Finite volume methods for incompressible flow
Darryl Whitlow
TL;DR
This work develops two finite-volume schemes for incompressible flow on triangular meshes: a cell-centered, piecewise-constant method for Laplace-type problems and a vertex-centered least-squares method that delivers higher accuracy and flexible triangulations. The cell-centered scheme yields a straightforward, flux-conserving stencil suitable for potential-flow problems, while the least-squares approach minimizes a residual-based functional via Newton’s method, extending naturally to vector systems and to boundary-layer equations. The methods are validated on canonical inviscid problems (via the Cauchy-Riemann equations) around cylinders and airfoils and on Prandtl boundary-layer flow, with comparisons to analytical solutions such as the Blasius solution and exact circulations. The results demonstrate robust performance on both structured and unstructured meshes, with clear orders of accuracy and explicit boundary-condition handling, and the work outlines pathways to extend these ideas toward Navier–Stokes and 3D applications.
Abstract
Two finite volume methods are derived and applied to the solution of problems of incompressible flow. In particular, external inviscid flows and boundary-layer flows are examined. The firstmethod analyzed is a cell-centered finite volume scheme. It is shown to be formally first order accurate on equilateral triangles and used to calculate inviscid flow over an airfoil. The second method is a vertex-centered least-squares method and is second order accurate. It's quality is investigated for several types of inviscid flow problems and to solve Prandtl's boundary-layer equations over a flat plate. Future improvements and extensions of the method are discussed.