*4.2.3 Trailing edge pivot*

**4.2 Case 1: Pitch ramp** *<sup>α</sup><sup>o</sup>* <sup>¼</sup> **<sup>45</sup>***<sup>o</sup>*

**Figures 9**–**11** show the lift coefficient response for a ramp maneuver with an amplitude of 45*<sup>o</sup>* at three different pivot locations. At the leading edge pivot

location; shown in **Figure 8**, at the beginning of the ramp (*τ* ¼ 0 � 1), Theodorsen model based on Fourier series model shows higher lift coefficient than all the other models as well as the experimental results. For the ramp upstroke, all the models showed a decrease in the lift coefficient compared to the experiment results preserving the same slope until the start of the second event then a continuous increase in lift response which appears as over predicted values compared to the experimental results presented by Ramesh et al. [9]. The UVLM model pertained the same lift pattern and all proposed models show a large discrepancy compared to experimental results. In addition, the UVLM model results show a good agreement with experiments at the ramp-up then starts to deviate with an increase in lift coefficient by 48% at hold-on and ramp down regimes. Furthermore, the quasi steady model shows a high lift coefficient at the end of the ramp-upstroke compared to the

**Figure 10** shows the lift coefficient for a ramp amplitude of 45*<sup>o</sup>* at half chord pivot location. The proposed models show a good match at the ramp-up regime then an over predicted lift coefficient occurs after the ramp-hold and ramp-down regimes compared to the experimental results except for Theodorsen FFT based model and the quasi-steady model. Again, Theodorsen FFT based model gives an attenuated response, and the quasi-steady model shows a magnified response (qualitatively similar to the results presented in **Figure 7** of a pitch ramp amplitude of 25°). During the ramp-hold phase, the UVLM model matches well with small discrepancy compared to all other models. At the final phase (ramp-down), all

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

experiments, followed by a sharp decrease at the ramp-down stroke.

*4.2.1 Leading edge pivot*

*Biomimetics*

*4.2.2 Half chord pivot*

**Figure 11.**

**116**

*amplitude* 45° *at trailing pivot location.*

**Figure 11** presents the lift coefficient for a ramp amplitude of 45*<sup>o</sup>* at trailing edge pivot location. By comparing **Figure 11** along with **Figure 8**, the two models (Thoedorsen FFT based and UVLM) show a very good prediction with the experimental results for the two phases (ramp-up and ramp hold), then show an increase in the lift coefficient at ramp down. On the contrary, all other models record an over predicted lift coefficient compared to the experimental results for all events preserving the lift response pattern. The quasi steady models for the two ramp cases (0°-25°-0° and 0°-45°-0°) at the same pivot location (trailing edge). **Figure 8** and **Figure 11**, do not show any sharp peak for lift coefficient for the ramp transition regimes. This is expected due to the lack of inclusion of wing stall and rotational effects.

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 ( sin ð Þ *α* ) that is induced by the no-penetration boundary condition in the UVLM model. Consequently, at this range of AoA (25°) (attached flow), the dominant effect for the LES and nonlinearity associated with the ramp maneuver appears to be matched well with the results of Ramesh et al. [9]. At high angle of attack maneuver (45°), this effect no longer exists as the flow separates and became more pronounced [43]. Recall that rotational lift is proportional to the distance between the pivot and three quarter chord point (Giacomelli and Pistolesi theorem [44]), which attains and preserves its largest value for a leading edge pivot. The UVLM results match the experimental results with small

**Figure 12.**

*Shedding of trailing vortices and wake convection downstream for* 25° *amplitude ramp maneuver. (a) Leading edge pivot. (b) Half chord pivot. (c) Trailing edge pivot.*

**Figure 13.**

*Shedding of trailing vortices and wake convection downstream for* 45° *amplitude ramp maneuver. (a) Leading edge pivot. (b) Half chord pivot. (c) Trailing edge pivot.*

nuances even for large amplitude (45°) at the ramp-up regime and partially at the ramp-hold only. At ramp-down regime, the UVLM results deviate from the experimental results and appeared to be over predicted.

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 semiempirical models based on potential flow theories and experiments that can predict

*V*

<sup>2</sup> ð Þ

*c*

unsteady forces of ramp maneuvers.

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

*b* airfoil semi-chord (*c=*2) *c* airfoil semi-chord (*c=*2)

*h* plunging displacement (mm)

*P* non-dimensional Laplace operator *q* non-dimensional pitch rate,*<sup>α</sup>*\_*<sup>c</sup>*

*S* distance traveled in semi-chords,<sup>2</sup>*Vt*

**Nomenclature**

\_

€

*CL* lift coefficient *C k*ð Þ lift deficiency factor *f* frequency (Hz)

*h* plunging velocity

ℓ wing span (m)

*Re* Reynolds number

*U*<sup>∞</sup> free stream velocity *Urel* free stream velocity *α<sup>o</sup>* airfoil mean angle of attack *αeff* effective angle of attack *α*\_ angular pitch velocity ð Þ *rad=s α*€ angular pitch acceleration *rad=s*

*T* time period

**Greek variables**

*ϕ* phase angle

*ρ* Air density

**Abbreviations**

**119**

*AoA* angle of attack *circ* circulatory

*RHS* right hand side

*FFT* Fast Fourier Transform

*UVLM* unsteady vortex lattice method

Γ total flow circulation

*γ<sup>b</sup>* elementary bound flow circulation *γ<sup>w</sup>* elementary wake flow circulation *ω* angular frequency,(rad/s) *σ* heaviside function variable *τ* Non-dimensional time

*h* plunging acceleration *k* reduced frequency *πfc=U*<sup>∞</sup>

**Figures 12** and **13** show the Shedding of trailing vortices and wake convection shape downstream for 25° and 45° amplitudes ramp maneuvers. All the figures show the same convection pattern for the three pivot locations with an increase in the y axis vortex location with increasing the ramp amplitude.

### **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 (25° and 45°) and three pivot location on the airfoil chord (*c=*4,*c=*2, 3*c=*4). The analytical results were compared to the experimental data and the comparison revealed an acceptable agreement at the pitch ramp amplitude of 25*<sup>o</sup>* compared to the results presented by the 45*<sup>o</sup>* ramp amplitude case. Thus, those models can be considered as promising aerodynamic models for predicting lift coefficient for such manoeuver at a ramp amplitude up to 25*<sup>o</sup>* only. Along the four analytical models, the UVLM showed very good results for the two ramp amplitude cases. It should be noted that, the UVLM captures all geometric nonlinearities, wake deformation, rolling wake, leading edge suction and post stall without the inclusion of leading edge vortex effects. Duhamel and the state space models appear to have the same behavior which asserts that the state space model shares the same physical base and obtained the same results compared to Theodorsen's model.

**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.

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


**Table 2.** *Proposed models output parameters for solving pitching maneuvers.* *Unsteady Aerodynamics of Highly Maneuvering Flyers DOI: http://dx.doi.org/10.5772/intechopen.94231*

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 semiempirical models based on potential flow theories and experiments that can predict unsteady forces of ramp maneuvers.
