A fully-coupled balanced-force numerical framework for two-phase flows with surface tension on arbitrary collocated meshes is presented, including a novel method to evaluate the curvature from volume fractions. The presented framework reduces the imbalances at the interface to solver tolerance and provides stable and reliable results for density ratios of 10⁶ and larger. The new method to evaluate the curvature is based on a least-squares fit, providing better or equal accuracy compared to implementations found in the literature, which are generally limited to structured meshes, and the accuracy on Cartesian and tetrahedral meshes is shown to be comparable.

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most attractive for describing the behaviour of non-spherical particles. However, there are a number of drawbacks when using unit Quaternions: the necessity of applying rotation matrices in conjunction to facilitate the transformation from body space to world space, and the algorithm integrating the Quaternion should inherently conserve the length of the Quaternion. Both drawbacks are addressed in this paper. The present paper derives a new framework to transform vectors and tensors by unit Quaternions, and the requirement of explicitly using rotation matrices is removed altogether. This means that the algorithm derived in this paper can describe the rotation of a non-spherical particle with four parameters only. Moreover, this paper introduces a novel corrector-predictor method to integrate unit Quaternions, which inherently conserves the length of the Quaternion. The novel framework and method are compared to a number of other methods put forward in the literature. All the integration methods are discussed, scrutinised and compared to each other by comparing the results of four test cases, involving a single falling particle, nine falling and interacting particles, a 2D prescribed torque on a sphere and a 3D prescribed torque on a non-spherical particle. Moreover, a convergence study is presented, comparing the rate of convergence of the various methods. All the test cases show a significant improvement of the new framework put forward in this paper over existing algorithms. Moreover, the new method requires less computational memory and fewer operations, due to the complete omission of the rotation matrix in the algorithm.

A generalized finite-volume framework for the solution of fluid flows at all speeds in complex geometries and on unstructured meshes is presented. Starting from an existing pressure-based and fully-coupled formulation for the solution of incompressible flow equations, the additional implementation of pressure–density–energy coupling as well as shock-capturing leads to a novel solver framework which is capable of handling flows at all speeds, including quasi-incompressible, subsonic, transonic and supersonic flows. The proposed numerical framework features an implicit coupling of pressure and velocity, which improves the numerical stability in the presence of complex sources and/or equations of state, as well as an energy equation discretized in conservative form that ensures an accurate prediction of temperature and Mach number across strong shocks. The framework is verified and validated by a large number of test cases, demonstrating the accurate and robust prediction of steady-state and transient flows in the quasi-incompressible as well as subsonic, transonic and supersonic speed regimes on structured and unstructured meshes as well as in complex domains