Models
1D heat release model
This 1D model intends to only model the heat release without representing the details of combustion. It relies on an additional source term in the energy equation. Following assumptions/simplifications are used:
Because it is a very empirical model and 2 unknown parameters are used, a calibration is required.
Mixing
Chemistry take places at small scales, and thus, require good mixing for efficient combustion. However, at high-speeds, like encountered within a scramjet combustor, the convective time scales are much larger than the diffusive ones. Therefore, achieving complete mixing within the combustor length is critical. This is also why predicting mixing accurately is very important.
The properties of the mixture depends on the mixture fraction Z through linear mixing rules. The thermodyanmic properties of the mixture, i.e., enthalpy and heat capacity, depends on its composition and the temperature. They are computed as a weighted sum of the single species enthalpies and the species mass fractions. The species thermodynamic properties themselves are only dependent on the temperature and are computed with the NASA polynomials. Molecular and thermodynamic transport coefficients of the mixture, i.e., dynamic viscosity and thermal conductivity, are computed using empiric mixture rules on basis of the properties of the components. The viscosity and conductivity of the single species rely on the elementary gas model.
In the RANS solver, the scalar turbulent fluxes are approximated with the classical gradient assumption and a constant turbulent Schmidt number. However, simulations of the jet in supersonic cross-flow experiment (see above) has demonstrated that this model is inaccurate and fails to capture key characteristics of the combustion close to the wall which are observed in the experiment. On the other hand, LES captures the larger scales and only model the smaller ones. In addition, the use of dynamic subgrid scale models provide a much better prediction of scalar mixing. Such higher resolution LES can then be used as basis for improved RANS scalar turbulent diffusion models.
FPVA-based model
While the 1D heat release model is computationally very efficient, it fails at capturing any of the combustion details. Therefore, it inherently entails large model-form uncertainties. To reduce those uncertainties, a more physics-based model is required.
The vast majority of computational work in supersonic turbulent combustion has so far relied on simplified/reduced mechanisms and the explicit transport of the involved species (Bray 1996). Such approaches require the closure of the chemical source term in the species transport equation. This can be achieved, for example, with simpler but low-accuracy models such as Arrhenius law (Davidenko et al. 2003), which neglects closure, the Eddy Dissipation Concept model (Chakraborty, Paul & Mukunda 2000), or with closure based on assumed (Baurle & Girimaji 2003; Karl et al. 2008) or transported (Baurle, Hsu & Hassan 1995) probability distribution functions (PDF). Some authors have also used the Linear Eddy Mixing model (LEM) (Genin, Chernyavsky & Menon 2004). But due to the strong non-linearity of the source term and the wide range of time scales associated with the chemistry, those equations are very stiff and difficult to solve. Moreover, due to very short residence times in such high speed flows, flame stabilization mechanisms are governed by auto-ignition. It is critical to model accurately such ignition and extinction phenomena in order to predict the stability of scramjet combustion. Therefore, prediction of flame stabilization requires detailed chemical kinetics. While a model transporting all involved species can easily be extended to more detailed chemical mechanisms, it quickly becomes computationally intractable, especially when complex fuels must be considered.
An alternative approach is based on the flamelet concept (Peters 2000; Pitsch 2006), which assumes that the chemical time scales are shorter than the turbulent time scales so that the flame can be approximated as one-dimensional. The flamelet approach allows the computation of the chemistry to be performed independently of the combustor simulation and stored in tabulated form as function of a limited number of scalars. During the actual scramjet simulation, the quantities of interest are read and interpolated, thus, dramatically decreasing the computational cost and allowing the use of complex chemical mechanisms. However, the implementation of the flamelet model is based on a low Mach number assumption, explaining the still very limited number of studies of high-speed flows using this approach (Berglund & Fureby 2007).
In the low Mach number flamelet implementation, the temperature and the species mass fraction are assumed to depend only on a transported scalar, traditionally the mixture fraction Z, and its dissipation rate. Chemical tables are then constructed assuming constant background pressure. This formulation can also be extended to better reproduce the unsteady character of combustion by replacing the scalar dissipation rate with a progress variable (Pierce & Moin 2004). However, the low Mach number assumptions do not hold anymore at supersonic speed and compressibility effects start to play an important role. Therefore, the flamelet implementation has been reformulated where temperature is no longer given by a chemistry table but computed from the total energy, thus, better accounting for compressibility effects (Birbaud & Pitsch 2008). Further, the model is extended for the auto-ignition regime with arguments similar to the model developed by Cook et al. (2007) for ignition in HCCI engines.
Groups
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