# Source of the Recirculation Induced by Nanosecond Discharges

Edouard Roger, P Mariotto, Christophe O Laux

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Edouard Roger, P Mariotto, Christophe O Laux. Source of the Recirculation Induced by Nanosecond Discharges. AIAA SCITECH 2025 Forum, American Institute of Aeronautics and Astronautics, Jan 2025, Orlando (Florida), United States. 10.2514/6.2025-2824. [hal-04934431](https://HAL Id: hal-04934431)

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# Source of the Recirculation
## Induced by Nanosecond Discharges

Edouard Roger<fnref n="1" />, Pierre Mariotto<fnref n="2" /> and Christophe O. Laux<fnref n="3" />

Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C,
91190 Gif-sur-Yvette, France

Nanosecond spark discharges generate nonequilibrium plasmas whose chemical, thermal, and hydrodynamic effects are of great interest for various applications, including plasma-assisted combustion and carbon dioxide conversion. This article focuses on the post-discharge recirculating flow, one of the hydrodynamic effects induced by the discharge. This effect has promising industrial applications because it may enhance the mixing of the plasma products with the surrounding environment. Understanding the origin of recirculation will enable the optimization of its benefits. This study presents a detailed investigation of vorticity generation during the post-discharge phase of nanosecond discharges through computational fluid dynamics (CFD) simulations. We show that recirculation originates from the non-uniformity of the shock expansion. The baroclinic torque and viscous dissipation play a secondary role. Finally, we propose a new dimensionless number $\Pi^*$ that characterizes the intensity of the shock-induced vorticity, from which we develop a criterion to predict the occurrence of the recirculation regime induced by nanosecond discharges. We validate this criterion against several experimental and simulation studies from the literature.

## I. Nomenclature

$c_v^{tr-rot}$ = trans-rotational heat capacity of the gas [J/K/kg]

$dt'$ = time step [s]

$dx$ = mesh size [m]

$\delta$ = Dirac delta function

$\Delta X$ = difference between the values of the physical quantity X on the two sides of the shock discontinuity

$\vec{e}_\theta$ = azimuthal unit vector

$E_{uh}$ = portion of pulse energy leading to ultrafast heating [J] (defined in [1])

$\gamma_k$ = adiabatic index of the hot plasma kernel after the discharge

$\gamma_{out}$ = adiabatic index of the surrounding gas

$H$ = Heaviside step function

$I_{mech}$ = cumulative integrated vorticity due to the mechanism referred to as “mech” [1/s]

$L$ = length of the interelectrode gap [m]

$M(r, z)$ = local Mach number of the upstream flow in the shock reference frame

$M_s$ = initial Mach number of the shock front

$M_k$ = molar mass of the hot plasma kernel after the discharge [kg/mol]

$M_{out}$ = molar mass of the surrounding gas [kg/mol]

$\vec{n}$ = unit vector normal to the shock front

$n_{freedom}$ = number of trans-rotational freedom degrees of the gas

$\vec{\nabla}_t$ = tangential gradient (along the shock front) [1/m]

$\omega$ = azimuthal vorticity [1/s]

$P$ = pressure [Pa]

<footnote n="1">¹ Ph. D. student.</footnote>

<footnote n="2">² Research scientist.</footnote>

<footnote n="3">³ Professor, AIAA Fellow.</footnote>
$P_k$ = pressure of the hot plasma kernel after the discharge [Pa]

$P_{out}$ = pressure of the surrounding gas [Pa]

$\Pi^*$ = characteristic number of the flow recirculation (defined in this article, Eq. (11))

$\Pi$ = characteristic number of the flow recirculation (defined in [2])

$(r, z)$ = radial and axial coordinates in a cylindrical coordinate system [m]

$\rho$ = density [kg/m³]

$\rho_0, \rho_1$ = density upstream/downstream of the shock front [kg/m³]

$R_d$ = discharge radius [m]

$R$ = perfect gas constant [J/K/mol]

$S_{mech}$ = instantaneous vorticity source due to any mechanism referred to as “mech” [1/s²]

$t, t'$ = time [s]

$\tau$ = stress tensor [Pa]

$\theta$ = azimuthal coordinate [rad]

$T_k$ = temperature of the hot plasma kernel after the pulse heating [K]

$T_{out}$ = temperature of the surrounding gas [K]

$t_{shock}$ = instant at which the shock reaches a specific point [s]

$\vec{u}$ = velocity of the flow [m/s]

$u_{0,n}$ = normal velocity of the upstream flow relative to the shock front [m/s]

$V$ = volume of the discharge [m³]

## II. Introduction

Nonequilibrium plasma discharges offer a promising solution for reducing the carbon footprint of certain industrial processes. They use electrical energy to convert inert gases into valuable products. In particular, Nanosecond Repetitively Pulsed (NRP) discharges are a highly efficient method for forming such nonequilibrium plasmas [3,4]. They have been used to assist combustion by reducing the lean blow-out limit or controlling instabilities for plasma-assisted combustion [5–7], to convert CO₂ into CO for the synthesis of low-carbon fuels [8,9], to produce nitric oxides and acidify organic waste to improve fertilizers [10], or for methane pyrolysis [11]. These discharges use high voltage pulses (1–25 kV) applied for a few nanoseconds at high-repetition frequency (1–100 kHz) between two electrodes separated by a few millimeters, see Fig. 1. They induce a plasma kernel that fills the interelectrode gap.

<figure><img src="image_2.png" /><figcaption>Fig. 1 Typical voltage profile of NRP discharges.</figcaption></figure>

Depending on the discharge regime (glow, non-thermal or thermal sparks), the plasma kernel can reach a wide range of pressures (1–100 atm), temperatures (1000–30,000 K), and dissociation-ionization degrees ($10^{-6}$–1) [12–14]. The formation of the kernel occurs on a timescale shorter than the hydrodynamic timescale and can be considered isochoric [1,15]. After the pulse, the energy stored in the kernel is released into the surrounding gas through the propagation of a blast wave [15] and diffusion. Experiments and CFD simulations showed that two counter-rotating recirculation cells may appear between the electrodes and collapse to expel the hot gas radially [2,16–18]. This is evidenced in Fig. 2 (taken from Ref. [19]), which represents Schlieren images recorded during the post-discharge of a nanosecond pulse. At 10 ns, the leading shock of the blast wave is attached to the plasma kernel. The images recorded between 2 and 8 µs show the propagation of the shock. The images obtained between 20 and 300 µs show a radially expanding torus of hot gases.
<figure><img src="image_3.png" /><figcaption>Fig. 2 Schlieren images of a single nanosecond spark (disharge and post-discharge) in air, at atmospheric pressure. The distance between the pins is 1 mm. The applied voltage is 36 kV. The filament structure of the spark is clear at 10 ns. The shockwave propagates on a microsecond timescale. The discharge-induced radial flow is visible after 20 µs.</figcaption></figure>

The physics behind the radial expulsion of hot gases into a torus-shaped volume has already been studied in different works, mostly converging to the same explanation. After the shock leaves the plasma kernel, vorticity is created near the electrode tips, and two opposite recirculation cells appear. They push the hot gases axially along the interelectrode axis while bringing cold ambient gases between the electrodes. The two opposite axial flows meet and radially expel a torus-like volume. In Ref. [20], the authors explained the global fluid movement as the result of two recirculation cells. In Ref. [18], recirculation is described as arising from the convergence of two vortex rings moving through self-advection [21]. However, the origin of the vorticity remains a subject of debate in the literature. Some researchers attributed it to an abrupt vorticity jump generated by the non-uniformity of the leading shock front [22]. Others proposed that the primary driver is the baroclinic torque, also induced by the shock wave [2,20]. Alternatively, some studies emphasized the role of viscous effects in generating recirculation in shear layers along the electrodes [23]. This lack of consensus motivates the need for a clearer understanding of the dominant mechanisms responsible for vorticity production. To our knowledge, no quantitative comparison have been made between all the vorticity sources for nanoseconds discharges. Addressing this knowledge gap will not only advance the fundamental understanding of the underlying physics but also support the optimization of reactor design by making the right adjustments to pulse frequency, energy input, and inter-electrode gap dimensions. [24].

<figure><img src="image_4.png" /><figcaption>Fig. 3 Sketch of the recirculating flow pattern induced by a nanosecond discharge. Left: Departure of the shock and creation of the vortices (~1 µs). Right: Radial expulsion of hot gases into a torus-like volume (~50 µs).</figcaption></figure>

To determine the primary source of vorticity in pin-to-pin nanosecond discharges, we first conduct CFD simulations of the post-discharge for a typical non-thermal spark configuration in air that results in a recirculation
flow (Section III). These simulations provide 2D velocity, pressure, and density fields at each time step. Second, we use these 2D fields to compute the magnitudes of all sources of vorticity, which we compare quantitatively (Subsection IV.A) to determine the dominant vorticity creation mechanism. Third, we use the previous conclusion to build a new characteristic number that quantifies the vorticity generated by the discharge (Subsection IV.B). We compute this number for many post-discharge flows taken from the literature (experiments and simulations), and we propose a criterion to predict the post-discharge flow regime (with or without recirculation).

### III. Numerical modeling

We perform a 2D axisymmetric Computational Fluid Dynamics (CFD) simulation of the interpulse flow between two pin electrodes. A schematic of the geometry for the study is given in Fig. 4. We restrict the calculation to the upper right quadrant of the domain because of the symmetry along the interelectrode plane and the interelectrode axis. We solve the non-reactive Navier-Stokes equation with the commercial solver Ansys Fluent 2023 [25]. The simulations start after the nanosecond pulse. Regarding the initial conditions, the surrounding gas is air at atmospheric pressure $P_{\text{out}}$ and at a temperature $T_{\text{out}} = 300$ K. The initial pressure and temperature between the electrodes are increased by the pulse and reach up to $T_k = 6000$ K, and $P_k = P_{\text{out}}T_k/T_{\text{out}} = 20$ bar inside an elliptic domain with a semi-minor axis of length $R_d = 100$ µm and a semi-major axis of length $L/2 = 1.5$ mm, where L is the interelectrode gap. The set of equations is solved with the finite-volume method and a first-order implicit temporal integration scheme. We use the Roe solver to calculate convective fluxes, and the second-order central differencing scheme for diffusive fluxes. We choose the MUSCL scheme to spatially reconstruct the fields at the cell boundaries. The mesh size is about $dx = 2$ µm with clustering near the electrodes. The fluid is assumed to be in Local Thermal Equilibrium (LTE) and its properties (heat capacity, thermal conductivity, viscosity) are computed with polynomial fits taken from the literature [26].

<figure><img src="image_5.png" /><figcaption>Fig. 4 Computational domain schematic. The initial conditions of the gases are indicated.</figcaption></figure>

## IV. Results and discussion

### A. Recirculation origin and shape

The resulting temperature fields are shown in the left column of Fig. 5 at several instants, from 0 to 240 µs. The flow pattern is consistent with the experimental images of Fig. 2. At $t = 200$ ns, an expanding elliptic shock propagates into the surrounding gases. Quickly (from $t = 3$ µs to $t = 10$ µs), a recirculation cell appears close to the tip of the
electrode. This cell progressively brings cold gases into the gap (from $t = 10$ µs to $t = 120$ µs). Later, the cell expels the hot gases into a torus near the mid-gap axis (at $t = 240$ µs).

<figure><img src="image_6.png" /><figcaption>Fig. 5 Two-dimensional maps of temperature (left) and vorticity (right) at several instants. The white dot indicates the instantaneous position of the tracked fluid particle, and the white circle indicates its initial position. The white line indicates its trajectory in time.</figcaption></figure>
We analyze the recirculation pattern by examining the azimuthal vorticity $\omega \equiv (\vec{\nabla} \times \vec{u}) \cdot \vec{e}_{\theta}$, shown in the right column of Fig. 5 at the same instants. The vorticity maps exhibit positive and negative vorticity zones. They correspond to counter-clockwise and clockwise rotations, respectively. At $t = 10$ µs and later, there is a strong core of negative vorticity where $|\omega| > 10^6$ s$^{-1}$. The center of the negative vorticity zone is located at each instant in the core of the global recirculation pattern observed on the temperature fields. The axial translation of the core of this vortex is consistent with the process of vortex ring self-advection described in Ref. [18].

Our strategy to identify the origin of the recirculation flow is as follows. First, we identify a particle of fluid belonging to the vortex ring's core observed in the simulation. Second, we compute the trajectory of the fluid particle with a Lagrangian tracking. Third, we compute the temporal profile of the particle's vorticity, which is negative as it belongs to the core of the vortex. Fourth, we compute the different source/sink terms (expansion, viscous dissipation, baroclinic torque, shock-induced) that influence the vorticity of the particle along its trajectory. Fifth, we compare the respective intensity of these various source/sink terms.

In the right column of Fig. 5, the position of the selected fluid particle is marked with a white dot. The particle's initial position is marked with the white circle, and its trajectory is indicated by the white line. The particle originates near the tip of the electrode and always stays inside the vortex ring. After about 100 µs, the particle is expelled radially. Before calculating vorticity and the different source terms along the trajectory, we briefly review the literature on the mechanisms that create or annihilate vorticity in the following paragraph.

The most common vorticity sources are the first four terms of the Right-Hand Side (RHS) of Eq. (1), referred to as the Helmholtz equation [27] (written here with a lagrangian formalism). These four terms account for mechanisms occurring in the continuous regions of the flow (i.e., neither interfaces nor shocks): the baroclinic torque, the viscous effect, the expansion term, and the stretching term (null in 2D). In addition to these four continuous source terms of vorticity, an instantaneous jump of vorticity $\Delta\omega$ may be generated downstream of a shock of non-uniform strength. The value of this jump was analytically determined by Truesdell [28] and generalized by Lighthill [29]. A non-uniform jump across the shock creates an entropy gradient downstream. Thus, it creates vorticity, according to the Crocco theorem. Later, Hayes relaxed the steady flow and isenthalpic flow assumptions [30], avoiding using the Crocco theorem in the demonstration. Instead, he only used the Rankine-Hugoniot jump conditions. With this approach, Hayes showed that a vorticity jump exists even in unsteady or non-uniform flows. We add this last source term to the RHS of Eq. (1). This term is non-zero at a given flow point only when the shock crosses that point; see Eq. (2).

$$ \frac{d\omega}{dt} = S_{\text{barocline}} + S_{\text{viscous}} + S_{\text{expansion}} + S_{\text{stretching}} + S_{\text{shock}} \quad (1) $$

with:

$$ S_{\text{shock}} = \Delta\omega \delta(t - t_{\text{shock}}) \quad (2) $$

$$ S_{\text{advection}} = -\vec{u} \cdot \vec{\nabla} \omega \quad (3) $$

$$ S_{\text{expansion}} = -(\vec{\nabla} \cdot \vec{u})\omega \quad (4) $$

$$ S_{\text{barocline}} = \frac{1}{\rho^2} (\vec{\nabla} \rho \times \vec{\nabla} P) \cdot \vec{e}_{\theta} \quad (5) $$

$$ S_{\text{viscous}} = \left( \nabla \times \left( \frac{1}{\rho} \nabla \cdot \frac{\tau}{\tau} \right) \right) \cdot \vec{e}_{\theta} \quad (6) $$

$$ S_{\text{stretching}} = 0 \quad (\text{in 2D}) \quad (7) $$

The computed temporal vorticity profile of the particle is plotted at the top of Fig. 6. For $t < 0.2$ µs, the vorticity is zero. When the shock reaches the particle (at $t = t_{\text{shock}}$), the vorticity abruptly falls from $0$ s$^{-1}$ to $-10^6$ s$^{-1}$. Then, the vorticity remains approximately unchanged until a few tens of microseconds. This duration corresponds to the time needed for the vortex ring to move axially along the interelectrode axis by self-advection. It increases slightly, up to $-10^5$ s$^{-1}$ after $t \approx 80$ µs. At this instant, the vortex ring reaches the interelectrode plane and collides with the counter-rotating vortex ring created in the other half-plane. We compare the contributions of the different mechanisms (shock-induced vorticity, baroclinic torque, viscosity, expansion) to vorticity variations over time by integrating their instantaneous values at the location of the particle $S_{\text{mech}}(t)$. We get the cumulative integrated vorticity $I_{\text{mech}}(t)$ that stores the vorticity at the point of interest created by the mechanism “mech” until the instant $t$, see Eq. (8).

$$ I_{\text{mech}}(t) \equiv \int_{0}^{t} S_{\text{mech}}(t') dt' \quad (8) $$
With this formalism, we can compare the continuous sink-source terms (Eq. (3) to (7)) with the shock-induced discontinuous source term (Eq. (2)). The temporal profiles of each mechanism are plotted at the bottom of Fig. 6. The shock-induced mechanism is found to be the primary contributor to vorticity. In comparison, the baroclinic torque and the viscosity effects are significantly weaker. Their contribution to the particle’s vorticity is one order of magnitude lower. The expansion term is also weaker, especially during the first tens of microseconds (including the vortex ring creation). After a few tens of microseconds, expansion becomes dominant compared to the baroclinic torque and viscous diffusion. It generates a positive cumulative integrated vorticity equivalent to half the shock-induced value with a characteristic timescale of 80 µs. Thus, expansion explains the final slow increase of vorticity noted above.

<figure><img src="image_7.png" /><figcaption>Fig. 6 Vorticity (top) and the absolute value of the cumulative integrated vorticity sources (bottom) as a function of time. The vorticity sources are those given in the RHS of Eq. (1). Negative and positive integrated sources are represented by solid and dotted lines, respectively. All quantities are computed at the probe location marked by the white dot in the right column of Fig. 5.</figcaption></figure>

## B. Influence of the pulse heating

In the previous Subsection, we showed that the dominant source of vorticity is the instantaneous shock-induced mechanism. Here, we demonstrate that the major factor driving the vorticity generation is the ratio between $T_k$ and $T_{out}$. We can decompose this reasoning into four steps:

1) The intensity of the shock-induced source at any point $\Delta\omega(r, z)$ of the domain crossed by the shock is directly related to the shock speed $u_{0,n}(r, z)$ by a formula derived by Hayes, see Eq. (9). This formula is valid only if the flow is normal to the shock, which is the case for a shock expanding in a medium at rest.

$$ \Delta\omega = \left[ -\vec{n} \times (\vec{\nabla}_t u_{0,n}) \left( \frac{\rho_1}{\rho_0} - \frac{\rho_0}{\rho_1} - 2 \right) \right] \cdot \vec{e}_{\theta} \quad (9) $$

$\vec{n}$ is the normal vector of the shock front, $\vec{\nabla}_t$ corresponds to the tangential gradient (along the shock front), $\rho_i$ is the density of the flow in the region $i$. Subscripts 0 and 1 refer to the flow upstream and downstream of the
shock, respectively. The upstream flow is also denoted by the subscript “out” in the other paragraphs of this paper, as the upstream domain corresponds to the outside domain in the studied flow configuration.

2) $\rho_0$ and $\rho_1$ are related to the local shock Mach number by the Rankine-Hugoniot jump conditions. In addition, the normal relative velocity of the shock $u_{0,n}(r, z)$ is proportional to the local Mach number $M(r, z)$ (and to the sound speed in the ambient gases). Thus, $\Delta\omega(r, z)$ is a function of $M(r, z)$.

3) In Ref. [31], the author developed a model for the expansion of non-planar shock-waves and suggested that the local Mach number $M(r, z)$ is proportional to the initial shock Mach number $M_s$ (with the proportionality factor depending on the radius of the shock and the shape of the shock front).

4) The initial Mach number of the shock $M_s$ is the implicit solution of the classical shock tube relation (see Ref. [31] and Eq. (10)). Thus, $M_s$ depends on the initial heating of the discharge $T_k/T_{out}$, the increase of pressure during the pulse $P_k/P_{out}$, and also the adiabatic indices and molar masses of the surrounding ambient gas $\gamma_{out}$, $\mathcal{M}_{out}$ and those of the plasma $\gamma_k$, $\mathcal{M}_k$. The pressure ratio is proportional to the temperature ratio and the molar masse ratio because the pulse heating is isochoric. The impact of the changes in adiabatic index and molar mass on $M_s$ can be neglected compared to the impact of the discharge heating $T_k/T_{out}$ [32]. Thus, the initial Mach number is an implicit function of $T_k/T_{out}$.

$$ \frac{P_{\text{out}}}{P_{\text{k}}} \left( \frac{2\gamma_{\text{out}} M_{\text{s}}^2}{\gamma_{\text{out}} + 1} - \frac{\gamma_{\text{out}} - 1}{\gamma_{\text{out}} + 1} \right) = \left( 1 - \frac{\gamma_{\text{k}} - 1}{\gamma_{\text{out}} + 1} (M_{\text{s}} - M_{\text{s}}^{-1}) \frac{\sqrt{\gamma_{\text{out}}(R/\mathcal{M}_{\text{out}})T_{\text{out}}}}{\sqrt{\gamma_{\text{k}}(R/\mathcal{M}_{\text{k}})T_{\text{k}}}} \right)^{2\gamma_{\text{k}}/(\gamma_{\text{k}}-1)} \quad (10) $$

The four steps of the above reasoning suggest that the temperature ratio between the plasma and the ambient gases governs the intensity of the instantaneous shock-induced vorticity and, consequently, of the recirculation created by the discharge. Thus, we propose a new characteristic number $\Pi^*$ (see Eq. (11)) to predict the post-discharge flow regime:

$$ \Pi^* \equiv \frac{T_k}{T_{\text{out}}} \qquad (11) $$

$\Pi^*$ is the ratio of the translational temperature of the plasma kernel (before the shock departure) to that of the surrounding gas. At first sight, this characteristic number looks similar to the $\Pi$ number proposed in Ref. [2] and reported in Eq. (12), which was derived using the Vaschy-Buckingham theorem:

$$ \Pi \equiv \frac{E_{\text{uh}}}{P_{\text{out}}V} \qquad (12) $$

where $E_{uh}$ is the portion of pulse energy leading to ultrafast heating [1], $P_{out}$ the ambient pressure and $V$ the discharge volume. Despite their similar-looking formulas, $\Pi^*$ presents some advantages over $\Pi$. First, $\Pi^*$ does not depend on $E_{uh}$ which is not well-known for all gas mixtures and discharge regimes (e.g., the thermal spark regime). Second, the $\Pi$ number depends on the discharge radius, which is difficult to measure and/or define in experiments. On the contrary, the discharge radius does not appear in the expression of $\Pi$. Third, $\Pi^*$ characterizes the shock-induced vorticity regardless of the nature of the gas contrary to $\Pi$. Indeed, using the volumetric energy instead of the post-heating temperature hides the dependency on the heat capacity. Thus, the same volumetric energy input inside a discharge of argon or methane would not result in the same temperature ratio $T_k/T_{out}$ (because there is a factor 2 difference between the number of trans-rotational degrees of freedom between the two gases). Consequently, the shock leaving the discharges is significantly stronger in the case of the argon discharge, and therefore the vorticity generated is higher. This distinction is not captured by the $\Pi$ number.

It is interesting to note that the $\Pi^*$ number can be related to the $\Pi$ number under certain conditions. This is the case if (1) the molar mass of the gas does not change during the pulse (i.e., $\mathcal{M}_k = \mathcal{M}_{out} = \mathcal{M}$), (2) the pulse heating is isochoric, and (3) the ultrafast heating induced by the pulse before the shock detaches from the plasma kernel is purely trans-rotational (the vibrational and electronic energy modes remain frozen during the pulse). Under these conditions, we get:

$$ c_v^{\text{tr-rot}} = \left( \frac{n_{\text{freedom}}}{2} \right) \times \left( \frac{R}{\mathcal{M}} \right) \qquad (13) $$

and:

$$ \Pi = \frac{V \rho_{\text{out}} (n_{\text{freedom}}/2) R \mathcal{M} (T_k - T_{\text{out}})}{\rho_{\text{out}} R \mathcal{M} T_{\text{out}} V} = \frac{n_{\text{freedom}}}{2} \left( \frac{T_k}{T_{\text{out}}} - 1 \right) \qquad (13) $$
Thus:

$$ \Pi^* = \frac{2}{n_{\text{freedom}}} \Pi + 1 \quad (14) $$

To test the robustness of $\Pi^*$ as an indicator of the recirculation intensity, we calculate this number for various discharges studied in the literature (experiments [20,33–36] and simulations [2,18,20,23,37–39]). We limit our selection to cases where the plasma temperature, the ambient temperature, and the post-discharge flow regimes are known a few microseconds after the pulse. The results are shown in Fig. 7. We also perform additional CFD simulations to complete this graph. The simulations correspond to the configurations described in Section III, varying the gap length or the initial plasma temperature. We identify in Fig. 7 a distinction between the cases for which $\Pi^* \gtrsim 10$ and the othersFor values of $\Pi^*$ above 10, the flow recirculates, and a torus of hot gases is expelled from the interelectrode axis. On the contrary, if $\Pi^*$ is below 10, no recirculating cells exist, nor any torus.

This result confirms that $\Pi^*$ is a parameter that effectively characterizes the vorticity generated after a nanosecond pulse. It can be used to predict the flow regime during the post-discharge of a nanosecond pulse with the criterion discussed above. Essentially, this criterion compares the intensity of the total shock-induced vorticity with that of the vorticity loss mechanisms (expansion and viscous dissipation). The threshold value $\Pi^* \approx 10$ corresponds to the scenario where the creation and loss terms are of equal magnitude. While the shock-induced vorticity source term does not depend on the gas composition, the viscous and expansion loss term do, given that viscosity, molar mass and adiabatic index are composition-dependent. Hence, the threshold value for $\Pi^*$ is expected to be gas-dependent. $\Pi^* \approx 10$ is the approximate threshold value for air of nitrogen mixtures.

Finally, it should be emphasized that recirculation (due to a high value of $\Pi^*$) does not necessarily result in the complete cooling of the interelectrode region nor the ejection of a torus. These also depend on the configuration parameters of the NRP discharges, such as the interelectrode gap length and the pulse frequency. For large gap lengths, recirculation cells will collapse into a torus only if the pulse frequency is low enough because the vortex rings need more time to travel across the interelectrode region.

<figure><img src="image_8.png" /><figcaption>Fig. 7 Dimensionless number $\Pi^* \equiv T_k/T_{\text{out}}$ predicting the transition between the recirculating and non-recirculating regimes. The shaded area ($\Pi^* = 8–12$) represents the uncertainty in the threshold value that separates the recirculating and non-recirculating regimes. The data points were obtained from CFD simulations conducted in this study, as well as CFD simulations [2,18,20,23,37–39] and experimental studies [20,33–36] reported in the literature. Post-discharges in the recirculating and non-recirculating regimes are represented by green and red symbols, respectively.</figcaption></figure>

## V. Conclusions

This study investigated the origin of the recirculation that sometimes follows nanosecond discharges. We identified the dominant mechanisms responsible for the generation and dissipation of vorticity. To this end, we performed a 2D CFD simulation of the post-discharge after a nanosecond pulse in air. We observed the formation of a vortex ring, which leads to the radial ejection of hot gases into a toroidal volume. Then, we tracked the trajectory of a fluid particle belonging to the vortex ring. We computed the vorticity along the trajectory and integrated the separate contributions
of the mechanisms that affect the vorticity. We observed an abrupt jump of vorticity at the instant at which the shock meets the fluid particle. In addition, we quantitatively compared the effects of the different vorticity source and sink terms. The shock-induced vorticity due to the non-uniformity of the shock curvature dominates the other mechanisms by one order of magnitude. The baroclinic torque and the viscous dissipation play a minor role. Our findings differ from the conclusions of several studies. We introduced a new dimensionless parameter, denoted as $Π^*$, which represents the ratio of the translational temperature of the plasma kernel before the shock departure to that of the surrounding gas. We demonstrated that the amount of vorticity generated by the shock is related to this parameter $Π^*$. Finally, we analyzed experiments and simulations from existing literature and showed that the transition from a non-recirculating regime to a recirculating regime occurs when $Π^*$ approaches 10.

## Acknowledgments

This work has received funding from the Ministerial Allocation for Polytechnique students (AMX) and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC GreenBlue grant agreement No.101021538).

A CC-BY-4.0 public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.

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