In the present section, the measurements of wind speed, wind direction, horizontal rain and driving rain are used to develop a model which estimates (predicts) driving rain intensities on the façade with given reference wind speeds, wind directions and horizontal rain intensities. Four steps will be made: definition of the model (section 5.3.1), obtaining the parameter values by fitting to the measurement data (section 5.3.2), estimating the driving rain with the measured reference quantities and, finally, comparison of the driving rain estimates with the measured driving rain amounts (section 5.3.3). Two models based on the empirical model described in section 2.3.2 will be formulated.
The first model is a very traditional model, based on [Lacy 1965],
which is similar to the approach implemented in the British
Standard 8401 [BSI 1992]. In section 2.3.2
we mentioned equations 2.29 and 2.35, which can be
put together as:
is defined as the wind velocity component perpendicular to the façade at roof height in the undisturbed approaching flow. In the case of the Main Building of the TUE, equals , measured on the mast on the Auditorium. Note that only negative will be taken into account, i.e. when the wind is blowing towards the façade. See the axis definition in figures 2.4 and 3.4.
The obstruction factor is obtained by division of the total sum of measured driving rain amounts by the total sum of 5-min products of . The total sums relate to total measurement period of 24 months, or to a year.
The second model is basically similar to the first model. It also
comprises horizontal rain intensity and wind speed, but wind direction
is now a distinct variable. Moreover, the values of the parameters
will be obtained by fitting the 5-min data points. The formula is:
The function is defined as for and as for the other values of . The values of range from 0 to 1 inclusive and the values of are restricted from 0 to 1, see figure 5.23a. The polar diagram of the function yields a lobe-like curve (figure 5.23b). The angle determines the angle of the axis of the lobe; the factor determines its width. When eq. 5.4 is fitted to the data, the angle will roughly correspond to the normal of the façade. Note that , and therefore formula 5.3 of model 1 is a special case of formula 5.4 of model 2 with , and . An advantage of model 2 is that follows automatically from the fitting of the data, and the a priori setting of (as in model 1) is not necessary.
Now the form of the function is given, we can explain its meaning in equation 5.4. Both and are meant to account for the position on the façade. On a particular position on the façade, driving rain can only come from a particular range of wind directions. Within this range the driving rain intensity will not be equal for every direction (e.g. the driving rain intensity will go towards zero when the wind blows more and more parallel to the façade). At a particular wind direction (i.e. at ) the driving rain intensity is maximal. In a horizontal plane, one can therefore assume a function like . Therefore, this function is intended to describe differences in driving rain intensities on a horizontal line on the façade. Differences depending on the height on the façade can be described in a similar way, but in that case the function has a three-dimensional form. We will only consider the two-dimensional form for the horizontal plane, because we only have two measurement positions, which are on the same height on the west façade of the Main Building (P4/5 and P6). Our approach was inspired by [Snape and Atkinson 1999] who drew lobe-shaped diagrams indicating the amount of driving rain on a building face as a function of wind direction. Moreover, in section 5.2.6 we observed from our measurements that wind direction is an important factor for the ratio of driving rain intensities on two façade positions.
The difference between models 1 and 2 is not only the applied formulae, but also the manner of obtaining the parameter values. Unlike model 1, the values of the parameters of model 2 are obtained by fitting 5-min data of reference wind speed (), reference wind direction (), reference horizontal rain intensity ( ) and driving rain intensity ( and ).
Tables 5.8 and 5.9 list the values of the parameters of model 1 and 2, respectively. The parameter values are listed for three time `blocks', namely the whole 24-month period (December 1997 to November 1999), the first 12-month period (December 1997 to November 1998) and the second 12-month period (December 1998 to November 1999). For position P6 the time blocks start at 2-3-1998 instead of 1-12-1997. An additional data selection criterion is mm h.
The parameter values of model 1 corresponding to the 24-month block are the average of those corresponding to the first and second 12-month blocks (table 5.8). For model 1, this is obvious because the parameter is calculated from a division of the total sum of driving rain amounts by the total sum of . The ratio between and is approximately 1.5 and resembles the average ratio presented in sections 5.2.5 and 5.2.6. The qualities of the fits, expressed by the coefficient of determination in table 5.8, are not very good.
The parameter values of model 2 in table 5.9 were obtained from fitting (by a least-squares method) the measured data. The table also lists the 95% confidence intervals and the coefficients of determination. The values of the parameters and of model 2 vary much between the 24-month block, the first 12-month block and the second 12-month block. The coefficient of determination for position P4/5 is good. Contrary to the second 12-month period, the coefficient of determination for position P6 is poor for the 24-month block and the first 12-month period. In order to relate the parameters of the two positions and to decrease the arbitrariness of the parameter values, we refitted the data with predefined, fixed values for some of the parameters. The results of this operation are tabulated in table 5.10. Only the parameters and were kept free during the refitting, and the parameters , and had the fixed values of 1.30 (or 1.00 for P6), 1.40 and 0.85, respectively. The value of was derived from the previous fit (table 5.9). We suggest that its value depends on the position on the façade: the more the position is near to the edge, the more it tends to 1. In section 6.4 (results of driving rain calculations with CFD) we will see that towards the edge the catch ratio becomes more and more constant for varying drop diameters. Therefore the driving rain intensity will become proportional to the horizontal rain intensity, and therefore goes to 1. The fixed value of (1.4) was taken from the average in table 5.9. We chose this value for of P6 too, because the second 12-month block for P6 yields a high value of (1.6) in combination with a high value of (0.85) in the previous fit, whereas the average of over the three blocks is about 0.95. The value of varies very much in the previous fits (table 5.9), but it seems not so critical and we fixed it to an average value of 0.85. Altogether, we cannot be very certain about the actual values of the parameters now, because we have measured data of only two positions on one particular building.
In the previous section 5.3.1 we asserted that accounts for the influence of wind direction on driving rain. At position P4/5 the value of is approximately 268. This is only 2 away from the normal of the façade (i.e. 270). Position P4/5 is on the southern half of the west façade, and therefore is (should be) inclined to the south-west. The value of at position P6 is approximately 282. In other words, at P6 is directed to the north-west because position P6 is at the northern half of the west façade.
Figure 5.24 shows how the parameterisation varies with the selected measurement data. The graphs are polar diagrams (as figure 5.23b) and depict 5-min driving rain intensities ( at P4/5 and P6) per wind direction (). The measured data were selected for two horizontal rain intensity intervals and for a certain wind speed interval. In the graphs, the area between the two lobe-shaped solid curves represents all possible driving rain intensities calculated with equation 5.4 of model 2 and with the parameterisation corresponding to the 24-month block for the given horizontal rain intensity intervals and a wind speed interval (table 5.10). The figure shows that most of the data points of the measured driving rain intensity fall within the larger lobe. Many measured data points, however, fall also within the smaller lobe, which should ideally contain no data. Despite of this, the measurement points do not seem to form a circle (i.e. ), and the obtained lobe-like shape ( ) seems to fit better. Moreover, the difference between the two positions P4/5 and P6, i.e. the sizes of the outer lobe comprising the measurement points, is quite well described by the model.
Estimates of 5-min driving rain intensities according to model 1 are calculated with equation 5.3, the parameter values of table 5.8 and the 5-min measurements of wind speed, wind direction and horizontal rain intensity. The estimates according to model 2 are calculated with eq. 5.4, the parameter values of table 5.10 and the same measurement data.
To compare the driving rain amounts estimated from the two models with the actually measured driving rain amounts, three representations will be considered:
The first representation is useful for a general comparison between the estimated and measured results. The histogram of differences and the list of the ten highest 5-min driving rain intensities give more detailed information.
Figure 5.25a shows the monthly estimated and measured driving rain amounts at position P4/5, cumulatively over time. The models obviously tend to overestimate the real driving rain amount. The three parameterisations of model 2 yield more accurate results than the three parameterisations of model 1. At the end of the 24-month period, the total driving rain amount estimated by model 1 deviates 29, 46 and 4%, respectively, from the measured total. The deviation of the results of model 2 is 17, 23 and 6%, respectively. Only one of the parameterisations yields a (slight) underestimation (-6%). This is the parameterisation of the second 12-month period of model 2, in which is lower than the other values of (table 5.10) and in which is better.
The measured and estimated cumulative driving rain amounts at position P6 are depicted in figure 5.25b. Here, the deviation of model 1 is 27, 34 and 17%, respectively. Model 2 yields deviations of 18, 8 and 33%, respectively. So, the estimates of model 1 at P6 tend to deviate generally more from the measurements than those of model 2.
Histograms of differences between measured and estimated 5-min driving rain intensities ( and respectively) are shown in figure 5.26. Only clock periods with non-zero horizontal rain intensities ( ) are taken into account. In other words, the histogram indicates the percentage of clock periods with rain with a particular absolute difference between and . From figure 5.26, we conclude that the estimates of model 2 for the two positions P4/5 and P6 are closer to the measurements than the estimates of model 1. Apart from this, the estimates for position P4/5 are closer to the measurements than the estimates for position P6. Approximately 75% (55%) of the driving rain intensities at P4/5 estimated with model 2 (model 1) differ less 0.02 mm h from the measurements. At position P6, the percentage is about 60% for model 2 (and about 50% for model 1).
Tables 5.11 and 5.12 list the ten highest measurements of 5-min driving rain intensities at P4/5 and P6, respectively, with the corresponding estimates from model 1 and model 2. The rank (``1'' means ``the highest'') and the number of occurrence of an actual value are also indicated. To avoid many ranking levels, the quantities were rounded to a tenth. If the number of occurrence of a particular value is more than one, this means that the same value occurred several times. Table 5.11 shows that the highest driving rain intensity measured at P4/5 was 29.3 mm h on 28-10-1998 at 10h30-10h35. The estimates according to model 2 for the same clock period are 23-29 mm h, which is a good result. The corresponding estimates of model 1 are very much lower, namely 5-8 mm h, and this model did not predict any higher values at all! An other good result of model 2 is the fact that the 5 highest estimates of driving rain intensities (7.2-28.7 mm h) fall within the range of driving rain intensities of the 7 highest measurements (7.2-29.3 mm h).
Table 5.12 shows that the highest 5-min driving rain intensity measured at P6 was 24.9 mm h on 29-10-1998 at 05h15-05h20. None of the corresponding estimates comes close to this value. Unfortunately, a rank of 1 is not present in the table for any of the estimates. An inspection of all results reveals that the overall highest estimated driving rain intensities are 10.5, 11.1, and 9.7 (model 1), 23.1, 22.3 and 25.4 (model 2). The estimates of model 1 are unsatisfactory; the highest estimated values of model 1 are only half of the highest measured value and the second highest estimated values (i.e. rank=2) are even below the 10th highest measured value. The results of model 2 are obviously better.
Altogether, from the comparison between the measured and estimated driving rain intensities it follows that model 2 gives more realistic results than model 1. Model 2 yields realistic data with respect to both cumulative driving rain amounts and 5-min driving rain intensities (see the histograms of driving rain intensity differences and the list of the ten highest intensity values). The estimated cumulative driving rain amounts after 24 months according to model 1 deviate 4-46% from the measurements at P4/5 and 17-34% at P6. The respective figures for model 2 are 6-23% at P4/5 and 8-33% at P6. Model 1 gives similar results with respect to cumulative driving rain amounts as model 2, but performs clearly worse in estimating actual 5-min driving rain intensities. An explanation of the latter observation may be the higher degree of turbulence at the building corner close to P6, and therefore the driving rain intensity may be more sensitive to changes in wind speed, wind direction and horizontal rain intensity than the applied models can cope with. In section 5.2.6 we already observed that the driving rain intensity correlation between the two positions is quite complicated.
© 2002 Fabien J.R. van Mook