Finite volume numerical simulation of viscoelastic flows in general orthogonal coordinates

This work reports on numerical simulations of unsteady two- and three-dimensional viscoelastic flows in complex geometries. The Navier–Stokes solver is designed for general orthogonal grids in two dimensions and a Cartesian description in the spanwise direction. The conservation equations are written in primitive pressure–velocity variables, making use of the physical curvilinear lengths and contravariant velocities [S.B. Pope, The calculation of turbulent recirculating flows in general orthogonal coordinates, J. Comp. Phys. 26 (1978) 197–217]. The space discretization is finite volume, with the pressure at the center of the control volumes and the velocity components staggered at the center of the cell faces. The conservative advective terms are discretized with a quadratic upwind scheme (QUICK). The extra advective terms induced by coordinate curvature are treated explicitly as integral volumes evaluated at the cell centers. The time advancement of the solution follows from an explicit decoupling procedure [F.H. Harlow, J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965) 2182–2189]. The Adams–Bashforth level 2 scheme is used to evaluate advection, curvature and viscous terms. This discretization produces a symmetric and positive definite matrix for the pressure. The overall formulation is second order accurate in space, first-order in time. To model the viscoelastic flow, the Phan-Thien–Tanner (PTT) constitutive equation is considered. Results are presented for 2D and 3D configurations (flow past a cylinder and flow through a U-curved channel), both presenting recirculations and secondary flows. This contribution demonstrates the versatility of the formulation for 2D and 3D flows with curved boundaries, and its easy implementation starting from a Cartesian description.