Evolution and Near-Wall Effect of the Vortex Structures Induced by In-Line Twin Synthetic Jets in a Crossflow

Abstract

This paper aims to further the understanding of the mixing process of in-line twin synthetic jets (SJs) and their impact in the near-wall region in a flat-plate laminar boundary layer. A numerical study has been carried out, in which colored fluid particles and the Q criterion are used to track the SJ-induced vortex structures at the early stage of the evolution. Interacting vortex structures at four selected phase differences are presented and analyzed. It is found that the fluid injected at the early stage of the blowing stroke mainly contributes to the formation of the hairpin legs, the fluid injected near the maximum blowing mainly contributes to the formation of the hairpin head, and the fluid injected at the late stage of the blowing stroke contributes very little to the formation of the hairpin vortex. It is also confirmed that, irrespective of the phase difference, the hairpin vortex issued from the upstream actuator is more capable of maintaining its coherence than its counterpart issued from the downstream actuator. The influence of the interacting vortex structures on the boundary layer is also studied through investigating excess wall shear stress. In all cases, a pair of streaks of high wall shear stress can be observed with similar size. Among them, the streaks have the strongest wall shear stress, with the largest gap at phase difference 0 when partially interacting vortex structures are produced. The findings can provide valuable guiding information for the applications of synthetic jets in heat transfer, mixing control, and flow control in a crossflow.

1. Introduction

A synthetic jet (SJ) actuator can produce a train of vortical structures by the periodical motion of an oscillatory diaphragm on a cavity [1,2,3]. These vortical structures can effectively enhance fluid mixing in the crossflow, thereby providing an effective mechanism of boundary layer flow control [4,5,6,7,8,9].

Previous works were carried out to investigate the impact of the SJ-induced vortical structures in the near-wall region of a boundary layer, where effective flow separation control and wall heat transfer are ultimately desired. Using dye visualization, Zhong et al. [10] and Jabbal and Zhong [11] looked at more detailed interactions and identified three key SJ-induced vortex structures, i.e., hairpin vortices, stretched vortex rings, and tilted vortex rings, under different jet operation and boundary layer flow conditions. Different thermal footprints on the wall are correlated to the vortical structures induced by SJs. Using high-frame-rate particle image velocimetry (PIV) experiments, Jabbal and Zhong [12] confirmed that the wall shear stress shares similar patterns with the thermal footprints, and found that hairpin vortices and stretched vortex rings are able to offer enhanced heat transfer in the near-wall region. Using numerical methods, Lee et al. [13] confirmed the effective wall cooling of single SJs in micro-channels. They also found that the flow structures induced by stronger SJs in the micro-channel can exert larger mixing regions and stronger convection at the heated wall surface.

To increase the impact region, SJ arrays are usually applied. For example, in the application of fluid cooling, Lee et al. [14] found that twin SJs in out-of-phase operation can significantly enhance wall surface cooling due to the enhanced fluid mixing. However, unlike single SJs, the interactions between multiple circular SJs and cross-flows, as well as among multiple SJs, have not been fully studied and understood. Very limited investigations have been reported on this topic in the literature, especially on the near-wall effect. Based on oil visualization, Watson et al. [15] examined the interaction of two neighboring SJs in a boundary layer flow, and pointed out that the twin SJs can perform in either a constructive (enhancing the strength of the SJs) or a destructive (weakening the SJs) manner under different configurations. Using hot-wire measurements, Liddle and Wood [16] and Liddle et al. [17] confirmed that the phase difference plays a very important role in the interaction of twin SJs in a crossflow. At a 270o phase difference, the SJ actuators in the upstream and downstream locations yielded a more concentrated movement of near-wall fluid towards the vortex core than at a 90o phase difference. With PIV measurements of in-line twin SJs in a channel flow, Honami and Motosuke [18] found that the twin SJs can have different vortex strengths with different configurations. Although these studies revealed useful physical insights, they are not enough to give a holistic picture of the flow structures induced by multiple circular SJs in a crossflow and their near-wall effect. Therefore, more detailed studies are highly desired.

Although these works have revealed some useful physical insights into the interaction of in-line twin SJs in boundary layers, the detailed vortex interaction and evolution of twin SJs are not well understood. Limited by the spatial and temporal resolution, it is difficult for current experimental methods to capture the three-dimensional flow structures of twin SJs. In addition, it is also difficult to examine the evolution of the interacting flow structure issued from different SJ actuators in Cartesian coordinates. Therefore, two research questions still remain: First, how does the mixing process of the twin-SJ-induced flow structures occur in detail? Second, what is the influence of these flow structures on the boundary layer flow, especially in the near-wall region? By examining the flow structures and their impact in the boundary layer, we can obtain results which are valuable for flow control and heat transfer applications. For example, strong vortices and high wall shear stress can produce high flow mixing in the boundary layer, delaying flow separation and enhancing heat transfer on the wall. Therefore, to address these questions in the present research, a numerical study is conducted. A cluster of colored fluid particles are used to track the evolution and mixing process of the single and twin SJs at the early stage. In the Lagrangian perspective, fluid particles are applied to examine the flow dynamical system rather than a continuum [19]. It is widely used to present high-quality visible details of time-dependent vortex structures, which may not be apparent in the classical Eulerian fields [20,21]. In addition, to capture the associated vortex structures, Q iso-surfaces are defined [22]. Finally, the excess wall shear stress is evaluated and compared.

2. Methodology

2.1. Problem Description

The twin SJ actuators used in the present numerical study are identical to those used in our previous experiment [23,24]. The SJ cavity has an orifice with a diameter of D_o = 5 mm. The centers of the two SJ actuator orifices are at a distance of d = 10 mm. Similar to previous experiments [23], the oscillation diaphragm of each actuator has a diameter of D_d = 45 mm, which oscillates like a piston, with velocity varying in a sinusoidal manner.

$$u_d\left(t\right)=\pi \Delta f\sin \left(2\pi ft\right)$$

(1)

where ∆ is the peak-to-peak displacement of the diaphragm and f the oscillation frequency. During an actuation cycle, the beginning of blowing is defined as t/T = 0 (and 1). In the present study, the working flow is water, which is incompressible. According to Tang and Zhong [25], the jet velocity Ūo averaged over an entire actuation cycle can be calculated as

$${\overline{U}}_o=\frac{1}{T}{{\displaystyle \int }}_0^{T/2}{u}_o\left(t\right)dt=\Delta f{\left(\frac{D_d}{D_o}\right)}^2$$

(2)

where the instantaneous jet velocity is u_o(t).

In current study, the bulk flow has a velocity of U_∞ = 0.055 m/s. The bulk flow has a boundary layer thickness of 15 mm, i.e., δ = 3D_o, at the actuator orifices. Based on Reyolds number, the momentum thickness is Re_θ ≈ 106. The actuators have a diaphragm displacement of Δ = 0.158 mm with a frequency of f = 1 Hz. To characterize the performance of SJs in a laminar boundary layer, the dimensionless stroke length L and velocity ratio VR are also defined according to Wen and Tang [24]. The stroke length L is defined as

$$L=\frac{L_o}{D_o}=\frac{{\overline{U}}_o}{f{D}_o}$$

(3)

where L_o is the jet column length during the SJ blowing stroke. The jet strength is characterized by the velocity ratio VR as

$$VR=\frac{{\overline{U}}_o}{U_{\infty }}$$

(4)

From Equations (2)–(4), the settings in the present study yield L = 2.6 and VR = 0.23, to produce hairpin vortices according to the parametric map revealed by Jabbal & Zhong [11]. These hairpin vortices can stay in the boundary layer and hence are very desired in SJ-based flow separation delay.

In the present study, the two SJ actuators are arranged in line with the crossflow and are separated by a fixed distance d = 2D_o. Phase difference between the two SJs are investigated by Δϕ = 0, π/2, π and 3π/2 (or −π/2). Here Δϕ = 0 means that the twin SJs are in phase, and the Δϕ = π/2 means the downstream actuator is π/2 behind, and so on.

2.2. Numerical Approaches

Figure 1 shows the computational geometry and boundary conditions, which are similar to those used in Wen and Tang [26] and Zhou and Zhong [27]. In the x (streamwise) direction, the simulation domain has a length of 55D_o. In the y (wall-normal) and z (spanwise) directions, the domain is 30D_o and 20D_o, respectively. The origin of the coordinate system is set at the midpoint of two actuator orifice centers, which are 15D_o downstream from the crossflow inlet. Although not presented in Figure 1, the cavities of the two SJ actuators are included in the simulation. The crossflow in-let on the left of the computational domain is set as the velocity inlet with a laminar Blasius velocity profile. The wall of flat plate in the boundary layer and the SJ actuators are set as no-slip conditions. Both lateral sides of the crossflow domain are set as periodic conditions. The upper face and the downstream boundary on the right of the crossflow domain are set as the pressure outlet. On the diaphragms of the actuators, a time-dependent velocity condition is set according to Equation (1).

To activate the SJ actuator, the motion of the diaphragm in the cavity is controlled by setting a time-dependent velocity boundary condition according to Equation (1). In this study, water is used, which is incompressible. The fluid is governed by mass con-servation and momentum conservation. The flow is laminar; therefore, no turbulent model is used. The second-order implicit scheme is used in time, and the second-order upwind scheme is used in space. For the pressure–velocity coupling, the pressure-implicit with splitting of operators method is used. The unsteady, three-dimensional incompressible Navier–Stokes equations are solved by a commercial CFD code, ANSYS FLUENT 2020 R2.

$$\frac{\partial u_i}{\partial x_i}=0$$

(5)

$$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+v\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}$$

(6)

where ρ is the fluid density and ν is viscosity, and the summation notion is applied. Sensitivity studies have been conducted to ensure the mesh number and time-step size are optimum. Three different mesh numbers, i.e. 2.1 × 106, 2.8 × 106, and 3.7 × 106, and three different time-step sizes, i.e. 1/240 s, 1/360 s, and 1/480 s, are examined. Mesh and time sensitivity studies have been conducted for the case of twin SJs operating in phase (i.e., Δϕ = 0). With different mesh sizes and time steps, the time-averaged velocity profiles along a spanwise, wall-parallel line (i.e., x/D_o = 5 and y/D_o = 2) within the boundary layer are compared. As shown in Figure 2, the simulation result seems independent of the selected mesh size and time step, confirming the mesh and time convergence in the current study. The total number of meshes used in the simulation is about 2.8 million. In one diaphragm oscillation cycle, there are 240 time steps as a compromise between computational accuracy and time.

In the present study, the SJ-induced vortex structures are identified using the Q criterion [22]. A vortex structure can be captured by an iso-surface of a positive Q value. To differentiate individual SJs in the present simulation, particles of two different colors are continuously and uniformly released from the two actuator orifices during the blowing cycle once the flow field reaches its steady state. The density of these particles is set to that of water, and the discrete particle model (DPM) is used to track them. Similar to our dye visualization experiment [23,24], the particles issued from the upstream actuator are colored in red, whereas those from the downstream actuator are in green.

To reflect the capability of the SJ-induced vortex structures in flow separation control, wall shear stress is calculated based on the least squares cell-based gradient discretization proposed by Anderson and Bonhaus [28] to ensure a second-order interpolation, with a spatial resolution of 0.5% of the boundary layer thickness or y+ = 0.2.

2.3. Validation

The present CFD framework has been well validated with a single SJ in our previous work [26]. To further validate this framework for the current twin-SJ configuration, experiments are conducted using color dye visualization and PIV measurement. In dye visualization, food dye with two different colors fills the two actuator cavities, i.e. red color for the upstream actuator and green color for the down-stream. Methanol is premixed with the dye to achieve a similar density to water. Two cameras are used to capture images from the side and bottom views. In the PIV measurement, the Dantec system is used. Polyamide seeding particles are used to track the flow, and a FlowSense 2M charge-coupled-device camera is used to capture the particle images. A double pulsed Nd:YAG laser is used to illuminate the seeding particles. The sampling frequency of PIV measurement is 10 Hz. For each case, 300 image pairs are taken to achieve convergence. The PIV measurement results are verified with Blasius solution. More details of the dye visualization and PIV measurement can be found in our previous work [23]. To validate the CFD framework, flow patterns represented by colored fluid particles in the simulation are first compared to those in the color dye visualization experiment. As shown in Figure 3, the simulated particle patterns match with the dye visualization in a good manner. As shown in Figure 4, instantaneous and time-averaged velocity profiles are also compared at x = 4Do, with a good agreement between the simulation results and the PIV data.

3. Evolution and Interaction of In-Line Twin SJ-Induced Flow Structures

3.1. Flow Structures of a Single SJ in a Crossflow

The single-SJ-induced flow structures visualized by the fluid particles, and the Q iso-surfaces are presented first to serve as the baseline case. As shown in Figure 5a, iso-surfaces of Q = 5 are superimposed over the particles, showing that the flow structures represented by the fluid particles coincide with the primary hairpin vortices captured by the Q iso-surfaces in terms of the shape, size, and location. This confirms that, in addition to the Q iso-surfaces, the SJ-induced hairpin vortices can also be examined through the pattern and motion of fluid particles. As shown in Figure 5b, where only the fluid particles are presented, each hairpin vortex is well represented by a strong head and a pair of slim, stretched legs. The hairpin head also carries a spiral of fluid particles when seen from the side cross-section view, indicating very strong rotation inside. However, no particles are observed in the induced secondary vortices that are captured by the Q iso-surfaces as shown in Figure 5a. This is not surprising because the particles are all released from the actuator orifice, while the secondary vortices form with ambient fluid that does not carry any particles. These secondary vortices are very close to the wall and hence play an important role in SJ-related flow separation control [26].

3.2. Evolution of Single and In-Phase Twin SJs

By using a sequence of snapshots of fluid particles, the evolution of the single SJ is presented in the upper two rows of Figure 6. It can be seen that, after expulsion from the actuator, the SJ-induced vortex ring gradually tilts downstream because of the velocity gradient in the boundary layer. Its upstream part is weakened by the boundary layer. As time advances, a hairpin vortex is formed by the shear stress in the boundary layer. It should be noted that the hairpin vortex is formed in a boundary layer with a flat plate. The wall-surface shape and obstacles in the boundary layer can significantly affect the evolution of hairpin vortices and horseshoe vortices, which is comprehensively discussed in previous works [29,30,31]. To further reveal the contribution of various SJ portions to the resulting hairpin vortex, three groups of fluid particles are marked in the snapshots, which are released from the orifice exit at three consecutive instants, i.e., t = 0.0625T (group one), 0.25T (group two), and 0.4375T (group three). Each group consists of four selected particles, representing the upstream, central, downstream, and side portions of the SJ cross-section, respectively. By tracking these three groups of particles, it is found that the hairpin head contains fluid particles mainly from group two, while the hairpin legs contain fluid particles mainly from group one. As for the fluid particles in group three, most of them are inhaled back into the actuator during the SJ ingestion process (t = 0.5T~T). This result indicates that the fluid injected at the early stage of the blowing stroke (t ≈ 0) mainly contributes to the formation of hairpin legs, the fluid injected near the maximum blowing (t ≈ 0.25T) mainly contributes to the formation of the hairpin head, and the fluid injected at the late stage of the blowing stroke (t ≈ 0.5T) contributes very little to the formation of the hairpin vortex.

If another SJ is introduced near the single SJ, the interaction between these two SJs will result in the formation of new flow structures. Take the in-phase, in-line twin SJs as an example, whose evolution is presented in the lower two rows of Figure 6. The SJ issued from the upstream actuator (in red, also called the trailing vortex in this case) is similar to the single SJ before it entrains its downstream counterpart, while the SJ issued from the downstream actuator (in green, also called the leading vortex) is significantly affected by its upstream counterpart. The interaction between these two SJs is evidenced by the lift-up of the leading vortex legs and their entrainment into the trailing vortex. Furthermore, since it is the leading vortex that is significantly affected by the interaction, the same three groups of fluid particles as in the single-SJ case are released from the downstream orifice exit. By tracking these particles and comparing their traces with those in the single SJ, it can be seen that during the blowing stroke (t = 0~0.5T), the evolution of these particles is similar to that in the single-SJ case, except that the particles released from the downstream portion of the SJ cross-section at the early stage of the blowing stroke (marked as “1D”) are stripped out from the main vortex structure under the influence of the preceding vortices. When the twin SJs start interacting with each other in the ingestion period (t = 0.5T~T), particles originally in the lower portion of the vortex leg, e.g. the particles “1U” and “3C”, are significantly lifted up by the trailing vortex, while particles originally in the vortex head, e.g. the particles in group two, still remain close in the highly rotating vortex head, indicating a well-maintained coherence in the head even under the strong interaction.

3.3. Impact of Phase Differences

With the simulated colored fluid particles, three different vortical structures are observed at the four selected phase differences. At Δϕ = π/2 (Figure 7b), the two SJs merge together to form one combined vortex, which is much stronger than single hairpin vortices and can pass through the boundary layer quickly. At Δϕ = 3π/2 (Figure 7d), the two hairpin vortices are separate from each other without obvious interaction, forming two completely separated vortices. At Δϕ = 0 (Figure 7a) and π (Figure 7c), the legs of one hairpin vortex are entrained into the head of the other, forming partially interacting vortex structures. As such, the twin SJs have distinct vortex strengths at various phase differences, as revealed in the authors’ previous PIV experiments [23,24]. Furthermore, it is seen that, no matter which type of interaction occurs, the green particles issued from the downstream actuator are mainly scattered in the outer layer of the interacting structures, whereas the red particles issued from the upstream actuator are mainly enclosed inside. The different particle distributions can be caused by the different strengths of the vortical structures issued from the twin SJ actuators. The flow structures issued from the upstream actuator can hold vortical structures better than those from the downstream. Therefore, the particles issued from the upstream actuator are more concentrated inside of the interacting flow structures. This is especially so when the interacting structures develop into the second actuation period, as shown in the plane view of Figure 7a–c. It is also seen that at Δϕ = 3π/2 (Figure 7d), unlike its counterpart, the head of the SJ issued from the upstream actuator is not affected at all by the near-wall particles. All these observations support our previous statement, i.e., in any case, the SJ issued from the up-stream actuator can better maintain its vortical structure than the SJ issued from the downstream actuator [23,24], and hence it dominates the interaction. The reason is obvious: the vortex produced by the upstream actuator has sufficient time to develop itself before the interaction, while the SJ issued from the downstream actuator is disturbed from the very beginning of its emergence.

The different capability of the two hairpin vortices in maintaining their own coherence can be further demonstrated in snapshots from the streamwise cross-section view. Figure 8 shows the snapshots from the Δϕ = 0, π/2 and π cases, at the same instant t/T = 0 as in Figure 7. In all three snapshots, the boundaries between the particles from different actuators are clear. At Δϕ = π/2 (Figure 8b), the head of the combined vortex includes two cores of fluid particles issued from the upstream actuator (in red) which are surrounded by scattered fluid particles issued from the downstream actuator (in green). At Δϕ = 0, the twin SJs produce partially interacting vortex structures (Figure 8a). The fluid particles issued from the upstream actuator still appear in two cores, forming the head of the trailing vortex, which is surrounded by the fluid particles issued from the downstream actuator. Moreover, two legs of the leading vortex with green particles appear near the mid-span plane, whose induced flow pushes the upper portion of the trailing vortex heads (two red cores) away from each other, which is prominent if compared with the combined vortex head shown in Figure 8b. In the partially interacting vortex structures at Δϕ = π (Figure 8c), the head of the trailing vortex (in green) no longer appears as two solid cores. Instead, it appears as two hollow tubes enclosing the two counter-rotating legs of the leading vortex with red particles, and its shape differs greatly from those in Figure 8a,b.

Particles issued from selected positions of the SJ actuator orifices are also tracked to study the interactions of SJ vortex structures, as shown in Figure 9. The single SJ and the twin SJs at Δϕ = π/2 and 3π/2 are selected for presentation. Compared to those in the single SJ, the particles issued at the center of the orifices at Δϕ = π/2 are lifted to a much higher position at x = 3Do outside of the boundary layer. This is because a stronger vortex is merged from the twin SJs. The two particles issued from different actuators have strong interactions and rotate around the merged vortex head. In addition, the particles issued at z/Do = 0.4 have a slightly faster rotation than that in a single SJ with a shorter traveling distance. On the other hand, the trajectories of particles at Δϕ = 3π/2 are similar to those in the single SJ because the two separated hairpin vortices are produced, with the exception that the particle issued at z/Do = 0.4 from the downstream orifice is trapped in the tip of the hairpin leg with a straighter trajectory.

Figure 10 shows the correlations of the streamwise and wall-normal velocity components at three selected stations in the mid-span plane. In the single-SJ case, the upwash flow of hairpin vortices can increase the velocity in the wall-normal direction and decrease the velocity in the streamwise direction in the boundary layer [23,24]. For the twin SJs at Δϕ = π/2, the disturbance of the boundary layer is much stronger, indicated by the larger correlation loop of the streamwise and wall-normal velocity components. The shape of correlation loop is similar to that in the single SJ, because only the stronger hairpin vortices are merged with similar vortex structures. On the other hand, the shape of the correlation loop becomes very different at Δϕ = 3π/2. At a phase difference of Δϕ = 3π/2, the SJ flow structures are produced alternately from the twin actuators with an almost equal time interval. Therefore, the twin SJs have a weak interaction but with a doubled frequency compared to single SJs. On the other hand, the twin SJs merge into a combined vortex at Δϕ = π/2 with a high strength but the same frequency to the single SJs. Differing from the single loop in the single-SJ case, two loops are observed with substantial differences. This indicates that the two separated hairpin vortices at Δϕ = 3π/2 also have notable differences in vortex strength. In addition, the smaller loops also imply that the hairpin vortices at Δϕ = 3π/2 are slightly weaker than that in the single-SJ case.

4. Impact of SJ Vortices on Wall Shear Stress

Previous studies have revealed that the SJ-induced near-wall streamwise vortices, i.e., the hairpin legs and induced secondary vortices, play an important role in flow separation control or enhancement of heat and mass transfer [12]. In this section, in order to reveal details of the evolution of these streamwise vortices, iso-surface of a smaller value of Q = 1 are colored by streamwise vorticity. To examine the impact of the SJ vortices on the boundary layer, the excess wall shear stress is presented and calculated as (τ_w − τ_w,nojet)/τ_w,nojet, which is a useful indicator of the SJ effectiveness in heat transfer enhancement [12].

4.1. Instantaneous Influence

Figure 11 shows the single-SJ-induced vortices and the corresponding contours of excess wall shear stress at the instant t/T = 0. Evolution of the primary hairpin vortices and secondary vortices has been discussed in our previous study [26]. These vortex structures influence the boundary layer as well as the wall shear stress. A pair of streaks of positive excess wall shear stress is observed outside of the SJs. This streak pair becomes wider near the regions between the legs of hairpin vortices and the secondary vortices (at about 8 < x/D_o < 10), where the downwash flow is strengthened by the interaction of the vortices.

The two SJs merge into a stronger vortex when their phase difference is Δϕ = π/2. As shown in Figure 12a, both the hairpin vortices and the induced vortices, including the secondary vortices and horseshoe vortex, become stronger compared to those induced by the single SJ. The evolution of these vortices presented in the upper two rows of Figure 12b reveals that, at t/T = 0.25 when the upstream SJ is at its maximum blowing, a vortex ring (labeled as A) is issued the upstream actuator. It lifts up the horseshoe vortex produced in the previous cycle. In the downstream part of this horseshoe vortex a pair of secondary vortices begin to form. At t/T = 0.5, the vortex ring A begins to evolve into a hairpin vortex due to its interaction with the crossflow. Meanwhile a new vortex ring (labeled as B) emerges at the downstream orifice, breaking the horseshoe vortex. In addition, the downstream secondary vortices become more obvious. At t/T = 0.75 & 1, the twin SJs combine into a stronger hairpin vortex, which is accompanied by stronger secondary vortices and a larger horseshoe vortex.

The enhancement of wall shear stress is more significant, especially near to the orifice (x/D_o < 12) in Figure 12a. However, the combined vortices are so strong that they quickly penetrate the boundary layer, resulting in significant decay of the excess wall shear stress in the far downstream. In addition, some deficit of the wall shear stress is found outboard of the secondary vortices, which is associated with the induced upwash flow [26]. Detailed examination in the near-orifice region shows that changing of the excess wall shear stress is obviously periodic during one actuation cycle, as shown in the bottom row of Figure 12b. At t/T = 0.25, besides the area around the upstream orifice, high excess wall shear stress is also found near the downstream orifice, which is induced by the legs of the preceding combined vortex and the associated secondary vortices. At t/T = 0.5, high excess wall shear stress is induced around the downstream orifice due to the emergence of the vortex ring B. The excess wall shear stress downstream of this orifice becomes weaker. At t/T = 0.75, as the two SJs begin to combine, the regions of high excess wall shear stress around the two orifices connect. At downstream of the orifices, a large area of negative excess wall shear stress forms between the new forming vortex and the preceding one. At t/T = 1, as the newly formed combined vortex convects downstream, positive excess wall shear stress recovers near the orifices.

Two completely separated vortices are produced at Δϕ = 3π/2. Primary hairpin vortex and outboard secondary vortices are induced as shown in left subplot of Figure 13a. However, their evolution processes start at different locations and instants. As shown in the upper two rows of Figure 13b, a pair of secondary vortices (labeled as #1) begins to form between the two orifices at t/T = 0.25. At t/T = 0.5, these secondary vortices become more obvious, as the horseshoe vortex is eliminated by the suction of the downstream actuator. They further grow at t/T = 0.75 and 1, and are sandwiched by the head of the SJ issued from the upstream actuator and the legs of the SJ issued from the downstream actuator. At t/T = 1, a vortex ring emerges from the downstream orifice and begins to evolve into a hairpin vortex at t/T = 1.25 (equivalently t/T = 0.25). By interacting with the preceding hairpin legs, it induces a pair of secondary vortices (labeled as #2) downstream. This pair of secondary vortices grows at t/T = 1.5 (equivalently t/T = 0.5), and finally becomes comparable to its counterpart (i.e., #1) at t/T = 1.75 (equivalently t/T = 0.75). Therefore, both the primary hairpin vortices and the induced secondary vortices double their appearance frequency (in other words, double their number) at far downstream. On the other hand, they are all shorter than those in single SJ case. The strip pair of enhanced wall shear stress also reflects this doubled frequency, as shown in the right subplot of Figure 13a. The strips are narrower compared to single-SJ case. Also, due to the doubled frequency effect, the strips in the near-orifice region can maintain the high excess wall shear stress, as revealed in the bottom row of Figure 13b.

Partially interacting vortex structures are produced at Δϕ = 0 and π. The Q iso-surfaces reveal that, no matter whether it is the trailing vortex (left subplot of Figure 14a) or leading vortex (left subplot of Figure 15a), the vortex produced from the upstream actuator dominates the interacting vortex structure in terms of vortex coherence. This is consistent to what has been revealed with the colored fluid particles in Section 3.3. For this reason, the vortex produced from the upstream actuator is also called the dominant hairpin vortex hereinafter, despite the phase difference. In both cases, a pair of much longer secondary vortices is induced by the interacting vortex structure, compared to the single-SJ case. Although appearing similar, these two partially interacting vortex structures show an interesting difference. That is, the lateral distance between the two legs of the partially interacting SJs at Δϕ = 0 is much larger than that at Δϕ = π. This is because in both cases the legs of the interacting vortex structure are from the dominant hairpin vortex. At Δϕ = 0, the leading vortex is entrained through the gap between the legs of trailing dominant vortex. Therefore, the gap is enlarged during the process. On the other hand, at Δϕ = π, the legs of dominant leading vortex is pushed together by the trailing vortex, resulting in a smaller gap. This difference also affects the pattern of excess wall shear stress. Although the patterns in both cases are similar to that in the Δϕ = π/2 case in far downstream, the deficit in the centerline low stress region in the Δϕ = 0 case is significantly larger (right subplot of Figure 14a). Furthermore, at Δϕ = 0, the side view reveals that the legs of the dominant hairpin vortex are gradually “pulled” down toward the wall as the reaction to their entrainment of the leading hairpin legs, which are then able to exert more influence on the wall shear stress, especially in the far-downstream region. As such, the two high stress streaks in the mid- and far-downstream regions are stronger than those in the near-downstream region, which is quite different from what is observed in the other cases.

4.2. Time-Averaged Influence

The time-averaged influence of the single and twin SJs on the wall shear stress is compared in Figure 16. It is seen that about 1D_o downstream of each operating actuator there exists an obvious region of negative excess stress, which is caused by the blockage of the SJs and the roll-up of the downstream branch of the SJ-induced vortex rings. Further downstream, the averaging removes the periodic patterns of the two streaks in all cases. Regardless the phase difference, the twin SJs can generate significantly more excess stress than the single SJ. But the persistency of the influence varies with phase difference. At Δϕ = π/2 the combined vortex structures induce a pair of wall shear stress streaks that is strong only within a short distance downstream from the orifices and starts decaying further downstream (see Figure 16c). This is because the combined vortices can penetrate the boundary layer deeply. The vortices move away from the wall as they travel in the crossflow, resulting in a fast decaying influence on the wall in the far-downstream region. This is evidenced by the particle tracking (Figure 7b) and instant flow structures (Figure 12) On the contrary, at Δϕ = 3π/2 the two completely separated vortices can stay close to the wall (Figure 7d and Figure 13), and produce a pair of high stress streaks that sustain their strength for a much longer distance. The partially interacting vortices at Δϕ = 0 induce two streaks of high wall shear stress with gradually increased strength as shown in Figure 11b, between which a much wider strip of low stress value is observed. However, the time averaged high stress streaks produced by the other partially interacting vortices at Δϕ = π (Figure 16d) appears similar to that in the Δϕ = π/2 case.

In Figure 17, the excess wall shear stress is time-averaged and compared at two positions, i.e., x = 4D_o and 18D_o. In all cases, the two high stress streaks are represented by two peaks. At x = 4D_o, the two peaks induced by the single SJ are symmetrically located about 0.8D_o away from the mid-span plane, whereas the two peaks induced by all the twin SJs are slightly closer. It is also seen that the Δϕ = 0 case has a peak value close to that in the single SJ case. On the other hand, the other three twin SJ cases all have a much larger peak value (at least 20% higher). This is because in the Δϕ = 0 case the legs of the leading vortex (issued from the downstream actuator) are lifted up immediately after their emergence (as shown in Figure 6 and Figure 7a), and hence the near-field wall shear stress is only affected by the legs of the trailing vortex (issued from the upstream actuator), while in the other three twin-SJ cases, i.e., Δϕ = π/2, π and 3π/2, the near-field wall shear stress is affected by the legs of both vortices.

At x = 18D_o that is far from the two orifices, due to the spreading of the vortex structures and their penetrating in the boundary layer, the two stress peaks in all cases are separated from each other with a larger distance, and, except for the Δϕ = 0 case, their peak values reduce quite significantly if compared with those at x = 4D_o. In the Δϕ = 0 case, wall shear stress gets larger as the flow structures traveling to downstream. This may be caused by the closer distance of the hairpin vortex to the wall and the increasing strength of the hairpin vortex during the merging process of the twin SJs. In addition, since the two legs of the dominant vortex is pushed away from each other by its counterpart vortex, the distance of the corresponding peaks becomes much larger than that in the other cases including the single SJ case. This is also consistent with what is observed in Figure 16. In addition, negative values of excess wall shear stress are also found at x = 18D_o in |z/D_o| > 2 regions, except at Δϕ = 3π/2. From the instantaneous patterns of excess wall shear stress (Figure 12a, Figure 14a and Figure 15a), these negative values are ascribed to the deficit of wall shear stress outboard of the secondary vortices.

The excess wall shear stress is time-averaged, and then spanwise averaged within a range of z = −1.6D_o~1.6D_o where most of the non-zero excess stress is covered as revealed in Figure 17, and the streamwise distribution of the newly calculated quantity is then plotted and compared in Figure 18. It is seen that this time- and spatial-averaged excess stress in the single SJ case reaches a gentle peak at about x = 3D_o, and then decreases along the streamwise direction. In all the twin SJ cases, the curve begins with a significant rise, which corresponds to the region of negative excess stress right next to the downstream orifice as shown in Figure 16. Compared to the single SJ, in the Δϕ = π/2 case the combined vortex structures produce a peak of more than 40% larger at about x = 3.5D_o, after which the mean excess stress decreases with a higher rate, reaching a value of 26% larger at x = 22D_o. The trend is similar in the Δϕ = 3π/2 case where the two completely separated hairpin vortices are generated. Because of the doubling frequency effect in this case, the mean excess stress becomes almost unchanged after x = 12D_o, which turns to be larger than that in the Δϕ = π/2 case.

Although some difference is expected between the two cases that generate partially interacting vortex structures, it is still surprising to see a huge difference. The curve in the Δϕ = π case follows a similar trend to that in the Δϕ = π/2 case, but with a slightly higher peak. But the curve in the Δϕ = 0 case varies quite differently: it reaches a plateau first at about x = 4D_o where the excess stress is only 15% larger than that in the single SJ, and then increases further and reaches a gentle peak at about x = 19D_o where the excess stress is about 50% greater. This huge discrepancy stems from the difference in the evolution of the dominant vortex structure in these two cases. At Δϕ = 0, the legs of the dominant hairpin vortex are gradually “pulled” down toward the wall, which can exert stronger influence on the wall shear stress, especially in the far-downstream region.

5. Conclusions

The interaction of in-line twin SJs is examined in a laminar boundary layer numerically, that operate at various phase differences along a flat plate. Fluid particles of different colors released from the actuator orifices are tracked so as to show the mixing process of SJs in the boundary layer. In addition, the Q criterion is used to identify the twin SJ-induced vortex structures including both the primary and secondary vortices. The impact of these vortex structures on the boundary layer is also investigated through studying the excess wall shear stress. The major findings are as follows:

(1)

The fluid injected at the early stage of the blowing stroke mainly contributes to the formation of hairpin legs, the fluid injected near the maximum blowing mainly contributes to the formation of hairpin head, and the fluid injected at the late stage of the blowing stroke contributes very little to the formation of hairpin vortex because most of them are inhaled back into the actuator during the ingestion process.

(2)

When the twin SJs generate a combined vortex at Δϕ = π/2 or partially interacting vortex structures at Δϕ = 0 and π, fluid particles issued from the downstream actuator are mainly scattered in the outer layer of the interacting structures, whereas those issued from the upstream actuator are mainly enclosed inside. When the twin SJs generate two completely separated hairpin vortices at Δϕ = 3π/2, unlike its counterpart, the head of hairpin vortex issued from the upstream actuator is not affected at all.

(3)

The trajectories of selected particles reveal that the fluid particle near the hairpin head rotates about the center of the vortex head, whereas the fluid particle near the hairpin leg is trapped in the leg and travels downstream with a helix path.

(4)

The SJ-induced vortex structures make great impacts on the wall shear stress. In all cases a pair of streaks of positive excess wall shear stress can be observed. Although both the Δϕ = 0 and Δϕ = π cases produce partially interacting vortex structures, the induced two excess stress streaks are significantly different, which stems from the difference in the evolution of the vortex structures, especially of the dominant hairpin vortex, in these two cases.