A POD-based variational multiscale method (VMM) for large eddy simulation (LES) of turbulent channel flows is proposed. The incompressible Navier–Stokes equations are solved using a residual-based VMM technique, and the residual-based model coefficient is adopted from Taylor et al. [1998] “Finite element modeling of blood flow in arteries,” Comput. Methods in Appl. Mech. Eng. 158(1–2), 155–196. A generalized-(Formula presented.) method is applied for the temporal integration. Numerical results show that four or higher wall-normal modes need to be applied for reproducing most of the turbulent characteristics. Results of large vortex structures and the low velocity streaks prove that the turbulent channel flows are well estimated by the proposed approach. The statistical analysis of the turbulent velocities confirms its reliability in turbulent channel flows.@en

A Goal-Oriented and Model-Constrained Optimization (GOMCO) approach was proposed as a discriminant technique for the Variational Geomano Method (VGM) in the coefficient determinations of variational multiscale Unresolved-Scale (URS) model in steady Stokes equations. Numerical implementations using both linear and nonlinear models were performed with both the GOMCO and VGM. Numerical results show that the coefficients determined by the GOMCO are scale-invariant, while they are scale-variant by the VGM. The GOMCO technique is found to be more appropriate for coefficient determinations in steady Stokes equations, as the VGM is sensitive to the computing procedures. Moreover, the GOMCO could provide reliable coefficients for the URS model.@en