# Velocity Induced by a Straight Vortex Section

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## Chapter 2 Aerodynamic Modeling

The aerodynamic model is designed to encompass low-speed, low-angle of attack,but large motions. The model is developed for inviscid, incompressible, and unsteady flow. The low-angle of attack, and low-speed assumption ensures that the flow separates at the trailing edge. To design a model, the vortex lattice method with vortex rings is implemented. An ideal model would take viscous effects into consideration, and would be based on the Navier-Stokes equations. Other ways of modeling the flow would be based on either vortex particle method, free vortex blob method, and computational fluid dynamics.CFD programs can simulate the case being studied with higher fidelity, as compared to vortex lattice methods or, vortex particle methods. More sophisticated codes available in the industry based on vortex lattice methods can also do a very good job of simulating the flow. However, the purpose of this study is to do a preliminary aerodynamic analysis. Since the flow parameters considered are inviscid; with small angle of attack, and pitch angle; and low speed; the vortex lattice method is a reliable method for preliminary analysis. Vortex lattice methods can be implemented
with a fixed wake or a free wake approximation. Since the method is being developed for cases of pitch and plunge; and has to be general enough to accommodate flapping wing effects for future work, the free-wake model’s wake shape works as a check point to validate whether the physics of the flight is being captured by the model. Also, the vortex stretching terms included in the free-vortex blob methods are not considered as the vortex stretching effects are insignificant for the cases being studied. The VLM is inexpensive medium order, medium fidelity model. The study and application of CFD methods, and sophisticated simulation softwares require experience, and are expensive. The Vortex Lattice Method is described below.

## Basic Concepts

Angular Velocity, Vorticity, and Circulation

The angular motion of a fluid element consists of translation, rotation, and deformation. The translation is caused by the local velocity of the fluid. Due to variations in the local velocity, the fluid element rotates and deforms. The components of the angular velocity of the fluid are given by [14],where, indices i, j and k represent the three directions of the cartesian frame, and The angular velocity can thus be written as,For convenience, the term vorticity is defined as For an open surface S with a closed curve C along its boundary shown in Figure 2.1, the vorticity on the surface S can be related to the line integral around curve C using Stokes’s theorem.where ~n is normal to the surface dS. This term is called the Circulation and is represented by the Greek letter Γ .The Circulation establishes the relationship with the velocity, and is later used in the computation of aerodynamic loads.

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Irrotationality

In the presence of viscous forces, a fluid particle will rotate like a rigid body On the other hand, in the absence of large viscous forces as being considered in the aerodynamic modeling, which exist in the region outside the boundary layer of a body in motion, the fluid is irrotational.

#### Biot-Savart Law

For an incompressible fluid, the continuity equation is in the form of Laplace’s equation The following steps are taken to establish a relationship in between the velocity and a known vorticity distribution. In a region where vorticity can exist, the velocity field is expressed as the curl of a vector field B such that,where r0 is the distance from the origin, with the vorticity at a distance of r1 and Q is the volume containing the vorticity The velocity due to a volume distribution of vorticity is derived below. Consider an infinitesimal vorticity element ζ, Figure 2.3 . The cross sectional area dS is selected such that the vorticity vector ζ is perpendicular to it. The direction of the filament dl is parallel to the vorticity vector.

1 Introduction and Overview
1.1 Motivation
1.2 Micro Air Vehicles (MAVs)
1.3 Aerodynamic modeling
1.4 Aero-Structures interation
1.5 Thesis Layout
2 Aerodynamic Modeling
2.1 Basic Concepts
2.1.1 Angular Velocity, Vorticity, and Circulation
2.1.2 Irrotationality
2.2 Biot-Savart Law
2.2.1 Velocity Induced by a Straight Vortex Section
2.3 Vortex Lattice Method
2.3.1 Spatial Conservation of Vorticity
2.3.2 Helmholtz’s theorem
2.3.3 Kelvin’s theorem
2.3.4 Discretized Vortex Sheet
2.3.5 Boundary conditions
2.3.7 Implementation of VLM
2.4 Theoretical Results
3 Structural Modeling And Aeroelasticity
3.1 Membrane theory
3.1.1 Static Membrane Deformation
3.2 Application of Fourier Series
3.3 Aeroelasticity
3.3.1 Time t = t1
3.3.2 Time ti > t1
4 Results
4.1 Validation of the 2-dimensional aspects of the Aerodynamic model
4.1.1 Constant Angle of Attack
4.1.2 Pitching wing
4.1.3 Plunging wing
4.2 Grid discretization studies
4.3 Comparison with lifting line theory
4.4 Validation of 3-dimensional aspects of VLM with Doublet Lattice method
4.4.1 Pitch
4.4.2 Plunge
4.5 Study of loads and deformations for constant angle of attack
4.5.1 Variation in the number of structural modes
4.5.2 Variation in stiffness
4.6 Study of loads and deformations for a plunging wing
4.6.1 Variation in the number of structural modes
4.6.2 Variation in plunge amplitude
4.6.3 Variation in reduced frequency
4.6.4 Variation in stiffness
4.7 Study of loads for a pitching and plunging wing at different phases
5 Conclusion
6 Bibliography

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