The proposed pH formulation and structure-preserving numerical methods can be extended to model more complex fluid flow phenomena, such as turbulence or multiphase flow, in energy networks by incorporating additional terms and operators in the governing equations. For turbulence modeling, the Navier-Stokes equations can be modified to include turbulence models like Reynolds-averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES) terms. These terms account for the turbulent fluctuations in the flow field and can be integrated into the pH framework to capture the effects of turbulence on energy transport and conversion processes.
In the case of multiphase flow, the pH formulation can be extended to include additional phases such as liquid or solid particles along with the compressible fluid phase. This extension would involve introducing new state variables, equations of state, and coupling conditions to describe the interactions between different phases. By incorporating multiphase flow models like the Euler-Euler or Euler-Lagrangian approaches, the pH framework can simulate the behavior of complex multiphase systems in energy networks.
The numerical methods used for solving the extended pH formulations would need to be adapted to handle the increased complexity of the models. This may involve developing specialized algorithms for solving the coupled equations, implementing efficient solvers for the additional terms, and validating the numerical schemes against experimental or computational data for turbulent or multiphase flows.