Analysis of Three-Dimensional Confined Laminar Flows with Multiple Flow Separation Regions

Abstract

This paper presents a three-dimensional analysis of the confined viscous flows encountered in many engineering systems. The present solutions are obtained with a numerical method developed by the authors for the accurate solution of the Navier-Stokes equations in confined laminar flows, which is based on a finite difference formulation and uses artificial compressibility. The method is successfully validated by comparison with the theoretical solutions for laminar flows in rectangular ducts and with the experimental results for the flows with multiple separation regions in a duct with a downstream-facing step. The present 3-D solutions confirm the effect of the lateral walls in the experimental configuration, which was suggested by several authors but never confirmed until now by theoretical or numerical 3-D solutions. I. Introduction he steady and unsteady fluid-structure interaction problems are present in numerous engineering fields such as in thermo-fluid systems, pumps, nuclear reactors, gas and hydraulic turbines and aeronautics. This explains why the analysis of the steady and unsteady confined flows, required in the study of fluid-structure interaction and flowinduced vibration problems, received a topical interest worldwide [1-8]. More recently, with the development of Micro-Electro-Mechanical Systems (MEMS), a research interest has been developed for steady and unsteady confined fluid flows at low Reynolds numbers (between 200 and 1200), with various engineering applications related to the cooling flows in miniature electronic devices [9] or to the aluminum continuous casting operation to a near-net shape [10]. The leading author and his graduate students contributed substantially to the study of unsteady confined flows for fluid-structure interaction problems by theoretical methods of solution and experimental studies, such as Mateescu et al. [2-3], and by numerical methods based on finite difference formulations [4-5] and on hybrid spectral formulations, such as [6]. Recent studies included the analysis of steady and unsteady flows at low Reynolds numbers in confined configurations and past airfoils [7-8]. In addition to the obvious engineering interest, this study is also motivated by an academic interest related to the steady laminar flows past downstream-facing steps. The two-dimensional numerical solutions for the flow separation and reattachment locations in this confined flow problem, such as those obtained by Gartling [12] and Mateescu & Venditti [4], were found to be not in good agreement with the experimental results obtained by Armaly et al. [11] and by Lee & Mateescu [5], especially for larger Reynolds numbers (between 700 and 1200). It was considered in [4, 12] that this disagreement between the 2-D numerical solutions and experimental results is due to the three-dimensional effect of the lateral walls in the experimental configuration, as opposed to the rigorous twodimensional numerical solution. However, up to now this explanation has not been scientifically confirmed by theoretical or numerical 3-D solutions for this problem. The aim of this paper is to present a three-dimensional analysis of the confined laminar flows, which is able to obtain accurate and efficient solutions for the flows with multiple separation regions. These solutions are obtained with a numerical method based on a finite difference formulation using artificial compressibility, which is second order accurate. This method is applied to obtain solutions for the confined laminar flows with separation regions past a downstream-facing step, with the aim to confirm the explanation mentioned above regarding the lack of good agreement between the 2-D solution and experiments. This method was successfully validated by comparison with experimental results for this flow case and with the theoretical solutions for laminar flows in rectangular ducts.