2.1. Geometry

The geometry investigated is shown in figure 1, and represents a simplified axisymmetric version of the torch and test chamber sections of the plasma wind tunnel. The plasma is generated by induction heating of a test gas injected into the torch through an annular inlet, and is blown in the test chamber forming a coaxial round jet. In practice, the swirling of the jet is prevented by the presence of a foam piece before the inlet, and simulations will consider a null azimuthal velocity. For the rest of this study, the coordinates $(r,\theta ,z)$ are used to describe namely the radial, azimuthal and streamwise directions. This work investigates different test cases obtained by varying the main driving parameters of the facility, respectively the static pressure in the test chamber noted $P_s$ and the power transmitted through induction by the coils to the flow noted $P_{el}$. A summary of the conditions studied is given in table 1. Pressures are kept above 100 mbar to maintain the flow in equilibrium, based on experimental observations of Cipullo et al. (Reference Cipullo, Helber, Panerai, Zeni and Chazot2014). Although the jet is horizontal, no buoyancy effects are observed in the facility for powers under 200 kW and below 250 mbar, and are therefore also ignored here. A third driving parameter is usually considered in experiments, namely the mass flow $\dot {m}$ injected at the torch inlet. In the present study $\dot {m}$ is kept constant at 16 g s$^{-1}$ and its influence is not considered. The reference length used throughout this work is the torch radius $l_{ref}=r_{torch}=0.08\ \textrm {m}$.

Figure 1.

Sketch of the computational domain used for Plasmatron simulations. The main dimensions and boundary conditions are indicated.

Table 1.

Parameters defining each test case studied, along with main quantities and non-dimensional numbers at the chamber inlet.

2.1.1. Simulations of the plasma wind tunnel

Simulations are carried out within the open source Computational Object-Oriented Library for Fluid Dynamics CoolFluid-ICP (Lani Reference Lani2008), relying on a second-order spatially accurate finite-volume solver, and an implicit Newton procedure to reach a steady state solution. Both the magnetic and fluid fields are considered by weakly coupling the Maxwell equations to the compressible Navier–Stokes equations. The resulting flow equations are summarized in their dimensional form below

(2.1a)\begin{gather} \frac{\partial}{\partial t}(\rho)+\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho \boldsymbol{u})=0, \end{gather}

(2.1b)\begin{gather}\frac{\partial}{\partial t}(\rho\boldsymbol{u})+\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol{u}\otimes\boldsymbol{u}) = -\boldsymbol{\nabla} p + \boldsymbol{\nabla}\boldsymbol{\cdot}\tau + \mathcal{F}_L, \end{gather}

(2.1c)\begin{gather}\frac{\partial}{\partial t}(\rho E)+\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho H\boldsymbol{u}) = \boldsymbol{\nabla}\boldsymbol{\cdot}\tau\boldsymbol{\cdot}\boldsymbol{u} - \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{q}+\mathcal{P}_J, \end{gather}

where $\tau$ is the viscous stress tensor, $E$ the total energy defined as $E=e+\frac {1}{2}\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {u}$ with the specific energy $e$, $H$ is the total enthalpy given by $H=E+{p}/{\rho }$ and $\boldsymbol {q}$ is the heat flux vector. The coupling is done through Joule heating $\mathcal {P}_J$ and the Lorentz force $\mathcal {F}_L$ respectively in the energy and momentum equations, and the temperature dependence of the gas electric conductivity. The reader is referred to Zhang, Lani & Panesi (Reference Zhang, Lani and Panesi2016) for a complete breakdown of the variables and electromagnetic equations. Details of the implementation are available in the reports of Honzatko (Reference Honzatko2006) and Prokop (Reference Prokop2007). Thermodynamic and transport properties of the gas are obtained using the Mutation$^{++}$ library developed by Scoggins & Magin (Reference Scoggins and Magin2014), assuming an 11-species air mixture including (N, O, NO, N2, O2, N$+$, O$+$, NO$+$, N2$+$, O2$+$, e$-$) in LTE. It is worth noting that such mixture allows for ionization effects on the flow properties.

An open boundary condition allowing for entrainment of gas is imposed at the top edge of the computational domain: homogeneous Neumann conditions are imposed on all quantities except for the pressure which is set to the desired $P_s$. A subsonic outflow condition is set at the outlet, no-slip isothermal walls are considered with $T_{wall}=350\ \textrm {K}$ to mimic the cooled walls of the facility. At the inlet, the streamwise velocity is imposed through the mass flow $\dot {m}$, the pressure is interpolated from the interior of the domain and the temperature is set to the wall value. The convergence of the solution is investigated by computing results on a series of meshes with increasing definition. For our set-up, a mesh of $23.10^3$ elements is enough to approach results of the finest mesh with precision of 0.03 % of the maximum temperature and 0.06 % of the maximum velocity. However, the solution does not provide second derivatives of flow quantities, and an algebraic model is fit to the simulations.

2.1.2. Base state

Our algebraic model is built by interpolating the non-dimensional streamwise velocity and temperature radial profiles at $177$ discrete streamwise positions along the test chamber. The interpolation is carried out in Matlab^® using the fit routine. The following expressions were found to reproduce the flow field for the range of conditions studied and streamwise positions from $z=0.5$ to $z=3.5\ \textrm {m}$:

(2.2a)\begin{align} W(r) &= a_WGN(r, 0, b_W) + c_W(1-\tanh[d_W(r - e_W)]) \nonumber\\ &\quad + f_W(1-\tanh[g_W(r - h_W)])GN(r,i_W,j_W)r^2, \end{align}

(2.2b)\begin{align} T(r) &= a_TGN(r, 0, b_T)+ c_T(1- \tanh[d_T(r - e_T)]) + f_T(1 - \tanh[g_T(r- h_T)]) \nonumber\\ &\quad + i_T(1 - \tanh[j_T(r - k_T)]), \end{align}

(2.2c)\begin{equation} GN(r,\mu,\sigma) = \frac{1}{\sigma\sqrt{2{\rm \pi}}}\exp({{-\left(r-\mu\right)^2}/{2\sigma^2}}),\end{equation}

where $W$ and $T$ denote the streamwise velocity and temperature along the non-dimensional radial coordinate $r$. Values of the coefficients ($a_W, b_W,\ldots$) resulting from the interpolation are then used to obtain the analytical derivatives of the profile quantities. The error introduced by the interpolation remains bounded by respectively $0.4\,\%$ and $0.15\,\%$ of the temperature and velocity at the chamber inlet centreline with respect to simulation results. Examples of the interpolation coefficients for different streamwise positions are given in tables 4 and 5 in appendix A.

Thermodynamic and transport properties involved in the stability equations are reconstructed from the temperature profiles and constant pressure $P_s$ by assuming the same mixture in LTE as in the simulations. Flow properties are computed in the Vesta toolkit as it also evaluates derivatives of the properties with respect to pressure and temperature. We have observed that derivatives of the Mutation$^{++}$ properties numerically reconstructed exhibit a noisy behaviour, which is found to greatly impact the stability results. The equilibrium composition is obtained from the Dekaf boundary layer solver (Groot et al. Reference Groot, Miró Miró, Beyak, Moyes, Pinna and Reed2018), which solves the equilibrium system based on partial pressure (Miró Miró et al. Reference Miró Miró, Beyak, Mullen, Pinna and Reed2018) while enforcing the ambipolar condition. Transport properties are obtained similarly to Scoggins & Magin (Reference Scoggins and Magin2014): using approximations to Chapman and Enskog's expansion using Laguerre–Sonine polynomials following Magin & Degrez (Reference Magin and Degrez2004). Collisional data required for such properties are obtained from the latest references included in Mutation$^{++}$, ignoring both frozen and reactive barodiffusion, together with elemental demixing. Thermal properties are retrieved from the NASA polynomial database (McBride, Gordon & Reno Reference McBride, Gordon and Reno1993), and the mixing rules are applied to obtain the mixture quantities. More details about these models are available in Miró Miró et al. (Reference Miró Miró, Beyak, Pinna and Reed2019).

In order to assess the influence of the LTE assumption on the stability for a given base state, a second set of properties is reconstructed from the simulations by assuming a calorically perfect gas (CPG). The viscosity is retrieved using Sutherland's law, and a constant Prandtl number is imposed $Pr=\mu c_p/\kappa =0.7$, where $c_p$ is the air isobaric specific heat capacity, $\mu$ is the viscosity and $\kappa$ is the thermal conductivity.

Quantities involved in the stability equations are made non-dimensional with respect to the local centreline quantities denoted by the subscript ‘$_c$’, except for the reference pressure chosen as $P_{ref}=\rho _cW_c^2$. The main non-dimensional numbers involved in the LTE stability equations are given by

(2.3a–c)\begin{equation} Re = \frac{\rho_c~W_c~l_{ref}}{\mu_c},\quad Pr=\frac{\mu_ch_c}{\kappa_c\,T_c},\quad Ec = \frac{W_c^2}{h_c}, \end{equation}

where $Re$, $Pr$ and $Ec$ are the Reynolds, Prandtl and Eckert reference numbers, the static enthalpy is noted $h$ and the density $\rho$.

Examples of dimensional velocity and temperature profiles obtained after the analytical interpolation are given in figure 2 for $P_{el}=100\ \textrm {kW}$ and $P_s=150\ \textrm {mbar}$, while corresponding profiles of the equilibrium flow properties are displayed in appendix A.2. Velocity profiles at the chamber inlet exhibit two streams and shear layers, similarly to coaxial jets studied in the literature (Balestra et al. Reference Balestra, Gloor and Kleiser2015). The main stream is caused by the heating at the jet centreline, while the secondary stream originates from the annular inlet. The bypass ratios defined as: $h_{\text {b}}=W_c/W_s$, where $W_s$ is the averaged velocity of the secondary stream, ranges from low to mid values ($h_{\text {b}}\in [0.18;\ 0.43]$) depending on the driving parameters. For all cases, the primary shear is much thicker than the secondary one. Humps observed in the temperature profiles are due to the evolution of the equilibrium gas composition with temperature, and are not linked to the velocity profiles as the viscous heating is negligible. Further downstream, the jet displays a single stream, with a large shear-layer thickness similar to the configurations studied by Lesshafft & Huerre (Reference Lesshafft and Huerre2007).

Figure 2.

Analytical profiles of Plasmatron velocity (a) and temperature (b) for $P_{el}=100\ \textrm {kW}$ and $P_s=150\ \textrm {mbar}$ along the test chamber, from $z=0.5\ \textrm {m}$ ($-$) to $z=3.5\ \textrm {m}$ (- -), with intermediate profiles ($\cdots$) linearly spaced between the inlet and outlet.

Although not displayed here, an increase of the electric power $P_{el}$ increases both the centreline temperature and velocity at the chamber inlet, while an increase in pressure $P_s$ is found to decrease the centreline velocity and to have little impact on the temperature field. In addition, increasing the power decreases the bypass ratio at the inlet which results in more slender velocity profile; in contrast, changes in the pressure are found to act as a scaling of the overall velocity profiles while having less influence of the bypass ratio.

The different test cases are summarized in the table 1, along with the main dimensional quantities and non-dimensional numbers taken at the inlet of the test chamber. The temperature ratio between the surrounding gas and the jet centreline is introduced as $S=T_{\infty }/T_c$, where $T_{\infty }=T_{wall}=350\ \textrm {K}$. One can note that the Mach number presented in table 1 is computed using the equilibrium speed of sound returned by the Mutation$^{++}$ library, taking into account high temperature effects.

2.2. Stability model

This study assumes that Plasmatron's jet is slowly evolving along the streamwise direction and can be locally approximated as a parallel flow. This is supported by the fact that derivatives of the base state quantities with respect to the radial dimension are in average two orders of magnitude larger than with respect to the streamwise direction. The local formulation of the LST is considered, where ($z, \theta$) are the spectral directions, while $r$ is the differential direction. The classical decomposition of the instantaneous flow field is performed: $Q(\textbf {x}, t) = \bar {Q}(\textbf {x}) + q'(\textbf {x}, t)$ where $\bar {Q}$ is the $\textit{O}^1$ base state and $q'$ is the related $\textit{O}^\epsilon$ perturbation quantity. The natural path to transition as defined by Morkovin, Reshotko & Herbert (Reference Morkovin, Reshotko and Herbert1994) is considered, and the perturbations are modally decomposed

(2.4)\begin{equation} q'(r,\theta,z,t) =\tilde{q}(r)\exp({\textrm{i}(\alpha z + m\theta -\omega t )}) +\textrm{c.c.}, \end{equation}

where $\tilde {q}$ is the complex amplitude function, $\alpha$ and $m$ are respectively the streamwise and integral azimuthal wavenumbers. The complex angular frequency is $\omega$ and c.c. denotes the complex conjugate necessary to retrieve real perturbations. Introducing (2.4) and the quantities’ decomposition into the flow equations, one obtains the stability equations, which reduce to a generalized eigenvalue problem once completed with the appropriate set of boundary conditions.

Stability equations are derived within the Automatic Derivation and Implementation module (Adit) of the Vesta toolkit (Pinna & Groot Reference Pinna and Groot2014) from the compressible Navier–Stokes equations in cylindrical coordinates. The influence of the magnetic field is ignored as it is mainly confined to the torch section of Plasmatron. The state vector considered is [$u', v', w', T', p'$], namely the components of the velocity along $(r,\theta ,z)$, the temperature and pressure. The stability equations include high temperature effects by expanding the transport properties, similar to the dynamic viscosity

(2.5)\begin{equation} \mu = \bar{\mu} + \frac{\textrm{d}\bar{\mu}}{\textrm{d}\bar{T}}T' + \frac{\textrm{d}\bar{\mu}}{\textrm{d}\bar{P}}p', \end{equation}

where the derivatives of the transport properties with respect to pressure and temperature are obtained as described in § 2.1.2. Stability equations for a jet in LTE are described in appendix B.1. The main differences between the current implementation of the LTE system with respect to more common low temperature compressible stability equations come from the introduction of the $F$ and $G$ variables introduced by Malik & Anderson (Reference Malik and Anderson1991) to modify the equation of state given in (2.6a).

(2.6a)\begin{gather} \tilde{\rho}=\frac{\bar{F}\bar{\rho}}{\bar{P}}\tilde{p} - \frac{\bar{G}\bar{\rho}}{\bar{T}}\tilde{T}, \end{gather}

(2.6b)\begin{gather}\bar{F} = 1 - \left.\frac{\partial \ln \zeta}{\partial \ln P}\right|_T, \quad \bar{G} = 1 - \left.\frac{\partial \ln \zeta}{\partial \ln T}\right|_P, \end{gather}

where $\zeta$ is the compressibility factor of the gas mixture in LTE. Unless specified otherwise, the stability equations solved in this work assume a flow in LTE. Stability analyses using the CPG assumption eliminate the dependence of the transport properties on the pressure and rely on the classical state equation.

Homogeneous Dirichlet conditions are imposed at the far field ($r\rightarrow \infty$) to yield vanishing perturbations. At the centreline, a set of boundary conditions is derived to cope with the apparent singular nature of the coordinates in a similar manner to Lesshafft & Huerre (Reference Lesshafft and Huerre2007). Equations for the perturbation centreline conditions depend on the value of $m$ and are given in appendix B.3.

Similarly to Huerre (Reference Huerre2000), the spatio-temporal formulation of the LST is used to study the absolute/convective nature of instabilities, thus, $\alpha$ and $\omega$ are both complex numbers. Absolute instabilities are detected when a localized impulse promotes self-sustained perturbations through a resonance mechanism in the flow. This is interpreted as an infinite growth of instabilities with null group velocity $v_{{g}}=x/t$ in the long-time limit after the impulse. In practice, Briggs and Bers provided a simple criterion to identify the nature of instabilities from isocontours of the spatio-temporal growth rate $\sigma =\textrm {Im}(\omega )$ in the complex wavenumber plane $(\textrm {Re}(\alpha ) \textrm {Im}(\alpha ))$. The sign of $\sigma$ where the contours describe a saddle point (noted ‘$\sigma _S$’) is related to the long-time limit of the perturbations at $v_{{g}}=0$. When $\sigma _S>0$, the flow is absolutely unstable, while for $\sigma _S\leq 0$ the flow is either stable or convectively unstable. However, to respect the criterion, valid saddles must be pinched between an $\alpha ^+$ and an $\alpha ^-$ branch. Starting at the saddle point, contours of growing $\sigma =\omega _i$ must reach respectively the $\textrm {Im}(\alpha ) > 0$ and $\textrm {Im}(\alpha ) < 0$ half-planes. Therefore the flow is absolutely unstable only if $\omega _i>0$ at a valid saddle point.

2.3. Numerical implementation

The stability equations reduced to a generalized eigenvalue problem are solved imposing the complex wavenumber $\alpha$ and integer $m$ in the Vesta toolkit (Pinna Reference Pinna2013), relying on the ZGEEV Lapack routine. Stability equations are discretized using the pseudo-spectral Chebyshev collocation method (Trefethen Reference Trefethen2000). The discretization in the radial direction is further modified by using a tangential mapping technique (Bayliss & Turkel Reference Bayliss and Turkel1992) described in appendix B.2 and implemented by Esposito (Reference Esposito2016). A number of collocation points set to $N=201$ is sufficient to reach a convergence of order $10^{-7}$ on the eigenvalues obtained for the conditions studied in this work. In practice, the construction of the eigenvalue problem matrices is $40\,\%$ slower in LTE than for the CPG problem. However, the two solvers take roughly the same time to compute the eigenvalue spectrum.

A local eigenvalue solver based on Newton's iterative method is used to obtain individual eigenvalues from a first guess (Pinna Reference Pinna2013). This solver is used to track eigenvalues across the complex wavenumber plane. Saddle points are found by approaching ${\partial \omega }/{\partial \alpha } = 0$ using the trust-region dogleg algorithm (Powell Reference Powell1968) integrated in the Matlab^®fsolve routine. Saddle points are tracked for variations of the streamwise position and group velocity using the same methods.

3.1. Analysis at the inlet

Stability features of the plasma jet are analysed in detail assuming $P_{el}=100\ \textrm {kW}$ and $P_s=150\ \textrm {mbar}$, corresponding to the case number $5$ of table 1. The global spectra of eigenvalues obtained for $N=201$ and $N=301$, imposing $\alpha =1$ and $m=0$ at the inlet of the chamber ($z=0.5\ \textrm {m}$) are displayed in figure 3. The present spectrum is composed of continuous branches, a set of stable discrete modes and a single unstable mode. The set of discrete stable modes is converged, and remains under the $\omega _r$ axis for real values of $\alpha$. The unstable mode is found at $\omega = 0.0473 + 0.1027i$ for the current wavenumber value, and it is the mode tracked with the local solver to draw the contour plots used in the spatio-temporal analysis.

Figure 3.

Spectrum of eigenvalues obtained by imposing $\alpha =1$ and $m=0$ for the test case 5 at $z=0.5$ m with $N=201$ () and $N=301$ () discretization points.

Contours of the spatio-temporal growth rate in the complex wavenumber plane at the chamber inlet are displayed in figure 5(a) for axisymmetric perturbations. At the inlet, the base state is akin to the family of coaxial jets studied by Balestra et al. (Reference Balestra, Gloor and Kleiser2015), although the core shear-layer thickness is significantly larger than that of the secondary shear. The saddle point with largest spatio-temporal growth rate at this station is found near the origin of the axis. This saddle is noted $S_0$ for the rest of this study, its absolute wavenumber is $\alpha _{S0}=0.4856 - 0.7902i$ and absolute non-dimensional frequency is $\omega _{S0}=0.0881 + 0.0739i$. As the saddle is pinched between an $\alpha ^+$ and an $\alpha ^-$ branch, while displaying a positive growth rate, the flow is absolutely unstable for this case and at this position. The absolute value of the perturbation functions related to $S_0$ are shown in figure 4. The pressure $\tilde {p}$ and streamwise velocity $\tilde {w}$ perturbations are amplified both at the centreline and in the secondary stream shear layer ($r=1$). Such shapes are similar to both the mixed modes described by Lesshafft & Huerre (Reference Lesshafft and Huerre2007) in top-hat jets with thick shear layers and outer modes described by Balestra et al. (Reference Balestra, Gloor and Kleiser2015) in coaxial jets. This is further explained by the fact that $S_0$ is pinched between an $\alpha ^-$ branch of jet-column modes located near the $\alpha _i$ axis, and an $\alpha ^+$ branch of shear-layer modes coming from the upper $\alpha _i>0$ half plane. The mixed mode at saddle $S_0$ exhibits a low dimensional frequency of $f_{S0}=\textrm {Re}(\omega _{S0})W_c/(2{\rm \pi} l_{ref})=25.62\ \textrm {Hz}$ and a large wavelength $\lambda _{S0}=2{\rm \pi} l_{ref}/ \textrm {Re}(\alpha _{S0})=1.035\ \textrm {m}$.

Figure 4.

Magnitude of $S_0$ perturbation functions with $m=0$ in case 5 at $z=0.5$ m. The perturbation variables are $\tilde {w}$ ($-$), $\tilde {T}$ (- -), $\tilde {p}$ (), and are normalized by $\max (|\tilde {w}|)$.

Following Huerre & Monkewitz (Reference Huerre and Monkewitz1990, p. 487), the absolute wavelength is compared to the development length of the flow $L = \theta (\partial \theta /\partial z)^{-1}$, where $\theta$ is the momentum thickness of the profile. It appears that the ratio $\epsilon =\lambda _{S0}/L$ is close to unity at the inlet of the jet, preventing a possible connection between local and global stability properties. However, results from the local stability analysis are still believed to be insightful on the presence of global modes, as the absolute regions discussed in the following sections extend to jet regions where $\epsilon \ll 1$.

A series of valid and non-valid saddle points is also observed in figure 5(a) near the $\alpha _i$ axis, however, displaying smaller growth rates than $S_0$ and thus are not considered any further. An additional saddle is present for low values of $\alpha _i$ and further away from the vertical axis, noted $S_5$, at $\alpha _{S5}=6.6581 -23.1098i$ and with $\omega _{S5}=1.3731 - 0.5581i$. This mode displays perturbation functions typical from the shear layer modes described by Lesshafft & Huerre (Reference Lesshafft and Huerre2007), at a higher frequency $f_{S2}=399.3\ \textrm {Hz}$ and shorter wavelength $\lambda _{S5}=0.075\ \textrm {m}$ than $S_0$, corresponding roughly to the radius of the secondary shear. Although $S_5$ is not linked to an absolute instability, it is highlighted here as it will play a role in the full impulse response described in § 5.

Figure 5.

Contours of constant spatio-temporal growth rate $\omega _i$ for the axisymmetric mode $m=0$ at $z=0.5$ m (a) and $z=2.086$ m (b). The contours are limited to $\omega _i\in [-1; 0.5]$ (a) and $\omega _i\in [-0.2; 0.175]$ (b) and the $\omega _i=0$ isocontour is highlighted (white dashed line). The least stable saddle is noted $S_0$, while $S_5$ and $S_1$ are the secondary saddles involved in the full impulse response.

Increasing the integral azimuthal wavenumbers to $m=[1,2]$ is found to decrease the growth rate of $S_0$ down to $\textrm {Im}(\omega _{S0}) = -0.1663$ at $m=2$, while $S_0$ moves to lower $\alpha _i$ values. Furthermore, for $m>0$, the shape of perturbations at $S_0$ changes to resemble shear layer modes. In contrast, the shear-layer mode associated with saddle $S_2$ is barely affected by the changes in $m$, in agreement with results of Lesshafft & Huerre (Reference Lesshafft and Huerre2007) for single top-hat jets. However, similar results for coaxial jets were not included in the work of Balestra et al. (Reference Balestra, Gloor and Kleiser2015) which was restricted to axisymmetric perturbations. One can note that no valid saddle was found to lie on the upper $\alpha _i>0$ half-plane in the results presented.

3.2. Influence of the streamwise position

As the jet diffuses in the test chamber, the primary and secondary streams merge to form a single jet with a thick shear layer around $z=1.5\ \textrm {m}$, which is found to impact the stability behaviour. Contours of the spatio-temporal growth rate are displayed after this station, at $z=2.086\ \textrm {m}$, in figure 5(b). No equivalent of the saddle $S_5$ is found at this station, however, a saddle similar to $S_0$ is still present at $\alpha _{S0}=0.6122 - 0.7533i$. The complex frequency at $S_0$ is $\omega _{S0}=0.2056 + 0.0982i$, which indicates that the flow is still absolutely unstable at this position. Saddles along the $\alpha _i$ axis still display lower spatio-temporal growth rates when compared to $S_0$. The valid saddle directly beneath $S_0$ noted $S_1$ in figure 5(b) at $\alpha _{S1}=1.0793 - 2.9858i$ and with $\omega _{S1}=0.2331 - 0.0090i$ is nonetheless singled out as it will play a role in the full impulse response at this streamwise position. Similarly to the inlet, no valid saddle is found in the upper $\alpha$-plane.

In order to better understand the structure of the absolutely unstable region, the saddle $S_0$ is tracked along the streamwise direction from $z=0.5\ \textrm {m}$ to $z=3.5\ \textrm {m}$. Streamwise positions beyond $z=3.5\ \textrm {m}$ are not considered, as they are not representative of the actual facility. The tracking is done for $m=[0,1]$, with both the CPG and LTE stability solvers. The resulting curves of absolute growth rate and dimensional frequency along the test chamber are displayed in figure 6. The symbols on the curves in figure 6 indicate saddles found by computing the full contours of the growth rate in order to confirm the accuracy of the tracking tool and to make sure no saddle playing a role in the stability behaviour is overlooked. Discontinuities observed in the curves are due to the local nature of the analytical interpolation used to reconstruct the base flow. This tracking confirms that $S_0$ in figures 5(a) and 5(b) are the same saddle.

Figure 6.

Streamwise evolution of the absolute growth rate (a) and dimensional frequency (b) associated with $S_0$, for axisymmetric $m=0$ ($-$) and first helical $m=1$ (- -) modes, with the LTE (thick line) and CPG (thin line) stability solvers. The absolute growth rate and frequency of $m=0$ () and $m=1$ () saddle points obtained from full contours are added to confirm the accuracy of the tracking tool.

Results for the axisymmetric modes are analysed first, after which the helical modes are discussed. Axisymmetric modes ($m=0$) display a positive growth rate along the whole domain considered, which yields a larger region of absolute instability than what is observed in the literature. The positive slope of the growth rate $(\partial \sigma /\partial z)$ at the inlet is in agreement with observations of Coenen & Sevilla (Reference Coenen and Sevilla2012) for jets displaying a small shear-layer thickness with respect to the nozzle diameter. This is the case for the inlet profiles studied here only when considering the secondary shear layer (around $r=1$ in figure 2), which coincides with the radial position where the amplification of disturbances associated with $S_0$ is maximum. The absolute growth rate reaches its maximum value $\max (\sigma _{S0})=0.1319$ in the chamber at $z_{max}=1.1657\ \textrm {m}$, and the frequency at the start of the absolute region is $f_{S0}=25.62\ \textrm {Hz}$.

Shapes of the perturbations affiliated with $S_0$ are given at four streamwise positions in figure 7. Saddle $S_0$ retains its mixed mode nature along the chamber as the pressure is amplified both at the centreline and shear layer, while the velocity perturbation becomes more amplified around the centreline as $z$ increases due to the decrease of secondary shear. As $z$ increases and the centreline temperature decreases, fewer dissociation reactions occur in the flow, which is seen by the vanishing number of peaks in the temperature perturbations. Finally, all perturbations are found to scale with the growing shear-layer thickness of the jet. Due to the extended length of the absolute region, the plasma jet should support a global mode (Lesshafft et al. Reference Lesshafft, Huerre, Sagaut and Terracol2006; Coenen & Sevilla Reference Coenen and Sevilla2012).

Figure 7.

Magnitude of the axisymmetric perturbation functions corresponding to saddle $S_0$ at 4 streamwise positions for the test case 5. The perturbation variables are respectively $\tilde {w}$ ($-$), $\tilde {T}$ (- -) and $\tilde {p}$ (), and are normalized by $\max (|\tilde {w}|)$ at each position.

In order to explain the non-monotonic behaviour of the absolute growth rate with respect to $z$, we recall two possible sources of absolute instabilities in jets: the baroclinic torque in heated jets (Lesshafft & Huerre Reference Lesshafft and Huerre2007) and the displacement of the maximum shear towards the low velocity areas (Lesshafft & Marquet Reference Lesshafft and Marquet2010). To promote the former, the positions of the maximum radial density gradient and maximum shear should be close to each other. However, the latter is also able to promote absolute instabilities even in the absence of baroclinic torque by reducing the group velocity at the maximum perturbation location. In the current cases, these two effects are in competition due to the specific shape of the plasma jet. As illustrated in figure 8(a), the velocity and density gradients are not aligned at the inlet but get closer as $z$ is increased, crossing each other around $z=0.92\ \textrm {m}$ and splitting further downstream. Accordingly, the absolute growth rate initially increases with $z$. However, its maximum value is reached beyond the position where the shears are aligned. The authors believe this offset to be due to the lower velocity at the maximum shear, observed as $z$ increases further. This is illustrated in figure 8(b), focusing on the secondary shear layer of the jet, where the velocity perturbation associated with $S_0$ is maximum. Moreover, the local velocity profiles of Plasmatron around $z=1\ \textrm {m}$ resemble greatly the profiles maximizing absolute instabilities obtained by Lesshafft & Marquet (Reference Lesshafft and Marquet2010). Further downstream, the absolute growth rate decreases while the Reynolds number, temperature ratio and shear-layer thickness all increase and $S$ remains of order $0.1$, in agreement with Lesshafft & Huerre (Reference Lesshafft and Huerre2007).

Figure 8.

Evolution of the alignment between the maximum shear and maximum density gradient (a) and evolution of the velocity profiles around the maximum shear location (b) along $z$ for case $5$. Profiles (solid lines) are given from $z=0.5\ \textrm {m}$ (dark) to $z=1.5\ \textrm {m}$ (light). The profiles are normalized by their maximum value at the inlet (see table 1 for velocity). The thick dashed line represents the profiles at maximum growth rate. The radial coordinate is normalized by the location where $\partial \rho /\partial r$ is maximum (a) and where $\partial W/\partial r$ is maximum (b) for each $z$ (thin dotted line).

However, jets with thick shear layers, characterized by $R/\theta <5$ where $\theta$ is the momentum thickness, remain convectively unstable when considering simplified TTP models according to Lesshafft & Huerre (Reference Lesshafft and Huerre2007). In order to explain why the present jet is still absolutely unstable at $z=3.086\ \textrm {m}$, we consider the stability of the analytical jet defined by

(3.1a)\begin{gather} \bar{w}=\frac{1}{2}\left(1 + \tanh\left[\frac{R}{4\theta} \left(\frac{1}{r} -r \right) \right] \right); \end{gather}

(3.1b)\begin{gather}\bar{h} = S_h + \frac{1-S_h}{2}\left(1 + \tanh\left[\frac{R}{4\theta} \left(\frac{1}{r-d} -r +d \right) \right] \right); \end{gather}

adapted from (3.1a) and (3.1b) given by Lesshafft & Marquet (Reference Lesshafft and Marquet2010). Original profile equations from Lesshafft & Marquet (Reference Lesshafft and Marquet2010) are adapted to high temperatures by imposing a static enthalpy profile rather than a temperature one. The temperature profile, the ambient-to-centreline enthalpy ratio $S_h = h_{\infty }/h_c$ obtained from $S$, and flow properties are reconstructed assuming LTE. The following values are used to approximate the plasma jet: $Re=250$, $R/\theta =4$, $S_h=0.0353$ (corresponding to $S=0.0715$), $M=0.0364$ and a radial offset of $d=0.125$ between the velocity and enthalpy shears. Resulting profiles are compared to the present jet in figure 9 in terms of velocity and density. The significant offset observed between the density profiles is partly due to the greater sensitivity of the density to low temperatures. By tracking absolute instabilities along $R/\theta =[20; 4]$, we find that for an equivalent jet thickness, the model adapted from Lesshafft & Marquet (Reference Lesshafft and Marquet2010) predicts a convectively unstable jet as plotted in figure 10. This indicates that the absolute nature of the thick shear-layer jet is not promoted by LTE effects, but is rather caused by the specific non-symmetric shape of the profiles, in agreement with conclusions from Lesshafft & Marquet (Reference Lesshafft and Marquet2010).

Figure 9.

Comparison of the velocity ($-$) and density () profiles of Plasmatron () against the adapted top-hat jet (). Density is scaled with $\rho _{\infty }$ for plotting purposes.

Figure 10.

Top-hat jet absolute growth-rate evolution with $R/\theta$ for $m=0$ () and $m=1$ (); and Plasmatron absolute growth rates at $z=3.086$ m ().

Helical modes are also studied along the chamber by imposing $m=[1,2]$. The streamwise evolution of the $m=1$ mode is plotted in figure 6(a), the $m=2$ mode is not plotted as its absolute growth rate remains below zero along the chamber. A region of absolute instability exists for the first helical mode and is contained inside the domain between $z=0.918$ and $z=2.838\ \textrm {m}$. The frequency at the start of the absolute region is $f=56.8\ \textrm {Hz}$ for $m=1$. In agreement with the conclusions of Lesshafft & Huerre (Reference Lesshafft and Huerre2007) for top hat jets in CPG, the absolute growth rate of helical mode remains smaller than the that of axisymmetric modes along the jet. The use of the jet model adapted from Lesshafft & Marquet (Reference Lesshafft and Marquet2010) also underestimates the absolute growth rate of the $m=1$ mode, as plotted in figure 10. Although not plotted, perturbation shapes of the first helical mode are similar to the axisymmetric ones, with the exception of the vanishing components of $\tilde {w}$, $\tilde {T}$, $\tilde {p}$ at the centreline imposed by the boundary conditions.

Assuming CPG properties and stability equations over-estimate both the absolute growth rate and frequency; as expected, the difference with respect to the LTE results is larger near the inlet of the test chamber, where the temperature is the highest. At the inlet, the CPG model over-estimates the growth rate by $21\,\%$ and the absolute frequency by $57\,\%$ of the LTE values for $m=0$. An interesting observation is that the CPG absolute growth rate evolves with the same qualitative trends as for the LTE case, despite ignoring the high temperature effects on the flow properties and therefore on the non-dimensional numbers. At high temperatures ($T>9000\ \textrm {K}$ at $P_s=150\ \textrm {mbar}$), the viscosity behaviour with respect to temperature of a gas in LTE is inverted due to the presence of electrons. Consequently, the Reynolds number is found to decrease with $z$ from the inlet until $T_c <9000\ \textrm {K}$, where it starts to increase. This confirms the importance of studying the shapes described by velocity and density profiles, rather than only considering non-dimensional numbers.