2.1. Ideal shock–refraction equilibrium

When an oblique shock wave impinges on a tangential discontinuity – such as a shear or mixing layer – it gets refracted, resulting in a transmitted shock and a reflected wave, as depicted in the sketch of figure 1(b). In general, the two post-shock streams (subscripts 3 and 4 for top and bottom, respectively) have different pressures and streamline deflections. Then an additional reflected wave (either an expansion or a shock) is responsible for providing additional deflection and pressure changes (subscript 5) to grant the mechanical equilibrium in the tangential discontinuity between regions $4$ and $5$ downstream (Landau & Lifshitz Reference Landau and Lifshitz1966). Depending on the initial values of the Mach numbers $M_1$ , $M_2$ , as well as the angle of inclination of the incident shock $\sigma _i$ , different regimes are found.

Figure 2 depicts the Mach number and shock angle domain, bounded by a lower limit of supersonic flow normal component ( $M_1 \sin \sigma _i = 1$ ) as a solid black line. For a given value of the upper-stream Mach number $M_1$ and oblique shock incident angle $\sigma _i$ , shown with a blue circle in figure 2(a), a family of possible combinations of $M_2$ enables the global mechanical equilibrium with different values of the transmitted shock angle $\sigma _t$ . Three different regular configurations can be attained: (i) a reflected expansion (points lying on the blue solid curve); (ii) a reflected shock (on the red dashed curve); or (iii) a neutral transmission with no additional reflected wave, either for $M_2 = M_1$ and $\sigma _t = \sigma _i$ or at the other transition point from reflected expansion to shock. Nevertheless, this group of regular solutions defines a critical Mach number $M_{\textit {cr}}$ (dash-dotted vertical line), below which a stable configuration cannot be reached. Irregular configurations may arise from the interaction between a shock wave and a mixing layer if, given a set $M_1$ and $\sigma _i$ , a value of $M_2\lt M_{\textit {cr}}$ is used at the other stream. In fact, an irregular structure is formed when the reflected shock or expansion fails to provide mechanical equilibrium of the tangential discontinuity downstream, developing an unstable solution with subsonic zones that grow with time (Martínez-Ruiz et al. Reference Martínez-Ruiz, Huete, Martínez-Ferrer and Mira2019).

Figure 2.

(a) Continuous set of ideal flow solutions with mechanical equilibrium (blue and red curves) of an oblique shock over a tangential discontinuity. (b) Combinations of upstream parameters $M_1,M_2,\sigma _i$ that provide post-shock Mach numbers equal to the base-case flow.

Considering that we will compare below the differences in reaction evolution between the shock-free base flow and the oblique shock impingement configurations, the post-shock conditions $M_3$ , $T_3$ and $M_4$ , $T_4$ are here matched to those of the free streams in the base-case mixing layer in absence of an incident shock wave ( $M_1=2$ , $T_1=1250$ K; $M_2=1.25$ , $T_2=545$ K) $_{\textit {base}}$ . Figure 2(b) shows the locus of input parameters $M_1$ , $\sigma _i$ (circles) that produce the desired post-shock $M_3 = 2$ and corresponding $M_2$ , $\sigma _t$ (crosses) that warrants $M_4 = 1.25$ with a regular equilibrium solution. These conditions are used for the inlet of the impinging oblique shock simulations, together with new upstream values of $T_1$ and $T_2$ that provide the desired $T_3$ and $T_4$ once each shock intensity is known. Finally, it should be noted that the rightmost sets of solutions involve greater incident angles $\sigma _i$ and Mach numbers of the free streams, which cause more intense reflected expansions that lower further the temperature and pressure of the final upper region $5$ . This cooling process – owing to the reflected rarefaction structure – is addressed in the following, where we will focus on its effect over the thermochemical problem.

For the sake of clarity, a first set of comparative simulations is presented in figure 3, whose numerical computation details are presented below. Figures 3(a,c) show a snapshot of the base-case reactive mixing layer (air–H $_2$ ) in a hot–cold configuration, once the ignition transient is produced. The instantaneous temperature (figures 3 a,b) and numerical schlieren (figures 3 c,d) distributions are presented to visualise the reactive layer and pressure wave structures, respectively. It should be noted that the sharp increase in temperature owing to the chemical reaction approximately indicates the characteristic ignition distance $x_{\textit {ign}}$ . Figures 3(c,d) depict the instantaneous pressure waves arising from the mixing layer and the reaction kernels, which are of acoustic type, $\Delta p \ll p \simeq 1$ atm.

Moreover, figures 3(b,d) show analogous distributions for the case with oblique shock impingement. The temperatures and Mach numbers of the incoming streams are selected to produce post-shock conditions that mimic the inlet in the shock-free reactive mixing layer. The ignition distance, given pre-shock mixing evolution and post-shock compressed conditions, is slightly reduced compared to the base case. In figure 3(d), the incident and transmitted shock waves are clearly detected as the most intense pressure gradient events, $\Delta p \sim p$ , including a reflected expansion of low intensity that cools down the upper stream. In turn, ignition- and vorticity-related acoustics can also be detected therein, starting at the point where a numerical forcing (vertical perturbation at $x\sim 0.01$ m) destabilises the layer. This disturbance physically displaces the shear layer, leading to local pressure fluctuations that propagate as acoustic waves.

Figure 3.

Mixing layer ignition (a,c) without and (b,d) with oblique shock impingement: (a,b) temperature and (c,d) numerical schlieren profiles. Inlet mixing layer profiles follow $\tanh$ -shape distributions.

These results enable the analysis of the isolated effects of ideal flow regular shock structures on the auto-ignition point and combustion dynamics. Nevertheless, the non-ideal effects and inner structure of the mixing layer are also of the greatest relevance, as the chemical reaction will take place wherever the reactants are heated up and mixed at a molecular level. For this reason, we will discuss in the following the prescribed profiles at the inlet of the numerical simulation domain.

2.2. Inlet profiles

The inlet values of each stream variable need to be provided through continuous profiles that represent a realistic upstream evolution of the diffusive processes. The domain is scaled with a characteristic vorticity thickness $\delta _\omega ^0$ , as depicted in figure 1. This thickness is a function of the distance to the virtual origin of the mixing layer, $x_0=(\delta _{\omega }^0/2)^2 \rho _1 u_1/\mu _1$ , with $\mu _1$ the upper-stream viscosity, $\rho _1$ the density, and $u_1$ the free-stream speed. In turn, the vorticity Reynolds number is defined as

(2.1) \begin{align} \textit{Re}_{\omega }^0 = \frac { \delta _{\omega }^0 \bar {\rho }\, \Delta u }{\bar {\mu }} , \end{align}

where $\Delta u = |u_1 - u_2|$ , $\bar {\rho }=(\rho _1+\rho _2)/2$ and $\bar {\mu }= (\mu _1+\mu _2)/2$ . Following previous works (Rogers & Moser Reference Rogers and Moser1994), the vorticity Reynolds number is set to $\textit{Re}_\omega ^0 = 3200$ , which defines the inlet vorticity thickness of the layer $\delta _\omega ^0$ . In addition, to induce Kelvin–Helmholtz instabilities, the aforementioned random perturbation is applied to the transverse velocity component $v$ at $(x,y)=(4\delta _{\omega }^0,0)$ .

Most common input distributions of flow variables in mixing-layer and shear-layer numerical studies involve hyperbolic tangent profiles, as a smooth characteristic transverse evolution of generic fluid variables,

(2.2) \begin{align} \varPhi = \frac {\varPhi _1+\varPhi _2}{2}+\frac {\varPhi _1-\varPhi _2}{2} \tanh \left (\frac {2y}{\delta _{\omega }^0} \right )\!, \end{align}

which is typically used for the streamwise velocity, temperature, and species. Other subsidiary magnitudes such as the Mach number can be obtained given the isobaric co-flow composition $Y_i$ . Nevertheless, a self-similar solution can be predicted owing to the lack of characteristic length in mixing-layer configurations. In fact, Huete et al. (Reference Huete, Urzay, Sánchez and Williams2016) showed that the self-similar solution of a laminar mixing layer can be calculated – Soret and Dufour effects included – as a function of the dimensionless similarity coordinate $\eta =y/[ (\mu _1/\rho _1 )x/u_1]^{1/2}$ . The dimensionless longitudinal velocity component ${U}(\eta )=u/u_1$ , and transverse velocity component $V(\eta )=v/u_1$ , temperature $\varTheta (\eta )=T/T_1$ and density $R(\eta )=\rho /\rho _1$ , are referred to the characteristic upstream values. Then the calculation of the self-similar profiles is performed with non-negligible Rayleigh viscous dissipation and temperature-dependent transport coefficients.

Consequently, the dimensionless mass, momentum, energy and species conservation equations for the self-similar problem can be written as

(2.3) \begin{align} -\frac {\eta }{2}\frac {\text{d}}{\text{d}\eta }(RU) + \frac {\text{d}}{\text{d}\eta }(RV) &= 0, \\[-28pt] \nonumber \end{align}

(2.4) \begin{align} R\left (V-\frac {\eta }{2} \right ) \frac {\text{d}U}{\text{d}\eta } &= \frac {\text{d}}{\text{d}\eta }\left (\mathcal{M}\frac {\text{d}U}{\text{d}\eta }\right )\!, \\[-28pt] \nonumber \end{align}

(2.5) \begin{align} RC_p\left (V-\frac {\eta }{2}U\right )\frac {\text{d} \varTheta }{\text{d}\eta } ={}&\frac {\text{d}}{\text{d}\eta }\left (\frac {\mathcal{M} C_p}{{Pr}}\frac {\text{d} \varTheta }{\text{d}\eta }\right )-\left (\mathcal{W}_2^{-1}-1\right )J\frac {\text{d} \varTheta }{\text{d}\eta } \nonumber \\ &{}-\frac {\alpha \left (\gamma -1\right )}{\mathcal{W}_2\gamma }\frac {\text{d}}{\text{d}\eta }\left (\mathcal{W}J \varTheta \right )+\left (\gamma -1\right )M_1^2\mathcal{M}\left (\frac {\text{d}U}{\text{d}\eta }\right )^2 \!, \\[-28pt] \nonumber \end{align}

(2.6) \begin{align} R\left (V-\frac {\eta }{2}U\right )\frac {\text{d}Y}{\text{d}\eta } &=-\frac {\text{d}J}{\text{d}\eta }, \\[0pt] \nonumber \end{align}

where the local composition is defined in terms of the mass fraction of fuel $Y$ . Here, $C_p=c_p/c_{p1}$ is the dimensionless specific heat at constant pressure, $\mathcal{M} = \mu /\mu _1$ is the dimensionless viscous coefficient, $\gamma$ is the adiabatic index, and $\alpha = -0.3$ is the thermal-diffusion factor, considered constant. The Prandtl number is also taken constant and equal to $Pr = \mu c_p/\lambda = 0.7$ , where $\lambda$ stands for thermal conductivity. In addition, $\mathcal{W}=W/W_1$ , where $W$ refers to mixture molecular weights, and $\mathcal{W}_2=W_2/W_1$ . Finally, $J$ is the fuel-diffusion flux non-dimensionalised with $\rho _1\mu _1u_1/x$ , and obtained explicitly from Sánchez & Williams (Reference Sánchez and Williams2014):

(2.7) \begin{align} J = - \frac {R \mathcal{D}}{{Pr} \,{Le}}\left (\frac {\text{d}Y}{\text{d}\eta }+\alpha \frac {Y(1-Y)}{{\varTheta }}\frac {\text{d} {\varTheta }}{\text{d}\eta }\right )\!, \end{align}

where $\mathcal{D}=D/D_1$ is the dimensionless binary diffusion coefficient. The system of equations must be completed with the equation of state

(2.8) \begin{align} RT=\mathcal{W}=\frac {1}{1+ \big(\mathcal{W}_2^{-1}-1 \big)Y}, \end{align}

and the expressions

(2.9) \begin{gather} \mathcal{M}=\frac {1+[(\mu _2/\mu _1)\mathcal{W}_2^{-1/2}-1]Y}{1+(\mathcal{W}_2^{-1/2}-1)Y}T^{\sigma _1}, \quad \mathcal{D}=T^{1+\sigma _1} \quad\text{and}\quad \! C_p=[1+(\mathcal{W}_2^{-1}-1)Y]T^{\sigma _2} \end{gather}

for the variations of the transport coefficients and specific heat with temperature and composition, taking the values $\sigma _1=0.7$ and $\sigma _2=0.2$ (Rosner Reference Rosner2012).

Figure 4.

Comparison between hyperbolic tangent (tanh) profiles (solid) and self-similar (s.s.) profiles (dashed) for (a) hot–cold and (b) isothermal H $_2$ –air mixing layers.

Figure 4. Long description

Two line graphs compare hyperbolic tangent and self-similar profiles for hot-cold and isothermal Hair mixing layers. Panel A: The line graph depicts the hot-cold mixing layer with the horizontal axis ranging from 0 to 1.5 and the vertical axis labeled with values from -5 to 10. The graph includes multiple lines representing different variables: R, Theta, M, YH2, and U. Solid lines indicate hyperbolic tangent profiles, while dashed lines indicate self-similar profiles. Panel B: The line graph depicts the isothermal mixing layer with the horizontal axis ranging from 0 to 2 and the vertical axis labeled with values from -5 to 5. Similar to Panel A, it includes multiple lines representing R, Theta, M, YH2, and U, with solid lines for hyperbolic tangent profiles and dashed lines for self-similar profiles.

Numerical integration of the system of equations between the far-field conditions of streams $1$ and $2$ yields the profiles of flow variables across the self-similar laminar mixing layer. Figure 4 depicts the difference between the hyperbolic tangent (solid) and self-similar (dashed) profiles obtained for the same simulation inputs in hot–cold scenarios and isothermal co-flow. Both the hyperbolic tangent and self-similar profiles are initialised with the same vorticity thickness $\delta _\omega ^0$ , ensuring that they start from an equivalent large-scale mixing condition.

The main intention of the comparison is to assess that ignition is sensitive not only to bulk parameters (e.g. vorticity thickness or global stream conditions), but also to the detailed thermochemical state contained in the profile structure. It should be noted that the widely used hyperbolic tangent is not representative of the particular thermal and mass diffusive processes of H $_2$ –air mixtures, where its characteristic diffusivity and disparate stream densities are playing an important role in the development of the inner region of the mixing layer. In fact, several effects should be highlighted here, such as the density gradient, the non-symmetric shift of temperature, and the non-monotonic Mach number distribution (figure 4 a), or the temperature increase through viscous dissipation in the isothermal case (figure 4 b), with major local variations inside the mixing-layer thickness $\eta \sim O(1)$ . The effect of the input profile on the ignition distance is also compared in the following sections, showing the great relevance of including detailed physical processes at the inlet of the domain.

4.1. General features of shock-free mixing layers

The temperature distributions of the base set of simulations, cases 1 and 2, are presented in figure 5 after $3.7$ residence times, when the statistically steady flow is already established. The panels offer a comparison of the different macroscopic flow features between the classical hyperbolic tangent inlet (figures 5 a,c) and the self-similar profiles (d) for the hot–cold (figures 5 a,b) and isothermal (figures 5 c,d) supersonic mixing layers. The first appreciative difference between the cases is the average thickness and vorticity scales appearing in the flow structure. This differential behaviour is controlled by stronger inertial effects at the fuel-side edge ( $y\lt 0$ ) of the self-similar case, where a larger density value is prescribed through incorporation of adequate mixing mechanisms (see discussion above). These changes produce a greater penetration of the vortical structures in the lower cold stream of figure 5(b) than in the $\tanh$ case of figure 5(a). In turn, the greater spread of reactant in the self-similar input also contributes to thicker reaction layers upon the ignition event. In the isothermal co-flow case 2 in figure 5(d), the temperature increase produced by the viscous dissipation mechanism brings the ignition event forward compared to the $\tanh$ case in figure 5(c), by approximately 15 % of the distance to the inlet of the computational domain.

Figure 5.

Temperature distributions of (a,b) hot–cold and (c,d) isothermal reactive mixing layers, with (a,c) $\tanh$ and (b,d) self-similar inlet profiles.

Figure 6.

Spatial evolution of the time-averaged vorticity thickness $\delta _\omega$ with respect to the inlet-profile thickness in the base mixing layers, for (a) case 1 and (b) case 2. Ignition distances are marked with vertical dashed lines.

In fact, these particular effects are quantified in figure 6, where the evolution of the vorticity thickness $\delta _\omega$ along the mixing layer is shown. The thickness value is computed as

(4.1) \begin{align} \delta _{\omega } = \frac {U_1 - U_2}{|\partial \langle u_1 \rangle _{f} / \partial y|_{\textit {max}}}, \end{align}

where the operator $\langle \bullet \rangle _f = \langle \rho \bullet \rangle / \langle \rho \rangle$ stands for Favre average. The growth of $\delta _\omega$ differs slightly in the hot–cold scenario in figure 6(a), with large differences in the transverse distributions of variables as prescribed by the inlets (presented in figure 4 a). Nevertheless, the isothermal results in figure 6(b) show a nearly identical evolution, with initial distributions that bear a greater resemblance (recall the profiles compared in figure 4 b).

Therefore, a first outcome of the present work on the comparison of inlet profiles is that the incorporation of major differences in density and momentum distributions, owing to diffusive mixing of hydrogen, does indeed affect the shape and size of the flow field, yet it barely modifies the time-averaged ignition distance marked in figure 6(a), given that the maximum injection temperature remains unaffected. Contrarily, when the thermal viscous dissipation effect is dominant in the isothermal case and is included via the self-similar inlet profile (figure 6 b), the reaction is enhanced and brought forward without great influence on the flow field or vorticity thickness scale.

To obtain the value of the ignition distance $d_{\textit {ign}}$ , a clear and unambiguous criterion of the ignition event is defined, ruling out popular temperature-increase criteria, which could be triggered here due to the aforementioned friction heating of supersonic shear layers with no relevant initiation of the chemical reaction. First, several alternative variables are considered to obtain the coordinates of the auto-ignition point, by namely the production rates of some of the radical species that control the chemical reaction $\dot {\omega }_{\textrm{H}}$ , $\dot {\omega }_{\textrm{O}}$ and $\dot {\omega }_{\textrm{OH}}$ . Moreover, it was shown by Sánchez et al. (Reference Sánchez, Fernández-Tarrazo, Boivin, Liñán and Williams2012) that the ignition process of hydrogen combustion was indeed controlled by the particular combination of radicals production–consumption rates $\dot {\omega }_{\textit {m}} = \dot {\omega }_{\textrm{H}}+2\dot {\omega }_{\textrm{O}}+\dot {\omega }_{\textrm{OH}}$ . Therefore, to calculate the instantaneous auto-ignition point for a given simulation case, we first evaluate the iso-contour of the composite reaction-rate value $\dot {\omega }_m$ that sits above a selected percentile level ( $99.9\,\%$ ), and then compute its nearest point to the inlet. The percentile threshold used to define the ignition point was selected after examining a range of high-percentile values. It was observed that the ignition distance converged to the same values for sufficiently high thresholds, and that values above $99.9\,\%$ (e.g. $99.95\,\%$ ) tended to capture isolated, non-representative peaks that appear downstream and do not correspond to the primary ignition event. Therefore, the $99.9\,\%$ level was chosen as a compromise between robustness and physical relevance. As this position can vary for each time step of the simulation, the ignition distance $d_{\textit {ign}}$ is averaged over the statistically steady temporal span, which is approximately 3.7 characteristic residence times of the numerical domain.

Figure 7.

Auto-ignition averaged (a,b) distance $d_{\textit {ign}}$ and (c,d) vertical coordinate $y_{\textit {ign}}$ , for different production-rate criteria $\dot {\omega }_i$ , in the (a,c) hot–cold and (b,d) isothermal cases.

Figure 7. Long description

The image contains four line graphs comparing the auto-ignition averaged distance and vertical coordinate for different production-rate criteria in hot-cold and isothermal cases. Panel A: This line graph shows the auto-ignition averaged distance (d_ign in meters) versus percentile for the hot-cold case (T1 not equal to T2). The x-axis represents the percentile ranging from 98.0 to 99.9, and the y-axis represents the distance (d_ign) in meters ranging from 0 to 0.06. The graph includes four data series represented by different symbols and colors: blue circles for ω_H, orange squares for ω_O, green triangles for ω_OH, and red diamonds for ω_m. Solid lines represent the tanh criterion, and dashed lines represent the s.s. criterion. Panel B: This line graph shows the auto-ignition averaged distance (d_ign in meters) versus percentile for the isothermal case (T1 equal to T2). The x-axis represents the percentile ranging from 98.0 to 99.9, and the y-axis represents the distance (d_ign) in meters ranging from 0 to 0.06. The graph includes four data series represented by different symbols and colors: blue circles for ω_H, orange squares for ω_O, green triangles for ω_OH, and red diamonds for ω_m. Solid lines represent the tanh criterion, and dashed lines represent the s.s. criterion. Panel C: This line graph shows the auto-ignition averaged vertical coordinate (y_ign in meters) versus percentile for the hot-cold case (T1 not equal to T2). The x-axis represents the percentile ranging from 98.0 to 99.9, and the y-axis represents the vertical coordinate (y_ign) in meters ranging from 0 to 0.0020. The graph includes four data series represented by different symbols and colors: blue circles for ω_H, orange squares for ω_O, green triangles for ω_OH, and red diamonds for ω_m. Solid lines represent the tanh criterion, and dashed lines represent the s.s. criterion. Panel D: This line graph shows the auto-ignition averaged vertical coordinate (y_ign in meters) versus percentile for the isothermal case (T1 equal to T2). The x-axis represents the percentile ranging from 98.0 to 99.9, and the y-axis represents the vertical coordinate (y_ign) in meters ranging from 0 to 0.0015. The graph includes four data series represented by different symbols and colors: blue circles for ω_H, orange squares for ω_O, green triangles for ω_OH, and red diamonds for ω_m. Solid lines represent the tanh criterion, and dashed lines represent the s.s. criterion.

Figures 7(a,b) show the values of the most $x$ -advanced position of the reaction-rate criteria for increasing percentiles (nearly maximum values) based on different radical reaction rates. Note that all the criteria explored herein converge for sufficiently high percentiles, and that the maxima are avoided owing to singularities and outliers that can appear in diverse reactive kernels far downstream from the auto-ignition event. Additionally, the comparison of the results for different inlet profiles is offered for hyperbolic tangent (solid) and self-similar (dashed) inlet cases. The results for hot–cold layers in figure 7(a) show that the ignition distance for the 99.9 % level is nearly 6 cm and does not vary for the different input profiles. However, the ignition position above the mixing-layer centreline $y_{\textit {ign}}$ shown in figure 7(c) is doubled for the self-similar inlet condition, pointing at the greater relevance of realistic reactant mixing, and probably indicating that the unchanged ignition distance is thermally controlled in this configuration, as predicted by Treviño & Liñán (Reference Treviño and Liñán1995). Our results support the asymptotic analyses of Treviño & Liñán (Reference Treviño and Liñán1995) for self-ignition in the simplest iso-velocity mixing-layer configuration. Specifically, when the temperature difference between streams is sufficiently large, the ignition location is controlled mainly by thermal diffusion and chemical induction, whereas moderate velocity variations play only a minor role. A similar conclusion applies in the opposite limit of fully developed turbulent mixing layers. The analysis of Im et al. (Reference Im, Chen and Law1998) shows that at sufficiently high temperatures, ignition is determined primarily by the interplay between chemical induction and scalar mixing time scales. In this regime, turbulence mainly affects the spatial distribution and intermittency of ignition kernels, while having only a limited impact on the overall ignition delay under the conditions considered.

On the other side, isothermal configurations show a reduction of the auto-ignition distance in figure 7(b) while the vertical position of the ignition point is unaffected by the type of inlet profile in figure 7(d). This fact was previously attributed to the friction-related temperature increase in the self-similar inlet condition with detailed mechanisms. Moreover, these preheated-fuel isothermal layers show a consistent reduction of the ignition distance when compared to the hot–cold cases of approximately $0.2$ cm, and a further reduction of more than $1$ cm in the self-similar case, with the initial extra temperature bump owing to thermal viscous dissipation. The approximately $50$ K temperature rise generated in the self-similar case, superimposed on the $1250$ K outer-stream temperature and absent from the prescribed inlet conditions of the hyperbolic-profile case, results in an approximately 25 % reduction in ignition distance. This decrease is broadly consistent with the reduction in homogeneous auto-ignition time reported in Sánchez & Williams (Reference Sánchez and Williams2014).

This behaviour differs from both the iso-velocity configuration examined in Treviño & Liñán (Reference Treviño and Liñán1995) and the turbulent configuration investigated in Im et al. (Reference Im, Chen and Law1998). Accordingly, an important conclusion of this subsection is that velocity gradients, when viscous dissipation is consistently accounted for, generate non-negligible temperature fluctuations that fundamentally alter the ignition dynamics of externally isothermal configurations. Moreover, beyond their impact on ignition analyses based on homogeneous assumptions and the associated reduction in ignition delay times, these effects also alter scalar transport by the Soret effect, thereby contributing further to the reduction in ignition distance. Additional details are provided below.

4.2. Specific results for non-isothermal streams

The ignition mechanism for case 1, involving $T_1=1250$ K and $T_2=545$ K, is analysed in figure 8 with the detailed comparison between the tanh (figure 8 a) and self-similar (figure 8 b) inlet profiles. The two-dimensional temperature field distribution at $t=0.0006$ s is presented with superimposed trajectories, for the fluid particle that is detected as the most-advanced ignition point at different instants. Each pathline, showing the history of the evolving fluid particle, is integrated backwards from the detected ignition spot using the stored velocity fields from the whole simulation. The detected igniting particle at $t=0.0006$ s is shown in red.

Figure 8.

Case 1 temperature fields and superimposed pathlines of fluid particles used to detect auto-ignition, for (a) $\tanh$ and (b) self-similar inlet conditions.

Figure 9.

Case 1 (a) temperature and (b) mixture fraction variation along the pathlines. (c) Time evolution of the mixture fraction value at the auto-ignition point $Z_{\textit {ign}}$ .

First, we point out that the temporal dispersion of detected pathlines, in both inlet subcases, is small. Moreover, the ignition points are also contained in a well defined ignition distance. However, the nature of the ignition on the self-similar condition shows in figure 8(b) a clear shift towards the hot-side stream, which can be explained by comparing the inlet profiles of figure 4(a): greater $y$ -coordinate positions are required to find high temperature values, and in turn sit on larger amounts of fuel owing to the consideration of realistic mass diffusion. This temperature-controlled shifting behaviour is hindered by the smaller mixing thickness in the hyperbolic tangent case in figure 8(a), which can be traced above in figure 6(a), and as a result, the igniting fluid parcels follow paths well contained in the roller region.

The nature of each auto-ignition regime can be explored through the information presented in figure 9. First, the temperature $T$ evolution over each detected ignited flow particle trajectory (figure 9 a) shows a coherent build-up with the reaction initiation, which starts from slightly hotter conditions in the hyperbolic tangent case. However, the mixture fraction $Z$ evolution along the pathlines (figure 9 b) shows a very different behaviour in each scenario. The mixture fraction definition used here is

(4.2) \begin{align} Z=\frac {S\hat {Y}_{\text{H}_{2}} - \hat {Y}_{\text{O}_{2}} +1}{S+1}, \end{align}

where $\hat {Y}_{\text{H}_{2}}$ and $\hat {Y}_{\text{O}_{2}}$ are normalised values of the fuel and oxidiser mass fractions, referred to upstream values of the bottom and top streams, respectively:

(4.3) \begin{align} \hat {Y}_{\textit{H}_{2}}=\frac {Y_{\text{H}_{2}}}{Y_{\text{H}_{2},0}\big |_2}, \quad \hat {Y}_{\text{O}_{2}}=\frac {Y_{\text{O}_{2}}}{Y_{\text{O}_{2},0}\big |_1}, \quad S=\frac {sY_{\text{H}_{2},0}\big |_2}{Y_{\text{O}_{2},0}\big |_1}. \end{align}

In particular, $s = 8$ is the amount of oxygen consumed per unit mass of hydrogen burnt, and the mass fractions for vitiated air $Y_{\text{O}_{2}} = 0.2448$ , $Y_{\text{N}_{2}} = 0.5799$ and $Y_{\text{H}_{2O}}=0.1748$ that correspond to the molar fractions $X_{\text{O}_{2}}=0.201$ , $X_{\text{N}_{2}}=0.544$ and $X_{\text{H}_{2O}}=0.255$ . Therefore, $S = 32.68$ and the stoichiometric value of the mixture fraction is $Z_{\textit {st}} = 1/(S+1) \simeq 0.0297$ , considerably higher (+50 %) than the values found at the detected ignition $Z_{\textit {ign}}$ displayed along time in figure 9(c). It should be noted that the standard deviation of the mixture fraction value (shaded regions) is much larger for the hyperbolic tangent inlet, where the relevant thermochemical region sits in the central part of the unstable shear layer, strongly entangled with the roller motion.

Figure 10.

Case 2 temperature field and ignition pathlines for different inlet conditions.

Figure 10. Long description

The image contains two line graphs labeled as Panel A and Panel B, each depicting temperature fields and ignition pathlines for different inlet conditions. Panel A is titled ‘tanh’ and Panel B is titled ‘s.s.’ Both graphs plot y (m) on the vertical axis and x (m) on the horizontal axis. The x-axis ranges from 0.040 to 0.070 meters, and the y-axis ranges from -0.002 to 0.002 meters. In both panels, dashed lines represent different ignition pathlines, and a red line indicates a specific pathline. The background of each graph shows a grayscale representation of the temperature field. In Panel A, the red line shows a relatively smooth path compared to the dashed lines, which exhibit more variation. In Panel B, the red line and dashed lines show more pronounced deviations, indicating different flow dynamics under the specified inlet conditions. The graphs illustrate the differences in combustion mechanisms, with Panel A showing a more controlled wave mechanism and Panel B showing a turbulent mixing process.

4.3. Specific results for isothermal streams

The shock-free case 2 is conformed by a co-flow of air and fuel at the same temperature $T_1=T_2 = 1250$ K. The results show a much larger spatial dispersion of the igniting particles paths, given that the hot region is not constrained to one side of the layer. The self-similar case in figure 10(b), including a viscous-dissipation temperature increase (see figure 4 b), provides shorter ignition distances than the hyperbolic tangent condition in figure 10(a), as already stated above. On top of that, the pre-heated hydrogen stream promotes even higher density and velocity changes across the layer for the same outer-region Mach numbers $M_1$ and $M_2$ compared to the hot–cold configuration of case 1. This fact enhances the hydrodynamic destabilisation of the shear layer, thus promoting the formation of intense vortical structures.

The temperature evolution over the trajectories plotted in figure 11(a) shows a much larger irregularity than case 1 solutions, with localised temperature decrease below 1250 K owing to the dynamics of the shear-vortex layer and its non-negligible local expansion. In turn, the disparity in flow-particle history renders a wide range of $Z$ values (from 0 to 1) at the origin of the ignition-detection pathlines shown in figure 11(b), along which mixing must also build up fast. Finally, figure 11(c) allows us to identify an equally disperse $Z_{\textit {ign}}$ oscillation in time for both inlet conditions. Unlike what was observed in figure 9 associated with non-isothermal streams, the standard deviation of the mixture fraction value (shaded regions) is barely affected by the choice of the inlet profile. As a result, it seems that the relevance of the inlet-profile information in a highly disturbed hydrodynamic configuration is primarily lost, showing very comparable trends in terms of auto-ignition character variability for both hyperbolic tangent and self-similar cases.

Figure 11.

Case 2 (a) temperature and (b) mixture fraction variation along the pathlines. (c) Time evolution of the mixture fraction value at the auto-ignition point $Z_{\textit {ign}}$ .

However, we should highlight the two key competing effects from the upstream boundary condition: (i) inertia variations driven by advanced description of temperature and composition of the co-flow mixing layer, and (ii) temperature increase owing to thermal viscous dissipation that accelerates the initiation of chemical reactions. From the results analysed herein, the former seems to promote homogenisation through flow destabilisation, while the latter acts strongly to reduce the ignition distance $d_{\textit {ign}}$ .

5.1. Incident shock angle comparison

The variation of the oblique shock angle implies a relative and consistent change of $T_1$ and $T_2$ to achieve the aforementioned values of $T_3$ and $T_4$ . In turn, the reflected expansion arising from the resulting configuration of mechanical equilibrium has a wide range of intensities depending on the selection of $\sigma _i$ , leading to relevant changes of temperature and Mach number behind this reflected wave ( $M_5, T_5$ ) along the upper stream. Therefore, the cooling effect of the sudden expansion over the initiation of the reaction must be considered in shock-assisted ignition studies. The range of $\sigma _i$ angles in this work is constrained to the surroundings of the limiting value that restrains ignition, $\sigma _i \simeq 36^\circ$ . In fact, the temperature fields of figure 12 show that the reactive mixing layer is ignited for the $\sigma _i=35^\circ$ configuration, while it is unable to react in the given domain length for $\sigma _i = 37.5^\circ$ . The black solid lines parallel to the deflected mixing layer placed at an unperturbed position are used to compare the temperature evolution along the upper stream behind the shock, represented in figure 13(a) for various oblique shock angles. It should be noted that the same post-shock temperatures are achieved in all cases ( $T_3 = 1250$ K), but the reflected expansion is more intense as the incident shock angle is increased. This behaviour leads to an upper stream that is further cooled down; dashed lines represent the ideal-flow solution behind the expansion, which eventually does not enable the mixture to achieve auto-ignition in the given distance. In fact, the secondary expansions arising in the flow modify the expected post-shock stream temperature, and render invalid first-order estimates of ignition based on ideal shock mixture temperature.

Figure 12.

Temperature distribution with oblique shock impingement: (a) case 4 with $\sigma _i = 35^\circ$ , and (b) case 5 with $\sigma _i = 37.5^\circ$ , for hyperbolic tangent inlet profiles.

Finally, the computed ignition distances are gathered in figure 13(b) for cases 3–5 with either $\tanh$ or self-similar inlet profiles, using the different species production–consumption criteria presented above with a 99.9 $\,\%$ percentile threshold. Note that the ignition distance decreases when employing the self-similar profile, whereas increasing $\sigma _i$ progressively delays the ignition event, leading to failure (i.e. no ignition within the computational domain) for the hyperbolic tangent profile with $\sigma _i = 37.5^\circ$ . Case 6, with $\sigma _i = 40^\circ$ , is not presented here since ignition does not occur for either inlet profile configuration.

Figure 13.

(a) Post-shock temperature evolution of the upper stream in the simulated cases (solid lines) and the corresponding ideal solution (dashed line). (b) Auto-ignition averaged distance $d_{\textit {ign}}$ for different production-rate criteria.

Figure 13. Long description

The image contains two line graphs comparing temperature evolution and auto-ignition distance for different initial angles. Panel A: This line graph shows the post-shock temperature evolution of the upper stream in simulated cases. The x-axis represents distance (x) in meters, ranging from 0 to 0.12 meters. The y-axis represents temperature (T) in Kelvin, ranging from 1200 K to 1300 K. Three different initial angles (sigma_i) are compared: 32.5 degrees (blue line), 35.0 degrees (orange line), and 37.5 degrees (red line). Solid lines represent the simulated cases, while dashed lines represent the corresponding ideal solution. The graph shows that the temperature initially drops sharply for all angles and then stabilizes, with slight variations depending on the initial angle. Panel B: This line graph shows the auto-ignition averaged distance for different production-rate criteria. The x-axis represents the initial angle (sigma_i) in degrees, ranging from 32.5 to 37.5 degrees. The y-axis represents the auto-ignition distance (d_ign) in meters, ranging from 0.030 to 0.045 meters. Four different production-rate criteria are compared: omega_H (blue circles), omega_OH (green stars), omega_O (orange squares), and omega_m (red diamonds). Solid lines represent the tanh criteria, while dashed lines represent the steady-state (s.s.) criteria. The graph shows that the auto-ignition distance generally increases with the initial angle for all criteria, with some variations depending on the specific criteria used.

The results of case 5 for a limiting shock incident angle $\sigma _i = 37.5^\circ$ with different inlet profiles are compared in figures 14(a,b). The computational data in this case induce us to think that the self-similar effects do constitute a requirement for achieving ignition. In the hyperbolic tangent case, the auto-ignition point cannot be calculated because a continuous ignition zone has not yet been formed, but weak isolated reactive kernels that are washed away are produced. In fact, the mean temperature and H $_2$ mass fraction profiles along the marked solid black line perpendicular to the mixing layer, placed before the auto-ignition position, are plotted in figure 14(c). This comparison shows an slightly greater amount of averaged H $_2$ concentration in the hot side of the self-similar case. However, given the barely appreciable difference between the two cases, figure 14(d) displays the mixedness of these profiles along the mixing layer, following the definition in Yu et al. (Reference Yu, He, Zhang and Liu2020) and Li et al. (Reference Li, Wang, Yu, Zhang and Liu2020):

(5.1) \begin{align} \xi (x) = \frac {1}{Y_{{\textrm {H}_2,0}}^{2}H}\int _{H_y} 4 Y_{{\textrm {H}_2}} \left ( Y_{{\textrm {H}_2,0}}-Y_{{\textrm {H}_2}} \right ) \text{d}y, \end{align}

where $H$ is the height of the selected subdomain. Thus it can be noticed that the mixing in the self-similar case develops from a higher value than in the hyperbolic tangent configuration. As the latter advances in space, it achieves the same mixing as in the self-similar case with a certain delay. However, it does so under further developed conditions of vorticity and stretching, which prevents the flame from effectively remaining anchored at a fixed ignition distance.

Figure 14.

Temperature distribution of case 5 with (a) $\tanh$ and (b) self-similar inlet profiles. (c) Comparison of time-averaged temperature and mass fraction profiles over the marked black transverse line across the mixing layers. (d) Mixedness evolution along the longitudinal coordinate.

Figure 15.

(a) Density fields of case 5 at the dotted subdomain marked in figures 14(a) and 14(b) before the completion of the numerical ignition transient, and (b) their associated snapshot temperature–density scatter plot.

Figure 15. Long description

The image contains two panels. Panel A shows two density fields labeled as tanh and s.s. The density fields are visualized with a color gradient ranging from light to dark brown, indicating varying densities from 0.0684 to 0.9409. Panel B presents a scatter plot of temperature versus density, with temperature in Kelvin on the vertical axis ranging from 400 to 1200 K, and density on the horizontal axis ranging from 0.25 to 0.75. The scatter plot includes two datasets, one in orange labeled tanh and the other in blue labeled s.s. The scatter plot shows a distribution of data points indicating the relationship between temperature and density for the two different cases.

We will then inspect the flow field during the transient regime, between the inert flow initial condition and the start of the chemical reaction. The density distributions of the post-shock region in the $\tanh$ and self-similar cases are presented in figure 15(a), showing small differences to visual inspection. The subdomain of analysis constitutes a fraction of the computational domain, which is highlighted in figures 14(a) and 14(b) with dotted rectangles. The lower level of mixing quality of the hyperbolic tangent case can also be observed in the density–temperature scatter in figure 15(b), extracted from the density fields in figure 15(a). For the same levels of temperature, a wider range of densities is observed in the $\tanh$ case. This indicates a stronger persistence of unmixed H $_2$ (low density) and air (higher density) streams, which later fail to ignite, in contrast with the self-similar reaction success.

In light of these results, the conclusion regarding the primary influence of inlet-prescribed mixing is reinforced. Subtle variations in the selected far-field characteristic conditions and in the inner-region profiles of the incoming mixing layer significantly affect the qualitative behaviour of the chemical reaction in supersonic hydrogen–air co-flows. These effects are fundamentally governed by enhanced mass diffusion and viscous dissipation, both of which must be consistently represented at the inlet.

5.2. Effect of the shock impingement distance

From an external inert perspective, the distance at which the shock impinges on the mixing layer does not affect the post-interaction equilibrium properties of the streams. However, when considering the inner region of the reactive mixing layer, with the mixing-layer thickness increasing and where reactions are promoted by mass diffusion and heat flux, it becomes evident that the shock impingement distance can directly influence the aerothermal properties inside the layer that are relevant to the local auto-ignition process. This effect is of paramount importance in scramjet design, where the combustion chamber configuration must ensure reliable ignition, so a robust performance must be maintained not only under varying shock angles but also for different impingement distances.

To measure the impact of this effect, the impingement distance $d_{\textit {imp}}$ , defined as the distance from the inlet domain and normalised by the initial vorticity thickness $\delta ^{0}_{\omega }$ , is varied while keeping the external thermodynamic conditions unchanged. The results are shown in figures 16(a–c) for $\sigma _i = 35^\circ$ and $d_{\textit {imp}}=20\delta ^{0}_{\omega }$ , $d_{\textit {imp}}=50\delta ^{0}_{\omega }$ and $d_{\textit {imp}}=70\delta ^{0}_{\omega }$ , respectively, where the two-dimensional temperature fields are displayed for comparison. It can be observed that the further the impingement, the smaller the ignition distance $d_{\textit {ign}}$ , measured from the shock intersection. The ignition location is computed again with the 99.9 % percentile of $\dot {\omega }_m$ , and presented in the $d_{\textit {ign}}^{99.9}{-}y_{\textit {ign}}^{99.9}$ plane in figure 16(d), with a confirmed trend of post-shock ignition distance reduction for increasing impingement distance. Note that although the simulations shown in figures 16(a–c) are all prescribed by the hyperbolic tangent profile, the computed distances also show the results of the case studies for the self-similar inlet profiles with empty markers, with both inlet conditions showing equivalent trends. The comparison between the two inflows shows that the auto-ignition distance remains virtually unchanged for $d_{\textit {imp}} = 20\,\delta ^{0}_{\omega }$ , with relative error 1.6 $\,\%$ when changing the inlet profile. In the intermediate $50\,\delta ^{0}_{\omega }$ case, the difference in the auto-ignition distance depending on the profile rises to 7.25 $\,\%$ , which remains comparable to the $70\,\delta ^{0}_{\omega }$ case, where the difference does not exceed 7.6 $\,\%$ .

This can be readily connected to the analysis of previous sections and explained by the fact that for longer impingement distances, the mixing layer has more time to develop before the shock compression. The streamwise distribution of the mixedness factor $\xi (x)$ is presented in figure 16(e), where the pre-shock behaviour is identical for all the cases, and the post-shock contraction triggers more intense mixing dynamics for further impingement distances. As a result, the post-shock temperature rise occurs under better-mixed conditions, reducing the post-shock ignition distance and thereby partially compensating the overall auto-ignition process.

Figure 16.

Temperature distribution of reactive mixing layers with oblique shock at $\sigma _i = 35^\circ$ impinging at (a) $20\,\delta ^{0}_{\omega }$ , (b) $50\,\delta ^{0}_{\omega }$ and (c) $70\,\delta ^{0}_{\omega }$ . Comparison of (d) auto-ignition averaged distances $d_{\textit {ign}}$ and (e) streamwise evolution of mixedness.