algorithm for the numerical solution of the complete fluid-structure interaction prolem. S ~²

5 Ç Ã Ñ Ã ß S Ç Ã ß * Ò Ò S TERACTO OF COMPRESSBLE FLUD AD A MOVG ARFOL Discretization of the flow prolem Discretization in time. We construct an equidistant partition of the time interval, formed y time instants 5š> ¾/ S ¾ d did n ¾, ÀÁ, x» where is a time step. We use the approximation 8 Ã@Ä$ Ã, *8šÃÄW* Ã of the exact solution and, ÃÄ Ã of the domain velocity at time Ã. On each time level Ã žs, under the assumption that the domain 8ÅgÆ is already known, using the second-order ackward difference formula, we otain the prolem to find functions Ã žs < 8ÅgÆ?AEY and * Ã žs < 8ÅgÆ?AEY, such that (3.1 Ã žs héèê Ê Ã RS P, Ã žs h Ã žs dif T Ã žs div Ã žs in 8ÅiÆ h+j l Ã žs f * Ã žs This system is considered with the oundary conditions (.1, (.13, and (.14. The symols Ê and Ê mean the functions Ã and Ã RS transformed from the domain 8Å and Ã RS 8ÅÌË to the domain 8ÅiÆ using the ALE mapping. This means that Í 5 }Î Î R S ÅgÆ. 3.. Discretization in space. The starting point for the approximate solution is the weak formulation of prolem (3.1. Because of simplicity we shall use the notation w 7 ÅiÆ (3. We introduce the forms W Ã žs *Ï:* Ã žs and!ð¼ ÅgÆ. The appropriate function spaces are ÓÒ [,9 (H FÔiÕÉ- Ñ Ö } Ø}}Ž ÚÙ}ÓÛ ÕXµ ¼Ü J Ý ß G 8 Õ j f f Õ, h Ã žs (3.3 dif» Õ háà* f«d Õ f«d á U! G ÈÚÊ ḧ Ê Ã RS Õ ãahkœ }ä * ³¹šº Õ d â where Q8 *` - Ñæå Û. K8 *ç - Ñèå Û.=ß ³á K8Õ éå Û, and d d denotes the scalar product in the spaces Ü (, Ü J, and Ü (. The weak solution is defined as a couple $F u *ç, such that it satisfies the conditions ÌêÚ «- Ñëå Û. Ý ³ß íì ß G¼U! F Há v 7- å Û. (3.4 and satisfies the oundary conditions (.1 and (.14. ow we define an approximate Ñ+ Ò Ñ HÒ ³Û solution. Û The spaces c finite-dimensional suspaces, ï-œ ï >g, ï >, where Ò FÔ Õð- ÑÉ Ö Ø} Ž ÙG Õ=µ The approximate solutions is defined as a couple F, 7- Ñ å Û Ý J ³ß ß GU! ñìéß F,Õ Há Ò 7- å Û (3.5 and are approximated y their satisfies a suitale approximation of the oundary conditions (.1 and (.14., such that Ñ+ ³Û n the construction of the spaces we assume that the domain is a polygonal approximation of the computational domain at time ³Ã žs. By ò (ï¼-«ï >g we denote a triangulation of with standard properties from the finite element method see, e.g., [. Ñð Û Then and are defined as continuous piecewise polynomial functions satisfying the Bauška-Brezzi condition see [1. Here we use the well-known Taylor-Hood 1 Ì1 elements over a triangulation ò in, quadratic on each element óô-lò of. This means that the velocity components are continuous, and the pressure is continuous piecewise linear.

6 Ü A ß 1 Ü Ã 1 j A ß ~ * ù ~ 18 M. RŮŽČKA, M. FESTAUER, J. HORÁČEK, AD P. SVÁČEK 3.3. Stailization of the finite element method. For high Reynolds numers approximate solutions can contain nonphysical spurious oscillations. n order to avoid them, we shall apply the streamline-diffusion and div-div stailization ased on the forms (3. J ù (3.7 where ³ß = = ö{ iø v ö{ iø v J ³ß Ç X khkjlm,è Ê h Ê Ã R S ö{ gø f«d dgf» Qdif»Õ f[d Õ f * dif»õ Há $F8 *çd «8 *ç F,Õ «ú are suitale parameters, ¼ h Ã žs is the transport velocity, and d d is the scalar product in the space Ü (óé or Ü (óé. The solution of the stailized discrete prolem is F, 7- Ñðûå Û, such that the component satisfies the oundary conditions (.1 on and (.14 on, and Ý J ³ß ³ß J ³ß ß G¼U ß (3.8 ìéß [,Õ ³á Ò 7- ûå Û f we solve prolem (3.8, we otain an approximate solution at time HÃ žs, i.e., Ã žs and * Ã žs +* defined in the domain 8ÅiÆ ¼. ow, we descrie how to choose parameters and. We follow the works [7 and [1. The magnitude of the velocity field varies in different sudomains of. That is why we split the domain into two sudomains. The diffusion component dominates on the first sudomain and the convective component on the second. On oth sudomains we choose these parameters in a different way. The parameter is ased on the transport velocity and the viscosity j. We put ï ü ü ýoþ (3.9 where (3.1 ï ü ü ý þ is the so-called local Reynolds numer and ï is the size of the element ó measured in the direction of. The function d is non-decreasing in dependence on in such a way that for local convective dominance g and for local diffusion dominance ¾Qg. The parameter - is chosen suitaly. The function d can e defined, e.g., y (3.11 G The parameters are defined y (3.1 ï =µ µ n practical computations we use the values } and -+ F.

7 È > À ˆ Ç À * Ç TERACTO OF COMPRESSBLE FLUD AD A MOVG ARFOL Simulation of the flow induced airfoil virations. n the solution of the complete coupled fluid-structure interaction prolem we apply the following algorithm: 1 Assume that the approximate solution.«8 of the flow prolem (3.8 at time levels šã R S and Ã is known and the force ˆ and torsional moment Š are computed from (.8 and (.9. Extrapolate ˆ Š and on the time interval Ã šã žs. 3 Compute the displacement 9 and angle at time HÃ žs as the solution of system (.7. 4 Determine the position of the airfoil at time Ã žs, the domain 8ÅiÆ, the ALE mapping 8ÅgÆ and the domain velocity Ã žs. 5 Solve the discrete stailized prolem (3.8 at time šã žs. Compute ˆ Š and from (.8 and (.9 at time Ã žs and interpolate ˆ Š and on Ã Ã žs. 7 s a higher accuracy needed? YES: go to 3 O: < (4.1 where and go to. The nonlinear flow prolem (3.8 is solved with the aid of the Oseen iterations Ý J ß ¼U! À žs ³ß J ß ß for all - À žs ß Ò å Û is defined on the asis of the approximate solution on the previous time level. The ALE mapping is constructed in such a way that the reference domain > is divided in three sudomains y two ellipses with center at the elastic axis of the airfoil. n the interior ellipse containing the airfoil the ALE mapping is defined as the rigid ody motion, outside of the exterior ellipse the ALE mapping is identity. n the domain etween oth ellipses the ALE mapping is defined y interpolation. The knowledge of the ALE mapping at time instants HÃ RS šã J À žs ³ß šã žs allows us to approximate the domain velocity with the aid of the second-order ackward difference formula (4. Ã žs ('GG ' hðè Å R S ÅgÆ ('G ÅÌË RS ÅiÆ ('G ':- 8ÅiÆ FG Anisotropically adapted mesh. 5. umerical solution. The descried method was applied to the simulation of flow induced virations of a profile (shown in Figure 5.1 inserted into a channel (wind tunnel in the case of the following data: j ô d R m ô Úz! z{z$% m/rad x{y7 kgm, s/m s, x{yxy /rad x} zy { kg Fh s/m yxy { Ç ms/rad } y7 m ' length of the profile chord m. {z /m x} { kgm i s/rad {y

8 7 7 j R R 13 M. RŮŽČKA, M. FESTAUER, J. HORÁČEK, AD P. SVÁČEK alpha [. -.5 H [mm time [s time [s FG. 5.. (*,+ and -.*,+ for the inlet velocity of the air / The computation was carried out for several values of the inlet velocity. We define the corresponding Reynolds numer y (5.1 The triangulation of the domain is realized y the method and software AGEER [4, [5, which can e used for the construction of an initial isotropic triangulation and also for an anisotropic adaptive mesh refinement see Figure 5.1. By the numerical solution of the complete prolem we otain the velocity and pressure fields and also the time development of the displacement 9 and the rotation angle. From this information we derive frequency characteristics otained with the aid of the Fourier transform 7 (5.,8!Ã G.Œ9 >: with 9 or and 8!Ã lc8ð- : : rectangle formula (5.3 Here is the imaginary unit and H length. 8ÃOG FE RS À > : 8 < DzÌ 8 ÀÌ< Å BA lc8k ÅÌ,G Hz, approximated y the is the numer of time steps in the interval with The main virational frequencies U S and U points of the function µ µ corresponding to ¼9 and W, respectively. : : fields for a fixed profile in the disequilirium position > and 9 > Ç are defined in our computations as maximum The numerical simulation started y the computation of the flow velocity and pressure mm. After a short time the airfoil was released, i.e., the motion of the airfoil starts to ehave according to system (.7 with initial conditions formed y the aove data!> 9½>, S, and 9 S cf., (.15. n what follows we present the dynamic response 78 9:, of the fluid-structure system in time domain for different inlet flow velocities. nlet velocity J RS hkv. For and 9 we otained the signals $! shown Ç in Fig- Hz and Hz. ure 5.. The frequency analysis gives main frequencies of the signals

9 È TERACTO OF COMPRESSBLE FLUD AD A MOVG ARFOL 131 RS Ç {z{ nlet velocity È J hka. We otained the results $ $ shown in Figure 5.3. The frequency analysis gives us main frequencies of the signals Hz and Ç È Hz. 4 1 alpha [ O H [mm time [s time [s nlet velocity J FG (*,+ and -L@*,+ for the inlet velocity of the air M/O >3P5. R S hkqï%è z{zz. We otained the signals shown in Figure 5.4. The frequency analysis gives us main frequencies of the signals Hz a Ç Hz. $ alpha [ O H [mm time [s time [s FG (*,+ and -L@*,+ for the inlet velocity of the air RS/O >3P5. { RS z{z nlet velocity J hm. The signals and 9 are shown Ç in Figure 5.5. The frequency analysis gives us only one main frequency of the signals È Hz. The second one is very hard to detect. n all cases studied the viration amplitudes decrease in time and the system is stale. Figure 5. shows the comparison of the computed main frequencies with wind tunnel experiments descried in [8 and [9.. Conclusion. n this article we derived a procedure for otaining the numerical solution of the interaction of a moving airfoil inserted in a channel with a running fluid. We used this approach for solving a particular prolem, which was studied experimentally in a wind tunnel [8, [9. The computational results show that the presented method is sufficiently roust and useful for a given type of prolem. The computed and experimentally otained main frequencies of the flow induced virations of the profile for several values of the inlet flow velocity are in good agreement see Figure 5.. A further goal will e the implementation of a turulence model to the description of the flow and the validation of the method in situations

10 13 M. RŮŽČKA, M. FESTAUER, J. HORÁČEK, AD P. SVÁČEK alpha [ O H [mm time [s time [s FG (@*,+ and -L@*,+ for the inlet velocity of the air TS/O 3P5. main frequency [Hz f 1 experiment f experiment f 1 numerical experiment f numerical experiment 4 8 velocity [m/s FG. 5.. Main frequencies in dependence on the inlet flow velocity. with large displacements of the airfoil and a higher inlet flow velocity, when the system may loose aeroelastic staility. Acknowledgments. This research was supported under the grant of the Grant Agency of the Academy of Sciences of the Czech Repulic o. AA713. t was also partly supported under the Research Plan MSM 1839 (M. Feistauer of the Ministry of Education of the Czech Repulic and the project o. 487 of the Grant Agency of the Charles University in Prague (M. Růžička. REFERECES [1 F. BREZZ AD R. S. FALK, Staility of higher-order Hood-Taylor methods, SAM J. umer. Anal., 8 (1991, pp [ P. G. CARLET, The Finite Element Method for Elliptic Prolems, orth-holland, Amsterdam, [3 T. A. DAVS, UMFPACK V4., University of Florida. Availale at

11 TERACTO OF COMPRESSBLE FLUD AD A MOVG ARFOL 133 [4 V. DOLEJŠÍ, Anisotropic mesh adaptation technique for viscous flow simulation, East-West J. umer. Math., 9 (1, pp [5 V. DOLEJŠÍ, AGEER V3., Faculty of Mathematics and Physics, Charles University Prague, Prague,. Availale at dolejsi/angen/angen3.1.htm/ [ E. H. DOWELL, A Modern Course in Aeroelasticity, Kluwer Academic Pulishers, Dodrecht, [7 T. GELHARD, G. LUBE, M. A. OLSHASK, AD J.-H. STARCKE, Stailized finite element schemes with LBB-stale elements for incompressile flows, J. Comput. Appl. Math., 177 (5, pp [8 J. HORÁČEK, J. KOZÁEK, AD J. VESELÝ, Dynamic and staility properties of an aeroelastic model, in Engineering Mechanics 5, V. Fuis, P. Krejčí, and T. ávrat, eds., May 9-1, 5, nstitute of Thermomechanics of the Academy of Sciences, Prague, 5, pp [9 J. HORÁČEK, M. LUXA, F. VAĚK, J. VESELÝ, AD V. VLČEK, Design of an experimental set-up for the study of unsteady D aeroelastic phenomena y optical methods (in Czech, Research Report of the nstitute of Thermomechanics of AS CR Prague, o. Z 1347/4, ovemer 4. [1 G. LUBE, Stailized Galerkin finite element methods for convection dominated and incompressile flow prolems, Banach Center Pul., 9 (1994, pp [11 T. OMURA AD T. J. R. HUGHES, An aritrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid ody, Comput. Methods Appl. Mech. Engrg., 95 (199, pp

Interaction of Incompressible Fluid and Moving Bodies

WDS'06 Proceedings of Contributed Papers, Part I, 53 58, 2006. ISBN 80-86732-84-3 MATFYZPRESS Interaction of Incompressible Fluid and Moving Bodies M. Růžička Charles University, Faculty of Mathematics