Thrust and flow characteristic of double synthetic jet actuator underwater

Ocean Engineering 176 (2019) 84–96

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Thrust and flow characteristic of double synthetic jet actuator underwater Lingbo Geng a b

a,b,∗

a

, Zhiqiang Hu , Yang Lin

T

a

The State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, 110016, China University of Chinese Academy of Sciences, Beijing, 100049, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Double synthetic jet Thrust vectoring Numerical method

In this paper, underwater thrust vectoring method based on double synthetic jet actuator is studied. The thrust characteristic of the system under different nozzle distances and phase lags are numerically investigated. The numerical method is validated using data in the literature. The average normal thrust increases when nozzle distance increases. The average tangential thrust decreases when nozzle distance increases. The jet interacts with each other strongly under small nozzle distances. When the two actuators are far enough, the interaction between adjacent jets is negligible. The average normal thrust keeps nearly invariant when phase lag varies. The average tangential thrust increases when phase lag increases from 0 to π and decreases when phase lag increases from π to 2π. The variation of the thrust vectoring angle has the same tendency with the average tangential thrust. The velocity vector and the vorticity contour are given. The mathematical model of the average normal and average tangential thrust is also established.

1. Introduction With the repaid development of underwater missions, it poses much higher demands on the maneuverability of underwater robots. The traditional way for underwater maneuvering is using rudders. This method is very efficient at cruising speed. However, when the speed is low, the steering force provided by the rudders is significantly reduced. On the other hand, the protruded rudders can induce significant resistance. The other way for underwater thrust vectoring is using special designed mechanical system. These mechanical systems can provide the propulsor additional degree of freedom (Xin et al., 2013; Cavallo and Michelini, 2004; Kopman et al., 2012). So using only one propulsor, thrust in different directions can be realized. Ba Xin, Luo Xiaohui (Xin et al., 2013) proposed a vectored water jet propulsor adopting 3RPS parallel mechanism. Emanuele Cavallo and Rinaldo Michelini adopted spherical parallel mechanism to control the thrust in all directions (Cavallo and Michelini, 2004). Using multiple thrusters or special designed mechanisms will increase the weight and complexity of the propulsion system. It will also induce significant resistance. There are many marine creatures propelled by jets, such as squid and jellyfish. Inspired by nature, Krueger (Krueger and Gharib, 2003, 2005), Whittlesey (Robert and John, 2013; Lydia et al., 2011), Krieg & Mohseni (Krieg and Mohseni, 2008, 2010; Mohseni, 2006) proposed a new type of underwater propulsion

∗

technique called synthetic jet. The main merit of synthetic jet lies in its high efficiency. Lydia A. Ruiz (Lydia et al., 2011) found through experiment that pulsed jet has a propulsion efficiency 40% higher than that of steady jet, and the drag-based hydrodynamic efficiency increase more than 70%. Michael Krieg and Kamran Mohseni (Krieg and Mohseni, 2008, 2010) developed an underwater robot with synthetic jet actuator mounted inside. The experiment showed the synthetic jet actuator induced the robot rotation of 2.23r/min. The feasibility of synthetic jet in the steering of underwater robots is validated. The numerical simulation of synthetic jet is also intensively investigated. There are several methods can be used to simulate turbulent flow, including direct numerical simulation method (DNS), large eddy simulation method (LES), and Reynolds averaged numerical simulation (RANS). Ravi, Mittal and Najjar (Ravi et al., 2004) used DNS to study the effect of slot aspect-ratio on the formation and evolution of synthetic jets in quiescent and non-quiescent external flow. Rizzetta, Visbal and Stanek (Rizzetta et al., 1999) investigated two- and three-dimensional flow fields of finite aspect-ratio synthetic jets using DNS. Sau and Mahesh (2008) used DNS to study the effect of crossflow on the dynamics, entrainment and mixing characteristics of vortex rings issuing from a circular nozzle. Wang, Yuan (Wang et al., 2005) studied an active cooling substrate (ACS) device which implements synthetic jet to enhance thermal management using LES. You, Moin (You and Moin, 2008) studied and evaluated the effectiveness of synthetic jets as a separation control technique over an airfoil. Wu, Leschziner (Wu and

Corresponding author. The state key laboratory of robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, 110016, China. E-mail address: [email protected] (L. Geng).

https://doi.org/10.1016/j.oceaneng.2019.02.036 Received 13 June 2017; Received in revised form 12 November 2018; Accepted 10 February 2019 Available online 23 February 2019 0029-8018/ © 2019 Elsevier Ltd. All rights reserved.

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Synthetic jet actuator

Leschziner, 2008) investigated the physical processes involved in the injection of a synthetic jet in quiescent surroundings using LES. From the view point of computational cost, LES and DNS at high Reynolds number flows requires huge computing resources, although it could provide higher fidelity details of turbulent flow-fields. At the same time, URANS simulation combined with adequate turbulence models such as the k–w SST turbulent model could provide reasonably good solutions (Rumsey et al., 2004). The obvious attraction of RANS lies in its economy: it gives a phase-averaged representation of the flow, and thus allows far larger time steps to be used in the computational solution; and it permits a coarser mesh to be used, as the phase-averaged fields of statistical quantities vary much more smoothly than the field of instantaneous turbulent motions. Jain, Puranik (Jain et al., 2011) studied the influence of the geometrical parameters of the cavity and the nozzle on the flow characteristics of synthetic jet using RANS. The actuation frequency of the synthetic jet in this study is as high as 500 Hz. Lv, Zhang (Lv et al., 2014) studied the influence of the actuation frequency on the fluidic characteristics of synthetic jet using SST kw turbulence model. Xu, Zhou (Xu and Zhou, 2015) studied the pitching stability of a blended wing body aircraft controlled by synthetic jets using SST k-w turbulence model. Duvigneau, Visonneau (Duvigneau and Visonneau, 2006) simulated and optimized the stall control of an airfoil using synthetic jet through RANS method. For underwater propulsion, thrust is crucial. The thrust of synthetic jet underwater is previously studied by Anderson & DeMont (Anderson and DeMont, 2000), Krieg & Mohseni (Krieg and Mohseni, 2013, 2015) and Krueger (2005). Anderson & DeMont quantified the contribution of the inside fluid acceleration when studying the squid jetting. The pressure term was identified by Kruger and modeled by the potential of a flat plate moving with the vortex ring. The pressure term was also quantified with respect to jet velocity profiles in that study. The forming of the vortex plays a big role in the thrust of synthetic jet underwater. Many experimental findings showed that the existence of vortex can enhance the thrust compared with the steady jet. Krieg & Mohseni studied the contribution of radial velocity gradient to thrust enhancement of synthetic jet. The radial velocity gradient is actually induced by the formation of the vortex. Geng, Hu & Lin (Geng and HuYang, 2018) studied the thrust characteristic of synthetic jet underwater using both experiment and numerical method. The thrust of synthetic jet actuator can be accurately predicted using numerical method adopted. Geng, Hu & Lin also modeled the thrust based on the potential flow theory. The thrust is modeled using three different parts which is jet momentum, jet acceleration and pressure force respectively. In this paper, a novel underwater maneuvering method based on double synthetic jet actuator is studied. Through adjusting the phase lags between these two actuators, thrust vectoring can be realized. The main contribution of this paper includes: 1) The thrust characteristic of the double actuator system under different nozzle distances and different phase lags is numerically studied. The numerical method is validated using experimental data in the literature. The numerical result validates the feasibility of thrust vectoring based on phase differences between double adjacent jets. 2) The velocity vector and vorticity contour are given to assistant the analysis. 3) The model of the normal and tangential thrust is given. The variation of the normal and tangential thrust can be well predicted by the model.

Normal

Tangential

Fig. 1. Double synthetic jet actuator.

of the actuator will be mainly in the normal direction. As the motion of the diaphragm is unsteady, there exists fluid acceleration in the tangential direction. As a result, the tangential thrust will not be zero. If the cavity is circular, the total fluid acceleration in the tangential direction will be zero resulting in the zero tangential thrust. So the design of the cavity in this paper can generate nonzero tangential thrust. As a result, thrust vectoring can be realized. The thrust vectoring angle can be controlled through adjusting the phase lags between the two actuators. The diaphragm of the actuator locates at the bottom of the actuator. It is oscillating up and down in the sinusoidal mode:

V (t ) = 2πfr am cos(2πfr t )

(1)

am stands for the amplitude which is 5 mm, fr stands for the frequency which is 10 Hz. The main geometric parameters of the synthetic jet actuator include diameter and thickness of the orifice, as well as diameter and depth of the cavity. The definition of the geometrical parameters is given in Fig. 2. And the value of geometrical parameters is given in Table 1. The average velocity of the jet flow can be computed as (Xia and Mohseni, 2016):

Vjet =

2 fr L

(2)

L stands for the height of the fluid column ejected by the actuator in one cycle. Using (2), the Reynolds number is:

Rejet =

ρVjet D μ

(3)

Using geometrical parameters given in Table 1, the Reynolds number is 3.5 × 105. As the RANS method can achieve satisfying accuracy with relatively low computation burden, it is chosen in this paper to simulate the turbulent jet flow. The computation is carried out by using the commercial CFD software FLUENT coupled with the user definition function (UDF) describing the diaphragm movement. Pressure-implicit with splitting of operators (PISO) based on a higher degree of approximation between the iterative corrections for pressure and velocity is chosen. Second-order upwind spatial discretization is used for the momentum, turbulent kinetic energy, and turbulent dissipation rate. SST k–w turbulence model is used to account for the turbulent nature of the synthetic jet flows. The length of the double actuator system is much larger than the width of the nozzle. So the 3D problem can be computed using 2D mesh. The mesh adopted is given in Fig. 2 (b). The nozzle is resolved with quadrilateral elements. The bottom part of the cavity is meshed with quadrilateral elements to allow for relative displacement among the nodes on the piston. All the other parts are resolved with triangular elements. The domain near the nozzle has a big influence on the development of the jet. To better simulate the jet interaction near the nozzle, a refinement region near the nozzle is created. The width and height of the refinement region is l1 and l2 respectively as given in Fig. 2 (a). The value of l1 and l2 is given in table using the nozzle diameter D. The domain far away from the nozzle has little influence on jet development. Meshes in this region can be coarser. When jet is ejected outside of the cavity, the boundary layer near the nozzle wall will evolves into vortexes. To accurately simulate the formation and development of vortexes, the element height near the nozzle wall is carefully designed. The element height of the boundary layer can be

2. Numerical method The double synthetic jet actuator system is depicted in Fig. 1. The traditional synthetic jet actuators are mostly circular with the nozzle locating at the center of the cavity. The synthetic jet actuator in this paper is rectangular and the nozzle locates at one side of the cavity. This design is mainly based on the consideration of thrust vectoring. The normal and tangential directions are defined in Fig. 1. The normal direction is parallel with the jet velocity. As a result, the thrust 85

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b2 Pressure outlet

Pressure outlet

Pressure outlet

l1

b1 l2

d h

Wall

Wall

D H

Wall

Moving wall

Dcav

(a)

Quadrilateral elements

Boundary layer

Deforming region

Quadrilateral elements

(b) Fig. 2. (a)Geometry of the actuator (b)Boundary condition and geometrical parameters (c)Mesh.

computed using dimensionless wall distance y+, which is given as y+ = Δyρμτ / μ . For SST k–w turbulence model, y+ should be smaller than 2. y+ < 2 is just a criterion to compute the initial guess of the element height. The real value is designed through trial and error. In this paper, after several trials, the first element's height of the boundary layer is confirmed to be 0.01 mm. The boundary condition is depicted in Fig. 2 (a). The driving diaphragm is modeled using moving wall. The motion of the moving wall is governed by (1) using UDF (User Defined Function). The boundary connecting to the moving wall is set to be deforming wall to allow for relative displacement. The height of the deforming wall and the elements in the deforming region can adjust automatically based on the motion of the moving boundary. The AUV surface is modeled with noslipping wall. The other boundaries of the outer flow domain are set to be pressure outlet. The relative static pressure of the outlet is set to be 0

Table 1 Geometrical parameters (mm). Parameter

Value

D h Dcav H b1 b2 l1 l2

10 D 5D 4D 80 D 100 D 40D 30D

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Sout

S refine Sbl

Fig. 4. Normal and tangential thrust under different grid resolutions. I

Snozzle

IV

III

Fig. 3. Mesh size in different zones.

(b)t=0

(e)t=T/2

Fine Coarse

Srefine

Sout

1 2

0.5 1

0.005 0.01

1 2

4 8

II

IV

(f) t=T/2

III

II

IV

(g)t=3T/4

III

IV

(h) t=3T/4

Fig. 5 (a)、(c)、(e)、(g) are results given in Ref. 28 and (b)、(d)、(f)、 (h) are results in this paper. Fig. 5(a) and (b) gives vorticity contour when t = 0. There are four pairs of vortexes in the flow field. Vortex I and vortex II are outside of the cavity. The position and the structure of these two vortexes given by Sajad Alimohammadi is close to numerical result in this paper. The tilting of the vortex I is accurately predicted. Vortex III and IV is inside of the cavity. Vortex III and vortex IV belongs to the current cycle. The vortex far from the centerline is much larger in size than the vortex close to the centerline. The room for the vortex near the centerline is very limited. As a result, the development of this vortex is restrained. The vortex contour when t = T/4 is given in Fig. 5(c) and (d). It can be seen the vortexes of the two adjacent actuators strongly interact with each other. There are three pairs of vortexes outside of the cavity. Each vortex pair is composed of two vortexes with opposite vorticity. Vortex pair II and III merges with each other forming a larger vortex. The interaction between these two vortexes and the structure of the interacted vortexes are close in Fig. 5(c) and (d). The two pairs of vortexes inside of the cavity also resemble each other. When t = T/2, the interaction between the two pairs of vortexes stopped. And the vortex of the left actuator, vortex I in Fig. 5(e) and (f) almost disappears. The tilting of vortex II is accurately predicted in Fig. 5 (f). The structure of the two pairs of vortexes inside the cavity is also close in Fig. 5(e) and (f). When t = 3T/4, the vortex of the right actuator starts to detach from the nozzle as given in Fig. 5(g) and (h). The tilting and the structure of the vortex are generally accurate. From results and discussion given above, it can be known the prediction accuracy of vortexes far away from the nozzle and vortexes with weak intensity is generally poor. Despite of this, the tilting, the structure and the interaction of the main vortexes can all be predicted with satisfactory accuracy. The thrust characteristic of the isolated synthetic jet actuator is intensively studied. The thrust of isolated synthetic jet actuator can be

Table 2 Grid size of the fine and coarse mesh (mm). Sbl

I

Fig. 5. Vorticity (ωD/U) contour when phase lag is π/3. (a)(c)(e)(g) is results given in Ref. (Alimohammadi et al., 2016) and (b)(d)(f)(h) is results in this paper.

The dependence of the numerical method on the grid resolution is studied in this paper. Two types of grids with different grid resolution are used to compute the tangential and normal thrust of the system. The grid size in different zones is defined in Fig. 3. Scav, Snozzle, Sbl, Srefine, Sout are the grid size of the cavity, the nozzle, the boundary layer, the refinement region and the far region respectively. The grid size of the fine and coarse mesh is given in Table 2. The tangential and normal thrust under different grid resolutions is depicted in Fig. 4. It can be seen, the thrust under different grid resolutions is close to each other. So the influence of the grid resolution on the result can be ignored. The numerical simulation of the isolated synthetic jets has been intensively investigated. When two synthetic jets are placed side by side, the flow field of these two adjacent synthetic jets will interact strongly with each other. As a result, the flow field of two adjacent synthetic jets is more complicated than that of single isolated synthetic jet. In order to validate the effectiveness of the numerical method adopted in the simulation of adjacent synthetic jets, the vorticity contour is compared with result given by Sajad Alimohammadi (Alimohammadi et al., 2016). Sajad Alimohammadi et al. studied the flow vectoring phenomenon in adjacent synthetic jets using CFD and PIV. For comparison, the actuator with the same geometrical parameters and boundary conditions with Ref. 28 is simulated using numerical method in this paper. The vorticity contour when phase lag is 60。is used for comparison.

Snozzle

III

(d) t=T/4

I

II

IV

V

(c)t=T/4 I

II

2.1. Numerical method validation

Scav

IV IV

V

III

I

III

III

III

IV

(a)t=0

to simulate the ambient environment which is far away from the nozzle. The turbulence intensity of the outlet is set to be 1%. The remaining boundaries of the actuator are set to be no-slipping wall. The default under-relaxation parameters are kept at 0.3, 1.0 and 0.7 for pressure, density, and momentum respectively. The default of 120 maximum iterations per time step is kept. A time step of 1/(200f) is chosen based on the actuation frequency to allow for 200 time steps per cycle. At the beginning of the simulation, the flow field has not convergence. So, in this paper, all the data are sampled after three cycles of computation.

II

II

II

II

Scav

I

I

I

87

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Fig. 6 (a), the thrust at t = T/4 and t = 3T/4 increases in magnitude when nozzle distance increases. As fluid interaction is stronger under small nozzle distance, this means the fluid interaction between the adjacent jets will reduce the acceleration effect. The thrust peak at t = T/2 and t = T is generated through jet velocity. The thrust can be expressed as:

well predicted using numerical method in this paper as validated in our previous studies (Geng and HuYang, 2018). The thrust of synthetic jet was measured using 6 DOF force sensor. The force measured by the force sensor includes not just the thrust of the jet but also the inertial and friction force of the system when it is running. To compute the thrust of the jet, the experiment is carried both in air and underwater (only part of the cavity is underwater). The inertial and friction force of the system generated by the reciprocal motion of the piston is independent of the medium. The thrust of the jet in the air can be neglected. In this case, the difference between the force in the air and underwater is the thrust of synthetic jet underwater (Geng and HuYang, 2018). As modeled by Geng Lingbo et al. (Geng and HuYang, 2018), the thrust of synthetic jet can be expressed as:

FT = k Qm V + M a cc +

1 k|Qm V| 2

T = 0.5k (|Qm V|)l→ n + (Qm V)l + (Qm V)r + 0.5k ′ (|Qm V|)r→ n

Qm is the mass flow rate and V is the average jet velocity. k is a → parameter describing the velocity distribution at the nozzle exit. n is → the outer normal vector of the nozzle plane. k (|Qm V|) n stands for the pressure force. ()l stands for the left actuator and ()r stands for the right actuator. As the geometrical and actuation parameters of these two actuators are identical, the mass flow rate and average velocity are the same in magnitude but reverse in direction. So, (Qm V)l + (Qm V)r is 0. (6) can be simplified as:

(4)

V and acc stands for the average velocity and average acceleration at the nozzle exit. Qm is the average mass flow rate. M is the total mass accelerated by the actuator. k is the parameter describing the velocity distribution at the nozzle exit. The interaction between jets will alternate the velocity distribution at the nozzle exit. As a result, the value of k changes. The velocity distribution mainly depends on the near field and the main vortex structures. The vortex far away from the nozzle and vortex with weak intensity has little effect on the velocity distribution. As results given in Fig. 5, the numerical method in this paper can accurately predict the vortex in the near field. In this case, the thrust of adjacent synthetic jets can also be accurately predicted.

T = 0.5k (|Qm V|)l→ n + 0.5k ′ (|Qm V|)r→ n

(7)

The mass flow rate Qm and the average velocity V don't depend on nozzle distance. So, thrust at t = T/2 and t = T is close in different cases. Using (5) and (7), the normal thrust can be expressed as:

T = 0.5k (|Qm V|)l→ n + 0.5k ′ (|Qm V|)r→ n + Ml accl + Mr accr

(8)

From Fig. 6 (a), the direction of the normal thrust never change in → one period. The direction of n doesn't change with time. So the first two parts of (8) has constant direction. The last two parts of (8) change direction every half period. This means the normal thrust mainly depends on the pressure force. This is evident in Fig. 6 (a), as the thrust at t = T (t = T/2) is much larger than thrust at t = T/4. The average normal thrust is given in Fig. 6(c). The average normal thrust increases in magnitude with nozzle distance. The variation of the average normal thrust is mainly induced by the acceleration term. The interaction between the adjacent jets will undermine the acceleration effect. The tangential thrust under different nozzle distances is given in Fig. 6(b). Tangential thrust has two peaks locating at T/4 and 3T/4 respectively. From (1), at t = T/4 and t = 3T/4 the acceleration is largest. This means the tangential thrust is mainly generated through fluid acceleration. As a result, the tangential thrust has just two peaks locates at 0.25T and 0.75T respectively. The first peak at t = T/4 decreases in magnitude with nozzle distance. The variation of the second peak at t = 3T/4 is not obvious. The average tangential thrust is given in Fig. 6(d). The average tangential thrust decrease with the nozzle distance. When nozzle distance is small, the interaction between jets is strong. This means the fluid interaction between the adjacent jets can enhance the tangential thrust. To better understand the mechanism behind this, velocity vector under different nozzle distances is given in Fig. 7–10. The velocity vector when nozzle distance is 5 mm is given in Fig. 7. When d = 5 mm the interaction between the two adjacent jets is very strong. There are two pairs of vortexes in the flow field. Vortex I and II belongs to the left jet, SJ-A for simplicity. Vortex I′ and II′ belongs to the right jet, SJ-B for simplicity. When vortex I′ and II′ forms, Vortex I and II flows far away from the nozzle. When SJ-B expels fluid, SJ-A is in suction cycle. The suction of SJ-A will deflect vortex I′ and II′ towards SJ-A as depicted in Fig. 7(b). The suction of SJ-A will strengthen the intensity of vortex II’. So, the strength of vortex II′ is much stronger than that of vortex I′ when these two vortexes flows downstream. At t = 2T/ 3, vortex II′ locates at the left side of SJ-A. At this time, SJ-A starts to expel fluid and vortex I and vortex II starts to form. The direction of vortex I and vortex II′ is the same. In this case, vortex I and vortex II′ will merge as given in Fig. 7(e). On the other hand, vortex I′ almost disappear. Because of the existence of vortex II′, fluid is continuously drawn from SJ-A into vortex II’. Meanwhile, SJ-B also draws fluid from SJ-A as SJ-B is in its suction cycle. The suction of vortex II’ and SJ-B is

3. Results and analysis 3.1. The effect of nozzle distance The effect of the nozzle distance is studied in this paper. The parameter d is defined to be the distance between the centerline of nozzles. The definition of the geometrical parameters is depicted in Fig. 2. The normal and tangential thrust under different nozzle distances is depicted in Fig. 6. The phase lag between these two actuators is fixed to be π. The peak value of the normal thrust changes significantly, about 50%, when the distance increases from 10 mm to 15 mm. When the nozzle distance continues to increase from 15 mm, the peak value of the normal thrust changes slowly. The normal thrust constitutes of four peaks in one period locating at t = 0, t = T/4, t = T/2 and t = 3T/4 respectively. From (1), the velocity of the piston is zero at t = T/4 and t = 3T/4. The velocity of the piston is largest at t = T and t = T/2. So, the thrust peak at t = T/4 and t = 3T/4 is generated through fluid acceleration. From Fig. 7 (a), the peak at t = T/4 and t = 3T/4 increase in magnitude when nozzle distance increases. This means the contribution of fluid acceleration to normal thrust increase when nozzle distance increases. The phase lag between these two actuators is π, which means the jet velocity is exactly reverse in direction. The normal thrust at t = T/4 and t = 3T/4 can be expressed as:

T = Ml accl + Mr accr

(6)

(5)

Ml 、Mr is the mass of fluid accelerated by the left and right actuator, V˙l 、V˙r is the average fluid acceleration of the left and right actuator respectively. accl and accr are of reverse direction. From Fig. 6 (a), the thrust at t = T/4 and t = 3T/4 is nonzero. This means the net effect of fluid acceleration for this dual synthetic jet actuator is not zero. As the two actuators are reverse in phase, this means the fluid acceleration during suction cycle and stroke cycle is not simply reverse of direction. In other words, the flow field during stroke cycle and suction cycle is not symmetrical. As the two actuators have identical geometrical and actuation parameters, accl will become accr after half period. So, the period of the acceleration force is T/2. This can be seen from Fig. 6 (a). In different cases, the thrust at t = T/4 and t = 3T/4 is very close. From 88

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(a)Normal thrust

(b)Tangential thrust

(c)Average normal thrust

(d)Average tangential thrust

Fig. 6. Normal and tangential thrust under different nozzle distances.

for the dual actuator system. When nozzle distance increases to 80 mm, the fluid interaction between these two actuators is negligible. The velocity distribution for these two actuators is nearly the same, as can be seen from the right plot of figure and the left plot of Fig. 12. From velocity vectors given above, it is evident that the jet will deflect towards the actuator leading in phase. In this paper, the actuator first ejects fluids and then switches to suction cycle. When one actuator, SJ-A for simplicity, is leading in phase, it will switches to suction cycle earlier than the other one, denoted as SJ-B. In this case, the jet ejected by SJ-B will be sucked by SJ-A. Jet B will be deflected as a result. The vortex of jet B will also deflect toward SJ-A. When SJ-A begins its stroke cycle and SJ-B switches to suction cycle, the vortex of jet B ejected during the last stroke cycle will restrain the deflection of jet A. In this case, the deflection of jet A is weaker than that of jet B. The net deflection of the jet is toward jet A, the one leading in phase. When nozzle distance is small, the net deflection angle is larger. The deflection of the jet will contribute to the tangential thrust besides the acceleration effect. In this case, the tangential thrust is larger, as given in Fig. 6 (b). The decrease of the average normal thrust is mainly induced by the acceleration term as can be seen from Fig. 6(a). Thrust at t = 0 and → t = T/2 is generated through pressure force, k (|Qm V|) n . On the other hand, thrust at t = T/4 and t = 3T/4 is generated through fluid acceleration. From Fig. 6(a), thrust at t = 0 and t = T/2 doesn't change while thrust at t = T/4 and t = 3T/4 increases with nozzle distance especially when nozzle distance is small. From velocity vector given,

reverse in direction. As a result, the deflection of SJ-A is reduced. The deflection angle of SJ-A is much smaller than that of SJ-B. So, the net deflection angle of these two adjacent jets is not zero. This is consistent with the result given by Sajad Alimohammadi (Alimohammadi et al., 2016). From result given by Sajad Alimohammadi, the jet always deflects towards the same direction despite the suction of the other actuator. As the jets always deflect toward the same direction, the net tangential thrust generated by the jet deflection is not zero. This is the mechanism behind the generation of net tangential thrust. When nozzle distance increases to 20 mm (Fig. 8), the interaction between the two adjacent jets decreases. The deflection angle of SJ-A is still smaller than that of SJ-B. The net deflection angle and the net tangential thrust generated by jet deflection are nonzero. When nozzle distance increases to 50 mm (Fig. 9), the interaction becomes very weak. The deflection angle of the jet becomes very small. The tangential thrust generated by jet deflection will be neglected. In this case, the tangential thrust is mainly generated through fluid acceleration. The velocity distribution at t = T/2 and t = T is given in Figs. 10–12. The left plot in Fig. 10 gives the velocity distribution of the left actuator which is now in its suction cycle. The right plot in Fig. 10 corresponds to the stroke cycle of the right actuator. When t = T the left actuator switches to stroke cycle while the right one is in suction cycle. When nozzle distance is small, the velocity distribution of the left and right actuator is different, as can be seen from right plot of Fig. 10(a) and left plot of Fig. 10(b). This means the flow field is not symmetrical 89

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(a)t=T/6

(b)t=2T/6

(d)t=4T/6

(c) t=3T/6

(e)t=5T/6

(f) t=T

Fig. 7. Velocity vector when d = 5 mm.

away from each other, the interaction is negligible. Every actuator will accelerate part of fluid independently. The total accelerated fluid increases.

when nozzle distance is small, the jet of SJ-A will be sucked by SJ-B. The smaller is the distance, more jet will be sucked. Synthetic jet is pulsed jet. The fluid is frequently accelerated/decelerated by the actuator. The acceleration force is proportional to the total mass accelerated by the dual synthetic jet actuator. When jet A is sucked by jet B, the total accelerated fluid decreases. As a result, the thrust at t = T/4 and t = 3T/4 decrease in magnitude. When the two actuators are far

3.2. The effect of phase lag In this section, the effect of phase lag is studied. The nozzle distance

(a)t=T/6

(b)t=2T/6

(c) t=3T/6

(c)t=4T/6

(d)t=5T/6

(e) t=T

Fig. 8. Velocity vector when d = 20 mm. 90

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(a)t=T/6

(c)t=4T/6

(b)t=2T/6

(d)t=5T/6

(c) t=3T/6

(e) t=T

Fig. 9. Velocity vector when d = 50 mm.

(a)t=T/2

(b)t=T Fig. 10. Velocity distribution when d = 5 mm. 91

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(a)t=T/2

(b)t=T Fig. 11. Velocity distribution when d = 20 mm.

in this section is fixed to be 10 mm. The thrust variation and vorticity contour under different phase lags are given. Thrust model which can precisely predict thrust variation is established. The thrust is modeled using potential flow theory. Thrust generation of double synthetic jet actuator is complicated with physical processes including vortex generation, fluid acceleration and jet interaction. All these processes are difficult to model precisely. The focus of this study is to investigate the characteristic of the double synthetic jet system. The thrust model only serves to predict the thrust variation. So, it is established using potential flow theory with many simplifications. Only the main physical parameters affecting the thrust variation is modeled. The thrust model can precisely predict the thrust variation as given in the following paragraphs. However, it can't be used to compute the magnitude of the thrust. The variation of the normal and tangential thrust under different phase lags, denoted as θ, is given in Fig. 13 and 14 respectively. The average thrust of synthetic jet can be computed as (Krieg and Mohseni, 2013, 2015):

π3 T¯dou = 2T¯ = ρ D 2fr2 L2 8

(9) is the average thrust of circular synthetic jet. L is the height of the fluid column expelled in one period by the actuator. fr is the driving frequency of the actuator. In this paper, the actuator is rectangular. In this case, the average thrust should be computed as:

π2 T¯dou = ρ fr2 L2Dl en 2

(10)

D stands for the nozzle width and len stands for the nozzle length. All the data points in this paper are normalized using (10). From Fig. 13, the peak of the tangential thrust first increase when phase lag increase from π/6 to π and then decrease when phase lag increase from π to 2π. The tangential thrust of the system can be expressed as:

T = Ml accl − Mr accr = M (sin(2πfr t ) − sin(2πfr t + θ))

(11)

The variation of (11) with M = 1 is depicted in Fig. 15. When θ and t vary, the value of (11) composes a 3-D surface. The red color represents positive thrust. While the blue color represents negative thrust. The Z -θ plane of the 3D surface is given in Fig. 15 (a). The amplitude of the tangential thrust corresponds to the chord length of the curve in Fig. 15 (b). It can be seen, the chord length increases when 0 < θ < π, and decreases when π < θ < 2π. This is consistent with the variation of

(9) 92

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(a)t=T/2

(b)t=T Fig. 12. Velocity distribution when d = 50 mm.

Fig. 14. Normal thrust under different phase lags.

the amplitude of the tangential thrust. The normal thrust can be expressed as:

T = (Qml Vl + k|Qml Vl |→ n + Qmr Vr + k|Qmr Vr |→ n) = k ′ (Qml Vl + Qmr Vr ) V

V

= k ′ [cos2 (2πfr t ) ‖Vl ‖ + cos2 (2πfr t + θ) ‖Vr ‖ ] l

= k ′ cos2 (2πfr t ) Fig. 13. Tangential thrust under different phase lags.

cos(2πfr t ) ‖ cos(2πfr t )‖

r

cos(2πf t + θ )

+ k ′ cos2 (2πfr t + θ) ‖ cos(2πfr t + θ)‖ r

(12)

The variation of (12) with k ′ = 1 is depicted in Fig. 16. Similar to 93

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(a)3D surface

(a) 3D surface

plane

(b)

Fig. 15. (a) All the data points constitute a 3-D surface (b) The variation of tangential thrust amplitude can be represented by the chord length of the curve in the Z-θ plane.

plane

Fig. 16. (a) All the data points constitute a 3-D surface (b) The variation of normal thrust amplitude can be represented by the chord length of the curve in the Z-θ plane.

(11), all the values of (12) constitute a 3D surface as depicted in Fig. 16 (a). The amplitude of the normal thrust corresponds to the chord length of the curve depicted in Fig. 16 (b). It can be seen, the chord length decrease when 0 < θ < π, and increase when π < θ < 2π. This is consistent with the variation of the amplitude of the normal thrust as depicted in Fig. 14. The variation of the average thrust is also studied in this paper. The total momentum variation of the system in one period is negligible based on the fact that the driving profile of the diaphragm is symmetrical. So the average thrust of synthetic jet is mainly generated through pressure force. As a result, average normal thrust of the double actuator system under phase lag of θ can be expressed as:

Table 3 Normalized average thrust and vectoring angle under different phase lags.

T 1 T¯nor = ∫0 k ( 2 |Qm V|) dt T

θ(degree)

T¯nor

T¯nor

α

0 π/6 π/3 π/2 2π/3 5π/6 π 7π/6 4π/3 3π/2 5π/3 11π/12

0 0.0265 0.173 0.207 0.309 0.360 0.382 0.332 0.307 0.181 0.16 0.08

−1.88 −1.853 −1.849 −1.837 −1.822 −1.814 −1.816 −1.828 −1.849 −1.867 −1.881 −1.876

0 0.82 5.34 6.43 9.65 11.22 11.88 10.3 9.42 5.54 4.86 2.4

=2ρπ 2fr2 am2 Spis k∫0 [cos2 (2πfr t ) + cos2 (2πfr t + θ)] dt 1

= 2 ρπ 2fr am2 Spis k π2

T /2 T T¯tan = ∫0 (Ml accl − Mr accr ) dt + ∫T /2 (Ml accl − Mr accr ) dt

π2

=kρ 2 fr am2 σS = kρ 2 fr am2 σlen D

(13)

T /2

=∫0

M (sin(2πfr t ) − sin(2πfr t + θ))

T + T /2 M′ (sin(2πfr t ) − sin(2πfr t + θ)) M =− 2πf [cos(2πfr t ) − cos(2πfr t + θ)]T0 /2 r

Spis is the area of the piston. S is the area of the nozzle. σ is the area ratio of the piston to the nozzle. len is the length of the nozzle and D is the width of the nozzle. The derivation is based on the relation Qm = ρ × Spis × V. (13) shows the average normal thrust is independent of θ. The average normal thrust under different phase lags is given in Table 3. Similar to thrust in Figs. 13 and 14, all the data points in table are normalized using (10). From Table 3, it can be known the average normal thrust is close to each other. The difference between the largest and the smallest value is less than 5%. So the average normal thrust can be taken as invariant under different phase lags. This is consistent with (13). The average tangential thrust can be expressed as:

∫

M′ 2πfr

−

[cos(2πfr t ) − cos(2πfr t + θ)]TT /2

= 2πf″ (2 − 2 cos θ) M

r

(14)

M is the mass of fluid accelerated during suction cycle and M′ is the mass of fluid accelerated during stroke cycle. From (14), it can be known when θ = 0, the average tangential thrust is 0. The average tangential thrust increases when 0 < θ < π. When phase lag is larger than π, the average tangential thrust starts to decrease. This is 94

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(a)

0

(e)

(b)

(c)

(f)

(d)

(g)

Fig. 17. Vorticity (ωD/U) contour under different phase lags.

using red arrow in Fig. 14. This means the contribution of the acceleration force increases. The thrust peak also decreases in magnitude indicating the weaker contribution of the fluid acceleration than the jet velocity. The magnitude of the thrust peak is smallest when phase lag is PI. The vorticity contour under different phase lags is given in Fig. 17. When phase lag is 0, the jet never deflects. The flow field is symmetrical about the centerline of the dual actuator system. With the increase of the phase lag, one jet will deflect due to the suction of the other actuator. The deflection angle of the jet leading in phase is smaller than the other one. Thus the net deflection angle is not zero. The jet deflects towards the actuator leading in phase. The larger is the phase lag, the larger is the deflection angle as can be seen from Fig. 17. The deflection angle is largest when phase lag is PI. When phase lag is larger than PI, with the increase of the phase lag, the deflection angle starts to decrease. And the deflection angle will decrease to be 0 when phase lag is 2π. This is consistent with the variation of the thrust vectoring angle. As the net deflection angle of the jet is always towards the same direction, the thrust vectoring angle is also towards the same direction.

consistent with that given in Table 3. The thrust vectoring angle can be expressed as:

α=

T¯tan T¯nor

(15)

The value of α under different phase lags is given in Table 3. The thrust vectoring angle increases when 0 < θ < π and decreases when π < θ < 2π. The maximum thrust vectoring angle realized by the double actuator system is 11.88°. This validates its effectiveness in underwater thrust vectoring. The tangential thrust is mainly generated through fluid acceleration. The jet deflection also contributes to part of the tangential thrust. When two actuators work in phase, the flow field is symmetrical. In this case, the jet deflection and the fluid acceleration are negligible. The tangential thrust will be 0. With the increase of the phase lag, fluid acceleration and jet deflection are enhanced. Tangential thrust increases as result. From Fig. 13, when phase lag is small, the peak of the tangential thrust locates near t = T. When t = T, jet velocity is significant while fluid acceleration is small as can be seen from (1). This means when phase lag is small, contribution of jet deflection is larger than fluid acceleration. With the increase of the phase lag (until θ = π), the peak of the tangential thrust moves towards t = T/4 and t = 3T/4 as marked using red arrow in Fig. 13. This means the contribution of fluid acceleration increase and become dominant in tangential thrust generation. When θ is larger than π, with the increase of the phase lag, the thrust peak starts to move far away from t = T/4 or t = 3T/4. This means the contribution of fluid acceleration starts to decay. The pressure force, the jet momentum and the fluid acceleration all contributes to the normal thrust. The variation of the normal thrust in figure is mainly induced by these three components. When two actuators work in phase, the peak of the normal thrust locates at t = T/2 or t = T. At this time, the jet velocity is highest while the fluid acceleration is very small. This means when phase lag is 0, normal thrust de→ → pends on k (|Qm V|) n and Qm V . k (|Qm V|) n always has the same direction despite the direction variation of V . On the other hand, Qm V → reverse its direction every half period. When k (|Qm V|) n and Qm V has → the same direction, the normal thrust is Qm V + k (|Qm V|) n . Otherwise, → the normal thrust is Qm V − k (|Qm V|) n . As a result, the positive and negative thrust peaks are different in magnitude. In this paper, the negative peak is larger than the positive one. With the increase of the phase lag, the thrust peak starts to move towards t = T/4 as marked

4. Conclusions A novel thrust vectoring method based on double synthetic jet actuator is studied using numerical simulation. The thrust and flow field of the double synthetic jet actuator under different nozzle distances and phase lags are investigated. Thrust of synthetic jet includes three parts, fluid acceleration, jet momentum and pressure force respectively. The net deflection angle of the jet is always towards the actuator leading in phase despite the suction of the other actuator. The deflection of the jet contributes to the tangential thrust. Both the nozzle distance and phase lag affect the jet deflection. The jet deflection angle decreases monotonously when nozzle distance increases. The jet deflection angle increase when phase lag increases from 0 to π and decreases when phase lag continues to increase from π to 2π. Fluid acceleration also contributes to the tangential thrust. And the variation of fluid acceleration with phase lag has the same tendency with jet deflection. When phase lag is small, contribution of jet deflection is much larger than fluid acceleration. With the increase of the phase lag, the contribution of fluid acceleration increases and becomes dominant in tangential thrust generation. The variation tendency of the tangential thrust with phase lag is the same with the jet deflection and fluid 95

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acceleration. Fluid acceleration, jet momentum and pressure force all contributes to normal thrust. When nozzle distance is small, the interaction between the adjacent jets is strong. Part of the jet A will be sucked by jet B. in this case, the total accelerated fluid decreases. The contribution of fluid acceleration decreases. When nozzle distance is large enough, the interaction between jets can be neglected. The acceleration force increases. With the increase of the phase lag, the contribution of the acceleration force increases. The thrust also decreases in magnitude indicating the weaker contribution of the fluid acceleration than the jet momentum and pressure force. The average normal thrust mainly depends on the pressure force. The pressure force is not affected by the nozzle distance and phase lag. As a result, the average normal thrust is independent of the phase lags. In this case, the variation of the thrust vectoring angle is the same with the average tangential thrust.

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