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Problem B

Team 763

November 2014

Abstract

In this paper, we will investigate a angular symmetric hollow-cone water fountain with fixed initialparameters. All the stages of the phenomenon (the hollow-cone water sheet is ejected into the air, water

ligaments gets separated from the water sheet and breakups into water droplets, and water droplets travel

in the air to land) are treated. The spreading of the water on the land is assumed to be due the gradual(over time and space) formation of droplets from the ligaments. Numerical analysis by Matlab yields aprobability distribution for water landing for all points on the ground. The probability spread of waterlanding is found to be quite narrow (non-zero probability between radial distances 5.077mand 5.249m)and has a sharp peak at 5.077m. A discussion of the strengths and weaknesses of the method follows at

the end of the paper.

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1 Introduction

Water fountains have long been a major aesthetic achievement of the human race. People gathering inin a square, doing all sort of things and enjoying the water beautifully shaped into the air from a waterfountain placed in the centre, is an usual and great scene. As diverse as humans beauty can be, thereare many water fountain arrangements and designs that can capture ones heart. In this paper, we willinvestigate the phenomenon of a water fountain (located on a pond) from which water is ejected straight

up and spreads out into a cone shape with the axis of the cone being vertical. We will build a model of thewater fountain and seek the three following objects:

1. A distribution of probability of being hit by any water molecule from the ejected water for all pointson the pond.

2. The most probable landing point.

3. The median circle, i.e. the circle around the water fountain inside which exactly half of the ejectedwater lands.

Before getting in details, let us mention the three dimensionless quantities which will be used in thefollowing analysis. Presented here, these quantities already take in some input relevant to our problem.The first one is Reynolds number, which actually is the simplified ratio of inertial force over viscous force:

Re=a|w|L

a(1)

where, besides the constants listed in Table 1, w is the relative velocity of water to air, and L is thecharacteristic length relevant to the magnitude of the inertial force. We will need the Reynolds number todetermine the air drag force. The second quantity is Weber number:

We=2a|w|2R

(2)

wherer is the radius of the water droplet. Here we have taken the diameter of the water droplet (2r) asthe characteristic length used in the Weber number formulae. The Weber number is essentially a measureof importance of the fluids inertia over its surface tension, and therefore is important in droplet breakup

and formation analysis. A table of needed constants follows below. The numerical values of the constantsare collected from Wikipedia.

Table 1: Table of constants

Quantity Notation Numerical value

density of air a 1.29(kg.m3)

dynamic viscosity of air a 3.5(kg/m)air temperature T 20density of water w 1000(kg.m3)

surface tension of water 0.073(N/m)(in contact with air)

2 The Model of the Water Fountain

2.1 The water-shooting system

Lies at the heart of every water fountain is a system aiming to shoot the water into the air in the desiredflow. The desired flow can be a flow of water droplets, or can be a connected, thin sheet of water. Here

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we assume the latter type: the water coming out of the water-shooting system forms a thin, hollow-coneshaped sheet.

The nozzle, the device set up at the exit of water from the shooting system to have the final controlover the water flows direction and characteristics, is assumed to have a ring shape, with radius R0. Wealso assume that the nozzle can rotate, thus being able to give the water a velocity along the tangentialdirection.

2.2 Observations about and the model of the flight path of waterThe hollow-cone-shaped water sheet rises high and spreads out wide from the small ring-shaped nozzle.Observation of the real water fountains shows that at some points high and wide enough, the water sheetwill be disintegrated into smaller water ligaments. The water ligaments have the following shape: imaginetwo cross-sectional cuts that are normal to the hollow-cones axis and are very near each other, the partbounded by the two cuts shares the same shape with the water ligament. These ligaments also quicklydisintegrate into small water droplets as they continue to travel in the air.

We could understand the process of breakup qualitatively as following. When the water sheet is shotinto the air, it experiences non-uniform aerodynamics forces, which disturb its surface. The disturbed waterreacts by trying to change to its equilibrium shape (to minimize its surface tension energy). However, dueto the inertia of shape changing, it will always go past its equilibrium shape. This process will happenagain and again, resulting in waves populated over the water sheet in all the directions. We can think ofthe waves as having two perpendicular propagating directions: s

direction and n

direction, with the

two directions are indicated in Figure 1.

Figure 1: The connected hollow-cone water sheet

In the later analysis, it would be clear that the magnitude of the waves would grow over time. Thedisturbance energy therefore grows over time until it can exceed the bonding energy of nearby watermolecules, i.e. the part of the water sheet which disturbance waves have large enough amplitudes wouldbreak apart from the water sheet. As the water velocitys upward component along the cones surfaceis significantly higher than that along the tangential, horizontal direction, the downward aerodynamicforces (relative to the water) along the cones surface will be significantly stronger than those tangential.

Therefore, the amplitudes of the waves propagating along sdirection are higher than the amplitudes ofthe waves propagating alongndirection, resulting in the water sheet breaking up into ring-like, horizontalligaments before it could be tore vertically due to the ndirection waves.

Since the ligaments are disconnected from the water sheet and have the widths of about the wavelengthof the waves that separate them, there cannot exist any residual sdirectionwaves that attempt to breakit more apart. However, the ligaments still inherit from the water sheet some ndirection waves whichmagnitudes constantly grow over time. It is these ndirectionwaves that break the ligaments into smaller

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parts which sizes in all directions are not too different. It is easy then for these small parts to changetheir shapes near to that of a sphere to minimize the surface tension energy, i.e. they are essentially waterdroplets. If the water droplets are large and fast enough, they will also breakup into multiple smallerdroplets. However, later we will show that this is not the case for water droplets formed by typical waterfountain.

We are going to assume a certain attributes of the water fountain; their numerical values are presentedin Table 2. If the model of the water fountain is built to a good extent of generality, and the dynamics ofit can be understood well, an analysis of only one water fountain with specific attributes is qualitativelyenough.

Table 2: Some numerical attributes of the water fountain

Attribute Notation Numerical value

initial spread angle 0 15initial thickness of the water sheet 0 5cm

radius of the nozzle R0 10cminitialsdirection water velocity u0 8m/sinitialkdirection water velocity ut0 3m/s

For simplicity, we also make the following assumptions about the flight path of water:

1. The water fountain creates no change for the air, i.e. no change in air pressure, temperature, humidity,etc.

2. The water is incompressible

3 Dynamics of the Water

3.1 Dynamics of the connected water sheet

Using curvilinear body-fitted coordinates(s,n,k), Ibrahim [3]has developed a system of equations governingthe dynamics of hollow-cone water sheet. The curvilinear coordinates and some attributes of the watersheet are indicated in Figure 1. Of waters velocity, let us also denote the component along s coordinate

u, and the component along qcoordinateut. The gravitational acceleration is denoted as g .Reference [1]s assumptions about the waters characteristics (Newtonian, inviscid, incompressible) are

compatible with our assumptions about water. The four following differential equations can be derivedfrom basic mechanics equations using the chain rule and the definition of velocity, and the definition ofsurface tension. Specifically: equation3 from the continuity condition of fluid, equations 4, 5, 6 from theNewtons second law along the three curvilinear coordinates, and equations 7, 8 from the local geometricconditions of a hollow-cone shape. The Coriolis effect, the centrifugal forces and the surface tension in allspatial direction have been taken into account.

rdu

ds+ u

dr

ds+ur

d

ds= 0 (3)

wudu

ds wuT

2 sin

r =

au

2

20.791+150

rRes1/4wg cos (4)

wu2 d

ds wuT

2 cos

r =2

cos

r d

ds

+wg sin (5)

wuduTds

+wuuTsin

r =auT

2

2

0.79

1+150

r

ReT

1/4

(6)

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dr

ds= sin (7)

dz

ds= cos (8)

whereRes andRet are the two Reynolds numbers corresponding the relative motion of water to air along

thes and t coordinates, respectively. If we consider the case where there is no wind, then the velocity ofwater is also the relative velocity between water and air. Taking the initial thickness of the water sheet asthe characteristic length for Res andRet, we obtain:

Res=au0a

(9)

Ret=aut0

a(10)

3.2 The breakup of the connected water sheet to water ligaments

To describe the water sheet breakup process, we follow the approach taken by Drombrowski and Johns[5], where a differential equation governing the amplitude growth of one-dimensional sinusoidal wave in aninfinite viscous liquid sheet is derived by balancing the inertial, pressure, viscous, and surface tension force(equation (25) in their paper):

f

t

2+ 42

ww2

f

t 4

w(t)

au(t)2 2

2

= 0 (11)

where we have rewritten the equation according to our notation system. This equation describes the growth

rate of a sinusoidal wave of wavelength and dimensionless wave amplitude, defined asf= ln

AA0

, where

A0 and A, respectively, are the amplitudes at the beginning and at an arbitrary time of the wave. The flowvelocity u(t) is taken from the result obtained in the previous section. There are two solutions for ft inequation (12),

f

t = 2

2ww2

22ww2

2

+4

wau2 2

2 (12)We are only interested in the (+) square root solution, which may potentially give an increasing amplitude

f

t = 2

2ww2

+

22ww2

2+

4

w

au2 2

2

(13)

Two authors([5], [8]) experimentally found that the breakup occurs when the dimensional wave am-plitude reaches the critical value fcrit = 12 for plain jets and fan nozzle sheets. The wave mode whichf reachest to 12 the fastest will be the one that separates the ligaments from the sheet. We numericallyevaluate the time elapsedTelapseduntilfgrows to fcrit for various wave modes. The result is plotted inFigure 2

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Figure 2: Telapsedvs

The wave mode that gives the minimum Telapsed= Tlig= 0.270s is

sheetcrit = 16.3mm (14)

The thickness of the sheet at the breakup point:

(Tlig) = 6.7mm (15)

The breaking up occurs at the following radius and height:

rlig= 1.03m (16)

zlig= 1.69m (17)

The most probable points for breakup to occur are the crests and troughs of the wave; therefore, thewidth along thesdirectionof the ligaments aresheetcrit /2. The thickness of the ligaments are simply equalto the sheet thickness at the breakup point.

The ligaments will try to change to a shape to minimize its energy. Such shape has a circular crosssection; therefore each ligaments cross section will experience a process of damped oscillation to settle intoa circular shape. The assumption of waters incompressibility gives

lig

sheetcrit

2 =R

2

lig (18)

From which we can obtain the radius of the ligament

Rlig=

lig

sheetcrit

2 = 4.17mm (19)

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3.3 The breakup of water ligaments to water droplets

The ligaments formed at the end of the previous stage are tend to break up into smaller ligament fragmentsdue to the growth of k direction waves (along the length of the ligaments). The aerodynamic forcesacting on the ligaments are perpendicular to the length of the ligaments, so they do not contribute to thegrowth of the k direction waves. The critical ligament wavelength, corresponding to the fastest growingkdirection wave can be obtained from a different stability analysis[6]:

ligcrit= Rlig12+

3w2wRlig

= 18.5mm (20)

The ligament fragments will approximately have a cylinder shape, with length ligcrit/2and radius Rlig.They will change to the shape that minimizes the surface tension energy, i.e. sphere, by a process of dampedoscillation. Again, the assumption about waters incompressibility gives the characteristic droplet radius:

Rdroplet=

3R2lig

ligcrit

4

1/3= 6.22mm (21)

The number of droplets that are formed after a whole ligament has completely disintegrated into droplets

can then be calculatedN=

VligVdroplet

=(2rlig)(R

2lig)

43R

3droplet

575 (22)

Ren[6]and Wu[2]provide the total time for a ligament to completely break up into droplets

Tdroplets= 24

2w

1/2R3/2lig = 1.07s (23)

Ligament fragments will be formed not all at once, but gradually: after being formed, each ligamentfragment would also breakup into smaller ones, as long as they are not too small and sphere-like. Assumethat characteristic time for a ligament to break into two parts is nearly constant, the ligament fragmentsreproduce rate is proportional to the number of ligament fragments existing at a particular time:

dNligdt

=C1Nlig (24)

Nlig= Nlig(0)eC1(tTlig) (25)

whereC1 is a constant of proportionality. From the fact that after durationTdroplets= 1.07s, one initialligament forms 575 small ligament fragments, we find that C1= 5.93.

A droplet will be produced from a long ligament when the breakup occurs near one of the ligamentsends. Therefore, the rate of droplet production then will be proportional to the number of ligamentsexisting:

dNdropletdt

=C2Nlig=C2e5.93(tTlig) (26)

If we substitute t =Tlig+Tdroplets, the result must match with 575.

Ndroplets(Tlig+Tdroplets) =C2e5.93tdroplets = 575 (27)

From this we getC2= 1.009 (28)

Thus, the number of droplets produced as function of time is

Ndroplets(t) e5.93(tTlig) (29)

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3.4 Dynamics of the droplets

In their flight, the water droplets are assumed to be subjected only to gravitation and air drag force. Theair drag force will distort the droplet, transforming its shape from a sphere to a more disc-like shape. If thedistortion is slight, then the droplet would take approximately an oblate ellipsoid shape. If the distortionis strong, then the droplet would even become like a thin disc. If the drag force strong and is appliedcontinuously for a long enough duration, then the droplet would, after being distorted, finally break apart

into multiple smaller droplets.Schmehl (2000)[7] shows that the condition for a droplet to break apart is the Weber number mustexceed a critical value:

Wec= 12(1 + 1.077On1.6) (30)

whereOn is the Ohnesorge number:

On= w

2wR (31)

Here, again, we have taken the diameter of the droplet as the characteristic length. For a water dropletof radius2cm(which is very large according to Giulio [4]), travelling at speed 8.5m/s(the initial shootingvelocity of water), W e= 5.11, far below the critical value. We therefore conclude that the droplets formedby water fountain cannot break apart systematically due to aerodynamic forces. The event of a dropletbreaking apart due to aerodynamic forces therefore would be a very rare event, only due to some randomturbulence of the air, thus enabling us to neglect this effect.

Schmehl also develops a relation for the maximum distorted size of a droplet with the original sphericalradius of the droplet:

Rmax=R(1+0.19We) (32)

Since the typical Weber number for droplets formed by water fountain is far below the critical Webervalue, it is reasonable to guess that the distortion of the droplet would be slight, thus making shape of thedroplet approximately an oblate ellipsoid with Rmaxbeing the major-axis length. Our assumption of thein-compressibility of water requires the ellipsoids volume to be equal to that of the original sphere. Wethus obtain the minor-axis length:

b= R3

Rmax2 (33)

Reference[1] gives us the formulae of the drag force subjected on an ellipsoid object:

Fd= 6KaRmaxv (34)

where the drag force coefficient K is given by:

K=43(

21)(22)

(21)1/2tan1 (21)1/2 +

(35)

where=Rmax/b=Rmax3/R3

Applying the Newtons second law to a droplet then gives:

a= g 9KaRmaxv2wR3

(36)

4 Results

Based on our calculation in part 3.3 above, droplets start forming at t=Tlig= 0.27s and the number ofdroplets will stop increasing at t =Tlig+Tdroplets= 1.34s. This corresponds to the case where the initialligament ring is formed at the most probably forming location. Our goal in this section is to determinethe landing point distribution for droplets formed within this time interval. We assume that after ligamentfragments are formed, the ligament fragments will still travel as a group with velocity u(t) which evolves

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according to the differential equations listed in section 3.1, while each droplet after being formed from aligament will move independently according to equation (36).

Let us start by dividing the said time interval into a large number of sections and assuming that thereis only one droplet formed within each section. By evaluating the radial distance travelled by each dropletcorresponding to each section, We can obtain the distribution of landing points for all time sections. Using1000 time segments between t = 0.27s and t = 1.34s, the full trajectories of all water molecules ejected bythe fountain are described by the following graph

Figure 3: Full shape of the fountain

Many attributes of the probability distribution are readily to be read. The probability for any point ofradial distance smaller thanrmin= 5.077mto be hit by any water molecule is 0. The same situation occursfor any point of radial distance larger than rmax= 5.249m. The radial distance travelled by a droplets as

function of its formation time is described by the following graph

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Figure 4: Radial distance travelled by a droplet versus its time of formationtform

The number of droplets formed between tform and tform+ dtform can be found be differentiatingequation (34)

dNdroplets(tform) = 5.93e5.93(tformTlig)dtform (37)

All these droplets will land on radial distances betweenr(tform)and r(tform +dtform). Therefore, theprobability distribution of droplets landing points over the radial distance can be well represented by the

graph dNdroplets

dtform(tform) as a function ofr(tform). To normalize this probability distribution, we calculate

the integral rmax

rmin

dNdropletsdtform

(r)dr= 24.75ms1 (38)

Where the numerical value obtained from the code "distribution.m" given below. The normalized probability

distribution can then be calculated by dividing dNdroplets

dtformwith the result of the integral we just obtained.

The normalized distribution of landing points is plotted in Figure 4 :

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Figure 5: dNdroplets

dtform(tform)as function ofr(tform)

The graph of probability distribution clearly shows that the points that have the highest probability tobe hit by any water molecule are located on a ring of radius rmin= 5.077m. All these numerical valuesare calculated using the code "section.m" appended below. Next, to calculate the median circle radius, wewrite a Matlab code that attempts to calculate the same integral as in equation (38) by dividing the area

under the graph of dNdroplets

dtform(tform) as a function ofr(tform) into rectangular segments of equal widths.

The code then adds the area of these rectangles until the sum of area reaches half the value of integral inequation (38), the code is asked to stop. The value of radius at this stopping point can be extractedThemedian circles radius is found to bermedian= 5.0843m.

5 Discussion

In evaluating our method, the most obvious strength is that, qualitatively, we have not avoided any aspectof the water fountain phenomenon: from the breakup of the water sheet into water ligaments and theninto droplets, to the possibility of breakup of the droplets, to the deformation of the droplets due to dragforce. We have also always backed our qualitative descriptions and explanation by quantitative reasoning.This qualitative and quantitative comprehensiveness is extremely useful in this problem of probability dis-tribution of landing, since using our method, one can try in so various ways to further analyse how theprobability distribution will vary over some disturbance in some aspects of the phenomenon. For example,one can always add some distribution of inaccuracy to the water-shooting system. Or one can add somediscrepancy of the formation positions of the ligaments (the ligaments will be alternatively formed at the

crest and trough of the waves on the sheet), with the discrepancy approximately equal to the water sheetsthickness at the breakup position. Or one can add some wind conditions, and replacev in section 3.4by the relative velocity of the droplet and air. Any disturbance at any stage of the phenomenon can becalculated to see how it affects the final probability distribution if every stage of the the phenomenon istreated quantitatively.

Not only comprehensive, our method is also quite rigorous. We require little but reasonable assumptions.

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There exist also several weaknesses in our method. Since they are specific, we shall make a list of them:

1. In section 3.3, we determine the rate of ligament production as dNlig

dt Nlig. However, if we wantto be more strict, we should distinguish between ligaments that are able to breakup and ligamentsthat are not (those small enough to be considered as droplets). The proportional relation we usedcan still work well if only until the end of the time interval for droplets formation, the ligaments get

small enough to not be able to break any more, making our relation approximately correct for mostof the time. The same weakness occurs for the relation

dNdropdt Nlig.

2. Also in section 3.3, we mention that the droplets formed are initially the smallest possible ligamentsfragments, which are of length ligcrit/2. Our reason was that the crough and crest of a wave are thepositions easiest to be tore apart, and therefore a separated unit from a big ligament should havehalf of the wavelength. It is not true so. For a ligament of length a few times the wavelength, it isshort enough to not populate such wave; it will travel essentially as an unit and will not break apartby the growing magnitude of the wave. If it breaks, it will likely follow one of the breakup regimesmentioned in [7], depending on its Weber number, to break into multiple much smaller droplets. If itdoes not break, then we should have increased our droplets size in subsequent calculations.

References

[1] G. Ahmadi. Hydrodynamic forces. drag force and drag coefficient.

[2] Andre W. Marshall Di Wu, Delphine Guillemin. A modeling basis for predicting the initial sprinklerspray. Fire Safety Journal, 42:283294, 2007.

[3] McKinney T.R. Ibrahim, E.A. Injection characteristics of non-swriling and swriling annular liquidsheets. Las Vegas, Nevada, May 10-13, 2004. Jannaf Propulsion Conference.

[4] Giulio Lorenzini. Simplified modelling of sprinkler droplet dynamics. Silsoe Research Institute, 2003.

[5] W. R. Johns N. Drombrowski. The aerodynamic instability and disintegration of viscous liquid sheets.Chemical Engineering Science, 18:203214, 1963.

[6] Ying-hui Zheng Chi Do Andre W. Marshall Ning Ren, Andrew F. Blum. Quantifying the initial sprayfrom fire sprinklers. Chemical Engineering Science, pages 503514, 2008.

[7] S. Wittig R. Schmehl, G. Maier. Cfd analysis of fuel atomization, secondary droplet breakup and spraydispersion in the premix duct of a lpp combustor. Pasadena, CA, USA, July 2000. Eigth InternationalConference on Liquid Atomization and Spray Systems.

[8] Weber Z. Zum zerfall eines flussigkeitstrahles (on the disruptioin of liquid jets). Angew Math Mech,11:13654, 1931.

6 Codes

6.1 functiondroplet.m

function d y d t = f u n c t io n d r op l e t ( t , Y );

% E x t r ac t v a r i ab l e s a n d c o n s ta n t s

r = Y ( 1 ); z = Y ( 2 ); r d ot = Y ( 3 ) ; z d o t = Y (4 ) ; R = Y ( 5 ) ; g = Y ( 6 ); m u a = Y (7 ) ; r h o a = Y (8 ) ; r h ow = Y ( 9 ) ; s i g ma = Y ( 1 0 ) ; m u w

% E q ua t io n s ( 3 0) t o ( 3 5)

On=muw/sqrt(2*rhow*sigma*R );

We=12*(1+1.077*On^1.6);

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Rmax=R*(1+0.19*sqrt(We));

beta=(Rmax/R)^3;

K=(4*(beta^2-1)/3)/(beta+beta *(beta^2-2)*atan(sqrt(beta^2-1))/sqrt(beta^2-1));

% E q u a t io n ( 3 6 )

dr=rdot;

dz=zdot;

drdot=-9*K*mu a*Rmax*rdot/(2*rhow*R^3);

dzdot=-g-9*

K*

mu a*

Rmax*

zdot/(2*

rhow*

R^3);

d y dt = [ d r ; d z ; d rd o t ; d z do t ; 0 ; 0 ; 0 ;0 ; 0 ; 0 ;0 ] ;

6.2 eventfunctiondroplet.m

function [value,isterminal,direction]=eventfunctiondroplet(t,Y);


r = Y ( 1 ); z = Y ( 2 ); r d ot = Y ( 3 ) ; z d o t = Y (4 ) ; R = Y ( 5 ); g = Y ( 6 ); m u a = Y (7 ) ; r h o a = Y (8 ) ; r h ow = Y ( 9 ) ; s i gm a = Y ( 1 0) ; m u w =

% T el l M at l ab t o s to p c a lc u la t in g w he n z = 0( t he d r op l et h it s t he p oo l )

value(1)=z;

isterminal(1)=1;

direction(1)=-1;

6.3 functionsheettrajectory.m

function d y d t = f u n c t i on s h e e tt r a j e ct o r y ( t , Y );


s = Y ( : , 1 ); a l p ha = Y ( 2 ) ; r = Y ( 3 ); z = Y ( 4 ) ; u = Y ( 5 ) ; u T = Y ( 6) ; D = Y ( 7 ) ; f = Y ( 8 ) ; l a m b d a = Y ( 9) ; g = Y ( 1 0 ) ; r h o w = Y ( 1

% R e y n o ld s n u mb e rs , e q u at i o n ( 9 ) a n d ( 1 0 )

Res=abs((rhoa*D*u)/mua);

ReT=abs((rhoa*D*uT)/mua);

% E q ua t io n s o f m ot io n , e q ua t io n ( 3) t o ( 8)

ds=u;dalpha=u*((rhow*uT^2*cos(alpha)/r-2*sigma*cos(alpha)/(D*r)+rhow*g*sin(alpha))/(rhow*u^2-2*sigma/D));

dr=u*sin(alpha);

dz=u*cos(alpha);

du=u*((rhow*uT^2*sin(alpha)/r-(rhoa*u^2/(2*D ))*(0.79*((1+150*D)/r)*Res^(-1/4))-rhow*g*cos(alpha))/(rho

duT=u*((-rhow*u*uT*sin(alpha)/r-(rhoa*uT^2/(2*D ))*(0.79*((1+150*D)/r)*ReT^(-1/4)))/(rhow*u));

dD=-D*du/u-D*dr/r;

% A m p l i tu d e g r o wt h r a t e e q u a t i on ( 1 3 )

df=-2*pi^2*muw/(rhow*lambda^2)+sqrt((2*pi^2*muw/(rhow*lambda^2))^2+(4*pi*rhoa*u^2/lambda-8*pi^2*sigma/

d y dt = [ d s ; d a lp h a ; d r ; dz ; d u ; d uT ; d D ; d f ; 0 ; 0; 0 ; 0 ; 0 ;0 ; 0 ; 0 ;0 ; 0 ; 0 ];

6.4 eventfunctionsheettrajectory.m

function [value,isterminal,direction]=eventfunctionsheettrajectory(s,Y)


s = Y ( : , 1 ); a l p ha = Y ( 2 ) ; r = Y ( 3 ); z = Y ( 4 ) ; u = Y ( 5 ) ; u T = Y ( 6) ; D = Y ( 7 ) ; f = Y ( 8 ) ; l a m b d a = Y ( 9) ; g = Y ( 1 0 ) ; r h o w = Y ( 1

% T e l l M a t l a b t o s t op c a l c ul a t i ng w h en t h e d i m e n s i o nl e s s a m p l it u d e r e a ch e s

% t h e c r i t ic a l a m p l it u d e

value(1)=f-fc;

isterminal(1)=1;

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direction(1)=0;

6.5 sheettrajectoryloop.m

c l e ar a l l ;

clc;

% I n i t ia l C o n d it i o n s

%%

a l ph a 0 = pi / 12 ;

r0=0.1;

z0=0;

u0=8;

uT0=3;

D0=0.05;

s0=0;

t0=0;

t f =3 0 ;

f0=0;

%Constants

%%

g=9.8;

rhow=1000;

rhoa=1.225;

sigma=0.0728;

mua=1.81e-5;

muw=1.002e-3;

rd=r0;

% L a mb d a R a ng e s f r om 0 . 01 4 m t o 0 . 0 20 , w it h 1 00 0 d a ta p o in t s

lambda=linspace(0.014,0.020 ,1000);

%Main

%%

% S t o p p in g c o n d it i o n s

o p ti o n = o d es e t ('Events',@eventfunctionsheettrajectory);

tlist=[];

% I t e r at e f o r d i f f e r e nt v a l ue s o f l a m bd a s

fo r i=1:length(lambda)

% E v a l u at e t h e e q u a t i o n o f m o t io n u s i ng R u ng e - K u t t a m e t h o d

[ t ,Y , s E , Y E ]= o d e4 5 ( @ fu n ct i on s he e tt r aj e ct o ry , [ t 0 t f ], [ s0 , a lp ha 0 , r 0 , z 0 , u 0 , u T0 , D 0 , f 0 , l a mb d a (i )

,option);

% E x t r ac t t h e d a t a

s = Y ( : , 1 ); a l p ha = Y ( : , 2 ); r = Y ( : , 3 ) ; z = Y ( : , 4 ) ; u = Y ( : , 5) ; u T = Y ( : , 6) ; D = Y ( : , 7 ) ; f = Y ( : , 8 );

tlist(i)=t(end);

en d

plot(lambda,tlist,'b');

% F in d t he l a mb d a t ha t g i ve s t he m i ni m um v al u e o f t im e t u n ti l t e ar i ng

% o c cu r s , e q u a ti o n ( 1 4 )

[ M , I ] = m i n ( t l is t ) ;

lambda(I)

6.6 breakup.m

c l e ar a l l ;

clc;


%%

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a l ph a 0 = pi / 12 ;

r0=0.1;

z0=0;

u0=8;

uT0=3;

D0=0.05;

s0=0;

t0=0;

t f =3 0 ;f0=0;

%Constants

%%

g=9.8;

rhow=1000;

rhoa=1.225;

sigma=0.0728;

mua=1.81e-5;

muw=1.002e-3;

rd=r0;

% T h e m o s t u n s ta b l e l a m bd a

lambda=0.0163;

%Main

%%


o p ti o n = o d es e t ('Events',@eventfunctionsheettrajectory);


[ t ,Y , t E , Y E ]= o d e4 5 ( @ fu n ct i on s he e tt r aj e ct o ry , [ t 0 t f ], [ s0 , a lp ha 0 , r 0 , z 0 , u 0 , u T0 , D 0 , f 0 , l am bd a , g

,option);



plot(r,z,'b');

h ol d o n ;

% M i n i mu m t i me e l a ps e d

Tlig=t(end)

% n u m e r ic a l v a l ue f o r e q u a t i o n ( 1 5 )

Dlig=D(end)


rlig=r(end)


zlig=z(end)

% D e te r mi n in g t he d r op l et r a di u s e q ua t io n ( 18 ) t o ( 21 )


Rlig=sqrt(D(end)*lambda/(2*pi))


lambdalig=pi*Rlig/sqrt(0.5+3*muw/sqrt(2*rhow*Rlig*sigma))


Rdrop=(3*Rlig^2*lambdalig/4)^(1/3)

%%

% C o nt i nu e t o e v al u at e t he m o ti o n o f w at e r d r op l et s


o p ti o n = o d es e t ('Events',@eventfunctiondroplet);


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[ t , Y , t E , Y E ] = o d e4 5 ( @ f u n ct i o n dr o p le t , [ t E t f ] , [ r ( e n d ) , z ( e n d ) , u ( e n d )*s i n ( a lp h a ( e n d) ) , u ( e n d )*cos(alp


r = Y ( : , 1 ); z = Y ( : , 2 ); r d o t = Y ( : , 3) ; y d o t = Y ( : , 4) ; R = Y ( : , 5 );

plot(r,z,'r');

h ol d o n

% M o s t p r o ba b l e l a n di n g p o i ntRmostprobable=r(end)

6.7 Distribution.m

c l e ar a l l ;

clc;


%%

a l ph a 0 = pi / 12 ;

r0=0.1;

z0=0;

u0=8;

uT0=3;

D0=0.05;

s0=0;

t0=0;

f0=0;

%Constants

%%

g=9.8;

rhow=1000;

rhoa=1.225;

sigma=0.0728;

mua=1.81e-5;

muw=1.002e-3;

rd=r0;

lambda=0.0163;

%d=20*60 ;

%Main

%%

%Droplets' t i me s o f f o r m at i o n

tf=linspace(0.27,1.34,100);

rlist=[];

% I t er a te f or e ac h t im e o f f o rm a ti o n

fo r i=1:length(tf);


[ t , Y] = o de 4 5 ( @f u nc t io n sh e et t ra j ec t or y , [ t 0 t f (i ) ], [ s0 , a lp ha 0 , r 0 , z 0 , u 0 , u T0 , D 0 , f 0 , l am bd a , g , r ho



o p ti o n = o d es e t ('Events',@eventfunctiondroplet);

% P lo t t he i n it i al w a te r s he e t s h ap eplot(r,z,'b')

h ol d o n

plot(-r,z,'b')

h ol d o n

% D e t e r m in a t i on o f d r o pl e t s i z e

Rlig=sqrt(D(end)*lambda/(2*pi));

lambdalig=pi*Rlig/sqrt(0.5+3*muw/sqrt(2*rhow*Rlig*sigma));

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Rdrop=(3*Rlig^2*lambdalig/4)^(1/3);

% E v al u at e t he m o ti o n o f t he d r op l et

[ t , Y , t E , Y E ] = o d e4 5 ( @ f u n ct i o n dr o p le t , [ t ( e n d ) 1 0 0] , [ r ( e n d) , z ( e n d ) , u ( e n d )*s i n ( a lp h a ( e n d) ) , u ( e n d )*c o


r = Y ( : , 1 ); z = Y ( : , 2 ); r d o t = Y ( : , 3) ; y d o t = Y ( : , 4) ; R = Y ( : , 5 );

% P l o t t h e d r op l et'

s t r a j ec t o r yplot(r,z,'c')

h ol d o n

plot(-r,z,'c')

h ol d o n

rlist(i)=r(end);

en d

% M i n i mu m r a d ia l l a n di n g d i s t an c e

[ M m in , I m i n ] = m i n ( r li s t ) ;

rmin=rlist(Imin)

% M a x i mu m r a d ia l l a n di n g d i s t an c e

[ M m ax , I m a x ] = m a x ( r li s t ) ;

rmax=rlist(Imax)

figure()

% P lo t t he r a di a l d i st a nc e t r av e ll e d b y a d r op l et v e rs u s i ts t i me o f f o rm a ti o n

p l ot ( t f , r l is t , 'b')

% e q u a t io n ( 3 7 )

dNdt=5.93*exp(5.93.*(tf-0.27));

figure()

% p lo t d N dt v s t f

p l ot ( t f , d Nd t , 'b')

% r i n c r em e n t

dr=diff(rlist);

%Integral

intdNdtdr=0;

fo r j=1:(length(dNdt)-1);

intdNdtdr=intdNdtdr+dNdt(j)*abs(dr(j));

en d

% R e s u lt o f i n t e gr a t i on

intdNdtdrtotal=intdNdtdr

figure()

% p l o t t h e n o r m al i z e d p r o b a b i l i ty d e n si t y

dNdtnormalized=dNdt/intdNdtdr;

plot(rlist,dNdtnormalized,'b');

% D e t e r m in a t i on o f t h e m e d ia n c i r cl e

% R e c a l cu l a t e t h e i n t e gr a l

intdNdtdr=0;

j=1;

while intdNdtdr

upc_2014

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