**4.1 Case 1: Pitch ramp** *<sup>α</sup><sup>o</sup>* <sup>¼</sup> **<sup>25</sup>***<sup>o</sup>*

#### *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. [9] as follows: (i) onset of flow separation at the ramp start (*τ* ¼ 1), (ii) the formation of a leading edge vortex (*τ* ¼ 1 � 3), (iii) ramp hold (*τ* ¼ 3 � 4) and (iv) detachment of the leading-edge vortex (*τ* ¼ 4 � 6).

**Figure 6** shows the ramp pitch motion with an amplitude of 25*<sup>o</sup>* about a leading edge hinge. Almost all the theoretical models have the same jump during the transition of each event start and end positions during the whole ramp manoeuver compared to the experimental results. During upstroke, *τ* ¼ 1 � 3, a very good match is found between the experimental results and the UVLM results. The prediction of the quasi-steady model is higher compared to the experimental results which is expected as it lacks the dynamics of the flow and is based only the static behavior of the generated lift. On the other hand, all other presented models show an attenuated lift response during the ramp-up phase. During the ramp hold period, *τ* ¼ 3 � 4, a very good agreement between all the models and the experiments except the quasi-steady and Theodorsen FFT based model. During the ramp-down phase, the UVLM model matches very well with the experimental results preserving the lift dynamics. On the other hand, all models over predict the lift coefficient except the quazi-steady model shows a lower lift coefficient. As reported by Yu et al. [14], the reason for this discrepancy could be attributed to the sensitivity of these models to capture the LEV de-attachment and the lift decrease during this phase. Based on the observations of Ramesh et al. [9], this behavior points to an


important aspect, where the lift dynamics results in a considerable delay; i.e. the lift

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and*

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and*

**Figure 7** shows the ramp with amplitude of 25*<sup>o</sup>* at the mid chord hinge location. The results show a good agreement with the experimental results by having the same lift response slope except for Thoedorsen model based on FFT model

response does not depend on the past history (memory effects).

*4.1.2 Half chord pivot*

*amplitude* 25° *at the half chord hinge location.*

**Figure 7.**

**113**

**Figure 6.**

*amplitude* 25° *at the leading edge hinge location.*

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

**Table 1.**

*Classical aerodynamics proposed models for solving pitching maneuvers.*

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

#### **Figure 6.**

*F* !

apply simple analytical equation to solve such maneuver.

detachment of the leading-edge vortex (*τ* ¼ 4 � 6).

where Δ*S* is the area of each panel.

**4. Maneuver case studies results**

**4.1 Case 1: Pitch ramp** *<sup>α</sup><sup>o</sup>* <sup>¼</sup> **<sup>25</sup>***<sup>o</sup>*

*4.1.1 Leading edge pivot*

**Table 1.**

**112**

**3.5 Models comparison**

*Biomimetics*

*NK* ¼ �ð Þ Δ*p*Δ*S <sup>i</sup>*,*jn*

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

**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. [9] as follows: (i) onset of flow separation at the ramp start (*τ* ¼ 1), (ii) the

formation of a leading edge vortex (*τ* ¼ 1 � 3), (iii) ramp hold (*τ* ¼ 3 � 4) and (iv)

**Models Input motion Nonlinearity wake deformation Camber variation**

Theodorsen Harmonic Geometric Flat � Wagner Step input Geometric � � State space Arbitrary �� � UVLM Arbitrary √√ √

*Classical aerodynamics proposed models for solving pitching maneuvers.*

edge hinge. Almost all the theoretical models have the same jump during the transition of each event start and end positions during the whole ramp manoeuver compared to the experimental results. During upstroke, *τ* ¼ 1 � 3, a very good match is found between the experimental results and the UVLM results. The prediction of the quasi-steady model is higher compared to the experimental results which is expected as it lacks the dynamics of the flow and is based only the static behavior of the generated lift. On the other hand, all other presented models show an attenuated lift response during the ramp-up phase. During the ramp hold period, *τ* ¼ 3 � 4, a very good agreement between all the models and the experiments except the quasi-steady and Theodorsen FFT based model. During the ramp-down phase, the UVLM model matches very well with the experimental results preserving the lift dynamics. On the other hand, all models over predict the lift coefficient except the quazi-steady model shows a lower lift coefficient. As reported by Yu et al. [14], the reason for this discrepancy could be attributed to the sensitivity of these models to capture the LEV de-attachment and the lift decrease during this phase. Based on the observations of Ramesh et al. [9], this behavior points to an

**Figure 6** shows the ramp pitch motion with an amplitude of 25*<sup>o</sup>* about a leading

!

*<sup>i</sup>*,*<sup>j</sup>* , (36)

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and amplitude* 25° *at the leading edge hinge location.*

#### **Figure 7.**

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and amplitude* 25° *at the half chord hinge location.*

important aspect, where the lift dynamics results in a considerable delay; i.e. the lift response does not depend on the past history (memory effects).

#### *4.1.2 Half chord pivot*

**Figure 7** shows the ramp with amplitude of 25*<sup>o</sup>* at the mid chord hinge location. The results show a good agreement with the experimental results by having the same lift response slope except for Thoedorsen model based on FFT model

#### **Figure 8.**

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and amplitude* 25° *at the leading edge hinge location.*

conclusion was reported also by Yu et al. [14] in their recent work for examining the

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.4 and*

In a similar manner, **Figure 8** shows a comparison between experimental and theoretical predictions for 25*<sup>o</sup>* ramp case pitched about the trailing edge pivot. Lift coefficient comparison shows a qualitatively good agreement between the experiment and all the presented models (lift coefficient pattern). Of particular interest, taking the lift transition peaks during different ramp regimes. UVLM model results match very well for the ramp-up and ramp-hold regimes then decrease slightly at the ramp-down regime to give an damped lift coefficient values. Theodorsen model based on Fast Fourier Transform records an attenuated lift coefficient compared to experiments. On the other hand, all other presented models show an over predicted lift coefficient compared to the experimental results while preserving the same lift

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 (*α*), one could be interested in the transient response, however, the analytical expressions cannot be obtained and the discrete Fourier Transform (FFT) has to be used instead. Discrete Fourier Transform compared to the exact Fourier Transform ignores some frequency content mathematically. In addition to the aside notion of flow dynamics, the leading edge location experiences rich dynamics when compared to the half and trailing pivot chord locations reported by Ford and Babinsky [41]. This is due to the inclusion of rotational effects which increase by increasing the spacing between the hinge point and the three-quarter chord location. It is clear that all models capture the lift peaks during transition between pitch ramps except the quasi-steady model. This lift peak has been reported by previous studies due to a

effect of pivot locations on force and moment coefficients.

*4.1.3 Trailing edge pivot*

*amplitude* 45° *at half chord pivot location.*

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

**Figure 10.**

**115**

response pattern for all ramp regimes.

delay in stall and/or a delay in LEV formation [39, 42].

#### **Figure 9.**

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.4 and amplitude* 45<sup>∘</sup> *at leading pivot location.*

(attenuated response) and the quasi-steady model (over predicted). Of particular interest, Duhamel and state space models coincide on top of each other having the same lift magnitude. This is excepted as the two models have the same mathematical base. At the ramp hold phase, the value of the saturated lift coefficient is approximately 2 for all models except the quasi-steady and Theodorsen based on FFT. Here, the impact of shifting the pivot location towards the trailing edge conducive in reducing the rotational effects which in turn decreases the lift coefficient by 15% at the ramp-up event compared to the quarter chord location. This

#### **Figure 10.**

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.4 and amplitude* 45° *at half chord pivot location.*

conclusion was reported also by Yu et al. [14] in their recent work for examining the effect of pivot locations on force and moment coefficients.

#### *4.1.3 Trailing edge pivot*

In a similar manner, **Figure 8** shows a comparison between experimental and theoretical predictions for 25*<sup>o</sup>* ramp case pitched about the trailing edge pivot. Lift coefficient comparison shows a qualitatively good agreement between the experiment and all the presented models (lift coefficient pattern). Of particular interest, taking the lift transition peaks during different ramp regimes. UVLM model results match very well for the ramp-up and ramp-hold regimes then decrease slightly at the ramp-down regime to give an damped lift coefficient values. Theodorsen model based on Fast Fourier Transform records an attenuated lift coefficient compared to experiments. On the other hand, all other presented models show an over predicted lift coefficient compared to the experimental results while preserving the same lift response pattern for all ramp regimes.

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 (*α*), one could be interested in the transient response, however, the analytical expressions cannot be obtained and the discrete Fourier Transform (FFT) has to be used instead. Discrete Fourier Transform compared to the exact Fourier Transform ignores some frequency content mathematically. In addition to the aside notion of flow dynamics, the leading edge location experiences rich dynamics when compared to the half and trailing pivot chord locations reported by Ford and Babinsky [41]. This is due to the inclusion of rotational effects which increase by increasing the spacing between the hinge point and the three-quarter chord location. It is clear that all models capture the lift peaks during transition between pitch ramps except the quasi-steady model. This lift peak has been reported by previous studies due to a delay in stall and/or a delay in LEV formation [39, 42].

(attenuated response) and the quasi-steady model (over predicted). Of particular interest, Duhamel and state space models coincide on top of each other having the same lift magnitude. This is excepted as the two models have the same mathematical base. At the ramp hold phase, the value of the saturated lift coefficient is approximately 2 for all models except the quasi-steady and Theodorsen based on FFT. Here, the impact of shifting the pivot location towards the trailing edge conducive in reducing the rotational effects which in turn decreases the lift coefficient by 15% at the ramp-up event compared to the quarter chord location. This

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.4 and*

*Comparison for the proposed models and experimental work done by Ramesh et al. with ramp rate of 0.2 and*

**Figure 8.**

*Biomimetics*

**Figure 9.**

**114**

*amplitude* 45<sup>∘</sup> *at leading pivot location.*

*amplitude* 25° *at the leading edge hinge location.*
