Simulating a Fountain 209 Simulating a Fountain Lyric p. doshi Joseph Edgar Gonzalez hilip B. kidd Maggie L. Walker Governor's school for Government and International Studies Richmond, va Advisor: John A Barnes Introduction We establish the mathematical behavior of water droplets emitted from a ountain and apply this behavior in a computer model to predict the amount of splash and spray produced by a fountain under given conditions We combine height and volume of the fountain spray, making both functions of the speed at which water exits the fountain nozzle. We simulate water droplets launched from the fountain, using basic physics to model the effects of drag, wind, and gravity. The simulation tracks the flight of droplets in the air and records their landing positions, for wind speeds from 0 to 15 m/s and water speeds from 5 to 30 m/s. It calculates the amount of water spilled outside of a pool around the fountain, for pool radii from 0 to 40 m We design an algorithm for a programmable logic controller, located inside an anemometer, to do a table search to find allowable water speeds for given pool radius, acceptable water spillage, and wind velocity. We simulated sub- jecting a fountain with a 4-m pool radius to wind speeds from 0 to 3 m/s with an allowable spillage of 5%. We tested the model for accuracy and sensitivity to changes in the base variables Problem Analysis Wind The anemometer measures two main wind factors that affect the fountain speed which affects the force exerted on the water, and direction The UMAP Journal 23(3)(2002)209-219. @Copyright 2002 by COMAP, Inc. Allrights reserve 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
Simulating a Fountain 209 Simulating a Fountain Lyric P. Doshi Joseph Edgar Gonzalez Philip B. Kidd Maggie L. Walker Governor’s School for Government and International Studies Richmond, VA Advisor: John A. Barnes Introduction We establish the mathematical behavior of water droplets emitted from a fountain and apply this behavior in a computer model to predict the amount of splash and spray produced by a fountain under given conditions. We combine height and volume of the fountain spray, making both functions of the speed at which water exits the fountain nozzle. We simulate water droplets launched from the fountain, using basic physics to model the effects of drag, wind, and gravity. The simulation tracks the flight of droplets in the air and records their landing positions, for wind speeds from 0 to 15 m/s and water speeds from 5 to 30 m/s. It calculates the amount of water spilled outside of a pool around the fountain, for pool radii from 0 to 40 m. We design an algorithm for a programmable logic controller, located inside an anemometer, to do a table search to find allowable water speeds for given pool radius, acceptable water spillage, and wind velocity. We simulated subjecting a fountain with a 4-m pool radius to wind speeds from 0 to 3 m/s with an allowable spillage of 5%. We tested the model for accuracy and sensitivity to changes in the base variables. Problem Analysis Wind The anemometer measures two main wind factors that affect the fountain: speed, which affects the force exerted on the water, and direction. The UMAP Journal 23 (3) (2002) 209–219. 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
210 The UMAP Journal 23.3 (2002) Fountain The main components of the fountain are the pool and the nozzle. The factors associated with the pool are its radius, which remains constant within a trial, and the acceptable level of spillage, which describes the percentage of water that may acceptably fall outside of the fountain Nozzle Major aspects of the nozzle are the radius of the opening the angle relative to the vertical axis(normal axis), and the spread and speed of the water passing initgugh it. The angle of the nozzle relative to the vertical axis determines the initial trajectory of the water. The spread, described in standard deviations from the angle of the nozzle, determines the extent to which the initial trajectory of droplets differs from the angle of the nozzle. For a given water speed and nozzle radius, the flow of water through the nozzle may be determined from where f is flow, v is the water launch speed and r is the radius of the nozzle The radius is constant, so the flow and consequent volume are functions of the speed the dominant controllable factor affecting the height of the stream Assumptions about fountains The fountain is composed of a single nozzle located at the center of a circular The ledge of the pool is sufficiently high to collect the splatter produced by particles impacting the surface of the water Fountains with higher streams are more attractive than those with lower streams about the nozzle The nozzle has a fixed radius, but the speed of the water through it can be controlled The nozzle is perpendicular to the ground The nozzle responds rapidly to input from the anemometer The nozzle produces a normally distributed spread of droplets with a low standard deviation
210 The UMAP Journal 23.3 (2002) Fountain The main components of the fountain are the pool and the nozzle. The factors associated with the pool are its radius, which remains constant within a trial, and the acceptable level of spillage, which describes the percentage of water that may acceptably fall outside of the fountain. Nozzle Major aspects of the nozzle are the radius of the opening, the angle relative to the vertical axis (normal axis), and the spread and speed of the water passing through it. The angle of the nozzle relative to the vertical axis determines the initial trajectory of the water. The spread, described in standard deviations from the angle of the nozzle, determines the extent to which the initial trajectory of droplets differs from the angle of the nozzle. For a given water speed and nozzle radius, the flow of water through the nozzle may be determined from f = πr2v, where f is flow, v is the water launch speed, and r is the radius of the nozzle. The radius is constant, so the flow and consequent volume are functions of the speed, the dominant controllable factor affecting the height of the stream. Assumptions ... about Fountains • The fountain is composed of a single nozzle located at the center of a circular pool. • The ledge of the pool is sufficiently high to collect the splatter produced by particles impacting the surface of the water. • Fountains with higher streams are more attractive than those with lower streams. ... about the Nozzle • The nozzle has a fixed radius, but the speed of the water through it can be controlled. • The nozzle is perpendicular to the ground. • The nozzle responds rapidly to input from the anemometer. • The nozzle produces a normally distributed spread of droplets with a low standard deviation
Simulating a Fountain 211 about Water Droplets Because the droplets are small and roughly spherical, they may be treated The radii of droplets are normally distributed The density of water is unaffected by conditions and therefore remains con- stant among and within droplets The only outside forces exerted on a water droplet are gravity and the force exerted by the surrounding air, including drag and wind acceleration due to gravity is the same for all droplets The effect of air perturbations produced by droplets on other droplets is insignificant All droplets share the same constant drag coefficient Droplet interactions and collisions do not increase the overall energy of the system or increase the distance traveled by droplets bout the Anemometer and Control System The anemometer and control system can rapidly evaluate the wind speed apply a basic formula, and adjust the nozzle in changing wind conditions about the wind The wind speed is uniform regardless of altitude Wind blows parallel to the ground without turbulence or irregularities Basic Description of Model Water droplets are emitted from the nozzle and follow trajec gauding cal by wind and drag. The particles are tracked until they land, in culations of trajectories in case of changes in conditions, such as wind. The landing distance from the center of the fountain is recorded. Since the fountain pool is circular, only radial distance is important The model ignores wind direction( does not affect a circular fountain pool) and turbulence (insignificant and too complicated to model accurately We tested droplet collisions and found that they do not greatly affect the distance that droplets land from the center of the pool; so we ruled out in- corporating complex interactions into the model. Further physical analysis
Simulating a Fountain 211 ... about Water Droplets • Because the droplets are small and roughly spherical, they may be treated as spherical. • The radii of droplets are normally distributed. • The density of water is unaffected by conditions and therefore remains constant among and within droplets. • The only outside forces exerted on a water droplet are gravity and the force exerted by the surrounding air, including drag and wind. • Acceleration due to gravity is the same for all droplets. • The effect of air perturbations produced by droplets on other droplets is insignificant. • All droplets share the same constant drag coefficient. • Droplet interactions and collisions do not increase the overall energy of the system or increase the distance traveled by droplets. ... about the Anemometer and Control System • The anemometer and control system can rapidly evaluate the wind speed, apply a basic formula, and adjust the nozzle in changing wind conditions. ... about the Wind • The wind speed is uniform regardless of altitude. • Wind blows parallel to the ground without turbulence or irregularities. Basic Description of Model Water droplets are emitted from the nozzle and follow trajectories affected by wind and drag. The particles are tracked until they land, including recalculations of trajectories in case of changes in conditions, such as wind. The landing distance from the center of the fountain is recorded. Since the fountain pool is circular, only radial distance is important. The model ignores wind direction (does not affect a circular fountain pool) and turbulence (insignificant and too complicated to model accurately). We tested droplet collisions and found that they do not greatly affect the distance that droplets land from the center of the pool; so we ruled out incorporating complex interactions into the model. Further physical analysis
212 The UMAP Journal 23.3 (2002) supported that decision: Because of conservation of energy and momentum,a droplet could not travel significantly farther after a collision Finally, we combined fountain height and volume into speed of the water out of the nozzle, because they are directly determined by the speed Our simulation tries all combinations of 11 different water speeds from 5 to 30 m/s(at intervals of 0.5 m/s), with 16 wind speeds, from 0 to 15 m/s(at intervals of 1 m/s). Each combination is run for five trials of 10,000 droplets Spillage is logged for radii from 0 to 40 m(at intervals of 0. 1 m). The five trials are then averaged to construct an entry in a three-dimensional reference table The Underlying Mathematics The simulation uses basic physics equations to model the flight of water droplets through the ai Each droplet is acted on by three forces: gravity, drag, and wind. Drag calculated from the following equation [Halliday et al. 19931 d=3CpAu D is the drag coefficient, an empirically-determined constant dependent mainl on the shape of an object p is the density of the fluid through which the object is traveling, in this case A is the cross-sectional area of the object; and v=u is the speed of the object relative to the wind The drag coefficient of a raindrop is 0.60 and the density of air is about 1.2 kg/m[Halliday et al. 1993]. Drag acts directly against velocity, so the acceleration vector from drag can be found from Newton s law F= ma as a D可_CpA2司_是CpA可 n where a is the acceleration vector and m is mass We factor in gravity by subtracting the acceleration g of gravity at Earth's surface, 9.8 m/s, from the vertical component of the acceleration vector 2 PaJU Uz-g ext, we use the acceleration to find velocity beginning with the expression dCpA可 dt
212 The UMAP Journal 23.3 (2002) supported that decision: Because of conservation of energy and momentum, a droplet could not travel significantly farther after a collision. Finally, we combined fountain height and volume into speed of the water out of the nozzle, because they are directly determined by the speed. Our simulation tries all combinations of 11 different water speeds, from 5 to 30 m/s (at intervals of 0.5 m/s), with 16 wind speeds, from 0 to 15 m/s (at intervals of 1 m/s). Each combination is run for five trials of 10,000 droplets. Spillage is logged for radii from 0 to 40 m (at intervals of 0.1 m). The five trials are then averaged to construct an entry in a three-dimensional reference table. The Underlying Mathematics The simulation uses basic physics equations to model the flight of water droplets through the air. Each droplet is acted on by three forces: gravity, drag, and wind. Drag is calculated from the following equation [Halliday et al. 1993]: D = 1 2CρAv2, where D is the drag coefficient, an empirically-determined constant dependent mainly on the shape of an object; ρ is the density of the fluid through which the object is traveling, in this case air; A is the cross-sectional area of the object; and v = | v| is the speed of the object relative to the wind. The drag coefficient of a raindrop is 0.60 and the density of air is about 1.2 kg/m3 [Halliday et al. 1993]. Drag acts directly against velocity, so the acceleration vector from drag can be found from Newton’s law F = m a as a = −D m v |v| = 1 2CρA| v| 2 m v | v| = 1 2CρA| v| m v, where a is the acceleration vector and m is mass. We factor in gravity by subtracting the acceleration g of gravity at Earth’s surface, 9.8 m/s2, from the vertical component of the acceleration vector: az = − 1 2CρA| v| m vz − g. Next, we use the acceleration to find velocity, beginning with the expression d v dt = − 1 2CρA| v| m v = a
Simulating a Fountain 213 To circumvent the difficulties of solving a differential equation for each compo- nent of the velocity vector, we use Euler's method to approximate the velocity at a series of discrete points in time △≈△ta,1≈o+△tao We use a similar process to find the position of the droplet, resulting in 1≈Co+△tto With At=0.001 s, error from the approximation is virtually zero Now that we have equations for describing the droplet in flight, we gener ate its initial position and velocity. first we randomly select a value z from a standard Gaussian(normal) distribution(mean 0, standard deviation 1). We calculate the angle from a set mean u and standard deviation o of the distribu tion of possible angles a P=zo +u We randomly select another angle 6 between 0 and 2T radians to be the angle between the velocity vector and the a-axi Thus, the initial velocity vector of the droplet in spherical coordinates is (P, 8, o), where p is the magnitude of the velocity. Conversion to rectangular coordinates yields(psin cos 6, psin sin 0, p cos o) Ne also randomly select a starting location within the nozzle(whose diam eter is 1 cm) and create a radius for the droplet using a similar sampling from a normal distribution The mass of the droplet is then where p is the density of water, 998.2 kg/m at 20 C [Lide 1995]. In the basic simulation, the o distribution has a mean of 0 and a standard deviation of T /60 radians, and the radius distribution has a mean of 0.0015 m and a standard deviation of 0.0001 m In the basic simulation, the nozzle points straight up; however, we also test che effect of tilting the nozzle away from the wind. The program first rotates the nozzle a set angle away from z-axis(/18, T /9, or T/6 radians). The initial position and velocity vectors are changed by the formula for rotating a point t radians about the x-axis, from z towards negative y[ Dollins 2001] 0 cos t sint cos t Next, the program rotates the nozzle around the z-axis to point directly away from the wind (in spherical coordinates, the 8 of the nozzle is equal to
Simulating a Fountain 213 To circumvent the difficulties of solving a differential equation for each component of the velocity vector, we use Euler’s method to approximate the velocity at a series of discrete points in time: d v dt = a, ∆ v ≈ ∆t a, v1 ≈ v0 + ∆t a0. We use a similar process to find the position of the droplet, resulting in x1 ≈ x0 + ∆t v0. With ∆t = 0.001 s, error from the approximation is virtually zero. Now that we have equations for describing the droplet in flight, we generate its initial position and velocity. First, we randomly select a value z from a standard Gaussian (normal) distribution (mean 0, standard deviation 1). We calculate the angle from a set mean µ and standard deviation σ of the distribution of possible angles as φ = zσ + µ. We randomly select another angle θ between 0 and 2π radians to be the angle between the velocity vector and the x-axis. Thus, the initial velocity vector of the droplet in spherical coordinates is (ρ, θ, φ), where ρ is the magnitude of the velocity. Conversion to rectangular coordinates yields (ρ sin φ cos θ,ρ sin φ sin θ,ρ cos φ). We also randomly select a starting location within the nozzle (whose diameter is 1 cm) and create a radius for the droplet using a similar sampling from a normal distribution. The mass of the droplet is then m = 4 3πr3ρ, where ρ is the density of water, 998.2 kg/m3 at 20◦ C [Lide 1995]. In the basic simulation, the φ distribution has a mean of 0 and a standard deviation of π/60 radians, and the radius distribution has a mean of 0.0015 m and a standard deviation of 0.0001 m. In the basic simulation, the nozzle points straight up; however, we also test the effect of tilting the nozzle away from the wind. The program first rotates the nozzle a set angle away from z-axis (π/18, π/9, or π/6 radians). The initial position and velocity vectors are changed by the formula for rotating a point t radians about the x-axis, from z towards negative y [Dollins 2001]: x y z = 10 0 0 cost − sin t 0 sin t cost x y z . Next, the program rotates the nozzle around the z-axis to point directly away from the wind (in spherical coordinates, the θ of the nozzle is equal to