Large-Eddy Simulations of the Mascotte Test Cases Operating at Supercritical Pressure

Abstract

Liquid rocket, Diesel or aircraft engines may operate in the transcritical regime. In such thermodynamic conditions, the classical phase change that occurs at subcritical pressure disappears and the mixing layer between the dense and cold jet and the outer gaseous stream is characterized by large variations of density and thermodynamic properties. Fluids show strong departure from a perfect gas behavior and a real-gas formulation is needed to model the fluid state. The extension of the unstructured AVBP solver, jointly developed by CERFACS and IFPEN, to handle high-pressure thermodynamics is presented in details. It is then validated on the experimental coaxial injectors studied with the Mascotte test rig from ONERA that operate in the transcritical range, namely the LOx/GH2 cases A60 and C60 and the LOx/GCH4 configuration G2. The flame pattern observed in experiments is properly recovered, hence validating the numerical strategy. Numerical results are then discussed focusing on the role of the momentum flux ratio on the development of transcritical flames.

Appendix : Jacobian Matrices for TTG Schemes

The general expression for the Two-Step Taylor Galerkin schemes [76] used in AVBP are given by :

$$ \begin{array}{@{}rcl@{}} \overrightarrow{\tilde{w}}^{n} &=& \overrightarrow{w}^{n} - \alpha \varDelta t \overrightarrow{\nabla} \cdot \overline{\overline{F}}^{n} + \beta \varDelta t^{2} \overrightarrow{\nabla} \cdot \left (\overline{\overline{\overline{\mathcal{A}}}}(\overrightarrow{\nabla} \cdot \overline{\overline{F}}^{n}) \right ) \end{array} $$

(30)

$$ \begin{array}{@{}rcl@{}} \overrightarrow{w}^{n+1} &=& \overrightarrow{w}^{n} - \varDelta t \overrightarrow{\nabla} \cdot \left (\theta_{1} \overline{\overline{\tilde{F}}}^{n} + \theta_{2} \overline{\overline{F}}^{n}\right ) \\ &&+ \gamma \varDelta t^{2} \left (\epsilon_{1} \nabla \cdot \left (\overline{\overline{\overline{\mathcal{A}}}}(\overrightarrow{\nabla} \cdot \overline{\overline{F}}^{n}) \right ) + \epsilon_{2} \nabla \cdot \left (\overline{\overline{\overline{\mathcal{A}}}}(\overrightarrow{\nabla} \cdot \overline{\overline{\tilde{F}}}^{n}) \right ) \right ) \end{array} $$

(31)

where $\tilde { }$ corresponds to values computed at the intermediate step. The coefficients α, β, 𝜃₁, 𝜃₂, 𝜖₁ et 𝜖₂ are set to 0.49, 1/6, 0, 1, 0.01 et 0 for TTGC. The tensor $\overline {\overline {\overline {\mathcal {A}}}}= (\overline {\overline {A}}, \overline {\overline {B}}, \overline {\overline {C}})$ is the vector of the jacobian of the convective flux tensor, where $\overline {\overline {A}}$, $\overline {\overline {B}}$ et $\overline {\overline {C}}$ are the jacobian matrixes of the non-viscous fluxes given by :

$$ \overline{\overline{A}}(\overrightarrow{w}) = \frac{\partial \overrightarrow{f}}{\partial \overrightarrow{w}} \qquad \overline{\overline{B}}(\overrightarrow{w}) = \frac{\partial \overrightarrow{g}}{\partial \overrightarrow{w}} \qquad \overline{\overline{C}}(\overrightarrow{w}) = \frac{\partial \overrightarrow{h}}{\partial \overrightarrow{w}} $$

(32)

with $\overrightarrow {w}=(\rho u_{1}, \rho u_{2}, \rho u_{3}, \rho E, \rho Y_{k})^{T}$ and $\overline {\overline {F}}=(\overrightarrow {f},\overrightarrow {g},\overrightarrow {h})$ the inviscid flux matrix:

$$ \overrightarrow{f}=\left( \begin{array}{c} \rho {u_{1}^{2}} +P \\ \rho u_{1} u_{2} \\ \rho u_{1} u_{3} \\ \rho u_{1} H \\ \rho_{k} u_{1} \end{array} \right) \overrightarrow{g}=\left( \begin{array}{c} \rho u_{1} u_{2} \\ \rho {u_{2}^{2}} +P \\ \rho u_{2} u_{3} \\ \rho u_{2} H \\ \rho_{k} u_{2} \end{array} \right) \overrightarrow{h}=\left( \begin{array}{c} \rho u_{1} u_{3} \\ \rho u_{2} u_{3} \\ \rho {u_{3}^{2}} +P \\ \rho u_{3} H \\ \rho_{k} u_{3} \end{array} \right) $$

where H = E + p/ρ is the sensible total enthalpy and

$$ \begin{array}{@{}rcl@{}} \overline{\overline{A}} &=& \left (\begin{array}{ccccc} u_{1}(2-\varLambda) & -u_{2} \varLambda & -u_{3} \varLambda & \varLambda & {\varGamma}_{k} + \varLambda e_{c} -{u_{1}^{2}} \\ u_{2} & u_{1} & 0 & 0 & -u_{1}u_{2} \\ u_{3} & 0 & u_{1} & 0 & -u_{1}u_{3} \\ H - \varLambda {u_{1}^{2}} & -u_{1} u_{2} \varLambda & -u_{1} u_{3} \varLambda & u_{1} (1+\varLambda) & u_{1}(-H + {\varGamma}_{k} + \varLambda e_{c}) \\ Y_{k} & 0 & 0 & 0 & u_{1}(1-Y_{k}) \end{array} \right ) \end{array} $$

$$ \begin{array}{@{}rcl@{}} \overline{\overline{B}} &=& \left (\begin{array}{ccccc} u_{2} & u_{1} & 0 & 0 & -u_{1}u_{2} \\ -u_{1}\varLambda & u_{2}(2- \varLambda) & -u_{3} \varLambda & \varLambda & {\varGamma}_{k} + \varLambda e_{c} -{u_{2}^{2}} \\ 0 & u_{3} & u_{2} & 0 & -u_{2}u_{3} \\ -u_{1} u_{2} \varLambda & H - \varLambda {u_{2}^{2}} & -u_{2} u_{3} \varLambda & u_{2} (1+\varLambda) & u_{2}(-H + {\varGamma}_{k} + \varLambda e_{c}) \\ 0 & Y_{k} & 0 & 0 & u_{2}(1-Y_{k}) \end{array} \right ) \end{array} $$

$$ \begin{array}{@{}rcl@{}} \overline{\overline{C}} &=& \left (\begin{array}{ccccc} u_{3} & 0 & u_{1} & 0 & -u_{1}u_{3} \\ 0 & u_{3} & u_{2} & 0 & -u_{2}u_{3} \\ -u_{1}\varLambda & -u_{2} \varLambda & u_{3}(2- \varLambda) & \varLambda & {\varGamma}_{k} + \varLambda e_{c} -{u_{3}^{2}} \\ -u_{1} u_{3} \varLambda & -u_{2} u_{3} \varLambda & H - \varLambda {u_{3}^{2}} & u_{3} (1+\varLambda) & u_{3}(-H + {\varGamma}_{k} + \varLambda e_{c}) \\ 0 & 0 & Y_{k} & 0 & u_{3}(1-Y_{k}) \end{array} \right ) \end{array} $$

with the coefficients Ω_k and Λ given by:

$$ \begin{array}{@{}rcl@{}} {\varGamma}_{k} &=& \frac{C_{p} v_{k}}{C_{v} \beta} - \varLambda h_{k} \end{array} $$

(33)

$$ \begin{array}{@{}rcl@{}} \varLambda &=& \frac{\alpha}{\rho \beta C_{v}} \end{array} $$

(34)

In Eqs. 33 and 34, C_p and C_v are the heat capacities at constant pressure and volume, α is the thermal expansion coefficient and β is the isothermal compressibility coefficient. Partial-mass volume and enthalpy are v_k and h_k, respectively and $e_{c}=1/2 \sum \limits _{i=1}^{3} {u_{i}^{2}}$ is the kinetic energy.