Wind and Wate 235 Wind and waterspray Tate jarrow Colin landon Mike powell U.S. Military Academy West point n Advisor: David Sanders Introduction Given anemometer readings from a nearby building, the task is to devise an algorithm that controls the height of a fountain in an open square. Our mission is to keep passersby dry and yet have the fountain look as impressive as possible. With ever-changing winds, we must devise a scheme to regulate the flow of water through the fountain to ensure that the bulk of the water shot into the air falls back to the ground within the fountain basin boundary. Our model considers many factors and is divided into five basic parts The conversion of wind speed on top of the building to wind speed at ground level based on height and the force of drag The determination of initial velocity, maximum height, and time of flight from fountain nozzle characteristics, using Bernoulli's equation and the rate of flow equation of continui The assessment of the displacement effects of the wind on the water's ascent. The assessment of the displacement effects of the wind on the waters de- scent The calculation of the optimal flow rate by comparing the water's total hor izontal displacement to the radius of the fountain basir After creating this model in a Math CAD worksheet, we solved every func- tion involved in this model as a function of the water flow rate. This worksheet takes the input from several variables such as the nozzle radius, the maximum flow rate the fountain can handle, the dimensions of the building on which the The LIMAP Journal23(3)(2002)235-250. 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 for components of this work owned by others than COMAP must be ho permitted, but copyrights advantage and that copies bear this notice. Abstracting with credit is nored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
Wind and Waterspray 235 Wind and Waterspray Tate Jarrow Colin Landon Mike Powell U.S. Military Academy West Point, NY Advisor: David Sanders Introduction Given anemometer readings from a nearby building, the task is to devise an algorithm that controls the height of a fountain in an open square. Our mission is to keep passersby dry and yet have the fountain look as impressive as possible. With ever-changing winds, we must devise a scheme to regulate the flow of water through the fountain to ensure that the bulk of the water shot into the air falls back to the ground within the fountain basin boundary. Our model considers many factors and is divided into five basic parts: • The conversion of wind speed on top of the building to wind speed at ground level based on height and the force of drag. • The determination of initial velocity, maximum height, and time of flight from fountain nozzle characteristics, using Bernoulli’s equation and the rate of flow equation of continuity. • The assessment of the displacement effects of the wind on the water’s ascent. • The assessment of the displacement effects of the wind on the water’s descent. • The calculation of the optimal flow rate by comparing the water’s total horizontal displacement to the radius of the fountain basin. After creating this model in a MathCAD worksheet, we solved every function involved in this model as a function of the water flow rate. This worksheet takes the input from several variables such as the nozzle radius, the maximum flow rate the fountain can handle, the dimensions of the building on which the The UMAP Journal 23 (3) (2002) 235–250. 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
236 The UMAP Journal 23.3 (2002) anemometer is placed, and the dimensions of the fountain From theinputs, the model finds the maximum flow rate that keeps the water in the fountain basin As wind speed and direction vary, the model reacts to produce the optimal flow rate Testing the model shows that while the results are reasonable the main source of error results from our drag calculations due to the interaction between wind and the buildings. To solve this error, measurements should be taken at both the building roof and the fountain itself. Although future work would resolve this issue and improve the model, our current model still provides realistic results We provide in Table 1 a list of symbols used Problem Approach We break the overall problem down into several smaller pieces, solve the pieces separately, and put the pieces together to find the overall solution e How the wind is affected as it flows around the buildin How the wind varies with height off the ground How the buildings slow the wind How the wind affects the water from the fountain How the wind affects the water on the way up How the wind affects the water on the way down How to contain that total displacement within the basin Assumptions Overall assumptions The plaza has a fountain in the center with four surrounding buildings Other arrangements can be handled with slight modifications. The buildings are rectangular and have the same dimensions. Most build- ings are rectangular; for same-size buildings, we can use a single constant drag coefficient The distances from each building to the fountain are the same, so each building has the same effect on the fountain water The acceptable splash area is the radius of the fountain basin. A basin surrounds the water jet, and people walking outside the fountain do not want to get we
236 The UMAP Journal 23.3 (2002) anemometer is placed, and the dimensions of the fountain. From the inputs, the model finds the maximum flow rate that keeps the water in the fountain basin. As wind speed and direction vary, the model reacts to produce the optimal flow rate. Testing the model shows that while the results are reasonable, the main source of error results from our drag calculations due to the interaction between wind and the buildings. To solve this error, measurements should be taken at both the building roof and the fountain itself. Although future work would resolve this issue and improve the model, our current model still provides realistic results. We provide in Table 1 a list of symbols used. Problem Approach We break the overall problem down into several smaller pieces, solve the pieces separately, and put the pieces together to find the overall solution. • How the wind is affected as it flows around the buildings. – How the wind varies with height off the ground. – How the buildings slow the wind. • How the wind affects the water from the fountain. – How the wind affects the water on the way up. – How the wind affects the water on the way down. – How to contain that total displacement within the basin. Assumptions Overall Assumptions • The plaza has a fountain in the center with four surrounding buildings. Other arrangements can be handled with slight modifications. • The buildings are rectangular and have the same dimensions. Most buildings are rectangular; for same-size buildings, we can use a single constant drag coefficient. • The distances from each building to the fountain are the same, so each building has the same effect on the fountain water. • The acceptable splash area is the radius of the fountain basin. A basin surrounds the water jet, and people walking outside the fountain do not want to get wet
Wind and Waterspray Table 1 Table of symbols Symbol Meaning(units rate of flow of the fountain(m/s) Reynolds number flow speed(m/s) kinematic viscosity of the fluid force of drag(N) of the wind(kg/m) Wbh speed of the wind before the building at height h(m/s) Cd drag coefficient A surface area interacting with the wind(m2) vz wind speed measured by the anemometer at the height z(m) h height above ground(variable)(m) z height of the building(m) terrain constant number=0.105 hmax maximum height that the water reaches, a function of R(m) kinetic energy of the wind-building system before the wind hits the building o) Kf kinetic energy of the wind-building system after the wind passes the building 0) work done by nonconservative forces, drag of the building times the leng over which it is applied o) d distance over which drag acts, length and width of the building (m) width or half the length of one of the buildings(m) wh speed of the wind after it passes the building at a height h(m/s) mass of the air that interacts with the building in 1 s if the speed Ubh was constant over the face of the building(kg) angle at which the wind strikes the building(o) Ap cross-sectional area of the pipe at the nozzle tip(m2) f speed of the water as it leaves the nozzle(m/s) radius of the pipe at the nozzle tip(m) 9 acceleration due to gravity, 9.803 m/s rc radius of the column of water at a time t after leaving the nozzle with a rate of flow R(m) pressure on the water caused by the wind(N/s Ac surface area of the column of ascending water(m2) density of air(kg/m) mt total mass of the water in the air at a flow rate R(kg) TTotal total time that the water spends in the air with a flow rate R(s) Water density of water(kg/m horizontal acceleration of the water in the column with a flow rate of r and a wind of speed Uh(m/s2) Fc force on the column of water from the wind of speed vh(N) horizontal displacement of the ascending column of water with a flow rate R and vind speed Uh at a time t(m) ressure on a drop of water from wind of speed uh(N/m2) Fa force on the drop from wind of speed uh(N) area of a drop(m2) md mass of a drop of water(kg) horizontal acceleration of the drop of water as a function of rate of flow R and time aavg average horizontal acceleration of a drop during its descent at a rate of flow R and wind of speed vh(m/s2)
Wind and Waterspray 237 Table 1. Table of symbols. Symbol Meaning (units) R rate of flow of the fountain (m3/s) Re Reynolds number υ flow speed (m/s) d a relevant dimension (m) ν kinematic viscosity of the fluid FD force of drag (N) ρ density of the wind (kg/m3) vbh speed of the wind before the building at height h (m/s) Cd drag coefficient A surface area interacting with the wind (m2) vz wind speed measured by the anemometer at the height z (m) h height above ground (variable) (m) z height of the building (m) α terrain constant number = 0.105 hmax maximum height that the water reaches, a function of R (m) Ki kinetic energy of the wind-building system before the wind hits the building (J) Kf kinetic energy of the wind-building system after the wind passes the building (J) WNC work done by nonconservative forces, drag of the building times the length over which it is applied (J) d distance over which drag acts, length and width of the building (m) b width or half the length of one of the buildings (m) vh speed of the wind after it passes the building at a height h (m/s) m mass of the air that interacts with the building in 1 s if the speed vbh was constant over the face of the building (kg) θ angle at which the wind strikes the building (◦) Ap cross-sectional area of the pipe at the nozzle tip (m2) vf speed of the water as it leaves the nozzle (m/s) rp radius of the pipe at the nozzle tip (m) g acceleration due to gravity, 9.803 m/s2 rc radius of the column of water at a time t after leaving the nozzle with a rate of flow R (m) P pressure on the water caused by the wind (N/s2) Ac surface area of the column of ascending water(m2) ρ density of air (kg/m3) mT total mass of the water in the air at a flow rate R (kg) TTotal total time that the water spends in the air with a flow rate R (s) ρwater density of water (kg/m3) ac horizontal acceleration of the water in the column with a flow rate of R and a wind of speed vh (m/s2) Fc force on the column of water from the wind of speed vh (N) xc horizontal displacement of the ascending column of water with a flow rate R and wind speed vh at a time t (m) PD pressure on a drop of water from wind of speed vh (N/m2) Fd force on the drop from wind of speed vh (N)) Ad area of a drop (m2) md mass of a drop of water (kg) ad horizontal acceleration of the drop of water as a function of rate of flow R and time in air t (m/s2) aavg average horizontal acceleration of a drop during its descent at a rate of flow R and wind of speed vh (m/s2)
238 The UMAP Journal 23.3(2002) Figure 1. The fountain in the center of four buildings The fountain does not squirt water higher than the buildings, although shooting water over the roofs would indeed be spectacular The fountain shoots water straight into the air. This is important for our model so that we can predict how the water will flow up, how it will fall, and where it will fall The fountain nozzle creates a single sustained stream of water. This as- sumption enables us to neglect drag as the water reaches its peak height Furthermore, most fountains have a continuous flow of water Wind The pertinent wind flow is around the sides of the buildings, not over them. Since the fountain does not exceed the height of the buildings, it does not interact with wind that passes over the tops of the buildings. This assumption is important in calculating the drag caused by the building The flow of the wind continues in the same direction across the entire plaza. The wind flows through the plaza in a constant direction, goes around obstacles, and resumes the same direction of motion. The wind does not get stuck in the plaza nor react to cars, people, doors, or windows in the plaza Wakes caused by buildings are not factors. The wake that results when wind hits a building and goes around it does not change the velocity after the wake, so the wake force does not influence the wind s speed or direction The fountain is not in the wake of the buildings. With this assumption, there is no need to worry about wake in our model. This is important because wake is too complex to be modeled The change in wind velocity is due solely to drag. The reason that the wind decreases before and after hitting the building is because of drag. This
238 The UMAP Journal 23.3 (2002) Figure 1. The fountain in the center of four buildings. • The fountain does not squirt water higher than the buildings, although shooting water over the roofs would indeed be spectacular. • The fountain shoots water straight into the air. This is important for our model so that we can predict how the water will flow up, how it will fall, and where it will fall. • The fountain nozzle creates a single sustained stream of water. This assumption enables us to neglect drag as the water reaches its peak height. Furthermore, most fountains have a continuous flow of water. Wind • The pertinent wind flow is around the sides of the buildings, not over them. Since the fountain does not exceed the height of the buildings, it does not interact with wind that passes over the tops of the buildings. This assumption is important in calculating the drag caused by the buildings. • The flow of the wind continues in the same direction across the entire plaza. The wind flows through the plaza in a constant direction, goes around obstacles, and resumes the same direction of motion. The wind does not get stuck in the plaza nor react to cars, people, doors, or windows in the plaza. • Wakes caused by buildings are not factors. The wake that results when wind hits a building and goes around it does not change the velocity after the wake, so the wake force does not influence the wind’s speed or direction. • The fountain is not in the wake of the buildings. With this assumption, there is no need to worry about wake in our model. This is important because wake is too complex to be modeled. • The change in wind velocity is due solely to drag. The reason that the wind decreases before and after hitting the building is because of drag. This
Wind and Waterspray 239 assumption allows us to use the law of conservation of energy to predict the change in velocity The anemometer measures wind speed and direction at the top building before any effects of drag. The anemometer must be at the the building on the windward side, elevated above the height of the building so as not to measure any of the effects of the building. To simplify, we assume that it is at the height of the building The wind pattern is the same across the entire plaza as measured at th anemometer. If the pattern changed, the anemometer reading would be invalid The fountain is in a city or urban area. This assumption allows us to determine the effect of the ground on wind speed a given height The drag applied to wind at a certain height is equal to the average effect of drag, that is, to the total drag caused by the building at the velocity at that height divided by the height of the building. This is slightly inaccurate but still produces a reasonable model Water heig ht Water has laminar flow. Water has a constant velocity at any fixed point, regardless of the time. A fluid may actually have various internal flows that complicate the model, but we consider the flow as the jet of water ascends as constant so that we can model it as an ideal fluid Water has nonviscous flow. The water experiences no viscous drag force in the pipe or in the air. The outer edge of the column of water actually interacts with the air and loses some energy to due to the viscosity of both fluids; but since air and water both have a low viscosity, this loss is negligible Water is incompressible. The density of water is constant and does not change as the water moves up into the air and back down again Water Movement sideways The water jet upward flows as a cylinder. since the surface tension of the water holds it together unless it is acted upon by a force, the water should somewhat retain the dimensions of the nozzle from which it emerges The pressure of the wind is a force per area on the water column and on water drops. Wind and water are both fluids, so the interaction between them is a complex relationship of their viscosities; but we also know that wind creates a pressure difference that we can model. We model the force on the water as the pressure caused by a certain velocity of wind multiplied by the surface area of the body of water
Wind and Waterspray 239 assumption allows us to use the law of conservation of energy to predict the change in velocity. • The anemometer measures wind speed and direction at the top of the building before any effects of drag. The anemometer must be at the top of the building on the windward side, elevated above the height of the building so as not to measure any of the effects of the building. To simplify, we assume that it is at the height of the building. • The wind pattern is the same across the entire plaza as measured at the anemometer. If the pattern changed, the anemometer reading would be invalid • The fountain is in a city or urban area. This assumption allows us to determine the effect of the ground on wind speed a given height. • The drag applied to wind at a certain height is equal to the average effect of drag, that is, to the total drag caused by the building at the velocity at that height divided by the height of the building. This is slightly inaccurate but still produces a reasonable model. Water Height • Water has laminar flow. Water has a constant velocity at any fixed point, regardless of the time. A fluid may actually have various internal flows that complicate the model, but we consider the flow as the jet of water ascends as constant so that we can model it as an ideal fluid. • Water has nonviscous flow. The water experiences no viscous drag force in the pipe or in the air. The outer edge of the column of water actually interacts with the air and loses some energy to due to the viscosity of both fluids; but since air and water both have a low viscosity, this loss is negligible. • Water is incompressible. The density of water is constant and does not change as the water moves up into the air and back down again. Water Movement Sideways • The water jet upward flows as a cylinder. Since the surface tension of the water holds it together unless it is acted upon by a force, the water should somewhat retain the dimensions of the nozzle from which it emerges. • The pressure of the wind is a force per area on the water column and on water drops. Wind and water are both fluids, so the interaction between them is a complex relationship of their viscosities; but we also know that wind creates a pressure difference that we can model. We model the force on the water as the pressure caused by a certain velocity of wind multiplied by the surface area of the body of water
240 The UMAP Journal 23.3(2002) The largest particle of water that we want to contain is the size of average drop of water 0.05 mL. The column of water breaks into smaller particles at the peak of its ascent, and they descend individually. We estimate that particles smaller than that size would be acceptable to bystanders hit by them. Any larger particle would have more mass, hence a lower mass-to- surface-area ratio, so the pressure could not push it as far. Water drop behaves as a rigid body. Since a drop is small, internal currents have very little effect. Additionally, the pressure acts over the entire surface area of the drop and should accelerate it as a single body Model Design Effects of Buildings on wind velocit Because buildings surround the fountain the wind velocity at the anemome- ter on top of a building is different from that at fountain level. Buildings disrupt wind currents, slow the wind, and change its direction [Liu 1991, 62]. Buildings create areas of increased turbulence, as well as a wake--an area of decreased pressure--behind the building. Thus, the behavior of wind after it passes a building is so complex as to be almost impossible to model. Hence, we assume that the fountain is located outside of the wakes of the buildings Wind speed reduction The wind inside a group of buildings is less than that outside of the group the interaction between the wind and the buildings causes a decrease in speed The drag between the building and the wind decreases the kinetic energy of the wind and hence its speed Since the fountain is squirting water into the air in a symmetrical shape, the wind affects where the water lands in the same way regardless of the winds direction; so there is no need to find the wind direction after it hits the building rag Nevertheless, wind direction before the wind hits the building is an impor- tant factor. The angle at which the wind hits the building changes the surface area that the wind interacts with and drag changes with area. The drag force Fd Is given Fd where p is the density of air, Ubh is the speed of wind at height h, Cd is the drag coefficient, and A is the surface area interacting with the wind. Therefore, we
240 The UMAP Journal 23.3 (2002) • The largest particle of water that we want to contain is the size of average drop of water 0.05 mL. The column of water breaks into smaller particles at the peak of its ascent, and they descend individually. We estimate that particles smaller than that size would be acceptable to bystanders hit by them. Any larger particle would have more mass, hence a lower mass-tosurface-area ratio, so the pressure could not push it as far. • Water drop behaves as a rigid body. Since a drop is small, internal currents have very little effect. Additionally, the pressure acts over the entire surface area of the drop and should accelerate it as a single body. Model Design Effects of Buildings on Wind Velocity Because buildings surround the fountain, the wind velocity at the anemometer on top of a building is different from that at fountain level. Buildings disrupt wind currents, slow the wind, and change its direction [Liu 1991, 62]. Buildings create areas of increased turbulence, as well as a wake—an area of decreased pressure—behind the building. Thus, the behavior of wind after it passes a building is so complex as to be almost impossible to model. Hence, we assume that the fountain is located outside of the wakes of the buildings. Wind Speed Reduction The wind inside a group of buildings is less than that outside of the group; the interaction between the wind and the buildings causes a decrease in speed. The drag between the building and the wind decreases the kinetic energy of the wind and hence its speed. Since the fountain is squirting water into the air in a symmetrical shape, the wind affects where the water lands in the same way regardless of the wind’s direction; so there is no need to find the wind direction after it hits the building. Drag Nevertheless, wind direction before the wind hits the building is an important factor. The angle at which the wind hits the building changes the surface area that the wind interacts with, and drag changes with area. The drag force Fd is given by Fd = 1 2 ρv2 bhCdA, where ρ is the density of air, vbh is the speed of wind at height h, Cd is the drag coefficient, and A is the surface area interacting with the wind. Therefore, we
Wind and Waterspray 241 must know from which angle the wind approaches the building and how this affects the surface area perpendicular to the direction of the wind For a rectangular building with the narrow face to the wind, Cd MAcdonald 1975, 80] Figure 2 diagrams the plaza and fountain. No matter which way the wind blows, it interacts with a narrow edge of a building Wind from due east or west create a problem for this model, because of discontinuity in the the drag coefficient. Instead, we assume that the coefficient remains constant Figure 2. The plaza. Wind Speed at Differing Heights The speed of wind changes with the height from the ground because there is an additional force on the wind due to surface friction(dependent on the surface characteristics of the ground). The effect of this friction decreases the wind speed is measured from a greater distance to the ground creating faster speeds at greater heights Wind speed also varies because the temperature varies with heigh location. However, if we assume that temperature and ground roughness are constant, a mean speed at a certain height can be modeled by bh=Uz2 MAcdonald 1975, 47, where Ubh is the speed of the wind before it hits the building, Uz is the wind measure ed by the anemometer at the height z of the building, h is the
Wind and Waterspray 241 must know from which angle the wind approaches the building and how this affects the surface area perpendicular to the direction of the wind. For a rectangular building with the narrow face to the wind, Cd = 1.4 [Macdonald 1975, 80]. Figure 2 diagrams the plaza and fountain. No matter which way the wind blows, it interacts with a narrow edge of a building. Wind from due east or west create a problem for this model, because of discontinuity in the the drag coefficient. Instead, we assume that the coefficient remains constant. Figure 2. The plaza. Wind Speed at Differing Heights The speed of wind changes with the height from the ground because there is an additional force on the wind due to surface friction (dependent on the surface characteristics of the ground). The effect of this friction decreases as the wind speed is measured from a greater distance to the ground, creating faster speeds at greater heights. Wind speed also varies because the temperature varies with height and location. However, if we assume that temperature and ground roughness are constant, a mean speed at a certain height can be modeled by vbh = vz h z α [Macdonald 1975, 47], (1) where vbh is the speed of the wind before it hits the building, vz is the wind speed measured by the anemometer at the height z of the building, h is the
242 The UMAP Journal 23.3(2002) variable height of the water, and a is the terrain constant number. We use c=0.105, the value for ground roughness of a city center [Macdonald 1975, We assume that the greatest height of the water that the fountain hits, h does not exceed the height of the building, so we can neglect the drag from the buildings roof (since the wind that goes over the building does not interact with or affect the water in the fountain) Converting Drag to Work We need to convert the drag force into a form that will enable us to deter mine the actual loss of speed. Since drag is a nonconservative force(energy is lost during its application), we can use conservation of energy in the form that says that the initial kinetic energy Ki minus the work WNc done by the nonconservative force equals the final kinetic energy Kf, or Ki=Kf+wNc For the K terms, we use the kinetic energy equation K For Ki, we have abhi for Kf, we have Uh Work is the dot product of the force and the distance that the force is in contact with the surface, or d The work done is the drag force exerted by the building on the wind times the distance that the wind travels along the sides of the building 8 With substitution, we find pubc The drag coefficient Cd is for the entire building. However, we cannot have the entire buildings drag force working on the speed at a specific height or we will overestimate the influence of the drag. Instead we find the average drag per meter of the building. To do this, we divide (3)by the height z of the building, then substitute the result into(2) gmbh=imun+2pubh Caad uah时 us from solving for Uh: the mass m, the area 4, and th UP find Uh at he Mass of air The mass of wind that interacts with the building per second at height h is m= Ubh Apt It is reasonable for convenience to use the average mass over 1 s
242 The UMAP Journal 23.3 (2002) variable height of the water, and α is the terrain constant number. We use α = 0.105, the value for ground roughness of a city center [Macdonald 1975, 48]. We assume that the greatest height of the water that the fountain hits, hmax, does not exceed the height of the building, so we can neglect the drag from the building’s roof (since the wind that goes over the building does not interact with or affect the water in the fountain). Converting Drag to Work We need to convert the drag force into a form that will enable us to determine the actual loss of speed. Since drag is a nonconservative force (energy is lost during its application), we can use conservation of energy in the form that says that the initial kinetic energy Ki minus the work WNC done by the nonconservative force equals the final kinetic energy Kf , or Ki = Kf + WNC. (2) For the K terms, we use the kinetic energy equation K = 1 2mv2. For Ki, we have vbh; for Kf , we have vh. Work is the dot product of the force and the distance that the force is in contact with the surface, or WNC = Fd · d. The work done is the drag force exerted by the building on the wind times the distance that the wind travels along the sides of the building. With substitution, we find WNC = 1 2 ρv2 bhCdAd. (3) The drag coefficient Cd is for the entire building. However, we cannot have the entire building’s drag force working on the speed at a specific height or we will overestimate the influence of the drag. Instead, we find the average drag per meter of the building. To do this, we divide (3) by the height z of the building, then substitute the result into (2): 1 2mv2 bh = 1 2mv2 h + 1 2 ρv2 bhCdAd z . Using (1), we can find vbh at any height h. But the equation still has several unknowns that stop us from solving for vh: the mass m, the area A, and the distance d. Mass of Air The mass of wind that interacts with the building per second at height h is m = vbhAρt. It is reasonable for convenience to use the average mass over 1 s.
Wind and Wate 243 Surface Area Interacting with wind As shown in Figure 3, the surface area as it relates to the drag due to wind is the cross section of the building perpendicular to the wind b Wind Figure 3. Orientation of wind to building. wind strikes the building of width b, is found using trigonometry and give o Therefore, the surface are of the building based on the angle 8 at which th A=(b cos 01+2b sin AD) where z is the height of the building. We take the absolute value of the cosine and sine because we use the direction of the wind measured by the anemometer in terms of a 360 compass Distance The distance d that the wind goes over the building is 3b, the length of one side plus the width of the building because the wind will curve around the building Combining the equations Combining, solving for Uh, and using a=0.105 gives the speed uh at height h. [EDITOR'S NOTE: We do not reproduce the complicated expression Height of the Fountain We find a function to model the maximum height hmax(r)of the fountain as a function of the rate of flow R. we assume that the water acts as an ideal fluid
Wind and Waterspray 243 Surface Area Interacting with Wind As shown in Figure 3, the surface area as it relates to the drag due to wind is the cross section of the building perpendicular to the wind. Figure 3. Orientation of wind to building. Therefore, the surface are of the building based on the angle θ at which the wind strikes the building of width b, is found using trigonometry and gives A = (b| cos θ| + 2b|sin θ|)z, where z is the height of the building. We take the absolute value of the cosine and sine because we use the direction of the wind measured by the anemometer in terms of a 360◦ compass. Distance The distance d that the wind goes over the building is 3b, the length of one side plus the width of the building, because the wind will curve around the building. Combining the Equations Combining, solving for vh, and using α = 0.105 gives the speed vh at height h. [EDITOR’S NOTE: We do not reproduce the complicated expression here.] Height of the Fountain We find a function to model the maximum height hmax(R) of the fountain as a function of the rate of flow R. We assume that the water acts as an ideal fluid
244 The UMAP Journal 23.3 (2002) and that the fountain shoots water straight into the air in a single sustained stream Volume Flow rate and bernoulli s equation We have from Halliday et al. [2001, 334 R=Apuf, or uf (r)= where R is the rate of flow, uf is its speed, A is the cross-sectional area of the pipe, and Tp is the radius of the pipe Based on the effect that we want the fountain to have we make the water column(the radius of the pipe at the tip of the nozzle) have a 6-cm diameter, hence a radius of 0.03 m We use Bernoullis equation[Halliday et al. 2001, 336l, which relates forms of energy in a fluid to calculate the maximum height of the water as it shoots into the ai +p9y1=p2+5p2+p992, where p1 and p2 are the pressure of the water(both are zero since we are looking only at the water in the air) and g is the acceleration due to gravity. At the initial point, we consider the height of the nozzle as having zero gravitational potential energy, so the pressure head pgy1 equals zero. Additionally, the speed un is the speed from (1). At the endpoint, the water has height hmax and the kinetic energy is zero. Substituting and simplifying gives r2 hmax(R) 2 With the radius Tp constant, the height of the top of the water stream varies directly with the square of the rate of flow R. Figure 4 shows the heights for values of R between 0 and 0.04 m2/s of water. Whatever mechanism s the water must be able to vary the flow rate by small amounts, particularly for large R, to maintain the maximum height allowable for the wind conditions The Effect of wind on the water ascent Radius change in ascent Photos of fountains show that the water ascends as a slowly widening col- umn until it reaches its maximum height, then falls back on itself and scatters We can derive an expression that shows the change in the radius as the cylinder of water ascends; but since the change is very small, on the order of 1 mm, we use the initial radius at the nozzle, r, in our calculations
244 The UMAP Journal 23.3 (2002) and that the fountain shoots water straight into the air in a single sustained stream. Volume Flow Rate and Bernoulli’s Equation We have from Halliday et al. [2001, 334] R = Apvf , or vf (R) = R Ap = R πr2 p , where R is the rate of flow, vf is its speed, A is the cross-sectional area of the pipe, and rp is the radius of the pipe. Based on the effect that we want the fountain to have, we make the water column (the radius of the pipe at the tip of the nozzle) have a 6-cm diameter, hence a radius of 0.03 m. We use Bernoulli’s equation [Halliday et al. 2001, 336], which relates forms of energy in a fluid, to calculate the maximum height of the water as it shoots into the air: p1 + 1 2 ρν2 1 + ρgy1 = p2 + 1 2 ρν2 2 + ρgy2, where p1 and p2 are the pressure of the water (both are zero since we are looking only at the water in the air) and g is the acceleration due to gravity. At the initial point, we consider the height of the nozzle as having zero gravitational potential energy, so the pressure head ρgy1 equals zero. Additionally, the speed v1 is the speed from (1). At the endpoint, the water has height hmax and the kinetic energy is zero. Substituting and simplifying gives hmax(R) = R πr2 p 2 2g . With the radius rp constant, the height of the top of the water stream varies directly with the square of the rate of flow R. Figure 4 shows the heights for values of R between 0 and 0.04 m2/s of water. Whatever mechanism s the water must be able to vary the flow rate by small amounts, particularly for large R, to maintain the maximum height allowable for the wind conditions. The Effect of Wind on the Water Ascent Radius Change in Ascent Photos of fountains show that the water ascends as a slowly widening column until it reaches its maximum height, then falls back on itself and scatters. We can derive an expression that shows the change in the radius as the cylinder of water ascends; but since the change is very small, on the order of 1 mm, we use the initial radius at the nozzle, rp, in our calculations.