Figure  1(b) shows the simulation geometry used in this paper. The working gas flows into the domain from the boundary between the needle and the dielectric tube (i.e., AJ in Fig.  1(b)). The composition of ambient air is simplified into 80% N₂ and 20% O₂. The governing equations for the neutral gas flow model consist of the mass continuity equation, the momentum conservation (Navier– Stokes) equations and the species transport equation as follows:

(i) The mass continuity equation is

where ρ is the density (helium 0.1664  kg· m^{− 3}, air 1.293  kg· m^{− 3}), u and v are the velocity components in the axial and radial directions, respectively.

(ii) Momentum conservation (Navier– Stokes) equations can be written as

where p is the pressure, μ is the viscosity coefficient (helium 1.94 × 10^{− 5}  Pa· s and air 1.82 × 10^{− 5}  Pa· s), and U is the velocity vector of gas flow.

(iii) The species transport equation is

where Y_i is the local mass fraction of each species, and J_i is the diffusion flux of species i. When the flow is laminar, the diffusion flux can be written as

where D_i is the diffusion coefficient for species i in the mixture (helium in air 7.2 × 10^{− 5}  m²· s^{− 1}). Moreover, the gas flow state inside the tube which is laminar or turbulence flow is usually determined by the Reynolds number, and defined as

where υ is the average velocity of the gas and L is the characteristic length.

The governing equations (1)– (4) are solved using a stationary solver in COMSOL. The boundary conditions are summarized in Table  1, ^{[14, 15]} and the corresponding boundaries are also illustrated in Fig.  1(b). The radial uniform He velocity at inlet AJ is given, and the direction of inlet velocity is normal to the boundary AJ. Generally speaking, the natural convection velocity is about 0.1  m/s– 0.3  m/s in an aeration room with an air conditioner.^[14] Therefore, in our simulation, we assume the natural convection velocity to be 0.3  m/s at the air inlet GF to simulate the He flow more accurately.

Table 1.

Table 1.

Table 1. Boundary conditions for the neutral gas model. The symbols (A– J) correspond to the vertices in Fig.  1(b). The unit 1  atm = 1.01325 × 105  Pa. AJ BC EF GF AB, GHIJ, CE u 15  m/s ∂ u/∂ r = 0 ∂ u/∂ r = 0 0.3  m/s 0 v 0 ∂ v/∂ r = 0 ∂ v/∂ r = 0 0 0 YHe 1 ∂ YHe/∂ r = 0 ∂ YHe/∂ r = 0 0 n· YHe = 0a YAir 0 ∂ YAir/∂ r = 0 ∂ YAir/∂ r = 0 1 n· YAir = 0a P 1  atm ∂ P/∂ r = 0 1  atm 1  atm n· P = 0a a n is the normal vector pointing toward the surface.

Table 1. Boundary conditions for the neutral gas model. The symbols (A– J) correspond to the vertices in Fig.  1(b). The unit 1  atm = 1.01325 × 105  Pa.

Figure  2 shows the results of gas flow simulation in a steady state. As can be seen in Fig.  2, the helium mole fraction is almost 1 inside the tube. From the tube exit to the ground electrode, due to the mixing of the ambient air with the helium gas flow, a helium– air mixing layer is created, which is essential for the propagation of the plasma jet. The gas velocity of the axial component slows down after the helium gas leaves the dielectric tube, and the air mixture ratio increases with the distance from the tube exit. When the helium gas flow reaches the ground electrode, it flows radially along the surface of the electrode, which is similar to the simulation results presented in Ref.  [21].

Fig.  2.

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	Fig.  2. Results of gas flow calculation: (a) helium mole fraction, (b) gas velocity component in the axial direction, and (c) gas velocity component in the radial direction.

Figure  3 shows the Laplacian electric field and electric potential distributions on the axis of the discharge setup for DPE. Owing to the small radius of curvature, the electric field is very high near the needle tip and is positive due to the positive voltage applied. The electric field decreases rapidly along the axis and its value is zero at the position z = 1.3  cm. There are two peak values of electric field located at the edges of the ring electrode, and then the electric field decreases to zero. It is noted that the peak value of electric field at the upstream edge (left edge in Fig.  3) of the ring electrode is negative, while that at the downstream edge (right edge in Fig.  3) is positive. This can also be explained by the potential distribution.

Fig.  3.

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	Fig.  3. Laplacian electric field and electric potential distributions on the symmetry axis. The filled region represents the position of the ring electrode.

The plasma jet characteristic parameters (electron density, total ionization rate, and mean electron temperature) and absolute value of electric field distribution evolutions are shown in Fig.  4. During the first 8  ns, the discharge is ignited in the needle tip region first, where the Laplacian field strength is much higher than that at the other position as shown in Fig.  3, and starts to propagate inside the tube with a tubular structure towards the ring electrode. At t = 8  ns, the ionization front is between the needle tip and the ring electrode. The maximum mean electron temperature and absolute value of electric field appear in the ionization front with values about 11  eV and 40  kV/cm, respectively. Then at t = 40  ns, another intense discharge is located at the edges of the ring electrode, which is also observed in experiments.^{[22, 23]} Two positively charged ionization fronts propagate toward each other with the decrease of discharge radius between the needle tip and ring, and a positively charged ionization front (ring downstream plasma plume) propagates toward the ground electrode between the ring and ground electrode. The decrease of the discharge radius is due to the sheath formed near the dielectric tube inner surface, which can shield the plasma channel from the tube surface.^[20] At t = 120  ns, the ionization front of the needle merges with that of the ring, and a discharge channel is created. The dynamics of the mergence process has been investigated in a previous study.^[13] With the progress of the ring downstream plasma plume, the discharge front leaves the dielectric tube and propagates along the helium– air mixing layer, forming a plasma jet at t = 270  ns. The maximum mean electron temperature and absolute value of electric field appear in the ionization front with values of about 7  eV and 24  kV/cm, respectively.

Fig.  4.

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	Fig.  4. Discharge characteristic parameter evolutions of atmospheric pressure helium jet with DPE. The distributions of (a) electron density, (b) total ionization rate Se, (c) mean electron temperature Te, and (d) absolute value of electric field.

Figure  5 shows the calculated spatial distributions of species densities in the region outside of the dielectric tube at t = 270  ns after the start of pulse excitation. Because of the off-axis maximum of the ionization rate, it can be seen from Figs.  5(a)– 5(f) that the maximal densities of the species are shifted some distance from the axis for all the species, and they also tend to decrease along the axis when the discharge front leaves the tube exit. The constriction effect of the ring-shaped zone with increasing the distance from the tube exit can be attributed to air mixing, which has also been observed in experiment.^[19] Note that the densities of the metastable species (He* and N₂(C³π )) in the plasma jet are much higher than the charged species densities, which is consistent with the experimental results based on the optical emission spectrum.^[24]

Fig.  5.

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	Fig.  5. Calculated spatial density distributions of (a) He+ , (b) , (c) , (d) He* , (e) N2(C3π ), and (f) outside the dielectric tube, t = 270  ns. The blank regions in the figures represent the dielectric tube.

In this subsection, to investigate the influence of the needle electrode on the discharge characteristics, we first remove the needle electrode and simulate the plasma jet driven only by a single ring electrode under otherwise identical parameter conditions. The electron density and the total ionization rate are presented and compared with the results obtained with the needle electrode in Fig.  4. For simplicity, we ignore the change of the spatial distribution of helium mole fraction caused by removal of the needle.

Figure  6 shows the time evolutions of electron density and the total ionization rates of the plasma jet driven by the single ring electrode. Due to the absence of the needle, no discharge breakdown occurs in the first 8  ns. From 40  ns to 120  ns, the discharge is ignited at the edges of the ring and two ionization fronts propagate symmetrically outside the ring with the decrease of the discharge front radius. While in the case of DPE, the propagations of the two ionization fronts are asymmetrical, the intensity, and propagation velocity of the ionization front propagating upward (ring upstream plasma plume) decrease because of the effect of needle discharge.^[13] At t = 270  ns, the ring downstream plasma plume leaves the dielectric tube and propagates at a distance of about 3  mm from the tube exit. It should be noted that the plasma jet propagates at a distance of about 7.5  mm from the tube exit in the case of DPE at the same time moment as shown in Fig.  4(a). This implies that the existence of a needle can accelerate the propagation of the plasma jet.

Fig.  6.

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	Fig.  6. Discharge characteristic parameters of atmospheric pressure helium jet with single ring electrode at different moments. The distributions of (a) electron density, (b) the values of total ionization rate Se.

A comparison of species density distribution along the r axis at a distance of 0.2  cm from the exit of the nozzle (lines in Figs.  4(a) and 6(a)) between two kinds of electrode structures is given in Fig.  7. In both cases, the peaks of species densities are shifted some distance away from the axis, and nearly at the same position for both electrode configurations. The species densities are higher in the case of DPE, especially for N₂(C³π ) and , which are about 6 times as much as those in the case of the single ring electrode. It is also noted that the width of the radial distribution of species density for DPE is larger than that for the single ring electrode. It means that the existence of a needle can increase both the density and the discharge width of the plasma jet.

Fig.  7.

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	Fig.  7. Species distributions along the r axis at 0.2  cm from the exit of the nozzle for two kinds of electrode configurations: (a) electron density, (b) He+ density, (c) density, (d) density, (e) He* density, (f) N2(C3π ) density, and (g) density, t = 270  ns.

To further investigate the underlying mechanisms of a DPE plasma jet discharge, we plot the axial profiles of the electric field evolution and the propagation velocity of the ionization front during the propagation of the ring downstream plasma plume for two kinds of electrode configurations in Fig.  8. The propagation velocity of the discharge for DPE is much faster than that for a single ring electrode after 120  ns. It can be seen from Fig.  8(a) that the maximum field decreases more slowly in the case of DPE from 120  ns to 220  ns. As mentioned above, the ionization fronts of the needle merge with that of the ring and a discharge channel is thus created at time t = 120  ns as shown in Fig.  4. The existence of the needle can affect the discharge characteristics of the ring downstream plasma plume only after the needle discharge has been connected with the ring upstream plasma plume.

Fig.  8.

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	Fig.  8. (a) Axial profiles of the electric field at several time moments for two kinds of electrode configurations. The filled region represents the position of the ring electrode. (b) Propagation velocities of the ionization front as a function of time for the two kinds of electrode configurations.

To explain why the ionization front of the ring downstream plasma plume is enhanced when the needle discharge merges with the discharge at the ring electrode, the axial profiles of the space charge and the electric field at t = 70  ns (the time before mergence) and 115  ns (when it begins to merge) are plotted in Fig.  9. From Fig.  9(a), it can be seen that after the needle discharge interacts with the discharge at the ring, a large number of positive space charges collect at the mergence location (corresponding to the strong electric field at z = 2  cm in Fig.  9(b)), which can be seen as a “ virtual anode” . This “ virtual anode” will generate an electric field, which enhances the total electric field in the ionization front from the ring electrode, as shown in Fig.  9(b). Consequently, the ionization front is enhanced and the propagation velocity of the plasma jet increases.

Fig.  9.

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	Fig.  9. Axial profiles of (a) the positive space charge and (b) the electric field at 70  ns and 115  ns.

As mentioned above, we can see that the existence of the needle has a significant effect on the characteristics of the helium plasma jet. Therefore, it is of interest to study the influences of the needle parameters on the helium plasma jet to further optimize the design of a plasma jet source.

First, we study the influence of the needle radius on the discharge by changing the needle radius from 3  mm to 8  mm. Figure  10 shows the comparison among electron densities at t = 270  ns after the start of pulse excitation. As shown in Fig.  10(a), a smaller needle radius results in a higher electron density in the region close to the needle. However, it is a surprise that the radius of the needle insignificantly affects the structure and propagation of the plasma jet. The values of length, density, and discharge width of the plasma jets are basically the same for needles with different radii.

Fig.  10.

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	Fig.  10. Comparison of electron density among different needle radii at 270 ns after the start of pulse excitation: (a) two-dimensional distributions of electron density, (b) radial distribution of electron density at 0.2  cm from the exit of the nozzle (line in plasma jet of Fig.  10(a) indicates the position of the radial profile).

To explain this phenomenon, we plot the axial distributions of the electric field at several times for different needle radii (3  mm, 5  mm, and 8  mm). As can be seen from Fig.  11, the initial field (Laplacian field at t = 0  ns) close to the needle tip can be significantly larger for a smaller radius (similar with Ref.  [25]), which only affects the initial stages of the needle discharge. In general, the physical shape of the needle electrode has little influence on the propagation process of discharge, which is in agreement with the results obtained in Ref.  [26]. Varying the needle radius only has an effect on the electron density close to the needle tip in the dielectric tube, but has a relatively low influence on the characteristics of the helium plasma jet.

Fig.  11.

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	Fig.  11. Axial profiles of the electric field at several time moments for different needle radii, (a) 3  mm, (b) 5  mm, and (c) 8  mm.

Then, the influence of the needle-to-ring discharge gap on the plasma jet is investigated by changing the distance from 0.5  cm to 2  cm. The simulated electron density profiles for four cases are shown in Fig.  12, indicating that the needle-to-ring discharge gap has two principal effects on the characteristics of the plasma jet. First, a smaller needle-to-ring discharge gap results in a longer plasma jet in ambient air. As shown in Fig.  12(a), the plasma jet length is 4.2  mm in the 2-cm discharge gap case, 5.3  mm in the 1.5-cm case, 6.5  mm in the 1-cm case, and 7.7  mm in the 0.5-cm case, at t = 240  ns. The second effect of discharge gap is that the density and discharge width of the plasma jet increase with the decrease of the discharge gap. All these can be attributed to the less time required for the needle discharge front and ring upstream front to merge in the case of a smaller distance.

Fig.  12.

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	Fig.  12. Comparison of electron density among different needle-to-ring discharge gaps at 240  ns after the start of pulse excitation: (a) two-dimensional distributions of electron density, (b) radial distributions of electron densities at 0.2  cm from the exit of the nozzle (line in plasma jet of Fig.  12(a) indicates the position of the radial profile).