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dynamic Loads

The unsteady variation of aerodynamic forces and moments in response to change in a flow incidence angle happens due to lag in vortical flow readjustment on the wings. Hence, many of the features of vortical flow are reflected in to variations of aerodynamic forces and moments acting on a maneuvering aircraft. Therefore, understanding the nature of vortical flow is an essential starting point in the formulation of model structure. It is also important because the proposed model structure is expected to be generic enough to reproduce the variations in loads due to input-types not considered in the process of identification. Hence, many of the models for unsteady aerodynamics in literature are formulated using the features of vortical flow phenomenon.

The aerodynamics of delta-wing configurations at stall angles-of-attack and low Mach number is dominated by leading edge vortices on the wing [4]. This is also true for double-delta and strake-delta configurations. All the modern high-maneuverability, high- performance aircraft have one of these three wing-shapes. Hence, extensive experimental studies of vortical flow on delta wings are available in literature. These studies have been

(a) Vortex breakdown location on a wing in response to pitch-up and pitch-down motion at various non-dimensional rates

(b) Lift overshoot due to pitch-up motion of a wing starting from different initialα

Fig. 3.2. Features of variation in lift coefficient due to unsteady aerodynamics in the stall region [4].

appropriately reviewed from modeling perspective in [4, 6] and aerodynamics perspective in [74].

For a delta wing with leading edge sweep-angle in 45 to 75 degree range, the flow on the wing leading edge separates to form a flow structure called vortices at a certain moderate angle-of-attack αv. The vortex core has an axial flow component of very high velocity and low pressure. This creates a suction peak on the wings. It produces an aerodynamic normal force component in addition to that created by the potential flow. Variation of the normal force and pitching moment acting on the wings beyond αv, depends on these leading-edge vortices.

The strength of the vortices or the airspeed in vortex core increases with increase in angle-of-attack. At a certain angle-of-attack αb, the vortex structure breaks down

on the wings. This drastically changes the pressure distribution on the wings. As the angle-of-attack is increased beyond αb, the normal force does not change much and eventually starts decreasing. This nonlinear variation of normal force and pitching moment coefficients is commonly called Stall. Unlike airfoils, the variation in CZ and Cm for delta-wing aircraft is smooth in this region. There is no static hysteresis versus angle-of-attack either.

The vortex breakdown location on the wings is a function of angle-of-attack and direction of pitching, as shown in Fig.(3.2,a). One can notice that the vortex breakdown location is different in pitch-up and pitch-down motion at a particular angle-of-attack. This is because it shifts towards wing leading edge or away from it with a certain finite time-lag in response to sudden change in angle-of-attack. The variation in load in response to pitch-up input also depends on initial angle-of-attack, as evident from Fig.(3.2,b). These features imply that the pressure on the wing has a memory, that is, the instantaneous pressure on the wing depends on the aircraft motion history [4]. As seen in Fig.(3.2,a), the vortex breakdown location follows a different trajectory in pitch-up and pitch-down motions depending on the non-dimensional pitching rate. It is therefore necessary to perform dynamic experiments in a wind tunnel to capture the transient responses in variation of forces and moments due to change in flow-incidence angles.

Change in angle-of-attack (α) at zero sideslip (β) primarily effects the longitudinal loads acting on the aircraft. Change in sideslip effects both longitudinal and lateral- directional loads. α and β are fundamental flow angles. Hence, the response of aerodynamic loads to variation in α and β is sufficient to model unsteadiness of aerodynamic loads acting on the aircraft [75]. Alternatively, the effect of unsteady aerodynamics on all the aerodynamic coefficients can be simultaneously excited by considering pitching with non-zero sideslip as done in [76]; or by considering body- axis rolling motion as done in [77]. In a forced oscillation wind tunnel test, the model is subject to sinusoidal variation of one of the flow incidence angles _{{α, β, φ}. These} data are considered to be sufficient for modeling the unsteady variation of longitudinal and lateral-directional aerodynamic loads.

For modeling longitudinal loads experimental data is generated for inputs in (α, β), while for lateral-directional loads inputs in (φ, β) are used. This is based on an implicit assumption that the aerodynamic loads can be modeled separately. Although there is some coupling between longitudinal and lateral-directional loads due to change in any flow incidence angle, currently there are no methods for generating suitable data for modeling the loads in an integrated fashion. In this thesis, we focus only on the effect of change in angle-of-attack on the normal force and pitching moment.

Fig. 3.3. Power spectrum plots showing harmonics in the pitching moment coefficient responses due to harmonic inputs of different amplitudes [73].

in terms of amplitude and frequency of sinusoidal inputs. A forced oscillation test in pitching motion, shows that the aerodynamic loads also vary periodically. A plot of the normal force or pitching moment coefficient shows dynamic hysteresis with respect to their steady state values. Cunningham and Boer observed that for sinusoidal input of small amplitude of the order of 3o _{− 5}o _{variation of loads is also sinusoidal, while for} larger amplitude inputs higher harmonic components are also generated [72, 73]. This is evident from the power spectrum plots in Fig.(3.2), of the pitching moment coefficient of a double-delta wing configuration. For sinusoidal input of amplitude ∆α = 18o there are three harmonics of input frequency excited, while for ∆α = 1.75o there is only the first harmonic. In the same experiments, the pressure measurement data also showed the existence of second and third harmonics of the input frequency for large amplitude inputs. The presence of these harmonics implies that the variation of aerodynamic loads is nonlinear in nature.

The linear nature of unsteady variations is also evident from the linear relationship between the in-phase and out-of-phase derivatives estimated from small amplitude forced oscillation test data. Assuming the unsteady variation of aerodynamic loads to be linear for small amplitude inputs in angle-of-attack, it has been shown that the in-phase versus out-of-phase derivatives estimated for an aerodynamic coefficient have a linear relationship between them [22]. As seen in the plot of in-phase versus out-of-phase derivatives in Fig.(3.4), the solid-curve connecting the points for different frequencies is approximately linear for normal force and pitching moment coefficients. Thus, the linear nature of unsteady aerodynamics can be verified from this type of a plot. Such plots or the linear relation, have been reported to be true for many delta-wing configurations like X31 [78], F16XL [31], GTA , Delta-wing model with 65oleading edge sweep angle (Delta-60)

(a) For Normal force coefficient

(b) For Pitching moment coefficient

Fig. 3.4. In-phase versus Out-of-phase derivatives estimated from SAFO data forCZandCmof a double-

delta wing [73].

wing [22] etc. Therefore, the approximately linear nature of unsteady aerodynamic loads due to small amplitude inputs for delta-wing configurations is a fundamental feature of the aircraft unsteady aerodynamics.

The measurement of time-lag in readjustment of vortex breakdown location also provides evidence of the nonlinear nature of pressure and load variations. Resienthel.et.al. studied the effect of change in pressure gradient on vortex breakdown location by oscillation of the vertical fins of an aircraft model in wind tunnel tests [79]. It was found that the transient response of the vortex-breakdown location in response to change in external pressure gradient (which can be due to any one of angle-of-attack, rolling motion, Elevators, Fins etc.) follows a trajectory similar to that of the response of a linear first- order differential equation to step input. Also, the time-constant for movement of vortex break-down location upstream is higher than the time-constant of its motion down-stream

[79]. These features have been observed for variation of loads in many other experiments presented in literature [74, 3]. The difference in time-constant depending on direction of motion causes the dynamic hysteresis in vortex breakdown location to be nonlinear.

All these experimental evidences imply that that a nonlinear model structure is essential to represent the nonlinear nature of unsteady aerodynamic loads.