Reynolds averaged simulation of unsteady separated flow

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Abstract

The accuracy of Reynolds averaged Navier–Stokes (RANS) turbulence models in predicting complex flows with separation is examined. The unsteady flow around square cylinder and over a wall-mounted cube are simulated and compared with experimental data. For the cube case, none of the previously published numerical predictions obtained by steady-state RANS produced a good match with experimental data. However, evidence exists that coherent vortex shedding occurs in this flow. Its presence demands unsteady RANS computation because the flow is not statistically stationary. The present study demonstrates that unsteady RANS does indeed predict periodic shedding, and leads to much better concurrence with available experimental data than has been achieved with steady computation.

Introduction

Recent advances in computing power have spurred interest in simulating time dependent, Reynolds averaged flows for problems ranging from noise prediction to fluid/structure interaction. When the flow is not statistically stationary, Reynolds averaging is not synonymous with time-averaging. Hence, a proper Reynolds averaged Navier–Stokes (RANS) simulation must be time dependent. This increases computational expense substantially, but it is demanded by a proper application of RANS methodology.

When the flow is periodic in time, an unsteady RANS simulation must be averaged over one period to compare to time-averaged data. The computational cost and the resolution requirements are mainly related to the vortical flow structures shed by the geometry and wall layers. Despite the time dependence, and large vortical structures, unsteady RANS is not a simulation of the turbulence, only of its statistics. Turbulence modeling plays a crucial role in establishing and correctly predicting the complex behavior of such flows. Indeed, the remarkable accomplishment of RANS models is their ability to directly predict the underlying statistics of a highly irregular, turbulent flow.

Unsteady RANS should not be confused with large Eddy simulation (LES): indeed, comparisons to LES will be made herein. The latter employs spatial, not ensemble, averaging. Averaging is over a scale sufficient to filter small eddies, not resolved by the particular grid being used, but the stochastic nature of turbulent solutions to the full Navier–Stokes is retained. Hence, Reynolds averaged statistics must be evaluated by accumulating a large enough sample size. In a temporally periodic flow, the samples must be at a fixed phase in order to obtain statistics of the turbulent portion of the velocity.

The mesh and time-step requirements of RANS and LES are quite different. LES resolves the eddies of the turbulence itself, whereas unsteady RANS models the turbulence and resolves only unsteady mean-flow structures. Consequently, LES typically requires much higher spatial and temporal resolution, and is more costly. LES also requires very long integration time to build an ensemble averaged solution. On the other hand, a few shedding periods are usually enough for an unsteady RANS computation to converge to its limit cycle.

A question arises over whether RANS can predict flows with gross unsteadiness. If the unsteadiness is deterministic, then unsteady RANS is suitable; for instance, if a frequency spectrum shows a spike at a shedding frequency, amidst a broadband background of turbulence, then unsteady RANS is warranted––indeed demanded.

In the present work, flow around a square cylinder and over a surface mounted cube are computed. These flows exhibit characteristics common to all flows past bluff obstacles, including separation and large scale unsteadiness. The square cylinder is a much-studied case in which a coherent vortex street forms in the wake. It is included here for comparison with the flow over a surface mounted cube.

A surface mounted, square cylinder would not produce a vortex street because the wall acts like a splitter plate to destroy the anti-symmetry. But evidence from experiment and LES is that a coherent component exists to the unsteady flow round a cube. Are the poor agreements between RANS and data in previous studies of this flow due to erroneously computing it as statistically stationary? Will unsteady RANS produce a periodic solution? If the answer to the latter is yes, then there is likely to be a better agreement with experimental data.

For quantitative validation, reliable experimental databases are available in the literature for both geometries (Hussein and Martinuzzi, 1996; Lyn et al., 1995). Vortical structures in the surface mounted cube flow are sketched in Fig. 1, taken from Hussein and Martinuzzi (1996): a strong horseshoe vortex and an arch-shaped vortex in the near wake were inferred from analysis of oil-flow patterns on the wind tunnel floor. This qualitative view suggests flow features that should be captured in a simulation.

LES of these flows have been carried out with considerable success (Shah, 1998). These test cases were selected for the ‘Workshop on Large Eddy Simulation of Flows Past Bluff Bodies’ (Rodi et al., 1997). In the workshop, several LES calculations were compared to data, showing good agreement. Steady-state RANS computations with different variations of the kϵ model were also presented: in general, unsatisfactory agreement with experimental data was obtained (see Rodi, 1997, Rodi, 2002). This paper will show that the poor agreement was due to the assumption of statistical stationarity, not to the use of Reynolds averaged simulation.

Section snippets

Numerical model

Two- and three-dimensional steady and unsteady RANS simulations were carried out using the v2f turbulence model (Durbin, 1995); an analysis of the effect of turbulence modeling is out of the scope of the present paper, but a very comprehensive study can be found in Breuer et al. (1996), Lakehal and Rodi (1997) and Lakehal and Thiele (2001).

A commercial computational fluid dynamics (CFD) code, FLUENT 5.5, was used to solve the equations of motion. The v2f model was implemented via user defined

Results and discussion

The flow around a square cylinder at the Reynolds number investigated presents coherent vortex shedding, with a periodically oscillating wake. The length of the recirculation region and the surface loads are of primary interest. A summary of the present simulations and several experimental data are reported in Table 1. As expected, the predictions obtained under steady-state assumptions are incorrect, with an extremely elongated recirculation bubble. Time accurate results, on the other hand,

Discussion

This paper has shown that unsteady RANS provides good quantitative and qualitative agreement with experimental data when the flow is not statistically stationary. The present simulation of the three-dimensional vortex shedding behind a surface mounted cube is one of the most ambitions computations of this ilk, to date. It serves to highlight the need to apply RANS models in a manner that is consistent with the definition of the Reynolds (or ensemble) average, notwithstanding the increased

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    The accuracy of LES and the limitations of steady RANS have been reported by Blocken (2015b, 2014), Tominaga et al. (2008a), and Xie and Castro (2009), for instance. One of the steady RANS limitations is the difficulty to reproduce the periodic releases of vortexes (vortex shedding) due to the presence of several bluff bodies, which generates periodic low-frequency movements (Iaccarino et al., 2003; Mannini et al., 2010). Given that and the high computational costs of LES, the unsteady RANS (URANS) is an alternative for solving the RANS equations affording a transient solution at a low computational cost.

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