Comparison of maximum speed-up

Maximum speed-up can be defined in two ways. It is either the maximum value of the fractional speed-up ratio (∆Smax) or the value of the fractional speed-up ratio at the height where the increase in velocity attains its maximum (∆Umax). Both types will be reported here, but estimates based on linearized theories are only available for the latter. The heights where the maximum values occur are also reported when available. All the maximum values listed are approximately at the crest.

Maximum values of the fractional speed-up ratio found in the experiments and those calculated by the LSD approach and the ESDU method are listed in Table 8.6. All the LSD estimates discussed in this context are the results calculated as if φ ≥ 1, which is the correct choice for the steep slopes in Case I-IV. The heights where these maximum values occur are the lowest points in the calculations, and the experimental data agree quite well with this. The maximum values found with the LSD approach are in all four cases approximately equal to the maximum possible value with this approach (i.e. 1.3). This is due to the very steep hills. In contrast to the LSD approach, the maximum values for the fractional speed-up ratio found with the ESDU method are quite different for the individual cases. The ESDU method gives lower values than the experimental data in Case I-III, but higher in Case IV.

Chapter 8. Results and discussion

The agreement between ∆Smax found by the ESDU method and the experimental data is best in Case IV, and this is probably due to the fact that Case IV is exactly an embankment like the ones the ESDU method is based on. Even if the steepest slope in Case III and Case IV are approximately equal, the horizontal length scale in the rounded hill in Case III is much longer than for the straight hill in Case IV. This affects the ESDU result considerably.

Case H/L H/LH/2 Experiment LSD ESDU

∆Smax lmax/H ∆Smax lmax/H ∆Smax lmax/H

I 0.53 1.05 1.02 0.020 1.27 0.009 0.86 0.009

II 0.86 1.72 1.47 0.011 1.25 0.009 0.87 0.009

III 0.70 1.40 1.35 0.012 1.24 0.012 0.74 0.012

IV 0.98 1.97 1.43 0.012 1.28 0.012 1.32 0.012

Table 8.6: Maximum values of the estimated and observed fractional speed-up ratios, and the height where these occurs, in Case I-IV. All values reported are from a position approximately at the crest.

Table 8.7 lists the values of the fractional speed-up ratio at the heights where the increase in velocity is largest. Results are reported for experimental data, the LSD approach, the ESDU method, the linearized theory of Jackson and Hunt and another simple estimate (Equation 4.5). The percentage deviation of these values from the experimental values are given in Table 8.8. The heights where ∆S∆Umax occur are somewhat higher than the height of ∆Smax for all the estimation methods. The heights of ∆S∆Umax in the experimental results are lower than all the estimated heights in all four cases.

The recommended maximum value in the simple method given by Equation 4.5 is ∆S∆Umax = 1.25. This is valid for axisymmetric hills, and the maximum for escarpments should be even lower than this, since larger separation bubbles are more likely to form upstream of an escarpment. The results given by Equation 4.5 are said to be generally accurate within ±15 %, and to have even better agreement on low hills that fit the assumption of linear theory. If the upper recommended value of 1.25 is ignored, the results in Case II-IV are all within this uncertainty range, while the result in Case I is slightly higher than the upper limit. This is quite good for such a simple method considering the very steep slopes in question. It also indicates that the limitation of ∆S∆Umax < 1.25 is not necessarily correct since the estimated and measured results are higher than this and in good agreement.

The deviations from the experimental values for the fractional speed-up ratio for the LSD approach are −18.9 % (Case IV) to 15.7 % (Case I), for the ESDU method −50 % (Case III) to −14.7 % (Case IV) and for the linearized theory 98 % (Case I) to 178.3 % (Case IV). The poorest agreement between the ESDU method and the experimental data occur close to the ground at the crest, which is also the region with highest speed-up values. This is the reason why the ESDU method seems to give less good results based on Table 8.8. When it comes to these maximum values the LSD approach and the simple proportional relation with the slope given in

8.8. Comparison and validation

Equation 4.5 give the results which fit best with the experimental data. The ESDU method fits less good with these data in the near ground region at the crest. It is clear that the Jackson and Hunt method with linearized theory is not applicable for the steep slopes under consideration. The maximum speed-up values are extremely high, with values of ∆S∆Umax up to 4.

Case I II III IV H/L 0.53 0.86 0.70 0.98 ∆S∆Umax Experiment 1.02 1.43 1.28 1.43 LSD 1.18 1.17 1.12 1.16 ESDU 0.78 0.75 0.64 1.22 0.8φ 0.83 1.38 1.12 1.58 J&H 2.02 3.41 2.80 3.98 lmax/H Experiment 0.020 0.014 0.020 0.012 LSD 0.069 0.069 0.096 0.096 ESDU 0.043 0.043 0.060 0.020 J&H 0.039 0.025 0.031 0.024

Table 8.7: Values of the fractional speed-up ratio ∆S and the corresponding height lmax/H where ∆U attains its maximum. All values reported are from a position approximately at the crest. Values from the measured data, LSD approach, ESDU method, Equation 4.5 (0.8φ) and linearized theory by Jackson and Hunt (J&H) are reported. I II III IV H/L 0.53 0.86 0.70 0.98 LSD 15.7 -18.2 -12.5 -18.9 ESDU -23.5 -47.6 -50.0 -14.7 0.8φ -18.6 -3.5 -12.5 10.5 J&H 98.0 138.5 118.8 178.3

Table 8.8: Percentage deviation between the experimental values and the estimates for ∆S∆Umax given in Table 8.7.

The results given by the LSD approach were studied thoroughly in Section 8.8.1, and it was seen that this approach for φ > 1 results in larger speed-up values than found in the experiments for all levels and positions near the crest. The only exception is very close to the ground at the crest. This is in accordance with a conclusion by Miller and Davenport (1998) who state that the general trend for methods like the LSD approach is to over-predict the speed-up. All slopes studied by Miller and Davenport were more gentle than the slopes studied here, and the ridges studied have other ridges upstream, as the purpose of the study was to see the effect of complex terrain. The values listed by Miller and Davenport are the maximum values of the fractional speed-up ratio, like the ones in Table 8.6.

Chapter 8. Results and discussion

Studying only the maximum speed-up can give an incorrect impression of the flow pattern around terrain features. It is clearly seen by the comparison of the LSD approach and the ESDU method with the experimental data (Section 8.8.1 and 8.8.2) that the match of maximum speed-up is not typically representative for the accordance in the rest of the flow. The results in the current section present how much the different methods predict the velocity to increase at the crest, and at which height this typically occurs. This should not be considered as a general measure of how good the experimental data and the different predictions fit.