This is the second article of a series that deals with the calculation of the aerodynamic unsteady forces on lifting surfaces. It presents some new important details on the lifting surface theory that performs oscillations in subsonic flow. These features will be applied to the aerodynamic response to certain kind of gusts and to the flapping wing calculations

Valentin Adrian Jean BUTOESCU

BUTOESCU Valentin Adrian Jean

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INCAS BULLETIN

Aerodynamic Interference between Oscillating Lifting Surfaces and Fuselage Part 2: Some Applications of Oscillating Lifting Surface Theory

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INCAS BULLETIN, Volume 6, Issue 1/ 2014, pp. 3 – 11         ISSN 2066 – 8201  Aerodynamic Interference between Oscillating Lifting Surfaces and Fuselage Part 2: Some Applications of  Oscillating Lifting Surface Theory Valentin Adrian Jean BUTOESCU* *Corresponding author INCAS – National Institute for Aerospace Research “Elie Carafoli” B-dul Iuliu Maniu 220, Bucharest 061126, Romania vbutoesc@incas.ro DOI: 10.13111/2066-8201.2014.6.1.1  Abstract: This is the second article of a series that deals with the calculation of the aerodynamic unsteady forces on lifting surfaces. It presents some new important details on the lifting surface theory that  performs  oscillations  in  subsonic  flow.  These  features  will  be  applied  to  the  aerodynamic response to certain kind of gusts and to the flapping wing calculations Key Words: Lifting surface, integral equation, harmonic oscillations, generalised airforces, flutter, flapping wings, Fourier series. 1. INTRODUCTION In our previous articol we presented the calculation of the airforces on lifting surfaces that perform  hamonic  oscillations  of  small  amplitude  in  a  subsonic  flow.  The  present  paper continues the investigations in the domain of the oscillation lifting surface theory in subsonic flow with a property of the integral equation. This property will allow us to find the pressure distributions for both the harmonic and general periodic oscillations expressed in real terms. This considerations will be the basis of a further application of the present theory to the flapping wing case. Basically, we used the same references as in our previous paper, [1]. 2. MODES OF OSCILLATION AND GENERALISED FORCES The generalised forces are usually presented for planar configurations [14]. We shall extend here the definition of these forces for the non-planar wing case. Like  in  the  previous  paper,  we  consider  lifting  surface  (wing  or  empennage).  It  is denoted here by the symbol (W0) and is represented by the next parametric form: ) , ( 0 s M M     (1) In the above formula, λ is a chordwise parameter (=0 for the leading edge and 1 for the trailing edge) and s a spanwise parameter ( 1 1    s for symmetrical wings). Suppose now that  this  surface  executes  oscillations  of  small  amplitude.  Geometrically,  it  has  different instantanous shapes (W) given by the equation: Valentin Adrian Jean BUTOESCU  4  INCAS BULLETIN, Volume 6, Issue 1/ 2014 ) ; , ( ) , ( 0 t s O l s M M         (1’) In the above equation, l is a reference length and  ) ; , ( t s O  is the small displacement. Then  ) ; , ( t s O  <<1 is the amplitude of the oscillation.  Fig. 1 The wing coordinate systems oxyz and oxsz Suppose that  ) ; , ( t s O can be represented as a linear combination of a number of K modes of displacements:         Kkk k t q s o l t s O1, ) ; , (   (2) Here,    s ok , ,  k=1,2...K  represent  the  shapes  of  the  K  oscillation  modes,   , t qk  k=1,2...K denote the generalised coordinates. The vectors    s ok ,  are often (but not always) parallel to    s n , 0 , the local normal to (W0). Consider now the virtual displacement of the wing:        Kkk k q s o l O1,   (3) Here,  k q   k=1,2...K represent small and arbitrary increments. So  we  can  now  calculate  the  virtual  work  done  by  the  lifting  forces  per  unit  area  t s L ; ,   with the virtual displacement (3)                 Kk Wk kWdS n o t s L q l dS O n t s L W1 0 0; , ; ,     (4) On  the  other  hand    kKkk q t Q W     1,  where    t Qk ,  k=1,2...K  represent  the generalised force in the k’s mode and at time t. Taking into account that  k q   are arbitrary, there results:             0; ,Wk k dS n o t s L l t Q   (5) 5  Part 2: Some Applications of Oscillating Lifting Surface Theory  INCAS BULLETIN, Volume 6, Issue 1/ 2014 Suppose now that the wing oscillates harmonically, t ik k e q t q   0 ) (   (6) where  represents the circular frequency of the oscillations. In our previous work we have shown that the aerodynamic equations can be linearised, so that we can write:   t ikKkk e q s p U t s L        012 ,21) , , (   (7) In the above formula,    t pk  – the load, or pressure coefficient jump between the upper and  lower  sides  of  the  wing,  has  been  calculated  by  solving  the  lifting  surface  integral equation:                 etc. , , or, , , ; , ,, , , , , , ; , , , , ,  0 01 810s x x W W z y xz y x w dS p k M z y x KWk k               (8) where K1 is the kernel of the equation given in [1] (and too complicated to be reproduced here), and    t ik kk x k ke n nNNnUlk n o ik n w      1 00 1 ;     (9) In (9),  n is the unit normal vector to the oscillating surface in the k mode, with  k o insead of O,  0 n is the unit normal vector to (W0), and  k n1 is the infinite small vector (index 1) due to oscillating in mode k. Returning to (5), and using (7), we get:               Km Wk mt im k dS n o s pl be q bl U t Q102 2210,1    (10) In (10) b represents the wing span. We introduce a new notation        0,1,Wk m m k dS n o s pl bQ   (11) So we can write       Kmt im m k k e q Q bl U t Q10 ,2 221   (12) From (11) one can see that Qk,m is dimensionless. This number is called the generalised force coefficient of the load calculated with the mode m and with displacement after the mode k.    Valentin Adrian Jean BUTOESCU  6  INCAS BULLETIN, Volume 6, Issue 1/ 2014 3. SOME PROPERTIES OF THE INTEGRAL EQUATION AND THEIR CONSEQUENCES 3.1 System of integral equations For  a  better  understanding,  we  will  write  again  from  our  previous  article  [1]  the  two equations of the lifting surface that oscillates harmonically in subsonic flow:       0 01*1 1 810*0 0 81, , ), ( ) , , (        (II)          ) ; , , ( ) , , ( ) ; ; , , ; , , ((I)            ) , , ( ) , , ( )      ; , , ; , , (00W W z y xk z y x w dS p k M z y x Kz y x w dS p M z y x KWW                   (13) where  0 1 1 10 0n O ik n wn wxx      (14) are the normalwash functions, calculated from (2). The first integral equation corresponds to the steady flow (index 0), whereas the second integral equation corresponds to the oscillatory flow (index 1). 3.2 Negative frequency. Conjugate of the integral equation We observe that the next equations hold true:        k y x K k y x Kk z y x w k z y x w,...; , ,...; ,; , , ; , ,1 11 1    (15) Consequently,         00*1 1 811*1 1 811; , ,; , ,WWdS p K k z y x wdS p K k z y x w  (16) Moreover, all the equations in [1] are available for negative frequency, i.e for – ω or – k. 3.3 Harmonic oscillation, case 1 a) Suppose we have a certain harmonic oscillation of a wing:   t s o l t s Oc       cos , ) , , (   (17) We can write        t t tc e e s o l e s o l t s O                 , , Re ) , , ( 21     (18) So we have two oscillations, one with positive and the other with negative frequency:               t it ie t s o l t s Oe t s o l t s O, , , ,, , , ,2121  (19) 7  Part 2: Some Applications of Oscillating Lifting Surface Theory  INCAS BULLETIN, Volume 6, Issue 1/ 2014 We can now calculate the normalwash for each of the two oscillations:         w n o ik n ww n o ik n wxx0 2110 211     (20) Solving the two integral equations (16), we obtain  * * *; p p p p    . Now we have the two pressure coefficients distributions:                t s P e p e p e p t s Pe p t s Pt i t i t it i, , , ,                                                    , ,* * **  (21) The equations being linear, we can write             t s P t s P t s P t s P t s P t s P c , , Re 2 , , , , , , , , ) , , (                   (22) or,       t p t p t s P c        sin Im cos Re 2 ) , , ( * *   (23) b) Harmonic oscillation, case 2 For other harmonic oscillation of the wing   t s o l t s Os       sin , ) , , (   (24) We use again the Euler’s identity,        t tits e e s o l e s o l t s O                , , Im ) , , ( 21     (25) Finally, we get:         t p t p P t s P s          cos Im sin Re 2 Im 2 ) , , ( * *   (26) c) General periodic oscillation A general periodic oscillation of the wing is described by the following function:   ) ( , ) , , ( t q s o l t s O       (27) where q(t) is a periodic function of period T, that can be approximated by a Fourier series with N terms:           NN nt inNN nntTinNnn nn e c q e c q ntTb ntTaaq t q 0201002sin2cos2) (   (28) In the above relations, q0 is the amplitude of the oscillation, a0, an, bn and cn are the Fourier series coefficients and   N N n nTn ,    ,2     (29) We remind here that the coefficients cn and a0, an, bn are related by the next equations: Valentin Adrian Jean BUTOESCU  8  INCAS BULLETIN, Volume 6, Issue 1/ 2014  nn nnn nncb i aca cb i ac2                          20 0   (30) So, (27) becomes:                   s o c q l t s O t s Os o c q l e s o c e s o c q le s o c q l e c q s o l t s ONnn nNnt int inNN nt inNN nt inn nn n, , , , ,, , ,, , ) , , (0 01) ( ) (0 0100 0                                        (31) where                   t innt innnne t s o c q l t s Oe t s o c q l t s O, , , ,, , , ,0) (0) (  (32) Now we can calculate the normalwashes corresponding to the n-th and 0-th terms:                                      0 0) (0 1) () (0 1) (xnxnnxnn ww n o ik n ww n o ik n w    (33) We use  ) ( ) ( n n w w   and  ) ( ) ( n n w w    to solve the integral equations (15), and we get ) (n p  and  ) (n p . Then,                                         , , , ,                                                       , ,* **t s P e p e p t s Pe p t s Pn t i n t i n nt i n nn nn  (34) The third normalwash in (33) is used to solve the steady flow integral equation (13 I), and so we get the solution  0 p . Finally, we obtain the solution to our problem:                        01sin Im cos Re 2 , , p t p c t p c t s PNnnnn nnn   (35) 4. NUMERICAL EXAMPLE As  a  numerical  application,  we  considered  an  entomopter  rectangular  wing  (c=10mm, b=30mm), oscillating after a plunging saw teeth-shaped mode. This mode is represented in fig. 2 as a Fourier expansion with only 5 terms, of which only 3 are not equal to 0. The velocity U is 5m/s. 9  Part 2: Some Applications of Oscillating Lifting Surface Theory  INCAS BULLETIN, Volume 6, Issue 1/ 2014  Fig. 2 Plunging mode, T=0.1s The  angular  frequency  is  s rad/ 8 . 62     so  that  k=0.126.  Then  for  N=5,  only  3 frequencies are needed to be considered:           5 ; 3 ; 5 3 1  or  k k k k k k 5 ; 3 ; 5 3 1     The non-zero Fourier coefficients are:  008 . 0 ; 023 . 0 ; 203 . 0 5 3 1       c c c . Introducing the geometry and the three reduced frequencies into the DLM code, we get three complex values for the load distributions, i.e p1, p2 and p3. Their real and imaginary parts are represented in fig. 3, respectively 4, at 2y/b =0.705 section only. Then,  . 0 0  p   Fig. 3 Chordwise pressure distributions at 2y/b=0.705, real parts (for k=0.126, 0.377 and 0.628)  Fig. 4 Chordwise pressure distributions at 2y/b=0.705, imaginary parts  (for k=0.126, 0.377 and 0.628) 0.2 0.1 0 0.1 0.2101221 z  ( )0.2 0.2  0 0.5 10.500.51 0.5530.347 p12  p22  p32  0.925 0.025 x0 0.5 11050 0.051 6.988 p13  p23  p33  0.925 0.025 xValentin Adrian Jean BUTOESCU  10  INCAS BULLETIN, Volume 6, Issue 1/ 2014 Now we can use the formula (35). In fig 5 it is represented the load distributions at the same wing section over a period of time (t=0; T/4; T/2; 3T/4). The distribution for t=T is equal to the distribution at t=0, since the load function is periodic as well; so, it has not been represented  in  fig.  5.  The  same  load  distribution  is  depincted  in  fig.  6  at  30  different moments of a period (t=0;T/30;2T/30,3T/30....T).  Fig. 5 Chordwise pressure distribution (at 2y/b=0.705) on plunging wing, at t=0,T/4,T/2 and 3T/4  Fig. 6 Load distribution evolution with time, over a period, T and at 2y/b=0.705 0 0.2 0.4 0.6 0.8 10.30.20.100.10.20.3 0.2630.263 P1P2P3P40.925 0.025 xPRESS11  Part 2: Some Applications of Oscillating Lifting Surface Theory  INCAS BULLETIN, Volume 6, Issue 1/ 2014 5. CONCLUSIONS 1.  An extension of the generalised forces for non-planar configurations has been presented. 2.  We proved that the linearised equation used for the lifting surface theory presented in the previous paper enjoys the property that the load distribution corresponding to a “negative frequecy” is the conjugate of the load that corresponds to the “positive frequency”, if the planform,  angular  frequencies  modules  and  oscillation  modes  are  the  same.  This property can be used to obtain the real load distribution not only for the harmonic case, but also for  the general oscillation case. A numerical example has been given. This property will be aplied for the calculation of gust air loads and to the flapping wings entomopter aerodynamics. REFERENCES [1]  V.  A.  J.  Butoescu,  Aerodynamic  Interference  between  Oscillating  Liftin  Surfaces  and  Fuselage  Part1: Oscillatng Lifting Surfaces, INCAS Bulletin, Vol. 5, Issue 3, (online) ISSN 2247–4528, (print) ISSN 2066–8201, ISSN–L 2066–8201, DOI: 10.13111/2066-8201.2013.5.3.3, July-Sept. 2013, pp. 21-35. [2] E. Suciu, N. Stathopoulos, M. Dickinson and J. Glaser, A DLM-Based Msc Nastran Aerodynamic Flutter Simulator For Aircraft Lifting Surfaces, MSC.Software 2013 Users Conference Paper No. AM-CONF13-34. [3]  W.  P.  Rodden,  P.  Taylor,  S.  C.  McIntosh,  Jr.,  Improvements  to  the  Doublet-Lattice  Method  in MSC/NASTRAN, 2012. [3]  W.  P.  Rodden,  P.  F.  Taylor  and  S.  C.  McIntosh  Jr.  Improvements  to  the  Doublet-Lattice  Method  in MSC/NASTRAN.  MSC  1999  Aerospace  Users'  Conference  Proceedings. http://www.mscsoftware.com/support/library/, conf/auc99/p03799.pdf, Last retrieved: 15 Jan 2012. [4] P. -O. Persson, D. J. Willis, and J. Peraire, Numerical Simulation of Flapping Wings using a Panel Method and  a  High-Order  Navier-Stokes  Solver,  Int.  J.  Numer.  Meth.  Engng,  (01:1-20,  Prepared  using nmeauth.cls [Version: 2002/09/18 v2.02]), Vol 89, Issue 10, pp 1296–1316, 9 March 2012, Online ISSN: 1097-0207, DOI: 10.1002/nme.3288, 2011. [5] L. H. van Zyl, Robustness of the Subsonic Doublet Lattice Method, The Aeronautical Journal, Vol. 107, No. 1071, ISSN 0001-9240, May 2003, pp. 257-262. [6] V. N. Constantinescu, V. Butoescu, O extindere a teoriei suprafetei portante la evolutii pe traiectorii oarecare, cu viteze subsonice, S.C.M.A.,Tom 40, noiembrie – decembrie, 1981. [7] L. Morino, Ching-Chang Kuo, Subsonic Potential Aerodynamics for Complex Configurations: A General Theory, AIAA Journal, Vol. 12, No. 2, ISSN 0001-1452, EISSN 1533-385X, pp. 191-197, February 1974. [8] R. Dat, La theorie de la surface portante appliquee a l'aile fixe et a l'helice, Rech. Aerosp., No 4, Juillet - Aout 1973. [9] J. P. Giesing, T. P. Kalman and W. P. Rodden, Subsonic Steady and Oscillating Aerodynamics for Multiple Interfering  Wings,  Journal  of  Aircraft,  Vol.  9,  No.  10,  ISSN  0021-8669,  EISSN  1533-3868,  doi: 10.2514/3.59066, pp. 693-702, Oct.1972. [10]  Y.  Akamatsu,  R.  Dat,  Calcul,  par  la  méthode  du  potentiel,  des  forces  instationnaires  agissant  sur  un ensemble de surfaces portantes. (Calculation of the instationary forces which are effective on a ensemble of airfoils with the potential method). (French), Rech. Aérosp. pp. 283-295, 1971. [11] T. P. Kalman, W. P. Rodden, J. P. Giesing, Application of the Doublet  - Lattice Method to Nonplanar Configurations in Subsonic Flow, Journal of Aircraft, Vol. 8, No. 6, ISSN 0021-8669, EISSN 1533-3868, pp. 406-413, June 1971. [12]  ***  AIAA  Selected  Reprint  Series,  Vol  V,  Aerodynamic  Flutter,  Editor  I.  E.  Garrick,  NASA  Langley, Research Centre, Hampton, Va., March, 1969.  [13]  E.  Albano,  W.  Rodden,  A  Doublet  -  Lattice  Method  for  Calculating  Lift  Distributions  on  Oscillating Surfaces in Subsonic Flows, AIAA Journal ,Vol. 7, no. 2, February, pp. 279-285, 1969. [14] D. E. Davies, Calculation of Unsteady Generalised Airforces on a Thin Wing Oscillating Harmonically in Subsonic Flow, Reports and Memoranda No. 3409, August 1963. [15] H. Multhopp, Methods for Calculating the Lift Distributions of Wings (Subsonic Lifting - Surface Theory), A.R.C., R.&M.2884, 1950.  

Aerodynamic Interference between Oscillating Lifting Surfaces and Fuselage Part 2: Some Applications of Oscillating Lifting Surface Theory

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