Motif: A Modular Finite-Volume Framework for Transient Incompressible Flow

Abstract

This document presents the theory, implementation, and verification of Motif, a two-dimensional incompressible flow solver developed as a teaching and research framework. The governing equations are discretized using the finite-volume method on staggered Cartesian grids. Convection is treated explicitly using the second-order Adams--Bashforth method, diffusion implicitly using the Crank--Nicolson method, and pressure--velocity coupling using projection methods. Both the Standard projection and the approximate second-order projection of Perot (1993) are considered. Particular attention is given to the discrete operators, boundary conditions, temporal treatment of pressure, conservation properties, and numerical dissipation and dispersion. The implementation is verified using a sequence of periodic, wall-bounded, and inlet--outlet flow problems on uniform and smoothly stretched grids. Spatial and temporal convergence, discrete mass conservation, kinetic-energy behavior, enstrophy, and energy spectra are examined. The results demonstrate the expected accuracy of the spatial discretization and distinguish the temporal behavior of the two projection approaches. The document is intended both to describe Motif in sufficient detail for independent implementation and to examine the numerical properties relevant to its continued development toward direct numerical simulation.