A Semi-Implicit Variational Multiscale Formulation for the Incompressible Navier-Stokes Equations via Exact Adjoint Linearization

Abstract

A semi-implicit, residual-based variational multiscale (VMS) formulation is developed for the incompressible Navier-Stokes equations. The convection term is linearized using an extrapolated (Oseen-type) convecting velocity, producing a linear advection operator whose adjoint can be written exactly. Because of this exact adjoint, unresolved-scale contributions enter the weak form without spatial derivatives of the fine-scale velocity, thereby eliminating the case-by-case adjustments that often accompany nonlinear residual-based VMS implementations. The formulation is presented for a generalized linear convection operator encompassing the convective, skew-symmetric, and divergence forms. Since the discrete method is linear by construction and monolithic for velocity and pressure, each time step requires only one linear solve, reducing wall-clock time by a factor of $2$ to $5$ relative to fully implicit nonlinear formulations while maintaining comparable accuracy. Temporal convergence is verified, and validation is performed on the lid-driven cavity, flow past a cylinder, turbulent channel flow, and flow over a NACA0012 airfoil at a high Reynolds number, demonstrating the efficiency of the proposed approach on problems of practical scale.