Vortex Interaction and Roll-Up in Unsteady Flow past Tandem Airfoils

Transcription

1 Journal of Applied Fluid Mehanis, Vol. 9, No. 6, pp , 216. Available online at ISSN , EISSN Vortex Interation and Roll-p in nsteady Flow past Tandem Airfoils H. Aziz 1 and R. Mukherjee 2 1 Department of Mehanial and Nulear Engineering, Virginia Commonwealth niversity, Rihmond, Virginia, SA Department of Applied Mehanis, Indian Institute of Tehnology Madras, Chennai, India 636 Corresponding Author rinku@iitm.a.in (Reeived Deember 3, 214; aepted February 21, 216) ABSTRACT A disrete vortex model oupled with a vortex dissipation and vortex ore riteria is used to study the unsteady flow past two airfoils in onfiguration. The unsteady wakes of the airfoils are modeled by disrete vorties and time-stepping is used to predit the individual wake shapes. The oupled flow is solved using a ombined zeronormal flow boundary ondition and Kelvin ondition whih result in (2N + 2)X(2N + 2) equations. Results are presented showing the effet of airfoil-airfoil and airfoil-wake interation on the aerodynami harateristis of the onfiguration. The effet of relative veloity, rate of pithing and phase-lag are studied on airfoil performane and wake shape is predited. Keywords: Airfoil; Vortex interation; nsteady; Aero-dynamis; Numerial singularity. NOMENCLATRE C l oeffiient of lift C m oeffiient of pithing moment C d oeffiient of indued drag airfoil hord length h o amplitude of heaving motion ˆn unit vetor normal to a solid surfae O, o origin of fixed and moving referene frames free-stream veloity r distane between free vortex and solid surfae t V latest time step airfoil veloity veloity potential stream funtion f irulation of the airfoil W irulation of the wake σ radius of the vortex ore ω angular frequeny of heaving motion α angle of attak 1. INTRODCTION Studies of unsteady aerodynamis of pithing airfoils are in ruial need in wind turbines, more speifially in rotorraft dynamis. More reently, studies on unsteady aerodynamis of airfoils have been motivated by the prospet of utilizing the advantages of large un-steady aerodynami fores for aerospae appliations like inreased maneuverability of fighter airraft, use of miro-avs, whih are designed for very small payloads for remote-sensing operations in areas of limited aess and the use of multi-avs in formation for surveillane purposes to inrease their longevity and endurane. In addition to the knowledge required for unsteady flow-field aerodynamis of single airfoils, in most of the appliations mentioned above, the airfoils also operate in lose proximity to eah other so that their flow-fields overlap. Hene, an aurate estimate of the flow-field and the knowledge of flow patterns and strutures are inevitable for the suessful design and modeling of these appliations. The unsteady aerodynamis of airfoils has been studied by researhers both as a pratial problem as well as an important fundamental one. Wu, Wang and Tuner (1986) studied the dynami stall of an airfoil pithed rapidly at a onstant rate to large angles of attak omputationally. Greenwell (24) reviewed approahes to modeling the unsteady

2 aspets of the aero-dynami harateristis of maneuverable airraft. Hall and Clark (1993) presented a linearized Euler solver to alulate the unsteady flows in turbomahinery blade rows due to both inident gusts and vibrations of the blade. Capee and Fleeter (1986) studied the effets of steady loading and the detailed aerodynami foring on airfoil row unsteady aerodynamis at low redued frequeny values. The unsteady aerodynamis of pithing airfoils involves an additional unsteady lift, whih is due to the vortex that forms near the leading edge during the pithing motion and onvets downstream. Fulayter, Lawless and Fleeter (22) studied the phenomenon of dynami stall itself besides the aerodynamis of unsteady flight. Jose and Baeder (29) studied the 2D unsteady aerodynamis of trailing edge flap-tab airfoils with overhang and gap both analytially and using CFD. Bai (28) experimentally predited and tested the a-uray of the free flight unsteady aerodynamis of a pithing airfoil at high angles of attak. Fritz and Long (24) used an unsteady vortex lattie method to study osillating, plunging, pithing, twisting and flapping motions of a finite aspet-ratio wing. M. Radmanesh et al. (214) designed a novel simple strategy to hoose and aommodate an airfoil based on the effets of airfoil type and planform shape on the flight performane of a miro air vehile by studying its unsteady aerodynamis. A. Shokrgozar et al. (216) studied the 3D axisymmetri unsteady stagnation-point flow and heat transfer impinging on a flat plate when the plate is moving with variable veloity and aeleration towards the main stream or away from it. Husain, Abdullah and Yap (33) did a two-dimensional analysis of tandem/staggered arranged airfoils of the anard and wing of an Eagle 15 airraft using omputational fluid dynamis and also onduted aerodynami tests in an open-iruit wind tunnel. In this work, an in-depth study is undertaken to understand the effet of two airfoils exhibiting unsteady motion on eah other, the hange in irulation, the motion of the trailing edge vorties and their effet on the airfoils. The aerodynami harateristis are srutinized, for example the Cl vs α plot is srutinized to see what auses a spike, the orresponding hange in indued drag, loation of the trailing edge vortex shed from the leading wing and the hange in irulation. 2. NMERICAL PROCEDRE The urrent numerial method onsists of extending an unsteady analysis of a single airfoil and its wake using disrete vorties proposed by Katz (21) to inlude multiple airfoils and hene multiple wakes. The airfoils are simulated using a disrete lumped vortex model, where the zeronormal boundary ondition is applied on the atual amberline. Eah of the multiple unsteady wakes is simulated using trailing edge disrete vorties shed from the airfoils at every time-step as shown shematially in Fig. 1a. The flow field is fore-free and the strength of the wake vorties remains unhanged. However, the loation of the free vorties is updated every time instant using a timestepping method. The presene of multiple airfoils results in two types of numerial singularities: (a) Free wake vorties interat and ome infinitesimally lose to eah other. A vortex ore approah proposed by Chorin (1973) is used to prevent suh a senario. (b) Free wake vorties inter-at and ome infinitesimally lose to the airfoils. When this happens the vorties would be expeted to dissipate. However, sine a potential flow solution is used here, vortex dissipation is aounted for using a separate near-field vortexsolid surfae riterion. 2.1 nsteady Disrete Vortex Model An unsteady disrete vortex model is developed starting with potential flow on whih orretions for vortex dissipation are imposed and time-looping is employed to evaluate the time instant behaviour of the vorties being shed into the flow field. Flow past two airfoils in onfiguration is onsidered and the amberline of eah airfoil is disretised into segments, on whih lumped vorties are distributed. On eah segment a lumped vortex element is loated at the quarter hord and a olloation point, where the zero normal flow on a solid surfae boundary ondition shown in Eq. (2) is satisfied is loated at the three-quarter hord as shown in Fig. 1a shematially. For example, the veloity indued by a lumped vortex element of irulation, Γ loated at ( xo, z o), at an arbitrary point, Pxz (, ) are given as in Eq. (1): z zo u ( xxo) ( zzo) x x w o ( xxo) ( zzo) (1) Surfae normals as shown in Fig. 1a are generated at the olloation points and the zero-normal flow boundary ondition is satisfied at the olloation points on the atual amberline of the airfoil and not on the hord line. For a different angle of attak, the influene oeffiients are generated by using an updated definition of the surfaes normals and the segment definitions are not re-generated. An unsteady solution involves aounting for the hange in irulation of the lifting surfaes at every instant of time and hene update the strength of the bound vorties of the lumped vorties given in Eq. (1). This hange in irulation is due to the hange in the strength of the bound vorties as well as the indued veloities due to the free wake vorties shed by both airfoils. For example, at time-step = 1, a single wake vortex A of strength w,1 is shed behind airfoil A and another B single wake vortex of strength w,1 is shed behind airfoil B as shown in Fig. 1. Therefore, at time-step = 1, the strength of the bound vorties on both airfoils is affeted by A B w,1 and w,1. 388

3 Fig. 1. nsteady Lumped Vortex Model. A B At time-step = 2, w,1 and w,1 move along with the free stream making way for a new set of single wake A B vorties of strength w,2 and w,2, whih are shed be-hind airfoils A and B respetively, as shown in Fig. 1e. Therefore, at time-step = 2, the strength of A the bound vorties on both airfoils is affeted by w,1 B A B, w,1, w,2 and w,2 A B A B At time-step = 3, w,1, w,1, w,2 and w,2 move along with the free stream making way for a new set A B of single wake vorties of strength w,3 and w,3, whih are shed behind airfoils A and B respetively, as shown in Fig. 1g. Therefore, at time-step = 3, the strength of the bound vorties on both airfoils is A B A B A affeted by, w,1, w,1, w,2, w,2, w,3 and B w,3 It is to be noted that if any of the vorties shed from airfoil A, say Γ, 1 is ahead of the leading edge of air-foil B, it auses an upwash or enhanes lift as shown in Figs 1, 1e and 1g. On the other hand, if it rosses the leading edge of airfoil B, it auses a downwash or loss in lift as shown in Fig. 1d. Kelvin ondition is satisfied as shown in Eq. (3). Hene, at every time instant, the strength of the free wake vorties shed from earlier time-steps are known and the free wake vortex shed in that partiular time instant is alulated using the Kelvin ondition. The strength of the wake vorties already shed does not hange but their positions are updated at every time instant. Therefore, the unsteady disrete vortex model satisfies the Laplae equation or a steady-state potential flow solution at every instant of time, using updated values of the strength of the bound vorties. 2.2 Boundary Conditions and Influene oeffiients By definition, the Kutta ondition is satisfied by the lumped vortex elements and it is not inluded into the solution expliitly. The unknowns in this method are the strengths of the bound vorties of the two airfoils (2N in number) and the strengths of the most latest shed trailing vorties (2 in number) from the two air-foils. 389

4 Boundary onditions imposed are the zero-normal flow on a solid surfae boundary ondition given in Eq. (2), whih aounts for the strengths of the bound vorties and the Kelvin ondition given in Eq. (3), whih a-ounts for the strengths of the latest shed trailing vorties for eah airfoil in the onfiguration. Both these onditions are used in tandem at every time instant to generate the (2N + 2) (2N + 2) influene oeffiient matrix for the two airfoils in onfiguration, where N is the number of disrete vorties used to simulate eah airfoil. Finally, a matrix equation given in Eq. (4) is solved at every time instant for the strength of the bound vorties of eah airfoil (ΓN ) in the onfiguration and the strength of the latest wake vorties shed from eah airfoil (Γw,t). The subsript t denotes the latest time step. Then, F1ΓA is the influene of the bound vorties of airfoil A on itself. F1ΓB is the influene of the bound vorties of airfoil B on airfoil A. G1ΓA is the influene of the bound vorties of airfoil A on airfoil B. G2ΓB is the influene of the bound vorties of airfoil B on itself. ΓWA and ΓWB are the strengths of the latest shed trailing vorties from airfoil A and B respetively. F3ΓWA is the influene of the latest shed trailing vortex from airfoil A on airfoil A. F4ΓWB is the influene of the latest shed trailing vortex from airfoil B on airfoil A. G3ΓWA is the influene of the latest shed trailing vortex from airfoil A on airfoil B. G4ΓWB is the influene of the latest shed trailing vortex from airfoil B on airfoil B. The first two rows of the influene oeffiient matrix are a statement of the zero-normal flow boundary ondition on both the airfoils. The last two rows of the influene oeffiient matrix are a statement of the Kelvin ondition on both the airfoils individually. ( ) nˆ ; nˆ nˆ( X, Y, Z, t) (2) d d f () t d w (3) dt dt dt F1 F2 F3 F4 A G1 G2 G3 G4 B L1 L2 L3 L4 WA M1 M2 M3 M3 WB ( rhs) A ( rhs) B WA1WA2... WA, t1 WB1 WB2... WB, t1 where (4) f1,1f1, N f1, N 1f1,2 N F1 F2 fn,1 f N, N f N N N, N 1 f N,2N NN f1,2n 1 f1,2n 2 F3 F4 f N,2N 1 f N 1 N,2N 2 N 1 gn 1,1 g N 1, N 1 F1 A g2 N,1 g 2 N, N N N N N N gn 1, N 1g N 1,2N N 1 G 2 B g 2 N, N 1 g 2 N,2N N N 2N N 1 gn,2n 1 gn 1,2N 2 G3 G4 g 2 N,2N 1 g N 1 2 N,2N 2 N 1 WA ( WA, t ) N 1 WB ( WB, t ) N 1 L1 M 2 (1...1) 1N L2 M1 (...) 1N L3M 4 (1) 1N L4M 3 () 1N The pressure differene between the upper and lower th surfaes of an airfoils j element is given as: j pj ρ ( ( t) uw, W( t) ww) j. j l j x j ρ γ( x, t) dx t j or p j ρ ( ( t) uw, W ( t) ww) j. j l j ρ ( xj, t) t j or p j ρ ( ( t) uw, W ( t) ww) j. j l j j ρ k t k 1 Where, (t) and W(t) are the omponents of the free stream veloity, u W and w W are the veloity omponents indued by the wake vorties on the airfoil, j is the unit vetor tangential to the surfae of segment j of the airfoil, j is the strength of the bound vortiity on segment j. As seen above, the first two terms in the expression for pressure are for the steady state ase. The seond unsteady term is due to the aelerated flow and also ontributes to the pressure differene. This first order time derivative is alulated as: ( xj, t2) ( xj, t1) ( xj, t) t t j ( xk, t2) ( xk, t1) ; t t2 t1 k 1 t 39

5 Hene, the aerodynami loads are as follows. 1. Lift x N L ( p)osαdx osα pjl j j j 1 x j = ρ ( ( t) u, ( ), ). osα w W t ww j j dx l j x j + ρ γ(x, tdx ) osαdx t 2. Indued Drag x N D ( p)sinαdx osα pjl j j j 1 x j = ρ ( ( t) u, ( ), ). sinα w W t ww j j dx l j x j + ρ γ(x, tdx ) sinαdx t 3. Pithing Moment about the Leading Edge of the airfoil N M pjosα jljxj j1 Where α is the angle between free stream veloity and the unit normal to the segment j. Hene, the aerodynami oeffiients are: L D M C,, l C 1 di C 2 1 m ρ ρ ρ Near-Feld Vortex-solid Surfae interation With time, the free vorties shed into the wake travel and hange their original loations along with the free-stream. A time-stepping method is used to update the loation of the shed vorties at the end of eah time step. In the urrent problem sine multiple airfoils are present in the flow field, the wake vorties of the leading airfoil interat with the trailing airfoil before travelling with the free-stream as shown in Fig. 1f. Computation-ally, this ondition will lead to numerial singularities and physially, it is expeted that a free vortex will dissipate when it hits a solid surfae. This has been re-ported in literature as well, for example, it was observed by Fage and Johansen (1927) and Nakagawa (1988) that vorties whih are too lose to the surfae of a plate, dissipate by the ation of visosity. In the present work, the effet of dissipation of the vorties is taken into a-ount using the ondition given in Eq. (5), where r is the distane between a free vortex in the flow field and a solid surfae. The relation in Eq. (5) is arrived at by onduting numerial tests, whih show that if r is greater that 8%hord, the method is unable to apture the unsteady nature of the flow-field aurately. Also, if r is less than 8% hord, say, r =.5, it does not give any additional information about the flow field. Hene, r =.8 is used as a ut-off point. Whenever a wake vortex satisfies this ondition, its influene on the flow-field, i.e. the veloity indued by it on the bound vorties, wake vorties and hene, the wake shape is ignored sine it is onsidered dissipated. The strength of this vortex is however ontinued to be used to satisfy the Kelvin ondition even after this ondition is met as the flow-field is onsidered to be fore-free and the net vortiity of the flow-field remains onstant. r.8 (5) 2.4 Near-Feld Vortex-Vortex Interation In the urrent problem, multiple airfoils shed multiple set of free wake vorties into the flow field, whih travel freely with the free-stream and interat with eah other. When the distane between two suh free wake vorties beomes infinitesimal as shown in Fig. 1h, omputation-ally this ondition leads to numerial singularities. To ounter this ondition, a vortex ore approah is taken, where a radius σ is sooped out about the vortex and the stream funtion riteria suggested by Chorin (1973) is used as given in Eq. (6). This prevents any infinite veloities due to two vorties oming too lose to eah other. In the urrent work, σ =.8. r log log r r ; r 2 2 (6) Where r is the distane between two free vorties and σ is the radius of the ore sooped out about the free vortex. The vortex dissipation riteria in Eq. (5) and vortex ore riteria in Eq. (6) depend on the airfoil hord length and in our urrent study the suitable ranges of r and σ arrived at by inspetion are: (7-1)%. This same value is used for all results generated and for validation. 3. RESLTS Results are presented for a onfiguration onsisting of two airfoils as shown in Fig. 1b. The leading airfoil is denoted as A and the trailing airfoil is denoted as B. A fixed referene frame, (X, Y) is attahed to airfoil A and the origin, O (, ) of this referene frame is loated at the leading edge of airfoil A. A moving referene frame, (x, y) is attahed to the trailing airfoil B and the origin of this referene frame o(x, Y) is loated at the leading edge of airfoil B. Note that X and Y are alulated with respet to the fixed referene frame, (X, Y). 3.1 nsteady Wake Vorties of Tandem Airfoils Set Into Motion with Sudden Aeleration The () l C t from the urrent work for two NACA12 airfoils in onfiguration, eah of hord length, = 1 shown in Fig. 1b is ompared with the analytial results of Wagner (1925) for a single airfoil. The airfoils in the present analysis and that of Wagner are at an angle of attak of 5o and set into motion with a sudden aeleration. 391

6 Fig. 2. Airfoils in Configuration. Fig. 3. Wake Shape. Sine Wagner s analytial result is for a single airfoil, the distane between the two NACA12 airfoils in the present analysis is taken as 5. It is expeted that at suh a large distane, the influene of the airfoil bound vorties and the wake vorties on eah other will be minimal. C () The plot of l t Cl ( steady) vs t for both airfoils is C () shown in Fig. 2a. It is seen that the l t Cl ( steady) values of both airfoils exatly math eah other and the omparison with the analytial result is also good. There is how-ever, some disagreement of the urrent result with that of Wagner at very small time steps, i.e. t 1. This may be attributed to the loss in auray due to the onstraints of a typial numerial approah and de-reasing the time-step does not improve the result. This problem is not expeted in an analytial approah as that of Wagner. Fig. 2b shows the irulation distribution, Γ(t) vs t and Fig. 3 shows the vortex roll up at t 1 for both airfoils from the urrent analysis. As expeted the result shows that both airfoils are unaffeted by the presene of eah other. Results for the irulation and wake shape are not available from Wagner for omparison. 3.2 Wake Vorties set off by Non-Interating Airfoils Pithing in Tandem The plot of Cl α for both airfoils, pithing in tandem for the onditions shown below and as shown in Fig. 4a. t α 3 1sinω t,.1, 1 2 The pith axis is at the quarter hord, 4. The results from the present work are ompared with those of M-roskey (1981) and Maskew (1988) whih are for single airfoils. Hene, the present results are d generated for 5.It is seen that there is no differene in the performane of both airfoils and they exatly math. They behave like single operating airfoils as they do not interat sine the distane between them has been taken to be very large. The omparison with literature is good. The wake shapes an be seen in Fig. 4b and the result emphasizes that the airfoils do not interat given the distane between them. 3.3 Behavior of Wake Vorties of Tandem Non-Interating Heaving Airfoils The onfiguration of two NACA12 airfoils as shown in Fig. 1b is subjeted to heaving motion. Wake shape of both the airfoils predited using the present numerial method is ompared with the omputational and experimental results of Katz and Weihs (1978) for a single airfoil. In this ase too, sine the results of Katz and Weihs are for a single airfoil, the distane between the two airfoils in onfiguration is taken as 1. It is expeted that at suh a large distane, the influene of the airfoil 392

7 Fig. 4. Airfoils Pithing in Configuration. Fig. 5. Wake of Single and Tandem Heaving Airfoils. bound vorties and the wake vorties on eah other will be minimal. The parameters of heaving motion are as follows: ω t h 8.5; 1.56 / s; o.19; / s 2 The experimental and numerial results of Katz and Weihs are shown in Figs. 5a and 5b respetively. The wake shape from the urrent analysis for both airfoils are shown in Figs. 5 and 5d. It an been seen that there is no interation between the airfoil bound vorties and wake vorties due to the 1 distane between them and hene the wake shape of both airfoils are not different from eah other. 3.4 Effet of Trailing Edge Vorties on the Performane of Interating Tandem Airfoils A omparison between the results from the present study and the omputational results of Platzer (1993) for airfoil and wake interation is presented here. For a onfiguration of two NACA12 airfoils, ( Xo, Yo) (.5,.2 ), Platzer s numerial ode ould not aount for the interation of the vorties shed from the leading airfoil with the trailing airfoil surfae for suh minimal distanes. They had to therefore adjust the onfiguration in suh a way that the vorties do interat but they do not ome lose enough to indue infinite veloities in the numerial 393

8 Fig. 6. Tandem Airfoils Interating with their Wakes. solution. Hene, they used the on-figuration ( Xo, Yo) (.5,.2 ) where the airfoils are also undergoing a step hange in angle of attak from to 5. The results for this onfiguration without the step hange in angle of attak is generated with the present ode and presented in Fig. 6. The results show individual ases of different angles of attak for the airfoils in onfiguration. The present numerial sheme aounts for the interation of vorties with solid airfoil surfaes as well as with eah other at minimal distanes without induing any infinite veloities in the flow-field. Sharp hanges in Cl () t for t.3 are observed for both airfoils. For the leading airfoil, the usp of C l inreases with inrease in the angle of attak. In other words, the Cl drops to a minimum and then sees a sudden jump at t.3. For the trailing airfoil, the C l dereases but less than the leading airfoil and at t.3, it drops sharply. The drop in C l of airfoil A an be attributed to the fat that the strength of the trailing edge vortex shed by the leading airfoil A and the orresponding downwash aused by it on itself inreases with the inrease in its angle of attak. The same trailing edge vortex shed from airfoil A initially auses an upwash and hene an inrease in C l of the trailing airfoil B. Essentially, t for.3, the effet of this trailing edge vortex on both airfoils takes preedene over the effet of angle of attak of the individual airfoils. However, at t.3, the trailing edge vortex just rosses the leading edge of airfoil B. At this point, airfoil A sees a sharp spike in its C l as now the upwash of airfoil B and the effet of its own angle of attak dominates. This sharp spike in Cl of airfoil A auses a orresponding large downwash on airfoil B, whih therefore sees a sharp derease in its C l. The effet of the strong trailing vortex shed by airfoil A wanes with inreasing distane. In this ase, it is seen that the effet is minimum at t.3. After this, the effet of the angle of attak of individual airfoils and their effet on eah other (i.e. the downwash of airfoil A on airfoil B and upwash of airfoil B on airfoil A) take preedene. Platzer s results on the other hand are similar to that of a single airfoil ase of Wagner shown in Fig Effet of Relative Veloity of Tandem Airfoils on Wake Vorties As seen earlier, the effet of the trailing vorties wanes with inreasing distane from the airfoils. In this ase, the leading airfoil A is set into motion with sudden aeleration with a veloity ofva 15 m/ s. The trailing airfoil is also set into motion with a sudden aeleration but with a finite relative veloity with respet to airfoil A. Three suh veloities are onsidered, namely, VB 2,25,3 m/ s. The starting loation of the leading edge of the trailing airfoil B is ox ( o, Yo) (6, ) and both airfoils are at an angle of attak, αa αb 3. The origin O(, ) is loated at the leading edge of the leading airfoil A. The C l of airfoil A shown in Fig. 7a, whih onsists of three peaks and three troughs orresponding to the relative veloities of VB VA 5,1,15 m/ s. The highest and sharpest peak orresponding to the maximum relative veloity of 15m/s ours first at t 4 and the lowest and least sharp peak orresponding to the mini-mum relative veloity of 5m/s ours last at t 14. Given that the C l is alulated using the instantaneous irulation, the sharp peaks and troughs mean that d is signifiant. dt 394

9 Fig. 7. Airfoils with Relative veloity: Cl. Fig. 8. Airfoils with Relative veloity: wake strength. This shows up in the C l plot, espeially for the relative veloities of 1 and 15m/s, where there are signifiant C l maxima and minima. For airfoil B, three usps in the C l are seen orresponding to the three relative veloities as shown in Fig. 7b. Again, the large values of dγ show up in the C l plot of dt airfoil B as maxima and minima points. The evolution of the strength of the wakes with time is shown in Fig. 8. Corresponding to the inreasing C l, the indued drag also inreases as seen in Fig. 9a but around t 6, when the Cl sees a sharp drop, the indued drag also sees a sharp drop. These lowest indued drag points mean that at this point the airfoil is experiening a forward thrust, whih inreases with inrease in relative veloity. This an be explained further with the help of Fig. 1, where the loations of the three kinks in the C l plot orresponding to the relative veloity 5m/s are marked in Fig. 1a. The orresponding relative distanes be-tween the airfoils is shown in Fig. 1b. It an be seen from Fig. 1a that for the kink loated at A, the orresponding relative distane is 3 as shown in Fig. 1b. A shemati of the loation of the two airfoils in this position is shown in Fig. 1. Similarly, for the loation B, the relative distane is 1, whih means that the air foils are bumper to bumper as shematially shown in Fig. 1d. For position C, the relative distane is zero, whih means that the leading edges line up as shown in Fig. 1e. Suh kinks in the Cl plot are also present for the relative veloities of 1 and 15m/s and an be explained the same way. It an also be seen from Fig. 7 that till t 2 the urves merge but after this time, the slopes beome different for different relative veloities. In other words, slopes inrease with inrease in relative veloity. For example, for the urve orresponding to a relative veloity of 15m/s, the C l inreases till t 4. During this time, it an be seen from Fig. 8a, the trailing edge vortex has negative strength, whih auses an upwash and hene an inreased Cl. Around t 4, however, the strength sees a sharp 395

10 Fig. 9. Airfoils with Relative veloity: C d. Fig. 1. V A = 15m / s,v B = 2m / s. Fig. 11. Airfoils with Relative veloity: wake shape. 396

11 Fig. 12. Pithing: Tandem Airfoils vs Single Airfoil. positive spike whih reahes a maximum at t 6. It is at this time, the C l also sees a sharp drop sine now the trailing edge vortex auses a large downwash on the airfoil. At this time the airfoils are in position B shown in Fig. 1d. Soon after this, the trailing edge vortex dies as shown by its zero strength in Fig. 8a. Therefore, the airfoils are now dominated by themselves and not the trailing vortex. It is seen from Fig. 11a that the wake is not signifiant for airfoil A but there is a signifiant rollup of the wake vorties for the trailing airfoil B with the traes of a seondary roll-up as well. 3.6 Single Airfoil and Configuration of Two Airfoils: A Comparative Study of their Pithing Motion Sine the flow field around a single airfoil is different when it is in a onfiguration with another airfoil, it is expeted that there will be differenes in their aerodynami harateristis as well. Hene, a omparison of the same for both airfoils pithing is presented here for the following operating onditions. The pith axis is at the quarter hord,. 4 d Additionally, for the tandem airfoils, 1.5. α 3 1sinω t,.1; 15 m/ s;ω 3 / s 2 It is seen from Figs 12a and 12b that ompared to the single airfoil, the leading airfoil in the onfiguration generates greater irulation, Γ(t) and orresponding greater Cl () t for the yle starting at α 5, reahing a maximum of α 13 and yling bak to α 5. Both the leading and single airfoils have the same starting Γ(t) and Cl () t as expeted at α 5. The trailing airfoil, on the other hand, has a greater starting Γ(t) and Cl () t at α 5 ompared to both the single and leading airfoils. However, it generates lesser Γ(t) and orresponding lesser Cl () t during the yle between α 1 and α 5 ompared to both the single and leading airfoils. The hysteresis in Γ(t) as well as Cl () t is muh larger for the leading airfoil than the trailing airfoil. In fat, for the trailing airfoil, the hysteresis is almost negligible. It is also seen from Figs 12 and 12d that around α 6 and α 1 there are abrupt hanges in the oeffiients of pithing moment, C m and indued drag, C d for the trailing airfoil B. This an be 397

12 Fig. 13. Pithing: Tandem Airfoils vs Single Airfoil, Wake. Fig. 14. Only Leading Airfoil Pithing: C l and C d. explained using Fig. 13b as follows. It is seen in Fig. 13b that for α 6, there is a slight inrease in w() t, whih is the strength of the trailing vortex of airfoil B. The strength is still negative, whih indiates that the sense of the vortex is ounter-lokwise. This inrease in strength of w() t auses additional downwash on airfoil B, whih is an additional vertial omponent of veloity in the diretion opposite to lift, i.e. downwards. This auses a derease in the effetive angle of attak, α at α 6 and hene a derease in Cl () t as seen in 12b, whih in turn auses a derease in the nosedown or negative Cm() t as seen in 12 and a orresponding derease in the indued drag Cd () t as seen in 12d. At α 1 as seen in Fig. 13b there is a slight inrease in the positive or lokwise w() t. This auses an upwash on airfoil B, whih is an additional vertial omponent of veloity in the diretion of lift, i.e. 398

13 Fig. 15. Only Leading Airfoil Pithing: wake strength and roll-up. upwards. This auses an inrease in the effetive angle of attak, α at α 1 and hene an inrease in its Cl () t as seen in 12b, a orresponding inrease in the nose-down or negative Cm() t as seen in 12 and a large inrease in the negative Cd () t as seen in 12d. It is seen from 12d that unlike airfoil A and the single airfoil, the Cd () t of airfoil B hovers around zero and negative even for α=12, whih is a post-stall angle of attak for the single and leading airfoils. Then negative Cd () t indiates that the trailing airfoil is generating thrust at a post-stall angle of attak, whih an be a major advantage. Finally, as seen in Fig. 13a there is almost no wake roll-up for airfoil A, while airfoil B has roll-up similar to the single airfoil with traes of a seondary roll-up. 3.7 Effet of Only the Leading Airfoil Pithing on Configuration of Airfoils Results presented here are for a ase study of tandem airfoils, where only the leading airfoil is pithing: αa 3 1sinω t ; 15 m/ s;ωa 1 / s,3 / s The trailing airfoil is stationary at: d αb 5 ; 15 m/ s; 1.5 As seen in Fig. 14a the lift-hysteresis is nominal for the leading airfoil. On the other hand, it is interesting to note that although the trailing airfoil is not pithing, its C l undergoes a sinusoidal pattern. The C d in Fig. 14 shows onsiderable hysteresis for the leading airfoil while that for the trailing airfoil hovers around zero aompanied by negative spikes. As seen in Fig. 15a and 15b the leading airfoil has nearly zero wake roll-up while the trailing airfoil shows limited wake roll-up for ωa 1 rad / s but onsiderable wake roll-up for ωa 3 rad / s along with marked seondary roll-up. This an be explained using Figs 15 and 15d, whih shows large hysteresis in w for airfoil A. For airfoil B, w is not signifiant and hovers around zero for ωa 1 rad / s but for ωa 3 rad / s it is signifi-ant and marked by both positive and negative peaks. 4. CONCLSION A numerial study of the flow-field around tandem airfoils exhibiting unsteady motion is undertaken using an unsteady disrete vortex method. Various ases of un-steady motion are studied to gain insights 399

14 into the effet of the trailing edge vorties on the performane of the airfoils. Interation of free wake vorties with the air-foils as well as with eah other is studied. While a free wake vortex dissipates as it enounters an airfoil, numerial singularity arises when two suh vorties ome very lose to eah other. It is found that the ounter-lokwise trailing edge vortex shed from the leading airfoil in the immediate time-instant is very strong and its effet on both the air-foils dominates the flow field, namely ausing a strong downwash on the leading airfoil and a strong upwash on the trailing airfoil. As it moves downstream, its effet on the airfoils wanes. However, as it rosses the leading edge of the trailing airfoil, due to its position, the ounter-lokwise vortex now auses a strong downwash on the trailing airfoil. As a result, the irulation of the trailing airfoil hanges abruptly resulting in abrupt hanges in its C l and C d. Strong wake roll-up is seen for the trailing airfoil due to the additional irulation of the trailing edge vortex shed by the leading airfoil as well as that shed by the trailing airfoil itself. For the leading airfoil on the other hand, there is no signifiant wake roll-up. The effet of relative veloity, rates of pithing and phase lag on the strength of the trailing edge vortex and its movement is found to be signifiant. Hene, the method developed here is simple and effetive to study the flow-field around interating airfoils. REFERENCES Bai, M. (28). On predition of airraft trajetory at high-angles of attak: Preliminary Results for a Pithing Airfoil. In Pro AIAA Model Simul. Tehnol. Conf. Capee VR, Fleeter S. Wake Indued nsteady Aerodynami Interations in a Multi-Stage Compressor, AIAA Chorin, A. J. and P. S. Bernard (1973). Disretization of a vortex sheet, with an example of roll-up. J. of Computational Physis 13(3), Fage, A. and F. C. Johansen (1927). On the flow of air behind an inlined flat plate of infinite span. Pro. Roy. So A 116, 17. Fritz, T. and L. N. Long (24). Objet-Oriented nsteady Vortex Lattie Method for Flapping Flight. Journal of Airraft 41(6), Fulayter, R. D., P. B. Lawless and S. Fleeter (22). Rotor Generated Vane Row Off-Design nsteady Aerodynamis Inluding Dynami Stall: Part II: 3-D Rotor Wake Foring Funtion & Stator nsteady Aerodynami Response. AIAA Greenwell, D. G. (24). A review of unsteady Aerodynami Modeling for Flight Dynamis of Maneuverable Airraft. AIAA Hall, K. C. and W. S. Clark (1993). Linearized Euler Preditions of nsteady Aerodynami Loads in Casades. AIAA Journal 31(3), Husain, Z., M. J. Abdullah and T. C. Yap (25). Twodimensional analysis of tandem/staggered airfoils using omputational fluid dynamis. International Journal of Mehanial Engineering Eduation 33(3), 195. Jose, A. I. and J. D. Baeder (29). Steady and nsteady Aerodynami Modeling of Trailing Edge Flaps with Overhang and Gap using CFD and Lower Order Models. AIAA Katz, J. and B. Maskew (1988). nsteady low-speed aerodynami model for omplete airraft onfigurations. AIAA Paper ; J. Airraft, 25(4), Katz, J, and A. Plotkin (1988). Low-speed Aerodynamis. Cambridge niversity Press, 21. Nakagawa T. On nsteady Airfoil-Vortex Interation, ACTA MECHANICA 75, Katz, J. and D. Weihs (1978). Behaviour of vortex wakes from osillating airfoils. J. Airraft 15(12), MCroskey, W. J., K. W. MAlister, L. W. Carr, S. L. Pui SL, O. Lambert and R. F. Indergrand (1981). Dynami stall on advaned airfoil setions. J. Amer. Heliopter So. 26(3), 4-5. Radmanesh, M., O. Nematollahi1, M. Nili- Ahmadabadi1 and M. Hassanalian (214). A Novel Strategy for Designing and Manufaturing a Fixed Wing MAV for the Purpose of Inreasing Maneuverability and Stability in Longitudinal Axis. JAFM 7(3), Platzer, M. F., K. S. Neae and C. K. Pang (1993). Aerodynami analysis of flapping wing propulsion. AIAA Shokrgozar, A., A. B. Rahimi and H. Mozayyeni (216). Investigation of Three-Dimensional Axisymmetri nsteady Stagnation-Point Flow and Heat Transfer Impinging on an Aelerated Flat Plate. Journal of Applied Fluid Mehanis 9(1), Wagner, H. ber die Entstehung des Dynamishen Autriebes von Tragflugeln, Z.F.A.M.M., Vol. 5, No. 1, pp 17-35, Wu, J. C., C. M. Wang and I. H. Tuner (1986). nsteady Aerodynamis of Rapidly Pithed Airfoils. AIAA