A Pressure Projection Approach within the Variational Multiscale Framework for Navier–Stokes Equations

We have developed a pressure-projection Variational Multiscale (VMS) formulation for the Navier–Stokes equations, using a linearised convective term. The solver first advances an intermediate velocity field and then applies a projection step to satisfy the continuity constraint and update the pressure. For comparison, we also implement a fully nonlinear VMS solver based on Newton–Raphson iterations within the same finite-element framework. To validate the new formulation, we will perform simulations of turbulent channel flow and flow past a NACA airfoil, using identical meshes and time-integration schemes. We will compare the accuracy of mean velocity profiles, lift, drag, pressure, and skin-friction coefficients, and also the robustness of the formulation in terms of mass-balance error and sensitivity to time-step. We will also measure the computational cost using iteration counts and wall-clock time. We will also examine how VMS stabilisation interacts with the projection step and how the linearised convection treatment influences convergence at high Reynolds numbers. The results will highlight practical trade-offs between the linearised pressure-projection approach and the nonlinear Newton–Raphson method within a common framework for turbulent flow simulation.