Jet deflection by two side-by-side arranged hydrofoils pitching in a quiescent fluid Open Access

The thrust and lift produced by the birds can be controlled by using the flapping mechanism of their wings; optimal propulsion in fish can be obtained from the flapping motion of their pectoral and caudal fins.¹ The dorsal fin on the fish is helpful for achieving their maneuverability. The oscillatory motion of the fins and wings of these aquatic and flying animals is responsible for acquiring a highly effective fluid dynamic performance. Through taking an insight into these curious phenomena, many experts and scholars have grasped some crucial principles that are useful for developing novel bio-inspired autonomous underwater vehicles. Earlier research dates back to Knoller² who recognized first and Betz who recognized later independently³ that the thrust can be generated by a heaving airfoil in moving fluid. Mackowski and Williamson⁴ experimentally investigated the forces acting upon a NACA 0012 airfoil pitching about its quarter-chord point from the leading edge at a low Reynolds number. They showed that a pure pitching motion is a relatively inefficient propulsion mechanism, reaching a peak efficiency of only 12%. The mean thrust coefficient depends on the flapping frequency and increases with frequency. The vortical flow patterns in the wake of a NACA 0012 airfoil pitching at small amplitudes are studied by Koochesfahani⁵ in a low-speed water channel. It is shown that the wake structure can be controlled by controlling the frequency, amplitude, and shape of the oscillation waveform. The more focusing observation in their study is the presence of axial flow in the core of the wake vortices. This axial flow has a linear dependence on the oscillation frequency and amplitude. The wake pattern from a pitching isolated airfoil in two-dimensions leads to the symmetric reverse Bénard–von Kármán vortex street.^5–9 However, beyond (Strouhal number) St ≈ 0.4, the foil produces an asymmetric wake, i.e., a vortex street inclined to the free-stream. Some critical parameters such as the Strouhal Number, reduced frequency, and amplitude of oscillation decide the wake structure of the flapping foils. Godoy-Diana et al.⁹ showed how the wake structure changes from the symmetric normal Bénard–von Kármán vortex street to asymmetric reverse Bénard–von Kármán vortex street as a function of the Strouhal number. They suggested that the dipoles are responsible for the generation of an asymmetric wake. Godoy-Diana et al.⁹ conjectured that asymmetric wakes might be exploited by flying and swimming animals during maneuvering activities. Akhtar et al.¹⁰ and Boschitsch et al.¹¹ reported that fish swimming in an in-line configuration could obtain a hydrodynamic benefit, which may have important implications for the design of underwater vehicles. The case where the foils swim in a side-by-side configuration to achieve better hydrodynamic performance has also received some limited attention.¹² One of the earliest investigations into the hydrodynamics of fish swimming in pairs or schools was reported by Weihs.^13,14 Using an inviscid potential flow model, Weihs¹³ suggested that schooling fish could significantly enhance their thrust production. Dong and Lu¹⁵ computed the flow over wavy foils traveling in a side-by-side arrangement. They found a reduction in power consumption when the foils oscillated in-phase with one another and an enhancement in the fluid forces when the foils oscillated out-of-phase. It suggests that the phase difference between the foils plays a vital role in deciding the hydrodynamic performance in the case of a multiple foil arrangement. Boschitsch et al.¹¹ experimented on two foils arranged in an in-line configuration and reported that the thrust production and propulsive efficiency of the upstream foil differed from those of an isolated one only for relatively closely spaced foils. Bao et al.¹⁶ reported numerical studies on symmetric foils arranged in a side-by-side configuration. Their study reveals that the hydrodynamic performance also depends on the spacing between the foils. The wake interference becomes vital for the case with a smaller foil–foil gap and induces the inverted Bénard–von Kármán vortex streets. Their results show that the hydrodynamic performance of two anti-phase flapping foils can be significantly different from that of an isolated pitching foil. Dewey et al.¹⁷ stated that for in-phase oscillations, the foils exhibit enhanced propulsive efficiency at the cost of a reduction in thrust. For out-of-phase oscillations, the foils exhibit enhanced thrust with no observable change in the propulsive efficiency. For oscillations at intermediate phase differences, one of the foils experiences a thrust and efficiency enhancement, while the other experiences a reduction in thrust and efficiency. Wei and Zheng¹⁸ numerically studied the effect of vortices on the jet deflection angle in the case of only heaving motion. Their study suggests that the switching of the vortex pattern is found to be the primary reason that a deflected asymmetric wake reverses its deflection angle. They also found that the deflection angle increases with the strength of the vortex pairs, which depends on the heaving amplitude, frequency, and free stream Reynolds number. Yu, Hu, and Wang¹⁹ have done both numerical and experimental work on NACA 0012, which pitches sinusoidally for different reduced frequencies. They observed two types of wake transition processes, namely, the transition from a drag wake to a thrust wake and that from a symmetric wake to an asymmetric wake. The deflected wake is found to appear approximately at a Strouhal number of 0.31 and a reduced frequency of 15.1 for the pitching amplitude of 5⁰. As the Strouhal number increases, the dipole model of the vortex pair becomes more apparent than that of the vortex street, which is considered to be a vital element to form the asymmetric wake. Most of the studies until now reported only about the thrust, lift, and jet deflection obtained from an isolated airfoil through pitching, plunging, and combination of pitching and plunging (flapping).^5–9 In addition, few studies reported the propulsive performance of multiple foils arranged in different configurations.^15–17 However, to the best of our knowledge, limited literature is available, which deals with the jet deflection that can be obtained by changing the oscillation phase difference between the airfoils arranged in a side-by-side configuration. Dewey et al.¹⁷ emphasized only on the propulsive performance at different phase differences between the foils but not on the jet deflection that can be obtained at the intermediate phase difference. Only a few studies^20,21 deal with an airfoil oscillating in the still environment. However, there are several studies that deal with non-zero free-stream velocity.^23–26 In the present work, we consider the limiting case, where the free stream velocity (U∞) would be zero. Shinde and Arakeri²⁰ suggested that a weak jet is generated whose inclination changes continually with time when an isolated NACA 0015 airfoil pitches in a still medium. This meandering is observed to be random and independent of the initial conditions. Heathcote et al.²¹ experimentally studied the effect of flexibility on the airfoil plunging in a still medium, and they showed that the strength of the vortices, their lateral spacing, and the time-averaged velocity of the induced jet were found to depend on the airfoil flexibility, plunge frequency, and amplitude. It is interesting to observe that studies on multiple foils pitching in a still medium are scarcely reported. To understand the underlying physical mechanisms governing the propulsion and jet deflection of side-by-side foils in a still medium (water), we focus our efforts on three questions, in particular: (1) how do the propulsive characteristics vary as a function of the oscillation phase difference between the foils and the frequency of oscillation? (2) how do jet deflection angles vary as a function of oscillation phase difference and frequency of oscillation? (3) what are the wake dynamics of pitching foils arranged in a side-by-side configuration in a still medium? Hopefully, the present work imparts insights to understand the maneuverability of UAVs, which can be obtained from the jet deflection. This work is organized as follows: Section II discusses governing equations and problem formulation in solving this problem. Section III discusses numerical methods employed to solve Navier–Stokes and continuity equations. Section V discusses propulsive performance and jet deflection. Vortex dynamics is discussed in Sec. VI. A summary of the results and conclusions is given in Sec. VII.

To describe the surrounding flow dynamics of the model, 2D incompressible Navier–Stokes and continuity equations are employed and are written in the vector form as follows:

where U is the flow velocity vector with components (u, υ), respectively, in $x,y$ Cartesian coordinate directions; p and t are pressure and time, respectively; and ν is the kinematic viscosity of the fluid, respectively. The schematic view of the physical model is presented in Fig. 1. Two rigid NACA 0012 (this airfoil profile is selected because of its symmetry, and the side thrust acting on the foils will be minimum for symmetric profiles) airfoils having a chord length L (60 mm) for each foil are arranged in a side-by-side (parallel) configuration separated by a distance of 0.5L, forming a geometric configuration with a sharp trailing edge. Studies done by Bao et al.¹⁶ suggest that the propulsive performance of the foils arranged in a side-by-side configuration is optimum when the spacing between the foils is less. The spacing between the foils in the present case is restricted to 0.5L to avoid collision between the foils’ trailing edge. Since there is no free-stream flow in this case$U∞=0$⁠, where U_∞ is the free stream velocity, the reduced frequency$k=2πfL/U∞$ and wake width (W) based Strouhal number $St=fW/U∞$ are infinity. Hence, a modified chord length based Reynolds number Re = ρV_TEL/μ is used, where V_TE is the maximum velocity of the trailing edge.²⁰ The Reynolds number range covered in the present study is 8460–33 929, characterizing flapping based propulsion in nature.

A harmonic pitching motion that describes the kinematics of the flapping bodies with an amplitude of oscillation A, the phase difference between the airfoils Φ, and the frequency of oscillation f is given as follows:

where $Vut$ and V_l (t) are the instantaneous velocity of the trailing edge for the upper and lower foils, respectively, and ω is the angular velocity of the foils.

The upper foil is the leading foil with a lead of Φ, and the lower foil is the trailing foil. Both the foils are pitching about the pivot, which is located at a distance of L/4 from the leading edge. A wide range of kinematic parameter space $f,Φ$ is investigated, varying in the range of 0.5 Hz < f < 2 Hz and 0° < Φ < 180° for the given amplitude of oscillation (10°). The governing equations of the flow model are solved with the appropriate boundary conditions. The pressure outlet is assigned in the outer region or outer boundary; a no-slip boundary condition is prescribed at the surface of the two foils, and water is considered as the fluid medium. The computational domain size is 20L in the streamwise direction, from the leading edge. The upstream boundary is profiled as semi-circular with a radius of 12L, and the lateral boundaries are located at 12L from the symmetric line of the geometry. A grid size of 0.0016L along the foil surfaces is found to be accurate enough for our simulations. A two-dimensional unstructured mesh containing triangular elements has been created with 108 828 elements and 65 830 nodes. A combination of a rectangular and a semi-circular patch containing a fine grid is generated around the airfoils to capture the dynamic details of the wake evolution more accurately, as shown in Fig. 2.

When the flow domain moves or deforms in time due to a moving boundary, a fixed mesh becomes inconvenient because it requires precise tracking of the domain boundary. Therefore, the Arbitrary Lagrangian–Eulerian (ALE) formulation is used to discretize the flow equations on moving and deforming meshes. The momentum equation is reformulated in the Arbitrary Lagrangian–Eulerian (ALE) form to take into account the effect of moving boundaries. A field variable of the velocity vector, $U′=u′,v′$⁠, is introduced to modify the convective term in Eq. (1) such that $U−U′.∇U$ is the convective acceleration in the ALE reference framework, as proposed originally by Donea et al.²⁷ and Hughes et al.²⁸ $u′,v′$ are the velocity components of the moving computational domain in $x,y$ directions. This method incorporates and combines both Lagrangian and Eulerian frameworks. The Lagrangian contribution allows the mesh to move and deform according to the boundary motion, whereas the Eulerian part takes care of the fluid flow through the mesh. Because of the pitching of airfoils, foil boundary motion will be inevitable, and because of these moving boundaries, the quality of the mesh will deteriorate with time, and hence, re-meshing of the deteriorated mesh is adopted after every few time steps to ensure the quality of the mesh. An unsteady, incompressible, laminar viscous model is used to solve this problem. The diffusion algorithm is employed in this work to update the mesh system. This algorithm moves the mesh nodes in response to the displacement of boundaries by calculating the mesh velocity using a diffusion equation. For solving Navier–Stokes and continuity equations, the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) scheme is used, and for the transient formulation, the second-order upwind scheme is employed. ANSYS FLUENT is used for numerical calculation. The simulations are performed for a pitching period of 100 cycles for each case. The non-dimensional time step used for all simulations is 0.001.

The validation study is conducted by considering an isolated NACA 0012 airfoil pitching about the pivot, located at a distance of L/4, as reported in Ref. 22. The coefficient of thrust as a function of reduced frequency is compared at the chord-based Reynolds number Re_L = 12 000 reported in the literature.^4,22 As plotted in Fig. 3, the comparison of the thrust coefficient with reduced frequency shows very good agreement with the numerical and experimental results reported by Ramamurti and Sandberg²² and Mackowski and Williamson,⁴ respectively.

A grid independence test is also conducted for the present case of two foils in a still medium by considering different mesh resolutions on the foil surface to ensure that the solution obtained is independent of grid size. Table I shows the time-averaged values of the thrust coefficient acting on the upper foil for the frequency of oscillation 1.5 Hz, the amplitude of 10°, and the phase difference of 45° between the airfoils. The time-averaged values of the thrust coefficient from Table I suggest that the solution is independent of grid size.

From Fig. 4, it is evident that the solution obtained from the three different mesh resolutions is superimposed on each other with an insignificant error.

A time step independence test is also conducted by considering three different time steps to confirm that the solution is independent of the time step. Table II shows the time-averaged results of the thrust coefficient acting on the upper foil for a frequency of oscillation of 1.5 Hz, the amplitude of 10°, and the phase difference of 45°. The data for the averaging in every case is taken by removing initial transient fluctuations and considering only the periodic fluctuations (20 pitching cycles) where the data are consistent with time. The time-averaged values from Table II suggest that the solution is independent of the time step as the error with respect to case 2 (actual time step considered in all simulations) is within the acceptable range.

The average thrust and lift forces acting on the airfoils are calculated by

where $Fxt$ and $Fyt$ are the hydrodynamic force components acting in the horizontal and vertical directions at an instant, which is obtained by integrating both pressure and viscous forces around the surface of the airfoil, and T is the time interval. The flow time has been normalized by multiplying with the frequency of oscillation, and this normalized time is denoted as t^* = f × t, where t is the flow time. The corresponding time-dependent average thrust coefficient and lift coefficient are defined as

where F_x and F_y are the time-averaged force components calculated along the x and y axes obtained from Eq. (5).

For almost every case we considered, after removing the initial transient fluctuations, the whole flow time is divided into three zones, as shown in Fig. 5, which shows the variation of the thrust coefficient on the upper foil as a function of normalized time for a frequency of oscillation of 2 Hz and the phase difference of 30°. In the first zone, the jet is in a deflected position, or there is no complete interaction of lower foil vortices with upper foil vortices. The thrust force acting in this region is less when compared to other zones. The second zone is the transition zone, where the deflected jet starts to switch toward the centerline position from the deflected position and finally aligns along the centerline, which is termed zone 3. In this zone, the complete interaction of vortices from the lower foil and the upper foil has been observed; in addition, there is a gradual increase in the thrust force from zone 1 to zone 3. Zone 3 is where the jet finally aligns along the centerline, and the thrust force acting on the foils is found to be the highest in this zone, which can be observed from Fig. 5. The same trend has been observed for the lift force as well. Here, the propulsive performance and jet deflection for our entire work are calculated by considering the periodic cycles of zone 1 only, i.e., when there is no complete interaction of vortices of the lower foil with those of the upper foil.

The time-averaged thrust coefficient acting on both the foils for different phase differences between the airfoils for a frequency range of 0.5 Hz < f < 2 Hz and oscillation amplitude of 10° is shown in Fig. 8. This thrust force acts in the negative x-direction, and thrust values are taken as positive in that direction. The results depend strongly on the phase difference between the airfoils. It can be observed from Fig. 8 that there is no drag producing (no negative values of thrust coefficient) region in any of the phase differences for all the frequency ranges we had considered. For a constant phase difference between the foils, the thrust force increases with an increase in the oscillation frequency. Godoy-Diana et al.⁹ and Bao et al.¹⁶ reported that there is a wake transition from momentum less (drag) to momentum gain (thrust), and this transition is a function of the Strouhal number. They also reported that at a particular Strouhal number, there is a transition from the normal BvK (Bénard–von Kármán) vortex street to reverse BvK vortex street. However, in our case, there is no free stream velocity; hence, no such transition is observed, i.e., there is no formation of normal BvK vortices, and only reverse BvK vortices have been observed in all the cases, which represents thrust.

The maximum total thrust is achieved at a phase difference of 150° for the frequencies of 0.5 Hz and 1 Hz. However, for the frequencies of 1.5 Hz and 2 Hz, the maximum thrust is achieved at 120° phase difference, as shown in Fig. 6.

At the above-mentioned phases (i.e., at Φ = 120°, Φ = 150°), the circulation strength of the vortices convected from the trailing edge is found to be maximum, thereby inducing a high momentum jet (because the induced velocities depend on the circulation strength of vortices), which leads to maximum thrust. From Fig. 7, it is clearly shown that at Φ = 120°, the distance through which the vortices convected is maximum when compared to other cases, which suggests that the circulation strength (so the induced velocities) is maximum at Φ = 120°.

Minimum thrust is generated when the phase difference is 180° (i.e., out-of-phase oscillation) for the frequencies of 1.5 Hz and 2 Hz and at 0° (in-phase oscillation) for the frequencies of 0.5 Hz and 1 Hz. As the phase difference increases for a constant frequency of oscillation (2 Hz) and amplitude (10°), the thrust coefficient increases and becomes maximum at 120°, and then, it decreases. This decrease in thrust force beyond Φ = 120° is because when the foils move toward each other, a low-pressure region is created near the trailing edge in the spacing between the foils (see Fig. 8, Φ = 120°–180°).

This creates the high-pressure difference between the upstream (low-pressure region in the spacing between the foils) and downstream regions (relatively high-pressure region at a considerable distance from the trailing edge), hence causing retardation of the high momentum jet (loss in the kinetic energy because of the pressure difference), whereas in remaining cases, though there are negative pressure regions in the spacing between the foils, this pressure difference is not high enough to create strong retardation force, and hence, the thrust force keeps on increasing from Φ = 0° to Φ = 120°.

The same trend is observed for the individual foils also, i.e., upper and lower foils with a maximum thrust at 120°, which can be seen in Fig. 9. As the jet is inclined in a counter-clockwise direction from the centerline, a higher thrust is observed in the upper foil (leading foil) than in the lower foil, which can be seen in Figs. 9(a) and 9(b).

Phase portraits of the thrust coefficient (C_t) acting on the upper foil relative to the transverse distance (Y_u) of the trailing edge at different phase differences for f = 1 Hz and A = 10⁰ are shown in Fig. 10. Except for in-phase oscillation, at every phase difference, the oscillation frequency of the thrust coefficient is two times that of the pitching oscillation frequency. For in-phase oscillation, the pitching frequency and thrust oscillation frequency are the same. As the phase difference increases, because of the change in wake characteristics, the symmetry in the phase portraits is broken and distorted for out-of-phase oscillation.

There is not much change that has been observed in the phase portraits at different frequencies for a phase difference of 45° and an oscillation amplitude of 10°, which can be seen in Fig. 11. This indicates that with an increase in pitching oscillation frequency, only the magnitude of thrust changes. However, the variation of the oscillation curve of the thrust coefficient in one pitching cycle is insensitive to that of the pitching oscillation frequency.

The variation of the lift coefficient acting on the upper foil and lower airfoil as a function of the phase difference between the airfoils for an oscillation amplitude of 10° is shown in Fig. 12. It shows that the lift forces on the two foils act in the opposite direction, i.e., in the negative y-direction on the upper foil and positive y-direction on the lower foil.

There is formation of a periodic low-pressure region in the spacing between the airfoils over time. This low-pressure region makes the y-direction forces to act in opposite directions on the airfoils, as shown in Fig. 13. The lift force acting on both the foils increases with the phase difference for all frequency ranges and becomes maximum at a phase difference of 180° (out of phase). Hence, there is a trade-off between the maximum lift (Φ = 180°) and the maximum thrust (Φ = 120°). The phase difference can be a useful parameter to control the maneuverability of underwater vehicles to get the desired effect.

To achieve the maneuverability of the UAVs, it is essential to create unbalanced moments in the vehicle.¹⁷ These unbalanced moments can be obtained if the propulsive jet is inclined at some angles with respect to the centerline position. The corresponding forces act in those angles, creating the unbalanced moments, which can be helpful for the change in the vehicle’s direction. Therefore, jet deflection may play a vital role in the maneuvering of the UAVs. The time-averaged velocity magnitude contours for a frequency of 1.5 Hz and the oscillation amplitude of 10⁰, indicating a jet deflection (this deflection is observed only in zone 1 in Fig. 5) from the centerline position, are shown in Fig. 14. It is clearly seen that the jet is aligned along the centerline at 0⁰ phase difference, and the jet deflection angle increases with phase difference until 45° and then keeps on decreasing beyond 45°, and finally, the jet aligns along the centerline again at 180° phase difference.

The time-averaged jet deflection angles obtained at different phase differences between the airfoils for a frequency range of 0.5 Hz–2 Hz and an oscillation amplitude of 10° are shown in Fig. 15. It shows that for in-phase and out-of-phase oscillation, there is no jet deflection. In contrast, for the intermediate phase difference, there is an inclination of the jet. This inclination ranges from 0° to 28°, and the maximum deflection is obtained at 45° phase difference for a frequency of 0.5 Hz. Beyond 45° phase difference, the deflection angle starts to decrease, and finally, the jet is aligned with the centerline at 180° phase difference.

The methodology employed to calculate the time-averaged jet deflection is described as follows: Initially, time-averaged velocity magnitude values (over the time interval at which there is a jet deflection, explicitly considering the pitching cycles of zone 1 in Fig. 5) are calculated, and from that, the maximum velocity at different chord lengths from the trailing edge (0.5L, 1.5L, 2L, 2.5L, 3L) along the downstream is calculated. The dots of the jet deflection trajectory curve in Fig. 14 represent the maximum velocities along the y-direction at different chord lengths. Upon joining these points and also considering the point (which is the projection of the pivot point on the x axis) as a reference from the x axis, a deflection curve is obtained. To calculate the slope of the deflected jet, a linear fit is employed over the curve that is obtained by joining all those points, which can be clearly seen in Fig. 16 (black line). From there, the slope () of the line is calculated, which is defined as a jet deflection angle. The trajectory of the maximum velocity along the downstream of the airfoil is not linear. However, it is approximated as linear to find the slope of the deflected jet.

Figure 17 shows the time-averaged x-direction (streamwise) velocity profiles and velocity contours along the downstream at different chord lengths from the trailing edge for a phase difference of 0°, 45°, and 180°, and frequency of 1.5 Hz. The x-direction velocity is normalized by the velocity at the trailing edge V_TE. It is evident from the velocity profiles [see Figs. 17(a)–17(c)] that the jet decelerates as the velocities decrease with an increase in the distance from the trailing edge. Shinde and Arakeri²⁹ reported their undulating flow with wavy motion due to the additional flap attached to the rigid pitching foil in a still medium. However, no such undulating flow with wavy motion has been observed in our case. The maximum velocity in their case is observed at a noticeable distance from the trailing edge instead of near the trailing edge. The maximum velocity in our case is observed at the trailing edge, which is in good agreement with Ref. 16, where they have free stream velocity, and the velocities keep on decreasing along the downstream direction. Therefore, the flow features, especially the velocity behavior along with the downstream of the foils, both in a still medium and a free flow medium are the same. There is a single peak along the y-direction in the velocity profile at all the chord lengths we considered for the phase difference of 45° [see Fig. 17(b)]. This suggests that the jet from the lower foil merged with the upper foil very close to the trailing edge and evolved as a single jet. The peak of the velocity profile for the phase difference of 45° shifts toward the left and in the upward direction. It signifies that the jet is in the deflected position, which can be seen in Fig. 17(b), and the deflection can also be seen in Fig. 17(b^*). The same trend is observed for the intermediate phase difference, except for the in-phase [Fig. 17(a^*)] and out-of-phase [Fig. 17(c^*)] cases. Two separate jets have been observed, even at a far distance from the trailing edge, and subsequently, two separate peaks [see Figs. 17(a) and 17(c)] along the y-direction are observed similar to the one reported in Ref. 16. As two separate jets are formed in the case of in-phase and out-of-phase oscillation, there is no merging of jets, unlike the intermediate phase difference cases, where the merging takes place close to the trailing edge. It is also observed that the jet’s merging distance for the entire intermediate phase difference is almost the same with insignificant differences, making it difficult to distinguish the difference of merging distance accurately. This difficulty comes because in our case, there is no free stream velocity, and hence, the convection velocities are not so high enough to merge at more considerable distances. However, exceptionally in the case of 180° phase difference, this merging of two separate jets is observed comparatively at a larger distance from the trailing edge, which can be seen in Fig. (17c). The velocity profiles have a single peak beyond a distance of 0.5L from the trailing edge instead of two peaks, which signifies that two jets from the lower and upper foils have merged into one jet beyond 0.5L.

The direction of jet deflection depends on the leading foil position. The deflected jet will be in the counter-clockwise or upward direction when the upper foil is the leading one and will be in the clockwise or downward direction when the lower foil is the leading foil, which can be seen in Fig. 18. Shinde and Arakeri²⁰ reported that the direction in which the jet deflects depends on the initial conditions. However, they did not mention precisely those initial conditions. In our case, it is observed that the direction of jet deflection is independent of the initial direction in which the foils oscillate (start of the motion), i.e., either clockwise or anti-clockwise.

Interestingly, we observed that, for a constant frequency of oscillation and phase difference, the time-averaged thrust force acting on the upper foil (leading foil) in Fig. 18(a) and on the lower foil (Leading Foil) in Fig. 18(b) is equal. In addition, there is a symmetry in the wake region at a particular instant of time, which can be seen in Figs. 19(a) and 19(b). This suggests that the propulsive characteristic acting on the respective leading and trailing foils remains the same, irrespective of the position of the leading and trailing foils.

The propulsive performance and jet deflection reported in Sec. V can be understood if the behavior of the vortices formed beyond the trailing edge is known. The wake region beyond the trailing edge has been investigated to find the possible reasons for the jet deflection. The instantaneous wake region of the foils operating at a phase difference of 0° and 45° at a frequency of 1.5 Hz, and the oscillation amplitude of 10°, is shown in Fig. 20. The following observations are made out of this wake region analysis. (1) No vortex street is formed unlike the flow over airfoil reported in Ref. 16 (in which there is a vortex street because of the free stream velocity in their case); instead, vortex dipoles without any shear layer are formed, which can be seen in Figs. 20(a) and 20(c), and the same dipole formation was observed in Refs. 6 and 9 at higher Strouhal numbers with free stream velocity. (2) The vortices (from both the foils), which are induced near the trailing edge, eventually interact with each other and evolve as more strong vortices far away from the trailing edge, which can be seen in Fig. 21(b). (3) Vortex interaction changes with time. (4) From the start of motion, reversed BvK vortices are formed from both the foils, which give the momentum gain flow beyond the trailing edge. The same can be observed in Fig. 8, where the thrust force acts only in one direction (negative x-direction) for all the cases, which suggests that there is no drag producing region unlike the one reported in Ref. 16. (5) The circulation strength of the vortices increases with an increase in the frequency of oscillation. Subsequently, as a result of this high strength of circulation, the thrust force is higher at higher frequencies, which can be observed in Figs. 6, 9(a), and 9(b).

For in-phase oscillation, there is convection of vortices from the two foils. Two separate straight jets are formed from the individual foils, as shown in Fig. 20(a). Initially, there is no interaction of vortices from individual foils. However, after few pitching cycles, the likewise (same direction vortices) vortices start to interact with each other and form as a single jet, which is aligned in a straight line, as shown in Fig. 20(b). The formation of a single jet comes under zone 3 in Fig. 5, which was already discussed in Sec. V, which leads to thrust improvement.

However, in the case of the intermediate phase difference, instead of forming two separate jets at a considerable distance, a single jet [Fig. 20(c)] is formed from the start of the motion. Because the merging distance is very less from the trailing edge, unlike the in-phase oscillation where there is no merging of jets. In the case of out-of-phase oscillation, though the same trend is observed as in the in-phase oscillation, however, in addition to that, a reverse flow (negative x-direction) at the leading edge is observed, which can be seen inside a square dotted box shown in Fig. 21(b). However, no such reverse flow is observed for in-phase oscillation, which can be seen in Fig. 21(a) or in any other intermediate phase differences.

In one pitching cycle for the out-of-phase oscillation, there will be formation of a low-pressure region near the trailing edge (see Fig. 13) when the foils move toward each other (the impact of this low-pressure region is already explained in Sec. V). However, when the foils move away from each other, and when they are in their extreme positions, a high-pressure region is created in the spacing between the foils, which can be seen in Fig. 22(a). This high-pressure region is formed close to the leading edge, and because of this high pressure, a pressure difference is established between the foils’ spacing and upstream region, causing the flow to move in the upstream direction in the case of out-of-phase oscillation. Although similar kinds of high-pressure regions are observed in the remaining phase difference cases, those high-pressure regions are not concentrated in the spacing between the foils. Instead, it is distributed over the surface of foils, which can be seen in Fig. 22(b), and hence, no such reverse flows are observed in other phase difference cases.

For all the intermediate phase difference cases, initially, only from the upper foil, the vortices convect at some angle. In contrast, from the lower foil, only a part of the vortex (partial interaction) (vortex d2) from Fig. 23(b) [which is formed due to breakdown of the counter-clockwise vortex (d) by the clockwise vortex (b)] interacts with the upper foil vortex. After the partial interaction, the remaining vortices diffuse at a very smaller distance from the lower foil trailing edge without any further convection. This partial interaction of vortices comes under zone I from Fig. 5. After a few pitching cycles, the likewise vortices [(a) and (b)] start to interact completely with each other from the two foils and form a single and stronger vortex, which can be seen in Fig. 23(c). After this complete interaction only, the jet starts to switch from its inclined position and tries to reach toward the centerline position, which corresponds to zone 2 (transition zone) in Fig. 5; when this complete interaction of vortices continues for few more cycles, the jet is aligned along the mean or centerline position, which corresponds to zone 3, just like in-phase and out-of-phase oscillations, which can be seen in Fig. 20(b).

Interestingly, after the jet is aligned along the centerline, again, no switching of the jet is observed in our case, unlike the jet switching mentioned in Ref. 20, for pitching of the single airfoil. This has been confirmed by continuing the simulation for 100 more pitching cycles. The exact reason for switching of the jet from the deflected position to the centerline due to the complete interaction of vortices is unknown and needs further experimental investigation.

In the present case, there is no free stream velocity; the disturbances/perturbations generated during initial pitching cycles will not get convected. We conjecture that this could be a possible reason for why in our case the initially deflected jet moves toward the centerline position after a finite number of pitching cycles. These disturbances facilitate the interaction of likewise vortices from the upper and lower foils, which leads to the alignment of the jet along the centerline. The jet aligned along the centerline may be a more stable state for such a kind of flow configuration. However, this needs to be explored in detail.

To understand the role of vortices on the jet deflection, the Biot–Savart law is used. Here, the vortices shed from the trailing edge in the near wake region are of most interest rather than those in the far wake region, which has less impact on propulsive performance.¹⁸ The velocity induced by one vortex over its neighboring vortices can be found using Eq. (7) (Biot–Savart law) by assuming the vortices to be circular,

where Γ is the average strength of the vortex, r is the distance (points where maximum circulation exists) between the two vortices, and Ui is the velocity induced by one vortex on the other. The direction of the induced velocity on the individual vortices depends on the rotational sense, i.e., velocity on the two vortices acts in the same direction if they rotate in the same sense (i.e., either in the clockwise or anti-clockwise direction). The direction of the velocities is different if the sense of rotation is different (i.e., one rotating in the clockwise direction and the other rotating in the anti-clockwise direction). In the present case, as there is no free stream velocity, there is no effect of the free stream velocity on the vortices shed from the trailing edge. Therefore, the only effect on the vortices is the induced velocity. With higher frequencies, the thrust increases (see Fig. 8) because at higher frequencies, the circulation strength of the vortices increases, and hence, induced velocities between the vortices increase, leading to increased high momentum jets at higher frequencies. Although the circulation strength Γ has a significant impact on induced velocities, the distance r between the vortices also plays a vital role in deciding the magnitude and direction of induced velocities. From the Biot–Savart law, if the circulation strength of the individual vortices at any instant and also their maximum strength location are compared, then the magnitude and direction of the velocities acting on those individual vortices can be estimated. The average circulation strength Γ of the vortices is obtained by calculating the surface integral of the vorticity over the vortex area. The distance r can be easily found by using the points of maximum circulation strength.⁷ By using the above-mentioned method, we estimated the effect of neighboring vortices on a single vortex, the direction, and the resultant velocity induced by other vortices on a single vortex.

To calculate the induced velocities, we considered only four vortices near the wake region. However, there is an effect of downstream vortices on the upstream four vortices, but that effect can be neglected because of insufficient circulation strength of vortices at the far region from the trailing edge. The reason for this low strength is because in the far wake region, viscous dissipation is very high, and the vortices lose their strength, and their effect can also be neglected because of the reciprocal-distance influence of the Biot–Savart law, which was reported in Ref. 18. The direction and magnitude of induced velocities between the vortices are calculated using the above method, which can be seen in Fig. 24. From Fig. 24(a), it is evident that the induced velocity direction on upper vortices is toward the centerline. In contrast, the direction on lower vortices does not maintain any particular pattern. This signifies that these lower vortices do not allow the breakdown of the symmetry, i.e., they hold the upper vortices that try to deviate from the centerline and try to maintain the symmetry.¹⁸ This results in no deflection of the jet in the case of 0⁰ phase difference. However, in the case of Fig. 24(b), the case when the phase difference is 45⁰, the direction of induced velocities on all vortices acts at some angles with respect to the horizontal axis, which results in an upward jet deflection. After some pitching oscillations, the deflected jet starts to switch toward the centerline position, and finally, it aligns along the centerline due to the re-orientation of vortices, which is already discussed in Sec. V.

In the present study, no switching of the jet in the downward direction is observed even after 200 cycles of oscillation, which is already mentioned above. It is observed that the switching of the jet occurs or is initiated when there is complete interaction of lower vortices with upper vortices. For the intermediate phase difference, when the oscillation starts, there is always a partial interaction of lower foil vortices with upper foil vortices, as shown in Fig. 23(b), but after some pitching cycles, lower vortices start to interact completely with upper vortices, and then, the switching starts. The exact reason for this switching is not known, which needs further investigations.

Figure 25 shows the variation of C_T and C_L with normalized time at f = 2 Hz, Φ = 45°, and A = 10°. The vertical line in Fig. 25 represents the time instant at which the switching of the jet toward the centerline from the deflected position (which is observed in the case of all intermediate phase difference) has started. From the vorticity field, we observed that there is a complete interaction of lower vortices with higher vortices at this instant of time. The right-hand side region from the vertical line in Fig. 25(a) suggests an increase in the thrust force acting on the upper foil. The same thing can be observed in Fig. 25(b) where there is an increase in the lift force acting on the upper foil after the jet has started to switch toward the centerline position. The possible reason for this increase in thrust force is because of this complete interaction of one vortex from the lower foil with that of another vortex from the upper foil, leading to a stronger and larger vortex of more circulation strength than that of the individual vortices. Because of this high strength, the induced velocities increase and eventually result in a high momentum jet, increasing the thrust force. The same trend is observed in the case of the lower foil as well.

The propulsive characteristics of unsteady foils oscillating in a side-by-side (parallel) configuration were found to depend on both the frequency of oscillation and the phase difference between the foils. Four general observations were made: (1) The maximum total thrust is obtained for a phase difference of 120° at a frequency of 1.5 Hz and 2 Hz and for a phase difference of 150° at a frequency of 0.5 Hz and 1 Hz. (2) Minimum thrust is generated when the phase difference is 180° (i.e., out-of-phase oscillation) for the frequencies of 1.5 Hz and 2 Hz, and 0° (in-phase oscillation) for the frequencies of 0.5 Hz and 1 Hz. (3) There is a jet deflection at the intermediate phase difference between the foils for a given frequency and amplitude of oscillation, and the high deflection angles are observed in the range of 30°–45° phase difference. The maximum deflection angle is found to be 28° at a frequency of 0.5 Hz for the phase difference of 45° between the foils. (4) For all the frequencies we consider, the lift force acting on the foils is maximum for out-of-phase oscillation, i.e., at Φ = 180°. These observations are made for fixed spacing between the airfoils, i.e., 0.5L.

These observations suggest that both propulsion and maneuvering of UAVs can be achieved just by altering some critical parameters. The phase difference between the foils, which are arranged in a side-by-side configuration, and the frequency of oscillation are the few significant parameters that affect the propulsive performance and maneuvering. The choice of the orientation of the jet depends on the position of the leading foil and is independent of the initial direction of motion of foils (i.e., clockwise or anti-clockwise direction). We believe that these simulation results will form the basis of maneuvering in underwater vehicles and will give some insight to researchers who are working on bio-inspired underwater vehicles.

Vortex analysis indicated that the wakes shed by foils are complex and depend critically on the phase difference between the foils. Two separate jets for considerable distances have been observed in the case of in-phase and out-of-phase oscillation, whereas a single jet is observed for all the intermediate phase differences. Jet switching toward the centerline from the deflected position is observed only when there is complete interaction of lower foil vortices with upper foil vortices. This switching of the jet has a significant amount of impact on the propulsive performance.

The authors would like to thank Dasari Chaitanya and Swapnil Jagadale of Indian Institute of Technology (IIT) Bhubaneswar for their assistance and support throughout the course of this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.