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A data assimilation approach is coined that enables the discovery of forcing functions in Lagrangian, point-particle models from limited measurements of trajectory coordinates. Central to the proposed formulation of this inverse problem is the expression of the forcing function in terms of modal basis functions that are dependent on the relative velocity difference between a known carrier flow and the particle solution weighted with coefficients that are known within confidence intervals. The probability density function of the random forcing coefficients is inferred using a combination of the forward, particle model and its adjoint dynamics, which calculates the gradient of the cost function defined as the distance between the measured and predicted particle locations. To ensure convergence of the gradient-based optimization, multiple measurements may be required. If the measurements are noisy, samples of the forcing model within an assumed Gaussian distribution of the confidence interval of the measurement are computed using a Hamiltonian Monte Carlo method. The method is verified to correctly infer the forcing function of particles traced in the Arnold–Beltrami–Childress flow and a homogeneous isotropic turbulence. The confidence interval of the inferred forcing function with respect to a flow condition is improved if the particle is exposed more frequently to the flow condition. The forcing coefficients adapt the model to flow conditions that are outside of the limited range for which the point-particle models are typically known only empirically or within confidence intervals. 

This paper is on an Eulerian-Eulerian (EE) approach that utilizes Godunov’s scheme to deal with a running shock that interacts with a cloud of particles. The EE approach treats both carrier phase (fluid phase) and dispersed phase (particle phase) in the Eulerian frame. In this work, the fluid equations are the Euler equations for the compressible gas while the particle equations are based on a recently developed model to solve for the number density, velocity, temperature, particle sub-grid scale stresses, and particle sub-grid scale heat fluxes. The carrier and dispersed phases exchange momentum and heat, which are modeled through incorporating source terms in their equations. Carrier and dispersed phase equation form a hyperbolic set of differential equations, which are numerically solved with Godunov’s scheme. The numerical solutions are obtained in this work for a two-dimensional normal running shock interacting with a rectangular cloud of particles. The results generated by the EE approach were compared against the results that were generated by a well-stablished Eulerian-Lagragian (EL) approach that treats the carrier phase in an Eulerian frame, while does the dispersed phase in a Lagrangian framework where individuals particles are traced and solved. For the considered configuration, the EE approach reproduced the EL results with a very good accuracy.