NUMERICAL SIMULATION OF FLOW AROUND A NACA 0012 AIRFOIL IN TRANSIENT PITCHING MOTION USING IMMERSED BOUNDARY METHOD WITH VIRTUAL PHYSICAL MODEL

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1 NUMERICAL SIMULATION OF FLOW AROUND A NACA 0012 AIRFOIL IN TRANSIENT PITCHING MOTION USING IMMERSED BOUNDARY METHOD WITH VIRTUAL PHYSICAL MODEL José Eduardo Santos Olvera Laboratory of Heat and Mass Transfer and Flud Dynamc Unversty Federal of Uberlânda Mechancal Engneer College Bloco 1M Av. João Naves de Ávla, CEP e-mal: jeolve@mecanca.ufu.br Ana Lúca Fernandes de Lma e Slva e-mal: alfslva@mecanca.ufu.br Arsteu da Slvera Neto e-mal: arsteus@mecanca.ufu.br Abstract. In the present work the numercal smulaton of turbulent flow over arfols n transent ptchng movng s presented. The flow s smulated through the numerc soluton of the Naver-Stokes equatons usng the Immersed Boundary Method wth Vrtual Physcal Model. Ths methodology allows the modellng of complex geometres mmersed n the flow. Wth the use of two ndependent meshes there s no restrcton as the movement or deformaton of the body. In ths way, the smulaton of the flow on movng boundares does not present any addtonal dffculty, as t happens n the classc methods where the meshes must be reconstructed. It s presented the two-dmensonal smulaton of flow past a NACA 0012 arfol profle n oscllatng ptchng for two dfferent movng speeds n order to evaluate the transent effect on the flow. Prelmnary numercal results presented a good physcal coherence. Keywords: Immersed Boundary Method, Vrtual Physcal Model, Arfols, Movng Boundares 1. Introducton Dynamc stall s a term used to descrbe a process where the lft force sudden drop, whle the arfol ncreases the attack angle through a ptch moton. The stall phenomenon for thn arfols s often assocated wth the formaton of a leadng-edge vortex and s generally preceded by lamnar boundary-layer separaton near the arfol nose. The vortex travel along the arfol surface as t grows, and fnally separates from the arfol at ts tralng edge. Dynamc stall s an unsteady phenomenon and dfferently of statc stall the flow separaton occurs at hgh angle-of-attack. A complete understandng of the dynamcs of leadng edge separaton on a ptchng arfol s essental to desgn of compressor blades, wngs of modern fghters, helcopters, etc. The dynamc stall phenomena was frst study by the helcopter ndustres, where large torsonal oscllatons of the blades was observed and attrbuted to the perodc stallng and unstallng of each blade on the retreatng sde of the rotor dsk (Akbar and Prce, 2003). To model the movng arfol mmersed n the flow s represented by mmersed boundary method (IB). The IB method was proposed by Peskn (1977) to smulate blood flow n heart valves, a hgh complex problem that nvolves movng boundares and complex geometres. The man dea of the Peskn method s to use two meshes. The Euleran mesh s used to solve the flud flow equatons whle an ndependent Lagrangan mesh represents the mmersed body. The nteracton between the two meshes s made through a force source term added to the Naver-Stokes equatons. Ths procedure allows to model complex geometres mmersed n a flud flow, avodng the use of complex meshes to ft the grd to the mmersed body. As a geometrc dependence does not exst between these two meshes, the mmersed boundary method can easly handle movng boundary problems wthout regeneratng grds n tme. The man ssue n the methods based on the concept of mmersed boundary s how to compute the force term. In the present work, a new model named Vrtual Physcal Model (VPM) proposed by Lma e Slva et al. (2003), s used. It s based on the calculaton of the force feld over a sequence of Lagrangan ponts, whch represent the nterface, usng the Naver- Stokes equatons. 4 In ths work s presented prelmnary numercal results of an oscllatng arfol at Reynolds number 10. The arfol was gven a harmonc ptchng moton about the md-chord axs at two reduced frequences: 0.5, and The mean ncdence and the oscllatng ampltude were 5 o and 10 o. The turbulence modelng methodology used was the Large Eddy Smulaton (LES) wth the Smagornsky algebrac sub-grd scale model to calculate the turbulent vscosty. A qualtatve comparson s made between the two smulated cases evdencng the effect of reduced frequency on the aerodynamc force coeffcents and flud flow behavor.

2 2. Mathematcal Formulaton The man dea n the IBM/VPM methodology s to solve the flow over mmersed bodes usng two ndependent meshes: a fxed Euleran mesh for the flud doman and a Lagrangan mesh to represent the sold-flud nterface. There are no dffcultes to represent complex mmersed bodes, related to the mesh buldng Mathematcal Model for Flud Doman A rectangular doman was used for the present smulatons and t was dscretzed wth an Euleran grd. The Naver- Stokes equatons for a vscous ncompressble flud can be wrtten as: ( u ) ( uu ) j p u u ρ + = + ( ν + ν ) j t + + f, t xj x j xj x (1) u x = 0. (2) It must be stressed that Eq. (1) s already the fltered equaton. Also, the Boussnesq hypothess was used to model the subgrd Reynolds stress tensor. These equatons are solved on the Euleran mesh and the couplng between the two meshes are made by the force source term f that s dfferent from zero only over the mmersed boundary. Equaton (3) models the nteracton between the mmersed boundary and the flud, by the dstrbuton of the Lagrangan force on the flud: f x F x, t x x dx, ( ) = ( ) δ ( ) Ω k k k where F s the Lagrangan force densty placed on x k ponts over the nterface and δ(x x k ) s a Drac delta functon. In order to dscretze Eq. (3), the Drac delta functon s replaced by the dstrbuton functon D j. Ths functon acts lke a Gaussan weght functon wth a untary ntegral over the nterval [, + ]. Therefore, Eq. (3) s replaced by: where ( ) = j ( k ) ( k ) 2 f x D x x F x, t s, s s the dstance between two Lagrangan ponts Sold-Flud Interface Model The VPM performs the dynamc evaluaton of the force exerted by the flud flow over the mmersed body. The force densty F(x k,t) s calculated over the Lagrangan ponts usng the movement equaton. The Lagrangan force can be expressed by: (3) (4) ( uu j ) uk ufk p u u j F ( x,t k ) = ν +. t x xj x j xj x (5) The dfferent terms on the rght sde of Eq. (5) are referred as: acceleraton force, pressure force, nertal force and vscous force. The four components of force densty F(x k,t) are calculated on a control volume centered at a Lagrangan pont. To evaluate the dfferent terms descrbed by Eq. (5) the pressure p( x,t ) and the velocty V(x,t) felds must be known prevously. These felds are calculated on the Euleran grd whle the force terms must be calculated over the nterface. One of the possble ways to do that s to nterpolate V(x,t) and p( x,t ) over approprate auxlary Lagrangan ponts near the nterface, as llustrated by Lma e Slva et al. (2003). For the pressure feld, only grd ponts external to the nterface and at a dstance less than or equal to 2 x are used n the nterpolaton process. For the velocty felds the external and nternal ponts are taken, because the nternal velocty feld helps to model vrtually the no-slp boundary condton. The pressure and velocty dervatves that appear n Eq.(5), are calculated usng a second order Lagrange polynomal approxmaton.

3 3. Turbulence Modellng The Naver-Stokes equatons are able to smulate, wth farly good agreement, a wld range of engneerng problems ncludng hgh complex unsteady turbulent flows. However, t s necessary to solve all degrees of freedom of the flow, 9 / 4 whch s proportonal to Re. Ths technque s called DNS (Drect Numercal Smulaton) and t s obvously restrcted to low Reynolds number due to the hgh mesh resoluton requred, consderng the nowadays computers. An alternatve way to handle ths problem s the use of Reynolds decomposton or general flterng process of Germano (1986). The governng equatons are sutably fltered and ths procedure gves rse to the closure problem. Ths closure problem s nowadays solved usng turbulence models. Dfferent methodologes have been employed n the turbulence modelng,. e., LES (Large Eddy Smulaton). In the LES methodology the largest turbulent structures are solved from the fltered equatons and only the smallest structures are modeled. These small structures are more homogeneous and sotropc (Slvera-Neto, 2003). The scale of the small structure s evaluated from the mesh used to solve the fltered equatons,.e., the flter wdth becomes a functon of the grd. The turbulent structures that are smaller than the grd resoluton are modeled by the so-called subgrd-scale models. In the present work the Smagornsky subgrd-scale model was employed. The formulaton of ths model s dscussed n the followng secton Smagornsky Sub-grd Scale Model In the present work LES methodology proposed by Smagornsky (1963) s appled. The sub-grd scale model, whch s based on the balance of producton of sub-grd scale turbulent knetc energy and dsspaton of sotropc turbulence energy, s used. The turbulent vscosty s computed as functon of stran rate ( S j ) and the length scale ( ): ν = (C ) 2S S, (6) 2 t s j j where C = 018. s the Smagornsky constant and = x y s the characterstc sub-grd length. S 4. Numercal Method The governng equatons, Eq. (1) and (2), were dscretzed usng the central second-order fnte dfference method n space and a Runge-Kutta second-order scheme n the tme. The pressure-velocty couplng was performed usng a pressure correcton method, as proposed by Chorn (1968). The lnear system for the pressure correcton was solved usng the teratve solver MSI procedure (Modfed Strongly Implct) of Schneder and Zedan (1981). The nterfacal force feld calculaton and the momentum equaton soluton are performed n an explct way. All the smulatons were carred out on a non-unform grd. The calculaton doman has a length of 10C and a wdth of 8C, where C s the arfol chord. A non-unform grd of 278x198 ponts wth three dstnct regons n each drecton was used, as can be observed n Fg. (1). On the x drecton the frst secton has 50 ponts and s extended untl 2.7C. The last secton has 5.8C of length wth 102 ponts. In the y drecton the two non-unform sectons are dentcal wth a length of 3.82C and 84 ponts. The arfol s placed nsde a rectangular box wth a unform mesh. The box has dmensons of 1.5C x 0.36C. Fgure 1. Vew of the calculaton doman.

4 The arfol s placed at 3.3C from the left boundary and centered vertcally at 4C and the ptchng axe s located at 0.5C. A constant velocty profle U was mposed at the doman nlet, n such a way that the flud goes from the left to the rght boundary. Neumann condton was mposed for the velocty at all other faces. For the pressure correcton, null dervatve was employed at the doman entrance and t was set as zero n the other faces. It must be stressed that even wth the ptchng movement, the Euleran mesh remans unchanged and only the Lagrangan mesh s reconstructed at each angle-of-attack. The angle-of-attack as a functon of tme s gven by: () t cos( t) α = α α Ω, (7) where α s the angle of attack at the tme t, α s the mean angle-of-attack, α s the ampltude of the ptchng oscllaton and Ω = 2π f, where f s the frequency of oscllaton. Usually f s wrtten n terms of the reduced frequency, defned as: π f C κ =. (8) U 5. Results and Dscussons In ths secton the prelmnary results are presented for the two-dmensonal unsteady flow past a ptchng NACA 0012 arfol. The mmersed arfol profle was represented by IBM/VPM methodology. The Smagornsky 4 turbulence model (to perform LES) was employed to smulate two cases at Re = 10, usng the reduced frequences of κ = 05. and κ = In ths work, only one statc case was smulated for α = 5 o. Ths case was used to ntalze all the ptchng smulatons. The ptchng movement starts from α = 5 o and than the angle s ncreased n accordance wth Eq. (7). The smulaton results are present n order to evaluate the effect of the reduced frequency (κ ) n the stall phenomena. Results reported by Akbar and Prce (2003) predcted the statc stall angle at α 15 o for a Reynolds 4 number of 10. Fgure (2) shows the drag ( C D ) and lft ( C L ) coeffcents for the frst test case. Ths numercal smulaton has a mean angle of α = 15 o, an ampltude of α = 10 o and a reduced frequency of κ = 05.. The smulatons were carred out for 4s seconds of tme that s suffcent to the arfol to performs sx cycles of oscllatons. The lnes of Fg. (2) were colored wth the tme scale to dentfy each cycle of movement. As can be vsualzed n the Fg. (2), there s a hysteress loop n the aerodynamc force coeffcents. Durng the downstroke the lft force coeffcent s smaller than durng the upstroke. Ths s due to the dfference between the flow over the arfol durng these two arfol movements, whch affects sgnfcantly the vortex sheddng dynamcs and manly the detached/reattached behavor of the flow. Durng the decrease of the angle, the flow s completed separated and the lft coeffcent s reduced. Near α = 65. o the lft force ncreases whch s concdent wth the flow reattachment on the upper surface. The dfferences between two consecutve cycles of the hysteress loop s due to the unsteady nature of the flow. (a) (b) Fgure 2. The (a) drag and (b) lft coeffcents versus angle of attack over sx oscllaton cycles (4 s) for the arfol at α = 15 o, α = 10 o and κ = 05. The tme evoluton of vortcty feld ( 150 ω z 150 )for the frst cycle of oscllaton of the arfol s presented n Fg. (3). The behavor of aerodynamc forces over the arfol, showed n Fg. (2), can be assocated wth the dynamc of

5 vortex sheddng presented n Fg. (3). Durng the upstroke movement s observed a constant ncrease of the lft force wth the ncrease of the angle-of-attack, and no stall behavor was observed. It can be explaned observng the flow over the arfol, where no apparent ndcaton of flow separaton s detected up to α = o (Fg. 10g). It s mportant to note that the statc stall occurs at α = 15 o. The effect of ptchng moton at ths reduced frequency nhbts the stall phenomena for ths range of angle of attack. (a) t= 0.00 s, α = 5.0º (b) t= 0.04 s, α = 5.8º (c) t= 0.08 s, α = 8.0º (d) t= 0.12 s, α = 11.4º (e) t= 0.16 s, α = 15.3º (f) t= 0.20 s, α = 19.2º (g) t= 0.24 s, α = 22.4º (h) t= 0.28 s, α = 24.4º () t= 0.32 s, α = 25.0º (j) t= 0.36 s, α = 24.0º (k) t= 0.40 s, α = 21.5º (l) t= 0.44 s, α = 18.0º (m) t= 0.48 s, α = 14.1º (n) t= 0.52 s, α = 10.3º (o) t= 0.56 s, α = 7.2º (p) t= 0.60 s, α = 5.4º Fgure 3. Vortcty feld vsualzaton for the NACA 0012 arfol durng the ptchng movement at several angles of attack at α = 15 o, α = 10 o and κ = 05. At α = o (Fg. 3h) a leadng-edge separaton bubble starts formng over the arfol. Around ths tme nstant the arfol starts the downstroke movement, whch forces the shed of the leadng-edge vortex, promotng the detachment of the boundary layer and consequently the drop of the lft force, as shown n Fg. (2). The hysteress effect n the aerodynamc forces durng the downstroke s observed. At α = o (Fg. 3m) the growth of a great tralng-edge vortex start. Ths vortex s shed and nhbts the reattachment of the flow. The structures are transported downstream by the flow and around α = 65. o occurs the reattachment of the flow and the lft force suddenly ncreases (Fg. 2b). When α reaches 54. o (Fg. 3p), durng the downstroke, the flow s already attached at the leadng-edge and the tme cycle s completed. Fgure (4) shows the drag and lft coeffcents versus the angle-of-attack for the reduced frequency of κ = The ntal condtons such as angle of attach and ampltude are the same n relaton to the prevous smulatons. The smulaton was accomplshed for 4s and the arfol performed three cycles of oscllaton. For ths case the lft coeffcent also ncreases durng the upstroke movement up to α = 20 o. At ths angle the lft coeffcent suddenly drops characterzng the stall pont. Note that n the prevous case, κ = 05., the stall phenomena was not observed. We can conclude that ncreasng the reduced frequency the stall angle s delayed to a hgh value of angle of attack. It s well know that LES turbulence models are better to descrbe flow structures and consequently the transent effects of vortces sheddng nduces hgh oscllatons n the aerodynamc forces coeffcents, when compared wth results from RANS turbulence models whch descrbe the average behavor of the flud flow. It s nterestng to note

6 that the flow structures formed for a reduced frequency of κ = 025. are bgger and more chaotc than the structures formed n the prevous smulaton ( κ = 05. ). As a consequence the oscllatons present on the aerodynamc coeffcents are bgger, as can be observed n Fg. (4). Therefore, consderng the turbulence modelng employed (LES), a better procedure for ths reduced frequency would be to smulate more ptchng cycles and then apply a tme average over the data, reducng the oscllaton. (a) (b) Fgure 4. The (a) drag and (b) lft coeffcents versus angle of attack over three oscllaton cycles (4 s) for the arfol at α = 15 o, α = 10 o and κ = 025. (a) t= 0.00 s, α = 5.0º (b) t= 0.08 s, α = 5.8º (c) t= 0.16 s, α = 8.0º (d) t= 0.24 s, α = 11.4º (e) t= 0.32 s, α = 15.3º (f) t= 0.40 s, α = 19.2º (g) t= 0.48 s, α = 22.4º (h) t= 0.56 s, α = 24.4º () t= 0.64 s, α = 25.0º (j) t= 0.72 s, α = 24.0º (k) t= 0.80 s, α = 21.5º (l) t= 0.88 s, α = 18.0º (m) t= 0.96 s, α = 14.1º (n) t= 1.04 s, α = 10.3º (o) t= 1.12 s, α = 7.2º (p) t= 1.20 s, α = 5.4º Fgure 5. Vortcty feld vsualzaton at several angles of attack for a ptchng NACA 0012 arfol, at α = 15 o, α = 10 o and κ = 025.

7 The vortcty felds for the frst cycle of oscllatons at reduced frequency of κ = 025. are presented n Fg. (5). Observng Fg. (5e) t s easy to see that the flow remans attached to the arfol surface. At α = o (Fg. 5f) the flow detaches from the arfol, one leadng-edge vortex already shed and ntates the growth of separaton bubble. When the arfol reaches α = o (Fg. 5g) the flow over the upper surface of the arfol s completely separated, the second leadng-edge vortex s fully formed and s shedded from the arfol at about the md-chord poston. Durng the downstroke other hgh complex structures are formed from the upper surface of the arfol due the nteracton between leadng edge and tralng edge vortces (Fg. 5-l). These structures prevent the flow reattachment. At the end of the cycle these structures are beng advected downstream but the flow s stll detached and small vortces are sheddng (Kelvn-Helmholtz nstabltes) from the leadng edge (Fg. 5m-o). Comparng the two smulated cases we can observe some dfferences n the flow feld between the frst case, κ = 05., and the second case, κ = In the frst case the flow remans attached to arfol durng the most part of the upstroke (Fg. 3b g), the leadng-edge vortex s formed and released entrely durng the downstroke, and only one tralng-edge vortex s shedded per cycle of oscllaton. In both smulated cases t s nterestng to note that the flow feld remans quet durng the most part of the upstroke and much of the separaton and vortex sheddng actvty occurs n the downstroke. Ths behavor s due to the nfluence of the arfol movement that reduces the adverse pressure gradent n the upstroke cycle and ncreases t n the downstroke cycle. 6. Conclusons In ths work the IB/VPM (Immersed Boundary Method/Vrtual Physcal Model) was used to smulate ptchng oscllatons of a NACA 0012 arfol, prelmnary results are presented showng good coherence wth the physcal phenomena. It was performed two numercal smulatons at two reduced frequences 0.5 and 0.25, n order to evaluate the effect of the reduced frequency on the flud flow. It was observed that ncreasng the reduced frequency of the ptchng moton the flow separaton was delayed to hgher ncdences compared to the statc stall case. Separaton was observed to start from the leadng-edge, followed by the formaton and advecton of vortces along the surface of the arfol, tralng-edge vortces were also observed to form and shed from the arfol. Large ampltude oscllatons was observed n the force coeffcents for the smulaton n the reduced frequency 0.25, ths behavor was attrbuted to the LES turbulence model used, t s necessary more tme of smulaton to establsh a trustworthy statstcal data set. 7. Acknowledgements The authors acknowledge CNPq and Fapemg for the fnancal support and the School of Mechancal Engneerng of the Federal Unversty of Uberlânda for the techncal support. 8. References Akbar, M. H. and Prce, S. J. (2003). Smulaton of dynamc stall for a NACA 0012 arfol usng a vortex method. Journal of Fluds and Structures, 17, Chorn, A. (1968). Numercal soluton of the Naver-Stokes equatons. Math. Comp. 22, Germano, M. (1986). A proposal for a redefnton of the turbulent stresses n fltred naver-stokes equatons. Phys. Fluds 29(7), Lma e Slva, A. L. F., Slvera-Neto, A. and Damasceno, J. J. R. (2003). Numercal smulaton of two dmensonal flows over a crcular cylnder usng the mmersed boundary method. Journal of Computatonal Physcs 189, Peskn, C. S. (1977). Numercal analyss of blood flow n the heart. Journal of Computatonal Physcs 25, Schneder, G. E. and Zedan, M. (1981). A modfed strongly mplct procedure for the numercal soluton of feld problems. Numercal Heat Transfer 4, Slvera-Neto, A. (2003). Introdução Turbulênca dos Fludos, Apostla do Curso de Pós-Graduação em Eng. Mecânca. LTCM/FEMEC/UFU. Smagornsky, J. (1963). General crculaton experments wth prmtve equatons. Monthly Weather Revew 91, Responsblty notce The authors are the only responsble for the prnted materal ncluded n ths paper.