## 1. Introduction

The increasing use of the unmanned aerial vehicles (UAVs) in different industries has led to the development of the common platforms used in the design of UAVs. The preferable of these platforms is the quadrotor due to its advantages such as an easier control in flight for drone operators, a rapid takeoff, and maneuverability. In the design of a quadcopter, it is necessary to estimate the performance of the propellers used in the vehicle. This chapter presents two of the simplest and more used theories to achieve a preliminary approximation to the performance of the selected propeller; also, a CFD model is used to study the aerodynamics of a commercial propeller used in a quadcopter of the Colombian company ADVECTOR.

Momentum theory is the simplest theory to study a propeller. A propeller produces thrust by the acceleration of the column of air that passes through the rotor plane, and with the use of the laws of conservation of mass, momentum, and energy, it is possible to obtain a relation of the induced velocity from the propeller to the air and the thrust that the air produces on the propeller [1].

Blade element theory applies the aerofoil theory to the rotating blade [1], this theory studies the forces actuating in a two‐dimensional (2D) section of the blade; thus, it is necessary to have a detailed description of the geometry of the blades of the propeller. It includes both the blade chord and the pitch angle distributions along the blade span; the lift and drag coefficients are assumed to be known or can be estimated for each section of the blade. The integration of the contributions of the sections along the radius yields the total thrust and torque.

The research community has been studying quadcopters in the last years, most of them developing control systems for these vehicles, but the aerodynamic effects of the rotorcraft vehicles have been often ignored [2]. The wake of a rotorcraft has a significant effect on the overall flow field and on the performance of the vehicle. The wake is characterized by induced velocities and regions with intense vortical flow which interact with rotor blades and fuselages [3]. Using computational fluid dynamics (CFD), it is possible to analyze the rotor wake and tip vortices and also to predict the performance of the propeller.

## 2. Momentum theory

The simplest theory to estimate the performance of a propeller is the momentum theory. This model uses an ideal fluid, that is, the air is incompressible, not viscous and irrotational, whereas the propeller is modeled as a disc with the same area that is described by the rotation of the propeller. This disc produces a pressure jump uniformly distributed to the flow passing through it [1]. **Figure 1** shows the sketch of the model.

Applying the Bernoulli equation to the fluid before the disc,

where

where
**Figure 1**. From Eqs. (1) and (2),

But,

where

Using momentum conservation, it is easy to see that the total increase on the momentum of the air must be equal to the thrust, and from **Figure 1**,

Using Eq. (4),

And with Eq. (3),

Which means that half of the velocity induced to the fluid is done before the propeller and half after this, therefore,

### 2.1. Induced power

The power induced on the air is the change in its kinetic energy per unit of time:

where
*figure of merit* (FM), and it is one of the forms to express the efficiency of a rotor in hover [4]:

Small propellers used in the UAV’s are currently capable of an FM up to 0.65 [4].

## 3. Blade element theory

This theory analyzes aerodynamic forces on a 2D section of a blade, in order to find the contribution of each section to the total thrust; for this, it is necessary to know the chord and pitch angle distributions along the span of the blade. The *Lift Coefficient*

**Figure 2** shows a blade of a propeller with an elementary section of blade at a distance *r* from the axis of the propeller, with width
**Figure 3** shows the section of blade.

The angular velocity of the propeller is

where
**Figure 3**.
**Figure 2**.

The blade pitch angle is

But assuming that

The incidence angle of the blade section is

For the blade section, the lift and drag forces are

where

From **Figure 3**,

where

If

With Eq. (21),

Or for many blades

where

For a thin section,

As in the momentum theory, it is necessary to use momentum conservation in the axial direction, that is, that the change in the moment of the fluid is equal to the thrust of the blade. For a section of blade in hover,

But from the momentum theory, Eq. (9):

But remembering that for hover

The solution method using the blade element theory requires detailed information about the geometry of the blade, that is, subdivide the blade in many sections of small width, and for each section, measure the pitch angle
**Figure 4** shows a flow diagram describing the iterative method of solution.

## 4. Computational method

This study was performed using the multiple reference frame (MRF) model in steady state. The conservative equations for a MRF for an incompressible flow are [6]

where

Two turbulence models have been widely used to model external flows over airfoils, wings, and rotors; these models are Spalart‐Allmaras and *k*‐*ω*. In wall‐bounded flows, the vorticity and the strain are of similar magnitude, but this is not true in a vortex core where the vorticity is high, but the strain is low. In these models, the turbulence production term is based on the velocity gradient; to include the effects of rotation, it is necessary to have some modifications to the classical models.

Many authors have studied the effect of the turbulence model used in the CFD simulations of rotors [7–10]. But though many models have been used, the Spalart‐Allmaras model is the most used model to simulate rotors; this is a one equation model that solves a modeled transport equation for the turbulent viscosity [11]

where

A drawback of the Spalart‐Allmaras model is that this model is insensitive to streamline curvature and system rotation [6]. A modification to the turbulence production term is available to sensitize this model to the effects of streamline curvature and system rotation. An empirical function is used as a multiplier of the production term to account for these effects; detailed information can be found in Ref. [12].

The simulations were performed using the CFD software ANSYS FLUENT v17.0. The turbulence model used in the model was the Spalart‐Allmaras model with curvature correction; the SIMPLE pressure‐velocity coupling solver was used, with second‐order upwind discretization for convective and diffusive terms. The simulations were performed increasing the rotational velocity with small discrete steps starting with 1 rpm up to a maximum velocity of 6547 rpm. For each increase in angular velocity, the model was run until the variation of residuals was very small; in this case, the convergence criteria were adjusted to

### 4.1. Geometry and computational domain

The actual rotor blade installed in the quadrotor model Araknos V2 of the Colombian company ADVECTOR is used for this study. A three‐dimensional (3D) scanning was performed to obtain a CAD of the propeller; see **Figure 5**.

The diameter of the propeller (DP) is 360 mm, the root chord is 20.77 mm, the tip chord is 4.33 mm, and the chord at 75% of span is 25.31 mm. The computational domain is a vertical cylinder, with 3.3 DP in diameter and 7 DP in length; the propeller is centered in the domain and at 2.7 DP from the upper surface. The generated mesh is an unstructured grid of tetrahedral elements but with a refined grid of pyramidal elements near the propeller surface.

The simulations were performed on the server clustergate of Universidad de los Andes, with Intel X86_64 processors with a velocity of 2.4 MHz using 32 cores and 64 GB of RAM memory. The simulation for the final grid took 168 h (**Figure 6**).

A grid convergence analysis was performed using different meshes with total number of elements varying from 0.8 to 16 million cells. Total torque was used as convergence criteria at 5221 rpm. It was found that for an increase of the number of elements from 9.6 to 16.5 million, the change in the force predicted was 0.2%; it was concluded that the mesh with 9.6 million cells was sufficient.

The boundary conditions for the upper and lateral surface of the cylinder are set as pressure inlet, while the lower surface is set as a pressure outlet and the propeller boundary is a non‐slip wall.

## 5. Experimental methodology

Flight test was performed with the quadcopter ARAKNOS V2, see **Figure 7**; the flying test was in hover at a height of 15 m from the ground, with an atmospheric pressure of 748 hPA and an average temperature of 24°C. Power for the motors was supplied only from one battery, whereas other batteries were used as ballast. These flights were monitored using the software Mission Planner, which provided real‐time information on battery voltage and control output to the engines. Flight data collected from the IMU, barometer, voltage and current sensor, and control output were recorded by the autopilot and downloaded for analysis. The data of rotational speed and electrical power recorded during hovering flight were analyzed.

The main features of the motor used in the quadcopter are as follows:

## 6. Results

### 6.1. Experimental measurements

The test was performed at an average velocity of 4430 rpm, the air density was

### 6.2. Momentum theory

As indicated in Ref. [4], the FM for the early propellers used in UAVs was in the range of 0.4–0.45; now, the currently used propellers in these vehicles develop FM up to 0.65. With this consideration, a conservative value of

From Eq. (14), it is possible to obtain the value of the induced power, taking the value of power measured in the flight test

The diameter of the disc is 0.36 m and the air density is

The value obtained with this model is close to the measured value. It was necessary to assume the FM for the propeller.

### 6.3. Blade element theory

To apply the blade element theory, it was necessary to split the blade of the propeller in sections with a

As explained in **Figure 4**, it is necessary to give an initial guess to

The initial guess of

A simple iterative code was written in Matlab to solve the thrust using the blade element theory, and the initial value of

The code needs the values of the radii, the chord, and the pitch angle for each section; also, it is necessary to indicate the rpm, in this case 4430 rpm; the density of the fluid; the width of the section; the number of blades; and the initial guess for

Adjusting the maximum residual of

Seventy‐one iterations were necessary to achieve the convergence of the method.

The thrust estimated using the blade element theory is close to the thrust measured in the experimental tests. Thought the iterative method was simple, it was easy to achieve convergence; this was possible thanks to the use of the momentum theory to estimate the initial value of

### 6.4. Computational results

**Figure 8** shows the wake developed by the propeller at 4500 rpm; the criterion to visualize the vortex is the *Q*‐criterion with a threshold of 200, and the structure of the wake is similar to that described in Ref. [1], where it is clear that the strong vortex is created at the tip of the blade, also the tip vortex and the inner vortex descend below the rotor following a helical path.

There are various methods used to visualize vortex; some of these methods are based on the tensor gradient of velocity as, for example, the *Q*‐criterion and

**Figure 9** shows streamlines colored by velocity magnitude and the vortex contours with a threshold for *Q*‐criterion of 50,000; the model shows the increase of the magnitude of the velocity down the propeller, also the streamlines swirl around the axis of the propeller.

**Figure 10** shows the contours of pressure for the upper and lower surfaces of the propeller. As expected, the pressure is lower at the upper surface of the propeller.

Finally, the thrust predicted by the model is

The model is able to predict the wake, pressure, velocity fields, and thrust of the propeller at hover and for different operational rotational velocities using the MRF method in steady state. The model overpredicts the thrust; this difference could be due to the interaction of the four rotors between them and also with the fuselage of the aircraft in the experimental measurements.

## 7. Conclusions

The momentum and blade element theories are useful tools to achieve a preliminary estimation on the performance of a propeller in hover, though it is important to understand the strong simplifications that each of these theories imply. For the momentum theory, simplifications are strong, for example, the flow is ideal, the simplification of the disc implies that the propeller has an infinite number of blades, also the induced velocity is uniform along the span of the propeller, and for the solution it was necessary to estimate the efficiency of the propeller in hover; this could lead to unrealistic results. For the blade element theory, the first simplification is that the flow is studied as a 2D flow, and effects of a 3D flow are ignored, for example, the induced velocities by the tip vortex, also linear aerofoil properties are used for the sections of the blade, that is, to estimate

A CFD model was implemented to study the flow around a propeller in hover. The model is able to predict the wake, pressure, velocity fields, and thrust using the MRF method in steady state. The predicted wake is in agreement with the literature.

The theoretical approximations for thrust are closer to the measurement of the experimental test, while the computational method overestimates the thrust. **Table 1** shows that the maximum percentage of error is 18.7% for the CFD simulation.

Method | Experimental | Momentum theory | Blade theory | Computational simulation |
---|---|---|---|---|

Thrust [N] | 5.89 | 5.58 | 5.07 | 6.99 |

Error | 5.3% | 13.9% | 18.7% |

For a better validation of the models analyzed in this chapter, a test bench to a more accurate measurement of thrust is necessary.

The control of a quadcopter is based on the thrust and torque of the propulsion system; therefore, an adequate estimation of the aerodynamic parameters is very important for the control of the vehicle. The methods described have some characteristics that are important to apply in some of the stages of the design, for example, the moment theory is important for a first estimation of the size of the rotor; **Table 2** shows the advantages and disadvantages of each method.