Inference of forcing kernels from limited observations in particle cloud tracers

Understanding the forcing law in particle-laden flows is crucial for accurately predicting and controlling these flows, especially under extreme conditions such as shock waves. Challenges arise when only limited statistical data for a group of particles is available for inferring the forcing law. Traditional predictive modeling approaches rely heavily on model assumptions with empirical parameters and lack the capability for uncertainty quantification. To address this issue, we developed an adjoint-based optimization framework built upon the closed Subgrid Particle-Averaged Reynolds Stress Equivalent (SPARSE) model. This framework infers the forcing law from the cloud location and covariance at the final time. The discrete adjoint operator functions provide an accurate gradient of the cost function with respect to the forcing law, which is then used for cost minimization. We applied this framework to a one-dimensional shock-particle interaction and to passive particles in a two-dimensional flow around a circular cylinder. Using this approach, we successfully reconstructed the forcing law, enabling accurate reproduction of observations. The results also demonstrate the impact of different modes within the forcing law on the resulting particle flow map.