MODELING OF UNSTEADY AERODYNAMIC CHARACTERISTCS OF DELTA WINGS.

Transcription

1 IAS00 ONGRESS MODEING OF UNSTEADY AERODYNAMI HARATERISTS OF DETA WINGS. Jouannet hristoher, rus Petter inköings Uniersity eywords: Delta wings, Unsteady, Modeling, Preliminary design, Aerodynamic coefficient. Abstract An analytic rediction model is resented to estimate aerodynamic coefficient of delta wing undergoing dynamic motions. The model is based on the introduction of a state sace ariable describing the ortex breakdown. This model will be suitable for modeling of flight dynamics in the reliminary design hase. Introduction High angles of attack region becomes more accessible to modern aircraft. Therefore, mathematical modeling of the unsteady aerodynamic forces and moments lay an imortant role in aircraft dynamic inestigation and stability analysis at high angles of attack. The model concentrate on the determination of lift, normal force, drag and moment coefficient for slender delta wings. The resent method is comare with ublished data to ensure agreement between mathematics and test cases. The resent method is based on Polhamus leading edge suction analogy, with integration of the work realized by Traub [3] onentional mathematical model based on aerodynamic deriatie concet is widely used in flight dynamic, but limitations of this method hae been shown by Greenwell []. This aroach is therefore based on the statesace reresentation used by Goman []. basically can be described by three states, that by themseles are rather well understood. These are otential lift, ortex lift and fully searated flow. Therefore, a critical factor in aerodynamic modeling is the transitions between these states that are described by internal state-ariables. These transitions describe the stall behaior and much of the unsteady behaior of the wing.. State-Sace ariable The nature of the flow field oer slender delta wing, characterized by two ortices that aear at the aex, is suitable for using internal state sace ariables. The ortex breakdown oer the slender delta wing can be defined by a nondimensional coordinate x [ 0, ]. Where x is reresented by the following first order differential equation: τ x = x0 ( α τ ) () t and t define the transient aerodynamics effects, t is the relaxation time constant and t is the total time delay. The driing function x 0 (a) can be obtain by wind or water tunnel measurements, in the resent model the following function is used: x 0 ( α) = ( ) σ ( α α ) e Method The methodology used in this aer is based on the hyothesis that the aerodynamic roerties of a delta wing, lift, drag, and itching moment, 73.

2 Jouannet. rus P. state sace ariable for the ortex breakdown is therefore introduced to couled the ortex lift coefficient to the burst osition. The ortex lift is finally exressed by: Figure Steady-state searation function for α * =40 = x Sin α osα (7) As shown in [5] a third comonent can be introduced to describe the lift coefficient when the flow is fully searated, no more ortex oer the wing. This term is called fully searated lift and is gien by: ( ) cos α sinα fs = fs x (8). ift coefficient The lift coefficient is based on Polhamus [4] leading-edge suction analogy, with some ariation exosed by Jouannet [5]. Polhamus describe the total lift oer slender delta wings with two arameters, otential lift and ortex lift, the total lift coefficient is determined by (x) = (3) Where is the otential lift coefficient defined by: k cos α sinα (4) = is the otential lift constant, witch can be determined from any lifting surface theory. Traub[3] and Jone[6] resented different ealuation method for. The resent model will use: = 4 Tanε for AR>0,5. As shown in Jouannet [9] the otential lift is limited to moderated angle of attack, therefore will be describe by: cos = x α sinα (5) Where x is the attenuation of the otential lift. The other term is the ortex lift obtain by: Figure ift coefficient decomosition = k cosα sin α (6) τ x = x0( α τ And is the ortex lift coefficient fixed to in the resent model. The ortex lift is affected by the ortex burst osition [7],[8], the internal Where x 3 describe the attenuation of the otential flow. And fs is obtain as half the alue of the drag coefficient of a delta wing at 90 degree of angle of attack. fs can be set a 0.7. fs = Df / The total lift is describe by: = 0 (9) The difference between the dynamic and static case is the internal state-sace ariable. For static case the state-sace ariable is defined by the following first order differential equation: τ x = x0( α) (0) During dynamic motions, the differential equation become: α ') fs () 73.

3 MODEING OF UNSTEADY AERODYNAMI HARATERISTIS OF DETA WINGS t and t can be aroximated by c = and 5c τ = according to Goman V V and hrabo[]. τ.3 Normal force coefficient The lift coefficient in Polhamus leading edge suction analogy deriate from the normal force, therefore the normal force coefficient can be decomosed in three coefficients in the same way as the lift coefficient. The normal force coefficient is defined by: = () N N N Nfs Where N is the otential comonent, and is exressed by: N = osα Sinα (3) is the same otential coefficient as before. In order to include the normal force decency on the ortex burst osition the state-sace ariable is included in the otential coefficient and gies the following exression: N = x 3 osα Sinα (4) The second term is related to the normal force due to the ortex. Polhamus defined it as following: T N = (5) osλ Where T is the leading edge thrust coefficient, and it is defined by: T = ( ) Sin α (6) i is defined by: i = i osλ (7) and are the same as the one used for lift coefficient. Including i in (x) gies the following exression for N : N = Sin α os (8) Λ osλ In order to include the deendency of the burst osition and the normal force due to the ortex the state-sace ariable reresenting the ortex breakdown osition is include in the reresentation of N and gies: N = x4 Sin os Λ osλ (9) α The third comonent is defined as the comonent due to fully searated flow, when the ortex breakdown has reached the aex, and it is reresented by the following equation: Nfs = Sinα (0) Df Where Df is the drag coefficient of a flat triangular lat erendicular to the flow. Df is set to.4 in this model. This comonent of the total normal force coefficient is only actie when the burst osition reaches the aex, therefore the final exression of Nfs is gien by: Nfs = ( x5 ) Sinα () Df Where x describe the state of the ortex breakdown. The final exression for the normal force coefficient is: = () N N N Nfs In the case of dynamic motions the equation is the same, only the first order differential equation reresenting the ortex breakdown state is changed from: τ x = x0( α) to τ x = x0( α τ α' ). The same change as for the lift coefficient..4 Drag oefficient The drag coefficient is defined from the lift coefficient and the normal force coefficient. D osα N = for a 0. (3) Sinα If a=0 the drag coefficient will be defined by D

4 Jouannet. rus P..5 Pitching moment coefficient The itching moment is defined from the lift coefficient and the osition of the center of ressure. m ( x x ) = (4) cg c with exerimental studies. From low angles of attack u to stall angles the agreement between ublished data and the resent redictions shows ery good agreement. At higher angles of incidence, the resent model underestimates the ublished data in the max region and in early ost stall behaiour. But the trend is still in accordance with the test data. 3 Static ases This section comare ublished wind tunnel data with the resent models. The main test set is based on the work realized by Wentz [7], often resented as a reference. 3. ift oefficient The static results from delta wing with swee angles from 45 to 55 degree are resented in figure (3). The resent method allows behaiour from low angles of incidence to ast stall region to be well catured. At low angles of attack, u to 0 degrees, the resent model agreement with the test data is ery good. Then u to the stall angle the resent rediction oer estimates the lift coefficient. For higher angles of attack agreement with exerimental data is seen to be ery good u to ost stall region. In all three cases max is seen to bee well estimated. Figure 4 omarison of theory and resent method for 60 to 70 degree delta wing. The resent model match exerimental data u to highest angle of attack with aailable data. 3. Normal force oefficient The normal force coefficient has only been comare and comuted for a 70-degree delta wing. The resent model shows ery good agreement with test data, figure(5). At low angles of attack the resent method oer estimates the normal force coefficient. Figure 3 omarison of theory and resent method for 45 to 55 degree delta wing. For delta wing with swee angle between 60 and 70 degree, resented in figure (4), the resent method shows ery good agreement Figure 5 omarison between test data and resent model for the normal force coefficient of a 70-degree delta wing. 73.4

5 MODEING OF UNSTEADY AERODYNAMI HARATERISTIS OF DETA WINGS At high angles of attack, the resent model is underestimating the normal force coefficient, but the trend in ost-stall region is in accordance with test data. 3. Drag oefficient The drag coefficient has been comuted for a 70-degree delta wing and comare with ublished wind tunnel test from Soltani[5]. Figure 6 omarison between test data and resent model for a 70-degree delta wing. Figure (6) illustrates this comarison. The resent model shows good agreement with test data. At low angles of attack the resent model oer estimates the drag coefficient, as for the lift and normal force coefficient. This is a direct result from the modelisation of the drag coefficient. At high angles of attack, oer 5 degree, the resent model shows good agreement, but fail to ick u the maximum alue at 45 degree. This difference could be seen as the difference obsered in the lift coefficient for a 70 degree delta wing in figure(4). 3.3 Pitching Moment oefficient The itching motion in static case has only been comuted for a 70-degree delta wing. At low angles of attack the resent model underestimates the itching moment coefficient, this is mainly due to the modelisation of the location of the center of ressure. At higher angles, oer 5 degree the resent method resent good agreements with the wind tunnel test data. Figure 7 Pitching moment for a 70-degree delta wing. The accuracy of the model is mainly deendent on the center of ressures model. Imroing this one would reduces the differences at low angles of attack. 4 Dynamic Motions omarison with test cases has been limited to the study roduced by Soltani et al. [9] on 70-degree delta wings. 4. ift oefficient Test cases are itching motions from 0 to 55 degree with different frequencies. Parameters used in the resent model are indeendent from the shifting, and are based on water tunnel results [9]. Figure (8) show comarison between the resent model and itching motions from 0 to 55 degrees of angles of attack with a reduced frequency k=0.07. In figure (8) all the comonent of the lift coefficient use the same time constant t and t. During the u stroke motion the resent model resent a ery good agreement with the exerimental data, but the maximum alue and the stall angle are not well matched. During the down stroke the resent model underestimated the lift coefficient. 73.5

6 Jouannet. rus P. During the down stroke the resent model under estimates the coefficient. The trend is resected but there is a deiation from the alue. Figure 8 omarison between exerimental data and resent method with a reduce frequency k=0.07. omarison with other reduced frequency, and ariable time constant are resented in [5]. The use of different time constant for the three different state-sace ariable resent ery accurate results with the test case, illustrated in figure (9) Figure 0 Normal force coefficient for a reduced frequency k=0.07 In figure () the normal force is comuted for a reduced frequency k=0.65. The same henomenon as in for the lift coefficient is obsered, the resent model accuracy increased with increasing reduced frequency. In figure () the resent method shows ery good agreement with the test data. The maximum alue is underestimated, but the trend is resected. Figure 9 omarison between test case and the resent method with k=0.07. Different time constant for the different inut. The same results hae been found for other reduced frequency[5], where the use of different time constant indicates more accurate results. 4. Normal Force oefficient The normal for determined by the resent method hae been comare with wind tunnel test resent by Soltani[8]. In Figure (0) the normal force coefficient is comuted without changing the arameters from the static case, only the itching rate has been taken into account. The resent model oer estimates the normal force coefficient during the u stroke, and misses the maximal alue. Figure Normal force coefficient for a reduce frequency k=0.65. As the reduced frequency increased there is less need for different time constant to increase the accuracy of the model. 4.3 Drag oefficient omarison between test data and the resent method is resented in figure (3) for a 70-degree delta wing at a reduced frequency k=

7 MODEING OF UNSTEADY AERODYNAMI HARATERISTIS OF DETA WINGS Figure Drag coefficient for a reduce frequency k=0.098,for a 70 degree delta wing. At low angles of attack the resent model indicates oor correlation with the test data. As the angles of attack increase the resent model shows better agreement on the u stroke, and accetable agreement on the first art of the itch down motion, from maximum angles of attack to 40 degree. Under the correlation is oor. The drag coefficient is calculated from the lift and the normal force coefficient. As obsered for the normal force and the lift coefficient the resent model is less accurate at low angles of attack, under 5 degrees, and on the down stroke. This is directly reflected in the drag determination. The drag coefficient accuracy will be imroed by imroing lift and normal force coefficients. 4.4 Pitching Moment oefficient Figure (3) illustrates the comarison between resent model and wind tunnel test data from Soltani on a 70 degree delta wing, with a reduced frequency of k=0.07. At low angles of attack the resent model underestimates the itching moment, as the angle of attack increases the model become more accurate. The resent model fails to ick u the maximum. The ery simle model for the center of ressure is one of the main exlanations for those differences. Figure 3 Pitching moment coefficient for a reduced frequency k=0.07. Figure (3) illustrate the comarison between the resent model and wind tunnel test data for a reduce frequency k=0.65. With higher reduced frequency the accuracy of the resent model increase, figure (4). At low angles of attack, u to 5 degree, the resent model underestimates the alue. This is resulting from the reresentation of the center of ressures osition. At low angles of attack, before the burst reaches the trailing edge, the osition of the center of ressure does not agree with reality. Figure 4 Pitching moment coefficient for a reduced frequency k= Obseration Prediction of aerodynamic coefficient in static case shows great agreement with ublished test data, for delta 70-degree delta wing. And could be used for delta wing from 45 degree swee u to 70 degree swee, results exosed for the lift coefficient. During dynamic itching motions the accuracy of the resent method increased with increasing reduced frequency. Present models 73.7

8 Jouannet. rus P. show good agreement with test data under itching motion, besides from the drag coefficient, in less magnitude for the itching motion. ift coefficient and normal force coefficient roduce ery good correlation with ublished test data in both static and dynamic motion. Showing that the use of steady state ariables is a simle and efficient way to roduce simle and accurate model for lift, normal force and itching moment coefficient. Induced drag has been determined from normal force and lift coefficient. This method shows accetable alue in the static case. During dynamic motion oor agreement is obtain, mostly at low angles of attack and during down stroke motions. This is a direct consequence from the model based on normal force coefficient and lift coefficient. The drag coefficient can be imroed by imroing both the lift coefficient or the normal force coefficient, or consider other methods to determined it. Determination of itching moment has shown good agreement with test data at high alha in static condition and during dynamic motions. At low angles of attack the resent model has oor agreement with test data, this is mostly deending on the modelisation of the center of ressure. 6 onclusion A simle model has been resented that redicts the lift, normal force, itching moment and drag coefficient for a ariety of lanforms. The resent method allows determination of these characteristics, suitable for use in flight dynamic simulation at the reliminary design stage, u to ery high angle of attack, including the unsteady behaiour of these characteristics, with an excetion for the drag coefficient. It must be noted that the method is based on seeral simlification and its limitations should be recognized. Aerodynamic coefficients were determined using the leading edge suction analogy with emirical modification and addition of steadystate ariables to describe the ortex breakdown and unsteady behaiour. The method was comare to ublished exerimental results, with ery good accuracy being demonstrated in static cases. The unsteady behaiour of aerodynamics coefficients from delta wings hae been resent and indicated that simle models can describe the trend of the unsteady behaiour and be suitable for concetual design studies with good confidence. Those models should not been seen as models for use when wind tunnel or flight data are aailable. The model does, howeer, contain a sufficient number of degrees of freedom so that the arameters can be adjusted to fit almost any set of data. 7 Summary The following equation hae been determined and used in this aer. = 0 x cos α sinα ( x ) cos α sinα x Sin α osα fs = x osα Sinα ( x ) N 3 x4 = x x D 4 3 x fs os Λ osλ osα ( x ) 3 cos α 3 ( x ) cos α osλ x Df osλ 5 Df Sin α Sinα os α 5 Sinα Sinα τ x = x0( α) x 0 ( α) = σ ( α α ) e The following coefficient hae been used for the 70 degree delta wing: Df =.4 s,,4 = 0.3 s 3 = 0.5 s 5 = 0. a * = 33 a *,3,4= 39 a * 5=

9 MODEING OF UNSTEADY AERODYNAMI HARATERISTIS OF DETA WINGS References [] Greenwell, D. I. Difficulties in the Alication of Stability Deriatie to the Maneuering Aerodynamics of ombat Aircraft, IAS Paer , the st ongress of the International ouncil of the Aeronautical Sciences, Set 998, Melbourne, Australia. [] Goman, M. G. And A. N. hrabo. State-Sace Reresentation of Aerodynamic haracteristics of an Aircraft at High Angles of Attack, Journal of Aircraft, Vol. 3, No.5, Set.-Oct. 994, [3] Traub,. W., Prediction of Vortex Breakdown and ongitudinal haracteristics of Swet Slender Planforms, Journal of Aircraft, Vol. 34, No 3, May- June 997. [4] Polhamus E.., A oncet of the Vortex ift of Shar-Edge Delta Wings Based on eading-edge Suction Analogy, NASA TN D-3767, Oct. 966 [5] Jouannet,. and rus, P. ift oefficient Predictions for Delta Wings Undergoing Pitching Motions, AIAA , 3 nd AIAA Fluid Dynamic onference, June 00, St ouis, USA. [6] Jones, R. T., Proerties of ow Asect Ratio Wings at Seeds Below and Aboe the Seed of Sound, NAA Ret. 835, 946, [7] Wentz, W. H., Jr & ohlman D.., Wind Tunnel Inestigation of Vortex Breakdown on Slender Shar-Edged Wings, NASA Research Grant NGR , 968 [8] Soltani, M. R. & Bragg, M. B. & Brandon, J. M., Exerimental Measurement on an Oscillating 70- degree Delta Wing in Subsonic Flow, AIAA P, 998 [9] Jouannet, h., A Water Tunnel Study of Vortical Flows oer Delta Wings, inköings Uniersitet, ith-ip-ex-786,