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

TL;DR

This work tackles instabilities in standard FEM discretizations of the incompressible NSE by introducing a semi-implicit residual-based VMS that linearizes convection via an extrapolated Oseen-type velocity ${\mathbf{a}}$. The key advance is the exact adjoint of the linearized convection operator, which enables a derivative-free representation of fine-scale effects and a single linear solve per time step. The method is tested on canonical laminar and turbulent flows, showing robust accuracy across convective, skew-symmetric, and divergence forms ($s\in\{0,\tfrac{1}{2},1\}$) with significant computational speedups (2–4×) over fully implicit nonlinear VMS. Overall, the approach offers a transparent, implementation-friendly stabilization framework with strong performance for parameter studies and long-time simulations, while highlighting boundary-condition sensitivity for the $s=1$ form in purely Dirichlet settings.

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.

Figures (27)

• Figure 1: Sketch of a general domain with different conditions applied at different boundaries. Neumann conditions are prescribed on $\Gamma_N$, and Dirichlet conditions on the velocities are prescribed on $\Gamma_{D_1}$ and $\Gamma_{D_2}$.
• Figure 2: $s=0$
• Figure 3: $s=1/2$
• Figure 4: $s=1$
• Figure 6: $s=0$

Theorems & Definitions (6)

• Remark 2.1
• Remark 2.2
• Proposition 1
• proof
• Remark 3.1
• Remark 3.2