2.1. Governing Equations and Solution Details
All simulations for this study solved Reynolds-averaged Navier–Stokes solutions. The flow equations were solved using the finite volume method, and the sliding mesh technique was applied for the interface between rotational and stationary domains. This led the computational domain to update the deformed mesh but change in time. Hence, the moving mesh integrated over a control volume was coupled with continuity and momentum equations, which can be written as follows:
where,
is the fluid density,
is the fluid velocity vector which is based on the Cartesian coordinates,
is the moving grid velocity,
is the molecular momentum transport tensor,
indicates the pressure gradients term,
is the mesh cell control volume,
n is the normal vector to the control volume surface, and S is the control volume surface area.
The
transition model of Menter et al., 2006 [
19], coupled the SST model with transport equations for the intermittency and
(i.e., momentum-thickness Reynolds number). The major improvement of this transition model was that it did not rely on nonlocal parameters, hence it was more suitable for modern CFD codes and complicated transitional-flow simulations. Further, a special modification to the intermittency was included to allow for separation-induced transition prediction. The SST
has been used widely at both low [
20] and high Reynolds numbers in aerodynamic applications. This paper adopted the
transition model, which is proportional to the maximum strain-rate Reynolds number, i.e., presents the advantage of being a local property. The vorticity Reynolds number (
) is defined as:
The
and
in Equation (3), are the vorticity and the wall normal distance, respectively, and the maximum value of
is dependent on the
. The momentum-thickness Reynolds number transport equation was used to capture the nonlocal effect of freestream-turbulence intensity and pressure gradient at the boundary-layer edge, which indicates where transition onset occurs, and is defined as:
The transport equation for intermittency is used to trigger the transition process (
) and is defined as:
However, when the boundary separates, the modification of intermittency to unity is defined as:
In the above equations, (5),
, and
are two key functions, in which the former controls the transition extent and the later determines the onset of transition. The source term
is activated when the local strain-rate Reynolds number exceeds the local transition-onset criterion. The destruction source term
enables the relaminarization prediction when the transition-onset criterion is no longer satisfied, and vanishes in the fully turbulent regime. A complete description of the model is available in the article by Menter et al., 2006 [
19]. This model has been used by numbers of researchers for low Re transitional flows. For example, a detailed study on two specific parameters (
are used in the intermittency equation for controlling the length of transition region and onset location of transition, respectively) was shown by Suluksna [
21]. Benyahia [
22] conducted a validation study for the model for low Re number flows. According to the comparison between the numerical and experimental data, it was shown that the
model accurately predicted the location and extent of the two-dimensional laminar separation bubble. Counsil [
23] also studied the two-dimensional airfoils using the transition model, showing that the transition model was accurate in the preturbulent regions. A comparative study of four airfoils at low Reynolds numbers using the different transition modeling methods, namely,
and
, was performed by Seyfert [
24].
A blended first- and second-order accurate scheme was implemented for the spatial discretization, which switched from the latter to the former in regions of steep spatial gradients, based on the boundedness principle of Barth and Jespersen [
25]. SIMPLEC algorithm [
26] with a staggered grid and the second order-accurate central difference method was enforced, and such pressure–velocity coupling used coordinated under-relaxations for the momentum and pressure corrections to improve the convergence which was inherently slow in the original SIMPLE method. An implicit second-order quadratic backward approximation with an iterative procedure was realized for temporal discretization. The nonlinear coefficients were updated within each inner loop while the outer loop advanced the solution in time.
The entire computational domain consisted of a rotational zone containing the propeller and a stationary zone containing the MAV wing and fuselage, as shown in
Figure 1. The multiple zones were connected with each other through non-conformal mesh [
27]. The rotating domain contained a cylinder-type boundary, and the central axis which was coincident with the rotational axis of the propeller. The height and radius of the rotational domain were determined based on the propeller diameter (D
ia), indicating 2.55D
ia and 7.6D
ia, respectively. The stationary domain, on the other hand, was set as a cubic block with a distance being roughly about 12.75D
ia, 20.5D
ia, and 12.75D
ia for upstream, downstream, and height, respectively; see
Figure 1e,f. Structured mesh for both rotational and stational domain was generated, shown in
Figure 1a,b. Mesh with high quality was considered and generated due to the importance of interpolation relationship between the interface surfaces, and a mapped mesh topology was proceeded between the interface boundaries. For the rotating domain, two O-topologies were created to cover the propeller and the center spinner segments. The O-grid included 10 cells normal to the propeller wall surface with a first cell distance of 2 × 10
−5 m. There were 30 grid points in propeller radial direction and 56 grid points in the circumferential direction. A cylindrical wake block was used to ensure a good resolution of the blade wakes and the tip vortices, which have a significant influence on the MAV domain. For the outer domain, a similar mesh topology as we showed in the validation case section was used. The total size of the mesh was about 8 × 10
6 nodes.
The standard characteristic boundary conditions were applied on the farfield boundaries. For the case of low Reynolds number flow, total pressure, incoming freestream velocity, freestream turbulent intensity (), and the turbulence length scale were imposed at the inlet, whereas pressure was prescribed at the outlet boundary. Furthermore, at the solid wall, the non-slip boundary condition was applied. The turbulent kinetic energy was set to zero, and the pressure on the wall had zero normal gradients. High performance computing (HPC) with a parallel-processing implementation over 48 partitions and efficient message-passing interface between the partitions was adopted.
2.2. Specifications of Validation Cases
A three-dimensional Zimmerman wing planform was selected for the validation purposes, and the geometry was investigated by Torres and Mueller experimentally [
9]. It has a zero camber and the aspect ratio (AR) is two, and the aerodynamic mean chord is 0.1725 m which gives a corresponding Reynolds number of 100,000. The model has a thickness-to-chord ratio of 1.96% and 5-to-1 elliptical leading- and trailing-edges. The aerodynamic center was assumed to be at the 25% point of the mean aerodynamic chord. To minimize the boundary condition effects, two different types of boundary settings were tested, and results were compared with the experiment data. Model 1 had an H-type mesh topology and the boundary conditions were set as velocity-inlet, pressure-outlet, and a non-slip wall boundary was applied on the wing. The farfield boundary condition was applied for the outer boundary. On the other hand, Model 2 was set as wing tunnel model boundaries. It has a velocity-inlet, pressure-outlet, and a wall boundary was applied for the wing and the outer domain faces, respectively (listed in
Table 1). Model 2 was set as the wing tunnel model which had a contraction ratio of 20.6 to 1 and a rectangular inlet contraction cone. The freestream turbulence intensity in the test section was measured to be less than 0.05%. The test section was 182 cm long with a square cross-sectional area of 61 cm × 61 cm [
9].
An O-grid mesh topology was applied around the wing planform to capture the detailed flow field near the boundary layer for Model 1 and Model 2 configurations. Therefore, the grid points were concentrated near the wing planform edge, as shown in
Figure 2d. For the wind tunnel setting case, a symmetric boundary condition was applied at the wing root plane, as shown in
Figure 2c. The no-slip wall boundary condition was enforced on the wing surface. To minimize the farfield boundary condition effects, Model 1 had the domain which is set at 25
upstream, 35
downstream, and the upper and lower boundaries were placed at 25
away from the airfoil leading-edge. However, Model 2 was set as wing tunnel model boundaries which had exactly same dimensions as the real wind tunnel.
The mesh sensitivity was studied for the three-dimensional cases for α = 4° and the results are shown in
Table 1. The baseline mesh (G1) had about 1.5 million mesh elements and model with a fine mesh (G2) had a doubled size of 3 million. Close to the wall, there were about 60 grid points within the boundary layer, and in the turbulent region, the y
+ value of the first cell distance was ensured to be in the order of O(1). The stretching ratio for the mesh was less than or equal to 1.2. Validation results are summarized in
Table 1. The aerodynamic results comparisons, in
Figure 2e,f, showed that Model 2 (wind tunnel settings) gave a better lift coefficient as compared with the experimental data. The results from Model 1 were close to the experimental data at low incidences but under-predicted at high incidences. The drag values showed a similar conclusion, showing potentially the stronger wall interference at higher incidences (Mueller [
9]). From the mesh sensitivity study, the larger mesh size showed a reasonably better comparison than the coarse mesh. Therefore, for the propeller-wing-fuselage model presented later in this paper, a similar mesh topology, as shown for the validation case, was chosen and applied on the wing-fuselage part. The mesh for the propeller was integrated with the wing-fuselage mesh. The general topology is that an O-grid mesh was used around the propeller, and an H-type mesh was on the outer zone inside the rest of the rotating domain (details are shown in
Figure 1).
2.3. Present Investigation Case
The model used in the current study was based on the flying wing MAV developed at the University of Sheffield [
8,
28]. This model is composed of a Zimmerman planform wing, a fuselage, a vertical stabilizer, and a propeller in a tractor configuration. The model is shown in
Figure 3c with a coordinate system with x in the chordwise direction, y normal, and z spanwise. The wing has a mean aerodynamic chord of 0.221 m and an aspect ratio of 2.12. The freestream speed is 8.4 m/s, and the corresponding Reynolds number is 1.3 × 10
5.
The airfoil used in this investigation was a simple cambered thin airfoil, as shown in
Figure 3b. It included both positive and negative cambers and the relevant parameters are listed in
Table 2. The positive camber was designed to have better aerodynamic performance whereas the reflex camber was designed to maintain stable level flight, resulting in decreased flight times. Not only the static stability but also the dynamic stabilities (for control handling) were important for the design. Similar to Torres and Mueller [
9], the lift center location was calculated from the normal force and pitching moment taken from the
/4 location of the wing. Although the location of lift center was close to
/4 point at low angles of attack, it shifted towards the trailing edge as the incidence increased. The reason behind this is from both the increasing trailing edge separation and the strengthening wing tip vortices. In this investigation, the mean aerodynamic chord,
, was used (
Figure 3a).
The wing had a constant thickness of 2 mm and the shape of the cross section was rectangular with no sharp leading- and trailing-edge, and the airfoil is shown in
Figure 3b. The camber was defined as maximum convex camber (h
1/
), maximum concave and reflex camber (h
2/
), maximum concave camber location (d
1/
), and the maximum reflex camber location (d
1/
); further details are shown in Ref. [
11].
Considering the MAV configuration, the fuselage was a substantial component to accommodate the payload and propulsion devices, such as battery, motor, and servos, etc.
Figure 3d shows the fuselage dimension for our model. It had a front height, h
f, of 0.059 m, rear height, h
t, of 0.021 m, and a total length h
L = 0.216 m (
Table 2). This design of the fuselage was dictated by the size and placement of the components. The fuselage layout affects the center of gravity margin and hence the static stability. For this purpose, the battery was designed to be movable to adjust the center of gravity. Another interesting point is how the fuselage affects the overall aerodynamics. Brion [
29] simulated the fuselage and wing separately and the relevant aerodynamic forces are shown and discussed. However, the authors did not mention anything about the interaction between the wing and the fuselage. Ramamurti’s [
30] numerical results showed MAV with fuselage reduced the lift-to-drag ratio dramatically and the drag for all configurations considered was nearly the same. The effects of the fuselage were also investigated in the present study.
The slow-flyer propeller was chosen for the current MAV study. The propeller was installed at a distance d
t = 0.068 m ahead of the wing planform. It had a diameter of 8 in, the pitch was 4 in, and the hub diameter was 0.014 m.
Figure 3e shows the propeller geometry and the blade azimuth angle. The propeller rotated in an anti-clockwise direction viewing from the front, and
Figure 3f shows the propeller geometric characteristics. The aerodynamic balance determined a horizontal component X and a vertical component Y of the total force acted on the MAV model (
Figure 3d). To obtain the overall lift and drag force on the model, horizontal and vertical components (i.e., X and Y components) were transferred into L and D components (i.e., based on the incoming freestream coordinates), as shown in Equation (7).
A time step of 1.5724 × 10
−5 s with 30 sub-iterations was applied for this study (is equivalent to 0.5 degrees per time step). To have reasonable numerical results, the y
+ value of the first grid point in the order of O(1) was required.
Figure 4 shows the numerical aerodynamic force coefficients versus the blade azimuth angle. It shows that periodic pulses were produced. This type of signature was found to be relatively independent of the advance ratio and appeared to be mainly associated with the local loading on the propeller itself.
A steady state flight condition is defined as one for which all motion variables remain constant with time relative to the body-fixed axis system XYZ. Mathematically speaking, steady state flight conditions are
, and
, and implies that MAV does not have any acceleration in any direction (i.e.,
), and that the roll, yaw, and pitch rates are zero. Therefore, a propeller with rotational speed of 555 rad/s satisfied the steady state flight condition, as shown in
Figure 5. Namely, the lift equaled to the weight and drag equaled to the propeller thrust.