We begin by looking at the wake of stationary panels (Cases 1–5) and that of the flow around pitching panels (Cases 6–7). Numerous experimental studies had provided insight into the characteristics concerning circulation and flow separation, which provides a standard dataset for validation. Herein, the spatial coordinates and length scales are normalized using the panel chord length (c), such that ${x}^{+}=x/c$, ${y}^{+}=y/c$, and ${z}^{+}=z/c$.

#### 3.1. Stationary Panel

In order to have an optimum size for the grid and to sufficiently resolve the flow characteristics, grid independence analysis was also completed for the angle of attack (

$\alpha $) of

${30}^{\circ}$. Four different grids were selected and the ratio of minimum grid sizes in the streamwise (

$x-$) direction was kept constant, i.e.,

$\delta {x}_{1}/\delta {x}_{2}\approx $ 1.5 (

$\delta {x}_{2}$ is the minimum grid size for a fine mesh, while

$\delta {x}_{1}$ is the minimum grid size for the coarse mesh). The thickness of the panel was not altered for any cases, in which all panels constitute a zero-thickness plate. The four grids correspond to Grid-1, Grid-2, Grid-3 and Grid-4. The number of grid elements and mesh spacing parameters are defined in

Table 2, where relative error shows the discrepancy with respect to [

23].

$\Delta \overline{{C}_{d}}$ is calculated with respect to

$\overline{{C}_{d}}$ for a coarse grid, and for its subsequent finer grid.

The mean drag coefficient (

$\overline{{C}_{d}}$) from Grid-4 compares well with that obtained by Taira et al. [

23]. The relative error was less than 2%. The relative difference between the mean drag for two subsequent studies is shown in

Table 2. The difference is less than 1% for Grid-3 and Grid-4. Spatial convergence is also obtained with respect to the mean drag, when comparing results in

Table 2.

Therefore, the domain and overset mesh parameters of Grid-4 are sufficient for accurately simulating the wake. The results described here use the mesh from Grid-4.

Figure 5 presents the mean coefficient of drag for stationary panel at different angles of attack. A good agreement was observed in

$\overline{{C}_{d}}$ for all angles of attack, compared to Taira et al. [

23]. The maximum deviation was for the panel at

$\alpha ={60}^{\circ}$, although the value was within 6.5% error margin.

The accuracy of different overset interpolation schemes is studied using a two-dimensional inclined stationary panel (Case 3). Four main schemes are tested: Inverse-Distance, CellVolumeWeight, LeastSquares and trackingInverse-Distance. These were selected based on their total simulation time for completion and prediction accuracy. For simplicity, we compared the coefficients of drag (

${C}_{D}$) estimated for a two-dimensional stationary panel at

$\alpha $ = 30

${}^{\circ}$ (Case 3). The details for CPU time (s) required for each scheme is shown in

Table 3, while the variation of

${C}_{d}$ with respect to non-dimensional time

${t}^{*}=t{U}_{\infty}/c$ is depicted in

Figure 6.

The Inverse-Distance scheme incorporated the least amount of CPU time as observed in

Table 3. Particularly, the CellVolumeWeight scheme, although conservative, required 72% more CPU time compared to Inverse-Distance. The CPU time required for LeastSquares and TrackingInverseDistance schemes was also longer by 26% and 10%, respectively. Despite the higher order of accuracy offered by LeastSquares, we suspected that the time requirement is even higher for three-dimensional cases with more refined grids. The estimated

${C}_{d}$ using different interpolation schemes was also within 0.01% (

Figure 6), which suggested that using the Inverse-Distance scheme is appropriate for all the three-dimensional cases.

We also observed a good agreement between the current results for

$\overline{{C}_{d}}$ (

Figure 5) and the data obtained by Senturk et al. [

11] who used the IBM technique. This indicated that OGA implementation yields a similar accuracy in terms of modeling wake characteristics compared to other techniques. Aarnes et al. [

22] further showed that for the study of two dimensional flow over a circular cylinder using IBM implementation, the domain requires an approximately 18.1 times more grid points than that used in the Overset method to achieve a similar accuracy [

22]. The cost and time required for computation would therefore increase, as was observed in another study by Tay et al. [

36] for flapping airfoil simulations. They showed that using the Overset grid conforming method [

36], a reduction in the total number of cores (or processors) was possible, along with a reduction in computational time compared to IBM implementation [

36].

Wake visualizations further provide insight into the vortex features that may affect the drag characteristics of inclined stationary panels.

Figure 7 presents contours of the spanwise component of vorticity

$\left({\omega}_{z}^{*}\right)$ for the panel at

${30}^{\circ}$ (Case 3). Snapshots at

${t}^{*}=1,5$ and 10 are presented. At

${t}^{*}=1$, the formation of trailing edge and leading edge vortices are clearly identifiable. The viscosity and shear effects are high at

$Re=$ 300. This leads to a smeared distribution at later times near the surface of the panel. This characteristic was also well captured by Taira et al. [

23] for flow over a panel at the same

$Re$. The plots at

${t}^{*}=5$ and 10 are similar, and the latter corresponds to flow after reaching statistical convergence.

The generation and interaction of vortices are shown using iso-surfaces of vorticity magnitude in

Figure 8. The results correspond to Cases 3, 4 and 5, where the stationary panel was fixed at

$\alpha ={30}^{\circ}$,

${45}^{\circ}$ and

${60}^{\circ}$, respectively. The iso-surface of vorticity magnitude enables detection of a vortex sheet formed and separated from the panel surface [

23]. The results were well captured at

${t}^{*}=$ 1, 5 and 10, respectively.

The iso-surfaces in

Figure 8a (

$\alpha ={30}^{\circ}$) indicates that a leading edge vortex (LEV1) is first formed at

${t}^{*}=1$. A similar vortex structure was also noted by Taira et al. [

23]. Over time, the transient effects are overcome by viscosity and the entire visible span of the panel is dominated by a pair of tip vortices (TEV1). The tip vortices were counter rotating and appear to be very similar in structural configuration. The downward velocity induced by tip vortices retains the leading edge vortex sheet from detachment.

Figure 8b depicts the formation of leading and trailing edge vortices, LEV2 and TEV2 respectively, for the panel at

$\alpha ={45}^{\circ}$. The TEV2 structure observed at

${t}^{*}=1$ is quite similar in configuration to TEV1 in Case 3. The difference is observed at

${t}^{*}=5$, wherein TEV2 expands to form a horseshoe vortex, retaining its coherence close to the panel. At

${t}^{*}=10$, the horseshoe vortex TEV2 breaks down to form two counter rotating tip vortices. The results for the panel at

$\alpha ={60}^{\circ}$ are also presented in

Figure 8c. At

${t}^{*}=1$, the generation of a leading edge vortex LEV3 was similar to LEV2 of Case 4 (

Figure 8b), although a difference in the wake features was observed with respect to the trailing edge vortices. The horseshoe vortex TEV3 at

${t}^{*}=1$ appear to be forming closer to the panel surface compared to TEV2 and TEV1, which emerged at

$\alpha ={45}^{\circ}$ and

${30}^{\circ}$ respectively, thereby leading to higher drag. With further advection of vorticity downstream, the newly formed TEV4 entrains the TEV3 at

${t}^{*}=$ 10, leading to an interconnected structure.

This interconnection is also attributed to the high shear effects experienced at low

$Re$, due to which the vortices counteract breakage and separation, in localized regions (such as the trailing edge tips of the panel [

7]).

#### 3.2. Pitching Panels (Cases 6–7)

Three dimensional simulations of pitching panels were carried out using the OGA method, where the body has an oscillatory pitching motion. The centre of oscillation was 0.1

c from the leading edge of the panel. Two cases corresponding to

$Re$ of 500 (Case 6), and 2000 (Case 7), were selected. Since higher

$Re$ are generally associated with an unstable shear layer, and therefore greater turbulence effects near the panel surface, we decided to proceed with a moderate

$Re=$ 2000. Senturk et al. [

11] determined this

$Re$ to inhibit turbulence effects.

The net propulsive or Froude efficiency of the panel is calculated using

$\eta =\overline{{C}_{T}}/\overline{{C}_{P}}$, where

$\overline{{C}_{P}}$ represents the coefficient of power. Since

$\eta $ depends on the force characteristics of the pitching panel, we first present the pressure and shear stress distribution on the top surface of the square panel, in

Figure 9a,b and

Figure 10, for

$Re$ = 500 and 2000, respectively. Four instances were selected based on the phase angle of the panel

$\left(\varphi \right)$ in its pitching cycle. The phases correspond to

${0}^{\circ}$,

${90}^{\circ}$,

${180}^{\circ}$, and

${270}^{\circ}$, respectively. The panel positions at these phases are also shown for clarity. The pressure coefficient,

${C}_{p}$, and shear stress coefficient,

${C}_{f}$, were calculated based on:

Here,

p is the pressure on the surface of panel,

${p}_{ref}$ is the reference pressure and

${\tau}_{w}$ is the wall shear stress on the surface of the panel.

${C}_{p}$ distribution for

$\varphi ={0}^{\circ}$, indicate a higher pressure at the top surface, except for the extremities of panel where the shear stress distribution was maximum. This trend compares well with the observation of Senturk et al. [

12], who performed a similar parametric study at

$Re$ = 500. The regions of high pressure gradients were concentrated along the panel extremities, which is responsible for the thrust generation [

12,

37]. A difference in magnitudes of

${C}_{f}$ is noted in the cases corresponding to

$Re$ = 500 and 2000, respectively. Due to reduced viscous effects, the contribution of pressure to drag increases with increasing Re.

Figure 9 also shows a chordwise variation of

${C}_{p}$ and

${C}_{f}$ at midspan of the panel (Case 6). The leading and trailing edges had higher shear stress coefficient compared to mid chord regions, while pressure coefficient tend to drop at the leading and trailing edges. This is also consistent with the findings of Senturk et al. [

12]. Differences in drag was also observed between different phases of the pitching cycle. A higher pressure is generated as the panel starts pitching upwards, thus confirming that a vortex core in the form of a leading edge vortex is forming at the top surface of the panel. As the panel reaches its peak amplitude (i.e.,

$\varphi ={90}^{\circ}$), the vortex seperates from the panel leading edge and a low pressure region is formed on the surface. At

$\varphi ={180}^{\circ}$, the pressure and shear stress are low, except at the leading edge where shear stress gradient is still high. As the panel reaches

$\varphi ={270}^{\circ}$ and moves upwards, the pressure increases similar to the start of the pitching cycle. The higher magnitudes of

${C}_{p}$ indicate that drag production (or in other cases, thrust production) is dominated by pressure effects.

The low versus moderate Reynolds number effects on the production of drag at low

$St$ is further analyzed by looking at the instantaneous

${C}_{d}$ and

${C}_{l}$ in

Figure 11a,b. The normalization factor for time was assumed to be the oscillation period (

${T}_{o}$) for the pitching cycle (i.e.,

${t}^{*}=T/{T}_{o})$. The time-averaged drag coefficient for Case 7 (

$Re=2000$) was ≈70% smaller than Case 6 at

$Re=500$. The less significant viscous effects at high

$Re$ tend to decrease drag, and generally promote thrust generation [

12]. The higher magnitudes of

${C}_{f}$ observed along the panel edges (

Figure 9b) for Case 6 as compared to Case 7 (

Figure 10b), verified the increased viscous contribution to drag at lower

$Re$. Hemmati et al. [

32] also explained the effects of surface pressure variation on the nature of thrust and side (lift) forces produced by pitching panels with different trailing edge shapes. For cases presented in our study, we investigated the differences in surface pressure distribution for Case 6 and Case 7, at peak trailing edge positions of oscillation cycle. These correspond to

$\varphi ={90}^{\circ}$ and

${270}^{\circ}$ respectively. These specific phase positions are also depicted in

Figure 11 as Instant “1” and Instant “2” for Case 6, and Instant “3” and Instant “4” for Case 7. At each of these instants, the distribution of pressure coefficient (

${C}_{p}$) on top and bottom panel surfaces are depicted in

Figure 12, separately. This further allowed evaluation of any potential similarities or differences in surface pressure distribution at the two

$Re$s, and its corresponding effects on the nature of drag and lift force at specific phases of the pitching cycle.

The variation of drag produced by the panel does not appear to change as the panel reaches

$\varphi ={90}^{\circ}$ and

${270}^{\circ}$. However, the extrema of lift coefficient switched its sign from representing maxima, to a minima, ahead of instant “1” and instant “2” in

Figure 11a,b, respectively. This trend was similar at the same

$\varphi $ for both Case 6 and Case 7, although the magnitudes of maxima and minima increased at higher

$Re$. The change in the sign of the lift force extrema can be further linked to the surface pressure distributions at respective instances shown in

Figure 12. This depends on the development and detachment of leading and trailing edge vortex structures [

32]. At Instant “1” (

$\varphi ={90}^{\circ}$) for Case 6, we observe that a low pressure region LPA1 exist on the top surface of the panel due to the detachment of a trailing edge vortex structure when the panel reaches its maximum pitch amplitude. The bottom surface however corresponds to a high pressure region HPA1, where a new trailing edge vortex is still under development. The existence of such large streamwise pressure gradient across the top and bottom surface of panel leads to the local extrema (maxima) in

${C}_{l}$ ahead of Instant “1” in

Figure 11a. The corresponding detachment of the trailing edge vortex structures leads to transfer of momentum that was gained from downstream advection of trailing edge vortex structures to the pitching panel. Thus, there exists a minima for

${C}_{d}$ ahead of Instant “1”. At Instant “2” (

$\varphi ={270}^{\circ}$), the region of higher

${C}_{p}$ gets switched from bottom to the top surface of the panel due to the change in the direction of pitching. However, the streamwise pressure gradients across the surface are quite comparable to Instant “1”. This again led to a local extrema (minima) in both

${C}_{d}$ and

${C}_{l}$, after the detachment of the trailing edge vortex structures.

Case 7 had some noticeable differences from Case 6 in terms of surface pressure distribution (see

Figure 12b). At Instant “3” (

$\varphi ={90}^{\circ}$), we observed an increase in size of the low pressure area LPA3 on the top surface compared to LPA1 in

Figure 12a for Case 6. This would increase the integrated streamwise pressure gradient, which existed across the top and bottom surfaces of panel. Thus, there is an increase in magnitude of

${C}_{l}$ maxima ahead of Instant “3” (

Figure 11b) compared to Case 6. A similar observation was also made at Instant “4” (

$\varphi ={270}^{\circ}$), where the areal size of LPA4 had increased in comparison to LPA2 in

Figure 11a for Case 6. The only difference here was that the increase in streamwise pressure gradient across the panel surfaces promoted an increased negative minima for

${C}_{l}$ (ahead of Instant “4”) since the direction of pitching got reversed.

The wake structures are shown using iso-surfaces of vorticity magnitude in

Figure 13. The chord length and freestream velocity were used to normalize the vorticity magnitude (

${\omega}^{*})$. The iso-surfaces correspond to

$|{\omega}^{*}|=1$. The vortical pattern appear to be in good agreement with the vortex skeleton model proposed by Buchholz et al. [

7], and further verified by Senturk et al. [

11,

12] and Hemmati et al. [

32]. The vortex street in

Figure 13a resembles the von Kármán vortex street. The vortex loops are formed and shed from the trailing edge, which generates drag or thrust. However, at low St, the trailing edge vortices (TEVs) are merged with the side shear layers, which inhibits the transfer of momentum from TEVs to the panel. This process leads to production of drag rather than thrust. Interconnecting vortex rings are observed in the wake (

Figure 14 top), which is consistent with findings of Buchholz et al. [

7] at

$Re$ = 640. As the vortices travel downstream, the interactions weaken between the two vortices, leading to distortion of wake structures. This interaction is stronger for detaching vortices near the trailing edge.

The effect of separation and subsequent downstream movement of vortices are examined using contour of temporal mean velocity magnitude (

$|\overline{{u}^{*}}|$).

Figure 14 also shows the contour plots of

$|\overline{{u}^{*}}|$ on the streamwise and spanwise planes. The mean wake is symmetric in both planes and compares well with those of Senturk et al. [

11].

#### 3.3. Scalability Results

In order to assess the computational performance and parallel scalability of OGA implementation in OpenFOAM, three performance metrics were evaluated which are also useful in analyzing and comparing cost of different algorithms. These parameters included the parallel speed-up (S), total execution time (

${T}_{p}$) for the complete simulation in seconds and efficiency (

$\eta $) of parallel computations. These parameters are calculated as:

where,

${T}_{s}$ and

${T}_{p}$ denote the total execution time for serial computation and parallel computations respectively.

p represents the total number of processors used for parallel computation. The assessment was carried out by running a similar pitching panel case at

$Re=$ 2000, as discussed in the previous section, although a coarser grid was constructed for this analysis. Four computations were performed using 24, 48, 96 and 192 processors respectively.

Figure 15 presents the results for Speed-up (S) and total execution time for parallel computations (

${T}_{p}$) in seconds. Parallel scalability is observed using OGA as the speed-up increases with increasing

p. The increase in the speed-up from

p = 96 and 192 is only 44.15%, which implies that using processors beyond 192 would also require the problem to be scaled up in order to achieve a higher speed up. The reduction in total execution time with the increase in number of processors is also observed, where the curve approaches an asymptotic region after

$p=$ 96. The reduction in execution time for

p = 96 and

p = 192 is only 30.63%.

The efficiency for parallel computations is shown in

Figure 16. An increase in efficiency is observed as

p increases from 24 to 96, beyond which a reduction is observed for

p = 192. Therefore in case of the problem size considered here, the maximum efficiency could be attained for

p = 96. It is important to note that although the computational performance of the simulations increase if we use the newer versions of OpenFOAM, mainly v1912, the comparability of OGA with other numerical solvers, e.g., IBM, remains the same.