A Three-Dimensional, Third-Order Characteristics-Based Implicit-Unfactored Scheme for Compressible Viscous Flows

Abstract

This study presents the development and validation of a high-resolution, three-dimensional numerical framework for the steady and unsteady compressible Navier–Stokes equations. The core of the method is a fully implicit-unfactored time-marching scheme combined with a third-order characteristics-based upwind discretization. The formulation removes the severe CFL restriction of explicit schemes and is more robust in three dimensions than classical approximate-factorization methods. Numerically, the solver comprises a finite-volume discretization, an unfactored Newton method with point-by-point Gauss–Seidel relaxation, and a conservative-to-primitive preconditioner that keeps the discrete system well-posed during relaxation over a wide Mach-number range. Turbulence closure is provided by the Launder–Sharma low-Reynolds-number k-e model, which is integrated to the wall. In contrast to the authors’ previous two-dimensional unfactored solvers, the present formulation includes the z-direction fluxes, three-dimensional metrics, and six-neighbour Gauss–Seidel coupling, together with a sweep-wise conservative-to-primitive preconditioner for the 3D block operator. Performance is assessed on the NACA 0012 airfoil for subsonic flow at M=0.5, a= 7o and transonic flow at M=0.85, a= 0o. With CFL=25, the implicit scheme remains stable and yields a four- to six-fold reduction in wall-clock time relative to an equivalent explicit method. The computed pressure fields, shock locations, and spanwise relief effects agree closely with Harris’s experiments and the AGARD benchmarks, supporting the use of the framework for three-dimensional compressible viscous flows.