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# Anthromes displaying evidence of weekly cycles in active fire data cover 70% of the global land surface

## Abstract

Across the globe, human activities have been gaining importance relatively to climate and ecology as the main controls on fire regimes and consequently human activity became an important driver of the frequency, extent and intensity of vegetation burning worldwide. Our objective in the present study is to look for weekly cycles in vegetation fire activity at global scale as evidence of human agency, relying on the original MODIS active fire detections at 1 km spatial resolution (MCD14ML) and using novel statistical methodologies to detect significant periodicities in time series data. We tested the hypotheses that global fire activity displays weekly cycles and that the weekday with the fewest fires is Sunday. We also assessed the effect of land use and land cover on weekly fire cycle significance by testing those hypotheses separately for the Villages, Settlements, Croplands, Rangelands, Seminatural, and Wildlands anthromes. Based on a preliminary data analysis of the daily global active fire counts periodogram, we developed an harmonic regression model for the mean function of daily fire activity and assumed a linear model for the de-seasonalized time series. For inference purposes, we used a Bayesian methodology and constructed a simultaneous 95% credible band for the mean function. The hypothesis of a Sunday weekly minimum was directly investigated by computing the probabilities that the mean functions of every weekday (Monday to Saturday) are inside the credible band corresponding to mean Sunday fire activity. Since these probabilities are small, there is statistical evidence of significantly fewer fires on Sunday than on the other days of the week. Cropland, rangeland, and seminatural anthromes, which cover 70% of the global land area and account for 94% of the active fires analysed, display weekly cycles in fire activity. Due to lower land management intensity and less strict control over fire size and duration, weekly cycles in Rangelands and Seminatural anthromes, which jointly account for 53.46% of all fires, although statistically significant are weaker than those detected in Croplands.

## Introduction

Across the globe, human activities have been gaining importance relatively to climate and ecology as the main controls on fire regimes, and during the last few decades anthropogenic vegetation burning expanded its impact on land cover, ecosystem dynamics, and atmospheric greenhouse gas emissions from regional to global scales1,2,3,4. Human activity became a significant driver of the frequency, extent and intensity of vegetation burning worldwide, and the timing of ignitions is strongly dependent on human agency5,6. Fire temporal patterns are a function of climate dynamics and human activity, with periodic cycles found at annual scale in seasonally dry areas, and at the daily scale in response to underlying cycles in meteorological variables and land management activities6,7. Fire regimes have been modified to manage croplands, rangelands, forests, and wildlife, to protect infrastructure and settled areas8, as a form of political protest9, and to wage war10. Predictability of spatial and temporal fire patterns increases with the length of human presence and the degree of human dominance over the landscape11. Vegetation burning has been analyzed at time scales ranging from daily7 to annual12, centennial13, and millennial14,15. It is a large source of greenhouse gases and aerosols and constitutes a major factor controlling variability of atmospheric composition at the inter-annual scale16.

Approaches to isolate the impact of anthropogenic aerosol on clouds from natural cloud variability were reviewed by17 and included the detection of weekly cycles, because seven-day periodicities are highly unlikely due to natural forcing, and thus provide strong evidence of anthropogenic perturbations. Weekly cycles over broad geographical areas cannot be explained by the reduction in industrial and transportation emissions in urban areas alone. The possible anthropogenic influence on these cycles remains unclear, but different processes are required to explain their occurrence. The most plausible hypothesis to interpret large-scale weekly cycles is through direct and indirect effects of anthropogenic aerosols, whose emissions also tend to show weekly periodicity at large-scale18. A weekly cycle of aerosol optical depth (AOD), observed in the United States and in Central Europe, was stronger in urban sites than in rural sites. A different weekly cycle detected in the Middle East and India had lower AODs on Thursday and Friday, the weekend for Middle Eastern cultures19. Close synchrony between weekly peaks in anthropogenic aerosols and tornado activity over the eastern US during the summer was detected by20. They found a statistically significant weekly cycle in storm activity during summer for the period 1995 to 2009, unlikely to be due to natural variability. A connection with fire activity was provided by21 in their analysis of an historical tornado outbreak in the southeastern U.S., where they proposed that smoke from biomass burning in Central America might have increased tornado severity due to boundary-layer stabilization by soot and an increase in the optical thickness of lower tropospheric clouds. The assumption that vegetation burning is a natural aerosol source, and thus does not display weekly cycles, was adopted in the analyses of20 and21, as well as in the atmospheric chemistry modelling studies of22 and17. However,23 found significantly weekly cycles in sub-Saharan Africa cropland burning, with Sunday minima in predominantly Christian regions and Friday minima in mainly Muslim regions. They did not find significant weekly cycles in fire activity in densely settled areas, where vegetation burning is very limited, nor in rangelands, forests and wildlands, where fire management intensity is weaker than in croplands.

These continental-scale findings, combined with the recognition of the strongly anthropogenic nature of global vegetation burning24 motivated the hypothesis of weekly cycles in global fire activity, as observed with remotely sensed active fire data. The hypothesis of a Sunday weekly minimum in fire activity has two justifications: (i) international labour legislation stipulates a weekly day of rest, which in most countries is Sunday, and (ii) of the two major global religions (Christianity and Islam) that prescribe a weekly day of rest, Christianity prevails in more high fire incidence areas. We also assessed the effect of land use and land cover on weekly fire cycle significance by testing the aforementioned hypotheses separately for the Villages, Settlements, Croplands, Rangelands, Seminatural, and Wildlands anthropogenic biomes (anthromes)25, which map global patterns of sustained direct human interaction with ecosystems. Anthrome-specific analysis provides valuable information to elucidate the land management activities, regions, and times of the year contributing to the occurrence of weekly cycles in global vegetation burning.

Previous research by26 and27 also detected Sunday minima in global fire activity, using spatially aggregated fire counts data from the NASA Earth Observations website (http://neo.sci.gsfc.nasa.gov/view.php?datasetId=MOD14A1_M_FIRE). However, inconsistencies between the active fire numbers reportedly used by26 and by27, and their discrepancies relatively to other published sources justify revisiting the topic of weekly cycles in global active fire data. Their data differ from those published by various other sources in the number of MODIS active fires reportedly used, in the time series of annual active fire numbers, and in the weekly distribution of active fires. In comparison with all other published sources that have partially overlapping study periods and mention the numbers of active fires used,26 claim having worked with many more fires, while27, who used a time series three years longer than that of26, surprisingly report substantially fewer fires. Table 1 in Supplementary Information shows the numbers involved in these comparisons. The discrepancies between26 and27 relatively to other authors are largest for the first two years of their time series, 2001 and 2002, which they claim to have the most active fires. This is an obvious impossibility, since during all of 2001 and the first half of 2002, only the TERRA instrument, which has a morning overpass at 10:30AM/PM, was in orbit. AQUA was launched in mid-2002 with a 2:30AM/PM overpass and that more than doubled the amount of active fires detected because the AQUA daytime overpass observes the most active period of the diurnal cycle of vegetation burning. Figure 5 of28 displays the relative contribution of AQUA to the total MODIS active fire detections and clearly shows that AQUA detections account for more than half of all active fires. Therefore, the number of active fires in the MODIS data product (MCD14ML) for the years 2001 and 2002 is less than half of that in each subsequent year, as shown in figure 2a of29. Table 1 in Supplementary Information provides specific comparisons for these two years, taking into account differences in the MODIS sensors and in the active fire data collections used by the various authors. There are also differences between the weekly distributions of MODIS active fire counts shown in figure 2a of26 and in figure 4 of27, and the distribution portrayed in Fig. 1 below. In the original MDC14ML data that we used the relative difference between the numbers of active fires on Sunday and Monday is larger, i.e., Sunday is a more prominent weekly minimum and Tuesday is a more prominent weekly maximum, and thus the magnitude of the weekly cycle in the number of active fires is larger. In addition, there is a change in rank order of the number of active fires by weekday with Friday, rather than Monday, having the second fewest fires. These differences are likely to affect the results of a weekly cycle analysis.

The paper by26 also has some methodological issues. In our opinion, the statistical tests they used to detect and characterize weekly fire cycles, may not be adequate to answer two fundamental questions: (i) if there is evidence in the data that there are 7 day cycles (ii) If there is such evidence in the data, fire activity in each day of the week have different statistical properties, in particular, if average Sunday fire counts are less that the average counts in other days of the week. Daily counts of fire activity constitute a time series and by nature, they show strong and complex temporal dependence structures. Without accounting for such temporal dependence structures it is not possible to come up with correct conclusions on (i) and (ii). The authors26 base their results on statistical methods, which we believe, do not take into account proper time series analysis. Although their Monte Carlo tests may show the extent to which a weekly cycle is present under random conditions, and thus the probability that the weekly pattern of fire counts observed in the data occurs by chance, they do not test the shape, nor even the existence of such a cycle. Their subsequent Kolmogorov-Smirnov test only shows, for the global scale analysis, that the two days of the week displaying the largest and smallest number of fires (Tuesday and Sunday respectively) have distinct statistical distributions. This, by itself, does not provide evidence that there is a minimum on Sunday fire activity, neither shows that the activity on Sunday differs from the other days of the week.

Our goal in the present study is to reanalyse the topic of weekly cycles in vegetation fire activity at global scale, relying on the original MODIS active fire detections at 1 km spatial resolution (MCD14ML) with appropriate statistical methodologies to detect significant periodicities in time series data. Using Bayesian methodology, a simultaneous 95% credible band for the mean function was constructed. Sunday minimum is hence directly investigated by computing the probabilities that the mean functions of every week day (Monday to Saturday) are inside the credible band corresponding to mean Sunday fire activity.

## Data

Daily fire activity data were obtained from the NASA MODIS MCD14ML Collection 5 active fire product30 at 1 km spatial resolution, after screening for false alarms and non-vegetation fires according to the procedures detailed in29. We analysed the period from July 8, 2002 to July 29, 2012 (TERRA and AQUA sensors), for a total of 525 full weeks and 43,416,629 fire counts. Figure 1 shows the mean + 1 sd. of total fire counts per weekday. Land use/land cover data used are from the Anthropogenic Biomes of the World (anthromes) dataset, at 5′ (ca. 9 km) spatial resolution, dated from the year 2000. Anthrome data combine information on population density, land cover and land use to derive a map of global patterns of sustained direct human interaction with ecosystems25. Since a large majority of global vegetation burning is anthropogenic24, they were considered particularly appropriate for our analysis. The relative stability of spatial patterns depicted by the anthromes dataset further justifies its use in conjunction with global fire activity data for the period 2002 to 2012.

Each active fire was labeled with the anthrome class present at its geographical coordinates to create the datasets for the anthrome-specific fire weekly cycle analyses (see Table 1). Maps aggregated at 0.5° spatial resolution of the weekday with the fewest fires and of Global Anthromes (v2.0) using the modal class rule are shown in Figs 2 and 3 respectively.

## Methods and Models

### Preliminary data analysis and sampling model identification

To search for weekly cycles in vegetation burning, we analysed a 525-week time series of global daily MODIS active fire counts.

The sample auto correlation (acf) and the partial auto correlation function (pcf) together with the periodogram of the log-transformed daily global fire count data were used to search for periodical variations in the data. The periodogram was calculated to identify frequencies at which jumps occur and their significance was tested with Fisher’s test and Whittle’s test31,32.

The periodogram showed statistically significant jumps at the fundamental frequency wj = 2πj/365 corresponding to 1 cycle in 365 days and its first 3 harmonics, corresponding respectively to 1 cycle in 6 months, 4 months, and 3 months. De-seasonalizing the data by fitting an harmonic regression at these frequencies unmasked further jumps in the periodogram at frequencies corresponding respectively to 1 cycle in 7 days and 1 cycle in 3 and half days. Further application of Whittle’s test showed that these two jumps are also statistically significant. Figure 4, shows the periodogram of the deseasonalized series where these jumps are quite evident. Jumps on the right hand side of the periodogram correspond to high frequency noise.

Results of the spectral analysis clearly suggest that the log transformed global daily fire count time series Yt is non-stationary, and that its mean function Mt should include a harmonic sum with k = 6 harmonics corresponding to the angular frequencies of 1 cycle per year, per 6, 4, and 3 months, and per 7, and 3.5 days. We then suggest the model Yt = Mt + Xt, with Xt a stationary error term. Detailed study of the deseasonalized series by using the software ITSM31 suggested that the stationary component of the time series Xt is well represented by a zero mean stationary and invertible Gaussian Autoregressive model of order 7 (AR(7)).

The annual and multi-month cycles in fire activity result from global climate seasonality patterns, including the lagged dry seasons of the Northern and Southern hemispheres, as well as the occurrence of fire seasons in the tropical savannas of the Northern hemisphere during the boreal winter6,33. We hypothesize that the weekly cycle is an anthropogenic fingerprint, due to weekly days of rest determined by religious norms and labor regulations. Figure 1 shows that there are four consecutive weekdays with a mean number of fires under 12,000 (Friday to Monday), followed by three days with over 12,000 fire counts (Tuesday to Thursday). Since Friday is a religious holiday for Muslims in the extensively burned sub-Saharan region of Northern hemisphere Africa, Saturday is a civil day of rest, and Sunday is simultaneously a civil and religious holiday in most of the western world, the reduced fire activity during these days is understandable. The inclusion of Monday in this group of days may be a spillover effect from the previous days. The period Tuesday to Thursday is not affected by imposed weekly days of rest and thus the use of fire as a land management tool increases.

### Bayesian model and inference

Following the preliminary data analysis we assumed the following model for the time series of log-transformed daily global fire count data

$${Y}_{t}={M}_{t}+{X}_{t},$$
(1)

where the mean function Mt is

$${M}_{t}={a}_{0}+{a}_{1}t+\mathop{\sum }\limits_{i=1}^{6}\,[{c}_{i}\,\cos \,{\omega }_{i}t+{d}_{i}\,\sin \,{\omega }_{i}t],$$
(2)

with angular frequencies ωi, i = 1,2, .. 6 corresponding to the significant jumps identified in the periodogram and

$${X}_{t}=\mathop{\sum }\limits_{i=1}^{7}\,{\varphi }_{i}{X}_{t-i}+{Z}_{t}$$

is a stationary, zero mean process compatible with a stationary and invertible Gaussian AR(7) process, where, Zt is a zero mean, uncorrelated (white noise) process.

For the vector of parameters of the fixed effects (a1, ci, di, i = 1, ..., 6) we considered independent normal priors with zero mean and precision 0.001 and a flat prior for a0. For the parameters of the AR(7) process, (ϕ1, ..., ϕ7) we used, after a transformation of the parameter space, penalised complexity priors for the partial autocorrelation functions, as suggested by34.

We fitted this model within the R-INLA environment35, in which the mean function Mt and the stationary process Xt appear respectively as the fixed and random effects in the linear model specification.

From the analysis of the model, we observe from Table 2 that all the harmonic coefficients are significant, but there is no evidence of a linear trend.

### Comparing the mean functions

Once we establish that fire activity indeed displays weekly cycles, it is interesting to see in which day of the week the minimum fire activity is observed, in particular, one may want to test the hypothesis that the fire activity is minimum on Sundays. One way of answering this question is to compare the mean functions $${M}_{{t}_{i}}$$ in (2) with ti = i, i + 7, i + 14, ... and i = 1, ..., 7, where i = 1 refers to Monday and i = 7 refers to Sunday. That is compare the mean functions corresponding to the fire activity series of different days of the week.

Our strategy to test the hypothesis that the fire activity is minimum on Sundays will be based on calculating the 95% simultaneous credible interval (credible band) corresponding to Sunday mean function $${M}_{{t}_{7}}$$ and (i) check how likely is the mean function of other days of the week falling within this credible band. Additionally we compute the probabilities that the mean functions of other days of the week are above the estimated mean function of Sunday (ii). Significant lower probabilities of (i) and significant higher probabilities of (ii) give evidence that indeed fire activity is minimum on Sundays. We explain below the computational details.

The marginal posterior distribution for the mean function Mt, t = 1, ..., 3765 is approximately multivariate normal with posterior mean $${\hat{M}}_{t},t=1,\ldots ,3765$$ and posterior covariance matrix S35, which can be obtained from R-INLA using inla.make.lincomb function. Pointwise 100(1 − γ)% HPD (highest posterior density) intervals for each component of the vector Mt can also de obtained using the function inla.hpdmarginal. However, rather than obtaining pointwise credible intervals, one should seek for a simultaneous credible band for Mt of size 100(1 − α)%. For that we follow the suggestion given by36.

For the random vector {Mt, t = 1, ..., n} = (M1, ..., Mn), a 100(1 − α)% simultaneous credible band can be defined as a hyper-rectangular region

$$R=\{\{{M}_{t}\}:{\cap }_{i=1}^{n}({M}_{i}\in {I}_{i,\gamma })\}$$

where {Ii,γ}, i = 1, ..., n denotes a set of individual 100(1 − γ)% HPD credible intervals for each component of the vector Mt, with γ chosen so that P({Mt} R) = 1 − α.

Since the posterior distribution for the vector Mt is approximately multivariate normal with known mean vector and known covariance matrix, by trial and error, one can obtain a simultaneous 100(1 − α)% HPD credible band for the vector {Mt, t = 1, ..., n}, by computing, for every individual component of the vector, a 100(1 − γ)% HPD credible interval, varying γ till the overall probability is 1 − α.

$$P(\{{M}_{t},t=1,\ldots ,n\}\in R)=1-\alpha .$$

Actually this can also be obtained using the function excursions.simconf available in the R package excursions37.

Considering the fact that the linear trend was not significant and considering the cyclic nature of Mt, above calculations were reported for the first 52 weeks, rather than for the full period.

In Figure 5 we display the series of the log counts with the estimated mean function (posterior mean) $${\hat{M}}_{t}$$, and in Fig. 6 the estimated mean function for the first year, ie, $${\hat{M}}_{t},t=1,\ldots ,364$$ together with the 95% simultaneous credible band.

Since the posterior distribution of the mean function Mt is multivariate normal, Mt can be partitioned into $${M}_{{t}_{i}},i=1,\mathrm{...},7$$ (with 1 for Monday, 7 for Sunday), as explained before, to represent the mean functions for the different days of the week. Properties of the multivariate normal distribution imply that the posterior distributions of $${M}_{{t}_{i}}$$, ti = i, i + 7, ... and i = 1, ..., 7 are also approximately multivariate normal with means given by the $${\hat{M}}_{{t}_{i}}$$, ti = i, i + 7, ... and i = 1, ..., 7 and corresponding covariance matrices, which are sub-matrices of the overall covariance matrix. Also if we represent the 100(1 − α)% HPD credible bands for Mt by the two series LBt and UPt, they can similarly be partitioned into $$L{B}_{{t}_{i}}$$ and $$U{P}_{{t}_{i}}$$, for i = 1, ..., 7.

We propose as a methodology to test the hypothesis that there is, on average, smaller fire activity on Sundays, by computing the probabilities, for i = 1, ..., 6

$$P(L{B}_{{t}_{7}}\le {M}_{{t}_{i}}\le U{B}_{{t}_{7}}),$$

that is, the probabilities that the mean function for every day of the week is inside the credible bands corresponding to Sunday, ie, inside $$L{B}_{{t}_{7}}$$ and $$U{P}_{{t}_{7}}$$. These probabilities can be easily obtained using the function gaussint() from the R package excursions37.

In Table 3 we show these probabilities, with the associated error in brackets, computed using the first year. For the other years the results are very similar.

We also compute the posterior probabilities that the mean function for every other week day $${M}_{{t}_{i}},i=1,\mathrm{...},7$$ is above the estimated mean function for Sunday (the mean of the posterior distribution of the mean vector $${M}_{{t}_{7}}$$). These probabilities are given in Table 4.

From Table 3 we observe that, besides Saturday, the mean functions of all the other days of the week have lower probabilities of being inside the credible band for Sunday. Also from Table 4 we observe that except for Monday, all other days of the week have relatively high probabilities of being above the estimated mean function of Sunday. Hence there is some evidence that, in general, minimum fire activity is indeed on Sundays.

Figure 7 is an unfolded reproduction of Fig. 6, that is, the estimated mean function $${\hat{M}}_{t}$$ is partitioned into $${\hat{M}}_{{t}_{i}}$$, ti = i, i + 7, i + 14, ..., and i = 1, ..., 7 to represent the estimated mean function for the different days of the week. For ease of reading, only the first year is plotted. Other years have the same behaviour, due to the cyclic nature of the mean function, since the linear trend is not statistically significant.

It is clear from that plot that the estimated mean function corresponding to Sunday fire counts is smaller than the other estimated mean functions almost uniformly for every week. Looking in more detail we remark that the estimated mean function for Sunday shows always lower values, except for the period ranging from the second week of November to the first week of December and from the third week of May until the end of June, when it is always very close to the estimated mean function for Monday. The second minimum is basically on Saturday or Monday. This may indicate the existence of a weekend effect on daily fire counts. To understand how much smaller the estimated mean function for Sunday is from the other days of the week, we display in Fig. 8 the estimated mean function for Sunday together with the corresponding 95% simultaneous credible band and the estimated mean functions for each of the other days of the week. We observe that the estimated mean function for Saturday is always inside the credible band for Sunday and only during two periods of the year, namely, third week of August to third week of October and second week of December to the second week of April, the estimated mean functions for some of other days of the week are above the upper bound of the credible band for Sunday. (see detailed study in the technical report)38 (http://ceaul.org/wp-content/uploads/2018/10/NotaseCom_3_2016.pdf). This is an indication that there is evidence of a weekly cycle with minimum on Sunday, only during these periods of the year.

Further, the comparison of fire activity in each day of the week involves comparison and ordering of time series, We note that existence of ordering for mean functions does not necessarily imply stochastic ordering for the respective time series. However the following result in39 gives an indication when ordering in mean functions imply respective stochastic ordering: If Xt and Yt are two independent Gaussian processes with the same covariance structure and if E(Xt) ≤ E(Yt) uniformly for every t, then Xt is stochastically dominated by Yt. The precise definition of stochastic ordering can be found in39. There is strong evidence in the data that deseasonalized log fire activity for each day of the week follow a AR(1) process and although they are certainly not independent, this might suggest that the results on comparison on the means may extend to respective stochastic ordering. However, further detailed work is needed to reach such complete conclusion.

## Anthrome-level Analysis

The methodology shown in section 3.3 was applied to time series of the logarithm of the number of active fire counts described with each of the six anthromes shown in Fig. 3. Following their global analysis,26, searched for weekly cycles in regions of the globe displaying abundant fire activity. This approach designated40 post-hoc domain selection, and warn that it risks biasing the analysis, since it might be expected that regions experiencing many fires might exhibit large weekly cycles. In order to avoid the potential for this type of bias, we regionalised our analysis independently of any observed fire activity spatial patterns, based on the hypothesis that the likelihood of active fire weekly cycles vary by anthrome. Only for Croplands did Whitle’s test show strong evidence of significant jumps in the periodogram at frequency corresponding to weekly cycles, although some evidence is also observed for Rangelands and Seminatural anthromes. Table 5 and Fig. 9 show, for each anthrome, the probabilities that the mean function for every day of the week is inside the Sunday 95% credible band.

Complementary to these probabilities, we also calculated the probability that a mean function of a day of the week is above the estimated Sunday mean function. These probabilities are given in Table 6 (see also Fig. 10).

Comparing the probabilities between Sunday and other weekdays displayed on Tables 5 and 6 we can say that there is a strong evidence that for Croplands, the mean function for Sunday is statistically smaller than the mean function for every other day of the week. This evidence is not so strong for Rangelands and Seminatural since the probability that Saturday mean function is inside the 95% credible band for Sunday is relatively high. For other anthromes there is no statistical evidence of a difference between the mean functions corresponding to the different days of the week. Globally, albeit with varied strengths of evidence, three anthromes (croplands, rangelands, and seminatural areas), corresponding to 70% of the land surface and to 94% of the active fire data analysed display weekly cycles in fire activity.

To understand how much smaller the estimated mean function for Sunday in Croplands is from the other days of the week, we display in Fig. 11 the estimated mean function for Sunday together with the 95% simultaneous credible band and the estimated mean functions for each of the other days of the week.

For each anthrome, we identified the longest uninterrupted period during which the estimated mean function for the largest number of weekdays is above the 95% upper credible band for Sunday, representing the time of the year when the weekly cycle of fire activity is strongest. In the Cropland anthrome this period lasts from early September to early October, when vegetation burning takes place mostly in Southern hemisphere Africa, Brazil, Southeastern USA, Southwestern Russia and the Ukraine (see Fig. 12). The Seminatural anthrome weekly cycle is clearest at a similar time of the year (mid-September to early October) and has a partially overlapping geographical distribution (Southern hemisphere Africa and South America, Fig. 13). During this period, fire activity in Croplands during this strongly cyclic period displays a clear Sunday minimum and is concentrated in regions that are predominantly Christian and/or where labor regulations concerning a weekly period of rest are enforced23,26. Evidence of a weekly cycle is weaker in the Seminatural anthrome and not so much as a Sunday minimum, but more as a weekend effect (Table 5). In Rangelands, the time of the year when the weekly cycle in vegetation burning is most evident goes from late December to early February. Fire activity is concentrated in the tropical savannas of the Northern hemisphere, mainly in Muslim sub-Saharan Africa (see Fig. 14). Unlike at other times of the year, fire counts on Thursday and Friday, the religious and civil days of rest in the region, are inside the credible band for Sunday fire counts, suggesting the influence of religious practices on the weekly fire cycle23.

## Discussion and Conclusions

The present study revisited the topic of global weekly fire cycles, previously addressed by26 and27; We used original MCD14ML data, without any form of gridding or normalization of fire counts that appear to have caused the problems described in the Erratum of26. These problems were not fully corrected, as shown in the Supplementary Information Table 1, and cast doubt upon the reliability of their results. Our geographical stratification of the analysis was defined a priori, avoiding potential biases introduced by post-hoc spatial domain selection and relies on an explicit hypothesis that the likelihood and features of active fire weekly cycles are anthrome-specific. The anthrome-based spatial stratification provides insights into the influence of land use intensity on the occurrence of fire weekly cycles.

Employing statistical methodologies more appropriate for the detection of weekly cycles in time series of active fire counts, and for determining the weekday with the fewest fires, we reliably detected a weekly global fire activity cycle with a Sunday minimum, and Table 3 shows that for Saturday there is a reasonable probability that the corresponding mean function is inside the credible band for Sunday. Also, Figs 7 and 8 show that the estimated mean function for Sunday is almost everywhere below the estimated mean function for the weekdays and almost always very close or even below the lower confidence bounds of the week days. However, contrary to the other days of the week, the estimated mean function for Saturday is always inside the credible bands for Sunday. This finding, associated with the reasonable high probability 0.7019 displayed in Table 3, indicates that rather than a Sunday minimum, there actually is a weekend minimum in global active fire counts.

Our research expanded previous continental-scale work on weekly cycles of fire activity when the data are disaggregated by anthrome23, by identifying Croplands (40.19% of all fires) as the anthrome with the strongest fire weekly cycles, which reaffirms the importance of land management intensity as a driver of the cycles. Table 6 shows strong evidence that the Sunday mean function for Croplands is statistically smaller than the mean function for every other day of the week including Saturday, revealing a Sunday effect on the decrease of cropland fire activity, rather than the weekend effect observed at global scale. Fire weekly cycles are weaker in Rangelands and Seminatural anthromes, which jointly account for 53.46% of all fires than in Croplands, due to lower land management intensity and less strict control over fire size and duration. Our results showing that anthromes exhibiting weekly cycles in fire activity cover 70% of the global land surface are in line with the findings of24 and41, who considered that anthropogenic vegetation burning clearly dominates vegetation fires at the global scale, and that fires started by lightning are responsible for a minor proportion of global biomass burning. According to24, exceptions are Canada, Russia, and the USA, where an estimated 85%, 49.0% and 67.6% of the burned biomass, respectively, is due to lightning fires. Their results indicate an amount of biomass burning due to anthropogenic fires ranging from 3.5 to 3.9 billion tons of dry matter per year (Pg dm/yr). Most of this biomass burning occurs in the savannas of Sub-Saharan Africa that, in the terms of the present study, correspond to a mixture of cropland, rangeland and seminatural anthromes.

Figures 9 and 10 show that there are no weekly fire cycles in the anthromes that accounf for a small percentage of global active fires (6.35%), and that are the most (Settlements and Villages) and least transformed (Wildlands) by human action. In Wildlands there is a much larger proportion of lightning-caused ignitions than in the other anthromes, and fires often burn without any attempts at suppression. This more natural pattern of burning is not conducive to the emergence of weekly cycles. In Settlements and Villages, high population density and landscape fragmentation lead to limited use of fire and, by definition, small amplitude in the number of fires per weekday. In conclusion, there is a weekly cycle in global active fire data, mostly due to the use of fire in croplands and, to a lesser extent, in rangelands and seminatural areas. Sunday is the weekday with the fewest fires at the global scale, because labour laws in most countries and Christianity, the largest global religion, prescribe it as the weekday of rest6 and33 showed the influence of human activity on the seasonality of global vegetation burning and that kind of influence is now evident at the weekly time scale. Fire-enabled dynamic vegetation models with daily temporal resolution and chemical weather forecast models will benefit from incorporating representations of weekly cycles in fire activity.

## Data Availability

All data used in our study are publicly available at https://earthdata.nasa.gov/c5-mcd14dl (active fires) and http://ecotope.org/anthromes/v2/data/ (anthromes map, version 2). The code to run the model in R-INLA can be obtained from the authors upon request.

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## Acknowledgements

We thank FCT (Fundação para a Ciência e a Tecnologia), Portugal, which funded this work through the projects UID/MAT/00006/2019 and UID/AGR/00239/2019.

## Author information

J.M.C. Pereira and K.F. Turkman wrote the main manuscript text; M.A. Amaral Turkman did all the computational work and prepared figures 4–11; D. Oom prepared the database and the figures 1–3 and 12–14. All authors reviewed the manuscript.

Correspondence to M. A. Amaral Turkman.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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# Thermodynamics

###### Question:

a.Steam at 35 bar and 600 K enters a throttling valve that reduces the steam pressure to 7 bar. Assuming there is no heat loss from the valve, what is the exittemperature of the steam and its change in entropy?
b. If air (assumed to be an ideal gas with Cp : 29.3 J/molK ) entered the valve at 35 bar and 600 K and left at 7 bar, what would be its exit temperature and entropychange?

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# True of false?

1. Apr 25, 2005

### quasar987

Every "spacially periodic" function [i.e. s.t there exist P s.t. f(x+P,t) = f(x,t)] of the form f(x,t) = X(x)cos(wt) is a solution of the wave equation.

Last edited: Apr 25, 2005
2. Apr 25, 2005

### James R

True..........

3. Apr 25, 2005

### jdavel

quasar,

False.

For one thing, if f(x,t) is periodic (and it doesn't have to be periodic), then there's a strict relation between the periodicity in x and in t. In other words, in an example using your format, if f(x,t) = sin(kx)cos(wt), then w/k = v.

It's not too hard to see what the wave equation is saying about f(x,t) if you think about it. Within a multiplicative constant, the two partial derivatives are the same. That means f has to depend on x and t in very similar ways. I think the most general form for f(x,t) that satisfies the W.E. is f(kx-wt). Certainly any function of that form will work. Although that's not really what you were asking.

4. Apr 25, 2005

### jdavel

Uh oh!

James R., I posted before I saw yours. Why do you say it's true?

5. Apr 26, 2005

### James R

Hmm... Here are my thoughts. I think I might have changed my mind!

At first, I thought that any function of the form f(x,t) = X(x)cos wt will describe a standing wave, and so it is necessarily a solution of the wave equation.

But then...

The wave equation, in 1 dimension, is:

$$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{v^2 \partial t^2} = 0$$

where v is a constant (the wave speed).

Using the function given, we have:

$$\frac{\partial^2 f}{\partial x^2} = \frac{d^2 X}{dx^2}\cos \omega t$$
$$\frac{\partial^2 f}{\partial t^2} = -\omega^2 X(x)\cos \omega t$$

Therefore, we require:

$$\frac{d^2 X}{dx^2} + \frac{X}{v^2} = 0$$

This restricts X(x) to harmonic functions of the form:

$$X(x) = A \sin kx + B \cos kx$$

where A and B are arbitrary constants, but k is restricted:

$$k=\omega / v$$

So, it seems that the most general functions of the given form which satisfy the wave equation are:

$$f(x,t) = [A \sin kx + B \cos kx]\cos \omega t$$

with the above restriction on k.

Does that seem right?

6. Apr 26, 2005

### Galileo

You don't you just work out the simplest case: $f(x,t)=\cos(wt)$.
We clearly have $f(x+P,t)=f(x,t)$ for any t (and any P).
The wave equation clearly doesn't hold in this case (unless $\omega=0$, but I understand $\omega$ can be arbitrary).


text:
# Matrix operation in dirac matrices

If we define $\alpha_i$ and $\beta$ as Dirac matrices which satisfy all of the conditions of spin 1/2 particles , p defines the momentum of the particle, then how can we get the matrix form ? $$\alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y \\ p_x+ip_y & -p_z \end{pmatrix} .$$

-
Dirac matrices in 4 dimensions are 4x4. You've written a 2x2 matrix. Where did you find this equation? – Michael Brown Jan 1 '13 at 4:05
Dirac matrices can be written as $2*2$ block form . check this link .nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_7/… – Unlimited Dreamer Jan 1 '13 at 6:40
I see... you're breaking the Dirac equation down to 2x2 blocks. This is the standard way of solving it. Where in the argument are you having trouble? Unfortunately the equations in your source aren't numbered, but I can see you mean $\vec{\sigma}\cdot\vec{p}$ rather than $\alpha_i p_i$. – Michael Brown Jan 1 '13 at 6:47

It's just a matrix manipulation. Let $\sigma_i$ pauli matrices. $$\alpha_i p_i= \begin{pmatrix} 0& \sigma_i \\ \sigma_i & 0 \end{pmatrix} p_i .$$ $\alpha_i p_i= \begin{pmatrix} 0& p_1 \sigma_1 \\ p_1\sigma_1 & 0 \end{pmatrix} + \begin{pmatrix} 0& p_2 \sigma_2 \\ p_i\sigma_2 & 0 \end{pmatrix} + \begin{pmatrix} 0& p_3 \sigma_3 \\ p_3\sigma_3 & 0 \end{pmatrix}$

But $\sigma_1 p_1 = \begin{pmatrix} 0& 1 \\\ 1 & 0 \end{pmatrix}p_1=\begin{pmatrix} 0& p_1 \\\ p_1 & 0 \end{pmatrix}$ ,

$\sigma_2 p_2= \begin{pmatrix} 0& -i \\\ -i & 0 \end{pmatrix}p_2=\begin{pmatrix} 0& -ip_2 \\\ ip_2 & 0 \end{pmatrix}$

$\sigma_3 p_3= \begin{pmatrix} 1& 0 \\\ 0 & -1 \end{pmatrix}p_3= \begin{pmatrix} p_3& 0 \\\ 0 & -p_3 \end{pmatrix}$

Now adding these we get ($1\rightarrow x$, $2\rightarrow y$ ,$3\rightarrow z$) , $$\alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y \\ p_x+ip_y & -p_z \end{pmatrix} .$$

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Your final formula is 4x4 on the left and 2x2 on the right. You should have a 4x4 matrix with what you've written as the upper right and lower left blocks, with zeros everywhere else. – Michael Brown Jan 2 '13 at 23:42
In the right $2\times 2$ makes the matrix $4 \times 4$ – Unlimited Dreamer Jan 3 '13 at 5:19
No, the answer you want is $\alpha_i p_i = \left(\begin{matrix} 0 & p_i \sigma_i \\ p_i \sigma_i & 0 \end{matrix}\right)$ which has four 2x2 blocks. Your right hand side is the 2x2 matrix $p_i \sigma_i$, not $\alpha_i p_i$. You had it right until the very last line! – Michael Brown Jan 3 '13 at 5:27

The equation you wrote only makes one choice that should answer all questions about this context: it chooses a representation of the $\alpha_i$ matrices with $$\alpha_i = \sigma_i$$ where $\sigma_i$ are the three Pauli matrices. You may check that if you substitute the Pauli matrices (particular $2\times 2$ matrices listed in the Wikipedia article linked in the previous sentence) for $\alpha_i$ on the left hand side of your equation, you obtain the right hand side.

If your formula had the Greek letter $\sigma$ instead of $\alpha$ on the left hand side, it would be uncontroversial. However, with $\alpha$, it is problematic. The $\alpha_i$ matrices are really $4\times 4$, not $2\times 2$, so all the equations above must be interpreted so that each matrix entry of the Pauli matrices is actually a block $$z \to \pmatrix {z&0 \\ 0&-z }.$$ We say that the Pauli matrices were tensor-multiplied by a $2\times 2$ unit matrix (in certain order). This extra tensor factor actually can't be the unit matrix because one couldn't find any matrix $\beta$ that anticommutes with all the $\alpha_i$ matrices. But it may be another $\sigma_z$, for example, in which case $\beta$ may be chosen to be ${\rm diag}(\sigma_x,\sigma_x)$, for example. Alternatively, you should ignore the source and learn some/all of the standard representations of the Dirac matrices.

At any rate, something is sloppy about the notation in which $\alpha_i$ were written as $2\times 2$ matrices and the simplest recipe to get $4\times 4$ matrices (tensor product with the $2\times 2$ unit matrix) doesn't work. So one should first see what $4\times 4$ matrices your source (if it is correct at all) actually means.

-


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## Introduction

In recent years, with the aim of achieving a cleaner and more sustainable energy model, wind energy has become an important source of electricity generation. Even so, an improvement in wind turbine performance is still required in order to compete with conventional energy sources in terms of energy production and associated costs. To solve this challenge, the implementation of both active, such as rotating microtabs, and passive, such as Gurney flaps, flow control devices is a widely used solution. Aramendia et al.1,2 extensively reviewed the available active and passive flow control devices for wind turbines.

Simulations by means of CFD tools are the most popular method for analyzing and optimizing the performance of airfoils and their flow control devices. Many authors have studied several different flow control devices applied on airfoils by means of CFD. For example, Fernandez-Gamiz et al.3 and Aramendia et al.4,5 performed parametric studies to analyze the effects of the implementation of passive microtabs and Gurney flaps, respectively, on the TE of the DU91W(2)250 airfoil.

In some cases, CFD simulations can be very demanding in terms of computational time and resources, especially when several simulations are necessary to optimize a flow control device or accurate turbulence modeling is required. For this reason, many authors have used alternative meshing models to reduce simulation time. Among these models, the cell-set model, introduced by Ballesteros-Coll et al.6 can be highlighted. In that work, different Gurney flaps were added to the TE of a DU91W(2)250 airfoil by means of the cell-set model. In further studies, Ballesteros-Coll et al.7,8 implemented microtabs and rotating microtabs on the same airfoil. Other authors, such as Portal-Porras et al.9,10 used this meshing technique to model the performance of three-dimensional Vortex Generators (VG) on a flat plate. All of them showed good agreements between the cell-set model, the fully-resolved model and the experimental data. Therefore, this meshing model is considered suitable for this kind of problems.

Despite the accurate predictions that can be obtained through CFD simulations, the increase in the computing speed of computers and the growth of Artificial Intelligence (AI) have led to an increasing number of studies in which Deep Learning (DL) techniques are used for flow prediction, obtaining a significant reduction in terms of computing time. For example, Ye et al.11 proposed a Convolutional Neural Network (CNN) to predict the pressure distributions around a cylinder based on the velocity field on its wake behind, Guo et al.12 and Ribeiro et al.13 designed different CNNs for two- and three-dimensional laminar flow prediction, Portal-Porras et al.14 used a CNN to predict turbulent flows on a channel, and Abucide-Armas et al.15 proposed a data augmentation technique to improve the predictions of the CNN proposed by Ribeiro et al.13 for unsteady turbulent flows.

Regarding airfoils, several machine learning algorithms have been proposed to study their performance. Initially, the most commonly used architecture was the Multi-Layer Perceptron (MLP) learning architecture. For example, Sekar et al.16 presented a two-step-consistent approach to predict the flow fields over an airfoil using deep learning techniques. In the first step the airfoil is parametrized into 16 airfoil parameters by means of a CNN, and in the second step a deep MLP network is used to predict the flow fields over airfoils. Even so, this architecture is not specifically designed to exploit spatial and temporal correlation that are intrinsic in many real-world problems. For this reason, the vast majority of current studies similar to the present one use other types networks, being the most common ones the CNNs. Yilmaz and German17 studied the airfoil pressure coefficient predictions provided by a CNN by varying its parameters, providing an initial approximation for a suitable CNN design. Thuerey et al.18 proposed a CNN to approximate the velocity and pressure fields obtained by Reynolds-Averaged Navier-Stokes (RANS)-based Spalart-Allmaras19 turbulence model on airfoils.

For the prediction of aerodynamic coefficients, there is a broader variety of architectures used, since these studies offer greater design flexibility than those of the fields. However, as stated by Zhang et al.20, although other networks can provide similar predictions, the CNN is the one that offers the greatest generalization capacity, allowing different geometries to be introduced as input in the network in a simple way. For example, in that study the lift coefficient of different airfoils are predicted with various network architectures. Chen et al.21 used a CNN to predict the drag ($${C}_{D})$$ and lift ($${C}_{L}$$) coefficients of different airfoils.

Regarding flow control devices, there are some studies in which deep learning techniques are used for a better understanding of the behavior of flow control elements in airfoils. For example, Rodriguez-Eguia et al.22 and Aramendia et al.5 used ANNs to predict the aerodynamic coefficients of an airfoil with flaps and Gurney flaps, respectively. However, in these studies the parameters are not predicted directly from the geometry and boundary conditions. Therefore, there are no deep learning studies applied to CFD in which the behavior of flow control elements is analyzed.

The present paper aims to evaluate the possibility of analyzing flow control elements applied to airfoils by means of deep learning techniques. For this purpose, velocity and pressure fields around different Gurney flaps and rotating microtabs implemented on the TE of the DU91W(2)250 airfoil by means of a CNN are predicted. In addition, a variation of the CNN is proposed to predict the $${C}_{D}$$ and $${C}_{L}$$ coefficients of the studied airfoil with each flow control device.

The remainder of the manuscript is structured as follows: “Methodology” provides an explanation of the methodology followed for preparing and running the CFD simulations and designing and training the proposed CNNs; “Results and discussion” shows qualitative and quantitative comparisons between the CFD results and CNN predictions; and “Conclusion” explains the conclusions reached from this study.

## Methodology

### CFD setup

With the purpose of obtaining data for training, validating and testing the proposed neural network, 158 CFD simulations of the DU91W(2)250 airfoil were conducted, each one under different conditions in terms of flow control device geometry and angle of attack (AoA). The selected profile, DU91W(2)250, is a profile extracted from a broadly referenced NREL horizontal axis wind turbine (HAWT), as stated by Jonkman et al.23. Star-CCM+ v2019.124 commercial code was used to run these simulations.

To perform the mentioned simulations, a two-dimensional structured O-mesh was generated, with the studied airfoil on its center. Following the study of Sørensen et al.25, the radius of this mesh (R) was defined as a function of the chord length (c) of the airfoil, $$R=32\cdot c$$. Therefore, as the chord length of the DU91W250 airfoil is equal to 1 m, R was set at 32 m. This mesh consists of around 207,000 cells. Non-slip conditions were assigned to the contour of the airfoil, and the first cell height ($$\Delta z$$) normalized with the chord length of the airfoil was set at $$\Delta z/c =1.35\cdot {10}^{-6}$$. On previous works, Fernandez-Gamiz et al.3 studied the mesh dependency of the used mesh, and showed a dependency below 4% for the calculation of $${C}_{L}$$ and drag $${C}_{D}$$ coefficients. Figure 1a provides a general view of the mesh, and Fig. 1b, a close-up view of the near-airfoil region.

In order to validate the results obtained with this mesh, the obtained results are compared with the experimental data results from the LM Low Speed Wind Tunnel obtained by Timmer26. For this comparison, the clean airfoil with different angles of attack is considered.

In addition, a mesh convergence study is performed considering the lift-to-drag ratio, in order to verify that the numerical solution is independent of the mesh resolution. With this purpose, the General Richardson Extrapolation27 method is used. To apply this method, three different meshes are generated, with a mesh refinement equal to 2. The fine mesh consists of around 207,000 cells, the medium mesh of around 103,000 cells and the coarse mesh of around 52,000 cells. Table 1 summarizes the results obtained for the grid convergence study, were RE is the solution of the Richardson Extrapolation, p the order of accuracy and R convergence condition. As the obtained R values are between 0 and 1, the solution is within the asymptotic range of convergence for all the tested angles of attack. Figure 2 compares the CFD results obtained with each mesh and the experimental data. As the results demonstrate, the results obtained with the fine mesh are close to the experimental ones, and the convergence criteria is fulfilled, which means that the mesh is suitable for these simulations.

Two different flow control devices were added to the TE of the airfoil: Gurney flaps and rotating microtabs. A total of 48 different cases with Gurney flaps were considered, with different lengths and angles of attack; and 105 different cases with rotating microtabs, with different lengths (L), orientations (β) and angles of attack. The remaining five cases consist of the airfoil without flow control devices. All the studied cases are summarized in Table 2.

These flow control devices were added to the previously-explained mesh by the cell-set model. This modelling technique consists of defining a geometry on an already-generated mesh, and then, splitting this geometry into a new region, and defining it as wall with no-slip conditions. As demonstrated by Ballesteros-Coll et al.6, this model is suitable for this kind of problems, since a global relative error of 3.784% of this model in comparison with the fully-resolved model was obtained in that study. Figure 3 illustrates an example of the cell-set model implementation for a rotating microtab.

Regarding the fluid physics, the dynamic viscosity of the air was set at $$\mu =1.855\cdot {10}^{-5} {\rm Pa} \cdot {\rm s}$$, and the density was set at $$\rho =1.2041 \text{kg/}{\text{m}}^{3}$$. The freestream velocity of the flow was set at $${U}_{\infty }=30 \text{m/s}$$, which means that the Reynolds number (Re) is equal to $$2\cdot {10}^{6}$$. For turbulence modelling RANS-based k-ω Shear Stress Transport (SST) model by Menter28 was chosen, which combines the k-ω model for the near-wall zones and k-ε model for the regions far from the walls. UpWind algorithm was employed for the pressure–velocity coupling and a linear upwind second order scheme was used to discretize the mesh.

### Convolutional neural network

#### Input and output layers

The domain is represented by four different 128 $$\times$$ 256 layers. The first two layers represent the geometry of the airfoil and the flow control device, and the other two layers represent the velocity components in both directions.

The layers describing the domain are generated by means of a binary representation, where the points belonging to the geometry are identified with a 1 and those which do not belong to the geometry are identified with a 0. One of these layers provides an overview of the airfoil, while the other provides a close-up view of the airfoil TE, showing the flow control device in detail. Figure 4a shows the zones represented by each layer, and Fig 4b,c display and example of these two layers.

The velocity layers provide the value of the velocity components. These layers are used to determine the AoA of the airfoil accurately, since in the above-explained binary-representation layers, slight variations in orientation may not be correctly represented. Figure 5a,b show the velocity layers, which represent the x and y components of the velocity, respectively.

Concerning the output of the networks, three layers and two scalars are considered. The three layers correspond to the velocity (both components) and pressure fields on the TE of the airfoil, and the scalars are the $${C}_{D}$$ and $${C}_{L}$$ coefficients.

To prepare the output layers, the values were first interpolated to fit into a 128 $$\times$$ 256 arrays. Then, the values of those arrays were normalized, following Expressions (1), (2) and (3).

$${u}_{x}^{*}=\frac{{u}_{x}}{{u}_{\infty }}$$
(1)
$${u}_{y}^{*}=\frac{{u}_{y}}{{u}_{\infty }}$$
(2)
$${p}^{*}=\frac{p}{\rho \cdot {u}_{\infty }^{2}}$$
(3)

where $${u}_{x}^{*}$$, $${u}_{y}^{*}$$ and $${p}^{*}$$ are the dimensionless variables.

Finally, all the values of the layers are ranged between 0 and 1 following Expression (4), in order to speed up and enhance the training process.

$${\Phi }^{\mathrm{^{\prime}}}=\frac{\Phi -\mathrm{min}(\Phi )}{\mathrm{max}\left(\Phi \right)-\mathrm{min}(\Phi )}$$
(4)

where Φ is replaced by each dimensionless variable.

This last step is also followed to range between 0 and 1 the input layers and the output scalars corresponding to the coefficients.

#### CNN architecture

In the present paper, two different CNN are considered, one for velocity and pressure field prediction, and another one for drag and lift coefficient prediction. This networks were designed and trained using MATLAB 2021a29 commercial code with its Deep Learning Toolbox30.

For velocity and pressure field prediction, an U-Net architecture31 is proposed, based on the previous works from Ribeiro et al.13 and Thuerey et al.18. The U-Net architecture is a special case of an encoder-decoder network. The proposed network consists of four encoder/decoder blocks. Each encoder block contains two convolutional layers. The first one is followed by a ReLU (Rectifier Linear Unit) layer, and the second one is followed by a ReLU layer and a Max Pooling layer. The kernel size of the first two encoding blocks is equal to 5, and strided convolutions are performed on those blocks, in order to reduce the data size for the training step. The kernel size of the last blocks is equal to 3. After each encoding block, the number of filters is doubled. The decoding blocks perform the reverse process of their symmetrical blocks of the encoding phase, and they are connected to the encoding blocks by concatenation layers. Figure 6 provides a schematic view of the explained network.

For lift and drag coefficient prediction, only the encoding part of the CNN is considered. In this case, a fully connected layer is added after the last layer. Unlike the complete structure, this network does not return a layer, it returns two scalar values, $${C}_{D}$$ and $${C}_{L}$$. Figure 7 shows a schematic view of this network.

For the network training, Adam32 optimizer is employed in both cases, with a batch size of 64. For field prediction, a learning rate of 0.001 and a weight decay of 0.0005 is selected; and for coefficient prediction a learning rate of 0.0001 and a weight decay of 0.0005. From the dataset of 158 samples, 21 are considered for testing the network. The other 137 are divided into 70% training and 30% validation for field prediction and 90% training and 10% validation for coefficient prediction.

These hyperparameters and data-splitting ratios were selected after training this network with 27 different configurations of these parameters. From the analyzed configurations, the selected ones provide the minimum Root-Mean-Square Error (RMSE) of the analyzed magnitude. Appendix A provides a summary of the considered configurations and the obtained RMSE for each magnitude and coefficient.

## Results and discussion

In order to determine the accuracy of the proposed CNN, the predictions of this CNN have been compared with those obtained by CFD. For this comparison, the 21 simulations of the test-set mentioned above are considered.

### Velocity and pressure field prediction

The velocity and pressure fields obtained by the CNN are both qualitatively and quantitatively compared in order to determine the accuracy of the proposed CNN for field prediction. For the qualitative comparison, four different cases are considered, each one with a different geometry and AoA. All these cases can be found in Fig. 8.

The results show that the proposed CNN is able to accurately predict the velocity and pressure fields around the flow control devices in all the tested cases. The most problematic area is the wake behind the flow control device in all the analyzed cases, especially when predicting $${u}_{x}$$. Some errors are also visible in the contour of the airfoil. In geometries which have surfaces perpendicular to the flow, i.e., Gurney flaps and rotating microtabs with high angles of orientation, slight errors of velocity fields appear at the front side of the flow control device. However, the CNN is able to reliably predict the flow characteristics, and all these mentioned errors are not considered significant.

In order to obtain a quantitative view of these results, data distribution histograms are made for each analyzed magnitude. In agreement with the qualitatively compared fields, the data distribution histograms show nearly equal shapes, being the ranges with the most data the only ones where differences between the two methods can be appreciated. Figure 9 shows data distribution histograms.

In addition, the arithmetic mean (µ) and standard deviation (σ) of both methods are calculated from the data distribution histograms. These two values, in accordance with all the results shown above, show almost equal values for all magnitudes, as shown in Table 3.

### Aerodynamic coefficient prediction

In order to evaluate the accuracy of the network for predicting aerodynamic coefficients, the predicted $${C}_{D}$$ and $${C}_{L}$$ coefficients are compared to the benchmark values obtained by CFD simulations. In addition, the lift-to-drag ($${C}_{L}/{C}_{D}$$) coefficient, which is calculated from the predicted coefficients, is also compared. The plots from Fig. 10 provide this comparison.

As the plots demonstrate, the CNN is able to reliably predict aerodynamic coefficients. In the case of microtabs, $${C}_{D}$$ values increase exponentially with low AoAs, and subsequently, for AoA > 6°, they decrease slightly. In the case of the Gurney flaps, this upward trend can also be seen throughout the analyzed range. In contrast, $${C}_{L}$$ values follow a linear trend with respect to AoA in all the cases considered. With microtabs, the $${C}_{D}/{C}_{L}$$ ratio shows a sharp rise for low AoA, followed by a less pronounced drop and a further stabilization for high AoA. The rise and the fall are more pronounced as the length of the microtab increases. When using Gurney flaps the $${C}_{D}/{C}_{L}$$ coefficient follows a logarithmic trend, tending to flatten out for AoA > 4°.

$${C}_{D}$$ predictions show small discrepancies between the two methods. With microtabs these differences become more noticeable as the microtab length increases and the AoA decreases, and with Gurney flaps as the AoA increases. In contrast, the predictions of the $${C}_{L}$$ are almost the same in all the studied cases. Nevertheless, the results show that the network is able to predict this trend, with values close to those obtained by CFD.

In order to quantify the results, the absolute and relative errors of the predictions are studied. These errors are shown in Table 4. As previously demonstrated, the errors of the $${C}_{D}$$ predictions are higher than those of $${C}_{L}$$, with a maximum relative error of 32.44%. However, the average relative error is 6.17%, which is considered acceptable. Low errors are observed in all the $${C}_{L}$$ predictions, with an average relative error of 0.827%.

### Performance analysis

The main objective of using neural networks to predict flows is to reduce the computational time required to run CFD simulations. Therefore, the computational time requirements for each method are compared. As shown in Table 5, neural networks clearly outperform CFD simulations in terms of computational time. As expected due to its simplicity, the CNN for aerodynamic coefficient prediction is the fastest one, being 16,148 times faster than the CFD simulations. However, the complete CNN used for field prediction is also considerably fast, being 7529 times faster than the CFD. Regarding the network training time, it took 12 h to train the CNN for field prediction and 5 h to train the CNN for coefficient prediction, which means a total of 17 h. This is slightly longer than the time required to perform a single CFD simulation. A single core of an Intel Xeon 5420 CPU was used for running CFD simulations and CNNs.

## Conclusions

In the present work, the implementation of flow control elements in airfoils using deep learning techniques is analyzed. With that objective, two different CNNs are proposed. One of them predicts the velocity and pressure fields around flow control devices implemented in the TE of the DU91W(2)250 airfoil, and the other one predicts the $${C}_{D}$$ and $${C}_{L}$$ aerodynamic coefficients of the airfoil for the same cases. These networks were trained and evaluated using the results obtained from CFD simulations, in which the cell-set model was used to implement the flow control devices. The dataset contains a total of 158 cases, with two different flow control devices, rotating microtabs and Gurney flaps, with different geometries and under different conditions.

Regarding the CNN for field prediction, the results indicate that the proposed network is able to predict the main flow characteristics around the flow control device, with very low errors, which mainly appear on the wake behind the flow control device and on the contour of the airfoil. This is attributed to the fact that the area of the wake behind the flow control device is the area in which the most differences appear between samples. Therefore, this area is the most conflicting one for the learning process of the neural network, and consequently, the area where most errors appear. Nevertheless, the network is able to correctly predict the main flow characteristics in this region.

With respect to aerodynamic coefficients, the CNN is also able to predict them accurately, with mean relative errors of 6.17% for $${C}_{D}$$ and 0.827% for $${C}_{L}$$. In both cases, the networks are sensitive to small changes of the geometry or the AoA, which is a key feature for geometry optimization.

In terms of computational time, the proposed networks clearly outperform the CFD simulations, reducing the computational time in four orders of magnitude.

Therefore, this paper demonstrates that flow control devices can be studied by means of neural networks, with acceptable errors and a significative reduction of required computational time and resources.