Numerical-experimental evaluation and modelling of aerodynamic ground effect for small-scale tilted propellers at low Reynolds numbers

Aerospace Science and Technology 126 (2022) 107625 Contents lists available at ScienceDirect Aerospace Science and Technology www.elsevier.com/locate/aescte Review Numerical-experimental evaluation and modelling of aerodynamic ground effect for small-scale tilted propellers at low Reynolds numbers Ambar Garofano-Soldado a,∗, Pedro J. Sanchez-Cuevasa,b, Guillermo Herediaa, Anibal Olleroa aGRVC Robotics Lab Seville, University of Seville, Spain bSpace Robotics Research Group, SnT, University of Luxembourg, Luxembourg a r t i c l e i n f o a b s t r a c t Article history: Received 1 February 2022 Received in revised form 25 April 2022 Accepted 7 May 2022 Available online 11 May 2022 Communicated by Damiano Casalino Keywords: Aerodynamic Effect Aerial Robot UAV CFD Propeller In recent years, aerial manipulators with fully-actuated capabilities are gaining popularity for being used in aerial manipulation operations such as critical infrastructure inspection or aerial manipulation tasks. Those scenarios usually demand the aerial platform to operate in constrained and narrow scenarios. It is well known that in these situations, the interaction of the wake generated by the propellers with the environment can significantly alter and change the performance of the rotors. Most studies have addressed this problem by considering the ground effect in hover conditions or during the landing maneuver for co-planar multirotor. However, few works analyze the behaviour of tilted rotors, which are used in fully actuated multirotor configurations thanks to their omnidirectional motion capabilities. This paper presents a numerical-experimental evaluation of the aerodynamic ground effect for smallscale tilted propellers at low Reynolds numbers. This aerodynamic effect has been experimentally evaluated through an extensive testing campaign in a testbench designed for this purpose which has been complemented by a CFD-based study. CFD results have been validated through a mesh independence study and a CFD-experimental propeller performance comparison. A numerical model has been also proposed to capture the dependence of thrust with distance to the ground and angle of inclination between the propeller and ground planes. We demonstrate that the proximity to the ground of tilted rotors decreases the thrust increment due to the ground effect as the tilt angle (θ) increases. This means that Cheeseman’s classical theory is inapplicable, as it only considers the distance from the ground without reference to how the thrust increment changes with the tilt angle. This outcome enables future aerial robotic applications that strongly demand accurate aerodynamic effect models to operate close to obstacles and narrow environments. ©2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Unmanned Aerial Vehicles (UAVs) and Micro Air Vehicles (MAVs) are significantly increasing their application range [1]. Typically, these aerial robots have been used as flying sensors to perform perceptual tasks such as remote surveillance and reconnaissance [2,3], visual inspection of large infrastructures [4,5]or filming activities [6,7]. However, emerging applications such as package delivery [8], contact inspections [9,10], warehouse inventory [11] and, in general, aerial robotic manipulation operations [12]require these rotorcrafts to fly close to obstacles and objects or within narrow environments. Propellers operating near obstacles experience different aerodynamic effects. This constrains the *Corresponding author. E-mail address: agar[email protected] (A. Garofano-Soldado). free development of the rotor wake and disturbs the propeller performance [13]. The ground effect on helicopters has been studied since the 1930s [14]to characterise the hovering aerodynamic performance [15]. In 1937, Betz [14]proposed a theoretical model based on replacing the helicopter rotor by a sink and applying the method of images. Knight et al. [16] considered two cylindrical vortex sheets located at a distance of 2zof equal strength and opposite direction. In 1947, Zbrozek [17] analysed the ground effect experimentally. Later, Cheeseman and Bennett [18]found that the flow pattern of a rotor in the vicinity of the ground was better matched with a source. Thus, [18]used blade element theory and the method of images to model the increase in thrust experienced by a rotor operating close to the ground under different conditions. Cheeseman and Bennett’s model is typically used because it allows results to be obtained with high accuracy compared to experimental evidence for large propellers such as helicopters or MAVs. In addition, https://doi.org/10.1016/j.ast.2022.107625 1270-9638/©2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Nomenclature θTilt angle of the propeller relative to the ground . deg zGround distance from propeller hub................ m αAngle of attack of a blade......................... deg βBlade pitch angle................................... deg φInflow angle........................................ deg UResultant velocity of a blade section.............. m/s UTTangential component of velocity in a blade section............................................. m/s UPNormal component of velocity in a blade section m/s Re Reynolds number ρAir density...................................... kg/m 3 μAir dynamic viscosity........................... kg/ms VωRotational velocity................................. m/s Angular velocity.................................. rad/s R75% Propeller radius at 3/4 of the centre of rotation.... m c75% Propeller chord at 3/4 of the centre of rotation.... m Fx,Fy,FzSensor force components........................... N Tx,Ty,TzSensor torque components....................... Nm XA,YA,ZAAxes in the absolute reference frame............ m XB,YB,ZBAxes in the propeller reference frame............ m CTPropeller thrust coefficient CTxHorizontal component of the thrust coefficient CTzVertical component of the thrust coefficient CQPropeller torque coefficient RPropeller radius...................................... m DPropeller diameter................................... m ARotor disk area..................................... m 2 TPropeller thrust...................................... N QPropeller torque.................................... Nm TIGE Thrust In-Ground-Effect.............................. N TOGE Thrust Out-of-Ground-Effect......................... N vIGE Induced velocity in ground effect................. m/s vi∞Induced velocity without ground effect........... m/s δviInduced velocity on the rotor caused by the image rotor................................................ m/s VhHover induced velocity............................ m/s fc(θ) Target function a0,a1,b1Coefficients proposed for the ground effect model βjModel coefficients calculated with the gradient descent algorithm J(βj)Cost function hβ(zi,θi)Hypothesis function yiThrust ratio TIGE/TOGE sNumber of samples γLearning rate Acronyms CFD Computational Fluid Dynamics ESC Electronic Stability Control IGE In-Ground-Effect MAVs Micro Air Vehicles MRF Multiple Reference Frame OGE Out-of-Ground Effect RANS Reynolds Average Navier-Stokes SST Shear Stress Transport UAVs Unmanned Aerial Vehicles BET Blade Element Theory the formulation of this model is relatively simple and depends only on the radius of the rotor (R) and the distance between the propeller and the ground plane (z). Other operating conditions, such as forward flight, were considered [18–20]. Since the first experimental tests by Zbrozek [17], some ground effect models have been derived from experimental evidence. Hayden (1976) [21]proposed an empirical model to analyse the ground effect in hover, which overpredicts the Cheeseman and Bennett model. Then, Curtis [22]presented the aerodynamic performance of a helicopter rotor when positioned close to the ground for low advance ratios. Other authors have focused on examining the rotor wake when it is constrained by the presence of the ground plane [23–27]. In the last decade, particular attention has been devoted to exploring the aerodynamic interactions of multirotor [28–32] and small scale rotors [33,34]. There is a need for in-depth knowledge of the behaviour of UAVs when approaching certain surfaces. In this way, proximity effects can be counteracted by incorporating a controller adapted to absorb aerodynamic disturbances [20,35,36]. In addition, the feasibility of tilting the propellers of a multirotor to perform tasks requiring direct contact between the aerial platform and the environment has been investigated. In [37], a new design of a fully actuated hexarotor with tilting propellers was presented and the application of control techniques to aid platform stabilisation. Other work has been carried out along the same lines [38], [39]. The aerodynamic effects of tilted rotors are not considered in these control laws as there is no thorough understanding of how the forces on a rotor change with inclination and distance from the ground. For instance, unlike co-planar aerial manipulators that classically accomplish the manipulation operation while they are hovering (and necessarily horizontal), a fully actuated aerial robot can perform a hover-flight while tilting with respect to the horizontal plane. This presents a completely new casuistry. According to [28], the disturbance that affects a co-planar multirotor is generating a “safe” behaviour cause it is always generating stabilization pitch and roll torques. However, due to fully-actuated robots usually have tilted propellers and can maintain an inclined hover, the aerodynamic ground effect will disturb the 6DoF, including the yaw axis. This situation will significantly impact the stability of these aerial robots, their safety, and the definition of their control allocation matrix. Moreover, a fully-actuated aerial manipulator will be affected by this effect mainly when flying close to an obstacle and specifically, when accomplishing the manipulation operation. The manipulation operation is usually when the accuracy and precision needed are higher. This means that any disturbance should be considered to guarantee the success and safety conditions during the operation. Though extensive previous works have characterised the aerodynamic performance of large and small-scale rotors parallel to the ground, there is little work on tilted rotors with respect to the ground or co-planar rotors operating at inclined surfaces. However, recent advances in aerial manipulation present the fully-actuated multirotor as the optimal candidate for certain contact-based operations [40]. These multirotor have full pose (position and orientation) tracking capabilities. Also, could land or hover very close to the ground or a different obstacle, maintaining a desired angle with the ground. In 1960, Fradenburgh [41]was the first to study the variation of the thrust ratio of a helicopter and annular jet considering the aerodynamic ground effect with different angles and heights above ground. Xin [42]proposed an extension of the inflow model for rotors at different heights above an inclined ground in hovering conditions. Different ground inclinations were analysed. The results were compared with the free-vortex/panel model. They also found that an asymmetry in the rotor wake appeared with the ground angle. More recent contributions were reported in [43], 2 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 1. Real and geometric model of the T-Motor propeller - 13 ×4.4. [44]. Platzer et al. examined the flow field of a rotor in the neighbourhood of an inclined surface using particle image velocimetry (PIV) and unsteady numerical simulations. Then, Pasquali et al. [45] used experimental and numerical techniques to analyse the evolution of the rotor wake on an inclined and parallel ground. The bounded domain method and method of images were employed and compared. However, there are no detailed studies showing the thrust augmentation of a small-scale rotor with tilt angle and distance to a parallel surface. Accordingly, neither theoretical nor empirical thrust models taking into account inclination have been developed. Due to the continuous development of computational fluid dynamics (CFD) tools, various simulation methods have been considered in the past to evaluate the influence of the ground on a rotor [43,46], [47–49]. Computational Fluid Dynamics is a handy tool for reproducing fluid behaviour around a propeller. Although several simulation techniques exist [50], Reynolds Averaged Navier- Stokes (RANS) models are less expensive and are used to estimate time-averaged parameters. In recent years, the number of investigations on propeller performance analysis with RANS models has increased [46–49,51–58]both high and low Reynolds numbers. Their suitability with experimental sets has been proven in several works [52,55,58]. Based on previous work, RANS models are considered capable of solving the proposed problem and will be used throughout this analysis. In this regard, the Multiple Reference Frame (MRF) method will be used to solve the 3D propeller flows. MRF has been adopted to examine flows around propellers by other authors both in free [51–54] and constrained environment [56,59]. Nevertheless, the adequacy of the MRF method has not been proved for inclined propellers, where the flow patterns are different from the co-planar case. In general, this study describes the ground effect of tilted rotors using small-scale propellers operating at low Reynolds numbers. So far, no ground effect model has been presented under these conditions. Considering the growth in the field of UAVs, it is necessary to analyse and quantify the effects that may appear when certain tasks are performed. In summary, the contributions of this research are listed as follows: •Investigation of the influence of inclination of rotor (θ) and distance to the ground (z) to quantify the thrust increase. The findings have been benchmarked against models developed in the past, highlighting the need to include the angle θin the ground effect analysis. Indeed, the parameter θhas not been accounted for in small-scale propellers until the current study. •A comprehensive 3D stationary CFD simulation-based study with a wide range of Reynolds numbers ([0.46 −2.2] ·105) and propeller diameters from 9 to 18 inches. Moreover, rotor inclinations of up to about 40◦and various ground clearances have been assumed. •Evaluation of a CFD-based methodology for flow simulation around propellers in confined environments. In particular, the MRF method has been applied. Although its reliability has been demonstrated for rotors parallel to the ground (see [59]), the fluid behaviour is distinct if the rotor is tilted. MRF has been verified with data from the propeller manufacturer in free environments and then with the experimental setup under IGE and OGE conditions. A method with a low computational cost that allows obtaining adequate results in even less time than the time required for bench testing. •Proposal of first ground effect model with tilted rotors in small-scale propellers is presented to capture the dependence on both tilt angle (θ) and ground distance (z). It has been derived a CFD simulation data-driven model using the gradient descent algorithm. For this purpose, a modification to the classical Cheeseman and Bennett theory has been made, where the first harmonic of the inflow models has been considered. The applicability of the proposed model has also been assessed. The remainder of the paper is organized as follows. Section 2 details the geometry of the different propellers used throughout this study, the Reynolds number range, the experimental setup and the fundamentals of Computational Fluid Dynamics (CFD). The computational domain, the boundary conditions, and the turbulent model are introduced. Section 3shows the validation of the CFD methodology. On the one hand, a grid independence study is presented. On the other hand, the propeller performance achieved by the proposed CFD methodology is evaluated with the manufacturer’s data. Section 4is devoted to presenting the ground effect results obtained from both the experimental setup and numerical simulations. The feasibility of the CFD method is discussed. Then, the flow field visualization is illustrated to understand the change in rotor performance with tilt. In Section 5, a data-driven ground effect model with tilted rotors is proposed and validated for several propellers. Section 6discusses the applications and limitations of the proposed model. Finally, it draws the conclusions from the ground effect analysis and gives ideas for future work along the same lines. 2. Materials and methods 2.1. Propeller geometry The first propeller selected is a small-scale T-Motor propeller with a diameter of 13 inches and a pitch of 4.4 inches. Fig. 1(a) and Fig. 1(b) present both the carbon fibre propeller used in the experimental study and the 3D geometry of the propeller employed in the numerical analysis respectively. Fig. 1also shows a detail of the geometry of a blade section according to the blade element theory (BET). Here, the angle of attack is defined as α=β−φ, where β is the pitch angle of the cross-section and φis the inflow angle, defined as φ=tan−1(Up/UT). Uis the resultant velocity and depends on the normal (Up) and tangential (UT) component to the plane of rotation. In the current study, freestream velocity is not taken into account as a result the velocity Upwill depend only on the induced velocities through the disk rotor. Moreover, the evolu- 3 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 2. Geometry, chord and twist angle distribution of the T-Motor propeller -13 × 4.4 used in experimental-numerical analysis. tion of the chord and the pitch angle of the propeller is shown in Fig. 2. Two extra propellers have been used to validate the proposed model and evaluate its applicability. These propellers are APC 9x4.4” and one T-Motor propeller 18x6.1” (see Fig. 3), the 3D geometrical model, the dimensions and chord and twist angle distributions of these propellers are shown. 2.2. Reynolds number range This section presents the range of Reynolds numbers evaluated in this work. It is known that in order to model the flow behaviour around a propeller, the Reynolds number, based on the rotational speed and the chord at 3/4 of the radius location, plays an important role. The performance of a propeller will change with the Reynolds number, as shown in previous studies [60]. Also, the Reynolds number in small-scale propellers will be much lower than in large-scale propellers used in helicopters. The Reynolds number when freestream velocity is not taken into account can be defined as follows [61]: Re75% =ρVωc75% μ(1) where the air density is ρ= 1.225 kg/m3, the dynamic viscosity is μ =1.7894 ·10−5kg/ms, the rotational velocity is Vω=R75%, is the angular velocity, R75% and c75% is the radius and chord of the propeller located at 3/4 centre of rotation, respectively. In this study, propellers are operating at low Reynolds numbers since the maximum value is 2.2·105[62]. It is well known that low Reynolds numbers are related to bubble separation, which is caused by adverse pressure gradients. For low advance ratios, a large number of cross-sections of the blade are in a stall condition. However, Coriolis effects, based on the rotation and the radial component of the velocity, delay the separation of the boundary layer and consequently the stall [63]. Table 1defines the Reynolds number of each propeller considered in this study. 2.3. Experimental setup Fig. 4shows the customized test bench used during the experiments in GRVC’s facilities. It is made up of an aluminum structure of 1mx 1mx 1mwhere an Axia80-M20 force/torque sensor has been mounted. This sensor allows measuring the six components Table 1 Reynolds Number for different propellers considered in this study. Propeller Reynold number APC 9x4.4 (0.46,0.74)·105 T-Motor 13x4.4 (0.88,1.1)·105 T-Motor 18x6.1 (1.5,2.2)·105 of force and torque with a resolution of 1/10Nand 1/200Nm, respectively. The sensing range is 500N for the forces Fx, Fy, 900N for Fzand 20Nm for Tx, Tyand Tzand the sensor frequency is 2kHz. An Arduino Mega 2560 is used to control the motor. The motor PWM input is sent from the computer to achieve a speed of 6300 revolutions per minute (rpm). Rotation speed is measured with an external tachometer during the experiments. The operating range of the motors used in multirotor is between 50% −60% throttle. The selected speed is within this range and is a typical value for the motor and propeller chosen in this analysis. Last, two 360W parallel power supplies have been used to power the sensor and the motor to maintain constant voltage conditions in all the experiments. A total of 40 experiments has been carried out. Fig. 5shows the two parameters considered to evaluate the aerodynamic ground effect. Unlike classical ground effect analysis where only the ground distance was taken into account [28,30,32,56], this study also considers the relative angle between the ground and the propeller (θ). In each experiment, five different measurements have been taken in order to take the mean value. The standard deviation of the force was calculated for each distance (z) and angle (θ), reaching a maximum value of 0.087Nand a minimum value of 0.002N. These results show that the sensor is highly accurate. The data collection process takes 85 seconds for each test. At first, the motor is switched off for 15 seconds. Then, the propeller is rotating at a constant speed for 10 seconds. Finally, the motor speed is reduced to a minimum speed for 60 seconds to avoid an increase in motor temperature before a new test was done. All experimental data have been collected in the absolute reference frame SA, defined by the axes XA, YA, ZA. In addition, the body frame SB= XB, YB, ZBis defined, which can be obtained by rotating the absolute reference frame SA. 2.4. CFD settings Computational fluid dynamics methods are typically used to analyze propeller performance and understand vortex generation mechanisms. For that, the finite volume method is used where the Reynolds Averaged Navier-Stokes (RANS) equations are solved on an unstructured mesh. The incompressible flow simplification is performed. According to the literature [51–55,59,64], there are two main CFD techniques to simulate of rotating flows: the multiple reference frame (MRF) and the sliding mesh methods. MRF is a steadystate approach. It divides the domain into several zones: a stationary zone, where the equations will be applied in a stationary frame, and a zone assigned to a rotational velocity relative to the propeller axis. These zones lead to the appearance of an interface that allows information to be exchanged between the stationary zone and the rotational zone. In contrast to other authors [49], this analysis considers the classical multi-zone MRF method. On the contrary, the sliding mesh method considers the propeller’s motion at each time-step and provides a time-dependent solution. The computational domain is defined similarly to the MRF method. In many cases, the output of the MRF method is taken, in fact, as the initial condition for the sliding mesh method. While the appli- 4 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 3. Geometry, chord and twist angle distribution of the propellers used to validate the proposed ground effect model. Left) APC propeller -9  ×4.4 used to validate the proposed ground effect model. Right) T-Motor propeller -18  ×6.1. Fig. 4. Test bench designed at Robotic, Vision, and Control Group (GRVC) to study ground effect with single tilted rotors. The structure is moved to locate the propeller’s centre of rotation at different distances from the wall (z). The inclination of the motor (θ) is achieved by using inclined printed plastic parts. Fig. 5. Absolute reference frame XA, YA, ZAand reference frame on blade axes XB, YB, ZBof the propeller and variables tested in the ground effect investigation. cation of transient techniques usually produces better results, the computational effort is prohibitively high. This paper uses MFR due to its simplicity, fastness, and fair accuracy. This method allows for analyse of the fluid flow around a propeller when it is close to the ground without the presence of external disturbances. In the past, other authors have analyzed the small and large scale propeller performance with this method in both free and confined environments [48,49,51,53,56]. However, as a tilted propeller approaches the ground, the interaction of the rotor wake with the ground surface causes an asymmetric pattern that becomes more noticeable as the tilt increases [42]. As a consequence, it is necessary to ensure that this stationary approach (MRF) allows modelling the flow behaviour to obtain the smallscale propeller performance. Silva et al. show how the MRF method is able to predict the wake breakdown and ring vortex using unstructured meshes. However, under IGE conditions, it is a challenge to capture these vortical structures accurately [49]. Fig. 6shows the computational domain set up to simulate the ground effect with a tilted propeller. It has three different domains. An external stationary domain with a diameter of 20R, a cylindrical inner domain with a diameter of 5R, and a rotating domain to enclose the propeller. The inner domain allows a mesh transition in order to capture the fluid flow behaviour accurately. In this domain, a flow with a rotational speed of 6300RPM is induced. The rotational domain and the propeller are rotated at an angle θ with respect to the ground. The vertical distance from the centre of rotation of the propeller to the ground plane is defined as z. The dimensions of the inner and outer domains change for each z value. Therefore, a parametric analysis has been performed where 5 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 6. Computational domain specified to analyse the ground effect on tilted rotors. The propeller is placed at a height z = 1R with respect to the ground. The boundary condition adopted to evaluate the aerodynamic performance is given. the parameters θand zare changed. This will generate a specific geometry and mesh for each configuration. The commercial software ANSYS-Fluent has been used for the calculations. The pressure-based coupled algorithm is used to solve the momentum and the pressure-based continuity equations. The coupled algorithm significantly reduces the number of iterations required to converge. However, the computational time is increased compared with a segregated algorithm, where each variable is solved sequentially. Green-Gauss Node-Based Gradient is employed to discretise the diffusive and convective terms of the conservation equation. The second-order scheme is chosen for Turbulent Kinetic Energy (k), Specific Dissipation Rate (ω), and pressure interpolation. The turbulence model used to solve the problem is the shear stress transport (SST) k −ωturbulence model proposed by Menter [65]. It combines the k −model in the remote region from the wall and the k −ωmodel in the near-wall region using blending functions. This is defined with two equations, one for the Turbulent Kinetic Energy (k)and one for the Specific Dissipation Rate (ω). Due to the Reynolds number range (see section 2.2) in which turbulent flows are found, it is considered that the SST k-w model should be applied [52]. The SST k −ωturbulence model has been used by other authors to analyze the aerodynamic performance of small-scale propellers [53–55,57,58,66]. Moreover, the simulations results were compared with experimental data obtaining a satisfactory agreement [52,55,58]. The turbulent intensity is set as 0.1%. This value has been achieved by taking measurements in an empty wind tunnel for all operating conditions [60,67]. Logarithmic wall functions are used to solve the boundary layer. In recent work, this approach has been considered in the near-wall region [52]. Additionally, a full ground effect case with a 20 degree inclination has been simulated with the Realizable k-e turbulence model. This turbulence model was used by [48]to analyse the stability of an aerial platform close to the ground plane. The differences found between the two turbulence models are negligible. However, the computational time and the instability of the solution with the Realizable k −model are increasing. Fig. 6shows the boundary conditions. In the upper boundary, the inlet is defined as a pressure inlet where the total gauge pressure is set to 0Pa. The outlet is set in the lateral boundary of the cylindrical domain as a pressure outlet with gauge pressure 0Pa. The ground plane and the propeller are defined as non-slip walls. Between the rotational and stationary domains, interfaces are generated that allow the separation of the two domains. An absolute velocity formulation has been used in this analysis, i.e. the equations are solved in each sub-domain. Therefore, no transformation is needed at the interface. This numerical analysis is divided into two parts: first, the simulation results will be compared with the experimental results to validate the defined CFD model. Thus, the angles analysed by simulation will be: 0◦, 10◦, 20◦and 30◦and dimensionless heights z/R =0.6, 0.75, 1, 1.5, 1.5, 2, 2.5, 3, 3.5, 4, 5. Second, when it is verified that the CFD model accurately fits the experimental results, this model will be extended to obtain results for other tilt angles such as 5◦, 15◦, 25◦, 35◦and 40◦. For angles greater than 25◦, it has not been possible to obtain results for heights less than z = 0.75R as the cylindrical rotational domain intersects with the ground plane. A total of 87 simulations has been carried out to characterise the aerodynamics ground effect with tilted rotors. 3. CFD validation 3.1. Mesh evaluation A mesh independence study was conducted to ensure that the solution does not depend on the mesh size. This process allows defining a mesh with enough quality to achieve reliable and accurate results, minimise computational time and achieve good convergence of the solution. In this analysis, four different meshes have been created where the cell size located on the surface of the propeller, the cell size of the propeller leading edge and the sizes of interfaces have increased from mesh 1to mesh 4. Additionally, automatic mesh refinement is applied. Fig. 7shows the four defined grids in the rotational domain and Table 2shows the number of nodes and elements in each of the domains and interfaces generated. As can be seen, Mesh 1 and 2are coarser with 1.06 y 2.2 million elements respectively, while Mesh 3 and 4are finer with 3.3 and 4.8 million respectively. In all cases, unstructured meshes were made with a refinement in the leading edge of the propeller and in the wake region. This study was carried out for a 20 degree tilt angle and for a height z=1R. In order to evaluate the accuracy of each mesh, the results of the thrust and torque coefficient of the propeller were calculated according to: CT=T ρA2R2(2) 6 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 7. On the left-hand side, definition of four different meshes (Mesh 1: coarse, Mesh 2: medium, Mesh 3: fine, Mesh 4: very fine) of the rotational domain. On the right-hand side, mesh chosen for the complete computational domain. Table 2 Number of elements and nodes in the stationary, rotational domain and at the interface separating the two zones for meshes 1 (coarse), 2 (medium), 3 (fine) and 4 (very fine). Mesh densities Rotating domain Stationary domain Interfaces Total Mesh 1 Nodes 13.210 171.549 2.848 184.759 Elements 70.166 993.442 12.396 1.063.608 Mesh 2 Nodes 30.653 354.682 27.288 385.335 Elements 163.357 2.076.988 136.696 2.240.345 Mesh 3 Nodes 202.980 385.787 80.416 588.767 Elements 1.110.512 2.243.860 414.278 3.354.372 Mesh 4 Nodes 440.121 418.441 133.352 858.562 Elements 2.409.443 2.421.935 694.222 4.831.378 Fig. 8. Thrust and moment coefficient versus number of elements (in millions) for each of the meshes proposed in the mesh independence study. They show the convergence of the solution as the mesh is refined. CQ=Q ρA2R3(3) where ρis the air density, A is the rotor disk area (m2), is the angular speed (revolutions per minute), T is the thrust of the propeller (N), and Qis the torque of the propeller (Nm). As can be seen in Fig. 8, from 3.3 million elements onward, the results barely vary. Between mesh 1 and mesh 3, the thrust coefficient changes by 11.63% and torque coefficient by 18.18%, while between mesh Table 3 Element size defined in the propeller and interfaces of computational domain for mesh 3, which is the best choice after grid independence analysis. Zone Element size (m) Propeller 0.001 Interfaces 0.005 Edge blade 0.0003 3 and mesh 4 the variation of both coefficients is minimum, being 0.4% for CTand 1.7% for CQ. For this reason, mesh 3 is the best option as it allows obtaining accurate results with lower computational costs than what would be achieved with meshes close to 5 million elements. Table 3lists the size of the selected element of mesh 3 at the propeller, blade edge, and computational domain interfaces. The mesh of the selected computational domain is shown on the right side of Fig. 7. In all cases, the mesh quality parameters have been evaluated, obtaining orthogonal quality values above 0.15 [55]. To ensure convergence of the solution, the simulations iterate until the change in propeller thrust is less than 0.01% in the last 1000 iterations, where the residual errors found are smaller than 10−6. Meshes similar to those proposed in this work are created by [52]. The average wall y+ value in mesh 3 is about 31. This value is similar to the one used in [57,68]. Although the full advantages of the turbulence model are not being taken advantage of, greater stability and convergence of the solution is achieved than if other models were used. On the contrary, 7 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 9. Static performance of the T-Motor 13x4.4 propeller for different rotational speeds (rpm): comparison of thrust and torque coefficients (CTand CQ) between CFD results and manufacturer’s data. Propeller used in experimental-numerical analysis. the flow separation is not accurately predicted. The minimum element size related to the wall value is 1 ·10−4m. 3.2. Propeller performance comparison A total of 22 simulations has been carried out to validate our CFD results. These simulations have assessed the static performance of the three propellers considered in this study (see section 2.1) with seven rotational speeds. Each simulation was performed in a free environment. The static performance results have been compared with the data provided by the manufacturer T- Motor and APC [69,70]. Figs. 9and 10 show the variation of thrust and torque coefficient with rotational speed (rpm) resulting from both CFD simulations and manufacturer’s data. These coefficients have been obtained by applying Eq. (2) and Eq. (3). It is observed that when the rotational speed (and as a consequence the Reynolds number) increases, the thrust coefficient increases while the torque coefficient remains practically constant. Regarding the thrust coefficient, it is noted that the CFD results overestimate the data found in [69,70] for the T-Motor 18x6.1 and APC 9x4.4 propellers. In contrast, the CFD torque coefficient is underestimated in all cases. However, the CTand CQerrors of the three propellers examined are less than 5% and 10%, respectively. Under static conditions, the fidelity of a rotor’s performance is related to the ability to capture the vortex structures of the blade tip [51]. Based on the results presented in this section, it is established that the CFD model is adequate to capture the forces on the propeller in a free environment. Also, it is assumed that the tip flow structures can be predicted although not as rigorously as in higher-order schemes. 4. Ground effect for small-scale tilted propellers 4.1. Experimental evaluation of the CFD-based method This section compares experimental and simulation results according to the procedures described in section 2.3 and section 2.4 respectively. In both procedures, the co-planar configuration, where the rotor is parallel to the ground (θ= 0◦), and three tilt angles (θ= 10◦, 20◦and 30◦) have been analysed. Furthermore, ten distances measured from the centre of rotation of the propeller to the wall (z/R =0.6, 0.75, 1, 1.5, 2, 2.5, 3, 3.5, 4, 5) were assessed. Fig. 11 shows the horizontal and vertical components of the thrust coefficient (CTxand CTz) versus dimensionless ground distance (z/R)for each rotor tilt angle (θ). The experimental results are shown in dotted lines and the CFD results in dashed lines. In addition, the triangular marker indicates the vertical component of the force, and the circular marker the horizontal component. It can be seen that the thrust coefficient of the rotor increases as it approaches the ground in all cases. However, as the angle of inclination increases, the vertical component of the force reduces and the horizontal component becomes larger. The numerical results overpredict the trend of the thrust coefficient curve compared to the experimental curves for all angles. An offset is observed between the simulation and experimental results for each of the configurations analysed. This offset remains practically constant from height 1.5Ronwards. However, the discrepancies between simulated and experimental results are considered acceptable taking into account the errors arising from both methods of data collection. In Table 4 the root mean square error (RMSE) between the simulation and experimental results of CTz, CTxare shown. To investigate the ground effect, the thrust ratio TIGE/TOGE is typically used, where TIGE is the thrust In-Ground-Effect and TOGE is thrust Out-of-Ground-Effect. This ratio gives the percentage increase in thrust experienced by the rotor when the ground is close. Accordingly, in Fig. 12, the thrust ratio of the experimental and simulation data is plotted versus the dimensionless distance z/R. Nevertheless, Fig. 11 has been presented to give an estimate of how each of the force components varies. For the case of a rotor parallel to the ground (θ=0◦) the results are compared with Cheeseman and Bennett’s classical helicopter theory [18]: TIGE TOGE =1 1−R 4z2(4) The results show that the ground effect is negligible when z/R is approximately greater than 2.5. For angles 0◦, 10◦and 20◦, the closest distance to the wall is 0.6R. In these cases, the thrust is increased by 13.3%, 9.6% and 8.7%, respectively. For the 30◦angle, the minimum distance at which simulations have been performed is 0.75R, since the rotational domain interfered with the ground plane for smaller distances. For this height, the thrust increases experienced by the rotor for angles 0◦, 10◦, 20◦and 30◦would be 10.4%, 7.8%, 6.1% and 5%, respectively. The CFD results fit quite accurately to those obtained by experimentation, reaching a maximum error of 5.1% for a height of 0.6Rand a tilt of 10◦. Regarding the observed discrepancies, they can be attributed to both numerical and experimental errors. On the experimental side, there are several sources of errors that could generate this kind of discrepancy. For instance, although the experiments were carried out thoroughly, the positioning parallel to the wall of the structure in Fig. 4is done manually. This could introduce errors if at some distance z it is not completely parallel. The ESC used to control the rpms of the rotor is a COTS ESC. In most cases, those control and regulate the speed of the electric motor by closing the control loop at the current measurement level. This means that the value of rpm could be not exactly the same as the one we have imposed in the simulations. The power supply used during the experiment is also a low-price system that could be sensitive to some environmental changes like the temperature [51]. Moreover, considering that the propeller is placed close to the plastic parts and sensor, this could create an extra interference with the flow field. However, it has been verified that there is no interference of the test bench on the propeller. For this purpose, after 8 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 Fig. 10. Static performance of the T-Motor 18x6.1 and APC 9x4.4 propellers for different rotational speeds (rpm): comparison of thrust and torque coefficients (CTand CQ) between CFD results and manufacturer’s data. Propellers used to validate the proposed ground effect model. Fig. 11. Vertical and horizontal component of the thrust coefficient versus z/R dimensionless distance for angles 0◦, 10◦, 20◦and 30◦derived by experiment and CFD simulations. Table 4 RMSE of the proposed CFD methodology for vertical and horizontal component of the thrust coefficient considering angles 0◦, 10◦, 20◦and 30◦. Angle Coefficients 0.6 0.75 1 1.5 2 2.5 3 3.5 4 5 0◦CTx7.27e-06 7.25e-06 6.63e-06 4.37e-06 2.90e-06 3.30e-06 2.31e-06 1.27e-06 1.70e-06 1.40e-06 CTz8.56e-05 9.49e-05 2.25e-05 1.58e-05 4.04e-05 3.84e-05 4.68e-05 3.92e-05 4.83e-05 3.60e-05 10◦CTx1.30e-05 2.82e-05 4.25e-05 4.76e-05 4.35e-05 4.49e-05 4.53e-05 4.51e-05 4.52e-05 4.45e-05 CTz1.3e-04 8.03e-05 5.18e-06 3.68e-05 2.29e-05 2.54e-05 3.27e-05 3.20e-05 2.46e-05 2.82e-05 20◦CTx6.13e-05 6.60e-05 7.86e-05 8.22e-05 8.05e-05 8.20e-05 8.29e-05 8.81e-05 8.63e-05 8.57e-05 CTz4.17e-05 7.25e-05 9.60e-05 1.07e-4 9.68e-05 9.73e-05 1.02e-04 1.14e-04 1.11e-04 1.10e-04 30◦CTx- 8.52e-05 9.82e-05 9.91e-05 9.49e-05 1e-04 1.02e-04 1e-04 1e-04 1.02e-04 CTz- 2.95e-07 2.94e-05 3.23e-05 3.82e-05 3.88e-05 3.81e-05 4.15e-05 3.38e-05 3.30e-05 9 A. Garofano-Soldado, P.J. Sanchez-Cuevas, G. Heredia et al. Aerospace Science and Technology 126 (2022) 107625 ratio decreases with respect to the classical theory as the angle increases. The developed model fits the results obtained from the numerical simulations with good accuracy. In future works, the proposed model for tilted rotors will be implemented in control laws in order to compensate for this effect when UAVs operate close to the ground. This will result in safer flights by avoiding instabilities in the aerial platform. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Ambar Garofano-Soldado reports financial support was provided by University of Seville. Ambar Garofano-Soldado reports a relationship with University of Seville that includes: employment. Acknowledgements This work has been supported by the H2020 RESIST (H2020- MG-2017-769066) and PILOTING (H2020-ICT-2019-2-871542) projects, funded by the European Commission and the ARTIC Project (RTI2018-102224-B-I00), funded by the Spanish Agencia Estatal de Investigación. 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