Revision as of 10:37, 31 March 2009

Boundary layer flow and dispersion over isolated hills and valleys

Application Challenge 5-05 © copyright ERCOFTAC 2004

Best Practice Advice for the AC

1. Key Fluid Mechanics

Quoting from AC5-05: 'Key aspects of the flow physics in this problem are those normally associated with flow of a rough-wall turbulent boundary layer over a surface-mounted obstacle immersed within it and the downwind turbulent dispersion of scalar sources within the flow. Thus, important features include the adverse pressure gradient region upstream of the hill and velocity acceleration on the windward side and deceleration on the downwind side; if the hill slope is large enough, the latter may be strong enough to lead to flow separation.' These features, along with the influence of the rough surface typical of atmospheric flows, all exert an influence, to a greater or lesser extent, on DOAPs like the maximum flow speed-up at the summit of the hill, or the extent of any separated region in the lee.

Furthermore, in terms of the dispersion from sources within the flow, 'Major features of the dispersion behavior include the possibility of entrainment into the separation region for certain source locations, the large variations of concentration that can occur within the separation region (in contrast to the common assumption that such a region will be well-mixed), the strong directional variability in effective diffusivities and the large increases in maximum ground level concentration caused by the hill'. So all these features effect DOAPS like, for example, the 'terrain amplification factor' - the factor by which the maximum surface concentration is amplified by the presence of the hill.

The Best Practice Advice contained in this document has been informed by the associated UFRs:

3-13: Flow and Dispersion over Isolated Hills and Valleys

3-17: Atmospheric Boundary Layer over Rough Wall (local scale)

2. Application Uncertainities

Uncertainties in the experiments which might make adequate comparison with CFD uncertain are in this case relatively minor. The upstream boundary layer was properly characterised, measurement errors were acceptably small and care was taken to ensure neutrally buoyant sources with negligible momentum. However, to ensure appropriate surface parameters the physical roughness used was rather larger (when scaled up) than typical field situations. This, arguably, makes parameters like the height above the summit at which the maximum velocity occurs rather difficult to scale up to the field situation and, indeed, to compare with CFD. Proper specification of rough surfaces in the latter is problematic, as explained in D30-AC5-05. The actual value of the maximum speed-up (a typical DOAP) will also be sensitive to the way the surface condition is applied in the CFD. Similarly, the precise location of separation and attachment points in the lee of the hill will be sensitive to the wall condition applied in the CFD, but the experiments identified these locations quite accurately.

In the context of a DAOP like maximum surface concentration, and its location, there is little experimental uncertainty, but naturally these will depend on the nature of the upstream boundary layer, the shape of the hill and the location of the source. The experiments identified the influence of the last two of these, but not the first.

None of these difficulties prevents the presentation of clear BPA for CFD in these cases.

Computational Domain and Boundary Conditions

Three-dimensional effects, particularly in cases where a large separated region exists, MAY have existed in the experiments, despite the relatively high aspect ratio used. Even for infinite aspect ratio, such effects may exist as a result of the nature of the flow, rather than boundary conditions. This is an inevitable issue which should be considered in any comparison between a nominally 2D experiment and a genuinely 2D computation. In the present case, however, any 3D effects were believed to be small in terms of their influence on all properties on the wind tunnel centre-line, which was where all measurements were made.

For comparison with AC5-05 experiments, computations can easily be done with the upper boundary specified at some height along which the (constant U) is known, provided some mass flux across this boundary is allowed, so that boundary layer growth can occur. Requirements for boundary locations which, assuming specified inlet conditions and zero-gradient outlet conditions, should be adequate are as follows:

1. Upper boundary. At least 10 hill heights, if this is outside the boundary layer and provided zero axial pressure gradient can be enforced along it.

2. Surface. This should be modelled as rough, but appropriate ways to do this are not obvious since, over much of the flow, standard log-law conditions do not apply. One should at the very least avoid using a formulation which makes the effective viscosity (in an eddy viscosity formulation) zero at separation and attachment points. More specific details are discussed in the AC5-05 document.

3. Upstream, the domain boundary should be sufficiently far from the hill to ensure that the latter's influence on the pressure field is negligible at that location. This normally means at least 20 hill heights, but will clearly depend on the axial aspect ratio (axial length/height) of the hill.

4. Location of the downstream boundary will depend to some extent on what is being required from the computation. In any case, it should be well beyond any separation region in the wake - at least 20 hill heights, say (with the same caveat as above). If downwind surface concentrations are of interest, considerably longer distances may be required.

The experiments showed that in one case the flow separation and subsequent attachment on the lee-side of the hill was intermittent, but non-existent in the mean. This could be taken as suggesting that there was a low-frequency, quasi-periodic instability but, in any case, would make the application of steady-flow CFD methods questionable. Amplification factors (at concentration field DOAP) obtained by CFD were certainly in much worse agreement with the laboratory experiment than in other cases.

Discretisation and Grid Resolution

1. Second-order differencing on the convective terms of all equations is crucial. (Formally 2nd order on a uniform grid, is what is meant here; such a scheme will usually become 1st order on a non-uniform grid, but provided grid expansion ratios are kept low - below 1.1, say - the resulting errors will be small). For practical cases this will usually require some kind of flux-limited mixture of central and 2nd-order up-winding. Regular 1st order up-winding should be avoided.

2. The best way to arrange the grid is to ensure that minimum mesh dimensions occur at the hill summit, with expansion from that point upstream, downstream and upward to the boundaries. One would normally use a boundary conforming mesh. Resolution tests (described in D30-AC5-05) indicate that with a smallest mesh of about 0.1H x 0.01H at the hill summit (in x,z directions, respectively, with x axis and z vertical) and total node numbers of at least 100 x 60 grid-independence results can be approximately achieved, provided a 2nd order scheme as discussed above is used. These numbers corresponded in that case to a domain running from -40H to 40H axially and about 14H vertically. Larger domains would obviously require more nodes, to keep expansion ratios suitably small.

If concentration data is required, care in setting conditions at the source is needed, since concentration gradients will always be very large there, even if gradients in flow variables are not. This may require a separate grid for the dispersion computations (with different grids for each source location) which is straightforward to implement if, as in AC5-05, the source is neutrally buoyant. Even better, an appropriate analytic solution to the convection-diffusion equation for a point source can be used as a 'patch' within the grid, to avoid having to create a very fine grid around the source. This was the approach used in AC5-05. What will usually NOT be adequate, is simple enforcement of a point source at the required location within the grid used for the flow solver, but 'extruded' spanwise for the third dimension. (Point sources, of course, require a 3D grid for solution of the resulting scalar field). This would normally lead to significant uncertainty in the calculation of DOAPs like terrain amplification factors, since the numerical inaccuracies around the source will depend significantly on the nature of the flow there.

3. In these flows, where adequate computation of the turbulence is a necessary pre-requisite to obtaining adequate mean-flow fields and/or dispersion fields, it is not sensible to relax the above grid requirements if only the latter properties are required.

Physical Modelling

The evidence is convincing that standard k-ε models are inadequate for these flows. At the very least, a modification to correct the response of the model to strong streamwise strains is required. Curvature corrections should ideally also be added. Provided such modifications are made, the evidence does suggest, however, that an eddy-viscosity-based k-ε model can yield reasonable agreement with experiment for the important flow variables and DOAPs like the summit speed-up. On the other hand, even rough agreement in local values of scalar concentration cannot normally be expected (because of the isotropic diffusion embodied within standard k-ε type models for the scalar equation), but normalised DOAPs like the terrain amplification factor can be obtained, perhaps to within 25-30%. This is usually quite adequate for air-quality regulatory purposes.

An important caveat to the above remarks is required for cases, as discussed earlier, where it is believed that intermittent separation/attachment may occur. Steady flow computations in such cases cannot be expected to yield satisfactory results and further work is necessary to discover the extent to which, say, unsteady RANS (or, better, LES) can handle such cases.

Recommendations for Future Work

For genuinely steady flow cases, there seems little merit in undertaking further computations with more sophisticated turbulence models, unless local values of scalar concentration (rather than normalised DOAPs like terrain amplification factor) are required. However, in practical cases the flows are almost always three-dimensional (like the hills!) and the extent to which modified k-ε can cope with such cases should perhaps be explored. There is some extant laboratory data which would allow careful comparison with CFD.

The whole issue of possible unsteadiness in these flows requires further work.

Contributors: Ian Castro - University of Southampton

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