Highly resolved numerical simulation of combustion downstream of a rocket engine igniter

Abstract

We study ignition processes in the turbulent reactive flow established downstream of highly under-expanded coflowing jets. The corresponding configuration is typical of a rocket engine igniter, and to the best knowledge of the authors, this study is the first that documents highly resolved numerical simulations of such a reactive flowfield. Considering the discharge of axisymmetric coaxial under-expanded jets, various morphologies are expected, depending on the value of the nozzle pressure ratio, a key parameter used to classify them. The present computations are conducted with a value of this ratio set to fifteen. The simulations are performed with the massively parallel CREAMS solver on a grid featuring approximately 440,000,000 computational nodes. In the main zone of interest, the level of spatial resolution is D/74, with D the central inlet stream diameter. The computational results reveal the complex topology of the compressible flowfield. The obtained results also bring new and useful insights into the development of ignition processes. In particular, ignition is found to take place rather far downstream of the shock barrel, a conclusion that contrasts with early computational studies conducted within the unsteady RANS computational framework. Consideration of detailed chemistry confirms the essential role of hydroperoxyl radicals, while the analysis of the Takeno index reveals the predominance of a non-premixed combustion mode.

Appendices

Appendix 1: Governing equations and multicomponent transport representation in CREAMS solver

CREAMS solves the three-dimensional Navier–Stokes equations written for a multicomponent mixture

$$\begin{aligned} \frac{\partial ^{}\varvec{q}}{\partial {t}^{}} + \frac{\partial ^{}\varvec{f}}{\partial {x_1}^{}} + \frac{\partial ^{}\varvec{g}}{\partial {x_2}^{}} + \frac{\partial ^{}\varvec{h}}{\partial {x_3}^{}} = \frac{\partial ^{}{\varvec{f}}_{\mathbf{v}}}{\partial {x_1}^{}} + \frac{\partial ^{}{\varvec{g}}_{\mathbf{v}}}{\partial {x_2}^{}} + \frac{\partial ^{}{\varvec{h}}_{\mathbf{v}}}{\partial {x_3}^{}} + \varvec{s} , \end{aligned}$$

(9)

where $\varvec{q}=\big ({\rho },{\rho } {u}_1,{\rho } {u}_2,{\rho } {u}_3,{\rho } e_\mathrm {t},{\rho } Y_1,\ldots ,{\rho } Y_{\alpha },\ldots ,{\rho } Y_{\mathcal {N}_\mathrm {sp}} \big )^T$ denotes the vector of conserved quantities, while the inviscid (Euler) and viscous flux vector components are given by

$$\begin{aligned} \varvec{f}= & {} {u}_1 \varvec{q} + {P} \left( 0,1,0,0,{u}_1,0,\ldots ,0\right) ^T , \nonumber \\ \varvec{g}= & {} {u}_2 \varvec{q} + {P} \left( 0,0,1,0,{u}_2,0,\ldots ,0\right) ^T , \nonumber \\ \varvec{h}= & {} {u}_3 \varvec{q} + {P} \left( 0,0,0,1,{u}_3,0,\ldots ,0\right) ^T , \end{aligned}$$

(10)

and

$$\begin{aligned} {\varvec{f}}_{\mathbf{v}}= & {} (0,\tau _{11},\tau _{12},\tau _{13},u_i\tau _{i1} -\mathcal {J}_1,-\rho Y_1 V_{1,1},\ldots ,\nonumber \\&\ldots ,-\rho Y_{\mathcal {N}_\mathrm {sp}} V_{\mathcal {N}_\mathrm {sp},1})^T , \nonumber \\ {\varvec{g}}_{\mathbf{v}}= & {} (0,\tau _{21},\tau _{22},\tau _{23},u_i\tau _{i2} -\mathcal {J}_2,-\rho Y_1 V_{1,2},\ldots ,\nonumber \\&\ldots ,-\rho Y_{\mathcal {N}_\mathrm {sp}} V_{\mathcal {N}_\mathrm {sp},2})^T , \nonumber \\ {\varvec{h}}_{\mathbf{v}}= & {} (0,\tau _{31},\tau _{32},\tau _{33},u_i\tau _{i3} -\mathcal {J}_3,-\rho Y_1 V_{1,3},\ldots ,\nonumber \\&\ldots ,-\rho Y_{\mathcal {N}_\mathrm {sp}} V_{\mathcal {N}_\mathrm {sp},3})^T , \end{aligned}$$

(11)

respectively. The vector $\varvec{s}$ corresponds to chemical source terms: $\varvec{s}= \left( 0,0,0,0,0,\rho \dot{\omega }_1,\ldots ,\rho \dot{\omega }_{\mathcal {N}_\mathrm {sp}} \right) ^T$.

In the above equations $\rho$ denotes the density, $\varvec{u}$ is the velocity vector, P is the pressure, $e_\mathrm {t}$ is the total energy per unit mass (defined below), $Y_\alpha =\rho _\alpha /\rho$ is the mass fraction of the $\alpha$th species ($\alpha = 1,\ldots ,{\mathcal {N}_\mathrm {sp}}$), with ${\mathcal {N}_\mathrm {sp}}$ the total number of chemical species, and $\dot{\omega }_\alpha$ its production rate. $\tau _{ij}$ is the Newtonian shear stress tensor. $\mathcal {J}_j$ and $V_{\alpha ,j}$ are the j-components of the heat flux and the species diffusion velocity, respectively.

The above system of conservation equations is complemented by the ideal-gas equation of state written for a multicomponent system ${P}={{\rho }{\mathcal {R}}{T}}/{\mathcal {W}}$, where $\mathcal {R}$ is the universal gas constant and $\mathcal {W}$ is the molecular mass of the mixture which is obtained as the sum of the molecular mass of the species $\mathcal {W}^{-1}= \sum _{\alpha =1}^{\mathcal {N}_\mathrm {sp}} Y_{\alpha } / \mathcal {W}_{\alpha }$. The total energy $e_\mathrm {t}$ is given by $e_\mathrm {t}={{{u}_i{u}_i}}/{2}+ \sum ^{\mathcal {N}_\mathrm {sp}}_{\alpha =1} {h}_\alpha Y_\alpha - {P}/{\rho }$, where $h_\alpha$ denotes the enthalpy of the $\alpha$th species

$$\begin{aligned} {h}_\alpha \left( T\right) = \varDelta {{h}}^0_{\mathrm {f}\alpha } + \int ^{T}_{{T}_0} {c}_{p\alpha } \left( {T^*}\right) \mathrm {d} {T^*}. \end{aligned}$$

(12)

The specific heat capacity at constant pressure of each species is standardly expressed as a polynomial form of the temperature T. Since the nonlinear dependency of the specific heat capacities on the temperature prevents the direct calculation of the latter given the total energy, a Newton–Raphson iterative algorithm is used to solve the resulting system.

The transport coefficients in multicomponent mixtures are not given explicitly by the kinetic theory and require solving large linear systems. Here they are evaluated following the approach described by Ern and Giovangigli [21, 22].

The shear stress tensor is given by

$$\begin{aligned} \tau _{ij}= & {} \kappa S_{kk} \delta _{ij} + 2{\mu }\left( {S}_{ij} - \frac{{S}_{kk}}{3} \delta _{ij} \right) ,\nonumber \\ {S}_{ij}= & {} \frac{1}{2}\left( \frac{\partial ^{}{u}_i}{\partial {{x}_j}^{}} + \frac{\partial ^{}{u}_j}{\partial {{x}_i}^{}}\right) , \end{aligned}$$

(13)

where $\kappa$ and $\mu$ denote the volume and shear viscosity, respectively.

The heat flux is expressed as follows

$$\begin{aligned} \mathcal {J}_j= \sum _{\alpha = 1}^{\mathcal {N}_\mathrm {sp}} \rho Y_{\alpha } V_{\alpha ,j} \left( h_{\alpha } + \frac{\mathcal {R}T \tilde{\chi }_{\alpha }}{\mathcal {W}_{\alpha }} \right) - \lambda \frac{\partial ^{}T}{\partial {x_j}^{}}, \end{aligned}$$

(14)

where $\lambda$ is the thermal conductivity of the mixture and $\tilde{\chi }_{\alpha }$ is the rescaled thermal diffusion ratios of the $\alpha$th species, which is defined such that $\sum _{\beta } X_{\beta } \tilde{\chi }_{\beta }=0$ and $\sum _{\beta } D_{\alpha \beta } X_{\beta } \tilde{\chi }_{\beta } = \theta _{\alpha }$, with $\alpha$ and $\beta \in \left[ 1, ..., \mathcal {N}_\mathrm {sp}\right]$. In the previous expression, $\theta _{\alpha }$ and $D_{\alpha \beta }$ denote the thermal diffusion vector and diffusion matrix, respectively, while $X_{\beta }$ denotes the molar fraction of the $\beta$th species.

The mass fluxes are represented by

$$\begin{aligned}&\rho Y_{\alpha } V_{\alpha ,j} = - \sum _{\beta = 1}^{\mathcal {N}_\mathrm {sp}} \rho \tilde{D}_{\alpha \beta } \left( \hbox {d}_{\beta ,j} + \frac{X_{\beta } \tilde{\chi }_{\beta }}{T} \frac{\partial ^{}T}{\partial {x_j}^{}} \right) , \nonumber \\&\hbox {d}_{\beta ,j} = \frac{\partial ^{}X_{\beta }}{\partial {x_j}^{}} + \frac{X_{\beta }-Y_{\beta }}{P} \frac{\partial ^{}P}{\partial {x_j}^{}} , \end{aligned}$$

(15)

where $\tilde{D}_{\alpha \beta }$ are the flux diffusion coefficients formed by the flux diffusion components $Y_\alpha D_{\alpha \beta }$. All the transport coefficients mentioned above, i.e., $\kappa$, $\mu$, $\lambda$, $\tilde{D}_{\alpha \beta }$, and $\tilde{\chi }_{\beta }$, are evaluated using the general purpose Fortran library EGLIB [22], which employs an iterative method to obtain an approximate solution to the linear system of transport coefficients derived from the kinetic theory. The molecular transport coefficients, as issued from the library EGLIB, satisfy mass constraints as well as positivity properties ensuring the positivity of entropy production.

The solver offers the opportunity to represent detailed chemical kinetics through $\mathcal {N}_\mathrm {r}$ elementary reaction steps involving $\mathcal {N}_\mathrm {sp}$ chemical species

$$\begin{aligned} \sum _{\alpha =1}^{\mathcal {N}_\mathrm {sp}} \nu _{\alpha ,j}' \mathcal {M}_{\alpha } \rightleftharpoons \sum _{\alpha =1}^{\mathcal {N}_\mathrm {sp}} \nu _{\alpha ,j}'' \mathcal {M}_{\alpha } , \quad j=1,\ldots ,\mathcal {N}_\mathrm {r}, \end{aligned}$$

(16)

where $\mathcal {M}_{\alpha }$ is the chemical symbol for the $\alpha$th species, $\nu _{\alpha ,j}'$ stands for the forward stoichiometric coefficients, while $\nu _{\alpha ,j}''$ denotes the reverse stoichiometric coefficients. The resulting chemical source terms are given by

$$\begin{aligned} \dot{\omega }_{\alpha }= & {} \sum _{j=1}^{\mathcal {N}_\mathrm {r}} (\nu _{\alpha ,j}''-\nu _{\alpha ,j}') q_j, \nonumber \\ q_j= & {} k_{\mathrm {f}j} \prod _{\alpha =1}^{\mathcal {N}_\mathrm {sp}} \left[ X_\alpha \right] ^{\nu _{\alpha ,j}' } - k_{\mathrm {r}j} \prod _{\alpha =1}^{\mathcal {N}_\mathrm {sp}} \left[ X_\alpha \right] ^{\nu _{\alpha ,j}''} , \end{aligned}$$

(17)

where $k_{\mathrm {f}j}$ and $k_{\mathrm {r}j}$ denote the forward and reverse rate constants of the jth elementary reaction, respectively, and $\left[ X_\alpha \right]$ is the molar concentration of the $\alpha$th species.

Appendix 2: Three-step reduced chemistry

Under the set of hypotheses presented in reference [39] the hydrogen–air chemistry can be reduced to the following three overall steps:

• (I) $3 \mathrm {H_2} + \mathrm {O_2} \rightleftharpoons 2 \mathrm {H_2O} + 2 \mathrm {H}$

• (II) $2 \mathrm {H} + \mathrm {M} \rightleftharpoons \mathrm {H_2} + \mathrm {M}$

• (III) $\mathrm {H_2} + \mathrm {O_2} \rightleftharpoons \mathrm {HO_2} + \mathrm {H}.$

The above description includes a branching reaction (I), a recombination reaction (II), and an initiation reaction (III). Five reactive species $\mathrm {H_2}$, $\mathrm {O_2}$, $\mathrm {H_2O}$, $\mathrm {H}$, $\mathrm {HO_2}$ (plus nitrogen diluent) are considered. Chemical-kinetic steady-state approximations have been retained for $\mathrm {O}$, $\mathrm {OH}$, and $\mathrm {H_2O_2}$. According to Boivin et al. [39] the application of a correction factor $\varLambda$, see (6), significantly improves the performance of the above reduced chemistry.