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A novel free-energy stable discontinuous Galerkin method is developed for the Cahn-Hilliard equation with non-conforming elements. This work focuses on dynamic polynomial adaption (p-refinement) and constitutes an extension of the method developed by Manzanero et al. in Journal of Computational Physics 403:109072, 2020, which makes use of the summation-by-parts simultaneous-approximation term technique along with Gauss-Lobatto points and the Bassi-Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates nonconforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free-energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaption is introduced based on the knowledge of the location of the interface between phases. The adaption methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We solve a steady one-dimensional interface test case to initially examine the accuracy of the adaption. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free-energy are recovered with a 35% reduction of the degrees of freedom for the two-dimensional case considered and a 48% reduction for the three-dimensional case.

We propose an invariant feature space for the detection of viscous-dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous-dominated, rotational region (boundary layer and wake region) and an inviscid, irrotational region (outer flow region). We have tested the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at $Re=40$ and $Re=3900$ and a laminar flow around an airfoil at $Re = 1 × 10^5$. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method (DGSEM). The results obtained are analysed to show that Gaussian mixture clustering provides an effective identification method of viscous-dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors.

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We present the latest developments of our High-Order Spectral Element Solver (HORSES3D), an open source high-order discontinuous Galerkin framework, capable of solving a variety of flow applications, including compressible flows (with or without shocks), incompressible flows, various RANS and LES turbulence models, particle dynamics, multiphase flows, and aeroacoustics. We provide an overview of the high-order spatial discretisation (including energy/entropy stable schemes) and anisotropic p-adaptation capabilities. The solver is parallelised using MPI and OpenMP showing good scalability for up to 1000 processors. Temporal discretisations include explicit, implicit, multigrid, and dual time-stepping schemes with efficient preconditioners. Additionally, we facilitate meshing and simulating complex geometries through a mesh-free immersed boundary technique. We detail the available documentation and the test cases included in the GitHub repository.