First, Second, and Third Order Finite-Volume Schemes for Navier-Stokes Equations

In this paper, we present first-, second-, and third-order implicit finite-volume schemes for solving the Navier-Stokes equations on unstructured grids based on a hyperbolic formulation of the viscous terms. These schemes are first-, second-, and third-order accurate on irregular grids for both the inviscid and viscous terms and for all Reynolds numbers, not only in the primitive variables but also in the first-derivative quantities such as the velocity gradients for the incompressible Navier-Stokes equations and the viscous stresses and heat fluxes for the compressible Navier-Stokes equations. Numerical results show that the developed schemes are capable of producing highly accurate derivatives on highly-skewed grids whereas a conventional scheme suffers from severe oscillations on such grids. Moreover, these first-, second-, and thirdorder schemes enable the construction of implicit solvers that converge intrinsically faster in CPU time than a conventional second-order implicit Navier-Stokes solver with the acceleration factor growing in the grid refinement.