Numerical Prediction of Turbulent Spray Flame Characteristics Using the Filtered Eulerian Stochastic Field Approach Coupled to Tabulated Chemistry

The Eulerian stochastic fields (ESF) method, which is based on the transport equation of the joint subgrid scalar probability density function, is applied to Large Eddy Simulation of a turbulent dilute spray flame. The approach is coupled with a tabulated chemistry approach to represent the subgrid turbulence–chemistry interaction. Following a two-way coupled Eulerian–Lagrangian procedure, the spray is treated as a multitude of computational parcels described in a Lagrangian manner, each representing a heap of real spray droplets. The present contribution has two objectives: First, the predictive capabilities of the modeling framework are evaluated by comparing simulation results using 8, 16, and 32 stochastic fields with available experimental data. At the same time, the results are compared to previous studies, where the artificially thickened flame (ATF) model was applied to the investigated configuration. The results suggest that the ESF method can reproduce the experimental measurements reasonably well. Comparisons with the ATF approach indicate that the ESF results better describe the flame entrainment into the cold spray core of the flame. Secondly, the dynamics of the subgrid scalar contributions are investigated and the reconstructed probability density distributions are compared to common presumed shapes qualitatively and quantitatively in the context of spray combustion. It is demonstrated that the ESF method can be a valuable tool to evaluate approaches relying on a pre-integration of the thermochemical lookup-table.

Validation of an Eulerian Stochastic Fields Solver Coupled with Reaction–Diffusion Manifolds on LES of Methane/Air Non-premixed Flames

The Eulerian stochastic fields (ESF) combustion model can be used in LES in order to evaluate the filtered density function to describe the process of turbulence–chemistry interaction. The method is typically computationally expensive, especially if detailed chemistry mechanisms involving hydrocarbons are used. In this work, expensive computations are avoided by coupling the ESF solver with a reduced chemistry model. The reaction–diffusion manifold (REDIM) is chosen for this purpose, consisting of a passive scalar and a suitable reaction progress variable. The latter allows the use of a constant parametrization matrix when projecting the ESF equations onto the manifold. The piloted flames Sandia D–E were selected for validation using a 2D-REDIM. The results show that the combined solver is able to correctly capture the flame behavior in the investigated sections, although local extinction is underestimated by the ESF close to the injection plate. Hydrogen concentrations are strongly influenced by the transport model selected within the REDIM tabulation. A total solver performance increase by a factor of 81% is observed, compared to a full chemistry ESF simulation with 19 species. An accurate prediction of flame F instead required the extension of the REDIM table to a third variable, the scalar dissipation rate.

Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7 spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. We obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.