A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids

Abstract

This paper presents a new unstructured-grid upwind finite-volume algorithm for accurate numerical simulation of incompressible flows and convection heat transfer on unstructured grids. It is an upwind method at both the differential equation level and discretized equation level, based on the method of characteristics. This is made possible with the introduction of Chorin's (J. Comput. Phys. 2 (1967) 12–26), artificial compressibility formulation to the incompressible flow equations. Flow variables are calculated along characteristics and their initial values are interpolated based on the signs of the corresponding characteristic speed. In addition, an upwind-biased interpolation method of third-order accuracy is used for interpolating flow variables on unstructured grids. With these inherent upwinding techniques for evaluating convection fluxes at control volume surfaces, no artificial viscosity is required. And the method is accurate and less sensitive to the grid orientation, thus suitable for unstructured grid application. The use of the finite-volume method combined with unstructured grids makes the scheme very flexible in dealing with very complex boundary geometries, while remaining to be fully conservative at the cell as well as global level. The discretized equations are solved by an explicit multistage Runge–Kutta time stepping scheme which is found to be efficient in terms of CPU and memory overheads. A computer code has been developed which uses the numerical methods presented, and a highly efficient edge-based data structure for flux calculation and data storage. A number of well-documented test cases, including 2D laminar flow in a channel with a backward-facing step, forced and natural convection flows, have been calculated in order to validate the code and evaluate the performance of the numerical algorithm. Good agreement between the computed and measured results have been obtained for all the cases. To demonstrate the accuracy of the third-order scheme, flow in another channel with a backward-facing step studied by Gartling (Int. J. Numer. Meth. Fluids (1990) 11) is simulated with both the third-order and first-order schemes. It is shown that the third-order scheme can produce a velocity profile across the channel that has perfect agreement with that of Gartling (local citation), whilst the first-order scheme cannot even predict the separation near the upper wall. The proposed numerical methods are also used to calculate 3D incompressible inviscid flows, and results show that the method is accurate and robust for general 3D simulation. The 3D Navier–Stokes code is later used to study a lid-driven cavity flow at a Reynolds number of 400. The unstructured grid used has 97,336 nodes and 455,625 tetrahedral elements with strong grid clustering near the walls. The convergence rate is found to be satisfactory, which shows that the method is also robust for 3D Navier–Stokes simulation. The results obtained agree well with those reported by Jiang et al. (Comput. Meth. Appl. Mech. Eng. 114 (109) (1994)). Results obtained using the second-order central scheme with artificial viscosity are also presented for comparison. It is observed that the third-order characteristics scheme is more accurate than the latter, thus confirming its third-order accuracy.