A Foul-Weather Fountain 251 a Foul-Weather fountain lyan K Card R Ernie e esser Jeffrey H Giansiracusa University of Washin Seattle Wa Advisor: James Allen Morrow Introduction We devise a fountain control algorithm to monitor wind conditions and ensure that a fountain at the center of a plaza fires water high enough to be dazzling while not drenching the pedestrian areas surrounding the fountain We construct a model of a fountain based on the physics of falling water under various wind conditions through computer simulation. Using complex analytic techniques, we model the wind flow through the plaza and estimate how anemometer readings from a nearby rooftop relate to plaza conditions We construct four algorithms-two intelligent algorithms, a conservatiy approach, and an enthusiastic system-to control the fountain We devise a measure of unacceptable spray levels outside the fountain and use this criterion to compare performance. First, we examine the behavior of these algorithms under general abstract wind conditions. Then we construct a wind signal generator that simulates the conditions of several major cities from meteorological database data, and we compare the performance of our control systems in each city Simulations show that the Conservative and enthusiastic algorithms both perform unacceptably in realistic conditions. The Weighted Average Algorithm works best in gusty cities such as Chicago, but the Averaging Algorithm is superior in calmer cities such as Los Angeles and Seattle The control algorithm cannot possibly respond to changes in conditions at anything below the 10 s scale, since wind is highly variable and the response of the anemometer is somewhat slow [Industrial Weather Products 2002]. The goal is therefore to design the algorithm to operate on a time scale of 10 s up to a couple of hours and adapt the height of the fountain to a maximum safe level The UMAP Journal 23(3)(2002)251-266. Copyright 2002 by COMAP, Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
A Foul-Weather Fountain 251 A Foul-Weather Fountain Ryan K. Card Ernie E. Esser Jeffrey H. Giansiracusa University of Washington Seattle, WA Advisor: James Allen Morrow Introduction We devise a fountain control algorithm to monitor wind conditions and ensure that a fountain at the center of a plaza fires water high enough to be dazzling while not drenching the pedestrian areas surrounding the fountain. We construct a model of a fountain based on the physics of falling water droplets considered as a particle system. We examine the behavior of a fountain under various wind conditions through computer simulation. Using complex analytic techniques, we model the wind flow through the plaza and estimate how anemometer readings from a nearby rooftop relate to plaza conditions. We construct four algorithms—two intelligent algorithms, a conservative approach, and an enthusiastic system—to control the fountain. We devise a measure of unacceptable spray levels outside the fountain and use this criterion to compare performance. First, we examine the behavior of these algorithms under general abstract wind conditions. Then we construct a wind signal generator that simulates the conditions of several major cities from meteorological database data, and we compare the performance of our control systems in each city. Simulations show that the Conservative and Enthusiastic algorithms both perform unacceptably in realistic conditions. The Weighted Average Algorithm works best in gusty cities such as Chicago, but the Averaging Algorithm is superior in calmer cities such as Los Angeles and Seattle. The control algorithm cannot possibly respond to changes in conditions at anything below the 10 s scale, since wind is highly variable and the response of the anemometer is somewhat slow [Industrial Weather Products 2002]. The goal is therefore to design the algorithm to operate on a time scale of 10 s up to a couple of hours and adapt the height of the fountain to a maximum safe level. The UMAP Journal 23 (3) (2002) 251–266. c Copyright 2002 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
252 The UMAP Journal 23.3 (2002) Model of the water jet Ne model the spray from the fountain as a particle system. As water droplets spew forth from the nozzle, they are subjected to forces(gravity, air drag, turbulence, etc. ) We formulate a simplified differential equation governing the motion and then numerically integrate to find the trajectory for each droplet This equation is based on a physically realistic model of small droplets(around 1 mm radius)and we scale it up to an effective model for larger clumps of water(up to 10 cm across) because the physics of turbulence and viscosity at he larger scale cannot be computed accurately We need the following assumptions The drag force is proportional to the square of the speed and to the square f the radius [ Nasa 2002 Droplets break into smaller droplets when subjected to wind. Breakup rate is proportional to relative wind speed and surface area [INobauer 1999 When a droplet breaks turbulence causes the new droplet fragments to move slightly away from their initial trajectory. Modeling a Single Droplet We formulate the motion of a water droplet a m dt=-mgi+mlwul-ir where vis the velocity w is the wind velocity relative to the motion of the droplet (wind vector minus velocity vector), m and r are the droplet's mass and radius, and n is a constant of proportionality. According to the Virtual Science Center Project Team[2002 a raindrop with radius 1 mm falls at a terminal velocity of 7 m/s; so we determine that n=0.855 kg/m. Large drops fall quickly; very tiny drops fall very slowly, mimicking a fine mist that hangs in the air for a long time We assume droplet breakup is a modified Poisson process, with rate λoulr If the breakup rate did not depend on variable parameters w and r2, the process would be a standard Poisson process. We l determine Ao by fitting the water ream of our fountain to the streams of two real fountains: the jet D'Eau of Geneva, Switzerland, and the Five Rivers Fountain of Lights in Miami, Florida When a breakup occurs, we split the droplet into two new droplets and divide the mass randomly, using a uniform distribution. Air turbulence tends to impart to the two new droplets a small velocity component perpendicular to the relative wind direction w. This effect causes a tight stream of water to spread
252 The UMAP Journal 23.3 (2002) Model of the Water Jet We model the spray from the fountain as a particle system. As water droplets spew forth from the nozzle, they are subjected to forces (gravity, air drag, turbulence, etc.). We formulate a simplified differential equation governing the motion and then numerically integrate to find the trajectory for each droplet. This equation is based on a physically realistic model of small droplets (around 1 mm radius) and we scale it up to an effective model for larger clumps of water (up to 10 cm across) because the physics of turbulence and viscosity at the larger scale cannot be computed accurately. We need the following assumptions: • The drag force is proportional to the square of the speed and to the square of the radius [Nasa 2002]. • Droplets break into smaller droplets when subjected to wind. Breakup rate is proportional to relative wind speed and surface area [Nobauer 1999]. • When a droplet breaks, turbulence causes the new droplet fragments to move slightly away from their initial trajectory. Modeling a Single Droplet We formulate the motion of a water droplet as m dv dt = −mgzˆ + η|w| 2wrˆ 2, wherev is the velocity, w is the wind velocity relative to the motion of the droplet (wind vector minus velocity vector), m and r are the droplet’s mass and radius, and η is a constant of proportionality. According to the Virtual Science Center Project Team [2002], a raindrop with radius 1 mm falls at a terminal velocity of 7 m/s; so we determine that η = 0.855 kg/m3. Large drops fall quickly; very tiny drops fall very slowly, mimicking a fine mist that hangs in the air for a long time. We assume droplet breakup is a modified Poisson process, with rate λbreakup = λ0|w|r2. If the breakup rate did not depend on variable parameters |w| andr2, the process would be a standard Poisson process. We l determine λ0 by fitting the water stream of our fountain to the streams of two real fountains: the Jet D’Eau of Geneva, Switzerland, and the Five Rivers Fountain of Lights in Miami, Florida. When a breakup occurs, we split the droplet into two new droplets and divide the mass randomly, using a uniform distribution. Air turbulence tends to impart to the two new droplets a small velocity component perpendicular to the relative wind direction w. This effect causes a tight stream of water to spread
A Foul-Weather Fountain 253 out as it travels, even under zero-wind conditions. We let this velocity nudge have magnitude 2% of the particle' s speed relative to the air and a random perpendicular direction. We give the two drops equal and opposite nudges Putting water drops Together to make a Fountain We define the waterjet as a stream of large water drops. Their size is roughly the size of the nozzle and they leave with an initial velocity equal to the nozzle output velocity( Figure 1) oO A real four Representati。n。 fa fountain Figure 1. A continuous water jet is approximated by a discrete stream of water blobs The water blobs leave at a rate such that the flux of water is equal to the flux given by a nozzle-sized cylindrical stream moving at the same speed To model the turbulence in the jet as the water leaves the nozzle, we give each water blob a normal distribution of radius and initial speed . The standard deviation of blob radii is 10% of the nozzle size The standard deviation of initial speeds is 5% of the initial speed The blobs leave with an angular spread of 3, consistent with industrial high-pressure nozzles [Spray Nozzles 2002] Wind drag in particle streams is significantly reduced for particles follow ing one another closely (nASCar drivers and racing cyclists are intima familiar with this phenomenon). These effects are already incorporated into the dynamics of large water blobs(which can be thought of as representing many small drops moving together
A Foul-Weather Fountain 253 out as it travels, even under zero-wind conditions. We let this velocity nudge have magnitude 2% of the particle’s speed relative to the air and a random perpendicular direction. We give the two drops equal and opposite nudges. Putting Water Drops Together to Make a Fountain We define the water jet as a stream of large water drops. Their size is roughly the size of the nozzle, and they leave with an initial velocity equal to the nozzle’s output velocity (Figure 1). Figure 1. A continuous water jet is approximated by a discrete stream of water blobs. The water blobs leave at a rate such that the flux of water is equal to the flux given by a nozzle-sized cylindrical stream moving at the same speed. To model the turbulence in the jet as the water leaves the nozzle, we give each water blob a normal distribution of radius and initial speed: • The standard deviation of blob radii is 10% of the nozzle size. • The standard deviation of initial speeds is 5% of the initial speed. • The blobs leave with an angular spread of 3◦, consistent with industrial high-pressure nozzles [Spray Nozzles 2002]. Wind drag in particle streams is significantly reduced for particles following one another closely (NASCAR drivers and racing cyclists are intimately familiar with this phenomenon). These effects are already incorporated into the dynamics of large water blobs (which can be thought of as representing many small drops moving together)
254 The UMAP Journal 23.3(2002) Fitting the Fountain The Five Rivers Fountain of Lights in Daytona, Florida, is one of the largest fountains in the world. It consists of several water jets, and on low-wind days each propels a water stream 60 m high and 120 m out. The jet D Eau in geneva Switzerland, anoth pressive fountain, shoots a 30 cm-diameter stream of water at 60 m/s straight up. The water reaches a height of 140 m and on an average breezy day (wind speed 5 m/s)returns to earth approximately 35 m downwind from the nozzle [micheloud cie 2002(Figure 2 45d-西 30 m Figure 2. The jet D Eau and the Fountain of lights To determine Xo, we first match our geometry to the Five rivers fountain of Lights. We fix Ao so that with an initial velocity such that the stream reaches height of 60 m, it returns to the ground at a distance of just over 100 m. Too large a Ao results in the water breaking up too quickly into tiny droplets, which have a much lower terminal velocity and thus fail to reach the desired distance; if the value is too small, then an unrealistically small amount of spray is produced and the water blob travels too far. The results are summarized in Table 1 We set Ao 5000. The results are highly insensitive to this parameter varying o by a factor of 2 cause only a 15% changes in the distances. Therefore, even though our method for determining this parameter is fairly rough, the important behavior is much more strongly affected by other parameters Table 1 Comparison between real fountains and our model Jet DEa eal model model Height(m) 140 121 Distance() 3 120 100 We conclude from this comparison that our model reproduces the spray patterns of extreme fountains accurate to within about 15%%. We expect that for
254 The UMAP Journal 23.3 (2002) Fitting the Fountain The Five Rivers Fountain of Lights in Daytona, Florida, is one of the largest fountains in the world. It consists of several water jets, and on low-wind days each propels a water stream 60 m high and 120 m out. The Jet D’Eau in Geneva, Switzerland, another impressive fountain, shoots a 30 cm-diameter stream of water at 60 m/s straight up. The water reaches a height of 140 m and on an average breezy day (wind speed 5 m/s) returns to earth approximately 35 m downwind from the nozzle [Micheloud & Cie 2002] (Figure 2). Figure 2. The Jet D’Eau and the Fountain of Lights. To determine λ0, we first match our geometry to the Five Rivers Fountain of Lights. We fix λ0 so that with an initial velocity such that the stream reaches a height of 60 m, it returns to the ground at a distance of just over 100 m. Too large a λ0 results in the water breaking up too quickly into tiny droplets, which have a much lower terminal velocity and thus fail to reach the desired distance; if the value is too small, then an unrealistically small amount of spray is produced and the water blob travels too far. The results are summarized in Table 1. We set λ0 = 5000. The results are highly insensitive to this parameter; varying λ0 by a factor of 2 cause only a 15% changes in the distances. Therefore, even though our method for determining this parameter is fairly rough, the important behavior is much more strongly affected by other parameters. Table 1. Comparison between real fountains and our model. Jet D’Eau Five Rivers Fountain real model real model Height (m) 140 121 60 62 Distance (m) 35 30 120 100 We conclude from this comparison that our model reproduces the spray patterns of extreme fountains accurate to within about 15%. We expect that for
A Foul-Weather Fountain 255 a plaza-sized fountain, our model will be more accurate, since our formulas for breakup and drag force are derived under less extreme conditions Wind Flow Through the Plaza Buildings and other structures in an urban environment can cause signif- icant disturbances to wind flow patterns; rooftop and street-level conditions can often be quite different. To model the plaza wind, we assume There are no significant structures between the buildings beside of the plaza The plaza is large, so effects caused when wind flow leaves the plaza negligible at the plaza center; the significant effects are entirely caused at inward boundary passage The air flow is smooth enough so that turbulent vortices are negligible Formulation We approximate the geometry of the plaza as in Figure 3 and use complex analytic flow techniques[ Fisher 1990, 225 Plaza Figure 3. Schematic representation of the relevant features of the plaza With a Schwarz-Cristoffel mapping of a smooth horizontal flow from the upper half of the complex plane onto the region above the plaza, we obtain a flow function for the wind as it enters the plaza area r0={(+(2-1y12+kg(++(+i92-1y/)} where t parametrizes a streamline for each value of c. These streamlines are plotted in Figure 4, where the acceleration of the wind as it passes over the building edge and the decreased velocity in the plaza are both clearly visible The flow velocity v is inversely proportional to the streamline spacing, so the horizontal component of it is O dc
A Foul-Weather Fountain 255 a plaza-sized fountain, our model will be more accurate, since our formulas for breakup and drag force are derived under less extreme conditions. Wind Flow Through the Plaza Buildings and other structures in an urban environment can cause significant disturbances to wind flow patterns; rooftop and street-level conditions can often be quite different. To model the plaza wind, we assume: • There are no significant structures between the buildings beside of the plaza. • The plaza is large, so effects caused when wind flow leaves the plaza are negligible at the plaza center; the significant effects are entirely caused at the inward boundary passage. • The air flow is smooth enough so that turbulent vortices are negligible. Formulation We approximate the geometry of the plaza as in Figure 3 and use complex analytic flow techniques [Fisher 1990, 225]. Figure 3. Schematic representation of the relevant features of the plaza. With a Schwarz-Cristoffel mapping of a smooth horizontal flow from the upper half of the complex plane onto the region above the plaza, we obtain a flow function for the wind as it enters the plaza area: Γc(t) = h0 π (t + ic) 2 − 1 1/2 + log t + ic + (t + ic) 2 − 1 1/2 , where t parametrizes a streamline for each value of c. These streamlines are plotted in Figure 4, where the acceleration of the wind as it passes over the building edge and the decreased velocity in the plaza are both clearly visible. The flow velocity v is inversely proportional to the streamline spacing, so the horizontal component of it is vx = Im ∂Γc ∂c .
256 The UMAP Journal 23.3(2002) Figure 4. Streamlines for wind flow entering the plaza; decreased wind speed at the plaza level is apparent. Note the highly increased wind speed near the edge of the building The horizontal velocity profile for a streamline that passes about 3 m above the building roof (corresponding to c=0.6)is plotted in Figure 5; 3 m is reasonable height for an anemometer mounting. From these graphs, one can see that the wind speed through the plaza center(at a distance of 30 to 40 m from edge) is approximately half of the rooftop wind speed 0 0.05 102030 Figure 5. Horizontal velocity profile for the streamline corresponding to c=0.6. This streamline passes above the building s roof at a height of 3 m, a reasonable anemometer mounting height This calculation is validated by its excellent agreement with the findings of Santamouris and Dascalaki [2000], who report that in flows perpendicular to a street the ground-level speeds are between zero and 55% of the free-stream Results We conclude from this flow model
256 The UMAP Journal 23.3 (2002) Figure 4. Streamlines for wind flow entering the plaza; decreased wind speed at the plaza level is apparent. Note the highly increased wind speed near the edge of the building. The horizontal velocity profile for a streamline that passes about 3 m above the building roof (corresponding to c = 0.6) is plotted in Figure 5; 3 m is a reasonable height for an anemometer mounting. From these graphs, one can see that the wind speed through the plaza center (at a distance of 30 to 40 m from edge) is approximately half of the rooftop wind speed. Figure 5. Horizontal velocity profile for the streamline corresponding to c = 0.6. This streamline passes above the building’s roof at a height of 3 m, a reasonable anemometer mounting height. This calculation is validated by its excellent agreement with the findings of Santamouris and Dascalaki [2000], who report that in flows perpendicular to a street the ground-level speeds are between zero and 55% of the free-stream speeds. Results We conclude from this flow model:
A Foul-Weather Fountain 257 Placement of the anemometer is important! It should be mounted near the center of the rooftop to minimize disturbances from the roofs edge The anemometer reports a wind speed that is highly biased! Ground-level wind moves approximately half as fast as the roof-level wind Wind speeds are fairly independent of height. If the fountain is not signif- seems to icantly higher than the surrounding buildings, then it is safe to assume that contradict the wind is spatially constant precious laim Modeling wind variation Over Time The control system must be able to handle a range of weather conditions, from calm up to strongly gusty. We abstract the wind patterns into three gen- eralized types of increasing complexity: algorithms ability to judge the proper height for a given wind spee st the Type 1: A low intensity constant breeze of a few m/s, meant to te Type 2: a breeze varying smoothly over a timescale of a couple of minutes. We use a sinusoidal oscillation in magnitude and direction, with a constant term to reflect the prevailing wind direction of the hour. This type tests the algorithm's ability to adapt to slowly changing conditions Type 3: Sudden unexpected wind gusts, with a few seconds duration and very high intensity. We model the occurrence of a gust as a Poisson process and distribute the gust durations and intensities normally. The mean and variance are chosen to produce reasonable results. This is perhaps the most important test, since the gusty scenario can easily fool a naive algorithm Generating a realistic wind signal We parametrize the wind profile of a location by four numbers The mean steady wind steady The mean gust strength gust, where a gust is defined to be variation on the sub-15 s timescale The mean gust duration gust The gust deviation ogust From Webmet data [2001], we estimate these characteristic numbers for ome major U.S. cities(Table 2) le construct realistic wind signals from these characteristic numbers to correspond to our types
A Foul-Weather Fountain 257 • Placement of the anemometer is important! It should be mounted near the center of the rooftop to minimize disturbances from the roof’s edge. • The anemometer reports a wind speed that is highly biased! Ground-level wind moves approximately half as fast as the roof-level wind. • Wind speeds are fairly independent of height. If the fountain is not signif- seems to contradict previous claim!! icantly higher than the surrounding buildings, then it is safe to assume that the wind is spatially constant. Modeling Wind Variation Over Time The control system must be able to handle a range of weather conditions, from calm up to strongly gusty. We abstract the wind patterns into three generalized types of increasing complexity: • Type 1: A low intensity constant breeze of a few m/s, meant to test the algorithm’s ability to judge the proper height for a given wind speed. • Type 2: A breeze varying smoothly over a timescale of a couple of minutes. We use a sinusoidal oscillation in magnitude and direction, with a constant term to reflect the prevailing wind direction of the hour. This type tests the algorithm’s ability to adapt to slowly changing conditions. • Type 3: Sudden unexpected wind gusts, with a few seconds duration and very high intensity. We model the occurrence of a gust as a Poisson process and distribute the gust durations and intensities normally. The mean and variance are chosen to produce reasonable results. This is perhaps the most important test, since the gusty scenario can easily fool a naive algorithm. Generating a Realistic Wind Signal We parametrize the wind profile of a location by four numbers: • The mean steady wind µsteady. • The mean gust strength µgust, where a gust is defined to be variation on the sub-15 s timescale. • The mean gust duration tgust. • The gust deviation σgust. From WebMET data [2001], we estimate these characteristic numbers for some major U.S. cities (Table 2). We construct realistic wind signals from these characteristic numbers to correspond to our types:
258 The UMAP Journal 23.3 (2002) Table 2 Characteristic parametrization of several major U.S.cities. These parameters specify the plaza ind conditions, which are slightly milder than the free-stream conditions Agust (m/s)(m/s)(s)(m/s) Seattle, wa Chicago, IL 2.0 4.0 2.3 4.2 Los Angeles, CA Nashin DC 1.3 34 3.0 1.0 Type 1: constant wind of strength 3asteadyr Type 2: sinusoidal oscillations of amplitude sTeady, and Type 3: a gust signal with mean amplitude gust, amplitude standard devi- ation gust, duration mean gust, and deviation 2tgust Figure 6 shows a comparison of wind signals for Seattle and Chicago; the extreme gustiness of the"Windy City"is apparent 0八八 Figure 6. Horizontal velocity profile
258 The UMAP Journal 23.3 (2002) Table 2. Characteristic parametrization of several major U.S. cities. These parameters specify the plaza wind conditions, which are slightly milder than the free-stream conditions. µsteady µgust tgust σgust (m/s) (m/s) (s) (m/s) Seattle, WA 1.2 2.25 6.0 0.7 Chicago, IL 2.0 4.0 3.0 4.0 Boston, MA 2.3 4.2 4.0 2.2 Los Angeles, CA 1.7 2.0 3.0 0.7 Washington DC 1.3 3.4 3.0 1.0 • Type 1: constant wind of strength 2 3µsteady, • Type 2: sinusoidal oscillations of amplitude 2 3µsteady, and • Type 3: a gust signal with mean amplitude µgust, amplitude standard deviation σgust, duration mean tgust, and deviation 1 2 tgust. Figure 6 shows a comparison of wind signals for Seattle and Chicago; the extreme gustiness of the “Windy City” is apparent. Figure 6. Horizontal velocity profile
A Foul-Weather Fountain 259 Fountain Control algorithms We also create a"Hurricane Floyd " wind profile by multiplying a Chicago wind signal by a factor that damps the wind to zero early on(the calm before the storm) and then amplifies it to hurricane level over a period of 10 min The goal of the control algorithm is to respond to the anemometer data b maximizing the height of the fountain while minimizing the probability of the laza area outside the fountain pool being drenched. The control algorithm has access to anemometer readings and direct control over the nozzle spee The control system must have some knowledge of how the water spray travel distance relates to nozzle speed and wind speed. We develop two com plementary measures of spray distance and tabulate the relationship between them and nozzle/wind conditions. The algorithms that we develop combine this table with an estimate of possible future wind speed (based on the current wind and a stored recent history) to decide on a good nozzle speed Measures of Water Spray distance Our measures of spray distance are the radius within which 99%o of sprayed water lands, and a threshold for acceptable water density outside the fountain that corre sponds roughly to a light rain: 1 cm in 10 h (2.8 x 104 mm/s) n simulations over a suitably long time period, we find that these two measures to agree to within 1% We evaluate the performance of our control algorithm by measuring how the spray distance compares to the actual radius of the pool. If the spray radius goes beyond the pool radius, then people might become unacceptably wet However, if this radius is significantly less than the pool radius, then we are not getting as much height out of the fountain as we could Constructing the Control System We begin with a few useful assumptions Variation in wind direction can be safely ignored. We use the triangle in- equality: If the wind pushes a drop first in one direction and then in another, it will necessarily land nearer to the fountain than if it had been pushed in one direction continuousl The algorithm has access to real-time anemometer data averaged over 10-s intervals as well as at least a 10-min history of measurements. Even if the anemometer responds faster than 10 s, it is nonsensical to vary the fountain any faster than this, because the water requires approximately this much time to complete its flight
A Foul-Weather Fountain 259 Fountain Control Algorithms We also create a “Hurricane Floyd” wind profile by multiplying a Chicago wind signal by a factor that damps the wind to zero early on (the calm before the storm) and then amplifies it to hurricane level over a period of 10 min. The goal of the control algorithm is to respond to the anemometer data by maximizing the height of the fountain while minimizing the probability of the plaza area outside the fountain pool being drenched. The control algorithm has access to anemometer readings and direct control over the nozzle speed. The control system must have some knowledge of how the water spray travel distance relates to nozzle speed and wind speed. We develop two complementary measures of spray distance and tabulate the relationship between them and nozzle/wind conditions. The algorithms that we develop combine this table with an estimate of possible future wind speed (based on the current wind and a stored recent history) to decide on a good nozzle speed. Measures of Water Spray Distance Our measures of spray distance are • the radius within which 99% of sprayed water lands, and • a threshold for acceptable water density outside the fountain that corresponds roughly to a light rain: 1 cm in 10 h (2.8 × 10−4 mm/s). In simulations over a suitably long time period, we find that these two measures to agree to within 1%. We evaluate the performance of our control algorithm by measuring how the spray distance compares to the actual radius of the pool. If the spray radius goes beyond the pool radius, then people might become unacceptably wet. However, if this radius is significantly less than the pool radius, then we are not getting as much height out of the fountain as we could. Constructing the Control System We begin with a few useful assumptions: • Variation in wind direction can be safely ignored. We use the triangle inequality: If the wind pushes a drop first in one direction and then in another, it will necessarily land nearer to the fountain than if it had been pushed in one direction continuously. • The algorithm has access to real-time anemometer data averaged over 10-s intervals as well as at least a 10-min history of measurements. Even if the anemometer responds faster than 10 s, it is nonsensical to vary the fountain any faster than this, because the water requires approximately this much time to complete its flight
260 The UMAP Journal 23.3 (2002) For concreteness, we focus on the plaza configuration of Figure 7. Most importantly, the fountain is at the center of a circular pool of radius 5 m 50m Figure 7. The layout of our hypothetical model plaza We display the spray distance as a function of wind speed and nozzle speed in Figure 8. An estimate of how far a water droplet can travel starting at height zo, falling at its terminal velocity vt, and moving at horizontal wind speed w is distance≈ The smallest droplets that our simulations produce have radii of about 1 mm with corresponding terminal velocity 7m/s. For specific heights and winds,we find that this rough estimate is usually within 30% of the corresponding min- imum safe distance shown in Figure 8, a good indication that our simulations produce reasonable results The Control algorithms We formulate four control algorithms Averaging Algorithm: This algorithm considers an average of the previous 10 min of wind data and the sample variance The worst-case scenario is estimated to be a wind strength of one standard deviation above the average
260 The UMAP Journal 23.3 (2002) • For concreteness, we focus on the plaza configuration of Figure 7. Most importantly, the fountain is at the center of a circular pool of radius 5 m. Figure 7. The layout of our hypothetical model plaza. We display the spray distance as a function of wind speed and nozzle speed in Figure 8. An estimate of how far a water droplet can travel starting at height z0, falling at its terminal velocity vt, and moving at horizontal wind speed w is distance ≈ z0w vt . The smallest droplets that our simulations produce have radii of about 1 mm with corresponding terminal velocity 7 m/s. For specific heights and winds, we find that this rough estimate is usually within 30% of the corresponding minimum safe distance shown in Figure 8, a good indication that our simulations produce reasonable results. The Control Algorithms We formulate four control algorithms: • Averaging Algorithm: This algorithm considers an average of the previous 10 min of wind data and the sample variance. The worst-case scenario is estimated to be a wind strength of one standard deviation above the average
