Unsteady Aerodynamics of Highly Maneuvering Flyers

*Mohamed Yehia Zakaria*

## **Abstract**

In this chapter, a set of analytical aerodynamic models, based on potential flow, that can be used to predict the unsteady lift response during pitching maneuvers are presented and assessed. The result examines the unsteady lift coefficients experienced by a flat plate in high-amplitude pitch ramp motion. The pitch ramps are chosen based on two ramp pitch maneuvers of a maximum amplitudes of 25 and 45 degrees starting from zero degree. The aim is investigate the use of such classical models in predicting the lift dynamics compared to a full physical-based model. Among all classical methods used, the unsteady vortex lattice method (without considering the leading edge vortex) is found to be a very good predictor of the motion lift dynamic response for the 25° ramp angle case. However, at high pitch maneuvers (i.e.,the 45° ramp angle case), could preserve the response pattern with attenuated amplitudes without high computational burden. These mathematical analytical models presented in this chapter can be used to obtain a fast estimate for aircraft unsteady lift during pitch maneuvers instead of high fidelity models, especially in the early design phases.

**Keywords:** canonical maneuvers, pitching maneuvers, unsteady aerodynamics, unsteady lift response

### **1. Introduction**

Loops, barrel rolls and pitch maneuvers are impressive aerial stunts. But even during the most intense in-air aerobatics, most planes are still constrained by aerodynamics. The air flowing over their wings gives them the lift to stay aloft and they control their movement by altering the surfaces that air flows over. The quick the rate of movement for the control surface, a fast response from the aircraft to change attitude. Pilots can pull off moves with precise control in conditions that would leave other aircraft hopelessly plummeting towards the ground. For fighter aircraft, there are numerous maneuvers can be done by the pilot to increase the aircraft maneuverability. These maneuvers such as, Cobra, Mango flip, high pass alpha that can save pilot's life during a dog fight (see **Figure 1**). Nowadays, unmanned aeriel vehicles autopilots can perform these maneuvers to an extent. Consequently, in order assure that UAVs could perform such maneuvers, one may need to relax the quasi-steady modeling to an unsteady nonlinear model to deal with these abrupt changes in attitude. Prediction of dynamic lift response of Harsh maneuvers for flying vehicles necessitate a compact aerodynamic modeling. For instance, pitching

For classical unsteady aerodynamic models, Theodorsen [5], Wagner [6] and others have been studied extensively the classical theories of unsteady aerodynamics to be employed in the aeroelasticity field. However, aerodynamic models of harsh maneuvers characterized by sharp pitch rates and amplitudes still present a challenge in modeling. While advances in computational fluid dynamics and experimental methods have opened the study of these maneuvers as such a low-fidelity analytical modeling for rigorous prediction is still forthcoming. Roderich et al. [4] performed experiments for touchdown to take-off for a very basic glider as shown

*Unsteady Aerodynamics of Highly Maneuvering Flyers DOI: http://dx.doi.org/10.5772/intechopen.94231*

In the last two decades, there have been several efforts exerted on unsteady aerodynamic modeling based on potential flow theories as well as modified thin airfoil theory to simulate the wing motion for an arbitrary input [7, 8]. The AIAA

A tremendous work was done based on nonlinear unsteady reduced order modeling to solve flow at high frequencies [8, 15–19]. The recent work done by Yuelong et al. [20] examined the unsteady forces and moment coefficients obtained by a thin airfoil in a pitch ramp high-amplitude motion. Wind tunnel experiments have been conducted at Reynolds number ( *Re* <sup>¼</sup> 45 x 104), using a rigid flat-plate

*Example of bird perching and successful experiments based on perching manoeuver. (a) A wire-tailed swallow feeding a re-cently edged chick [3] (a) A wire-tailed swallow feeding a re-cently edged chick [3]. (b) A basic*

Fluid Dynamics Technical Committee's (FDTC) Low Reynolds Number Discussion Group introduced some cases for the assessment of experimental efforts [9], on large amplitude pitching maneuvers. The proposed motions are used as a benchmark for obtaining analytical and phenomenological models, in which a ramp up, hold, and ramp-down motions are analyzed using theory and numerical computations [10–14] Theodorsen's and Wagner's Inviscid theories are purely proper only for small amplitude oscillations associated with planar wakes. However, a tremendous work has shown that these methods remain substantially accurate even at moderate amplitudes and high frequencies. The results obtained by Ramesh et al. [9] during the hold and downstroke show that the aerodynamic forces are dominated by a deep-stall as well as leading edge vortex (LEV). The shedding effects were seen from the vorticity and dye injection plots from his experimental results. These results proved that viscous state indicate that the inviscid assumptions are insufficient for modeling the hold and downstroke portions of the motion and adequate for capturing the lift time history during the

in **Figure 2(b)**.

ramp phase.

**Figure 2.**

**103**

*glider, manually thrown and con-trolled by perching [4].*

**Figure 1.** *High alpha Fighter's aircraft maneuvers.*

maneuvers for fighter aircrafts (ex. F35 - SU-57) with specified handling qualities stimulate the idea to impose new modeling techniques to be applied on UAVs. The unsteady lift response plays an important role to control the vehicle at such low speeds. Escaping from a flying threat, first performed by Soviet test pilot Viktor Pugachoyov in 1989, the maneuver that would go on to be called "Pugachev's Cobra" is one of the building blocks that makes up many other more complicated supermaneuvers. During flight, the pilot pulls back to an absurd angle of attack, taking the nose of the aircraft completely vertical or even beyond. From here, one of two things can happen. In a plane without thrust vectoring but with a thrust-toweight ratio higher than one, the drag towards the tail of the plane can be used to pitch the nose forward again. If the plane does have thrust vectoring, that can help the re-orientation even more. But either way, the engines are firing hard enough the entire time to maintain the jet's altitude despite the loss of speed and lift.

After few years, a German test pilot Karl-Heinz Lang performed the Herbst Maneuver in 1993. The Herbst Maneuver is basically Pugachev's Cobra with a bit of a twist. Instead of just pulling up and going forward again, the Herbst Maneuver has the pilot roll the plane (experimental X-31) a bit while its nose is pointed at the sky, so that when the nose comes back down, the plane is pointed in a different direction. On the other hand, such maneuvers are also possessed by birds and flapping insects. They can twist their wings at high angles of attack while flapping their wings without approaching stall. This is known as non-conventional lifting mechanisms invoked from biomemetics in order to perform such maneuver with a stabilized flight (i.e. vibrational stabilization). In preliminary design of UAVs, potential flow models are used as a start point to ensure acceptable estimates for aerodynamic forces and moments. A recent motivation is devoted towards designing flight control systems that can achieve harsh maneuvers such as perching and sudden landing for fixed wing MAV's [1, 2]. Bird perching is considered one of the most fascinating landing and decelerating maneuvers. **Figure 2(a)** shows a tailed swallow feeding a chick by pitching its wing at high angle of attack. For specific missions, such maneuver is useful for both flapping-wing and fixed-wing MAVs.

*Unsteady Aerodynamics of Highly Maneuvering Flyers DOI: http://dx.doi.org/10.5772/intechopen.94231*

For classical unsteady aerodynamic models, Theodorsen [5], Wagner [6] and others have been studied extensively the classical theories of unsteady aerodynamics to be employed in the aeroelasticity field. However, aerodynamic models of harsh maneuvers characterized by sharp pitch rates and amplitudes still present a challenge in modeling. While advances in computational fluid dynamics and experimental methods have opened the study of these maneuvers as such a low-fidelity analytical modeling for rigorous prediction is still forthcoming. Roderich et al. [4] performed experiments for touchdown to take-off for a very basic glider as shown in **Figure 2(b)**.

In the last two decades, there have been several efforts exerted on unsteady aerodynamic modeling based on potential flow theories as well as modified thin airfoil theory to simulate the wing motion for an arbitrary input [7, 8]. The AIAA Fluid Dynamics Technical Committee's (FDTC) Low Reynolds Number Discussion Group introduced some cases for the assessment of experimental efforts [9], on large amplitude pitching maneuvers. The proposed motions are used as a benchmark for obtaining analytical and phenomenological models, in which a ramp up, hold, and ramp-down motions are analyzed using theory and numerical computations [10–14] Theodorsen's and Wagner's Inviscid theories are purely proper only for small amplitude oscillations associated with planar wakes. However, a tremendous work has shown that these methods remain substantially accurate even at moderate amplitudes and high frequencies. The results obtained by Ramesh et al. [9] during the hold and downstroke show that the aerodynamic forces are dominated by a deep-stall as well as leading edge vortex (LEV). The shedding effects were seen from the vorticity and dye injection plots from his experimental results. These results proved that viscous state indicate that the inviscid assumptions are insufficient for modeling the hold and downstroke portions of the motion and adequate for capturing the lift time history during the ramp phase.

A tremendous work was done based on nonlinear unsteady reduced order modeling to solve flow at high frequencies [8, 15–19]. The recent work done by Yuelong et al. [20] examined the unsteady forces and moment coefficients obtained by a thin airfoil in a pitch ramp high-amplitude motion. Wind tunnel experiments have been conducted at Reynolds number ( *Re* <sup>¼</sup> 45 x 104), using a rigid flat-plate

**Figure 2.**

maneuvers for fighter aircrafts (ex. F35 - SU-57) with specified handling qualities stimulate the idea to impose new modeling techniques to be applied on UAVs. The unsteady lift response plays an important role to control the vehicle at such low speeds. Escaping from a flying threat, first performed by Soviet test pilot Viktor Pugachoyov in 1989, the maneuver that would go on to be called "Pugachev's Cobra" is one of the building blocks that makes up many other more complicated supermaneuvers. During flight, the pilot pulls back to an absurd angle of attack, taking the nose of the aircraft completely vertical or even beyond. From here, one of two things can happen. In a plane without thrust vectoring but with a thrust-toweight ratio higher than one, the drag towards the tail of the plane can be used to pitch the nose forward again. If the plane does have thrust vectoring, that can help the re-orientation even more. But either way, the engines are firing hard enough the

**Figure 1.**

*Biomimetics*

**102**

*High alpha Fighter's aircraft maneuvers.*

entire time to maintain the jet's altitude despite the loss of speed and lift.

maneuver is useful for both flapping-wing and fixed-wing MAVs.

After few years, a German test pilot Karl-Heinz Lang performed the Herbst Maneuver in 1993. The Herbst Maneuver is basically Pugachev's Cobra with a bit of a twist. Instead of just pulling up and going forward again, the Herbst Maneuver has the pilot roll the plane (experimental X-31) a bit while its nose is pointed at the sky, so that when the nose comes back down, the plane is pointed in a different direction. On the other hand, such maneuvers are also possessed by birds and flapping insects. They can twist their wings at high angles of attack while flapping their wings without approaching stall. This is known as non-conventional lifting mechanisms invoked from biomemetics in order to perform such maneuver with a stabilized flight (i.e. vibrational stabilization). In preliminary design of UAVs, potential flow models are used as a start point to ensure acceptable estimates for aerodynamic forces and moments. A recent motivation is devoted towards designing flight control systems that can achieve harsh maneuvers such as perching and sudden landing for fixed wing MAV's [1, 2]. Bird perching is considered one of the most fascinating landing and decelerating maneuvers. **Figure 2(a)** shows a tailed swallow feeding a chick by pitching its wing at high angle of attack. For specific missions, such

*Example of bird perching and successful experiments based on perching manoeuver. (a) A wire-tailed swallow feeding a re-cently edged chick [3] (a) A wire-tailed swallow feeding a re-cently edged chick [3]. (b) A basic glider, manually thrown and con-trolled by perching [4].*

model. Forces have been measured for reduced pitch rates ranging from 0.01 to 0.18 reduced frequency (*k* ¼ *ωc=*2*U*∞) along with four maximum pitch angles (30°; 45°; 60°; 90°) at different pivot axis locations. The results show that the unsteady aerodynamics is limited to a delayed stall effect for reduced pitch rates lower than k = 0.03. At higher pitch rates, the unsteady aerodynamic response is associated with a formation of circulation, which in turn increases with the pitch rate and the distance between the pivot axis and the 3/4-chord location. An enhanced response was noted in the normal force and moment coefficients due to these circulatory effects. These overshot is slightly reduced for a flat plate with a finite aspect ratio near eight compared to two-dimensional configuration. The authors proposed a new time-dependent model for both lift and moment coefficients. The model based on the Wagner function and a time-varying input along with nonlinear variation of the quasi steady aerodynamics. A satisfactory results for 0° to 90° pitch ramp motions were compared with experiments for different pivot locations and various circulation intensity based on pitch rates.

**2. Motion kinematics**

*Unsteady Aerodynamics of Highly Maneuvering Flyers DOI: http://dx.doi.org/10.5772/intechopen.94231*

**Figure 3.**

**Figure 4.**

*respectively.*

**105**

*and a = 1 is trailing edge pivot).*

In order to explore the non-periodic motions of wings rapid manouevers, the ramp-hold-return motions were proposed by the AIAA FDTC Low Reynolds Number Discussion Group [25]. The smoothed ramp motion proposed by Eldredge's canonical formulation [12] is used in this work as a reference case for comparison. Here, the experimental work done by Ramesh et al. [13] is considered as a benchmark. Variations of this motion are considered by varying the pitch amplitude (25° and 45°) at a Reynolds number of 10,000. **Figures 2** and **3** show a schematic of the pitch motion variables and the two studied maneuvers versus the non-dimensional time, respectively. **Figure 4** shows the ramped motion for a maximum amplitude of 25° versus the corresponding effective angle of attack and the local angle of attack at

To avoid any numerical instabilities, (e.g., dirac-delta function spikes in the calculation of the added mass force) all motions are smoothed based on a smoothing parameter introduced by Elderedge [12]. For a ramp going from 0 degrees angle of attack to 25 or 45 degrees, the first 10% (2.5 or 4.5 degrees) can be replaced with a sinusoidal tangent to the baseline ramp, and similarly in approaching the "hold" portion at the maximum amplitude angle of attack, consequently again on the downstroke. This treatment avoids a piece-wise linear fit which has discontinuities

*Pitching motion nomenclature and motion variables (a = 1 is the leading edge pivot, a = 0 is the mid chord pivot*

*The proposed ramp maneuver with a maximum amplitudes of* 25*<sup>o</sup> and* 45*<sup>o</sup> and pitch rates of 0.2 and 0.4,*

the 3*=*4 chord location as suggested by Pistolesi theorem [26].

in the angle derivatives. The smoothing function G(t) is defined as:

On the other hand, fluid structure interaction modeling became essential for solving flow around vibrating and rotating structure [8, 21–23]. Modeling such moving bodies requires aerodynamic unsteady nonlinear models to assure accuracy in modeling results rather than using quazi-steady models. Carlos et al. [24] work discuses modeling and analyzing procedures of the non-linearities induced by the flow-structure interaction of an energy harvester consisting of a laminated beam integrated with a piezoelectric sensor. The cantilevered beam and the piezoelectric lamina are modeled using a nonlinear finite element approach, while unsteady aerodynamic effects are described by a state-space model that allows for arbitrary nonlinear lift characteristics.

The major contribution about the classical unsteady formulations discussed in the literature is the inefficacy to account for a non-conventional lift curve, such as LEV effects and dynamic stall contributions. Taha et al. [7] developed a state space model that captures the nonlinear contributions of the LEV in an unsteady fashion. However, their underpinning dynamics is linear: convolution with Wagner's step response. Consequently, there is a considerable gap in the literature for consolidating low fidelity models for predicting accurate lift forces associated with these large-amplitude maneuvers. An analytical unsteady nonlinear aerodynamic model that can be used to characterize the local and global nonlinear dynamic characteristics of the airflow is a mandatory task for aerodynamicists. Developing such a model will be indispensable for multidisciplinary applications (e.g., dynamics, control and aeroelasticity).

The chapter investigates and assesses relevant classical analytical models in solving lift response for pitching maneuovers. In doing so, Theodorsen, Wagner and Unsteady vortex lattice methods are used to predict the lift dynamics, then the results are compared with the experimental data presented by Ramesh et al. [9]. Also, the work proposed a simple time-dependent model in order to predict the lift response for a two dimensional wing performing rapid pitch motion. In addition, the results provide a comparison with numerical simulation using the unsteady vortex lattice method. The aerodynamic system receives the time histories of angle of attack, quasi-steady lift as inputs and produces the corresponding total unsteady lift as output. In the following sections, each presented model will be explained in detail. The chapter is organized as follows. The adopted motion kinematics are presented in Section 2. Aerodynamic classical models are reported in Section 3, along with the effect of reduced pitch rate and pivot axis location. In Section 4, the effect of pitch amplitudes on the unsteady lift coefficient is investigated by comparing the obtained results using two different pitch amplitudes with the experimental results [9].
