The fountain that math built 221 The Fountain that math Built lex Mccauley Josh michener Adrian Miles North Carolina School of Science and Mathematics Durham, NC Advisor: Daniel J. Teague Introduction We are presented with a fountain in the of a large plaza, which we wish to be as attractive as possible but not to passersby on windy day Dur task is to design an algorithm that controls the flow rate of the fountain given input from a nearby anemometer. During calm, the fountain sprays out water at a steady rate. When the wind picks up, the flow should be attenuated so as to keep the water within the fountains pool; in this way, we strike a balance between esthetics and comfort We consider the water stream from the fountain as a collection of different sized droplets that initially leave the fountain nozzle in the shape of a perfect cylinder. This cylinder is broken into its component droplets by the wind, with smaller droplets carried farther. In the reference frame of the air, a droplet is moving through stationary air and experiencing a drag force as a result; since the air is moving with a constant velocity relative to the fountain, the force on the droplet is the same in either frame of reference Modeling this interaction as laminar flow, we arrive at equations for the drag forces. From these equations, we derive the acceleration of the droplet, which we integrate to find the equations of motion for the droplet. These allow us to find the time when the droplet hits the ground and-assuming that it lands at the very edge of the pool-the time when it reaches its maximum range from the horizontal position equation. Equating these and solving the initial flow rate, we arrive at an equation for the optimal flow rate at a given constant wind speed. Since the wind speeds are not constant, the algorithm must make its best prediction of wind speed and use current and previous wind speed measurements to damp out transient variations The UMAP Journal23(3)(2002)221-234. 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 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, republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
The Fountain That Math Built 221 The Fountain That Math Built Alex McCauley Josh Michener Jadrian Miles North Carolina School of Science and Mathematics Durham, NC Advisor: Daniel J. Teague Introduction We are presented with a fountain in the center of a large plaza, which we wish to be as attractive as possible but not to splash passersby on windy days. Our task is to design an algorithm that controls the flow rate of the fountain, given input from a nearby anemometer. During calm, the fountain sprays out water at a steady rate. When the wind picks up, the flow should be attenuated so as to keep the water within the fountain’s pool; in this way, we strike a balance between esthetics and comfort. We consider the water stream from the fountain as a collection of differentsized droplets that initially leave the fountain nozzle in the shape of a perfect cylinder. This cylinder is broken into its component droplets by the wind, with smaller droplets carried farther. In the reference frame of the air, a droplet is moving through stationary air and experiencing a drag force as a result; since the air is moving with a constant velocity relative to the fountain, the force on the droplet is the same in either frame of reference. Modeling this interaction as laminar flow, we arrive at equations for the drag forces. From these equations, we derive the acceleration of the droplet, which we integrate to find the equations of motion for the droplet. These allow us to find the time when the droplet hits the ground and—assuming that it lands at the very edge of the pool—the time when it reaches its maximum range from the horizontal position equation. Equating these and solving the initial flow rate, we arrive at an equation for the optimal flow rate at a given constant wind speed. Since the wind speeds are not constant, the algorithm must make its best prediction of wind speed and use current and previous wind speed measurements to damp out transient variations. The UMAP Journal 23 (3) (2002) 221–234. 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
222 The UMAP Journal 23.3(2002) Our final solution is an algorithm that takes as its input a series of wind speed measurements and determines in real time the optimal flow rate to max imize the attractiveness of the fountain while avoiding splashing passers excessively. Each iteration, it adds an inputted wind speed to a buffer of pre- vious measurements. If the wind speed is increasing sufficiently, the last 0.5 s of the buffer are considered; otherwise, the last l s is. The algorithm computes a weighted average of these wind speeds, weighting the most recent value slightly more than the oldest value considered. It uses this weighted veloc ity average in the equation that predicts the optimal flow rate under constant wind. The result is the optimal flow rate under variable wind knowing only current and previous wind speeds A list of relevant variables constants and parameters is in Table 1 Table 1 Relevant constants, variables, and parameters Physical constants Description Value Viscosity of air 1849×10-5kg/ms e1999] Density of wate 1000kg/m3 Density of air 12×10-6kg/m3 Situational constants Cross-sectional area of fountain nozzle fm Maximum flow rate of fountains pump m3/s R Radius of fountain pool Radius of smallest uncomfortable water dt Sampling interval of anemometer k Situational variables Instantaneous wind speed m/s Instantaneous flow rate of water n=g/k+f/A Dynamic variables r(t),y(t) Droplet's horizontal and vertical positions vr(t), Uy(t) Droplet's horizontal and vertical speeds ar(t), ay(t) Droplet's horizontal and vertical accelerations m/s2 Situational parameters Default sample wind velocity buffer time Buffer time for quickly increasing sample wind velocities Weight constant dimensionless
222 The UMAP Journal 23.3 (2002) Our final solution is an algorithm that takes as its input a series of wind speed measurements and determines in real time the optimal flow rate to maximize the attractiveness of the fountain while avoiding splashing passersby excessively. Each iteration, it adds an inputted wind speed to a buffer of previous measurements. If the wind speed is increasing sufficiently, the last 0.5 s of the buffer are considered; otherwise, the last 1 s is. The algorithm computes a weighted average of these wind speeds, weighting the most recent value slightly more than the oldest value considered. It uses this weighted velocity average in the equation that predicts the optimal flow rate under constant wind. The result is the optimal flow rate under variable wind, knowing only current and previous wind speeds. A list of relevant variables, constants, and parameters is in Table 1. Table 1. Relevant constants, variables, and parameters. Physical constants Description Value ηa Viscosity of air 1.849 × 10−5 kg/m·s [Lide 1999] ρw Density of water 1000 kg/m3 ρa Density of air 1.2 × 10−6 kg/m3 Situational constants Units A Cross-sectional area of fountain nozzle m2 fmax Maximum flow rate of fountain’s pump m3/s Rp Radius of fountain pool m r Radius of smallest uncomfortable water m droplet dt Sampling interval of anemometer s k k = 9ηa/2ρwr2 Situational variables va Instantaneous wind speed m/s f Instantaneous flow rate of water m3/s from the fountain n n = g/k + f/A m/s Dynamic variables x(t), y(t) Droplet’s horizontal and vertical positions m vx(t), vy(t) Droplet’s horizontal and vertical speeds m/s ax(t), ay(t) Droplet’s horizontal and vertical accelerations m/s2 Situational parameters τd Default sample wind velocity buffer time s τi Buffer time for quickly increasing sample s wind velocities K Weight constant dimensionless
The Fountain That Math Built 223 Assumptions Passersby find a higher spray more attractive Avoiding discomfort is more important to passersby than the attractiveness of the fountain The water stream can be considered a collection of spherical droplets, each of which has no initial horizontal component of velocity Every possible size of sufficiently small water droplet is represented in the water stream in significant numbers Water droplets remain spherical The interaction between the water droplets and wind can be described as non-turbulent, or "laminar, "flow. There exists a minimum uncomfortable water droplet size; passersby find it acceptable to be hit by any droplets below this size but by none above When the wind enters the plaza, its velocity is entirely horizontal. The wind speed is the same throughout the plaza at any given time The pool and the area around it are radially symmetric, so there is no pre- ferred radial direction We can neglect any buoyant force on the water due to the air, since the error fluids involved, on the order of 10-3, which is negligible of densities of the introduced by this approximation is equal to the ratio The anemometer reports wind speeds at discrete time intervals dt Analysis of the problem For a water stream viewed as a collection of small water droplets blown from a core stream, the interaction between the droplets and the air moving past them can best be described in the inertial reference frame of the moving air. In this frame, the air is stationary while the droplet moves horizontally through the air with a speed equal to the relative speed of the droplet and wind, Ur= va -vr. In the vertical direction, Ur Wu since the wind blows horizontally In the air's frame of reference, the water droplet experiences a drag force opposing ur. Assuming that the air moves at a constant velocity, this force is the same in both frames of reference. In the frame of the fountain then the droplet is being blown in the direction of the wind. The smaller water droplets are carried farther, so we need only consider the motion of the smallest
The Fountain That Math Built 223 Assumptions • Passersby find a higher spray more attractive. • Avoiding discomfort is more important to passersby than the attractiveness of the fountain. • The water stream can be considered a collection of spherical droplets, each of which has no initial horizontal component of velocity. • Every possible size of sufficiently small water droplet is represented in the water stream in significant numbers. • Water droplets remain spherical. • The interaction between the water droplets and wind can be described as non-turbulent, or “laminar,” flow. • There exists a minimum uncomfortable water droplet size; passersby find it acceptable to be hit by any droplets below this size but by none above. • When the wind enters the plaza, its velocity is entirely horizontal. • The wind speed is the same throughout the plaza at any given time. • The pool and the area around it are radially symmetric, so there is no preferred radial direction. • We can neglect any buoyant force on the water due to the air, since the error introduced by this approximation is equal to the ratio of densities of the fluids involved, on the order of 10−3, which is negligible. • The anemometer reports wind speeds at discrete time intervals dt. Analysis of the Problem For a water stream viewed as a collection of small water droplets blown from a core stream, the interaction between the droplets and the air moving past them can best be described in the inertial reference frame of the moving air. In this frame, the air is stationary while the droplet moves horizontally through the air with a speed equal to the relative speed of the droplet and wind, vr = va − vx. In the vertical direction, vr = vy, since the wind blows horizontally. In the air’s frame of reference, the water droplet experiences a drag force opposing vr. Assuming that the air moves at a constant velocity, this force is the same in both frames of reference. In the frame of the fountain, then, the droplet is being blown in the direction of the wind. The smaller water droplets are carried farther, so we need only consider the motion of the smallest
224 The UMAP Journal 23.3 (2002) uncomfortable water droplets, knowing that bigger droplets do not travel as rar The water droplet initially has a vertical velocity uy(O) that is directly related to the flow rate of water through the nozzle of the fountain This initial vertical velocity component can be controlled by changing the flow rate. The droplets motion causes vertical air resistance, slowing the droplet and affecting how long(tu) the droplet is in the air Since the vertical and horizontal components of a water droplet's motion are independent, tu is determined solely by the vertical motion. Knowing this time allows us to find the horizontal distance traveled which we wish to constrain to the radius of the pool When the wind is variable, however, we cannot determine exactly the ideal flow rate for any given time. We must instead act on the current reading but also rely on previous measurements of wind speed in order to restrain the model from reacting too severely to wind fluctuations. We need to react fast to increases in wind speed, since they result in splashing which is weighted more Design of the model For our initial model, we assume that va is constant for time intervals on the order of tw, so that any given droplet experiences a constant wind speed We model the water stream as a collection of droplets that are initially co- esive but are carried away at varying velocities by the wind. The distances that they travel depend on the wind speed va and the initial vertical velocity of the water stream through the nozzle, uy(O). Since the amount of water flowing through the nozzle per unit time is f=vy,(0)A, we have vy(0)=f/A. The dynamics of the system, then, is fully determined by f and va. First, we find the equations of motion for the droplet Equations of Motion for a droplet For laminar flow, a spherical particle of radius r traveling with through a fluid medium of viscosity n experiences a drag force Fp such that (6丌mr) Winters 2002 Since a spherical water droplet has a mass given by m=p(r3), the acceleration felt by the droplet is given by Newtons Second Law as the total force over mass. Since there are no other forces acting in the horizontal
224 The UMAP Journal 23.3 (2002) uncomfortable water droplets, knowing that bigger droplets do not travel as far. The water droplet initially has a vertical velocity vy(0)that is directly related to the flow rate of water through the nozzle of the fountain. This initial vertical velocity component can be controlled by changing the flow rate. The droplet’s motion causes vertical air resistance, slowing the droplet and affecting how long (tw) the droplet is in the air. Since the vertical and horizontal components of a water droplet’s motion are independent, tw is determined solely by the vertical motion. Knowing this time allows us to find the horizontal distance traveled, which we wish to constrain to the radius of the pool. When the wind is variable, however, we cannot determine exactly the ideal flow rate for any given time. We must instead act on the current reading but also rely on previous measurements of wind speed in order to restrain the model from reacting too severely to wind fluctuations. We need to react fast to increases in wind speed, since they result in splashing which is weighted more heavily. Design of the Model For our initial model, we assume that va is constant for time intervals on the order of tw, so that any given droplet experiences a constant wind speed. We model the water stream as a collection of droplets that are initially cohesive but are carried away at varying velocities by the wind. The distances that they travel depend on the wind speed va and the initial vertical velocity of the water stream through the nozzle, vy(0). Since the amount of water flowing through the nozzle per unit time is f = vy(0)A, we have vy(0) = f/A. The dynamics of the system, then, is fully determined by f and va. First, we find the equations of motion for the droplet. Equations of Motion for a Droplet For laminar flow, a spherical particle of radius r traveling with speed v through a fluid medium of viscosity η experiences a drag force FD such that FD = (6πηr)v [Winters 2002]. Since a spherical water droplet has a mass given by m = ρw 4 3 πr3 , the acceleration felt by the droplet is given by Newton’s Second Law as the total force over mass. Since there are no other forces acting in the horizontal
The Fountain That Math Built 225 direction, the horizontal acceleration ar is given by d 2r ar(t) 9na where k=9na/2puur2 The droplet experiences both air drag and gravity in the vertical direction, so the vertical acceleration is t + g With constant va, we use separation of variables and integrate to find vz(t)and vy(t), using the facts that vz(0)=0 and vy(0)=f/A. The results are 1()=t(1-c-k),()=ne-k-8 where n=g/k+f/A Integrating again, and using a(0)=y(0)=0, we have 2()=k(kt+e--1) (1 Determining the Flow Rate Because f is the only parameter that the algorithm modifies, we wish to find the flow rate that would restrict the smallest uncomfortable water droplets to ranges within Rp, so that they would land in the fountains pool After a time tu, the droplet has fallen back to the ground. Thus, y(tu)=0 This equation is too difficult to solve exactly, so we use the series expansion for e-kt and truncate after the quadratic term: e-kt a 1-a+x2/2. Solving y(tu)=0, we find We know that the maximum horizontal distance r(tu)must be less than or equal to Rp, with equality holding for the smallest uncomfortable droplet. For that case, using the same expansion for e-kt as above, k Rp=a(tu)21ktw-1+l-ktw+ Solving for tu and equating it to the earlier expression for tw, we get 2R
The Fountain That Math Built 225 direction, the horizontal acceleration ax is given by: ax(t) = d2x dt2 = 9ηa 2ρwr2 vr = k(va − vx), (1) where k = 9ηa/2ρwr2. The droplet experiences both air drag and gravity in the vertical direction, so the vertical acceleration is ay(t) = − 9ηa 2ρwr2 vy + g = −k vy + g k . With constant va, we use separation of variables and integrate to find vx(t) and vy(t), using the facts that vx(0) = 0 and vy(0) = f/A. The results are vx(t) = va 1 − e−kt , vy(t) = ne−kt − g k , where n = g/k + f/A. Integrating again, and using x(0) = y(0) = 0, we have vx(t) = va k kt + e−kt − 1 , vy(t) = 1 k n 1 − e−kt − gt. Determining the Flow Rate Because f is the only parameter that the algorithm modifies, we wish to find the flow rate that would restrict the smallest uncomfortable water droplets to ranges within Rp, so that they would land in the fountain’s pool. After a time tw, the droplet has fallen back to the ground. Thus, y(tw)=0. This equation is too difficult to solve exactly, so we use the series expansion for e−kt and truncate after the quadratic term: e−kt ≈ 1 − x + x2/2. Solving y(tw)=0, we find tw ≈ 2 k 1 − g nk . We know that the maximum horizontal distance x(tw) must be less than or equal to Rp, with equality holding for the smallest uncomfortable droplet. For that case, using the same expansion for e−kt as above, Rp = x(tw) ≈ va k ktw − 1+1 − ktw + (ktw) 2 2 = vak 2 t 2 w. Solving for tw and equating it to the earlier expression for tw, we get 2Rp vak = tw = 2 k 1 − g nk .
226 The UMAP Journal 23.3(2002) Recalling that in this equality only n is a function of f, we substitute for n and solve for f. The result is f(ua) 20,k Rp As va-kRp/2, this equation becomes singular( see Figure 2). At lower values of va, it gives a negative flow rate. These wind speeds are very small; at such speeds, the droplets would not be deflected significantly by the wind (2)assumes that the flow rate can be made arbitrarily high it is unrealistic invalid in application. To make the model more reasonable, we modify include the maximum flow rate achievable by the pump, fn min F(va) nax? va>krp/2 vak k max, ta≥kRp/2. An algorithm can use the given constants and a suitable minimal droplet size to determine the appropriate flow rate for a measured va. However, (3) assumes that the wind speed is constant over the time scale tu for any given droplet. A more realistic model must take into account variable wind speed Variable wind speed When wind speed varies with time, the physical reasoning used above be- omes invalid, since the relative velocity of the reference frames is no longer tant. Mathematically, this is manifested in the equation for velocity horizontal acceleration; integrating is now not so simple, and we must resort to numerical means to find the equations of motion. Additionally, the algorithm can rely only on past and present wind data to find the appropriate flow rate Our model needs to incorporate nese wina data to make a reasonable predic tion of the wind's velocity over the next tu and determine an appropriate flow rate using ( 3) a gust is defined to be a sudden wind speed increase on the order 5 m/s that lasts for no more than 20 s; a squall is a similarly sudden wind speed increase that lasts longer [Weather Glossary 2002]. Our model should account for gusts and squalls, as well as for"reverse"gusts and squalls, in which the wind speed suddenly decreases. Since wind speeds can change drastically and unpredictably over the flight time of a droplet, our model will behave badly at times and there is no way to completely avoid this- -only to minimize its effects
226 The UMAP Journal 23.3 (2002) Recalling that in this equality only n is a function of f, we substitute for n and solve for f. The result is f(va) = Ag 2vak Rp − k . (2) As va → kRp/2, this equation becomes singular (see Figure 2). At lower values of va, it gives a negative flow rate. These wind speeds are very small; at such speeds, the droplets would not be deflected significantly by the wind. Since (2) assumes that the flow rate can be made arbitrarily high, it is unrealistic and invalid in application. To make the model more reasonable, we modify (2) to include the maximum flow rate achievable by the pump, fmax: F(va) = min Ag 2vak Rp − k , fmax , va > kRp/2; fmax, va ≥ kRp/2. (3) An algorithm can use the given constants and a suitable minimal droplet size to determine the appropriate flow rate for a measured va. However, (3) assumes that the wind speed is constant over the time scale tw for any given droplet. A more realistic model must take into account variable wind speed. Variable Wind Speed When wind speed varies with time, the physical reasoning used above becomes invalid, since the relative velocity of the reference frames is no longer constant. Mathematically, this is manifested in the equation for velocity-dependent horizontal acceleration; integrating is now not so simple, and we must resort to numerical means to find the equations of motion. Additionally, the algorithm can rely only on past and present wind data to find the appropriate flow rate. Our model needs to incorporate these wind data to make a reasonable prediction of the wind’s velocity over the next tw and determine an appropriate flow rate using (3). A gust is defined to be a sudden wind speed increase on the order 5 m/s that lasts for no more than 20 s; a squall is a similarly sudden wind speed increase that lasts longer [Weather Glossary 2002]. Our model should account for gusts and squalls, as well as for “reverse” gusts and squalls, in which the wind speed suddenly decreases. Since wind speeds can change drastically and unpredictably over the flight time of a droplet, our model will behave badly at times and there is no way to completely avoid this—only to minimize its effects
The fountain that Math built 227 The model's reaction to wind speed is not fully manifested until the droplet lands, after a time tw(approximately 2s). By the time our model has reacted to a gust or reverse gust, therefore, the wind speed has stopped changing. Without some type of buffer, in a gust our model would react by suddenly dropping flow rate as the wind peaked and then increasing it again as the wind decreased; the fountain would virtually cut off for the duration of any gust, which would release less water and thus seem very unattractive to passersby. Additionally, the water released just before the onset of the gust would be airborne as the wind speed picked up, splashing passersby regardless of any reaction by our model We exhibit an algorithm for analyzing wind data that makes use of (3) Because velocity now varies within times on the order of tu, we do not want to directly input the current wind speed but rather a buffered value, so that the model does not react too sharply to transient wind changes. The model should react more quickly to sudden increases in wind than to decreases, because increases cause splashing, which we weight more heavily than attractiveness The model, therefore, has two separate velocity buffer times: one, Td, the default, and another, Ti, for when the wind increases drastically. We also weight more-recent values in the buffer more heavily since we want the model to react promptly to wind speed changes but not to overreact. We weight each value in the velocity buffer with a constant value K plus a weight proportional te its age: Less-recent velocities are considered but given less weight than more recent ones. The weight of the oldest value in the buffer is K and that of the most recent is K+1, with a linear increase between the two with the constraint that the weights are normalized (i.e., they sum to 1), the equation for the ith weight factor is K+ K+ The speeds are multiplied by their respectivenormalized weights and summed at a given time. We use Ti rather than Ta when m late flow rate for the fountain the wind speed increases suf- ficiently over a recent interval, but not when it increases slightly or fluctuates rapidly. We switch from Ta to Ti whenever the wind speed increases over twe successive 0.2 s intervals and by a total of at least 1 m/s over the entire 0. 4 s interval Our algorithm follows the flow chart in Figure 1 in computing the current flow rate We wrote a C++ program to compute this algorithm, the code for which is included in an appendix. [EDITOR'S NOTE: We omit the code
The Fountain That Math Built 227 The model’s reaction to wind speed is not fully manifested until the droplet lands, after a time tw (approximately 2 s). By the time our model has reacted to a gust or reverse gust, therefore, the wind speed has stopped changing. Without some type of buffer, in a gust our model would react by suddenly dropping flow rate as the wind peaked and then increasing it again as the wind decreased; the fountain would virtually cut off for the duration of any gust, which would release less water and thus seem very unattractive to passersby. Additionally, the water released just before the onset of the gust would be airborne as the wind speed picked up, splashing passersby regardless of any reaction by our model. We exhibit an algorithm for analyzing wind data that makes use of (3). Because velocity now varies within times on the order of tw, we do not want to directly input the current wind speed but rather a buffered value, so that the model does not react too sharply to transient wind changes. The model should react more quickly to sudden increases in wind than to decreases, because increases cause splashing, which we weight more heavily than attractiveness. The model, therefore, has two separate velocity buffer times: one, τd, the default, and another, τi, for when the wind increases drastically. We also weight more-recent values in the buffer more heavily, since we want the model to react promptly to wind speed changes but not to overreact. We weight each value in the velocity buffer with a constant value K plus a weight proportional to its age: Less-recent velocities are considered but given less weight than more recent ones. The weight of the oldest value in the buffer is K and that of the most recent is K +1, with a linear increase between the two. With the constraint that the weights are normalized (i.e., they sum to 1), the equation for the ith weight factor is wi = K + i τ − dt dt K + 1 2 τ . The speeds are multiplied by their respective normalized weights and summed. This sum, v∗, is then used in (3) to find the appropriate flow rate for the fountain at a given time. We use τi rather than τd when the wind speed increases suf- ficiently over a recent interval, but not when it increases slightly or fluctuates rapidly. We switch from τd to τi whenever the wind speed increases over two successive 0.2 s intervals and by a total of at least 1 m/s over the entire 0.4 s interval. Our algorithm follows the flow chart in Figure 1 in computing the current flow rate We wrote a C++ program to compute this algorithm, the code for which is included in an appendix. [EDITOR’S NOTE: We omit the code.]
228 The UMAP Journal 23.3 (2002) Input Decide um weighted Current whether CITes In t Setv equal to to this sum f(v") Sum weighted velocities in td Figure 1. Flow chart for computing flow rate with variable wind speed Testing and Sensitivity Analysis Sensitivity of Flow Equation nourequation for flow rate, two variables can change: minimal droplet size and wind speed. While the minimal droplet size will not change dynamically, its value is a subjective choice that must be made by the owner of the fountain The wind speed, however, will change dynamically throughout the problem, and the purpose of our model is to react to these changes We examined( 3)for varying minimal drop sizes(Figure 2)and wind speeds (Figure 3). We used a fountain with nozzle radius 1 cm, maximum flow rate 7.5L/s, and pool radius 1.2 m (This maximum flow rate is chosen for illustra tive purposes and is not reasonable for such a small fountain. 0006 f(002,可a) f(001,可a)0004 f(0005,Ta 0 Figure 2. Graphs of flow rate f vs. wind speed va for several values of radius r of smallest uncomfortable droplet
228 The UMAP Journal 23.3 (2002) Figure 1. Flow chart for computing flow rate with variable wind speed. Testing and Sensitivity Analysis Sensitivity of Flow Equation In our equation forflow rate, two variables can change: minimal droplet size and wind speed. While the minimal droplet size will not change dynamically, its value is a subjective choice that must be made by the owner of the fountain. The wind speed, however, will change dynamically throughout the problem, and the purpose of our model is to react to these changes. We examined (3) for varying minimal drop sizes (Figure 2) and wind speeds (Figure 3). We used a fountain with nozzle radius 1 cm, maximum flow rate 7.5 L/s, and pool radius 1.2 m. (This maximum flow rate is chosen for illustrative purposes and is not reasonable for such a small fountain.) Figure 2. Graphs of flow rate f vs. wind speed va for several values of radius r of smallest uncomfortable droplet
The fountain that Math built 229 At any wind speed, as the acceptable droplet radius decreases, the flow rate o decreases. At higher wind speeds, this difference is less pronounced; but at lower speeds, acceptable size has a significant impact on the flow rate. At very low wind speeds, the fountain cannot shoot the droplets high enough to allow the wind to carry them outside the pool, regardless of drop size. Our cutoff, fmax, reflects that the fountain pump cannot generate the extreme flow needed to get the droplets to the edge of the pool in these conditions 0006 f(r,25) f(x,5)0004 f(r,10 0251045-1047s10740001000125000150001750002022500025 Figure 3. Graphs of flow rate f vs radius r of smallest uncomfortable droplet for several values of wind speed va For any droplet size, as the wind speed increases, the flow rate must decrease to keep the droplets in the pool. For large r, a change in wind speed requires a greater absolute change in flow rate than for small r. For very small droplets, the drag force dominates the force of gravity, and an increase in flow also increases the drag force to such an extent that the particle spends no more time in the air. This behavior is readily apparent in(1) as r approaches zero to discomfort passersby and thus are not significant to our moder are unlikely These extremely small values of r, though, describe droplets that Sensitivity of Flow algorithm The results of the algorithm depend on the parameters Ti, Td, and K, which determine the size of the buffer and weights of the velocities in the buffer. To
The Fountain That Math Built 229 At any wind speed, as the acceptable droplet radius decreases, the flow rate o decreases. At higher wind speeds, this difference is less pronounced; but at lower speeds, acceptable size has a significant impact on the flow rate. At very low wind speeds, the fountain cannot shoot the droplets high enough to allow the wind to carry them outside the pool, regardless of drop size. Our cutoff, fmax, reflects that the fountain pump cannot generate the extreme flow needed to get the droplets to the edge of the pool in these conditions. Figure 3. Graphs of flow rate f vs. radius r of smallest uncomfortable droplet for several values of wind speed va. For any droplet size, as the wind speed increases, theflow rate must decrease to keep the droplets in the pool. For large r, a change in wind speed requires a greater absolute change in flow rate than for small r. For very small droplets, the drag force dominates the force of gravity, and an increase in flow also increases the drag force to such an extent that the particle spends no more time in the air. This behavior is readily apparent in (1) as r approaches zero. These extremely small values of r, though, describe droplets that are unlikely to discomfort passersby and thus are not significant to our model. Sensitivity of Flow Algorithm The results of the algorithm depend on the parameters τi, τd, and K, which determine the size of the buffer and weights of the velocities in the buffer. To
230 The UMAP Journal 23.3(2002) test sensitivity to these parameters and to find reasonable values for them we created the set of simulated wind speeds shown in Figure 4, including sma random variations, on which to test our algorithm. This data set does not reflect typical wind patterns but includes a variety of extreme conditions Time(s) Figure 4. Simulation of wind speed for 3 min. We wish to create a quantitative estimate of the deviation of our flow al- gorithm from ideal performance and then test the algorithm with different combinations of parameters to find the set that produces the smallest deviation under simulated wind conditions To measure how"bad"a set of flow choices is, we consider only the droplets that fall outside the pool. The"badness"is the sum over the run of the distances outside the pool at which droplets land To determine the distance, we need to know how droplets move through e air in varying wind speeds. Describing this motion in closed form is math- ematically impossible without continuous wind data, so we approximate the equations of motion with an iterative process Since the time that a particle spends in the air, tu, is not affected by the wind ed we know tu for each particle. We step through the time tw in intervals computing the particles acceleration, velocity, and position as ai=k(u +a-1d +vidt When we reachtw, the droplet has hit the ground and we compare its horizontal position to the radius of the pool. We do this for each droplet and keeping track of both the largest absolute difference and the average difference To test the flow algorithm, we ran our program with each combination of parameters on each set of flow data. The parameter values that produced the
230 The UMAP Journal 23.3 (2002) test sensitivity to these parameters and to find reasonable values for them, we created the set of simulated wind speeds shown in Figure 4, including small random variations, on which to test our algorithm. This data set does not reflect typical wind patterns but includes a variety of extreme conditions. Figure 4. Simulation of wind speed for 3 min. We wish to create a quantitative estimate of the deviation of our flow algorithm from ideal performance and then test the algorithm with different combinations of parameters to find the set that produces the smallest deviation under simulated wind conditions. To measure how “bad” a set of flow choices is, we consider only the droplets that fall outside the pool. The “badness” is the sum over the run of the distances outside the pool at which droplets land. To determine the distance, we need to know how droplets move through the air in varying wind speeds. Describing this motion in closed form is mathematically impossible without continuous wind data, so we approximate the equations of motion with an iterative process. Since the time that a particle spends in the air, tw, is not affected by the wind speed, we know tw for each particle. We step through the time tw in intervals of dt, computing the particle’s acceleration, velocity, and position as ai = k(va,i − vi), a0 = kva,0; vi = vi−1 + ai−1dt, v0 = 0; xi = xi−1 + vidt, x0 = 0. When we reachtw, the droplet has hit the ground, and we compare its horizontal position to the radius of the pool. We do this for each droplet and keeping track of both the largest absolute difference and the average difference. To test the flow algorithm, we ran our program with each combination of parameters on each set of flow data. The parameter values that produced the
