Procedia Engineering 31 (2012) 16 – 22
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.984
International Conference on Advances in Computational Modeling and Simulation
Large eddy simulation of turbulent flow downstream of a
backward-facing step
Wenquan Wang
a*, Lixiang Zhang
a, Yan Yan
aaFaculty of Civil Engineering and Architecture, Kunming University of Science and Technology, Kunming 600051, China
Abstract
large eddy simulation is used to predict turbulent flow in the separated and reattached flow regions downstream of a backward-facing step.Simulations were carried out at a Reynolds number of 28 000 (based on the step height and the upstream centreline velocity) with a channel expansion ratio of 1.25. Two subgrid-scale models were tested, namely the dynamic eddy-viscosity, and the dynamic Vreman model. Both models showed good overall agreement with available experimental data. It is conjectured that the peak in these fluctuations is caused by an impingement mechanism, in which large eddies, originating in the shear layer, impact the wall just upstream of the mean reattachment location.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology
Keywords: Large eddy simulation; Dynamic global model; Turbulent flow; backward-facing step
I Introduction
The separation and reattachment of turbulent flows are common features in engineering devices. They occur in many components of turbines, diffusers and combustors, as well as in external flows, including flows around buildings and aircraft. Reynolds-averaged turbulence modelling (without significant tuning of the models) has been found to be generally incapable of an accurate prediction of the fluid dynamics and heat transfer in these types of flow, of which the backward-facing step is a well-studied, simplified example.
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In large eddy simulation (LES), the Smagorinsky subgrid-scale (SGS) eddy viscosity model with a constant model coefficient has been widely used, especially due to its robustness and simplicity. However, the constant adopted for the Smagorinsky model coefficient is not universal. To overcome this problem, a
i u filtered velocity p filtered pressure Q kinetic viscosity ij W SGS stress tensor ij S SGS strain-rate tensor
' characteristic length scale of the largest SGS eddies
Cs Smagorinsky model coefficient. CX Vreman model coefficient
m
' characteristic filter width in the mth direction
few approaches have been proposed, e.g., dynamic localization model [2] and Lagrangian dynamic model [3]. The employments of these models in actual LES have proven their applicability to complex flows [4]. Vreman [5] suggested a SGS eddy viscosity model denoted as the Vreman model which guarantees theoretically zero SGS dissipation for various laminar shear flows. This property of the Vreman model is certainly superior to that of the Smagorinsky model. Park et al.[6] recently showed that by a priori and a posteriori tests, the optimal Vreman model coefficient predicting accurate flow statistics is not universal and needs to be dynamically determined according to the flow. These dynamic global models successfully predicted various turbulent flows, such as forced isotropic turbulence, turbulent channel flow, flows over a circular cylinder and a sphere, and flow over a simplified three-dimensional model vehicle [6,7]. Therefore, in the present study, we apply the Lagrangian dynamic model [3] and the dynamic global models based on the global equilibrium[6] to LES of turbulent flows in Francis turbine to further test the capability of dynamic global model.
The present simulation aims to resolve two issues. First, large-eddy simulation are carried out at a Reynolds number high enough to allow for direct comparison with experimental data. Second, the results of these simulations are used to investigate the mechanisms of turbulent flow in the recirculation and reattachment zones.
2 Numerical method
Applying a filter operation to the continuity equation and incompressible Navier-Stokes equations, the filtered equations are obtained as
0 j j u x w w , (1)
where u pi, and Q are the filtered velocity, the filtered pressure and the kinetic viscosity respectively,Wij u ui ju ui jthe SGS stress tensor. A standard Smagorinsky model for SGS stresses is written as: 2 3 ij ij kk t ijS G W W Q , (3)
2 1 2 2 t Cs S Sij ij Q ' . (4) Where Sij is the SGS strain-rate tensor, ' the characteristic length scale of the largest SGS eddies and Csis the Smagorinsky model coefficient.
2 1 2 ij ij h s ij ij h L M C M M ' , (5) whereLij u uki ju ui j, k 2 ij ij ij M ' S S S S ' § · ¨ ¸ © ¹, and the caret ( )k denotes a quantity filtered using the test filter ' 2'ˈand hdenotes the instantaneous averaging over homogeneous direction (denoted as DSSG hereinafter).
On the other hand, the Vreman model [6] determines Qt in the following form: t ij ij II C E X Q D D , (6) j ij i u x D w w , (7) 11 22 22 33 33 11 12 23 31 IIE E E E E E E E E E , (8) 3 2 1 ij m mi mj m E
¦
' D D , (9) where CX is the Vreman model coefficient and 'm is the characteristic filter width in the mth direction. where CX is the Vreman model coefficient and 'm is the characteristic filter width in the mth direction.The dynamic global model based on the Germano identity suggested by Park et al. [6] denoted as 1 2 ij ij V ij ij V L M C M M X (10) Where k ij i j i j L u u u u (11)
k ij ij ij ij ij ij II II M E S E D D D D (12) Where V denotes the instantaneous volume average and /' ' 2is assumed. We call this model as the dynamic Vreman model with the Germano identity (denoted as DVMG hereinafter). Note that this global constant CX is the function of time only. Although the dynamic procedure to get the model coefficient is similar to that of DSM, DVMG requires no homogeneous direction in the flow field for averaging.
3 Computational details
The simulation was carried out in the same configuration as the experiments of Vogel and Eaton [8]. Further data on these experiments are presented in [9, 10]. A schematic layout of the simulation domain is shown in Fig. 1. The channel expansion ratio was 1.25, with a Reynolds number of 28 000 (based on the freestream velocity and step height, h). The experiment was carried out with an inflow condition consisting of two developing boundary layers separated by a relatively undisturbed core. These boundary layers had a measured thickness of /G h =1.1. The total domain size used for the computations was 22h × 5h × 3h, which included an entry length of 2h. A grid containing 192 × 128 × 128 nodes was used, which was stretched in the wall-normal and streamwise directions using hyperbolic tangent functions to cluster grid points at the step edge and in the wall boundary layers. The grid stretching can be observed in Fig. 1.
Due to the need to supply a time-varying turbulent inflow condition, a time-series obtained from a separate periodic channel flow simulation was used at the inflow plane. A forcing method [11] was used to force the periodic channel flow simulation to match the experimental results for the mean and fluctuations of streamwise velocity. A convective boundary condition [12] was used at the exit plane. Statistics were averaged in the homogeneous spanwise direction and over 60 flow-through times.
4 Rresults and discussion
Tab. 1 summarizes the time-averaged mean reattachment lengths obtained for the two grid simulations, the simulation of Akselvoll and Moin [13] and the experimental data. The present results using the dynamic
model containing 192 × 128 × 128 nodes and the results obtained by Akselvoll and Moin [13] are within the estimated experimental error bounds.
Tab. 1. Mean reattachment lengths
/
x h Deviation from experiment[10] Experiment[10] 6.7r0.1
LES [13] 6.74 0.6%
Present, LES, FVM, Grid(192×128×128) 6.78 1.2%
The computed coefficient of friction along the lower wall is compared to the experimental results in Fig.2. The coefficient of friction is defined as 2 2
f w c
C W UU , where Ww is the shear stress at the wall and c
U is the freestream velocity. There are no known differences between the two grid simulations. The results show a similar agreement with the experimental results as the simulations by Akselvoll and Moin [13]: good agreement upstream of x/h = 2 and from reattachment to x/h = 16, but poor agreement in both the recirculation zone and downstream of x/h = 16. The reason for the poor agreement downstream of x/h = 16 is probably the effect of the outflow boundary condition. In the recirculation zone, it is unclear why all the LES simulations predict a larger negative value of Cf. Akselvoll and Moin [13] noted that this could be caused by either the inflow generation method used or by inadequate grid resolution in this region.
Fig. 3 shows the mean streamwise velocity at a number of locations downstream of the step. Both grid simulations show generally good agreement with the experimental results with the major differences occurring downstream of reattachment and in the recirculation region. The DVMG model results show a slightly stronger reversed flow in the recirculation zone comparing with the DSGS model, but both model simulations show generally a little lower of the experimental results. Near reattachment region both subgrid-scale models show smaller velocities than the experiments, which it is attributed to the flow gaining momentum as it passes downstream (due to side-wall boundary-layer growth) in the experiments, however, downstream of reattachment region, the LES results of both model are showed that velocities are good agreement with the experimental results in this region. The primary discrepancy with the experimental results is in the recirculation region (at x/h=3.2 and 5.9), where both subgrid-scale models predict a weaker backflow. 0 5 10 15 20 -0.002 -0.001 0.000 0.001 0.002 <C f > x/h Present 192x128x128 Akselvoll and Moin [13] Adams et al [10]
5 Conclusion
In the present study, Vreman model is as a basic eddy viscosity model, and the dynamic approach based on the global equilibrium between the subgrid-scale dissipation and the viscous dissipation with a global model coefficient are applied to large eddy simulation of turbulent flow in the separated and reattached flow regions downstream of a backward-facing step. Distributions of skin friction coefficient, streamwise velocity profile and mean reattached length are gained, which is good overall agreement with available experimental data and suggested that the SGS model based on the global equilibrium is valid for turbulent flow in complex geometries.
Acknowledgements
The authors thank the National Natural Science Foundation of China (NSFC) [Grant no.11002063 and
50839003], the Natural Science Foundation of Yunnan Province of China (Grant no. 2009ZC035M and 2008GA027) for financial support of this research.
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Fig.3 Compared with mean streamwise velocity profiles between computation and experiment
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