chord point, AxA axis is directed forward of the aircraft along the mean aerodynamic chord, AzA axis is perpendicular to AxA and is directed downward.
THIRD SEMINAR ON RRDPAE'98 NUMERICAL ANALYSIS OF RAPID, HIGH ANGLE OF ATTACK MANOEUVRES Tomasz Goetzendorf-Grabowski Warsaw University of Technology
Abstract This paper presents numerical analysis of rapid, high angle of attack manoeuvres. The physical and mathematical models of an aircraft motion were shown. Paper contains selected results of the following manoeuvres: water bomb dropping from the fire fighting aircraft PZL-106 Kruk, theoretically predicted Cobra manoeuvre for the jet combat aircraft Skorpion (project developed in PZL-Okęcie). Calculations for Cobra manoeuvre were performed in the extended range of the angle of attack (from 0 to 90). 1. Introduction The rapid manoeuvres of the aircraft are the most difficult and dangerous phase of the flight. It requires, from a pilot, a strong concentration and very fast reaction. These manoeuvres are normal during aerobatics and are known for many years. However, sometimes ordinary aircraft, which normally are not used in aerobatics, do very rapid manoeuvres. In such manoeuvres, flight parameters change abruptly and reach very high values. Modern combat aircraft do normally very rapid manoeuvres in which flight parameters, especially angle of attack have very high values. The analysis of these manoeuvres needs aerodynamic data for wide range of angle of attack. Also, the control function necessary to realise correctly rapid manoeuvres should be known. The aerodynamic data were taken from the wind tunnel investigation and numerical calculation. The control function was determined from the real manoeuvres analysis. 2. Physical and mathematical model The manoeuvres considered in the analysis do not require a full (three dimensional, six degrees 3 of freedom) model of aircraft dynamics . Therefore, the following assumptions have been used to establish the physical model: the aircraft is considered as a rigid body having three degrees of freedom: two linear
displacements (x0, y0) in vertical plane of symmetry and the pitch angle , horizontal elevator is movable and can be used for control, but it is weightless and cannot vibrate, the air is calm, windless, and undisturbed; air parameters change according to the Standard Atmosphere, 2
Dynamic equations of motion were derived by use the fundamental laws of mechanics: the balance of momentum theorem (1) and the balance of the moment of momentum theorem (2).
F, t
(1)
K o K o Vo M o . t
(2)
Dynamic, nonlinear equations of motion for aircraft of three degrees of freedom have been written in the body axis system. The origin of the axis system coincides with the mean wing quarterchord point, AxA axis is directed forward of the aircraft along the mean aerodynamic chord, AzA axis is perpendicular to AxA and is directed downward. The equations of motion has the following form:
Q W) mz c Q mx c Q 2 m (U ) X T mg sin X( , Q, w Q U) mx c Q mz c Q 2 m (W
(3)
) Z T mg cos Z( , Q, w mz Q W J y Q mx c W mx c UQ mz c U c ) M T mgz c sin mgx c cos M A ( , Q, w where U, W are velocity components in the body frame of reference, Q - angular velocity, - pitch angle, MA - pitching moment about the mean quarter-chord point (1/4 MAC point), xc, zc - mass centre coordinates in the body frame of reference, w - acceleration along Az axis. Aerodynamic forces X,Z and pitching moment MA depend on
angle of attack , pitch rate Q and acceleration w . XT,ZT,MT denote the generalised forces coming from engine trust. Equations of motion are completed with kinematic relation between body axis system AxAyAzA and gravity axis system 0x0y0z0. The kinematic relations are as follows:
x 0 U cos W sin z 0 U sin W cos Q
directly the aerodynamic characteristics of the aircraft, especially the lift, drag and pitching moment.
hopper
XA A
(4)
0
Numerical analysis consists in numerical flight simulation. The equations of motion presented above were numerically integrated in the time domain. 3. Analysis of water bomb dropping Over the last years there have been many forest fires in the world. It happens that fire brigades need a small fire fighting aircraft, which could carry about 1500-2000 l of water. The fire fighting aircraft drops a water bomb very fast, in about 1s (Fig. 1). The mass of the water bomb is equal approximately to 40% of the total mass of the aircraft. Additionally, a water drop shifts the gravity centre more than 10% of MAC (Fig. 2). The numerical simulation was used to investigate the 1 reaction of the aircraft during the water dropping .
Vo
Change of gravity center position
ZA
FIG.2. Side view of fire fighting aircraft PZL-106 Kruk
Numerical simulations have been performed for the following initial conditions: speed V0 = 40 m/s and flight altitude H = 100 m. For such initial conditions the state of equilibrium has been found: angle of attack , angle of elevator deflection H and required thrust T have been computed. It was assumed that water release lasted 1 s according to the function presented at Fig. 3. 3
total mass [kg x 10 ] 4.00
3.50
3.00
2.50
2.00 4.00
5.00
6.00
7.00
time [s] FIG. 3. Change of aircraft mass versus time during water release
FIG. 1. Water bomb dropping. If the hopper is designed properly, its whole capacity can be drained in about 1 s and about 90 % of water is concentrated in narrow, almost vertical column of liquid. (Courtesy of PZL Warszawa-Okęcie)
The additional assumption has been used to establish the physical model of an aircraft dynamics in this case - the water, being released from the hopper, influences the mass and moments of inertia only; it does not influence
The following various models of water release have been considered: The aircraft is not controlled. Elevator is fixed at the angle of deflection corresponding to the state equilibrium at initial condition; The aircraft is controlled, Fig. 4. Elevator is being deflected from the value corresponding to the state of equilibrium at the initial condition to the value corresponding to state of equilibrium after water bomb dropping. Elevator deflects simultaneously with water release and according to the same, harmonic, timedependent function;
The aircraft is controlled but with a time-lag (Fig. 4). The pilot does not start reacting before the whole hopper is empty; -2.00
[deg]
not controled (stick fixed)
-3.00
simultenous control
angle of attack [deg]
H
delayed control
-4.00
elevator deflection
4) the angle of attack decreases and stabilises very quickly. Fig.7 presents the change of pitch rate versus time. For not controlled and delayed controlled cases (case 1 and 4) pitch rate changes rapidly. All controlled cases result in quick decreasing of the pitch rate. 30.00
-5.00 25.00
-6.00 -7.00
20.00
-8.00 5.00
6.00
7.00
not controled (stick fixed)
time [s]
elevator control (simultenous)
15.00
elevator control (delayed)
FIG. 4. Control functions
elevator and throttle control
The aircraft is controlled. Elevator is being deflected and thrust of engine is being changed simultaneously with water release and according to the same, harmonic, timedependent function.
C
D M 0.40
2.0
5.00 0.00
5.00
10.00
15.00
20.00
25.00
time [s]
FIG. 6 - Angle of attack after water bomb dropping versus time
C C
L
10.00
pitch rate [rad/s]
1.6
0.30
0.32
not controled (stick fixed) elevator control (simultenous) elevator control (delayed)
1.2
0.24
0.8
0.16
0.20
CL
elevator and throttle control
0.10
CD
0.4
0.08
0.0 -10
CM
0.00
0.00 0
10
20
30
40
Angle of attack [deg]
-0.10
-0.4 -0.08 Fig.5 Basic aerodynamic coefficients of PZL-106 -0.20
Fig.5 presents the basic aerodynamic coefficients of PZL-106 versus angle of attack. They are given in wide range of angle of attack (>30). Data were taken from a wind tunnel investigation. Results of numerical simulations are presented at Fig.6-8. Fig.6 shows the change of angle of attack in time of water release. If the aircraft is not controlled (case 1) the angle of attack slightly decreases, then rapidly increases to cross the critical value. If the aircraft is controlled (cases 2-
0.00
5.00
10.00
15.00
20.00
25.00
time [s]
FIG. 7. Pitch rate versus time after water release
Fig.8 presents the flight path and corresponding aircraft attitude (pitch angle) in delayed control case. The time-lag of the elevator deflection with respect to water release results in the heavy altitude oscillation. The pitch angle also reaches high values.
180
Altitude [m]
160
140
120
100
Water dropping 80 0
200
400
600
800
1000
Range [m]
FIG.8. Flight altitude (with corresponding aircraft attitude) versus flight distance of flight after water bomb dropping
4. Analysis of Cobra manoeuvre The analysis of Cobra manoeuvre was realised for PZL-230 Scorpion (project) combat aircraft 7 (Fig.9) . The mass and geometric data are as follows: mass - 4900 kg, wing span - 10 m, 2 main wing surface - 33 m 2 horizontal tail surface - 10.5 m 2 canard surface - 5.7 m , mean aerodynamic chord (MAC) - 3.3 m.
extreme manoeuvres (as in Cobra manoeuvre, for example) all flight parameters change rapidly and usually are highly distinct from that of the steady flight. There are two possibilities to tackle with this problem in order to obtain the time-dependent aerodynamic forces - a problem which is the key point in numerical simulation of any extreme manoeuvre. One way is to solve synchronously the fluid dynamic equations (for example NavierStokes equations) using a selected CFD code together with flight dynamic equations of motion. The other approach is to use the time-dependent forces and moments, known in the whole range of angle of attack and being the functions of Mach number, Reynolds number, pitch rate, reduced frequency and normal acceleration. Such functions should be measured in wind tunnel tests. However, both methods are very costly to obtain, 8,9 need highly specialised laboratory equipment and computers with unattainable today power. An example of so-called „reaction surfaces” is 10,11,12 presented by Rückemann . Aerodynamic data used in this paper for numerical simulation were partially computed (in linear range), partially measured in wind tunnel and partially estimated (interpolated or extrapolated) on the bases of 8,9 results obtained by other authors . 2.00
1.50
1.00 Normal force - wing Drag Normal force - tail
0.50
Normal force - canard Pitching moment - wing+body
0.00
-0.50 -20
0
20
40
60
80
100
Fig.10 Aerodynamic characteristics as the functions of o o angle of attack in the range of -5 to 90
FIG.9. Plan view of an aircraft being analysed (PZL-230 Scorpion design). A - mean quarter-chord point (25 % of MAC); N - neutral point of stability (13 % of MAC)
Classical stability derivatives widely used in 4,5,6 flight dynamics can not be applied here to compute forces and moment. It is because that in
Fig.10 presents basic aerodynamic coefficients versus angle of attack (AOA) with extended range of AOA (from -5 to 90): normal force coefficients for main wing, horizontal tail and canard, drag coefficient and pitching moment coefficient for wing-body configuration. Fig.11 presents two meaningful stability derivatives: normal force and pitching moment with respect to pitching rate and ). normal acceleration ( w
30.00
The results of numerical integration were shown in Fig.13-15. Fig.13 presents angle of attack in time which was performed for different gravity centre position. The high value of AOA is attained only for configurations which are not static-stable (xc>10% MAC).
CN - Q CN - alpha_p CM - Q
20.00
CM - alpha_p
60 10.00
angle of attack [deg]
Center gravity position 0.00
7 % MAC
40
10 % MAC 13.7 % MAC
-10.00 -20
0
20
40
60
80
Fig.11 Normal force and pitching moment derivatives (excluding contribution of canard and horizontal tail) with respect to pitch rate and normal acceleration
20
time [s] 0 0
0.00
-2.00
-4.00
3
6
FIG.13. Increasing of angle of attack after dynamic entrance into the post-stall region for different position of the centre of gravity
Fig.14 presents angle of attack and pitch angle after rapid elevator deflection shown in Fig.12. Both variables reach high values: 65 for AOA and 125 for pitch angle. 160
Pitch angle, Angle of attack [deg]
Dynamic equations of motion (3-4) were integrated under the assumption that horizontal tail was deflected according to the function presented in Fig.12. In the beginning, elevator deflection corresponds to the state equilibrium at initial condition of flight. Next the elevator was deflected to -7 in 3 s according to cosine function. After next 2 s the deflection was increased up to -10 in the same way (cosine function). Then elevator was deflected very quickly to initial value and next again to -7. The final value corresponds to the state equilibrium for speed attained at the end of the manoeuvre.
elevator deflection [deg]
15 % MAC
100
pitch angle
120
angle of attack
80
40
0
-40 200
-6.00
400 500 Range [m]
600
700
FIG.14. Increasing of angle of attack and pitch angle after dynamic entrance into the post-stall region (the centre of gravity 13.7% of MAC)
-8.00
-10.00 0.00
300
4.00
8.00
12.00
16.00
time [s] FIG.12. - Elevator control function used to realise Cobra manoeuvre.
Fig.15 presents the flight path and corresponding aircraft attitude (pitch angle). The altitude is increasing by about 250m during manoeuvre. The pitch angle is stabilised in low values corresponding to the state of equilibrium for the flight speed at the end of the manoeuvre.
Altitude [m]
1300
1200
1100
1000
200
300
400
500
600
700
Range [m] FIG.15. Flight altitude (with corresponding aircraft attitude) versus range of flight after dynamic entrance into post-stall region of angles of attack
5. Concluding remarks In this paper two different examples of rapid manoeuvres were analysed in details. In these manoeuvres and in many other manoeuvres the flight parameters usually change abruptly. Numerical simulation of motion can deliver many detailed information about flight parameters. However, one should take into account that: aerodynamic characteristics should be given in o extended range of angle of attack (from 0 to o 90 ); aircraft dynamics is usually very sensitive to the change of the centre of mass position and control function; parameters of the state of equilibrium should be found before the dynamic equation of motion are integrated; these parameters could be found either from the full nonlinear equations of the state of equilibrium or from the simplified linear equations; control function, aerodynamic characteristics, and shifting of the centre of gravity function, should be smoothed because of the requirement of the numerical calculation. References [1] BŁASZCZYK P., GOETZENDORF-GRABOWSKI T., GORAJ Z., SZNAJDER J.: Modelling of Unsteady Flow about a Fire Fighting Aircraft Dropping the Water Bomb, 36th Aerospace Sciences Meeting & Exhibit, AIAA-98-0763 [2] ETKIN B., Dynamics of Flight - Stability and Control, John Wiley & Sons, Inc., New York 1982 [3] GOETZENDORF-GRABOWSKI T., Modelling of the aircraft motion - theory and application , in Optimalsteuerungsprobleme von HyperschallFlugzeugsystemen, Workshop des Sonderforshungsbereich 255, Greifswald-München 1997, pp.21-34 [4] GOETZENDORF-GRABOWSKI T., NumericaL Calculation of Stability Derivatives of an Aircraft,
Journal of Theoretical and Applied Mechanics, No.3, 32, 1994, pp.591-606 [5] GOETZENDORF-GRABOWSKI T., GORAJ Z., Calculations of stability characteristics of an aircraft in subsonic flow using panel methods (in Polish), Transactions of the Institute of Aviation Scientific Quarterly, 2/1996(145), Warszawa 1996, pp 31-49 [6] GORAJ Z., Calculations of equilibrium, maneuverability and stability of an aircraft in subsonic range of speed (in Polish), Warsaw University of Technology - "Zaklady Graficzne", Warszawa 1984 [7] GORAJ Z., High Angles of Attack Flight Dynamics of Contemporary and Prospective Fighters as a Function of Their Configuration and Aerodynamics, 21th Congress of the International Council of the Aeronautical Sciences ICAS-981.7.5, 13-18, September 1998, Melbourne [8] GRAFTON S.B., LIBBEY CH.E., Dynamic Stability Derivatives of a Twin-Jet Fighter model o o for Angles of Attack from -10 to 110 , NASA TN D-6091, Washington Jan.1971. [9] GRAFTON S.B., ANGLIN E.L., Dynamic o Stability Derivatives at Angle of Attack from -5 to o 90 for the Variable-Sweep Fighter Configuration with Twin Vertical Tails, NASA TN D-6909, Washington Oct.1972. [10] ORLIK-RüCKEMANN K., Dynamic Stability Testing of Aircraft - Needs Versus Capabilities, Progress in Aerospace Science, Vol.16, No.4, 1975, pp.431-447. [11] ORLIK-RüCKEMANN K., Aerodynamic Aspects of Aircraft Dynamics at High Angles of Attack, Journal of Aircraft, Vol.20, No.9, Sept.1983, pp.737-752. [12] ORLIK-RüCKEMANN K., Aerodynamic Coupling between Lateral and Longitudinal Degrees of Freedom, AIAA Journal, Vol.15, No.12, Dec.1977, pp.1792-1799.
