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 approach linearizes convection using an extrapolated (Oseen-type) convecting velocity, producing a linear advection operator at each time step. For this operator, the adjoint can be written exactly. Exploiting this exact adjoint yields a systematic derivative-transfer mechanism within the VMS closure. In particular, unresolved-scale contributions enter the weak form without spatial derivatives of the modeled fine-scale velocity. The resulting terms also avoid derivatives of coarse-scale residuals and stabilization parameters. This eliminates the boundary-condition-sensitive, case-by-case integrations by parts that often accompany nonlinear residual-based VMS implementations, and it simplifies implementation in low-order FEM settings. The formulation is presented for a generalized linear convection operator encompassing three common advection forms (convective-, skew-symmetric- and divergence-form). Their numerical behavior is compared, along with the corresponding fully implicit nonlinear VMS counterparts. Because the method is linear by construction, each time step requires only one linear solve. Across the benchmark suite, this reduces wall-clock time by $2$--$4\times$ relative to fully implicit nonlinear formulations while maintaining comparable accuracy. Temporal convergence is verified, and validation is performed on standard problems including the lid-driven cavity, flow past a cylinder, turbulent channel flow, and turbulent flow over a NACA0012 airfoil at chord Reynolds number $6\times 10^{6}$. Overall, the convective and the skew-symmetric forms remain robust across the test cases, whereas the divergence-form can become nonconvergent for problems with purely Dirichlet boundaries.