Numerical Investigation of Injection, Mixing and Combustion in Rocket Engines Under High-Pressure Conditions

The flow of reacting, multicomponent, compressible fluids is governed by the conservation equations of mass and momentum together with the transport equations for the energy and the different involved species. Let \(\rho \), t, \(\mathbf {u}\), \(\mathbf {\sigma }\), \(h_t\), \(\dot{\mathbf {q}}\), \(Y_k\), \(\mathbf {j_k}\), \(\dot{\omega }_k\) and \(N_c\) represent the density, time, velocity vector, stress tensor, total enthalpy, heat flux vector, mass fraction, mass flux vector, source term and the number of species, respectively. The governing equations read: andThe stress tensor \(\sigma \) can be expressed for Newtonian fluids aswhere \(\tau \) and \(\mu \) are the viscous stress tensor and the dynamic viscosity, respectively. In the energy conservation Eq. (3) the total enthalpy \(h_t\) represents the sum of the static enthalpy h and the kinetic energy \(\frac{1}{2} \left\Vert \mathbf {u}\right\Vert ^2\). By applying both Fourier’s and Fick’s law together with the unitary Lewis-number assumption, i.e., \(\text {Le} = \kappa \left( \rho c_p D \right) ^{-1} = 1\), for the deduction of the diffusion coefficient D, the heat flux vector \(\dot{\mathbf {q}}\) and the mass flux vector \(\mathbf {j_k}\) can be expressed in the following way: Here, \(\kappa \) and \(c_p\) denote the thermal conductivity and the specific heat at constant pressure, respectively. Under subsonic flow conditions—as it is the case here—the pressure contribution in Eq. (6) can be neglected and therefore the divergence of the heat flux \(\nabla \cdot \dot{\mathbf {q}}\) can be handled implicitly as the energy equation is written in an enthalpy explicit form. In Eqs. (1)–(4), Soret, Dufour and radiation effects are neglected.

The finite volume method together with a pressure-based solver formulation is used to discretize the governing equations. In the pressure-based approach, the pressure instead of the density is used as a primary variable. Therefore, the density is recalculated by means of a suitable equation of state (EoS). The pressure is solved with the help of a Poisson-like equation which can be derived from the momentum and continuity equations as  [7]:Here, the variables \(a_P\) and \(H_P\) result from the discretization of the momentum equation, for further details see, e.g., Ferziger and Peric  [7]. As Eq. (8) is elliptical in nature and therefore derived for the incompressible flow regime, slight modifications have to be made to account for the compressibility of the fluid. Following typical pressure correction approaches, the total differential of the density as function of pressure p, enthalpy h and species composition \(\mathbf {Y}\) can be expressed as:Recasting this equation into a Taylor series truncated after the first term, neglecting all variations except the pressure and applying the resulting equation in the time derivative of Eq. (8) yields the following adjusted pressure equation:Here, the superscript \(n-1\) refers to the last iteration/time step. For solving this pressure equation together with the other governing equations, a segregated solution algorithm is selected. In detail, a so-called PIMPLE approach is applied which is a combination of the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) and the Pressure-Implicit with Splitting of Operators (PISO) method. The solver is implemented in the open-source toolbox OpenFOAM  [1]. A more thorough description and discussion can be found in Traxinger et al.  [30].

For relating and calculating the different thermodynamic properties, appropriate state equations and relations are required. The density and pressure are coupled by means of cubic EoSs which are commonly applied due to their efficiency and acceptable accuracy. The pressure-explicit form of the cubic EoSs reads  [21]:Here, \(\mathcal {R}\) is the universal gas constant, T is the temperature and v is the molar volume. The parameters \(a \left( T \right) = a_c \alpha \left( T \right) \) and b account for the intermolecular attractive and repulsive forces, respectively, and u and w are model constants. Based on Eq. (11), the popular cubic EoSs of Peng and Robinson  [19] and Soave, Redlich and Kwong  [26] can be deduced as well as the ideal gas equation (\(a = b = 0\)).

For the consideration of multicomponent mixtures, the concept of a one-fluid mixture in combination with mixing rules is applied  [21]where \(z_i\) is the mole fraction of the i-th component. Pseudo-critical combination rules  [22] are employed to determine the off-diagonal elements of \(a_{ij}\). The caloric properties are derived consistently by applying the departure function formalism  [21]. In this concept, the respective property, e.g., the enthalpy h, is divided into an ideal gas (ig) and a real gas (rg) part, i.e.,where R is the specific gas constant. The real-gas contribution can be derived from the applied cubic EoS. The ideal gas part is determined using the seven-coefficient NASA polynomials  [9]. For the determination of the viscosity and the thermal conductivity, the empirical correlation of Chung et al.  [5] is employed in the real-gas case. In contrast, the approach of Sutherland  [27] is used for ideal gases. For considering multicomponent phase separation, a multiphase framework based on the cubic EoSs and the tangent plane concept of Michelsen  [13] is applied. This framework relies on the assumption of a local instantaneous thermodynamic equilibrium. For further details, please refer to Traxinger et al.  [28, 30].

In the applied tabulated combustion models, the mixture fraction f is introduced to describe the combustion progress by means of a passive scalar transport equationwhere Sc and \(Sc_t\) are the laminar and turbulent Schmidt number, respectively, and the subscript sgs denotes the contribution of the applied subgrid model due to filtering. Here and in the following, the finite-volume filter is indicated by a bar \(\bar{\star }\) and Favre-filtering is denoted by a tilde \(\tilde{\star } = \overline{\rho \star }/\bar{\rho }\). To model unresolved fluctuations of the mixture fraction, a transport equation for the variance of f is solved  [11]where \(\chi \) is the scalar dissipation rate modeled according to Domingo et al.  [6].

For the generation of the thermo-chemical database, the flamelet concept is applied  [20]. Under the assumption of a Damköhler number \(Da \gg 1\), the flamelet approach allows to describe the structure of a turbulent nonpremixed flame as a brush of laminar counterflow diffusion flames. By transforming the governing equations into the mixture fraction space and assuming both a constant pressure and a unity Lewis-number, a one-dimensional set of equations can be derived: High-fidelity reaction mechanisms like, for instance, GRI-3.0  [10] are employed to determine the reaction rates. Solving Eqs. (16)–(17) for different scalar dissipation rates up to extinction and a subsequent filtering by means of a presumed probability density function (PDF) yields a suitable library for the LESs of adiabatic combustion. Nonadiabatic effects can be introduced conveniently by assuming a frozen composition or by introducing a semi-permeable wall into the mixture fraction space, see, e.g., Zips et al.  [35, 36]. Under real-gas conditions, the pressure dependency of the thermodynamic properties has to be taken into account in the tabulation  [34].

In contrast to presumed PDF approaches, transported PDF methods allow for the evaluation of the PDF \(\mathcal {P}_{sgs}\) by means of a transport equation  [8]where \(\varPsi \) denotes the thermo-chemical state space. This transport equation is usually solved using statistic methods. In the present work, the Eulerian stochastic fields (ESF) method proposed by Valiño  [33] is employed. A more thorough description and discussion can be found in Zips et al.  [35].

Under supercritical pressure conditions, the dense-gas approach is widely-used in the LRE community which implies the assumption of a sole single-phase state. For the pioneering works with only a single component like, e.g., Zong et al.  [37], this assumption is perfectly valid. However, in multicomponent mixtures, as it is the case during injection and combustion under LRE conditions, the multiphase region is determined by a critical locus rather than a distinct critical point. Due to the nonlinear behavior of real-gas mixtures this locus can exceed the critical values of the pure components by orders of magnitude, especially with respect to the pressure, see Fig. 1 left.

A-posteriori investigations of the mixture states of a binary hydrogen/nitrogen test case  [18] at cryogenic temperatures showed first evidence of phase separation under initially supercritical conditions  [14]. In Fig. 1 right, simulation data of this test case are scattered into a temperature composition diagram and superimposed onto the mixture two-phase region. In the composition range of \(0.15 \le z_{\text {H}_2} \le 0.35\) the mixture states penetrate the vapor-liquid equilibrium (VLE) rendering the single-phase assumption invalid. As there is no profound evidence for this statement, additional test cases were defined in close cooperation with the ITLR at the University of Stuttgart where an appropriate experimental test facility is available  [2]. In this campaign, initially supercritical n-hexane (C\(_6\)H\(_{14}\)) was injected into a pressurized nitrogen atmosphere at three different total temperatures (\(T_{t,\text {C}_6\text {H}_{14}} = \left[ 480, 560, 600 \right] \) K). In the experiments, simultaneous shadowgraphy and elastic light scattering (ELS) was conducted to visualize both the jet structure and the phase separation process. LESs employing the multicomponent VLE model as thermodynamic closure have been used for the numerical investigation. Both the experiments and the simulations show similar phase separation phenomena at the respective temperature and a transition from a dense-gas mixture (\(T_{t,\text {C}_6\text {H}_{14}} = 600\) K) to a spray-like jet (\(T_{t,\text {C}_6\text {H}_{14}} = 480\) K) proving the presence of phase separation under high-pressure conditions depending on the injection temperature, see Fig. 2 and for further details Traxinger et al.  [29]. Applying the same VLE model, the single-phase stability in a high-pressure methane combustion case was investigated. The analysis showed strong phase separation phenomena on the oxidizer-rich side and a large spatial extent inside the flame, see Fig. 3. The phase separation process is triggered due to the low temperatures and the presence of water  [32] originating from the combustion and subsequent diffusion processes.

The increased interest in reusable, methane-fired engines sets new demands into the development and improvement of CFD tools. Although a lot of research effort has been invested in the field of nonadiabatic rocket combustion chamber modeling in recent years, most of the past studies investigate hydrogen combustion. Furthermore, the dimensions of the test chambers, e.g., the Pennstate pre-burner, were not really rocket-engine typical and therefore the results cannot be fully used as a blueprint for more application-relevant studies. Therefore, two new oxygen/methane combustion chamber test cases have been defined by the group of Prof. Haidn at the Technical University of Munich featuring more application-related chamber characteristics, e.g., element-element and element-wall distances: a single-element combustion chamber  [4] and a 7-element combustor  [25]. Using the single-element combustion chamber as a reference case, the influence of the turbulence wall model on the predicted wall heat flux has been investigated. In Fig. 4 left, a general impression of the temperature field is given at different axial positions. The quadratic chamber cross section clearly influences the axial development of the flame. The comparison of the predicted heat flux with the experiment, see Fig. 4 right, shows that all models capture the basic trend. Overall, the wall-modeled LES (WF-LES) shows the best results and was therefore employed for further studies on the influence of the combustion model. The 7-element test case was used to compare three different combustion models namely two presumed PDF approaches and one transported PDF approach. In Fig. 5 left, temperature contours at different axial positions are shown indicating the gradual flame-flame interaction with increasing axial direction. In terms of the predicted wall heat flux, see Fig. 5 right, the ESF method shows the best results followed by the nonadiabatic and the frozen flamelet model. In Fig. 6 scatter plots of temperature and selected mass fractions are shown for the different combustion models revealing clear differences. As expected, the frozen model reproduces the composition of the nominal flamelet. The nonadiabatic solution shows stronger scattering in terms of species mass fraction but is different to the ESF solution.

Finally, a 60 degrees section of a full-scale methane-fired thrust chamber demonstrator (TCD) defined by ArianeGroup has been investigated focusing on flame-flame interaction, see Fig. 7. The operating conditions of the TCD are inspired by the Prometheus engine. The LES result reveals strong flame-flame interaction starting at \(x/D \approx 3\). As a consequence, temperatures below 1500 K are only present up to \(x/D \approx 10\) which is very different to the investigation of a single injection element under identical operating conditions where flame-flame interaction is inherently missing  [31].