Suriya Dhakshinamoorthy

arXiv2026

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

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.

physics.flu-dyn↗

arXiv2024

FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometries

Simulating fluid flow around arbitrary shapes is key to solving various engineering problems. However, simulating flow physics across complex geometries remains numerically challenging and computationally resource-intensive, particularly when using conventional PDE solvers. Machine learning methods offer attractive opportunities to create fast and adaptable PDE solvers. However, benchmark datasets to measure the performance of such methods are scarce, especially for flow physics across complex geometries. We introduce FlowBench, a dataset for neural simulators with over 10K samples, which is currently larger than any publicly available flow physics dataset. FlowBench contains flow simulation data across complex geometries (\textit{parametric vs. non-parametric}), spanning a range of flow conditions (\textit{Reynolds number and Grashoff number}), capturing a diverse array of flow phenomena (\textit{steady vs. transient; forced vs. free convection}), and for both 2D and 3D. FlowBench contains over 10K data samples, with each sample the outcome of a fully resolved, direct numerical simulation using a well-validated simulator framework designed for modeling transport phenomena in complex geometries. For each sample, we include velocity, pressure, and temperature field data at 3 different resolutions and several summary statistics features of engineering relevance (such as coefficients of lift and drag, and Nusselt numbers). %Additionally, we include masks and signed distance fields for each shape. We envision that FlowBench will enable evaluating the interplay between complex geometry, coupled flow phenomena, and data sufficiency on the performance of current, and future, neural PDE solvers. We enumerate several evaluation metrics to help rank order the performance of neural PDE solvers. We benchmark the performance of several baseline methods including FNO, CNO, WNO, and DeepONet.

physics.flu-dyn↗