**2. Motion kinematics**

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

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

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

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

circulation intensity based on pitch rates.

nonlinear lift characteristics.

aeroelasticity).

*Biomimetics*

experimental results [9].

**104**

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

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

#### **Figure 3.**

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

#### **Figure 4.**

*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, respectively.*

$$\mathbf{G}(t) = \ln \left( \frac{\cosh \left( a U\_{\infty} (t - t\_1)/c \right) \cosh \left( a U\_{\infty} (t - t\_4)/c \right)}{\cosh \left( a U\_{\infty} (t - t\_1)/c \right) \cosh \left( a U\_{\infty} (t - t\_4)/c \right)} \right) \tag{1}$$

where *a* is the smoothing parameter and is taken to be 11, *t*<sup>1</sup> through *t*<sup>4</sup> are the transition times and a pitch amplitude angle *A*. As such, the smoothed angle of attack can be written as:

$$a(t) = A \frac{G(t)}{m \omega \,\, (G(t))}\tag{2}$$

• Fourier series approach

*LCirc* <sup>¼</sup> *πρU*<sup>2</sup>

• Fast Fourier transform

between �∞ to þ∞.

the static lift as:

velocity *U*∞.

**107**

considering Theodorsen function C(k) such that:

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

*cao* <sup>þ</sup> *<sup>ρ</sup>U*<sup>2</sup>

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

circulatory lift given after applying Fourier series is given by:

By applying Fourier series for the given effective maneuver angle of attack and

where *Ac k*ð Þ is the absolute value (amplitude) and *ϕ* is the phase angle. The

*LNon*�*Circ* ¼ �*πρb*<sup>2</sup>

*αeff*ð Þ¼ *ω*

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

2 *ρU*<sup>2</sup> *c* ð<sup>∞</sup> �∞

**3.2 Wagner step response and Duhamel superposition principle**

where the non-dimensional time *<sup>S</sup>* is defined as *<sup>S</sup>* <sup>¼</sup> <sup>2</sup>*U*∞*<sup>t</sup>*

*LCirc*�*FFT* <sup>¼</sup> <sup>1</sup>

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

ð<sup>∞</sup> 0

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

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 [6] 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, *W s*ð Þ, and Theodorsen function, *C k*ð Þ can be related together. Wagner [6] determined the circulatory lift due to a step change in the wing motion. The unsteady lift is then written in terms of

By knowing the indicial response for a linear dynamical system, the response due to arbitrary motion (input) can be described as an integral (superposition)

*αeff*ð Þ*t e*

�*iωt*

*αeff*ð Þ *w C k*ð Þ*e*

*iωt*

ℓðÞ¼ *s* ℓ*sW s*ð Þ (11)

*Ac k*ð Þ ¼ ∣*C k*ð Þ∣, *and ϕ* ¼ ∠ð Þ *C k*ð Þ (6)

*c AC k*ð Þ � �ð Þ *an cos*ð Þþ *<sup>ω</sup><sup>t</sup>* <sup>þ</sup> *<sup>ϕ</sup> bn sin* ð Þ *<sup>ω</sup><sup>t</sup>* <sup>þ</sup> *<sup>ϕ</sup>* (7)

½ � *Uα*\_ þ *α*€*ab* (8)

*dt* (9)

*dω* (10)

*<sup>c</sup>* for constant free-stream

### **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 [6], 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 [5]. The lift on a thin rigid airfoil undergoing oscillatory motion can be written as:

$$L = \underbrace{\pi \rho b^2 \left(\ddot{h} + U\_{\text{os}} \dot{a} + b a \ddot{a}\right)}\_{\text{Added\\_mass}} + \underbrace{2 \pi \rho U\_{\text{os}} b \left(\dot{h} + U\_{\text{os}} a + b \left(\frac{1}{2} + a\right) \dot{a}\right)}\_{\text{Quasi\\_steady}} \text{C}(k) \tag{3}$$

or in normalized form,

$$\mathbf{C}\_{L} = \frac{\pi b}{U\_{\infty}^{2}} \left( \ddot{h} + U\_{\infty} \dot{a} + ba \ddot{a} \right) + 2\pi \mathbf{C}(k) \left( \frac{\dot{h}}{U\_{\infty}} + a + b \left( \frac{1}{2} + a \frac{\dot{a}}{U\_{\infty}} \right) \right) \tag{4}$$

where, € *h* and *α*€ are plunging and pitching accelerations respectively. The first group of terms are the noncirculatory components which account for the inertia of fluid (added mass force). The second group of terms are the circulatory components, where C(k) accounts for the influence of the shed wake vorticity (lift deficiency factor). Since Theodorsen function necessitates a periodic motion for its input parameters (e.g. angle of attack or quasi steady lift), a Fourier transform should be applied to the pitch ramp maneuver under study. The effective angle of attack of the proposed ramp pitch motion can be written as:

$$a\_{\sharp\mathcal{Y}} = a + \dot{a}\left(\frac{1}{2} + a\right)\frac{b}{U\_{\infty}}\tag{5}$$

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

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

#### • Fourier series approach

*G t*ðÞ¼ *ln cosh aU* ð Þ <sup>∞</sup>ð Þ *<sup>t</sup>* � *<sup>t</sup>*<sup>1</sup> *<sup>=</sup><sup>c</sup> cosh aU* ð Þ <sup>∞</sup>ð Þ *<sup>t</sup>* � *<sup>t</sup>*<sup>4</sup> *<sup>=</sup><sup>c</sup>*

*<sup>α</sup>*ðÞ¼ *<sup>t</sup> <sup>A</sup> G t*ð Þ

attack can be written as:

*Biomimetics*

**3. Classical models**

**3.1 Theodorsen model**

*<sup>L</sup>* <sup>¼</sup> *πρb*<sup>2</sup> €

*CL* <sup>¼</sup> *<sup>π</sup><sup>b</sup> U*2 ∞ €

where, €

**106**

or in normalized form,

going oscillatory motion can be written as:


*h* þ *U*∞*α*\_ þ *baα*€ � �

> *h* þ *U*∞*α*\_ þ *baα*€ � �

attack of the proposed ramp pitch motion can be written as:

Theodorsen classical unsteady model as follows:

*<sup>α</sup>eff* <sup>¼</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>α</sup>*\_ <sup>1</sup>

where *a* is the smoothing parameter and is taken to be 11, *t*<sup>1</sup> through *t*<sup>4</sup> are the transition times and a pitch amplitude angle *A*. As such, the smoothed angle of

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.

The tremendous work done by Wagner [6], 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 [5]. The lift on a thin rigid airfoil under-

<sup>þ</sup> <sup>2</sup>*πρU*∞*<sup>b</sup>* \_

<sup>þ</sup> <sup>2</sup>*πC k*ð Þ \_

group of terms are the noncirculatory components which account for the inertia of fluid (added mass force). The second group of terms are the circulatory components, where C(k) accounts for the influence of the shed wake vorticity (lift deficiency factor). Since Theodorsen function necessitates a periodic motion for its input parameters (e.g. angle of attack or quasi steady lift), a Fourier transform should be applied to the pitch ramp maneuver under study. The effective angle of

*h U*<sup>∞</sup>

*h* and *α*€ are plunging and pitching accelerations respectively. The first

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>* � � *b*

Two approaches were undertaken to test the transformed input functions for

*U*<sup>∞</sup>

*<sup>h</sup>* <sup>þ</sup> *<sup>U</sup>*∞*<sup>α</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>1</sup>


� �

<sup>þ</sup> *<sup>α</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>1</sup>

! � �

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>* � �

> <sup>2</sup> <sup>þ</sup> *<sup>a</sup> <sup>α</sup>*\_ *U*<sup>∞</sup>

*α*\_

*C k*ð Þ (3)

(4)

(5)

*cosh aU* ð Þ <sup>∞</sup>ð Þ *t* � *t*<sup>1</sup> *=c cosh aU* ð Þ <sup>∞</sup>ð Þ *t* � *t*<sup>4</sup> *=c* � �

*max G t* ð Þ ð Þ (2)

(1)

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

$$A\_{c(k)} = |\mathbf{C}(k)|, and \,\phi = \mathcal{L}(\mathbf{C}(k))\tag{6}$$

where *Ac k*ð Þ is the absolute value (amplitude) and *ϕ* is the phase angle. The circulatory lift given after applying Fourier series is given by:

$$L\_{\text{Circ}} = \pi \rho U^2 c a\_o + \rho U^2 c \left( A\_{C(k)} \right) \left( a\_n \cos \left( a t + \phi \right) + b\_n \sin \left( a t + \phi \right) \right) \tag{7}$$

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

$$L\_{\text{Non}-\text{Circ}} = -\pi \rho b^2 [\mathbf{U}\dot{a} + \ddot{a}ab] \tag{8}$$

• Fast Fourier transform

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

$$a\_{\sharp\overline{\mathcal{Y}}}(o) = \int\_0^{\infty} a\_{\sharp\overline{\mathcal{Y}}}(t) e^{-i\alpha t} dt \tag{9}$$

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

$$L\_{\text{Circ}-FFT} = \frac{1}{2} \rho U^2 c \int\_{-\infty}^{\infty} a\_{\sharp \overline{f}}(w) \mathcal{C}(k) e^{i\alpha t} d\alpha \tag{10}$$

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 �∞ to þ∞.

#### **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 [6] 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, *W s*ð Þ, and Theodorsen function, *C k*ð Þ can be related together. Wagner [6] determined the circulatory lift due to a step change in the wing motion. The unsteady lift is then written in terms of the static lift as:

$$\mathcal{E}\left(\mathbf{s}\right) = \mathcal{E}\_s \mathbf{W}(\mathbf{s}) \tag{11}$$

where the non-dimensional time *<sup>S</sup>* is defined as *<sup>S</sup>* <sup>¼</sup> <sup>2</sup>*U*∞*<sup>t</sup> <sup>c</sup>* for constant free-stream velocity *U*∞.

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:

$$\mathcal{E}(s) = \pi \rho U^2 c \left( a(\mathbf{0}) W(s) + \int\_0^s \frac{da(\sigma)}{d\sigma} W(s - \sigma) d\sigma \right) \tag{12}$$

*X αeff*

*Y*

*Y* ¼ *b*<sup>2</sup> *αeff* � *aoX*<sup>1</sup> � *a*1*X*<sup>2</sup>

<sup>¼</sup> 0 1 �*ao* �*a*<sup>1</sup> � � *X*<sup>1</sup>

*y* ¼ ½ � *bo* � *b*2*ao b*<sup>1</sup> � *b*2*a*<sup>1</sup>

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

where *W*3*=*<sup>4</sup> is the normal velocity component and is given by:

*Lc*ðÞ¼ *t* 2*πρU*∞*b b*½ � *<sup>o</sup>* � *b*2*ao b*<sup>1</sup> � *b*2*a*<sup>1</sup>

**3.4 Unsteady vortex lattice method (UVLM)**

*<sup>W</sup>*3*=*<sup>4</sup> <sup>¼</sup> *<sup>U</sup>*<sup>∞</sup> *sin* ð Þþ *<sup>α</sup> <sup>α</sup>*\_ *<sup>b</sup>*

The unsteady Vortex lattice methods (UVLMs) are well suited to the bioinspired 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 [34–36]. 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

By writing these equation in a matrix form, we obtain

and to the output via:

then applying Laplace inverse we get:

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

then let *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>X</sup>* and *<sup>X</sup>*<sup>2</sup> <sup>¼</sup> *<sup>X</sup>*\_ .

*d dt*

*X*1 *X*2 � �

Also,

Hence,

**109**

<sup>¼</sup> <sup>1</sup>

*<sup>X</sup>* <sup>¼</sup> *<sup>b</sup>*2*P*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>P</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>0</sup>

*<sup>P</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>P</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>0</sup>

(20)

<sup>1</sup> (21)

*XP*<sup>2</sup> <sup>þ</sup> *Xa*1*<sup>P</sup>* <sup>þ</sup> *Xa*<sup>0</sup> <sup>¼</sup> *<sup>α</sup>eff* (22)

*<sup>X</sup>*€ <sup>þ</sup> *<sup>a</sup>*1*X*\_ <sup>þ</sup> *aoX* <sup>¼</sup> *<sup>α</sup>eff* (23)

*<sup>Y</sup>* <sup>¼</sup> *Xb*2*P*<sup>2</sup> <sup>þ</sup> *Xb*1*<sup>P</sup>* <sup>þ</sup> *<sup>b</sup>*0*<sup>X</sup>* <sup>¼</sup> *<sup>b</sup>*2*X*€ <sup>þ</sup> *<sup>b</sup>*1*X*\_ <sup>þ</sup> *boX* (24)

*X*2 � �

*X*1 *X*2 � �

� � <sup>þ</sup> *<sup>b</sup>*1*X*<sup>2</sup> <sup>þ</sup> *boX*<sup>1</sup> (25)

þ

*LQS* ¼ *ρU*∞Γ ¼ 2*πρU*∞*bW*3*=*<sup>4</sup> (28)

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>* � �

> *X*1 *X*2

!

0 1

� �*αeff* (26)

þ ð Þ *b*<sup>2</sup> *αeff* (27)

þ ½ � *b*<sup>2</sup> *W*3*=*<sup>4</sup>

(29)

We note that *W s*ð Þ can also be used as an indicial response to aerodynamic inputs other than the angle of attack. Van der Wall and Leishman [32] used it as an indicial response to the wing normal velocity, *w* ¼ *Uα*, in the case of time-varying free stream. For a relatively high angle of attack, the Duhamel superposition is performed using a more exact normal velocity *w* ¼ *U* sin *α*. Eq. (11) is then rewritten as

$$\mathcal{E}(\mathbf{s}) = \pi \rho U(\mathbf{s}) c \Big( U(\mathbf{0}) \sin a(\mathbf{0}) W(\mathbf{s}) + \int\_0^s \frac{d(U(\sigma) \sin a(\sigma))}{d\sigma} W(\mathbf{s} - \sigma) d\sigma \Big) \tag{13}$$

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:

$$\phi(s) = \mathbf{1} - A\_1 e^{-c\_1 s} - A\_2 e^{-c\_2 s} \tag{14}$$

where *A*<sup>1</sup> ¼ 0*:*165, *A*<sup>2</sup> ¼ 0*:*335,*c*<sup>1</sup> ¼ 0*:*0455,*c*<sup>2</sup> ¼ 0*:*3 and s is the reduced time parameter and is given by *U*∞*t=b*. By taking the Laplace transform with an operator P:

$$\phi(P) = \frac{1}{P} - \frac{A\_1}{P + \frac{c\_1 U\_\infty}{b}} - \frac{A\_2}{P + \frac{c\_3 U\_\infty}{b}} \tag{15}$$

the transfer function is then written as:

$$G(P) = \frac{Y(P)}{a\_{\rm eff}(P)} = \frac{\phi(P)}{\mathbf{1}/P} = \mathbf{1} - \frac{A\_1}{P + \frac{c\_1 U\_\infty}{b}} - \frac{A\_2 P}{P + \frac{c\_2 U\_\infty}{b}}\tag{16}$$

$$G(P) = \frac{\left(P + \frac{c\_1 U\_m}{b}\right)\left(P + \frac{c\_2 U\_m}{b}\right) - A\_1 P \left(P + \frac{c\_2 U\_m}{b}\right) - A\_2 P \left(P + \frac{c\_1 U\_m}{b}\right)}{\left(P + \frac{c\_1 U\_m}{b}\right)\left(P + \frac{c\_2 U\_m}{b}\right)}\tag{17}$$

$$G(P) = \frac{(\mathbf{1} - A\_1 - A\_2)P^2 + \left(\frac{c\_1 U}{b} \left(\mathbf{1} - A\_2\right) + \frac{c\_2 U}{b} \left(\mathbf{1} - A\_1\right)\right)P + \frac{c\_1 c\_2 U^2}{b^2}}{P^2 + (c\_1 + c\_2)\frac{U P}{b} + \frac{c\_1 c\_2 U^2}{b^2}}\tag{18}$$

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

$$\frac{Y}{a\_{\text{eff}}} = \frac{b\_2 P^2 + b\_1 P + b\_0}{P^2 + a\_1 P + a\_0} = \frac{Y}{X} \frac{X}{a\_{\text{eff}}} \tag{19}$$

where *X* is the internal states of the system, which is related to the input via these coefficients *ao* <sup>¼</sup> ð Þ *<sup>C</sup>*1þ*C*<sup>2</sup> *<sup>U</sup>*<sup>∞</sup> *<sup>b</sup>* , *<sup>a</sup>*<sup>1</sup> <sup>¼</sup> ð Þ *<sup>C</sup>*1*C*<sup>2</sup> *<sup>U</sup>*<sup>2</sup> ∞ *<sup>b</sup>*<sup>2</sup> , *bo* <sup>¼</sup> ð Þ *<sup>C</sup>*1*C*<sup>2</sup> *<sup>U</sup>*<sup>2</sup> ∞ *<sup>b</sup>*<sup>2</sup> , *b*<sup>1</sup> ¼ *C*1*U*<sup>∞</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>C</sup>*2*U*<sup>∞</sup> *<sup>b</sup>* � *<sup>A</sup>*1*C*2*U*<sup>∞</sup> *<sup>b</sup>* � *<sup>A</sup>*2*C*1*U*<sup>∞</sup> *b* � �, *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>A</sup>*<sup>1</sup> � *<sup>A</sup>*<sup>2</sup> as follows:

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

$$\frac{X}{a\_{\sharp f}} = \frac{1}{P^2 + a\_1 P + a\_0} \tag{20}$$

$$\frac{Y}{X} = \frac{b\_2 P^2 + b\_1 P + b\_0}{1} \tag{21}$$

and to the output via:

$$XP^2 + Xa\_1P + Xa\_0 = a\_{\sharp \sharp} \tag{22}$$

then applying Laplace inverse we get:

$$
\ddot{X} + a\_1 \dot{X} + a\_o X = a\_{\sharp \circ} \tag{23}
$$

then let *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>X</sup>* and *<sup>X</sup>*<sup>2</sup> <sup>¼</sup> *<sup>X</sup>*\_ . Also,

$$Y = Xb\_2P^2 + Xb\_1P + b\_0X = b\_2\ddot{X} + b\_1\dot{X} + b\_oX \tag{24}$$

Hence,

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 *W s*ð Þ can also be used as an indicial response to aerodynamic inputs other than the angle of attack. Van der Wall and Leishman [32] used it as an indicial response to the wing normal velocity, *w* ¼ *Uα*, in the case of time-varying free stream. For a relatively high angle of attack, the Duhamel superposition is performed using a more exact normal velocity *w* ¼ *U* sin *α*. Eq. (11) is then re-

> ð*s* 0

This equation is usually used in dynamic stall models where relatively high angles

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

where *A*<sup>1</sup> ¼ 0*:*165, *A*<sup>2</sup> ¼ 0*:*335,*c*<sup>1</sup> ¼ 0*:*0455,*c*<sup>2</sup> ¼ 0*:*3 and s is the reduced time

<sup>1</sup>*=<sup>P</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>A</sup>*<sup>1</sup> *<sup>P</sup>*

*<sup>b</sup>* ð Þþ <sup>1</sup> � *<sup>A</sup>*<sup>2</sup> *<sup>c</sup>*2*<sup>U</sup>*

To determine a second-order state-space realization of the transfer function in

*UP <sup>b</sup>* <sup>þ</sup> *<sup>c</sup>*1*c*2*U*<sup>2</sup> *b*2

�*c*<sup>1</sup> *<sup>s</sup>* � *<sup>A</sup>*<sup>2</sup> *<sup>e</sup>*

� *<sup>A</sup>*<sup>2</sup> *<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*3*U*<sup>∞</sup> *b*

*<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*1*U*<sup>∞</sup> *b*

� � � *<sup>A</sup>*2*P P* <sup>þ</sup> *<sup>c</sup>*1*U*<sup>∞</sup>

*<sup>b</sup>* ð Þ 1 � *A*<sup>1</sup> � �*<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*1*c*2*U*<sup>2</sup>

*b*

*b*

¼ *Y X X αeff*

*<sup>b</sup>*<sup>2</sup> , *bo* <sup>¼</sup> ð Þ *<sup>C</sup>*1*C*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

∞

� *<sup>A</sup>*<sup>2</sup> *<sup>P</sup> <sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*2*U*<sup>∞</sup> *b*

� � (17)

∞ *<sup>b</sup>*<sup>2</sup> , *b*<sup>1</sup> ¼

of attack are encountered, e.g., the Beddoes-Leishman dynamic stall model [33].

*ϕ*ðÞ¼ *s* 1 � *A*<sup>1</sup> *e*

1 *<sup>P</sup>* � *<sup>A</sup>*<sup>1</sup> *<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*1*U*<sup>∞</sup> *b*

*<sup>α</sup>eff*ð Þ *<sup>P</sup>* <sup>¼</sup> *<sup>ϕ</sup>*ð Þ *<sup>P</sup>*

*b*

� � � *<sup>A</sup>*1*P P* <sup>þ</sup> *<sup>c</sup>*2*U*<sup>∞</sup>

*<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*1*U*<sup>∞</sup> *b* � � *<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*2*U*<sup>∞</sup>

*<sup>P</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *<sup>c</sup>*<sup>1</sup> <sup>þ</sup> *<sup>c</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>b</sup>*2*P*<sup>2</sup> <sup>þ</sup> *<sup>b</sup>*1*<sup>P</sup>* <sup>þ</sup> *<sup>b</sup>*<sup>0</sup> *<sup>P</sup>*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*<sup>P</sup>* <sup>þ</sup> *<sup>a</sup>*<sup>0</sup>

where *X* is the internal states of the system, which is related to the input

*<sup>b</sup>* , *<sup>a</sup>*<sup>1</sup> <sup>¼</sup> ð Þ *<sup>C</sup>*1*C*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

*ϕ*ð Þ¼ *P*

the transfer function is then written as:

*b* � � *<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*2*U*<sup>∞</sup>

*G P*ð Þ¼ ð Þ <sup>1</sup> � *<sup>A</sup>*<sup>1</sup> � *<sup>A</sup>*<sup>2</sup> *<sup>P</sup>*<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*1*<sup>U</sup>*

*Y αeff*

*G P*ð Þ¼ *<sup>P</sup>* <sup>þ</sup> *<sup>c</sup>*1*U*<sup>∞</sup>

Eq. 17 can be written as:

*C*1*U*<sup>∞</sup> *<sup>b</sup>* <sup>þ</sup> *<sup>C</sup>*2*U*<sup>∞</sup>

**108**

via these coefficients *ao* <sup>¼</sup> ð Þ *<sup>C</sup>*1þ*C*<sup>2</sup> *<sup>U</sup>*<sup>∞</sup>

*<sup>b</sup>* � *<sup>A</sup>*2*C*1*U*<sup>∞</sup> *b* � �, *<sup>b</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>A</sup>*<sup>1</sup> � *<sup>A</sup>*<sup>2</sup> as follows:

*<sup>b</sup>* � *<sup>A</sup>*1*C*2*U*<sup>∞</sup>

*G P*ð Þ¼ *Y P*ð Þ

parameter and is given by *U*∞*t=b*. By taking the Laplace transform with an

� �

ð*s* 0

*dα σ*ð Þ

*d U*ð Þ ð Þ *σ* sin *α σ*ð Þ

*<sup>d</sup><sup>σ</sup> W s*ð Þ � *<sup>σ</sup> <sup>d</sup><sup>σ</sup>*

�*c*<sup>2</sup> *<sup>s</sup>* (14)

*b* � �

*b*2

� �

*<sup>d</sup><sup>σ</sup> W s*ð Þ � *<sup>σ</sup> <sup>d</sup><sup>σ</sup>*

(12)

(13)

(15)

(16)

(18)

(19)

*c α*ð Þ 0 *W s*ðÞþ

<sup>ℓ</sup>ðÞ¼ *<sup>s</sup> πρU*<sup>2</sup>

ℓðÞ¼ *s πρU s*ð Þ*c U*ð Þ 0 sin *α*ð Þ 0 *W s*ðÞþ

**3.3 State space finite model**

written as

*Biomimetics*

operator P:

$$Y = b\_2(a\_{\sharp \overline{f}} - a\_o X\_1 - a\_1 X\_2) + b\_1 X\_2 + b\_o X\_1 \tag{25}$$

By writing these equation in a matrix form, we obtain

$$
\frac{d}{dt} \begin{pmatrix} X\_1 \\ X\_2 \end{pmatrix} = \begin{bmatrix} 0 & 1 \\ -a\_o & -a\_1 \end{bmatrix} \begin{pmatrix} X\_1 \\ X\_2 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} a\_{\mathcal{G}\dagger} \tag{26}
$$

$$y = [b\_o - b\_2 a\_o \ b\_1 - b\_2 a\_1] \begin{pmatrix} X\_1 \\ X\_2 \end{pmatrix} + (b\_2) a\_{\sharp \overline{\!\!\!f}} \tag{27}$$

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

$$L\_{\mathbb{Q}\mathbb{S}} = \rho U\_{\curvearrowright} \Gamma = 2\pi \rho U\_{\curvearrowright} b \,\, W\_{\mathbb{S}/4} \tag{28}$$

where *W*3*=*<sup>4</sup> is the normal velocity component and is given by:

$$\begin{aligned} W\_{3/4} &= U\_{\infty} \sin \left( a \right) + \dot{a} \left[ \frac{b}{2} + a \right] \\ L\_{\epsilon}(t) &= 2 \pi \rho U\_{\infty} b \left[ b\_o - b\_2 a\_o \, b\_1 - b\_2 a\_1 \right] \binom{X\_1}{X\_2} + [b\_2] W\_{3/4} \end{aligned} \tag{29}$$

#### **3.4 Unsteady vortex lattice method (UVLM)**

The unsteady Vortex lattice methods (UVLMs) are well suited to the bioinspired 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 [34–36]. 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.

From Kelvin's circulation theorem, we have:

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

steps. As such, we obtain the following linear system,

*a*1,1 *a*1,2 … *a*1,*MN a*2,1 *a*2,2 … *a*2,*MN* ⋮ ⋮⋱ ⋮ *aMN*,1 *aMN*,2 … *aMN*,*MN*

> *<sup>K</sup>* � *<sup>n</sup>* !

0

BBBBB@

*RHSK* ¼ � *v*

and *v* !

equation,

*pl* � *pu ρ* � �

surfaces, respectively.

**111**

*K*

<sup>¼</sup> <sup>Δ</sup>*<sup>p</sup> ρ* � �

*∂ϕ<sup>u</sup> ∂t* � �

force on each panel is obtained from:

! <sup>þ</sup> *<sup>v</sup>* ! *w* � �

<sup>X</sup>Γ*tot* <sup>¼</sup> <sup>X</sup>*γ<sup>b</sup>* <sup>þ</sup> *<sup>γ</sup><sup>w</sup>* <sup>¼</sup> <sup>0</sup> ) <sup>X</sup>*γ<sup>b</sup>* <sup>þ</sup> *<sup>γ</sup>TE* ¼ �Γ*<sup>w</sup>* (31)

where Γ*<sup>w</sup>* is the sum of all wake vortices, which are know from previous time

Γ1 Γ2 ⋮ Γ*MN*

*<sup>w</sup>* is the velocity induced at the control point *K* by all the other vortex point in the wake created before the time *t*. In order to satisfy the unsteady Kutta condition, the wake is created at each instant of time at the trailing edge by shedding a new vortex that has an intensity equal to the bound vortex on the panel along the trailing edge. At each instant of time all the points in the wake generated in previous steps are convected downstream following the induced velocity generated by all the vortices on the surface and through the wake. The velocity induced by each vortex is computed by using the Biot Savart law. This induced velocity is inversely proportional to the distance between the vortex location and the control point where the velocity is calculated. Having solved the linear system [39] in the bound vorticity, the pressure difference through the bound vortex sheet is computed based on

9 >>>>>=

*RHS*<sup>1</sup> *RHS*<sup>2</sup> ⋮ *RHSMN*

! is the wind flow velocity relative to the surface

9 >>>>>=

>>>>>;

, (32)

8 >>>>><

>>>>>:

>>>>>; ¼

1

8 >>>>><

>>>>>:

where *aK*1,*K*<sup>2</sup> are the influence coefficients from the point vortex *K*<sup>2</sup> at the control point *K*<sup>1</sup> and it is equal to the normal velocity that the point vortex induces

at the control point Γ*<sup>K</sup>*<sup>2</sup> ¼ 1. Each element on the right hand side is

the unsteady Bernoulli's equation. More details can be found in [40].

*d dt* ð*c* 0

*<sup>u</sup>* � *<sup>V</sup>*<sup>2</sup> *l* 2 � �

*K* þ

where *p* denotes the static pressure, *V* is the tangent velocity, *ϕ* is the velocity potential, and the subscripts *u* and *l* are used to represent the upper and lower

the unsteady pressure difference on the *Kth* panel is given by,

<sup>¼</sup> *<sup>V</sup>*<sup>2</sup>

Δ*P x*ð Þ 1 <sup>2</sup> *<sup>ρ</sup>U*<sup>2</sup> ∞ ¼ *ρ*

*K*

From the definition of circulation, we have:

� *<sup>∂</sup>ϕ<sup>l</sup> ∂t* � �

*K*

<sup>¼</sup> *<sup>∂</sup>*Γ*<sup>i</sup>*,*<sup>j</sup>*

for *i* = 1, 2, ..., *M*, and by applying the Kutta-Joukowski theorem, the normal

*K*

The unsteady aerodynamic loads can be calculated from the circulation Γ*<sup>K</sup>* of the *Kth* panel and its time rate of change [40]. Using the unsteady Bernoulli

*γb*ð Þ *x dx* þ *U*∞*γb*ð Þ *x*

*∂ϕ<sup>u</sup> ∂t* � �

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>Γ</sup>*<sup>i</sup>*,*<sup>j</sup>*ðÞ�*<sup>t</sup>* <sup>Γ</sup>*<sup>i</sup>*,*<sup>j</sup>*ð Þ *<sup>t</sup>* � <sup>1</sup>

*K*

� � (33)

� *<sup>∂</sup>ϕ<sup>l</sup> ∂t* � �

*K*

<sup>Δ</sup>*<sup>t</sup>* , (35)

, (34)

*<sup>K</sup>*, where *v*

CCCCCA

Zakaria et al. [8] 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:


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 *γboundvortices*'s and *γatrailingedgevortex*), then (m + 1) equations are needed for closure. For the no-penetration boundary condition at m control points, we have:

$$\left.V\_{cp}^{(n)} = V\_{air}^{(n)}\right|\_{cp} \tag{30}$$

**Figure 5.**

*A schematic diagram showing the panels on the airfoil camber and the shedded vortices used in UVLM modeling.*

is now a powerfull tool in hand for aerodynamicists for its ease implementation

Zakaria et al. [8] 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:

• 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 *γboundvortices*'s and *γatrailingedgevortex*), then (m + 1) equations are needed for closure. For the no-penetration boundary

> *V*ð Þ *<sup>n</sup> cp* <sup>¼</sup> *<sup>V</sup>*ð Þ *<sup>n</sup> air cp*

*A schematic diagram showing the panels on the airfoil camber and the shedded vortices used in UVLM*

(30)

• No penetration boundary condition.

condition at m control points, we have:

**Figure 5.**

*modeling.*

**110**

even for complex shapes.

*Biomimetics*

From Kelvin's circulation theorem, we have:

$$
\sum \Gamma\_{\text{tot}} = \sum \gamma\_b + \gamma\_w = \mathbf{0} \Rightarrow \sum \gamma\_b + \gamma\_{\text{TE}} = -\Gamma\_w \tag{31}
$$

where Γ*<sup>w</sup>* is the sum of all wake vortices, which are know from previous time steps. As such, we obtain the following linear system,

$$\begin{pmatrix} a\_{1,1} & a\_{1,2} & \dots & a\_{1,MN} \\ & a\_{2,1} & a\_{2,2} & \dots & a\_{2,MN} \\ & \vdots & \vdots & \ddots & \vdots \\ & & a\_{MN,1} & a\_{MN,2} & \dots & a\_{MN,MN} \end{pmatrix} \begin{Bmatrix} \Gamma\_1 \\ \Gamma\_2 \\ \vdots \\ \Gamma\_N \\ \Gamma\_{MN} \end{Bmatrix} = \begin{Bmatrix} RHS\_1 \\ & RHS\_2 \\ & \vdots \\ & & \vdots \\ & & RHS\_{MN} \end{Bmatrix} \tag{32}$$

where *aK*1,*K*<sup>2</sup> are the influence coefficients from the point vortex *K*<sup>2</sup> at the control point *K*<sup>1</sup> and it is equal to the normal velocity that the point vortex induces at the control point Γ*<sup>K</sup>*<sup>2</sup> ¼ 1. Each element on the right hand side is *RHSK* ¼ � *v* ! <sup>þ</sup> *<sup>v</sup>* ! *w* � � *<sup>K</sup>* � *<sup>n</sup>* ! *<sup>K</sup>*, where *v* ! is the wind flow velocity relative to the surface and *v* ! *<sup>w</sup>* is the velocity induced at the control point *K* by all the other vortex point in the wake created before the time *t*. In order to satisfy the unsteady Kutta condition, the wake is created at each instant of time at the trailing edge by shedding a new vortex that has an intensity equal to the bound vortex on the panel along the trailing edge. At each instant of time all the points in the wake generated in previous steps are convected downstream following the induced velocity generated by all the vortices on the surface and through the wake. The velocity induced by each vortex is computed by using the Biot Savart law. This induced velocity is inversely proportional to the distance between the vortex location and the control point where the velocity is calculated. Having solved the linear system [39] in the bound vorticity, the pressure difference through the bound vortex sheet is computed based on the unsteady Bernoulli's equation. More details can be found in [40].

The unsteady aerodynamic loads can be calculated from the circulation Γ*<sup>K</sup>* of the *Kth* panel and its time rate of change [40]. Using the unsteady Bernoulli equation,

$$\frac{\Delta P(\mathbf{x})}{\frac{1}{2}\rho U\_{\infty}^{2}} = \rho \left(\frac{d}{dt}\int\_{0}^{\epsilon} \gamma\_{b}(\mathbf{x})d\mathbf{x} + U\_{\ast\ast}\gamma\_{b}(\mathbf{x})\right) \tag{33}$$

the unsteady pressure difference on the *Kth* panel is given by,

$$\left(\frac{p\_l - p\_u}{\rho}\right)\_K = \left(\frac{\Delta p}{\rho}\right)\_K = \left(\frac{V\_u^2 - V\_l^2}{2}\right)\_K + \left(\frac{\partial \phi\_u}{\partial t}\right)\_K - \left(\frac{\partial \phi\_l}{\partial t}\right)\_K \tag{34}$$

where *p* denotes the static pressure, *V* is the tangent velocity, *ϕ* is the velocity potential, and the subscripts *u* and *l* are used to represent the upper and lower surfaces, respectively.

From the definition of circulation, we have:

$$
\left(\frac{\partial \phi\_u}{\partial t}\right)\_K - \left(\frac{\partial \phi\_l}{\partial t}\right)\_K = \frac{\partial \Gamma\_{i,j}}{\partial t} = \frac{\Gamma\_{i,j}(t) - \Gamma\_{i,j}(t-1)}{\Delta t} \quad , \tag{35}
$$

for *i* = 1, 2, ..., *M*, and by applying the Kutta-Joukowski theorem, the normal force on each panel is obtained from:

$$
\overrightarrow{F}\_{N\_K} = - (\Delta p \Delta \mathcal{S})\_{i,j} \overrightarrow{n}\_{i,j} \quad , \tag{36}
$$

where Δ*S* is the area of each panel.

#### **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.

#### **4. Maneuver case studies results**
