Implicit subgrid modelling of incompressible flow and heat transfer using the discontinuity capturing Petrov-Galerkin/pressure stabilising Taylor-Galerkin (DCPG/PSTG) method

• Derivation of the Pressure Stabilising/Taylor-Galerkin (PSTG) method for circumventing the restrictions on the choice of interpolation spaces for velocity and pressure. • Introduction of the Discontinuity Capturing/Petrov-Galerkin (DCPG) method for implicit subgrid modelling of incompressible flow and heat transfer. • Definition and use of appropriate local time-steps as stabilisation parameters. • Numerical examples including free, mixed, and forced convection and Large Eddy Simulations (LES) of turbulent flows in 2D Cartesian and Axisymmetric problems. A new stabilised finite element method for the simulation of incompressible flow and heat transfer is presented. A velocity Taylor series in time is combined with the momenta balances and a Galerkin approximation of the mass conservation to derive an equation to compute pressure. The Pressure Stabilising Taylor-Galerkin (PSTG) method circumvents the Babuška-Brezzi restrictions on the interpolation spaces for velocity and pressure. To deal with strong convective flows, the Discontinuity Capturing Petrov-Galerkin (DCPG) method is introduced. It relies on the effective transport velocity, i.e., the projection of the flow velocity on the direction of the gradient of the quantity transported, introducing implicit subgrid modelling for Large Eddy Simulations (LES). Numerical examples, for both laminar and turbulent flows, demonstrate the good performance of the DCPG/PSTG method.