An unstructured finite volume method based on the projection method combined momentum interpolation with a central scheme for three-dimensional nonhydrostatic turbulent flows

Abstract

This paper presents a three-dimensional nonhydrostatic model to solve the Navier–Stokes equations using an unstructured finite volume method. The physical domain could be geometrically arbitrary. To avoid the checkerboard problem caused by non-staggered grids, a momentum interpolation method is used by introducing face-normal velocities at the mid-points of the cell faces. As the Large Eddy Simulation (LES) requires at least second-order accuracy in time and in space for all the terms, a central scheme combined with an explicit Adams–Bashforth scheme is proposed in this model. The projection method is applied to decouple the velocity field and pressure. Several benchmark test cases are used to validate the second-order accuracy, the numerical stability and the performance of the model. Analysis on divergence noise using an unstructured collocated triangular grid, as well as on the ratio between vertical and horizontal spacing steps have been done to show the reliability of the model. The proposed model has been used to simulate backward-facing step flows, lid-cavity flows, turbulent open channel flows and the turbulent flows around a vertical cylinder. The convergence of the linear solver is analyzed in terms of the iterations and CPU time. The results are fairly in agreement with the references in the literature. The proposed model is able to correctly reproduce the characteristic flow features in all the test cases.

Introduction

Due to the complex physics processes and the limitation of computing power, traditional numerical efforts devoted to the geophysical flow problems, are usually confined into two-dimensional (2D) modeling using the hydrostatic approach [1], [2]. These models can provide rough estimation for future scenarios but lack the capability to handle with three-dimensional (3D) near-field flows such as nonhydrostatic turbulent flows around obstacles in complex geometry. They usually fail in capturing velocity profiles and coherent structure of flows due to the hydrostatic approximation and the absence of correct turbulence modeling [3]. With the development of high performance computer, nowadays LES has been largely used to simulate high turbulent geophysical flows by massive parallel [4], [5], [6].

This paper proposes an accurate, robust and efficient solver for the incompressible Navier–Stokes (N-S) equations using LES in collocated unstructured grids. It is well-known that a collocated grid arrangement for incompressible flows could generate unrealistic pressure oscillations due to the pressure–velocity decoupling, which is known as the checkerboard problem. This problem can be solved using the Momentum Interpolation Method (MIM) [7], [8]. In the past few years, the finite volume method with collocated unstructured grid has been used for both steady and unsteady flows [9], [10], [11], [12]. In these work, the mass flux was calculated by introducing a face-normal velocity, defined at the mid-point of each cell face. This mass flux or the face-normal velocity was interpolated from the cell centers and later corrected by the pressure gradient, which is obtained by the least square method. Depending on the circumstances of different applications, both first order interpolation [13], [14], [15] and second order interpolation [9], [10], [11], [16] could be used.

The originality of this paper is based on the reconstruction of the Projection Methods (PM) using Adams–Bashforth scheme and the Momentum Interpolation Method (MIM) to determine the face-normal velocity combined with central schemes for both convection and diffusion terms. The approximation of the cross diffusion term is improved to handle non-orthogonal, unstructured grids with moderate skewness by introducing an additional correction term. Based on the foregoing method, the second-order accuracy in space and in time is insured in simulating flows at moderate Reynolds numbers.

The structure of the theoretical part is now organized as follows. First, the governing equations including the LES model are presented. The projection method, the time integration, and the finite volume discretization are described in the next section. The numerical results are organized in three sections: accuracy, validation and performance. In the accuracy section, analytical solutions are used to quantify the numerical error of the proposed numerical techniques including the divergence errors, the diffusive and convective terms approximations, and the overall accuracy in time and space. In the validation section, the proposed model is now tested by several well-known benchmarks in comparing the results with experimental solutions, or with other numerical formulations existing in the literature. The test cases include the backward facing step flow, the turbulent channel flow, the lid-driven cavity flow and the flow around a vertical cylinder at a moderate Reynolds number. In the performance section, the convergence of the linear solver is analyzed in terms of iteration numbers and CPU time. Finally, a brief summary and the proposition for future work are also included.