UFR 3-30 Best Practice Advice

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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. There impingement of the eddies on the wall is observed ("splatting effect"), and 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 curvilinear grid in the vicinity of the lower wall has to be high, especially close to the hill crest. Thus the application of appropriate elliptic grid generators delivering high-quality grids is highly recommended.

Computational Domain and Boundary Conditions

• Computational Domain

The dimensions of the domain are: L_x = 9.0 h, L_y = 3.036 h, and L_z = 4.5 h, where h denotes the hill height and x,y,z are the streamwise, wall-normal and spanwise direction, respectively. It consists of a single streamwise periodic segment and thus covers solely one complete hill with an upstream and a downstream region. In the present predictions the domain starts and ends at the hill crest. However, that is not necessarily required. The spanwise extension of the computational domain was recommended for LES or hybrid LES-RANS predictions based on investigations by Fröhlich et al. (2005) and Mellen et al. (2000), who tried other values for L_z and found that value as a good compromise between accuracy and computational effort.

• Boundary Conditions

Since the grid resolution in the vicinity of the wall is sufficient to resolve the viscous sublayer, the no-slip and impermeability boundary condition is used at both walls.

The flow is assumed to be periodic in the streamwise direction and thus periodic boundary conditions are applied. That represents a simple way out of the dilemma of specifying appropriate inflow boundary conditions for LES/DNS. Similar to the turbulent plane channel flow case the non-periodic behavior of the pressure distribution can be accounted for by adding the mean pressure gradient as a source term to the momentum equation in streamwise direction. Two alternatives exist. Either the pressure gradient is fixed which might lead to an unintentional mass flux in the configuration or the mass flux is kept constant which requires an adjustment of the mean pressure gradient in time. Since a fixed Reynolds number can only be guaranteed by a fixed mass flux, the second option should be chosen.

Furthermore, the flow is assumed to be homogeneous in spanwise direction and periodic boundary conditions are applied, too. For that purpose the use of an adequate domain size in the spanwise direction is of major importance in order to obtain reliable and physically reasonable results. To assure this criterion the two-point correlations in the spanwise direction have to vanish in the half-width of the domain size chosen. Based on the investigations by Fröhlich et al. (2005) and Mellen et al. (2000) a spanwise extension of the computational domain of L_z = 4.5 h was used in all computations presented. It represents a well-balanced compromise between spanwise extension and spanwise resolution. A detailed discussion on the implications can be found in Fröhlich et al. (2005).

Physical Modeling

A detailed analysis of physical issues 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 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.

The entire recirculation region of the hill flow case is dominated by large-scale energetic eddies with strong deformations and dynamics, which are ill-described by one-point turbulence models that assume a high degree of "locality" of turbulence (Fröhlich et al. 2005). Furthermore, the reattachment length strongly depends on the location of the separation, which is appears at a curved surface and thus demands greater care in the resolution and modeling of the near-wall region than for flows separating from sharp edges.

Since in LES the large energy carrying eddies are resolved by the numerical method, this methodology is well suited for the flow phenomena under investigation. At Re = 10,595 the Reynolds number is still moderate so that the largest part of the energy spectrum can be resolved easily. As a consequence, the subgrid-scale modeling in LES is of minor importance for this test case, at least at Re = 10,595. Thus, the classical Smagorinsky model with van Driest damping near solid walls as well as the dynamic model based on a Smagorinsky base model were both found to deliver reliable and nearly identical results.

Application Uncertainties

Application uncertainties are given by the following issues:

• Periodic boundary conditions in streamwise direction:

In the experimental setup periodicity is achieved by an array of 10 hills in streamwise direction. In order to keep the computational effort small, a single streamwise segment is applied combined with the assumption of periodicity of the flow field. The implications were carefully investigated by Temmerman (2004) and a detailed discussion about this issue can be found in Fröhlich et al. (2005). As a conclusion, the streamwise extension chosen was confirmed to be a good compromise. Nevertheless, it induces a low level of uncertainty.

• Periodic boundary conditions in spanwise direction:

The same applies for the other direction in which the flow is assumed to be periodic. Again the experimental set-up uses a large spanwise extent of the channel and thus a certain level of uncertainty remains. However, based on the investigations of Fröhlich et al. (2005) the implications are expected to be minor.

Furthermore, as remarked in Fröhlich et al. (2005), if used as a test case, the issue of fully adequate (optimal) spanwise extent only affects comparisons with experimental measurements and solutions based on the assumption of complete spanwise homogeneity, as is the case with two-dimensional RANS computations. If, in contrast, LES or DNS computations are undertaken with the same spanwise periodicity imposed, the comparison of the associated results is not affected.

• Time averaging:

To reduce statistical errors due to insufficient sampling to a reasonable minimum, the flow field was averaged in spanwise direction and in time over a long period which is also given in Table 1. Partially the averaging period covers a time interval of about 140 flow-through times. Nevertheless, the results are not completely free of uncertainties arising from the averaging process.

Recommendations for further work

• It would be highly interesting to extend this study to higher Reynolds numbers in the order of 100,000 or 1,000,000.

• Furthermore, the present test case is also well-suited for testing new hybrid LES-RANS approaches. Partially, that was already done, see e.g. Breuer et al. (2008) and Jaffrezic and Breuer (2008).

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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