Heat and momentum losses in ${\text {H}}_{2}$–${\text {O}}_{2}$–${\text {N}}_{2}/{\textrm{Ar}}$ detonations: on the existence of set-valued solutions with detailed thermochemistry

Abstract

The effect of heat and momentum losses on the steady solutions admitted by the reactive Euler equations with sink/source terms is examined for stoichiometric hydrogen–oxygen mixtures. Varying degrees of nitrogen and argon dilution are considered in order to access a wide range of effective activation energies, $E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}$, when using detailed thermochemistry. The main results of the study are discussed via detonation velocity-friction coefficient (D–$c_{\textrm{f}}$) curves. The influence of the mixture composition is assessed, and classical scaling for the prediction of the velocity deficits, $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, as a function of the effective activation energy, ${E}_{\textrm{a,eff}}/R_{\textrm{u}} T_{0}$, is revisited. Notably, a map outlining the regions where set-valued solutions exist in the $E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}\text {--}{\alpha }$ space is provided, with $\alpha$ denoting the momentum–heat loss similarity factor, a free parameter in the current study.

1 Introduction

One of the most promising steps toward reducing greenhouse gas emissions is the decarbonization of the economy via the widespread use of hydrogen (${\textrm{H}}_{2}$) [1, 2]. While different possibilities exist to promote its use, such as direct oxidation, ${\textrm{H}}_{2}$ fuel cells seem to have gained the most traction with the general public as a plausible approach for stationary and mobile applications [3]. The increased presence of ${\textrm{H}}_{2}$ fuel cells, however, augments the likelihood of fuel leaks, accidental combustion events, and explosions. In particular, ${\textrm{H}}_{2}$ storage and handling requires pressurized tanks (up to 70 MPa), which results in additional safety hazards in comparison with conventional hydrocarbons [4]. The small size of the ${\textrm{H}}_{2}$ molecule may lead to the embrittlement of piping [5] and subsequent fuel losses to the surrounding environment. Due to the wide flammability of ${\textrm{H}}_{2}$ in air ($\approx \,4{-}75$ vol%) [6], hot surfaces (ubiquitous in industrial settings) [7] or uncontrolled electric discharges could lead to the ignition of a flame that, under the right circumstances, may accelerate and transition to a detonation.

To control and/or mitigate these hazards and promote the deployment of green hydrogen as a renewable energy carrier, the prediction of limiting behaviors, both for flammability [8,9,10] and for detonability [11,12,13], continues to be a topic of active research; the current manuscript is concerned with the latter. Near the limits, phenomena that are typically discarded in the modeling of detonations, i.e., momentum and energy losses, need to be accounted for. In multidimensional simulations, this entails employing very fine computational grids near the walls so that the hydrodynamic and thermal boundary layers are properly resolved, in addition to the very stringent resolution requirements for resolving the detonation front itself. These computations are resource-hungry in that they require a tremendous amount of CPU time, memory, storage, and/or network bandwidth, thus rendering them impractical/inaccessible for detailed parametric studies, even if simplified descriptions of the chemistry are used [14]. One-dimensional (1-D) models arise as a natural alternative to seek order of magnitude estimates for detonability limits or to explain the velocity deficits ($D/D_{\textrm{CJ}} < 1$) observed for detonation propagation in narrow channels or small tubes; channels/tubes whose characteristic length scale is of the order of the average cell size, $\lambda$, of the mixture considered. In such models, the aforementioned losses are included as effective source/sink terms to the conservation laws and yield steady solutions at sub-CJ values. These models have been explored since the mid-1980s.

Fig. 1

Regions and critical point found in classical D–$c_{\textrm{f}}$ curves of interest for this work

Zel’dovich et al. [15] set up the eigenvalue problem. The model’s output is D–$c_{\textrm{f}}$ curves, which determine the combination of detonation velocity, D, and friction coefficient, $c_{\textrm{f}}$, that yield steady detonation wave propagation. These curves usually present a single turning point, $c_{\textrm{f,crit}}$, at a mixture-dependent velocity deficit—usually at the so-called quasi-detonation regime [16] (see Fig. 1)—which provides a first-order estimate of the minimum diameter capable of sustaining a detonation in tubes, and that compares reasonably well against experimental data [12, 17,18,19] given the simplicity of the model. Zel’dovich et al.’s [15] solutions were computed numerically in the limit of strong heat release for relatively high detonation velocities extending slightly below $c_{\textrm{f,crit}}$. Brailovsky and Sivashinsky [16] extended Zel’dovich’s work in the early 2000s to higher deficits by realizing that there are solutions that do not involve satisfying the CJ condition, thereby unveiling the co-existence of multiple steady solutions for a given value of $c_{\textrm{f}}$. The authors defined additional detonation propagation regimes, beyond the quasi-detonations referred to above, based on D–$c_{\textrm{f}}$ curves, namely choking and subsonic detonations. Thereafter, the same authors re-examined their regime diagram for non-adiabatic cases [20] and found that the addition of convective heat losses through the walls moves the detonability limits entirely within the supersonic domain (the quasi-detonation regime), in line with experimental observations, and strongly affects the low-speed regimes. For years, only the upper high-velocity branch of the D–$c_{\textrm{f}}$ curves would be thought as physical and stable [15, 21]. The middle branch including the choking regime was usually identified as a repeller, that is, the solutions either fail or jump toward the upper high-velocity solutions, and thus were often excluded and considered nonphysical. Higgins [22] provides a complete account of the progress in steady 1D detonations until 2012.

Recently, Semenko et al. [23] performed an extensive analysis of the non-adiabatic steady solutions in [20] and found a peculiarity in the vicinity of the low-velocity choking regime. Given a detonation propagation speed, D, a family of friction coefficients, $c_{\textrm{f}}$, satisfies the downstream boundary condition. These solutions cover a continuum region of D–$c_{\textrm{f}}$ space and are thus called set-valued solutions. Sow et al. [24] carried out unsteady 1-D simulations to investigate the long-time evolution using these set-valued solutions as initial conditions. The authors revealed interesting dynamics in which a transition to the upper high-velocity quasi-detonation regime takes place or the solution simply stabilizes into a neighboring point within the set-valued region; these results question the rigid convictions regarding the stability of the lower branch of the D–$c_{\textrm{f}}$ curves as the authors were able to sustain low-velocity detonations for long periods of time. These solutions were also argued to explain experimental observations of detonations in porous media [25], a regime likely to correspond to the choking regime that was thought to be unstable.

The applicability of the latter models or even simpler scalar analogs to deflagration-to-detonation transition (DDT) had not previously been suggested nor explored until Melguizo-Gavilanes et al. [26] pointed out that the transition between the aforementioned traveling wave solutions bears striking similarities with what is typically observed during DDT experiments in narrow channels.

Recent work in our group [27] examined the effect of chemistry modeling on D–$c_{\textrm{f}}$ curves. Qualitative and quantitative differences were reported when conventional simplified kinetics schemes (i.e., 1-step and 3-step chain-branching) were compared with detailed thermochemistry. Note that most (if not all) of the fundamental work cited above employed 1-step descriptions of the chemistry, and given the findings in [27] a natural question to ask is whether the set-valued solutions reported in [23] exist for detailed kinetics. To answer this question, a wide range of effective activation energies for stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ are investigated (by diluting with Ar or ${\text {N}}_{2}$) using the momentum–heat loss similarity factor, $\alpha$, as a free parameter.

The paper is structured as follows: Section 2 introduces a general mathematical formulation used to compute detonation waves with friction/heat losses and detailed chemistry. Section 3 discusses the main results of our study via D–$c_{\textrm{f}}$ curves; the influence of the mixture dilution is assessed, and classical scalings for the prediction of the velocity deficits $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$ as a function of the effective activation energy, $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$, are revisited, and a map outlining the regions where set-valued solutions exist in $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0 \text {--} \alpha$ space is provided. Closing remarks are given in Sect. 4.

2 Physical model

2.1 Governing equations

The effect of friction [22, 28] and heat losses [15, 20, 23] on one-dimensional detonations in tubes is modeled by including sink/source terms to momentum,

$$\begin{aligned} f = {\frac{P}{2 A \phi }} {\tilde{c}}_{\textrm{f}} \; \rho |u| u = c_{\textrm{f}} \rho |u| u, \end{aligned}$$

and energy,

$$\begin{aligned} q&= {\frac{P}{2 A \phi }} \alpha {\tilde{c}}_{\textrm{f}} \; \rho |u| \left[ c_{{p}} (T - T_{\textrm{wall}}) + u^2/2\right] \\&= \alpha c_{\textrm{f}} \rho |u| \left[ c_{{p}} (T - T_{\textrm{wall}}) + u^2/2\right] , \end{aligned}$$

in the reactive Euler equations. Here, T, $\rho$, u, and $c_{{p}}$ are the gas temperature, gas density, axial velocity in the laboratory frame, and specific heat at constant pressure, respectively; P represents the perimeter of the tube, $T_{\textrm{wall}}$ the temperature of the wall, A its cross-sectional area, and $\phi$ its porosity; the wetted area P/A is directly proportional to $\phi$ [23]. The variable ${\tilde{c}}_{\textrm{f}}$ denotes a dimensionless skin-friction coefficient representative of the roughness on the walls of the tube, and $\alpha$ is the momentum–heat loss similarity factor; $\alpha = 1$ reproduces the Reynolds analogy for heat and momentum losses, characteristic of turbulent flow in smooth tubes; values of $\alpha < 1$ violate this analogy and are argued to mimic rough tubes or packed beds. Note that the inclusion of the kinetic energy in the heat sink term q was considered negligible in previous studies [23]. The authors argued that discarding the kinetic energy would yield a simpler formulation while retaining the general nature of the solutions. Given the high flow speeds induced by the leading shock in detonations, both terms are retained in the present formulation since for a stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ mixture they have a similar order of magnitude:

$$\begin{aligned} {\text {(i)}}\;c_{{p}} (T - T_{\textrm{wall}})&\sim c_{{p,\textrm{vN}}} (T_{{\textrm{vN}}} - T_{\textrm{wall}}) \\&\sim 3000\times (1768.8-300) \sim 4.4 \times 10^6\;{\text {J/kg}}, \end{aligned}$$

where the subscript ${\text {vN}}$ denotes the von Neumann state, and

$$\begin{aligned} \text {(ii)}\;u^2/2 \sim u_{{\textrm{vN}}}^2/2 \sim 2330^2/2 \sim 2.7\times 10^6\;{\text {m}}^2/{\text {s}}^2. \end{aligned}$$

The system of equations written as a function of thermicity reads:

$$\begin{aligned}&\frac{{\textrm{D}} \rho }{{\textrm{D}} t} + \rho \frac{\partial u}{\partial x} = 0, \end{aligned}$$

(1)

$$\begin{aligned}&\rho \frac{{\textrm{D}} u}{{\textrm{D}} t} + \frac{\partial p}{\partial x} = -\frac{f}{\phi }, \end{aligned}$$

(2)

$$\begin{aligned}&\frac{\textrm{D}p}{\textrm{D}t} - a_{\textrm{f}}^2 \frac{{\textrm{D}} \rho }{{\textrm{D}}t} = (\gamma -1) \left( u\frac{f}{\phi } - \frac{q}{\phi }\right) + \rho a_{\textrm{f}}^2 {\dot{\sigma }}, \end{aligned}$$

(3)

$$\begin{aligned}&\frac{{\textrm{D}} Y_k}{{\textrm{D}} t} = \frac{W_k {\dot{\omega }}_k}{\rho },\;\; k = 1, \ldots , N. \end{aligned}$$

(4)

The material derivative, ${\textrm{D}}/{\textrm{D}}t$, the frozen speed of sound, $a_{\textrm{f}}$, and the thermicity parameter, ${{\dot{\sigma }}}$, are expressed as:

$$\begin{aligned} \begin{aligned} \frac{{\textrm{D}}}{{\textrm{D}} t}&= \frac{\partial }{\partial t} + u \frac{\partial }{\partial x}; \quad a_{\textrm{f}}^2 = \left. \frac{\partial p}{\partial \rho } \right| _{s,{{\textbf {Y}}}}; \\ {\dot{\sigma }}&= \sum _{k=1}^{N} \left( \frac{{\overline{W}}}{W_k} - \frac{h_k}{c_p T}\right) \frac{{\textrm{D}}Y_k}{{\textrm{D}}t}. \end{aligned} \end{aligned}$$

(5)

Here, p, x, and t are the gas pressure, the spatial coordinate, and time, respectively. The mass fraction, molecular weight, enthalpy, and net production/consumption rate per unit mass of species k are given by $Y_k$, $W_k$, $h_{k}$, and ${\dot{\omega }}_k$; $\gamma = c_{p}/c_{v}$ is the ratio of the specific heats, ${\overline{W}}$ the average molecular weight of the mixture, and the subscripts s and ${\textbf{Y}}$ denote that the derivative that defines $a_{\textrm{f}}$ is taken at constant entropy and fixed composition.

The system above is closed with the ideal equation of state:

$$\begin{aligned} p = \rho \frac{R_{\textrm{u}}}{{\overline{W}}} T; \quad {\overline{W}} = 1/\sum _{k=1}^{N} Y_k/W_k \end{aligned}$$

(6)

with $R_{\textrm{u}}$ the universal gas constant.

2.1.1 Shock-attached frame of reference

System (1)–(4) can be written for a frame of reference moving with the leading shock at a constant speed, $-D$, by introducing the following two variables, $\xi = x_{\textrm{s}} - x$ with $x_{\textrm{s}}$ the instantaneous shock position and ${w} = D - u$, which represent a new spatial coordinate and flow velocity measured relative to the leading shock. Substituting in the material derivative and seeking for steady solutions (i.e., $\partial /\partial t = 0$), the mapping is $\textrm{D}/\textrm{D}t \rightarrow {w}\textrm{d}/\textrm{d}\xi$. Furthermore, noting that ${w} = \textrm{d}\xi /\textrm{d}t$, the system above becomes:

$$\begin{aligned} \frac{\textrm{d} \rho }{\textrm{d} t}&= - \rho \left[ \frac{{\dot{\sigma }} + F_q - Q + (\eta -1)F}{\eta } \right] , \end{aligned}$$

(7)

$$\begin{aligned} \frac{\textrm{d} {w}}{\textrm{d} t}&= {w} \left[ \frac{{\dot{\sigma }} + F_q - Q + (\eta -1)F}{\eta }\right] , \end{aligned}$$

(8)

$$\begin{aligned} \frac{\textrm{d} p}{\textrm{d} t}&= - \rho {w}^2 \left[ \frac{{\dot{\sigma }} + F_q - Q - F}{\eta }\right] , \end{aligned}$$

(9)

$$\begin{aligned} \frac{\textrm{d} \xi }{\textrm{d} t}&= {w}, \end{aligned}$$

(10)

$$\begin{aligned} \frac{\textrm{d} Y_k}{\textrm{d} t}&= \frac{W_k {\dot{\omega }}_k}{\rho },\;\;k = 1, \ldots , N. \end{aligned}$$

(11)

The variable $\eta =1-M^2$ is the sonic parameter with $M={w}/a_{\textrm{f}}$ being the Mach number computed using the frozen speed of sound, $a_{\textrm{f}}$. The functions $F_q$, F, and Q are given by:

$$\begin{aligned} F_q&= \frac{\gamma - 1}{a^2_\textrm{f}} \frac{c_{\textrm{f}}}{\phi } (D - w)^2 |D - w|, \end{aligned}$$

(12)

$$\begin{aligned} F&= \frac{c_{\textrm{f}}}{\phi } \left( \frac{D}{\textrm{w}}- 1\right) |D - \textrm{w}|, \end{aligned}$$

(13)

$$\begin{aligned} Q&= \frac{(\gamma - 1)q}{a^2_\textrm{f} \phi \rho }. \end{aligned}$$

(14)

The implementation was done using the shock and detonation toolbox (SDT) [29] as a base, which only includes the ideal case in its standard distribution, by introducing $F_q$, F, and Q, into the ZND_solve subroutines; any chemical mechanism written in Cantera format [30] (i.e., .cti and/or .yaml files) can thus be investigated in this fashion. The Scipy-solveivp open-source Python package is used to solve the system of equations using an implicit multi-step backward differentiation formula with relative and absolute numerical tolerances of $10^{-6}$ and $10^{-12}$, respectively, minimizing any numerical errors.

2.2 Solution procedure for non-ideal cases

The system of equations above yield D–$c_{\textrm{f}}$ curves, which represent the locus of steady solutions that the model admits. Detonations with friction losses admit two (or more) steady solutions for a particular value of $c_{\textrm{f}}$. The boundary condition at the shock ($t = 0$ or $\xi = 0$), that is, the von Neumann (${\textrm{vN}}$) state, is computed using the shock jump conditions at a fixed initial pressure, temperature, and composition. The solution methodology consists in marching downstream from the shock until the condition $M = {w/a_{\textrm{f}}} = 1$ is satisfied. For this to occur, the numerator and denominator on the right-hand side of Eqs. (7)–(11) should vanish simultaneously. An initial closed interval is thus selected, $[c_{\textrm{f},\min },\, c_{\textrm{f},\max }]$, for a given detonation speed, D, and it is checked whether (and where) the latter condition is satisfied. Further details on the algorithm implemented to automate the aforementioned solution procedure and on the existence of solutions without a sonic point are included in the work of Mejía-Botero et al. [27]. Our solver has been used in the past to study detonations with curvature [31,32,33] and friction [27] losses. The ability of the model and the solver to estimate detonability limits was assessed by Mejía-Botero et al. [34], showing that it is in reasonable agreement with experimental data.

2.3 Mixtures of interest

Stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ mixtures with ${\text {N}}_{2}$ and Ar dilution ranging from 0–56% ($X_{\textrm{N}_{2}}=0{-}0.56$) and 0–70% ($X_{\textrm{Ar}}=0{-}0.70$), respectively, are used to access a wide range of effective activation energies, $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$. The initial conditions of the fresh mixture ahead of the shock are fixed to $T_0 = 300$ K and $p_0 = 100$ kPa. Figure 2 shows the effect of dilution on relevant thermodynamic parameters and ideal detonation properties using the chemical mechanism of Ó Conaire et al. [35] computed with the ZND solver available in the standard distribution of the SDT. Note that the quantitative differences between the mechanism of Mével et al. [36] and Ó Conaire’s are modest—see Veiga-López et al. [27]; the latter is used for simplicity as it already included Ar as a diluent.

Fig. 2

$p_{\textrm{vN}}$, $T_{\textrm{vN}}$, and $\gamma$ at initial (0), von Neumann (${\textrm{vN}}$), and Chapman–Jouguet (${\textrm{CJ}}$) conditions, effective activation energy $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$ calculated with a 2% variation of the gas temperature around the $\textrm{vN}$ state, the induction length $l_{\textrm{ind}}$, and the ideal detonation speed $D_{\textrm{CJ}}$ for varying molar fraction X of nitrogen (${\text {N}}_{2}$) and argon (Ar) computed with the SDT [29] and the chemical mechanism of Ó Conaire et al. [35]

A few features stand out from Fig. 2. As dilution is increased for ${\text {N}}_{2}$ mixtures: (i) the quantities $p_{\textrm{vN}}$ and $T_{\textrm{vN}}$ decrease by 16 and 13%, respectively; (ii) the ideal propagation speed, $D_{\textrm{CJ}}$, exhibits a stronger decrease to reach 70% of the undiluted value; (iii) $\gamma$ remains almost constant ahead of the shock, 0, at the ${\textrm{vN}}$ and ${\textrm{CJ}}$ states; (iv) the induction length, $l_{\textrm{ind}}$, increases fourfold; (v) $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$ increases from 28 to $\approx \,33$ for the maximum dilution considered.

For increasing Ar dilution, on the other hand: (i) $p_{\textrm{vN}}$ decreases by 13%, whereas $T_{\textrm{vN}}$ increases by $16\%$; (ii) $D_{\textrm{CJ}}$ exhibits a stronger nonlinear decrease when compared to ${\textrm{N}}_{2}$ to reach $60\%$ of the undiluted value; (iii) $\gamma$ shows a significant increase that lies between 11 and 17% depending on the level of dilution considered; (iv) $l_{\textrm{ind}}$ is not monotonic, showing an initial reduction from 41.3 to $34.9\,\upmu {\text {m}}$ for $X_{\textrm{Ar}}=0{-}0.25$, and an increase thereafter to $70.3\,\upmu {\text {m}}$ at $X_{\textrm{Ar}} = 0.7$; (v) $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$ decreases by $23\%$ for $X_{\textrm{Ar}} = 0.7$ being 21, the lowest value to be examined. The features just described will be helpful to interpret the results that will be presented in Sect. 3 in which momentum and heat losses are included.

3 Results and discussion

3.1 Effect of heat losses on D–$c_{\textrm{f}}$ curves

3.1.1 Undiluted mixtures

Figure 3a shows the curves obtained for varying similarity factors, $\alpha$. For the most part, they are qualitatively similar to those presented by Brailovsky and Sivashinsky [20] except that the curve without heat losses (i.e., $\alpha = 0$) shows a single turning point $c_{\textrm{f,crit}} = 419.5\,{\text {m}}^{-1}$ located within the quasi-detonation regime ($a_{\textrm{b}}< D < D_{\textrm{CJ}}$; $a_{\textrm{b}} = \sqrt{\gamma _{\textrm{CJ}} T_{\textrm{CJ}} R_{\textrm{u}}/{\overline{W}}_{\textrm{CJ}}}$ the equilibrium sound speed), with a velocity deficit of $D = 0.77D_{\textrm{CJ}}$. As it was already discussed by Veiga-López et al. [27], no second turning point is obtained at high velocity deficits (low shock speeds) for stoichiometric ${\text {H}}_{2}$–O$_{2}$ with detailed thermochemistry. That is to say, the branch at the choking regime ($D < a_{\textrm{b}}$) asymptotically reaches $c_{\textrm{f}} \rightarrow 0$ as $D \rightarrow a_0$, the sound speed in the fresh mixture. Note that the integration of the system was stopped for values of $c_{\textrm{f}} < 1\,{\text {m}}^{-1}$ since $c_{\textrm{f}} \rightarrow 0$ represents a smooth tube with no roughness; ${\tilde{c}}_{\textrm{f}} = 0$ or a tube whose wetted area $P/A \rightarrow \infty$ represent two asymptotically unrealistic cases. This arbitrarily defined value of $c_{\textrm{f}}$ was chosen so that $c_{\textrm{f}}/c_{\textrm{f,crit}} \approx \, 0.24\%$ and thus is negligible. While reducing this lower numerical limit might slightly vary the quantitative values for the boundaries defined hereafter, the main conclusions of the work remain unchanged.

Fig. 3

a D–$c_{\textrm{f}}$ curves and b integrated chemical times, $\tau _{\textrm{chem}}$, obtained with the detailed mechanism of Ó Conaire et al. [35]. The horizontal dotted line denotes the limit between the quasi-detonation and choking regimes. The round markers in (a) indicate the points at which the p, T, u, ${\dot{\sigma }}$, and $Y_k$ profiles are analyzed. The $D/D_{\textrm{CJ}}$ value at which the first turning point occurs for each mechanism is also included in (a). The intensity change of the lines in (b) represents the scaled friction coefficient, $c_{\textrm{f}}/c_{\textrm{f,crit}}$, for a given value of $D/D_{\textrm{CJ}}$

For $\alpha = 1$, characteristic of tubes for which the Reynolds analogy holds, the curve shows a similar qualitative behavior but the influence of heat losses is clear: (i) the turning point, $c_{\textrm{f,crit}}$, moves toward smaller $c_{\textrm{f}}$ values and lower velocity deficits ($D(c_{\textrm{f,crit}}) = 0.89D_{\textrm{CJ}}$); $c_{\textrm{f,crit}}=183.1\,{\text {m}}^{-1}$ (56% lower than for the adiabatic case); (ii) $c_{\textrm{f}} > 0$ are only obtained at the quasi-detonation regime which naturally implies the existence of a limiting similarity factor, referred to as $\alpha _{\textrm{crit}}$ here, for which the choking regime ceases to exist. The importance of $\alpha _{\textrm{crit}}$ will become apparent in Sect. 3.3 where set-valued solutions with detailed thermochemistry are discussed. The dependence of $c_{\textrm{f,crit}}$ and $D(c_{\textrm{f,crit}})$ as a function of $\alpha$ is summarized in Fig. 4. Figure 3b shows the effect of heat losses on the chemical times, $\tau _{\textrm{chem}}$, defined as the time required for the main heat release to take place, that is, the time to the maximum thermicity peak. Note that $\tau _{\textrm{chem}}$ is plotted as a function of $D/D_{\textrm{CJ}}$ following the D–$c_{\textrm{f}}$ curves shown in Fig. 3a. As expected, the heat loss through the walls slows down the chemical reactions; $\tau _{\textrm{chem}}$ is over an order of magnitude larger (16 times) for $\alpha =1$ ($7.25\,\upmu {\text {s}}$) than for the adiabatic case, $\alpha =0$ ($0.45\,\upmu {\text {s}}$), at $D=0.77D_{\textrm{CJ}}$. However, near ideal conditions, low velocity deficits ($D > 0.97 D_{\textrm{CJ}}$), irrespective of the value of $\alpha$, the differences are modest; this can also be visualized in the collapse of the D–$c_{\textrm{f}}$ curves in Fig. 3a.

Intermediate values of $\alpha$ (i.e., $0< \alpha < 1$) yield similar curves to those just described. The closer $\alpha$ is to unity, the lower the critical friction coefficient ($c_{\textrm{f,crit}} = 385\,{\text {m}}^{-1}$ and $247.8\,{\text {m}}^{-1}$ for $\alpha = 0.05$ and $\alpha = 0.5$, respectively), progressively appearing at lower velocity deficits ($D(c_{\textrm{f,crit}}) = 0.8D_{\textrm{CJ}}$ for $\alpha = 0.05$, and $0.87D_{\textrm{CJ}}$ for $\alpha = 0.5$); Brailovsky and Sivashinsky [20] reported similar trends using one-step bimolecular kinetics.

The special case in which friction losses are neglected, $f = 0$ in system (1)–(4), yields solutions for low velocity deficits only (i.e., high shock speeds $D > 0.92/D_{\textrm{CJ}}$; $c_{\textrm{f,crit}} = 393.2\,{\text {m}}^{-1}$ at $D(c_{\textrm{f,crit}}) = 0.95$) as heat losses readily suppress the chemical reactions. Note that heating due to friction extends the range/regimes over which steady solutions exist; see [27] for a complete discussion.

Fig. 4

a $c_{\textrm{f,crit}}$ and b $D(c_{\textrm{f,crit}}$) variation with the similarity factors $\alpha$ obtained with a stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ mixture

3.1.2 Diluted mixtures

The influence of ${\text {N}}_{2}$ and Ar dilution on the critical friction coefficient $c_{\textrm{f,crit}}$ and its corresponding velocity deficit $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, for different values of $\alpha$, are presented in Fig. 5. Figure 5a, c shows the results for ${\text {N}}_{2}$. The value of $c_{\textrm{f,crit}}$ decreases linearly at different rates (higher for values of $\alpha \rightarrow 0$) as dilution is increased. Furthermore, increasing $\alpha$ at high dilution levels leads to significantly smaller changes in $c_{\textrm{f,crit}}$ than those observed at low dilutions; variations of $58\%$ for $X_{\textrm{N}_2} = 0.29$ and $62\%$ for $X_{\textrm{N}_2} = 0.56$ are reported. The velocity deficits, $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, on the other hand, are rather insensitive to dilution for a fixed $\alpha$, but decrease as $\alpha$ increases for fixed dilution levels; variations on the order of $10\%$ are observed. The decrease in $D_{\textrm{CJ}}$, $p_{\textrm{vN}}$, and $T_{\textrm{vN}}$ as dilution increases, leads to a sharp increase in $l_{\textrm{ind}}$; note also the more modest increase in reduced effective activation energy, $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$, which results in large induction length-to-reaction zone ratios ultimately making the reaction zone more vulnerable to heat losses.

Fig. 5

$c_{\textrm{f,crit}}$–$D(c_{\textrm{f,crit}}$) variation with the molar fraction of a, c nitrogen and b, d argon for different similarity factors $\alpha$

Dilution with Ar leads to a completely different evolution. Figure 5b, d shows the results. The quantity $c_{\textrm{f,crit}}$ exhibits a nonlinear dependence as dilution is increased with a maximum at $X_{\textrm{Ar}} = 0.25$, subsequently decreasing to a minimum at $X_{\textrm{Ar}} = 0.7$ (highest dilution considered) irrespective of the value of $\alpha$. Similar to what was observed for the ${\text {N}}_{2}$ case, an increase in $\alpha$ at a fixed dilution leads to lower $c_{\textrm{f,crit}}$. Note that the velocity deficits, $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, behave differently than in ${\text {N}}_{2}$-diluted cases in that these decrease with increasing $X_{\textrm{Ar}}$, and are significantly more sensitive to dilution. The overall effect of heat losses on $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, however, is the same for both diluents, i.e., larger $\alpha$ results in smaller velocity deficits. The maximum deficit predicted in the absence of heat losses (i.e., $\alpha =0$) is $D(c_{\textrm{f,crit}})=0.6D_{\textrm{CJ}}$, whereas for $\alpha =1$ is $D(c_{\textrm{f,crit}})=0.8D_{\textrm{CJ}}$. Semenko et al. [23] obtained the same dependence using single-step kinetics, namely that a decrease in $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$ yields turning points at higher values of $c_{\textrm{f}}$, and larger deficits (lower $D/D_{\textrm{CJ}}$). Although it is not straightforward to compare the effect of $\gamma$ on the critical points of the curves between simplified kinetics and detailed thermochemistry (as $\gamma$ varies across the reaction zone), an attempt can be made for $\alpha =0$ and $X_{\textrm{N}_{2}} \approx X_{\textrm{Ar}}=0.5$. An increase in $\gamma$ yields a lower $c_{\textrm{f}}$ and a higher $D/D_{\textrm{CJ}}$, respectively, which is also in agreement with the findings of Semenko et al. [23].

3.2 $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$ scaling with $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$

Scalings for the velocity deficit as a function of the effective activation energy for single-step kinetics have been proposed in the previous work by Zel’dovich and Kompaneets [37] for the case of heat transfer without friction losses:

$$\begin{aligned} D(c_{\textrm{f,crit}})/D_{\textrm{CJ}} = \sqrt{1-R_{\textrm{u}}T_{\textrm{vN}}/E_{{\textrm{a}},\textrm{eff}}} \end{aligned}$$

(15)

and by Zel’dovich et al. [15] who suggested a more general form including both friction and heat losses:

$$\begin{aligned} D(c_{\textrm{f},\textrm{crit}})/D_{\textrm{CJ}} \approx 1-\epsilon \cdot R_{\textrm{u}}T_{\textrm{vN}}/E_{{\textrm{a}},\textrm{eff}} \end{aligned}$$

(16)

where the constant parameter $\epsilon \in [0.9,1.8]$ for physically relevant values of $E_{{\textrm{a}},\textrm{eff}}$ [22]. Note that both of the expressions above were derived in the limit of high activation energy. Figure 6 compares our results with the previous theoretical estimates. Detonations with frictional losses alone ($\alpha =0$) are more resilient to failure as they exhibit larger velocity deficits for the same activation energy as a result of frictional heating. From the conservation laws, the momentum loss (term f in Eq. 2) turns into a source term in the energy equation (work due to friction—term uf in Eq. 3) heating the gas in the reaction zone, which helps to sustain the chemical reactions. In contrast, detonations with heat losses alone ($f=0$) exhibit low velocity deficits for the same activation energy, thus providing an upper limit. Models including both friction and heat losses exhibit intermediate velocity deficits, as expected.

Fig. 6

$D(c_{\textrm{f,crit}}$) variation with activation energy for different similarity factors $\alpha$. The thick black lines depict the results obtained with the analytical expressions using high activation energy asymptotics developed by Zel’dovich et al. [37] and taken from Higgins’ chapter [22] for detonations with heat losses only (top) and a combination of heat and friction losses with $\epsilon = 1.5$ (bottom). The shadowed region indicates the upper/lower limit of the theoretical expression with the parameter $\epsilon \in [0.9, 1.8]$

Finally, although our results using detailed thermochemistry show some nonlinearities and quantitative discrepancies for increasing $E_{{\textrm{a}},\textrm{eff}}$, the overall qualitative trends are well captured by the theory. Note that the latter outcome holds even for intermediate values of activation energy which naturally lie outside of the assumptions made for deriving the expressions above (i.e., $E_{{\textrm{a}},\textrm{eff}} \rightarrow \infty$), therefore proving useful to give a priori estimates for upper/lower bounds for the velocity deficits induced by momentum and heat losses.

3.3 Existence of set-valued solutions with detailed thermochemistry

The existence of a low-velocity detonation region where several $c_{\textrm{f}}$ values fulfill the boundary conditions downstream, i.e., set-valued solutions, was first reported by Semenko et al. [23] for the D–$c_{\textrm{f}}$ model including both momentum and heat losses using simplified kinetics. These solutions are found in the absence of a sonic point in the detonation structure (i.e., at the choking regime $D < a_{\textrm{b}}$), provided that heat losses are accounted for (i.e., $\alpha > 0$). To our knowledge, these solutions have only been reported for one-step Arrhenius chemistry. In this section, their existence in realistic ${\text {H}}_{2}$–${\text {O}}_{2}$–${\text {N}}_{2}$/Ar mixtures is examined.

As mentioned above, since these types of solutions only appear within the choking regime, a critical similarity factor, $\alpha _{\textrm{crit}}$, can be defined as the value of $\alpha$ for which this regime vanishes for a given mixture composition. That is, when the friction factor approaches unity ($c_{\textrm{f}} \sim 1\,{\text {m}}\,^{-1}$) for a propagation speed of the order of the speed of sound in burnt gases ($D = a_{\textrm{b}}$). This is shown in Fig. 7a as the dash-dotted D–$c_{\textrm{f}}$ curve computed for an undiluted ${\text {H}}_{2}$–${\text {O}}_{2}$ mixture; for $\alpha = \alpha _{\textrm{crit}} = 0.345$, $c_{\textrm{f}} = 0.946\,{\text {m}}^{-1}$ when $D = a_{\textrm{b}}$.

The effect of the reduced effective activation energy, i.e., dilution, on $\alpha _{\textrm{crit}}$ is shown in Fig. 7b. On the one hand, adding ${\text {N}}_{2}$ to the mixture (brown) results in a reduction of $\alpha _{\textrm{crit}}$, from 0.345 (undiluted) to 0.12 (highest dilution of ${\text {N}}_{2}$), as the effective activation energy increases. Consequently, for $\alpha > 0.12$ the choking regime ceases to exist excluding the possibility of set-valued solutions for these mixtures. On the other hand, the addition of Ar (orange) decreases the effective activation energy and, therefore, moves $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$ toward lower values yielding D–$c_{\textrm{f}}$ curves that exhibit a choking regime. Figure 7b shows that a much higher $\alpha$ is required ($\alpha _{\textrm{crit}} = 0.675$ with the highest Ar dilution tested) to suppress the choking regime from the D–$c_{\textrm{f}}$ curves.

Fig. 7

a D–$c_{\textrm{f}}$ curves for the mixtures marked with big dots in (b) with $\alpha < \alpha _{\textrm{crit}} = 0.05$. The colored bands in the choking regime refer to the set values if applicable; width increased five times for clarity (consult text for additional information). The horizontal dashed lines mark the boundary—mixture-dependent—between the quasi-detonation and choking regimes. Furthermore, the dash-dotted line depicts the D–$c_{\textrm{f}}$ curve obtained with $\alpha = \alpha _{\textrm{crit}} = 0.345$. b Influence of the effective activation energy—dilution-dependent—on $\alpha _{\textrm{crit}}$

The shaded regions in Fig. 7a represent set-valued solutions obtained with detailed thermochemistry. The curves were computed for $\alpha = 0.05$, a value well below $\alpha _{\textrm{crit}}$ for the mixture compositions shown in the figure (undiluted ${\text {H}}_{2}$–${\text {O}}_{2}$, and highest Ar and ${\text {N}}_{2}$ dilutions). Note that the width of the set-valued region was widened by a factor of five for ease of visualization as follows: the extended maximum friction coefficient is obtained using $c_{{\textrm{f}},\max -{\textrm{ext}}} = c_{{\textrm{f}},\min -0} + (c_{{\textrm{f}},\max -0} - c_{{\textrm{f}},\min -0}) \times 5$, where subscripts $\min -0/\max -0$ refer to the minimum/maximum friction coefficient that satisfies the downstream boundary condition at a fixed D, below $0.51D_{\textrm{CJ}}$ and $0.52D_{\textrm{CJ}}$, for the undiluted and the Ar-diluted mixture, respectively. The set-valued solutions found with detailed thermochemistry seem to start slightly below the choking boundary, in contrast to the solutions observed by Semenko et al. [23] with single-step chemistry, which always started at the boundary itself. While being inside the choking regime is a necessary condition for set-valued regions to exist, it is not a sufficient condition as evidenced by the curve shown in Fig. 7a for the ${\text {N}}_{2}$-diluted mixture. Semenko et al. [23] related an increase in the effective activation energy to a decrease of the width of the set-valued region for simplified kinetics; a similar effect is reported in this work using detailed chemistry as a wider set-valued region is obtained when diluting with argon.

Fig. 8

a Choking regime of the D–$c_{\textrm{f}}$ curve obtained with the chemical mechanism of Ó Conaire et al. [35] for an stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ blend and $\alpha = 0.05$. The gray diamond indicates the case at which the set-valued solutions become distinguishable within our tolerances ($\varepsilon = 10^{-4}\,{\text {m}}^{-1}$). b and c Lab-frame velocity profiles for $D = 0.43 D_{\textrm{CJ}}$ and $D = 0.45 D_{\textrm{CJ}}$. The line-style shown in the insets of (a) represents the $c_{\textrm{f}}$ values used for the three given solutions for the case labeled with capital letters A and B. The insets are zooms of the expansion zones linked to the exothermic heat release

To verify the validity of the solutions obtained within the set-valued envelope, the D–$c_{\textrm{f}}$ curve obtained for an undiluted mixture in Fig. 7a is used an example. Figure 8a zooms in to the set-valued region, and the flow speed profiles in the laboratory frame of reference, u, for $D = 0.43 D_{\textrm{CJ}}$ and $D = 0.45 D_{\textrm{CJ}}$ are, respectively, shown in Fig. 8b and c; note that the detonations are propagating from right to left in these plots. The passage of the leading shock induces a flow behind it whose speed increases as it approaches the reaction zone marked by an abrupt speed decrease immediately after the chemical heat release. The latter is due to the gas expansion that results in a flow in the opposite direction, hence bringing the gas to rest or even reversing it (i.e., $u < 0$ m/s). The boundary conditions downstream are thus satisfied, rendering these solutions valid; similar velocity profiles were reported by Semenko et al. [23]. Note that the slower the detonations the further downstream this flow condition is achieved. Specifically, at $\xi _{u \rightarrow 0} = 2{-}6$ m for $D = 0.43 D_{\textrm{CJ}}$ and increasing $c_{\textrm{f}}$ (Fig. 8b), and at $\xi _{u \rightarrow 0} \approx 10{-}40$ m for $D = 0.45 D_{\textrm{CJ}}$ and increasing $c_{\textrm{f}}$ (Fig. 8c). For some cases, $D > 0.51 D_{\textrm{CJ}} < a_{\textrm{b}}$ (not shown in Fig. 8), only one friction coefficient satisfies the boundary condition downstream in contrast to Semenko et al. [23], in which set-valued solutions at detonation speeds slightly below $D = a_{\textrm{b}}$ were reported. For detailed chemistry, the sonic point does not necessarily correspond with chemical equilibrium due to the presence of recombination and endothermic reactions at the tail of the reaction zone. The choking boundary ($D=a_{\textrm{b}}$) predicted under chemical equilibrium is therefore not representative for detonations computed using realistic thermochemistry. Namely, for the undiluted case the sonic point disappears at $D = 0.52\,D_{\textrm{CJ}}$, a value lower than $a_{\textrm{b}} = 0.56\,D_{\textrm{CJ}}$—mixture-dependent, thus displacing the choking boundary toward higher velocity deficits (slower detonations). Flow reversal, a requirement for the existence of set-valued solutions, only occurs within the choking regime. This supports the finding that set-valued solutions can only be found at detonation velocities slightly lower than $D=a_{\textrm{b}}$ for detailed chemistry, that is, at their practical choking boundary. Similar but opposite behaviors were previously reported for pathological detonations computed with detailed chemistry [22, 38]. In some cases, the reaction zone structure shows non-singular solutions only at the sonic point yielding velocities greater than that predicted under chemical equilibrium, that is, $D > D_{\textrm{CJ}}$. For completeness, Fig. 9 shows the reaction zone structure (p, T, ${{\dot{\sigma }}}$, and $Y_i$) at $D = 0.45 D_{\textrm{CJ}}$ for $c_{\textrm{f}}$ values corresponding to the edges of the set-valued region. The differences among the plotted profiles are minor with only a slight delay in chemical heat release (i.e., the peak in ${{\dot{\sigma }}}$) when $c_{\textrm{f}}$ is smaller, but enough to modify the flow reversal conditions behind the main heat release stage; both solutions satisfy the boundary condition $u = 0$ downstream making them valid.

Fig. 9

p, T, u, ${\dot{\sigma }}$, and scaled $Y_k$ profiles obtained for a shock velocity $D = 0.45 D_{\textrm{CJ}}$, $\alpha = 0.05$, and friction factors that lie within the set-valued region of $c_{\textrm{f}} = 10.1\,{\text {m}}^{-1}$ and $c_{\textrm{f}} = 11.4\,{\text {m}}^{-1}$

Finally, Fig. 10 summarizes the results obtained with detailed thermochemistry in $\alpha \text {--}E_{\textrm{a},\textrm{eff}}/R_{\textrm{u}}T_0$ space which allows clear visualization of the regions where the different regimes exist. The map was obtained by calculating all the D–$c_{\textrm{f}}$ curves for the combinations of $\alpha$ and $E_{\textrm{a},\textrm{eff}}/R_{\textrm{u}}T_0$ indicated in the figure. Three distinct regions are identified: (i) quasi-detonation—circles, whenever the resulting curve lies within the quasi-detonation regime; (ii) choking—squares, if the calculated D–$c_{\textrm{f}}$ curve reaches the choking regime but no set-valued solutions are obtained, such as the case shown in Fig. 8a for a ${\text {N}}_{2}$-diluted mixture; and (iii) set-valued—triangles, for the cases in which set-valued solutions exist, defining $\alpha _{\mathrm {set-valued}} < \alpha _{\textrm{crit}}$ as the value of $\alpha$ for which the set-valued region appears given an effective activation energy, $E_{\textrm{a},\textrm{eff}}/R_{\textrm{u}}T_0$. The map shows a strong nonlinear dependence. In general, the lower the activation energy, the higher the likelihood of finding set-valued solutions. For $E_{\textrm{a},\textrm{eff}}/R_{\textrm{u}}T_0 \rightarrow \infty$, set-valued nor choking solutions are found, as the D–$c_{\textrm{f}}$ curves reach $c_{\textrm{f}} \rightarrow 0$ within the quasi-detonation regime. Stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ mixtures with high Ar dilution are thus good candidates for investigating the transition dynamics from the set-valued region to the quasi-detonation regime focusing on detailed kinetic-induced effects; a natural extension of the work of Sow et al. [24] is out of the scope of the present study.

Fig. 10

Detonation regimes as a function of the effective activation energy and the heat loss similarity factor $\alpha$. The lines indicate the boundaries between regimes in the said parametric space. The colors represent the mixtures selected, using green, brown, and orange for the undiluted, nitrogen, and argon mixtures, respectively

A direct comparison with the results of Semenko et al. [23] is not straightforward because the authors only analyzed detonation propagation in porous media ($\phi = 0.4$). For increasing $\phi$, a reduced probability in the appearance of set-valued solutions was observed. The general qualitative picture shown in Fig. 10 should remain unchanged. Moreover, the activation energy and ratio of specific heats are free parameters in their chemical model (single-step Arrhenius). Some of the values used are not accessible with the mixture and diluents examined in the current study. For Semenko et al.’s conditions, set-valued regions exist for a wide range of similarity factors ($\alpha \in [0.4{-}1]$) and friction factors (around eight times larger regions) and significantly wider ranges than those presented here. Detailed chemistry yields a much more complex map and strongly reduces the existence of set-valued regions to sufficiently low similarity factors, $\alpha$, in contrast to what was reported in the work of Semenko et al. [23] for one-step Arrhenius chemistry. It seems that the assumptions on their model (e.g., lack of an induction length, constant $\gamma$, single-step chemistry) make more likely the appearance of set-valued solutions within the choking regime at conditions where detailed thermochemistry predicts quenching of the wave since the locus of steady solutions is restricted to lower velocity deficits only; simplified kinetics are prone to miss information in the intermediate regimes. For additional discussion on the qualitative and quantitative differences brought about by the chemical modeling to D–$c_{\textrm{f}}$ curves (i.e., simplified vs. detailed kinetics), the reader is referred to [27].

Semenko et al. [23] and Sow et al. [24] used the set-valued solutions to explain the stabilization mechanism of quasi-steady low-velocity detonations. The authors postulated that the presence of a set of stable solutions around a given friction coefficient $c_{\textrm{f}}$ at the choking regime could rationalize the experimental observations of Lyamin et al. [25] and the rather long steady propagation at low velocities in Sow’s transient solutions [24]. Perturbations to the system would not directly move the solution to the upper branch of the D–$c_{\textrm{f}}$ curve but cause slow continuous passages to the neighboring solutions within the set-valued region. Due to the significant reduction of the region where set-valued solutions exist when using detailed thermochemistry, one may infer that the stabilization mechanisms of low-velocity detonations just described are indeed only a component of a bigger, yet unknown, picture when dealing with realistic mixtures.

4 Conclusions

The effect of heat and momentum losses on the steady solutions admitted by the reactive Euler equations with sink/source terms was examined for stoichiometric hydrogen–oxygen mixtures. Varying degrees of ${\text {N}}_{2}$ and Ar dilution were considered to access a wide range of effective activation energies, $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$, for detailed thermochemistry. The main results of the study, discussed using D–$c_{\textrm{f}}$ curves, were: (i) for undiluted mixtures, heat losses move the turning point, $D(c_{\textrm{f,crit}})$, toward lower velocity deficits ($D/D_{\textrm{CJ}} \rightarrow 1$) and yield lower $c_{\textrm{f,crit}}$ values; (ii) ${\text {N}}_{2}$ dilution (higher $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$) strongly reduces $c_{\textrm{f,crit}}$ and increases $D(c_{\textrm{f,crit}})$ whereas dilution with Ar (lower $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$) exhibits a nonlinear dependence on $c_{\textrm{f,crit}}$ but a linear decreasing trend on $D(c_{\textrm{f,crit}})$; (iii) classical scalings for the prediction of the velocity deficits, $D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$, as a function of the effective activation energy, $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$, were revisited; some nonlinearities and quantitative discrepancies for increasing $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0$ were observed but the overall qualitative trends are well captured by the theory. Finally, the existence of set-valued solutions was verified for detailed chemistry. That is, a family of $c_{\textrm{f}}$ values that always lie within the choking regime that satisfy the governing equations for a fixed detonation velocity, D. A map outlining the regions where the aforementioned solutions exist in $E_{\textrm{a,eff}}/R_{{\textrm{u}}}T_0\text {--}\alpha$ space was provided, suggesting that stoichiometric ${\text {H}}_{2}$–${\text {O}}_{2}$ mixtures with high Ar dilution are good candidates to investigate the transition dynamics from the set-valued region to the quasi-detonation regime. The latter will be the object of a future study focusing on how the aforementioned dynamics is affected by detailed kinetics, hoping to provide important clues related to DDT.