2.2. Validation of Numerical Method
The coordinate system used in this study has its origin at the intersection of the fan rotation axis with the blade quarter-chord line. All moments were referenced to this origin. The coordinate system followed the right-hand rule.
Figure 3 illustrates the coordinate axes and the positive directions of the velocity, angle of attack, forces, and moments.
The nondimensional coefficients defined by Equations (1)–(8) were used in the subsequent analysis:
The Multiple Reference Frame (MRF) method was used to simulate the fan rotation in this study. The MRF method required two separate zones connected by frame interfaces inside the computational domain: the rotating zone containing the block near blades and the stationary zone containing the rest block, as shown in
Figure 4. The computational domain was a cuboid with lengths of 100
D, 50
D, and 100
D in the flow, normal, and span directions, respectively. Out of the ground effect simulations, the most downstream farfield boundary was set as the pressure outlet, with a gauge pressure of 0 Pa relative to the reference pressure
p∞; the other boundaries were set as velocity inlets, which were specified to have the same velocity as the freestream. All surfaces of the ducted fan were set as no-slip walls. Trial simulations showed that if the pressure inlet was replaced by a velocity inlet of 0.5 m/s at the corresponding farfield boundaries, the discrepancies between the calculated thrust and torque were within 1%, but the order of magnitude of the maximum residual reduced from 10
−3 to 10
−5, as shown in
Figure 5. Thus, a velocity of 0.5 m/s was specified on the velocity inlet boundaries when simulating the ducted fan hovering in still air. The air was assumed to be ideal gas with a constant dynamic viscosity
μ = 1.7894 × 10
−5. The CFD software STAR CCM+ was adopted to perform the numerical simulations. The Realizable
k-ε turbulence model with two-layer all
y+ wall treatment was used in all simulations. The other options remained the default settings of the software [
38].
The experiment conducted by Grunwald [
9] was chosen for CFD validation, because the ducted fan studied in this paper has no experimental data yet. The ducted fan used in Grunwald’s experiment, hereafter labeled as Grunwald’s model, consists of a three-blade fan with a diameter of 0.381 m, a hub with a diameter of 0.109 m, and a duct with a chord length of 0.261 m. The tip clearance is 0.53%
Rb. Since the tip clearance is very small, to resolve it, the number of meshes as well as the difficulty of mesh generation is increased. According to the method proposed by Bunker [
39], ignoring the tip clearance will overestimate the ducted fan performance, in terms of propulsion efficiency, by about 0.9%. This discrepancy is acceptable for the present study. Therefore, the tip clearance is ignored in the subsequent simulations. To perform the grid independence study, three sets of polyhedral meshes were generated: the coarse, medium, and fine mesh with 4.44 million, 8.86 million, and 16.6 million cells, respectively. The grid refinement was mainly performed on the ducted fan surfaces and the spherical region with a diameter of 5
D surrounding the ducted fan. The flowfield near walls was calculated by a wall function, and thus the first layer height of the grid was taken to make the
y+ value between 30 and 200 [
38]. The fan rotational speed was set to
ω = 8000 rpm in simulations of Grunwald’s model. Firstly, the case of hovering was simulated. The thrust and torque coefficients calculated with different grid densities are plotted in
Figure 6. The discrepancy between the torque coefficient calculated by the coarse mesh and that calculated by the medium mesh is 7.1%. However, the discrepancies between the coefficients calculated by the medium mesh and those calculated by the fine mesh are all within 1%. The comparison between the CFD results of the medium mesh and the experimental values is shown in
Table 2. It can be seen that all discrepancies are within 8%, indicating that the CFD results agree well with the experimental values. The calculation accuracy in this study is comparable to that of Qing et al. [
40]. In the results of Qing, the discrepancy of
CT, total is 7.06%, and the discrepancy of
CQ is −3.05%.
Then, the case of
J = 0.595,
α = 30° was simulated with the medium mesh, and the results are shown in
Table 3. It should be noted that the coefficients in
Table 3 were calculated by the method defined in Grunwald’s report [
9], which means that
CL,
CX, and
CM were nondimensionalized based on the dynamic pressure of the freestream. The discrepancies between the calculated and experimental values of all coefficients are within 10%, and this means that the medium mesh can achieve acceptable accuracy. The results of these two cases also show that the numerical model established in this paper can be used to analyze the aerodynamic performance of ducted fans under different freestream conditions.
To further verify the appropriateness of the MRF method, the results of hovering out of the ground effect were compared with those obtained via the sliding mesh method (SMM). While performing the sliding mesh simulation, the implicit unsteady solver was selected; the temporal discretization was set to second order; the time step was set to Δ
t = 2.083 × 10
−5 s, corresponding to a rotation of 1° per time step; and the maximum inner iterations within a time step was set to 20. The comparison between the aerodynamic coefficients obtained via the MRF method, the sliding mesh method, and the experiment is shown in
Table 4. As can be seen, the results of the sliding mesh method are closer to the experimental values. However, the sliding mesh method took about 10 times more CPU time than the MRF method in this case.
Figure 7 shows the limiting streamlines on the suction side of the blade and streamlines on the cross section very close to the blade tip. As can be seen in
Figure 7a, the streamline patterns obtained by these two methods are only slightly different in the area immediately adjacent to the blade tip. This indicates that, for the current simulation, the flow field obtained via the MRF method is similar to that obtained via the sliding mesh method. In general, the accuracy of the MRF method is sufficient to support the flow field analysis in this study.
To the best of our knowledge, there are currently no experiments on the ground effect of ducted fans available for numerical method validation in the public literature, because it is difficult to accurately recreate the geometric models used in the experiments. The MRF-based RANS solver was used to calculate the aerodynamic performances of ducted fans in the ground effect in Han’s study [
36]. In Han’s study, the discrepancies between the calculated coefficients and the experimental values are less than 10%, considering the 5% uncertainties in the measurements. This indicates that the MRF-based RANS solver can achieve acceptable accuracy when calculating the aerodynamic performance of ducted fans in the ground effect.
To perform a grid independence study for the ducted fan studied in this paper, three sets of meshes for simulations out of the ground effect were generated using control parameters similar to those used in the meshing of Grunwald’s model. The cell numbers were 4.71 million, 8.79 million, and 16.8 million and were labeled as coarse, medium, and fine mesh, respectively.
Figure 8 shows some surface and cross section meshes of the medium mesh. The same surface meshes of the ducted fan were used to generate the volume meshes for the ground effect simulations. Another three sets of meshes with the number of 3.74 million, 6.25 million, and 14.9 million were obtained. The first-layer height of the grid was taken to make the
y+ value at the three-quarter radius of the blade the same as that of Grunwald’s model. The fan rotational speed was set to
ω = 3500 rpm. At this rotational speed, the blade tip Mach number was 0.35 and the Reynolds number based on the chord length at the three-quarter radius and tip speed was
Re = 5.2 × 10
5. The two cases of hovering in and out of the ground effect were simulated. The boundary conditions used while performing simulations of the ducted fan hovering in the ground effect are shown in
Figure 9: the bottom surface was set as the non-slip wall; the four side surfaces were set as pressure outlets with a gauge pressure of 0 Pa relative to the reference pressure
p∞; the top surface was set as the velocity inlet, and a velocity of 0.5 m/s was imposed on it to reduce the order of magnitude of the residuals, as mentioned above. The results are shown in
Figure 10. The maximum discrepancy between the coefficients calculated using the coarse mesh and those calculated using the fine mesh was 5.3%, while the discrepancies between coefficients calculated using the medium mesh and those calculated using the fine mesh were all within 1%. Therefore, the medium mesh was used for subsequent simulations, and the other meshes used in this study were generated according to the medium one.