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

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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.

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Support provided by ArianeGroup, the prime contractor of the Ariane launcher cryogenic propulsion system and CNES, the French National Space Agency, is gratefully acknowledged. The author also greatly acknowledge Robin Nez from EM2C, Bénédicte Cuenot and Gabriel Staffelbach from Cerfacs and Laurent Selle from IMFT for their contributions in the AVBP-RG project. This work was granted access to the HPC resources of IDRIS and CINES made available by GENCI (Grand Equipement National de Calcul Intensif) under the allocation A0042B06176. A part of this work was performed using HPC resources from the mesocentre computing center of Ecole CentraleSupélec and Ecole Normale Supérieure Paris-Saclay supported by CNRS and Région Ile-de-France.

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Appendix : Jacobian Matrices for TTG Schemes

Appendix : Jacobian Matrices for TTG Schemes

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

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

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 :

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

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 Ωk and Λ given by:

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

In Eqs. 33 and 34, Cp and Cv 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 vk and hk, respectively and \(e_{c}=1/2 \sum \limits _{i=1}^{3} {u_{i}^{2}}\) is the kinetic energy.

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Schmitt, T. Large-Eddy Simulations of the Mascotte Test Cases Operating at Supercritical Pressure. Flow Turbulence Combust (2020).

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  • Large-Eddy Simulation
  • Transcritical regime
  • Turbulent combustion