Research on the interaction mechanism of multiple vortices in biological wake flow field

The vortex-ring interaction of synchronous jets has been paid more attention by predecessors, while the research of asynchronous jets (i.e. there is a time difference between the start of jets) of parallel dual nozzles has been studied less. Therefore, the interaction between vortex rings and the vortex structure evolution of the dual-nozzle asynchronous jets with different time differences (Δt) is studied in this paper. Numerical results obtained can be divided into four intervals: 1. Synchronous jets (Δt = 0); 2. Critical interval (Δt < 0.1t, where t is the jet time), having similar vortex structure evolution modes and dynamic characteristics as the synchronous jet; 3. Strong interaction interval (0.2t ≤ Δt ≤ 0.4t), in which the main vortex ring shows obvious acceleration, and the streamwise vortex structure is found in the wake; 4. weak interaction interval (Δt ≥ 0.6t), in which the interaction between vortex rings is much less strong and the streamwise vortex structure also appears in the wake. In addition, it is found that pressure on the nozzle outlet plane increases and the vortex dissipation in the downstream flow field slows down significantly due to the vortex-ring interaction on the condition of asynchronous jets.


Introduction
In the wakes generated by the unsteady motions of organisms such as fish swimming, squid jets, and insect flight, there are complex multi-vortex interaction flow structures.Organisms can use multi-vortex interactions to achieve efficient transport of mass and momentum and capture of energy.Organisms propelled by jets (squid, jellyfish, salps, etc.) generate vortex rings through jets, the shape of which is a three-dimensional ring structure formed by the vortex rolled back.Vortex rings formed by jets have greater effects on the transport of mass and momentum than the same amount of direct flow, so aquatic organisms are propelled by this vortex ring to improve efficiency [1].The salps have another feature.They tend to gather in parallel to form a chain, and each individual in the chain can make the whole move efficiently by adjusting the intensity and timing of water jets.This may due to a mode of coupling effect that is beneficial to propulsion among these jets.According to the observation from Dabiri et al. on the water absorption and spraying process of jellyfishes, it is found that the multi-vortex interaction in the wake increases the probability of preying on plankton in the water [2].
Previous research on jet vortex rings mainly focused on the formation process and vortex structure evolution of single-nozzle jet vortex rings, and there are few studies on parallel vortex rings.Athanassiadis et al. investigate the loss of thrust and propulsion efficiency caused by the interaction of the vortex rings of the synchronous jets with parallel dual nozzles through experimental methods, and explain this as the interaction between the two vortex rings to reduce (straighten) the curvature of fluid streamline from the nozzle exit plane.This can also be regarded as the restriction of the lateral flow in the wake, resulting in the reduction of the overpressure (one of the sources of jet thrust) at the nozzle outlet, thus leading to losses of both trust and propulsion efficiency [3].Based on this, Athanassiadis et al. put forward a hypothesis that the thrust and propulsion efficiency can be gained by reasonably controlling the time interval of the dual-nozzle jets [3].Therefore, it is very meaningful to study the effect of the ejection time interval of the two vortex rings on the thrust, propulsion efficiency, and lateral transport effect of jet flow.
According to the existing research, this paper focuses on the influence of the asynchronous time difference on the evolution of the vortex structure and the interaction between the vortex rings and on the dynamics of the wake flow field.

Methodology
Before the numerical simulation, it is necessary to predict the flow state and determine whether the flow is laminar flow, turbulent flow, or there is a transition from laminar flow to turbulent flow.When considering this issue, the vortex formation number [4] and the Reynolds number are two dominant parameters.The vortex formation number L/D is the ratio of the piston movement distance to the inner diameter of the nozzle.
Figure 1 is the transition diagram of the jet vortex ring obtained by Glezer [5].The abscissa is half of the Reynolds number and the ordinate is the vortex formation number.The gray shadow is the transition boundary and the left side of it represents the laminar vortices while the right side represents the turbulent vortices.The simulation conditions of this paper are listed in Table 1.Then the position of the simulation condition is shown as the red dot in Figure 1 so it can be predicted that the formation and development of vortex rings in this study keep laminar, which can be solved by the laminar flow model.

Computational domain
The front view of the 3D computational domain is shown in Figure 2. Due to the symmetry of the parallel dual-nozzle jet flow field, in order to reduce the overall number of grids, only half of the flow field is used for numerical simulation, and the full flow field information is obtained by mirroring the results during post-processing.The diameter of the nozzle is D = 25.4 mm.The fluid enters the nozzle from the velocity inlet according to a given velocity profile, which simulates the movement of the piston.The simulated piston movement time is t = 0.7773s and the total flow time is T = 3s.

Vortex area determination
Intuitively, the vortex represents the rotational motion of the fluid, but there is still no widely accepted method to strictly define the vortex [6].In 1858, Helmholtz proposed the concept of vortex line and vortex tube and three Helmholtz's laws [7].In the subsequent research, the vorticity line or the vortex tube is mainly regarded as the vortex structure, and the magnitude of the vorticity is regarded as the strength of the rotational motion.This method of identifying vortex structures based on vorticity is classified as the first generation of vortex identification methods.Subsequent studies have successively put forward the second generation of vortex identification methods represented by methods such as Q,  2 , , and  ci [8][9][10][11].These methods have modified the concept that vorticity is equivalent to the vortex to some extent, which improves the accuracy to identify vortices.
Among the second generation vortex identification methods, Q criterion is the most widely used vortex identification method [12].The Q criterion is a vortex identification method based on the eigenvalues of the velocity gradient tensor.The velocity gradient tensor is expressed as: Its characteristic equation is Then, If three eigenvalues are represented by λ 1 , λ 2 and λ 3 , then Q can be obtained by: Q is one of the three Galilean invariants of the velocity gradient tensor ∇V.Hunt et al. [10] proposed to use the second Galilean invariant Q > 0 to indicate the vortex structure.Generally, a certain threshold ( a certain value from Q > 0) is selected as the isosurface to represent the vortex structure.

Grid independence test
As mentioned above, the laminar flow model is used for numerical calculation, so the height of the first grid layer is not determined by y + .Considering that the development of the vortex downstream of the nozzle outlet is studied, the grid growth rate and the densified area downstream of the nozzle outlet are very important.In order to find suitable parameters, a sample simulation of a single-nozzle jet is first performed and compared with the experimental results of Gharib et al. [4].The grid information is shown in Table 2. Experimental and numerical results are compared in Figure 3.The experimental results for comparison (Figure 3a) use the vortex formation number L/D = 3.8, and the main vortex is located at around 7D downstream of the nozzle exit plane.It can be seen that the deformation and distortion of the main vortex rings are serious for Case 1 (Figure 3b) and Case 2 (Figure 3c).While in Case 3, the main vortex ring maintains a relatively good shape, which is consistent with the experimental results (Figure 3d).Therefore, the subsequent grid adopts the same encryption method as Grid 3.  Then Grid 3 is used for appropriate scaling to obtain three sets of grids as shown in Table 3.In Figure 4, circulations of the upper half of the symmetry plane downstream of the nozzle outlet of Case 4, Case 5, and Case 6 are detected and circulations are calculated by Formula 3. It can be seen that the variation of circulations with flow time of Case 5 and Case 6 (red and yellow curves) are highly consistent, while the result of Case 4 (blue line) is quite different.Considering both restricting the number of meshes at a low level and the guarantee of simulation accuracy, the grids of the subsequent dual-nozzle jets are set according to the mode and parameters of Grid 5.

Simulation result verification
In order to detect the correctness of the simulation results, results from Case 5 are compared with the experimental results from O'Farrell et al. [13].The simulation adopts the same physical model and the vortex ring formation number (L/D = 12) as the experiment.Figure 5 shows the vorticity contour maps downstream of the nozzle outlet when the formation times are 1.5, 4.1, and 8.5 respectively.The abscissa is the radial distance to the nozzle outlet plane (in terms of the nozzle inner diameter D).The equation [14] for formation time calculation is: where U is the velocity of the piston in the nozzle, T is the actual flow time, and D is the inner diameter of the nozzle.The simulation results can be verified by intuitive comparative analysis.According to the grid independence test and simulation verification above, the grid mode and parameters are determined to obtain the parallel dual-nozzle jet grid as shown in Figure 6.

Vortex structure evolution
In this part, interactions of the vortices in the wake of jets with different time intervals (t) from two parallel nozzles and the evolution process of the vortex structure are analyzed.Taking the jet time t as the standard, simulations of synchronous jets and asynchronous jets with time differences of 0.1t, 0.2t, 0.3t, 0.4t, and 0.6t were carried out respectively.The Q criterion is used to identify the vortex structure, and the isosurface value of the vortex boundary is taken as Q raw = 180 or Q normalized ≈ 0.08.According to the interaction characteristics between the main vortex rings, results are divided into four intervals: synchronous jet (t = 0), critical interval (t ≤ 0.1t), strong interaction interval (0.2t ≤ t ≤ 0.4t), and weak interaction interval (t ≥ 0.6t).The main vortex ejected first is named as W 1 and the one ejected later is named as W 2 .
3.1.1.Vortex interaction and structure evolution of synchronous jets.Figure 7 shows the threedimensional structure of the main vortex in the wake of synchronous jets at the stage (flow time  = 0.68).It can be seen that when the two main vortices radially expand to a certain extent, they approach each other and interact with each other, causing the closer part of the two vortex rings to be decelerated (shown by the red circle in Figure 7).This is because this part of the vortex ring is affected by the induced velocity of the other one, which is in the opposite direction to the forward velocity of the vortex ring, thus producing a deceleration effect, which coincides with the phenomenon observed by Oshima et al. [15].Then the decelerated parts of the two vortex rings rupture and reconnect (indicated by the red arrows in Figure 8), and gradually merge into a larger twisted vortex ring.Figure 8 shows three views (the front view is from downstream to upstream) of the twisted vortex ring.It can be seen that it is similar to the vortex ring [15] observed in the experiment by Oshima et al.The front view is close to an ellipse, the top view is an upward horseshoe shape, and the whole is a twisted vortex ring structure connected in the middle and curved backwards.The morphology of the vortex ring after fusion is clearly demonstrated in the axonometric view in Figure 9.When the twisted vortex ring continues to move downstream, two vortex structures (indicated by the red arrow in Figure 9) are found at the front of the decelerated part.

Critical interval.
When the time difference between two jets is small ( = 0.1), the interaction between the vortex rings is strong, and the interaction mode is similar to the one of synchronous jets.The interaction still plays a relatively obvious role in delaying the part where the two vortex rings are close (as shown by the red circle in Figure 10).In this region, the induced velocity of another vortex ring still has a large component in the direction opposite to the forward direction of main vortex rings.There is one difference from the synchronous jet observed: the vortex structure distortion of the two vortex rings affected by the induced velocity is no longer symmetrical.The vortex ring W 1 is delayed, while the vortex ring W 2 is not only delayed but dragged toward W 1 .Then, at very early time (T < 1s), the delayed part of W 1 fractures (shown by the red circle and red arrow in Figure 11).Subsequently, the fractured part of the W 1 develops a new flow vortex structure towards the upstream side (as shown by the red circle and red arrow in Figure 12).Finally, this vortex structure departs away from the main vortex ring, forming an independent vortex structure (as shown by the red arrow in Figure 13).

Strong interaction interval.
When the time difference between the two jets is 0.2~0.4,the interaction is strong and the vortex structure evolution mode is obviously different from that of synchronous jets.After the two vortices begin to interact, in the part where the two vortex rings are close to each other,  1 is accelerated while the  2 vortex ring is decelerated, but the deceleration is weaker than that in the critical interval case (Figure 14, circled in red).Finally, when the flow time  approaches to 1.90, the side of  1 close to  2 breaks first due to the strong interaction (shown by the red circle in Figure 17).
When the flow time is T = 0.80s, in the upstream of W 1 , a streamwise vortex structure begins to appear (indicated by the red arrow in Figure 14), and gradually develops to the downstream and the center of W 1 .Then the upstream part of the streamwise vortex is broken (shown by the red circle in Figure 15), forming two upper and lower streamwise vortices (shown by the red arrow in Figure 15).These two streamwise vortices formed from fracture continue to develop towards the downstream side and the center of W 2 .During the process, another fracture occurs (red arrows and red circles in Figure 16 mark the fracture position).Finally, four streamwise vortex structures are formed behind the two main vortex rings.It is worth noting that in the strong interaction interval, the larger the jet time difference is, the earlier the flow-direction vortex structure appears.
In the results of the strong interaction interval, it can also be observed that due to the interaction between the vortex rings, when the two vortex rings develop downstream, the center positions of the two vortices will gradually deviate from the central axis of corresponding nozzles.In other words, based on the center symmetry plane of the two nozzles (indicated by the green dotted lines in Figure 14 ~ Figure 17), a radial offset to the side of W 1 occurs.

Weak interaction interval.
When the time difference between the two jets is large enough ( > 0.6), the interaction between the two main vortex rings is not obvious for a long period of initial flow time (i.e.before the streamwise vortex structure appears) (as can be seen in Figure 18).During the later development of the main vortex rings, the acceleration effect of the  1 is more obvious than those in the critical interval and the strong interaction interval, while  2 is weakly affected by the interaction (Figure 19).
Compared with the results in the strong interaction interval, the streamwise vortex in this case appears earlier and has a stronger structure at the same flow time (T = 0.80s, shown by the red arrow in Figure 19).The streamwise vortex structure also develops downstream and in the center of W 1 (indicated by the red arrow in Figure 20) and fractures later (indicated by the red circle in Figure 20).However, the upstream ends of the two streamwise vortex structures will not develop towards the center of the W 2 vortex ring at a later time, which is different from the strong interaction interval.
Another difference is that the first fracture part of W 1 is on the side away from the W 2 , that is also where the interaction between the two main vortex rings is weaker (indicated by the red circle in Figure 20).Afterwards the closer part breaks (as shown by the red circle in Figure 21).When the two main vortex rings develop downstream, they will also shift radially to the side of W 1 , but the offset is smaller than that of a smaller interaction interval for the interaction between vortex rings is weaker.

Effect of vortex ring interaction on vortex dynamic characteristics 3.2.1. Effect of vortex ring interaction on circulation.
According to the definition of circulation, the integral of circulation in a certain area of the flow field is a macroscopic representation of the rotation strength of the fluid in this area.Figure 22 compares the variation of the circulation in different intervals downstream of the two parallel nozzles with the flow time.The nozzle corresponding to  1 is named as the  1 , and the other one is named as  2 .It can be seen that different intervals have little effect on the slope of the increasing stage (when the jets enter the flow field from the nozzle) and the maximum value of circulations.The delay of time reaching the maximum value is also consistent with the time difference of intervals.It shows that the interaction between the vortex rings has little effect on the circulation at this stage.In the strong interaction interval and weak interaction interval, as the time difference increases, the absolute value of slope of the circulation after reaching the peak becomes smaller, indicating that the dissipation of vorticity in the flow field is getting slower.However, also with the increase of the time difference, the slowing speed of the decrease of circulation becomes smaller, which shows that the enhancement of the effect of slowing down the dissipation of vorticity is getting less obvious.In the critical interval, the flow time circulation drops below that of synchronous jet during T = 1.18s ~ 2.10s, but the overall change trend is similar to the strong interaction interval.Combined with the vortex structure evolution observed above, it shows that the critical interval not only has some similar characteristics of the vortex structure evolution with the synchronous jet (for example, the vortex ring ejected first breaks very early, and the fractured part is dragged upstream to form streamwise vortex structure), but have similar dynamic characteristics (i.e., circulation) to the strong interaction interval.and weak interaction interval (green curve).and weak interaction interval (green curve).
It can be seen in Figure 23 that at the downstream of N 1 , after the fluid stops entering the flow field from the nozzle, only the circulation of strong interaction interval is higher than that before the jet ends.This indicates when the vortex system develops freely, the overall vorticity is enhanced due to the strong interaction of the vortices.In the downstream of the N 2 (Figure 24), after the fluid stops entering the flow field from the nozzle, the circulation increases in both the critical interval and the strong interaction interval cases.

Effect of vortex ring interaction on nozzle outlet plane pressure and momentum flux.
According to Krueger and Gharib 's analysis of the control volume of jet vortex rings [16], the thrust () generated by the jet can be divided into two parts: the momentum flux   () and the pressure   ().Equations for calculating the total propulsion thrust of nozzle from the pressure and momentum flux are listed as follows: In which, u 0 and p 0 are normal velocity and pressure of the nozzle outlet plane respectively, which can be obtained directly from the numerical simulation results.
Figure 25 shows the pressure on the outlet plane of N 1 and N 2 in different intervals.It can be seen that when jets start and stop, sudden pressure changes occur on the plane of the nozzle outlet.Moreover, when N 1 produces a sudden change in the outlet plane pressure due to the start and stop of the jet, it will cause a small increase in the pressure on the outlet plane of N 2 exactly at the same time (marked by the black dotted line in Figure 25).Conversely, when the N 2 produces a sudden change in the outlet plane pressure due to the start and stop of the jet, a small increase in the outlet plane pressure of N 1 will occur as well.

Effect of vortex ring interaction on total momentum of flow field.
Figure 27 shows the total streamwise momentum in the three-dimensional flow field downstream of nozzles.In general, the time difference between  1 and  2 to start the jet will affect the growth rate of the total momentum at the initial stage (before the  1 stops the jet, i.e. flow time  < 0.7773) of the downstream flow field.The larger the time difference is, the slower the momentum grows and the lower the maximum value can be reached.The variation trend of the total momentum of the critical interval is similar to that of the synchronous jet, which means this trend is not affected by the jet of  2 .While for the strong interaction interval and the weak interaction interval, after  1 finishes the jet, the jet of  2 dominates the increase of the total momentum, so that the total momentum increases and then falls back before  2 ends the jet.After both jets of the two nozzles are accomplished (i.e.no new momentum is added into the downstream flow field and the flow time  >  + ), due to the interaction between the vortex rings, the greater the jet time difference is, the faster the increase of total momentum grows.

Conclusion
Previous studies paid less attention to the interaction mechanism between vortex rings and the vortex structure evolution of the asynchronous jets with parallel dual nozzles.Therefore, this paper uses numerical simulation to study this problem.
According to the interaction mode between vortex rings, this paper divides the results into four intervals according to start-up time differences of jets, which are synchronous jet, critical interval, strong interaction interval, and weak interaction interval.The numerical result of the synchronous jet coincides with the phenomena observed by predecessors in experiments well, including the deceleration effect and the fusion of two vortex rings into a twisted vortex ring.Although the criticalinterval is one of the asynchronous jet cases, the vortex structure evolution mode and vortex dynamics are similar to those of the synchronous jet and dissipation also plays a critical role.
In the strong interaction interval, with the increase of the time difference, the effect of the main vortex ring ejected first being accelerated is getting stronger and the effect of the main vortex ring ejected later being decelerated is getting less obvious.At the same time, the streamwise vortex structure is also found in the wake of the vortex ring, and the generation of this structure takes place earlier with the increase of the time difference.In the weak interaction interval, the interaction mode between the vortex rings is similar to the strong interaction interval, but the interaction strength is significantly lower.At the same time, the streamwise vortex structure in the wake of the vortex ring is also weaker.
In addition, compared with the synchronous jet and the critical interval, the interaction between the vortex rings in the strong interaction interval and the weak interaction interval significantly inhibits the rate of the vortex dissipation in the flow field downstream of the nozzles.The asynchronous jet will also cause the pressure on the nozzle outlet plane to increase at a certain time, so that the thrust has a certain gain compared with the synchronous jet case.The total momentum downstream of the nozzles also increases at a higher rate than the synchronous jet case due to the integration of pressure over time.
The physical model of the interaction between the vortex rings considered in this paper is idealized, so future research can consider more complex situations, such as the nozzle axis being set at different distances or different angles.In addition, the Reynolds number of the case studied in this paper is relatively small, so the jet produces a laminar vortex system.However, in the real jet propulsion scenario, the Reynolds number can be quite high, and the vortex system will develop into a turbulent vortex system.Therefore, the research on the interaction of the turbulent vortex system generated by the jet is also worthy of further exploration.

Table 2 .
Comparison of parameters of grids.Experimental result from Gharib et al. [4].(b) Numerical result of Case 1. (c) Numerical result of Case 2. (d) Numerical result of Case 3.

Figure 3 .
Figure 3.Comparison between experiments and three sets of grid numerical results (vorticity contours).
(a) Experimental results and numerical results, formation time T ̂= 1.5.(b) Experimental results and numerical results, formation time T ̂= 4.1.(c)Experimental results and numerical results, formation time T ̂= 8.5.

Figure 5 .
Figure 5.Comparison of experimental and numerical results.For each figure, upper parts are the experimental results [13], and lower parts are numerical results.
(a) Overall grid outlet of the symmetrical plane.(b) Grid near the two parallel nozzles.(c)Grid of the inlet plane.

Figure 7 .
Figure 7. Structure of main vortex rings at T = 0.68s (Experimental result from Oshima et al. is on the left [15]).

Figure 8 .
Figure 8. Three views of two vortex rings fused into the twisted vortex ring (Experimental result from Oshima et al. is on the left [15]).

Figure 9 .
Figure 9.The twisted vortex ring merged from the two main vortex rings.

Figure 10 .
Figure 10.Vortex-ring interaction and structure evolution in critical interval, flow time T = 0.68s.

Figure 11 .
Figure 11.Vortex-ring interaction and structure evolution in critical interval, flow time T = 1.00s.

Figure 12 .
Figure 12.Vortex-ring interaction and structure evolution in critical interval, flow time T = 1.10s.

Figure 13 .
Figure 13.Vortex-ring interaction and structure evolution in critical interval, flow time T = 1.28s.

Figure 14 .
Figure 14.Vortex-ring interaction and structure evolution in strong interaction interval, flow time T = 0.80s.

Figure 15 .
Figure 15.Vortex-ring interaction and structure evolution in strong interaction interval, flow time T = 1.36s.

Figure 16 .
Figure 16.Vortex-ring interaction and structure evolution in strong interaction interval, flow time T = 1.60s.

Figure 17 .
Figure 17.Vortex-ring interaction and structure evolution in strong interaction interval, flow time T = 1.90s.

Figure 18 .
Figure 18.Vortex-ring interaction and structure evolution in weak interaction interval, flow time T = 0.80s.

Figure 19 .
Figure 19.Vortex-ring interaction and structure evolution in weak interaction interval, flow time T = 1.52s.

Figure 20 .
Figure 20.Vortex-ring interaction and structure evolution in weak interaction interval, flow time T = 2.08s.

Figure 21 .
Figure 21.Vortex-ring interaction and structure evolution in weak interaction interval, flow time T = 2.14s.

Figure 22 .
Figure 22.Variation of circulation with flow time on the symmetry plane downstream of nozzles (encircled by yellow dashed lines) of synchronous jet (blue curve), critical interval (red curve), strong interaction interval (yellow curve) and weak interaction interval (green curve).

Figure 23 .Figure 24 .
Figure 23.Variation of circulation with flow time on the symmetry plane downstream of N 1 (encircled by yellow dashed lines) of synchronous jet (blue curve), critical interval (red curve), strong interaction interval (yellow curve) Figure 24.Variation of circulation with flow time on the symmetry plane downstream of N 2 (encircled by yellow dashed lines) of synchronous jet (blue curve), critical interval (red curve), strong interaction interval (yellow curve)

Figure 25 .
Figure 25.Variation of pressure with flow time on the outlet plane of nozzles, above for N 1 and below for N 2 .

Figure 26
Figure26shows velocity flux at the outlet plane of N 2 in different intervals.It is found that the jet of N 1 does not cause a change in the N 2 outlet velocity flux.That is, there is only a time difference between velocity flux images of the N 2 in different intervals, which is consistent with the jet start time difference.The result shows that either the synchronous jet or the asynchronous jet has no obvious influence on the variation and the momentum flux at the nozzle outlet.

Figure 26 .
Figure 26.Variation of velocity flux with flow time on the outlet plane N 2 of synchronous jet (blue curve), critical interval (red curve), strong interaction interval (yellow curve) and weak interaction interval (green curve).

Figure 27 .
Figure 27.Variation of total streamwise momentum with flow time in the flow field (indicated by green dotted lines) downstream of nozzles of synchronous jet (blue curve), critical interval (red curve), strong interaction interval (yellow curve) and weak interaction interval (green curve).

Table 3 .
Comparison of grids with different parameters.