LES of Reacting Flow in a Hydrogen Jet into Supersonic Crossflow Combustor Using a New Turbulent Combustion Model

Abstract

Modelling the complete flow physics and chemical kinetics of supersonic combustion is a particularly complex and daunting task that requires significant computational resources. To foster performance evaluation tools for future hypersonic vehicles, develo** accurate yet computationally efficient solution methods is of great importance. In this work, a new subgrid combustion model for large eddy simulations is derived and used in a three-dimensional in-house flow solver to provide simulations of experimental scramjet ground tests. In particular, this paper introduces a hybrid model closure with the reaction-rate approach to close the filtered chemical source terms in the governing equations for species mass fractions and total energy. The model developed here makes use of a linear bridging function, depending on the segregation rate of the mixture fraction, between a resolved contribution issued from a perfectly stirred reactor (PSR) estimation, and a subgrid-scale (SGS) contribution where a closure that approximates the Lagrangian trajectory in the composition space is retained. The new model considers the effect of fluctuations of compositions and can be extended to take into account, for example, the fluctuations of temperature. The new approach is tested using a hydrogen-fueled scramjet combustor from circular injector into a Mach 2 vitiated airflow for total pressure and temperature of 0.40 MPa and 1695 K, respectively. The selected operating conditions are representative of the LAPCAT-II dual-mode ramjet/scramjet combustion. Chemistry is described using a four-step reduced mechanism. The results obtained with the present modelling proposal are compared to those issued from numerical simulations performed with the quasi-laminar chemistry or PSR approach. These results do show that, even for a highly resolved computational mesh, the effects of composition fluctuations remain significant, especially in the vicinity of the injection where the SGS fluctuations of the scalar field are non-negligible.

Appendix: MIL Model

For the sake of completeness, a brief description of the $\texttt {MIL}$ turbulent combustion modelling framework is now provided. From a general point of view, it has been previously established to be robust, easy to implement, and it features small computational costs [36]. The MIL model is a Lagrangian model in the composition space which is based on the knowledge of two scalar variables, namely the mixture fraction $\xi$ and a progress variable $Y_c$ which follows the departures from chemical equilibrium, i.e., the progress of the chemical reaction. Based on the simplest micromixing closure, i.e., the one given by the IEM (Interaction par Echange avec la Moyenne) model of Villermaux & Devillon [37], the Lagrangian evolution of fluid particles issuing from either oxidizer or fuel stream is given by the following set of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\mathrm{d}\xi }{\mathrm{d}t}=\frac{( {{\widetilde{\xi }}-\xi } )}{\tau _\texttt {SGS}}\\ \frac{\mathrm{d}Y_c}{\mathrm{d}t}=\frac{( {{\widetilde{Y}}_c-Y_c})}{\tau _\texttt {SGS}}+\omega _{Y_c}^{\texttt {MIL}} \end{array}\right. }. \end{aligned}$$

(28)

The MIL model relies on the sudden chemistry assumption: it is assumed that Lagrangian particles first need a finite time to ignite and, after ignition occurred, they instantaneously reach chemical equilibrium conditions. This permits a strong but clearly stated functional dependence between the two scalars to be introduced: $Y_c=Y_c^{\texttt {MIL}}(\xi )$. As a result, the SGS joint scalar probability density function $\widetilde{\texttt {PDF}}({Y_c},\xi ;{\varvec{x}},t)$ can be simply expressed from the single knowledge of the SGS marginal mixture fraction PDF since we have $\widetilde{\texttt {PDF}}({Y_c},\xi ;{\varvec{x}},t)=\widetilde{\texttt {PDF}}(\xi ;{\varvec{x}},t)\texttt {PDF}({Y_c} | \xi ;{\varvec{x}},t)$ where the conditional $\texttt {PDF}({Y_c} | \xi ;{\varvec{x}},t)$ is obtained by considering the MIL trajectory in the composition space: $\texttt {PDF}({Y_c}|\xi ;{\varvec{x}},t) = \delta ({Y_c}-Y_c^{\texttt {MIL}}(\xi ))$. A classical $\beta$-function is retained to estimate the marginal mixture fraction $\widetilde{\texttt {PDF}}(\xi ;{\varvec{x}},t)$ while the conditional $\texttt {PDF}({Y_c} | \xi ;{\varvec{x}},t)$, i.e., the Lagrangian trajectory as given by $\delta ({Y_c}-Y_c^{\texttt {MIL}}(\xi ))$, is fully determined from the single knowledge of the two jump (or ignition) positions $\xi _{J^{-}}$ and $\xi _{J^{+}}$ (cf. Fig. 16). The corresponding values are obtained from a direct comparison between the mixing time scale $\tau _\texttt {SGS}$ and chemical time scales, which have been tabulated as a function of the mixture fraction, i.e., $\tau _{\text {chem}}(\xi )$, thus reflecting the competition that may exist between chemical reaction and scalar mixing. Once the joint $\widetilde{\texttt {PDF}}({Y_c},\xi ;{\varvec{x}},t)$ is estimated, the mean chemical production rate reads

$$\begin{aligned} {\widetilde{\omega }}_{Y_c} = \int _{\xi =0}^{\xi =1} \int _{Y_c=Y_c^u}^{Y_c=Y_c^b} \omega _{Y_c}^{ \text{ MIL }}({Y_c},\xi ) \widetilde{\texttt {PDF}}({Y_c},\xi ;{\varvec{x}},t) \, d{Y_c} \, \text {d}\xi , \end{aligned}$$

(29)

with the instantaneous chemical production rate $\omega _{Y_c}^{\texttt {MIL}}$ deduced from the Lagrangian trajectory

$$\begin{aligned} \omega _{Y_c}^{ \text{ MIL }}(Y_c,\xi ) = \frac{\left( {\widetilde{\xi }}-\xi \right) \frac{\mathrm{d}Y_c}{\mathrm{d}\xi }-({\widetilde{Y}}_c-Y_c)}{\tau _\texttt {SGS}}. \end{aligned}$$

(30)

The pre-tabulation of the chemical characteristic time scale is presently replaced by an analytical estimate based on the expression of the reactivity $\lambda$ as given by Eq. (11), which is evaluated using the local composition and temperature. According to Bridel-Bertomeu & Boivin [38], this should allow for a significant reduction of the computational costs.

Fig. 16

Conventional representation of the MIL model in the phase plane $(\xi ,Y)$: Comparison between chemical and mixing time scales: delineation of an ignition domain $[\xi _{J_{-}},\xi _{J_{+}}]$ in the mixture fraction space