UFR 3-30 Best Practice Advice
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 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).
It 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.
Interestingly, a small counterrotating
flow structure with positive averaged wall shear stress was detected within the main recirculation region at the falling edge of the hill between x/h = 0.6 and 0.8. This phenomenon is exclusively visible at this Reynolds number and provides an explanation for the variations of the reattachment length.
Based on the analysis of the Reynolds stresses and the behavior of the flow in the anisotropy-invariant map, it is obvious that the development of the shear layer past the hill crest is delayed and thus shifted downstream at Re = 700 compared to Re = 10,595. If this downstream shift is taken into consideration, similar states of turbulence can be found at both limiting Reynolds numbers with more distinctive extrema observed for the high-Re case. The similarity is especially pronounced for the `splatting' phenomenon of large-scale eddies originating from the shear layer and convected downstream towards the windward slope as described in Fröhlich et al. (2005). In this flow region the spanwise velocity fluctuations show nearly the same peak values and distribution for all Re studied. Nevertheless, in the remaining domain clear trends in the distributions of the mean velocities, Reynolds stresses, anisotropies and the wall shear stresses were found. As mentioned above that led for example to the observation of the tiny recirculation bubble at the hill crest at Re = 10,595.
Furthermore, the length scales appearing in the turbulent flow field at varying Re and the dynamic behavior of the flow were investigated taking even smaller Reynolds number into account. At Re = 100 the flow is found to be steady and two-dimensional. The situation changes completely at for which a three-dimensional instantaneous and chaotic flow field is observed. The corresponding spectrum at this and any higher Reynolds number considered comprises a fully continuous spectrum which extends towards higher frequencies with increasing $Re$. Three main flow structures can be detected already in the lowest-Re case. These are streaky structures close to the upper wall, streamwise vortices close to the concave wall in front of the second hill, and vortical structures induced by the Kelvin-Helmholtz instability of the free shear layer.
Numerical Issues
A very critical issue is the 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".
Computational Domain and Boundary Conditions
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.
Application Uncertainties
Recommendations for further work
It would be highly interesting to extend this study to higher Reynolds numbers.
Contributed by: (*) Christoph Rapp, (**) Michael Breuer, (*) Michael Manhart, (*) Nikolaus Peller — (*)Technische Universität München, (**) Helmut-Schmidt Universität Hamburg
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