UFR 3-30 Best Practice Advice

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

2D Periodic Hill

Underlying Flow Regime 3-30

Best Practice Advice

Best Practice Advice for the UFR

Key Physics

The flow over periodically arranged hills in a channel as proposed by Mellen et al. (2000) is a geometrically simple test case, which offers a number of important features challenging from the point of view of turbulence modeling and simulation. The pressure-induced separation takes place from a continuous curved surface and reattachment is observed at the flat plate. Thus, it includes irregular movement of the separation and reattachment lines in space and time. The shear layer developing past the hill is distinctively visible followed by the well-known Kelvin-Helmholtz instability. Large-scale eddies originating from the shear layer are convected downstream towards the windward slope of the subsequent hill ("splatting effect"), where the flow is strongly accelerated. Hence the spanwise Reynolds stress $\langle w\prime w\prime \rangle$ in the vicinity of the wall is high. That phenomenon was found to be nearly independent of the Reynolds number.

The series of predictions for the broad range of Reynolds numbers considered here shed new light on the flow (Breuer et al. 2009). In particular, the existence of a small recirculation at the foot of the windward face of the hill was confirmed for Re=10,595 but also exists for 200 < Re < 10,595. Besides, a tiny recirculation on the hill crest which has not been discussed before was found which solely exists at the highest Re (Re >= 10,595).

The separation and reattachment lengths vary as a function of the Reynolds number. The separation length past the hill crest was found to continuously decrease with increasing Re until it reaches at minimum at Re = 5600 and slightly increases again for Re = 10,595. The reattachment length decreases with increasing Re (with one exception).

In conclusion, the flow over periodically arranged hills is a very useful benchmark test case since it represents well-defined boundary conditions, can be computed at affordable costs and nevertheless inherits all the features of a flow separating from a curved surface and reattachment.

Numerical Issues

• Accuracy of the discretization

In order to perform DNS or LES predictions for this flow case some minimal requirements concerning spatial and temporal discretization are that both are at least of second-order accuracy. Since a wide range of different length scales have to be resolved, it is obvious that the numerical schemes applied possess low numerical diffusion (and dispersion) in order to resolve the scales and not to dampen them out.

• Grid resolution

A very critical issue is the grid resolution. That implies the near-wall region, the free-shear layers but also the interior flow domain. This topic was already discussed in the section "Test Case Studies / Resolution Issues". For wall-resolved LES the recommendations given by Piomelli and Chasnov (1996) should be followed or outperformed, e.g. $y^{+}<2$, $\Delta x^{+}<50-150$, and $\Delta z^{+}<15-40$. Since the point of separation in the vicinity of the hill crest strongly determines the flow development behind the hill, a sufficient resolution around the hill crest is of major importance. Using a curvilinear grid a grid consisting of about 1 million grid nodes was found to be sufficient to capture the main flow features at Re = 10,595 correctly (provided that the points are reasonably distributed).

• Grid quality

Besides the number of grid points the quality of the grid with respect to smoothness and orthogonality is a very important issue in the context of LES/DNS. In order to capture the separation and reattachment reliably the orthogonality of the grid in the vicinity of the lower wall has to be high, especially close to the hill crest.

Physical Modeling

A detailed analysis was carried out in Fröhlich et al. (2005) including the evaluation of the budgets for all Reynolds stress components, anisotropy measures, and spectra. The emphasis was on elucidating the turbulence mechanisms associated with separation, recirculation and acceleration. The statistical data were supported by investigations on the structural features of the flow. Based on that interesting observations such as the very high level of spanwise velocity fluctuations in the post-reattachment zone on the windward hill side were explained. This phenomenon revealed to be a result of the `splatting' of large-scale eddies originating from the shear layer and convected downstream towards the windward slope. That explains why RANS simulations even when applying second-moment closures can not capture the flow field accurately.

Recommendations for further work

It would be highly interesting to extend this study to higher Reynolds numbers.

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*)Technische Universität München, (**) Helmut-Schmidt Universität Hamburg

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