Classical aerodynamics proposed models for solving pitching maneuvers.

## 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 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-to-weight 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.

For classical unsteady aerodynamic models, Theodorsen [3], Wagner [4] 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. [5] 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 [6, 7]. The AIAA Fluid Dynamics Technical Committee’s (FDTC) Low Reynolds Number Discussion Group introduced some cases for the assessment of experimental efforts [8], 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 [9, 10, 11, 12, 13] 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. [8] 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 [7, 14, 15, 16, 17, 18]. The recent work done by Yuelong et al. [19] 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 (

On the other hand, fluid structure interaction modeling became essential for solving flow around vibrating and rotating structure [7, 20, 21, 22]. Modeling such moving bodies requires aerodynamic unsteady nonlinear models to assure accuracy in modeling results rather than using quazi-steady models. Carlos et al. [23] 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. [6] 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. [8]. 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 [8].

## 2. Motion kinematics

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 [11] is used in this work as a reference case for comparison. Here, the experimental work done by Ramesh et al. [12] is considered as a benchmark. Variations of this motion are considered by varying the pitch amplitude (

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

where

## 3. Classical models

In order to analytically describe the generated lift force due to pitching maneuvers, a well established models were introduced. In this section, a detailed description of these models is discussed and explained in a straight forward manner.

### 3.1 Theodorsen model

The tremendous work done by Wagner [4], Prandtl [27], Theodorsen [28] and Garrick [29] described some fundamental physical concepts in understanding and modeling the unsteady aerodynamics. These concepts are usually incorporated with a potential flow approach and small disturbance theory to obtain analytical expressions of flow quantities. The unsteady lift on a harmonically oscillating airfoil in incompressible flow has been studied by Kussner and Schwarz [30], but the most well known solution is due to Theodorsen [3]. The lift on a thin rigid airfoil undergoing oscillatory motion can be written as:

or in normalized form,

where,

Two approaches were undertaken to test the transformed input functions for Theodorsen classical unsteady model as follows:

Fourier series approach

By applying Fourier series for the given effective maneuver angle of attack and considering Theodorsen function C(k) such that:

where

The non-circulatory lift part [31] is given by:

Fast Fourier transform

The Fast Fourier Transform of the effective angle of attack is written as:

and the circulatory component of lift based on FFT is given by:

It should be noted that practically, this Fourier transform approach will be implemented numerically using discrete fourier transform. However, discrete Fourier transform in contrast with the exact Fourier transform (Fourier integral) will necessarily ignore some frequency contents due to the integration limits between

### 3.2 Wagner step response and Duhamel superposition principle

Using Wagner’s linear step response, the Duhamel principle can be used to include the unsteady effects in an exact form such as a finite-state aerodynamic models suitable for aeroelastic problems and flight mechanics simulations. Wagner [4] obtained the time dependant-response of the lift on a flat plate due to a step input (indicial response problem). Garrick [29] showed that by using Fourier transformation, Wagner function,

where the non-dimensional time

By knowing the indicial response for a linear dynamical system, the response due to arbitrary motion (input) can be described as an integral (superposition) using the indicial response and an input varies with time. The variation of the circulatory lift for an arbitrary change in the angle of attack is given by:

We note that

This equation is usually used in dynamic stall models where relatively high angles of attack are encountered, e.g., the Beddoes-Leishman dynamic stall model [33].

### 3.3 State space finite model

RT Jones proposed an approximate expression for Wagner function as follows:

where

the transfer function is then written as:

To determine a second-order state-space realization of the transfer function in Eq. 17 can be written as:

where

and to the output via:

then applying Laplace inverse we get:

then let

Also,

Hence,

By writing these equation in a matrix form, we obtain

then by applying the quasi-steady lift expression, we have;

where

### 3.4 Unsteady vortex lattice method (UVLM)

The unsteady Vortex lattice methods (UVLMs) are well suited to the bio-inspired flight problems because they can account for the circulation distribution variations on wings, the velocity potential time-dependency, and the shedding of wake downstream. Although they are considered low fidelity models, they may be extended to capture unconventional lift mechanisms such as leading edge vortex [34, 35, 36]. These discrete vortex models are widely used in modeling aerodynamics of aircraft and rotorcraft analysis, compared to computational fluid dynamics (CFD) models which are more computationally expensive [37]. The use of UVLM method is now a powerfull tool in hand for aerodynamicists for its ease implementation even for complex shapes.

Zakaria et al. [7] used UVLM to model the aerodynamic loading on different Samara leaves (Maple seeds) during their steady state flight. The results were verified with experiments. Parameters including the drop speed, angular velocity and coning angle for different sets of Maple Samaras were determined from experiments. The aerodynamic loads were calculated using UVLM against the forces required for maintaining a steady state flight as obtained from the experiment. Consequently, the UVLM approach yields adequate aerodynamic modeling features that can be used for more accurate flight stability analysis of the Samara flight or of decelerator devices inspired by such flight. Also, Simon et al. [38] showed that by imposing an arbitrary input as a control surface deflection to an unsteady VLM suitable for efficient aerodynamic loads analysis within aeroelastic modeling, analysis and optimization frameworks for preliminary aircraft design. By using a continuous time state space aerodynamic model is extended for accepting arbitrary motion, control surface deflection and gust velocities as inputs. Their results showed good agreement for a large range of reduced frequencies. Accepting arbitrary motion, control surface deflection and gust velocities as inputs.

The (UVLM) divides the lifting surface into panels. A point vortex is then associated with each of these panels. The center of this ring is set at the 1/4 of the panel chord length. One collocation point is set in each panel at the 3/4 of the panel length, and the panel normal vector is calculated in this point as shown in Figure 5.

The UVLM model is based on the following assumptions:

No penetration boundary condition.

Kelvin Circulation Theorem (Conservation of Circulation).

Vortices is convected by local velocities. (Wake deformation)

The velocity induced by all the vortex points, including the shed vorticies through the wake, is calculated at each control point and the no-penetration kinematic boundary condition is applied to calculate vortex intensity on each panel. At each time step, there are (m + 1) unknowns (m

From Kelvin’s circulation theorem, we have:

where

where

The unsteady aerodynamic loads can be calculated from the circulation

the unsteady pressure difference on the

where

From the definition of circulation, we have:

for

where

### 3.5 Models comparison

In order to summarize the merit of the proposed classical potential models for solving high pitch maneuvers, Table 1 is shown. Table 1 represents the key parameters for each model in the sense of input motion, nonlinearity, wake deformation and camber variation for flying vehicles. The merit of each model is how one can apply simple analytical equation to solve such maneuver.

Models | Input motion | Nonlinearity | wake deformation | Camber variation |
---|---|---|---|---|

Theodorsen | Harmonic | Geometric | Flat | |

Wagner | Step input | Geometric | ||

State space | Arbitrary | |||

UVLM | Arbitrary |

## 4. Maneuver case studies results

### 4.1 Case 1: Pitch ramp α o = 25 o

#### 4.1.1 Leading edge pivot

Figures 6–11 show a comparison between the proposed models discussed above for different ramp amplitudes and hinge locations. A physical interpretation for the jump and attenuated lift peaks show four flow events as reported by Ramesh et al. [8] as follows: (i) onset of flow separation at the ramp start (

Figure 6 shows the ramp pitch motion with an amplitude of

#### 4.1.2 Half chord pivot

Figure 7 shows the ramp with amplitude of

#### 4.1.3 Trailing edge pivot

In a similar manner, Figure 8 shows a comparison between experimental and theoretical predictions for

The common result in all pivot location cases (leading, half and trailing chord location), show that Theodorsen FFT model has a damped lift response compared to all the proposed models and experiments. This is because for a given AOA (

### 4.2 Case 1: Pitch ramp α o = 45 o

#### 4.2.1 Leading edge pivot

Figures 9–11 show the lift coefficient response for a ramp maneuver with an amplitude of

#### 4.2.2 Half chord pivot

Figure 10 shows the lift coefficient for a ramp amplitude of

#### 4.2.3 Trailing edge pivot

Figure 11 presents the lift coefficient for a ramp amplitude of

It is clear that a very good matching found between the UVLM model and experiments which can be attributed to the favor of leading edge suction inclusion as well as the nonlinear behavior (

Figures 12 and 13 show the Shedding of trailing vortices and wake convection shape downstream for

## 5. Conclusion

In this chapter, different classical analytical models were presented in a simple mathematical form based on potential flow to solve unsteady problems constrained by an input motion. A canonical pitch ramp motion is chosen to present the input motion for two different ramp amplitudes (

Table 2 discuses and concludes the output of each proposed model with the perspective of output response, pitch amplitudes, computational cost and the obtained loads.

Models | Response type | Large amplitude | Computational cost | Loads |
---|---|---|---|---|

Theodorsen | Steady state harmonic | force | ||

Wagner | Transient | Force | ||

State space | Full response | Force | ||

UVLM | Full response | Pressure |

The benefits of the UVLM compared to other methods is that is enabling aerodynamic modeling for arbitrary motion. An extension is easy to implement to include a formulation of the boundary conditions for arbitrary three-dimensional motion and control surface rotation. Furthermore the calculation of unsteady induced drag by a nonlinear extension of the force computation can be done. Furthermore the proposed UVLM method shows advantages in predicting unsteady aerodynamic forces of high frequency motion compared to other analytical models. In general, it can be said that the unsteady vortex lattice method is a powerful tool for modeling of incompressible and inviscid unsteady aerodynamics. A continuous time formulation in particular can be used to decrease the computational costs for aeroelastic simulations. The possibility of calculating unsteady loads without the need of approximations for time-domain simulation makes the method especially useful within aeroservoelastic optimization algorithms. Other models formulated in time domain (for example sensor and actuator models or control laws) can be easily integrated. Furthermore, the nonlinear aerodynamic state space formulation is suitable for the integration of further nonlinear aerodynamic correction models (e.g. stall models). This provides confidence towards the development of semi-empirical models based on potential flow theories and experiments that can predict unsteady forces of ramp maneuvers.

## Nomenclature

b | airfoil semi-chord (c/2) |

c | airfoil semi-chord (c/2) |

CL | lift coefficient |

Ck | lift deficiency factor |

f | frequency (Hz) |

h | plunging displacement (mm) |

ḣ | plunging velocity |

h¨ | plunging acceleration |

k | reduced frequency πfc/U∞ |

ℓ | wing span (m) |

P | non-dimensional Laplace operator |

q | non-dimensional pitch rate, α̇cV |

Re | Reynolds number |

S | distance traveled in semi-chords, 2Vtc |

T | time period |

U∞ | free stream velocity |

Urel | free stream velocity |

αo | airfoil mean angle of attack |

αeff | effective angle of attack |

α̇ | angular pitch velocity rad/s |

α¨ | angular pitch acceleration rad/s2 |

ϕ | phase angle |

Γ | total flow circulation |

γb | elementary bound flow circulation |

γw | elementary wake flow circulation |

ω | angular frequency,(rad/s) |

σ | heaviside function variable |

τ | Non-dimensional time |

ρ | Air density |

AoA | angle of attack |

circ | circulatory |

FFT | Fast Fourier Transform |

RHS | right hand side |

UVLM | unsteady vortex lattice method |