CFD aided investigation of a three-blade propeller in multirotor UAV applications

In the recent years a rapid increase of multirotor UAVs in the commercial market is observed resulting in a large number of motor/propeller concepts and thrust architectures. The limited availability of data for the aerodynamic performance of the motor/propeller system often leads to a non-optimal operation on multirotor UAVs design points. Since experimental investigations are both cost- and time-demanding, the accurate CFD modeling of UAV propellers is crucial and highly supportive in the early design phases of multirotor UAVs. In the current study, a CFD framework is employed for the performance investigation of a small-scale three-blade propeller on a lightweight micro quadrotor UAV, designed for indoor search and rescue operations. More specifically, two widely implemented methods for propeller modeling are examined, namely the Multiple Reference Frame (MRF) and the Sliding Mesh (SM). Several operating points are investigated, corresponding to different propeller rotating speeds (RPM) and Reynolds numbers. The accuracy of each method is evaluated by comparing the CFD results with those obtained from literature experimental data. Finally, the uncertainty of the computational methods is quantified through Richardson’s extrapolation method.


Introduction
In the recent years, the commercial use of unmanned aerial vehicles (UAV) has shown a rapid increase, due to several UAV components factors, such as the low-cost / high-performance and miniaturized electronic equipment, while are also ease of use.The current size of the UAV market is estimated at 15.4 billion USD and with a compound annual growth rate of about 14%, it is forecasted that will be almost doubled and will reach about 29.66 billion USD by 2028 [1].Both fixed-wing and multirotor UAV configurations can offer versatility in terms of size, platform design, and operational cost, with many choices for every different mission type.The multirotor configurations are mainly employed for short-endurance, as well as low range, both outdoors and in confined indoor spaces missions, thanks to their vertical take-off and landing capability and high maneuverability [2].Their typical operations cover a wide range, such as urban aerial deliveries, industrial facility inspections, and search-and-rescue operations.
In multirotor UAVs, the mechanism of lift generation is intertwined with the propulsion systems, namely the electrical motors and propellers.This renders their accurate numerical modeling of utmost importance, to optimize the UAV's flight performance and control characteristics [3].Currently, a great selection of different propellers and motors is available, and though it provides several alternatives for the designer, it also makes the final design selection more difficult.Usually, the experience of the designer plays a pivotal role in the propulsion system selection, supported by data from low-fidelity analyses conducted by the manufacturers.This fact further highlights the significance of accurate CFD modeling of propellers during the UAV design, in order to have accurate values for the propellers' performance.
In the present study, the three-blade propeller of a micro quadrotor UAV is numerically investigated at high rotational velocities, using CFD modeling.Two different modeling techniques are applied, the Multiple Rotating Frame (MRF) and the Sliding Mesh (SM), each of them providing a different level of fidelity and computational cost.The CFD results are compared against published measurements from the manufacturers' datasheets.Additionally, an orderly methodology is presented for the quantification of the errors associated with the computational mesh discretization, by performing both a Richardson extrapolation and a Grid Convergence Index (GCI) analysis.Lastly, different turbulence models are examined in the MRF cases, to assess their performance for computations on micro UAVs.

The MIDRES UAV platform
The investigations performed in this work are related to the MIDRES project [4], in which an innovative micro quadrotor UAV is designed with a primary focus on indoor search-and-rescue operations.The UAV features a compact design (Figure 1) and a Gross Take-Off Weight (GTOW) of 1.4kg.The inclusion of an RGBD camera introduces simultaneous localization and mapping (SLAM) capabilities, enabling the UAV to operate effectively in GPS-denied environments.The propulsion system includes four identical T-Motor 2203.5 1500KV electrical motors [5] equipped with 5.1-inch Gemfan three-blade propellers [6].The motor/propeller systems operate at rotational speeds between 12,000 and 36,000 RPM, and all analyses are performed for this speed range.The available manufacturer's data are based on a zero freestream velocity for the level horizontal flight.Additionally, the available from the literature experimental data, for various freestream conditions are very limited for such propeller sizes [5].Therefore, to have coherency with the experimental data, the CFD modeling is performed for zero freestream velocity, which corresponds to the hovering flight phase, only.

Propeller CFD modelling
When the computational cost must be kept low, the propeller is usually modeled as a momentum source disk, with a predefined pressure difference across this disk, and only the resulting thrust is calculated [7].However, this simplistic approach does not consider, neither the 3D rotational phenomena nor the unequal radial distribution of the flow quantities.To accurately reproduce the three-dimensional flow field created by the rotating propeller, two modeling methods can be employed: the Multiple Reference Frames (MRF) method and the Sliding Meshes (SM) method.For both methods, two cylindrical computational domains are required, a smaller one around the propeller on which the flow rotation due to the blades is imposed, and a larger one around the former, where the flow is considered stationary (Figure 2).For the outer larger domain, the imposed boundary conditions on the planar surfaces are pressure inlet and outlet, and on the curved surface, the zero gradient symmetry condition is imposed.
Between the two domains exists an interface, through which the information about the flow quantities is exchanged.One key difference between the MRF and the SM approaches, in terms of the computational domain setup, is the requirement of a conformal or non-conformal interface.For the MRF approach, both options can be employed, though the SM requires a strictly non-conformal interface.In this work, a non-conformal interface was employed in the CFD modeling computations, to allow the use of a single computational grid and to eliminate any possible discrepancy from the different grid generations.The employed computational grid is the result of the analysis presented in section 2.3, where the effects of grid refinement are quantified using the Richardson extrapolation and it is produced using the BETA ANSA pre-processing software (v21.0.1, Root, Switzerland).Though it is an unstructured grid, it features several structured layout regions (with hexahedral cells) with intermediate unstructured transitioning regions (with polyhedral cells).This hybrid hexahedral-polyhedral computational grid can provide high-quality cells, increased accuracy, and robustness, as well as high efficiency due to the reduced number of total required cells.Near the solid surfaces of the propeller, where the boundary layer develops, a structured zone of 20 cells in the normal to-the-wall direction exists.An appropriate first cell height and a growth ratio of 1.1 are selected, to ensure that the value is maintained everywhere below unity and that at least 5 computational cells lay within the viscous sublayer.Regarding the two modeling methods, the MRF, (also called as "the frozen rotor approach"), is considered as a low computationally cost method, since it only requires a steady modeling approach.More specifically, the rotational velocity components are imposed on the computational cells of the propeller domain, hence no actual rotation of the domain is necessary, and the requirement for unsteady analysis is eliminated.On the other hand, the SM approach involves the rotation of the propeller domain, with a sliding condition being applied to the domain interface.In this approach, the walls of the propeller are rotating at a predefined velocity, with respect to the global stationary reference frame of the larger outer domain, and the associated downward flow movement is realistically reproduced.However, the actual rotation of the propeller domain, imposes the additional requirement of transient modeling, with an appropriate timestep being selected, based on the rotational velocity.
Both the RANS and URANS analyses, of the MRF and SM methods respectively, are solved using a coupled pressure-based solver, a 2 nd order upwind discretization scheme for all the differential equations, and by employing the shear stress transport (SST) k-ω turbulence model for the turbulence modeling.The solution is considered converged when all RSM residuals have been reduced by at least three orders of magnitude and their values, together with the propeller thrust value, demonstrate a consistent behavior for a sufficient number of iterations.In this work the quality of the CFD results, when different turbulence models are adopted, has been also studied.This is important, due to the 3D characteristics of the flow along with the propeller's low Reynolds numbers.Towards that goal, four different turbulence models are investigated using the MRF method for the whole RPM range.More specifically, the commonly used one-equation Spalart-Allmaras model, the k-ω SST model that was also employed for the SM analyses also, the k-ω SST model coupled with the γ-Reθ transition prediction model, and the Baseline Reynolds Stress Model (RSM).The latter is an ω-based RSM model that adopts a blending function for modeling the transport of the turbulent dissipation which has been successfully used for the unsteady flow modeling of compressor blades [8]

Richardson extrapolation
The Richardson extrapolation [9] is a numerical technique that has attracted interest in the CFD community due to its potential to enhance the quantitative accuracy of numerical solutions and mitigate errors associated with numerical discretization.The method is based on the idea that by combining the results obtained from two simulations with different grid resolutions, the leading order error term in the assumed error expansion can be eliminated [10].The main advantage of the method lies in its independence of the discretized equations and the dimensions of the problem, as well as its ability to postprocess available computational results.In the Richardson extrapolation, a function is fitted to the parameter of interest (for example, the drag coefficient), with respect to the grid refinement (i.e. the representative cell length is used as the independent variable), and the desired parameter value can be calculated at an infinitely fine grid.The representative cell length of a grid is used instead of the total number of cells, since it also incorporates the local grid refinement effect.For three-dimensional problems this length calculated as the cube root of the average cell volume, (Equation 1), where is the total number of cells and is the volume of each cell.The equation can be reduced to if all cells are uniform cubes. (1) At the next step, the refinement ratio between the different grids is calculated (Equation 2).In this work, the recommendations of Celik et al. [11] are followed, with three significantly different grids that feature refinement ratios above 1.3.The subscript values in the following equations increase from the coarser grid to the finer grid. (2) If the parameter of interest is denoted , the extrapolated value at the infinitely fine grid can be calculated from Equation 3(Figure 4).Here the information obtained from the fine and medium grids is combined, also using the refinement ratio and the order of accuracy of the discretization schemes.Finally, the extrapolated relative error between the fine grid solution and the ideal solution is obtained through Equation 4.
(3) (4)  Having performed the Richardson extrapolation, it is worthwhile to also calculate the Grid Convergence Index (GCI), proposed by Roach [12,13], which is a popular alternative way to measure the relative error between two grid refinements.In Equation 5, represents a safety factor, which is equal to 3 if only two grids are used or 1.25 when more than two grids, with different refinements are examined. (5) Based on the methods presented above, the Richardson extrapolation is performed for the MRF case at 33,000 RPM.The results obtained are presented in Table 1, with a significant improvement being observed in the grid refinement between the Coarse and Medium grids.Contrarily, based on the relative error and GCI for the grid refinement between the Medium and Fine grids, we can observe that the result changes are relatively small.Moreover, the extrapolated error is also quite small, indicating that the Fine grid solution is very close to the ideal solution of an infinitely refined grid.

Results
The computational results from both the MRF and the SM analyses are presented in Figure 5.They are compared against the propeller's data provided by the manufacturer [5].For rotational speeds below 18,000 RPM, both the MRF and SM methods overpredict the generated thrust, up to 52 and 45gr respectively.In the range between 18,000 and 30,000 RPM, the SM method produces results in a very close agreement with the manufacturer's data, with a maximum deviation of 1.2%.However, the MRF method in the same RPM range underpredicts the thrust force by 4.3%.Finally, for rotational speed above 30,000 RPM, the MRF continues to underpredict the force value (though less, up to 1%) and the SM results in a 2.5% overprediction.Overall, the SM method provides more accurate results for a wider RPM range.It only slightly overpredicts the thrust force at the very low and very high rotational speeds.
Regarding the different turbulence models, the MRF results are shown in Figure 6.It is observed that all examined turbulence models overpredict the thrust force at 12,000 RPM by a significant percentage, though the absolute error value is only between 47 and 54gr.For the other RPM, the computational results when the k-ω SST γ-Reθ model is adopted, are the only ones where a higher thrust force than the data is computed.Both the simple k-ω SST and the Spalart-Allmaras turbulence models produce similar results, very close to the manufacturer's data, with the former performing better at the higher RPM values.The RSM model shows a similar trend as the previous two models, though the computed thrust force is significantly lower.Overall, the k-ω SST and the Spalart-Allmaras models provide the smallest average errors of 6.2% and 6.6% respectively, while the k-ω SST γ-Reθ and the RSM models feature average error values of 7.4% and 7.1% respectively.

Conclusions
In the present study, CFD modeling studies, on a three-blade propeller of a micro quadrotor UAV have been performed.The Multiple Reference Frame methods and the Sliding Mesh method were employed for the propeller modeling, and their results were compared against available from the propeller manufacturer data.Overall, the SM method provides accurate results for the examined rotational speed range, though the associated computational cost is significant.Additionally, four well-known turbulence models have been investigated using the MRF method, and their performance was assessed.It is concluded that the k-ω SST and the Spalart-Allmaras turbulence models are the ones in closest agreement with the propeller data.Finally, the quantification of the errors associated with the computational grid quality, in terms of grid density, was also examined, by using the Richardson extrapolation, as well as the Grid Convergence Index analysis.Concluding, for a small three-blade propeller operating at high rotational speeds, the use of the MRF method can provide very good results with a reasonable computational cost, and either the k-ω SST or the Spalart-Allmaras model can be adopted for the turbulence modeling and both can provide results with a satisfactory accuracy.

Figure 2 .
Figure 2. Computational domain and applied boundary conditions.

Figure 3 .
Figure 3.The computational grid and detail near the propeller surface.

Figure 5 .
Figure 5. Results of the MRF and SM analyses, compared to the manufacturer's data [5].

Figure 6 .
Figure 6.Relative error of different turbulence models, compared to the manufacturer's data [5].

Table 1 .
Results of the Richardson extrapolation for the thrust force.