**Coupled Solving**

**MultiFlow** consists ofÂ a coupled balanced-force numerical framework for single- and multi-phase flows at all speeds on unstructured meshes, including the incompressible, weakly compressible and fully compressible flow regimes. This framework enables the simulation of creeping, laminar and turbulent flows at any Mach number (sub-, trans-, super- and hypersonic flows) in complex geometries. The coupled numerical framework provides a strong, implicit pressure velocity coupling and accurately accounts for source terms using a novel balanced-force discretisation, which is of distinct advantages in flows with large source terms, such as multiphase flows or porous media, reduces errors caused by a force imbalance at the interface to solver tolerance and the fully-coupled methodology provides a strong pressure-velocity coupling.

We have successfully applied this pioneering numerical framework to a range of applications and flow regimes. In interfacial flows, for instance, the strong pressure-velocity coupling as well as the balanced-force discretisation allows us to accurately and robustly simulate fluid pairs with arbitrarily large density difference and surface tension, a currently unparalleled capability. With regards to compressible flows, the pressure-based formulation of the numerical framework enables us to simulate compressible flows of all Mach numbers on unstructured meshes and in complex geometries. In multi-phase flow problems, this enables the efficient, stable and accurate prediction of such flows in the Eulerian-Lagrangian or Eulerian-Eulerian frameworks.

**Unstructured and adaptive meshes**

Unstructured meshes are typically required to simulate flows in complex geometries, since structured meshes are not able to represent complex geometries. The discretisation of the governing equations on unstructured meshes, such as tetrahedral meshes or polyhedral meshes, involves additional difficulties compared to the discretisation on structured meshes, due to the additional topological complexity of unstructured meshes. The discretisation must account for geometric errors, such as mesh skewness and non-orthogonality, in order to provide a robust and accurate solution.

Our research on the application of unstructured meshes in CFD is multifaceted and includes the numerical framework (see above), the discretisation of advection and diffusion terms as well as mesh generation and adaptive meshing (adaptive tetrahedral and polyhedral meshes). As a result of our continuing research efforts, we have for instance developed a novel method to address numerical diffusion caused by the geometrical skewness of the computational mesh in donor-acceptor schemes, such as TVD differencing schemes or compressive VOF advection schemes, without compromising the computationally efficiency and the monotonicity of the solution.

**Interface capturing on arbitrary meshes**

We have developed a sophisticated interface capturing methodology that is applicable to arbitrary unstructured meshes and provides a similar accuracy on Cartesian, tetrahedral and polyhedral meshes. The interface is represented by using a Volume-of-Fluid (VOF) method and is advected based on the underlying flow using a linear hyperbolic advection equation, which is discretised using a specifically designed compressive VOF method. This compressive VOF method is based on algebraic discretisation schemes and actively adapts the coefficients of the governing equations to correct errors introduced by an adverse mesh quality on unstructured meshes, thereby increasing the accuracy of the numerical solution as well as the robustness of the numerical solving algorithm significantly. Once the interface has reached its new location, we evaluate the new topology (normal vectors and curvature) of the interface using a least-squares method called CELESTE (Curvature Evaluation with Least-Squares of Taylor Expansion), which we developed particularly for unstructured meshes. CELESTE constructs on overdefined system of equations based on Taylor expansion of the volume fraction field of each phase and solves this system of equations using a least-squares method. Applying a least-squares method to an overdefined system of equation mitigates spatial aliasing of the calculated first- and second-order derivatives of the abruptly varying volume fraction field, which is a well-known problem of VOF methods.

**Immersed boundary methods**

The immersed boundary method (IBM) is an alternative to boundary fitted fluid mesh in order to approximate moving solid boundaries, inherent in fluid-structure interaction and particulate flow problems. As it permits the use of a fixed fluid mesh, the IBM is easier in implementation and more efficient in computation. The method enforces the existence of the boundary which is non-conformal and internal to the fluid mesh by applying appropriate modifications to the momentum and continuity equations of the cells close to the boundary. Our current development of the IBM focus on extending the applicability of direct forcing schemes for unstructured meshes. While keeping a sharp interface approximation, an effort to reduce oscillations of the applied forces commonly observed in IBM simulations will further improve the accuracy achieved by local refinement of the unstructured meshes.

With regard to multiphase flows, the IBM can efficiently handle fully resolved simulations of particulate flows in order to study intricate and small scale features of the particle-fluid and particle-particle interaction. From this detailed understanding, new models and improvements to existing models can be devised for use in large-scale simulations.