Numerical simulation of flows over 2D and 3D backward-facing inclined steps

Highlights

• 2D and 3D simulations of backward facing step flows.

• Step perpendicular or inclined, top wall parallel or inclined.

• Used finite element and finite volume method.

• Used eddy viscosity and EARSM turbulence models.

• All results compared with experimental data.

Abstract

The work deals with the numerical solution of incompressible turbulent flow in a channel with a backward-facing step having various inclination angles. Also, the inclination of upper wall is considered. The mathematical model is based on the Reynolds averaged Navier–Stokes equations. The governing equations are closed by the explicit algebraic Reynolds stress (EARSM) model according to Wallin and Johansson or by linear eddy viscosity models (SST, TNT k–ω). The numerical solution is carried out by the implicit finite-volume method based on the artificial compressibility and by the finite-element method amd both approaches compared. The numerical simulations use as reference the experimental data by Makiola and Driver and Seegmiller in large aspect ratio channels. In these cases, the results are obtained by 2D and 3D simulations. Further narrow channel PIV experimental data are used as reference for 3D simulations.

Introduction

In this work, incompressible flow over different geometrical configurations of the backward facing step (BFS) are simulated numerically. The step is perpendicular or inclined and the top wall is considered not only parallel to the bottom wall but also inclined, superimposing a small constant pressure gradient to the expansion. The features of the flow then range from sudden expansion to one-sided diffuser.

The mathematical model is based on the Reynolds averaged Navier–Stokes equations for incompressible fluid. The turbulence models use the eddy viscosity approximation or an explicit algebraic Reynolds stress (EARSM) constitutive relation for the Reynolds stress tensor. For the prediction of turbulent scales, all of the models employ very similar k–ω systems of equations which enables, to some degree, the influence of constitutive relation to be isolated. The mathematical models are solved by in-house codes implementing either the finite-element method (FEM) or finite-volume method (FVM). The comparison of both methods with the same turbulence model then provides a qualitative estimate of the numerical diffusion of the numerical schemes used. In general the sensitivity of BFS flow to numerical diffusion can be large. Having more numerical approaches at hand, the influence of the numerical approximation can be distinguished from the influence of the turbulence model. Some grid refinement study is also reported.

The FEM and FVM methods are very different, among other aspects, in the way boundary conditions are imposed. The outlet boundary condition routinely used by FEM in simulations of internal flows within domains with one outlet is the so called “do-nothing” boundary condition. In a FEM weak formulation, it is satisfied “implicitly”. Formally this condition can be transferred to the finite-volume approach, which showed considerable advantage in simulations of BFS flow in the diverging channel.

The measurement of flow over a perpendicular backward facing step over a wide range of Reynolds numbers from laminar to fully turbulent regimes is given by Armaly et al. (1983). The step was relatively confined having the expansion ratio ER ≈ 2 (height of the channel behind the step divided by the height of the channel in front of the step). Driver and Seegmiller presented measurements of less confined flow over a perpendicular step with expansion ratio ER = 9/8 = 1.125 in their detailed study (Driver and Seegmiller, 1985). They also superimposed pressure gradients to the flow by deflecting the upper wall of the channel. In this work, the simulations of cases with adverse pressure gradient are considered. Another generalization of the geometry was considered by Makiola (1992) who measured flow over steps with an inclined wall. For the smallest inclination angle, 10°, the geometry could better be labeled as a one-sided diffuser than a step.

The above cases all use very wide channels so that the flow near the center-plane is considered 2D. The authors in earlier works (Kozel et al., 2006, Kozel et al., 2005, Kozel et al., 2004) indeed used the data for 2D simulations but arrived at the conclusion that eddy viscosity turbulence models were not capable of predicting consistent results going from 2D to 3D – e.g. the separation behind the step in the center-plane of a 3D channel is different than in the analogous 2D problem. A typical 3D effect that the eddy viscosity models are not capable of predicting are secondary corner vortices in rectangular channels. On the other hand, the EARSM turbulence model predicts the corner vortices very well and also 2D and 3D results of BFS flow are consistent. Nevertheless it is shown below that the EARSM model has some limitations concerning a too slow response to the expansion. This probably has more to do with turbulent scales estimation (k–ω system) than with the constitutive relation for the Reynolds stress itself.

In order to include also the influence of side walls, the PIV measurement of flow over the perpendicular step in a narrow channel by Uruba and Jonáš (2012) are used as the basis of one of present 3D simulations.

The flow over a circular ramp in a channel, which can be considered as another kind of one-sided diffuser, has previously been simulated by the authors in Louda et al. (2012).