Mario Vargas

Glenn Research Center, Cleveland, Ohio

Droplet Deformation Prediction With the Droplet 

Deformation and Breakup Model (DDB)

NASA/TM—2012-217705

September 2012

AIAA–2012–3131
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI 

Information Desk at 443–757–5803

• Phone the NASA STI Information Desk at

 443–757–5802

• Write to:

 STI Information Desk

 NASA Center for AeroSpace Information

 7115 Standard Drive

 Hanover, MD 21076–1320
Mario Vargas

Glenn Research Center, Cleveland, Ohio

Droplet Deformation Prediction With the Droplet 

Deformation and Breakup Model (DDB)

NASA/TM—2012-217705

September 2012

AIAA–2012–3131

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Atmospheric Space Environments Conference

sponsored by the American Institute of Aeronautics and Astronautics

New Orleans, Louisiana, June 25–28, 2012
Acknowledgments

Thanks to Dr. Andy Broeren and Dr. Jen Chin Tsao for their patience and encouragement to the author to fi nish the paper. The 

present work augments the experimental work performed in an international agreement between NASA and the Instituto Nacional 

de Técnica Aeroespacial (INTA). It is funded by the Atmospheric Environment Safety Technologies (AEST) Project, Aviation 

Safety Program (AvSP). Especial thanks to Dr. Ron Colantonio and Ms. Mary Wadel for their support of the effort.

Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217705 1 

Droplet Deformation Prediction With the Droplet Deformation 

and Breakup Model (DDB) 

Mario Vargas

National Aeronautics and Space Administration 

Glenn Research Center 

Brook Park, Ohio 44135 

Abstract 

The Droplet Deformation and Breakup Model was used to predict deformation of droplets 

approaching the leading edge stagnation line of an airfoil. The quasi-steady model was solved for each 

position along the droplet path. A program was developed to solve the non-linear, second order, ordinary 

differential equation that governs the model. A fourth order Runge-Kutta method was used to solve the 

equation. Experimental slip velocities from droplet breakup studies were used as input to the model which 

required slip velocity along the particle path. The center of mass displacement predictions were compared 

to the experimental measurements from the droplet breakup studies for droplets with radii in the range of 

200 to 700 m approaching the airfoil at 50 and 90 m/sec. The model predictions were good for the 

displacement of the center of mass for small and medium sized droplets. For larger droplets the model 

predictions did not agree with the experimental results. 

Nomenclature 

As Area of the deforming droplet 

a major semi-axis of oblate spheroid 

b minor semi-axis of oblate spheroid 

Bo Bond number 

Cd Drag coefficient 

DDB Droplet Deformation and Breakup Model 

R Droplet radius 

Re Reynolds number 

SLD Supercooled Large Droplets 

TAB Taylor Analogy Break-up Model 

Urel Slip Velocity, relative velocity between the droplet and the air

We Weber number 

Introduction 

Supercooled Large Droplet (SLD) breakup near the leading edge of large transport airfoils may alter 

the mass flux of water reaching the leading edge and be a factor that needs to be accounted for in the 

prediction of ice accretion formation in icing codes. The present research is part of current efforts to 

understand, measure and predict SLD breakup near the leading edge of large transport wings. The studies 

will help determine if icing codes need to be modified to account for droplet breakup when the size of the 

droplets is in the SLD regime. 

Recent experimental studies(Refs. 1 to 3) of Supercooled Large Droplet (SLD) deformation and 

breakup near the leading edge of airfoils have provided imaging data and measurements of important 

parameters along the droplet paths. The experimental data from these studies can be used to evaluate 
NASA/TM—2012-217705 2 

previously developed droplet breakup models (Refs. 4 to 7) and determine the limits of their applicability 

to the experimental configuration where droplets deform and break up near the leading edge of an airfoil. 

When there is a difference between the velocity of a droplet and the velocity of the air, called slip 

velocity, a flow field develops around the droplet and aerodynamic forces begin to deform it. As the 

viscous forces and pressure forces deform the droplet, its shape changes which, in turn modifies the flow 

field and the aerodynamic forces acting on the droplet. The flow field and the droplet deformation are 

coupled physical processes. In a strict sense the modeling of droplet deformation requires the 

mathematical equations describing the coupling between the two processes. This is a very complex 

situation because the droplet is a liquid elastic body held together by surface tension, and internal 

circulation of water inside the droplet may be present. General models of droplet deformation do not 

capture all of the physics but use simplifications to make the solution more tractable. These models are 

valuable because they capture the important physical processes and provide insight into the phenomena. 

The only way to know if a model may be used in a given experimental configuration is to test it against 

the experimental results and find the limits of its usefulness and applicability. 

Droplet breakup has been studied extensively in the past for experimental configurations(Refs. 4 to 7) 

where the droplet is subjected to a rapidly changing slip velocity. Current droplet breakup models were 

developed primarily for that type of configuration. For droplets which approach the airfoil along the 

stagnation line the droplets are subject to a gradual increase in the slip velocity that allows them to adjust 

their shape along the path and tolerate higher Weber numbers before beginning the actual breakup 

sequence. It is important to test the applicability of current droplet breakup models to this configuration 

because the models may help interpret experimental data, provide insights into the physics involved in the 

process and offer guidance in the planning of future experiments. 

The Droplet Deformation and Breakup (DDB) Model and the Taylor Analogy Break-up (TAB) 

Model more closely model the kind of deformation observed in droplet breakup studies near the leading 

edge of airfoils. The DDB was selected for the current study to predict the droplet deformation along the 

path of the droplet. A program was developed to solve the second order, non-linear ordinary differential 

equation that governs the model. A fourth order Runge-Kutta method was used in the numerical scheme 

to solve the equation. Experimental values of the slip velocity were used as input to the model. The 

predicted displacements of the droplet center of mass were compared to the experimental measurements 

from current droplet breakup studies on a DBKUP airfoil geometry. Comparisons were generated for 

airfoil speeds of 50 and 90 m/sec and droplet sizes from 200 to 700 m. 

Background on Droplet Deformation Breakup Model 

The DDB model was proposed in 1993 by Ibrahim, Yang and Przekwas (Ref. 6). The model assumes 

that “the liquid droplet is deformed from an initial spherical shape of radius R into an oblate spheroid of 

an ellipsoidal cross section with major semi-axis a and minor semi-axis b”. This assumption makes the 

model attractive for droplet deformation prediction near the leading edge of an airfoil because during 

experimental studies the droplets were observed to deform from a spherical shape to an oblate spheroid. 

For this reason the model was tested against the experimental data to determine its applicability. 

The DDB model tracks the vertical motion of the center of mass of a half droplet (Fig. 1). It is based 

on conservation of mass and energy for half the droplet as it deforms. In the rest of this section, the 

derivation of the fundamental equation representing the model is presented, following Ibrahim’s original 

derivation (Ref. 6). 

Assuming the droplet does not exchange heat with its surroundings and that the only forces involved 

in the deformation of the droplet are the viscous forces, the pressure forces and the surface tension forces, 

the energy equation for the half-droplet is: 

dt

dW

dt

dE   (1) 
NASA/TM—2012-217705 3 

where E is the energy and W is the work done on the droplet by the pressure and viscous forces. 

Therefore 

   

dt

dW

dt

dE

dt

dEkinetic potential pressure (2) 

where  is the viscous dissipation. Each term in Equation (2) will be evaluated next. 

Kinetic Energy: 















             

 



   2

2

kinetic 3 3

3

2

3

2

dt

d y

dt

dy R

dt

dv

v R v

dt

dv

m

dt

dE

l l (3) 

where m is the mass of half the droplet, l the density of water, R the radius of the droplet, v = dy/dt and 

dv/dt = d2

y/dt2 are the velocity and acceleration respectively of the center of mass of the deforming half 

droplet (Fig. 1). 

Potential Energy 





 



   dt

dA

dt

dE s

2

potential 1 (4) 

where  is the surface tension (energy/unit area), 1/2 is the factor for the half droplet energy balance, As is 

the area of the deforming droplet. The area of the deforming droplet is the area of the oblate spheroid with 

major semi-axis a and minor semi-axis b. It is given by: 

      2 2 A 2 a 2 b s (5) 

where 





 





 

  



  1

1

2

1 ln (6) 

and 

2

1







 





   b

a (7) 

In the model As is approximated by assuming  = 1: 

2 ( ) 2 2 A a b s    (8) 

For increasing values of a/b,  quickly increases from a value of 1, but the approximation is 

reasonable within the overall calculation of As. As the droplet deforms into an oblate spheroid, the value 

of the major semi-axis a increases while the value of the minor semi-axis b decreases. The first term in 

the Equation (5) becomes larger, the second term smaller, and the impact of the error in the calculation of 

the second term is not as large for the calculation of As. In Figure 2, the ratio of the approximation to the 
NASA/TM—2012-217705 4 

area divided by the exact value is plotted against the ratio a/b for five different droplets. The ratio 

decreases to a minimum at a/b = 2 and then increases. 

Before discussing the derivative with respect to time of Equation (8), two other terms that will be 

used are introduced. From the conservation of volume: 

2 3

3

4

3

4

a b  R (9) 

therefore 

2

3

a

R b  (10) 

The distance of the center of mass in terms of the major semi-axis a was approximated by the 

distance of the center of mass for an area of a semi-ellipse of altitude a: 

y a

  3

4 (11) 

which can also be noted as 

a  c  y (12) 

where 

4

3

c  (13) 

For half an oblate spheroid the distance of the center of mass in terms of the major semi-axis a is 

given by (3/8)a. The approximation given in Equation (11) introduces an error of 12 percent. The exact 

value of c = 8/3 was used when running the model for this study. 

Taking the derivative with respect to time of As and substituting in Equation (8) the expressions for b

and a from Equations (10) and (12) gives: 





 



























 



     



dt

dy

R

c y y dt

dAs 3 6

1 2

4

9 (14) 

Substituting Equation (14) into Equation (4) gives the expression for the potential energy: 





 



























 



      



dt

dy

R

c y y dt

dE 3 6

potential 1 2

8

9 (15) 

The rate of change of the work done by the pressure is given by: 





 



      dt

dy p A

dt

dW

p 2

pressure 1 (16) 
NASA/TM—2012-217705 5 

Where p is the pressure and Ap is the projected area of the droplet. The projected area of the deformed 

droplet is approximated by the original projected area of the spherical droplet: 

2 Ap   R (17) 

This assumption is very restrictive because it limits the applicability of the model to small droplet 

deformations. But since Ap is the projected area of the oblate spheroid and is given by a

2

, where a is the 

major semi-axis of the oblate spheroid, the model can be modified to avoid the approximation in 

Equation (17) by substituting a

2 for R2

. 

The value of the average pressure exerted on the front and back of the half oblate spheroid is assumed 

to be equal to the stagnation pressure of the droplet: 

p U rel g

2

2

1    (18) 

where g is the air density and Urel is the slip velocity. The approximation in Equation (18) is reasonable 

because it results from a dimensional analysis which implies that inertia force that deform the droplet is 

proportional to the stagnation pressure (Ref. 1). 

Substituting Equations (17) and (18) into Equation (16) gives the rate of work done by the pressure 

as: 





 



       dt

dy R U

dt

dW

g rel

pressure 2 2

4 (19) 

The negative sign in front of term indicates work is being done on the droplet by the pressure forces, 

following the standard convention. The rate of work done by the viscous forces is given by the viscous 

dissipation function (Ref. 6): 

2

viscous 3 1

3

8















       dt

dy

y

R

dt

dW

l (20) 

where l is the water viscosity and R the radius of the droplet. Substitution of Equations (3), (15), (19), 

and (20) into Equation (2) gives the basic equation that governs the model: 

2

2 2 3

3 6

2

2

3

1

3

8

4

1 2

8

9

3

2















       

 



      





 



























 



      















 



 





  



dt

dy

y

R

dt

dy R U

dt

dy

R

c y y

dt

d y

dt

dy R

g rel l

l

 (21) 

To non-dimensionalize the equation the following dimensionless variables are defined: 

R

y y *

 (22) 
NASA/TM—2012-217705 6 





 



   R

U

t t * rel

 (23) 

g

gUrelR

Re



  (24) 

where g is the density of the air and g is the absolute viscosity of the air. 



  U R

We g rel

2

 (25) 

where  is the surface tension of water. 

g

l K



  (26) 

Substituting Equations (22) to (26) into Equation (21) and re-arranging we get: 

  8

3 1 2

16

4 1 27 6 2

2 2

2





 



   



  













  * *

*

*

* *

*

y c y

dt We

dy

y Re

N

dt

d y K (27) 

For readability, the asterisk is dropped from the y and t terms in Equation (27). The equation that 

represents the DDB model is now written: 

    8

3 1 2

16

4 1 27 6

2

2 2

2

   



  













  y c y dt We

dy

y Re

N

dt

d y K * (28) 

This is a second order non-linear ordinary differential equation with two initial conditions given by: 

  3

4

y at t = 0 (29) 

where y and t are non-dimensional. This initial condition reflects the fact that the half-droplet is a 

hemisphere before any deformation begins. The second initial condition is: 

 0

dt

dy at t = 0 (30) 

where the dimensionless initial velocity of the center of mass is assumed zero. Equations (28), (29), and 

(30) constitute the initial value problem. In addition to the water and air properties, the radius of the 

droplet R and Urel are needed to solve Equation (28) together with its initial conditions. 
NASA/TM—2012-217705 7 

Numerical Solution of the Initial Value Problem 

The initial value problem constituted by the ordinary differential equation and the boundary 

conditions of Equations (28), (29), and (30) was solved numerically. The first step in the numerical 

approach was to use a transformation of variables and convert the original problem into a system of two 

coupled first order ordinary differential equations with corresponding initial conditions. 

   

0 0

3

4 0

8

3 1 2

16 We

4 1 27 6

2

2 2

2

  

   



  













 

( ) dt

dy y( )

y c y dt

dy

y

Re

N

dt

d y K *

0 0

8

3

4

3 1 2

16 We

4 1 27

3

4 0

2 6

2   



















 



    

    

  



IC : z( ) K

y

K

z

K Re y

N

dt

dz

z IC : y( ) dz

dy

 (31) 

A fixed time step, fourth order Runge-Kutta (RK) integrator was used to solve the system in 

Equation (31). A second order RK was used to estimate the truncation error. 

Input Parameters Needed for the DDB Model and the Numerical Solution 

Two types of input parameters are required to test the DDB model: (1) The input parameters needed 

for the DDB model and (2) The input parameters needed for the numerical scheme to solve the initial 

value problem. 

The input parameters for the DDB model are: density of air, viscosity of air, density of water, 

viscosity of water, surface tension of water, diameter of the droplet and the slip velocity. 

The input parameters for the numerical solution of the initial value problem are: the number of first 

order differential equations, initial and final value of the independent variable in the interval where the 

solution is evaluated, number of steps in the interval, step size, and error tolerance. 

Experimental Data Used With the DDB Model 

Experimental studies of droplets approaching the leading edge of an airfoil along the stagnation line 

have provided data on droplet deformation and breakup (Refs. 1 to 3). The experiments were conducted in 

a rotating arm facility (Ref. 2). An airfoil was placed at the end of the arm and rotated at a given velocity. 

Water droplets from a TSI-100 droplet generator were allowed to fall at a specific location in the path of 

the airfoil being rotated. High speed movies of the droplet deformation and breakup were recorded. 

Digital image processing programs developed for the analysis of the data were used to track the droplets 

and measure the main parameters involved. Figure 3 is a sequence of images captured during these 

experiments at different times as a droplet approached the airfoil. Initially the droplet was spherical as 

there was no difference between the velocity of the droplet and the air (slip velocity). As the droplet 
NASA/TM—2012-217705 8 

approached the airfoil, the air velocity decreased and the droplet’s slip velocity increased. The increase in 

the slip velocity generated aerodynamic forces that deformed the droplet initially to a symmetrical oblate 

spheroid (frame 40 in Fig. 3) and later to an asymmetrical oblate spheroid (frame 110 in Fig. 3), which is 

asymmetric with respect to the major semi-axis but rotationally symmetric with respect to the axis of 

motion. The image processing program measured the width and the height of the droplet along the path. 

Figure 4 shows the variation in width of the droplet along its path. The droplet width decreased to a 

minimum and then increased. The location at which the width of a droplet is a minimum defines where 

droplet deformation ends and droplet breakup begins (Ref. 3). This corresponds to frame 192 in Figure 3. 

As the droplet width decreased, its height increased. Figure 5 shows the height of the droplet along its 

path. The height of the droplet is the major a semi-axis (Fig. 1) of the oblate spheroid. The center of mass 

location is given by (3/8)a. The DDB model, which predicts center of mass of the deformed droplet, was 

only used to analyze the droplet deformation phase of the breakup and not the post droplet breakup phase. 

Data from five tracked droplets were used to test the DDB model. Table 1 presents the basic data for each 

droplet. 

Approach Taken and Main Assumption 

The approach taken to apply the DDB model to the deformation of the droplets was to assume that at 

each location along the trajectory the droplet is in quasi-steady equilibrium. Based on this assumption, the 

model was solved at each location along the path of the droplet. To solve the model equations at each 

location, the experimentally measured value of the slip velocity (Ref. 3) for that location was required as 

an input parameter for the model. The predictions of the displacement of the center of mass of the half 

droplet were compared to the experimentally measured values. 

Results and Discussion 

The DBB model, although highly nonlinear, behaved much like the classical spring-mass-friction 

case at a given location. The value of the center of mass oscillated around the final equilibrium value and 

as time increased it approached the equilibrium value (Fig. 6). The larger the experimental value of the 

slip velocity, the larger the amplitude of the initial oscillation, and the longer it took the system to reach 

equilibrium (Fig. 7). 

The DDB model was applied to five cases of droplet deformation. In each case the experimentally 

measured value of the slip velocity was used as part of the input for the model. The model was run at each 

location where experimental data was available up until the onset of breakup. Figures comparing 

computational results to experiment are presented for each case. A discussion of each case follows. 

Figure 8 shows the comparison between the model prediction and the experimental data for Case 1: 

Run 072211-1.8 droplet 4, droplet radius 199 m, airfoil chord 0.710 m and airfoil velocity 50 m/sec. At 

this low velocity the small droplet deforms very little as it approaches the airfoil. This is reflected in the 

nearly constant value of the experimental measurement of the center of mass location. The model 

prediction agrees well with the experimental data. In general small droplets resist deformation more than 

larger ones because the small radius of curvature translates into a larger surface tension force. 

Figure 9 shows the comparison between the model prediction and the experimental data for Case 2: 

Run 072611-18.6 droplet 1, droplet radius 287 m, airfoil chord 0.710 m and airfoil velocity 

90 m/sec. The droplet is also relatively small but the velocity of the airfoil is higher than for Case 1 

resulting in a larger slip velocity. This larger slip velocity results in an increase in the droplet center of 

mass over Case 1. The predictions for this case were also relatively good. 

Figure 10 shows the comparison between the model prediction and the experimental data for Case 3: 

Run 072611-16.10 droplet 2, droplet radius 439 m, airfoil chord 0.710 m and airfoil velocity 90 m/sec. 

The droplet is much larger than for Case 1 and 2. This results in a larger slip velocity and more 

 
NASA/TM—2012-217705 9 

deformation along the path. The model initially over predicts the value of the center of mass and then 

converges more closely to the experimental data starting at a distance of –100 mm from the leading edge 

of the airfoil. 

Figure 11 shows the comparison between the model prediction and the experimental data for Case 4: 

Run 072011-14B.1 droplet 1, droplet radius 516 m, airfoil chord 0.710 m and airfoil velocity 90 m/sec. 

The droplet is somewhat larger than for Case 3. Initially the model over-predicts the position of the center 

of mass by 13 percent. The difference between the model and experiment then decreases until the droplet 

reaches a distance of –50 mm from the leading edge. From there on the model increasingly over-predicts 

the position of the center of mass. 

Figure 12 shows the comparison between the model prediction and the experimental data for Case 5: 

Run 072611-15B.9 droplet 2, droplet radius 685 m, airfoil chord 0.710 m and airfoil velocity 90 m/sec. 

The droplet is the largest of the five cases to which the model was applied. The model over-predicts the 

position of the center of mass of the droplet over the entire range and the prediction worsens as the 

droplet approaches breakup. The over-prediction is 13 percent initially and is 75 percent at the breakup 

location. 

From the results it can be seen that the model does relatively well predicting the displacement of the 

center of mass for the small and medium size droplets. As the droplet radius increases the model 

prediction deviates more from the experimental data, and for the largest droplets the model greatly overpredicts the position of the center of mass. The fact that the model prediction deviates gradually from the 

experimental data as the droplet size increases seems to indicate that one of the explicit or implicit 

assumptions may not apply at larger droplet sizes. 

The assumption that the projected area of the droplet remains constant leads to under prediction of the 

area as the droplet deforms. This causes the model to under predict the pressure force and the 

displacement of the center of mass. 

The assumption that the average pressure over the half droplet can be approximated by the pressure at 

the stagnation point is a reasonable approximation as long as the shape of the droplet does not deviate 

significantly from the shape of a sphere. For a droplet distorted into an asymmetric oblate spheroid shape, 

no experimental data was found to help determine the soundness of the average pressure approximation. 

The approximation did not cause a problem for the predictions of the model for small and medium 

droplets. 

The assumption about the deformed droplet being an oblate spheroid breaks down as the droplet 

approaches initiation of breakup. This can be seen for the droplet in Figure 3 after frame number 100. 

This approximation did not affect the prediction of the displacement of the center of mass for the small 

and medium droplets. 

Large droplets show large oscillations after being released from the droplet generator until the slip 

velocity begins to deform the droplet into an oblate spheroid. The initial oscillation of the droplet did not 

change the initial condition for the velocity of the center of mass as expressed in Equation (30). This was 

confirmed by studying the graphs of the major semi-axis against time for the cases studied. 

Large droplets begin deforming closer to the airfoil and the time lapse from the initiation of 

deformation to the beginning of the breakup is shorter than the small and medium droplets but the 

difference is not significant enough to question the quasi-steady assumption. During deformation the large 

droplets follow the same pattern as the smaller ones. 

The force of gravity was included in the model to improve the prediction. The following 

non-dimensional gravitational term was added to the right hand side of Equation (28): 

g U

K R

rel

2

  (32) 

where K is defined in Equation (26), R is the radius of the droplet, Urel is the slip velocity and g is the 

acceleration of gravity. The effect of gravity was too small to make a difference in the model prediction. 
NASA/TM—2012-217705 10 

The results indicate that the DDB model can be used for small and medium droplets. The model will 

be used in the analysis of the data from previous droplet breakup experiments and in the scaling of the 

results to large airfoils. 

Conclusions 

The Droplet Deformation Breakup (DDB) model developed by Ibrahim, Yang and Przekwas (Ref. 6) 

is used to predict the deformation of droplets approaching the leading edge of an airfoil along the 

stagnation line. The experimental data used in the evaluation of the model came from experiments 

conducted in a rotating arm facility (Refs. 1 to 3). The following conclusions can be stated about the 

results: 

 For small and medium size droplets (radius between 200 and 500 m) the model prediction agrees 

with the experimental data. 

 For large droplets (radius larger than 500 m) the model over-predicts the displacement of the center 

of mass by a large margin. 

 The increasing deviation between the model prediction and the experimental data as droplet size 

increases seems to indicate that one or more of the model assumptions are invalid for large droplet 

sizes. 

 The quasi-steady assumption needed to apply the DDB model works well for small and medium size 

droplets. 

 The model can be used in the analysis of the deformation of small and medium droplets from 

previous experiments. 

References 

1. Vargas, M. and Feo, A., “Experimental Observations on the Deformation and Breakup of Water 

Droplets Near the Leading Edge of an Airfoil,” NASA/TM—2011-216946, AIAA Paper 2010-7670, 

August 2010. 

2. Feo, A., Vargas, M. and Sor, S., “Rotating Rig Development for Droplet Deformation/Breakup and 

Impact Induced by Aerodynamic Surfaces,” SAE paper number 2011-38-0087, 2011. 

3. Vargas, M., Sor, S. and García Magariño, A., “Mechanism of Water Droplet Breakup Near the 

Leading Edge of an Airfoil,” AIAA publication, presented at the 4th AIAA Atmospheric and Space 

Environments Conference, New Orleans, June 25-28, 2012, to be released July 2012. 

4. O’Rourke, P.J. and Amsden, A.A., “The TAB Method for Numerical Calculation of Spray Droplet 

Breakup,” SAE paper number 872089, 1987. 

5. Reitz, R.D., “Modeling Atomization Processes in High-Pressure Vaporizing Sprays,” Atomization 

and Spray Technology 3, 1987, pp. 309-337. 

6. Ibrahim, E.A., Yang, H.Q. and Przekwas, A.J., “Modeling of Spray Droplets Deformation and 

Breakup,” AIAA Journal of Propulsion and Power, vol. 9, no. 4, 1993, pp. 652-654. 

7. Taylor, G.I., “The Instability of Liquid Surfaces When Accelerated in a Direction Perpendicular to 

Their Planes,” in Batchelor, G.K. (ed) “The Scientific Papers of Sir Geoffrey Ingram Taylor,” vol. 3, 

Cambridge University Press, 1963, pp. 532-536. 
NASA/TM—2012-217705 11 

TABLE 1.—DATA FOR DROPLETS USED TO TEST THE DDB MODEL 

[Airfoil DBKUP 001—Chord size 0.710 m.] 

Run number Droplet 

number 

Airfoil 

velocity, 

m/sec 

Radius, 

m 

Time at which 

breakup begins, 

sec 

072211.1.8 4 50 199 ------ 

072611.18.6 1 90 287 2200 

072611.16.10 2 90 439 2300 

072011.14.B.1 1 90 516 2600 

072611.15.B.9 2 90 685 1600 

Figure 1.—Diagram of deforming half droplet. The center of 

mass and the axis of the oblate spheroid are indicated.

Figure 2.—Ratio of the surface area approximated by As-app = 2(a2 + b2) 

over the exact value As-exact = 2(a2

 + b2 ) plotted against the ratio of 

the large semi-axis a over the small semi-axis b, for the droplets: 

072211 Run 1.8 Dp#4 R = 199 m, 072611 Run 18.6 Dp#1 R = 

287 m, 072611 Run 16.10 Dp#2 R = 439 m, 072011 Run 14B.1 

Dp#1 R = 516 m, 072611 Run 15B.9 Dp#2 R = 685 m. 
NASA/TM—2012-217705 12 

Frame # 1 10 20 30 40 50 60 70 80 90 

Time 

(μsec) 

0 120 253 387 520 653 787 920 1053 1187 

x-Distance 

(mm) 

–238.6 –227.9 –215.9 –204.0 –192.1 –180.1 –168.2 –156.3 –144.3 –132.4 

Urel

(m/sec) 

14.5 15.3 16.2 17.3 18.4 19.6 20.9 22.3 23.8 25.5 

We 3.6 4.0 4.5 5.1 5.8 6.5 7.4 8.5 9.7 11.1 

Re 1019 1075 1142 1213 1291 1375 1466 1565 1673 1792 

Bo 1.62 1.68 1.74 1.81 1.88 1.97 2.08 2.22 2.41 2.69 

Cd 0.60 0.56 0.51 0.47 0.44 0.40 0.37 0.35 0.33 0.32 

Frame # 100 110 120 130 140 150 160 170 180 190 

Time 

(μsec) 

1320 1453 1587 1720 1853 1987 2120 2253 2387 2520 

x-Distance 

(mm) 

–120.5 –108.6 –96.7 –84.8 –72.9 –61.0 –49.1 –37.3 –25.5 –13.8 

Urel

(m/sec) 

27.4 29.5 31.9 34.7 38.0 41.9 46.7 52.7 60.3 70.0 

We 12.8 14.9 17.4 20.6 24.6 30.0 37.2 47.4 62.0 83.7 

Re 1925 2073 2242 2438 2668 2944 3281 3702 4235 4921 

Bo 3.13 3.82 4.97 6.89 10.13 15.66 25.11 41.32 69.16 117.05 

Cd 0.33 0.34 0.38 0.45 0.55 0.70 0.90 1.16 1.49 1.86 

Frame # 191 192 193 194 195 196 197 198 199 200 

Time 

(μsec) 

2533 2547 2560 2573 2587 2600 2612 2627 2640 2653 

x-Distance 

(mm) 

–12.6 –11.5 –10.3 –9.2 –8.0 –6.8 –5.7 –4.5 –3.4 –2.3 

Urel

(m/sec) 

71.1 72.3 73.5 74.7 75.9 77.2 78.5 79.8 81.2 82.6 

We 86.4 89.2 92.2 95.2 98.4 101.7 105.2 108.8 112.6 116.4 

Re 4998 5080 5163 5246 5335 5424 5515 5610 5705 5803 

Bo 123.42 130.15 137.26 144.76 152.68 161.04 169.87 179.19 189.03 199.42 

Cd 1.90 1.94 1.99 2.03 2.07 2.11 2.15 2.20 2.24 2.28 

Figure 3.—Sequence of deformation and initiation of breakup for Run 072011.14B.1 droplet #1, droplet radius = 

516 m, airfoil chord = 0.710 m, airfoil velocity = 90 m/sec. The table below the sequence gives the 

corresponding values for several of the parameters at each stage in the droplet deformation. 
NASA/TM—2012-217705 13 

Figure 4.—Droplet width (semi-axis b) in pixels against distance from the 

leading edge of the airfoil in millimeters. Run 072011.14B.1 droplet #1, 

droplet radius = 516 m, airfoil chord = 0.710 m, airfoil velocity = 

90 m/sec. 

Figure 5.—Droplet Height (semi-axis a) in pixels against distance from the 

leading edge of the airfoil in millimeters. Run 072011.14B.1 droplet #1, 

droplet radius = 516 m, airfoil chord = 0.710 m, airfoil velocity = 

90 m/sec. 
NASA/TM—2012-217705 14 

Figure 6.—Center of mass oscillation when the DDB model is solved at 

location 100, distance from the leading edge of the airfoil = –120.5 mm, 

Run 072011 14B.1 droplet #1, droplet radius = 516 m, airfoil chord = 

0.710 m, airfoil velocity = 90 m/sec. 

Figure 7.—Center of mass oscillation when the DDB model is solved at 

location 170, distance from the leading edge of the airfoil = –37.3 mm, 

Run 072011 14B.1 droplet #1, droplet radius = 516 m, airfoil chord = 

0.710 m, airfoil velocity = 90 m/sec. 
NASA/TM—2012-217705 15 

Figure 8.—DDB Model prediction compared to the experimental data for 

Run 072211 1.8 droplet #4, droplet radius = 199 m, airfoil chord = 

0.710 m, airfoil velocity = 50 m/sec. 

Figure 9.—DDB Model prediction compared to the experimental data for 

Run 072611 18.6 droplet #1, droplet radius = 287 m, airfoil chord = 

0.710 m, airfoil velocity = 90 m/sec. 
NASA/TM—2012-217705 16 

Figure 10.—DDB model prediction compared to the experimental data for 

Run 072611 16.10 droplet #2, droplet radius = 439 m, airfoil chord = 

0.710 m, airfoil velocity = 90 m/sec. 

Figure 11.—DDB Model prediction compared to the experimental data for 

Run 072011 14B.1 droplet #1, droplet radius = 516 m, airfoil chord = 

0.710 m, airfoil velocity = 90 m/sec. 
NASA/TM—2012-217705 17 

Figure 12.—DDB Model prediction compared to the experimental 

data for Run 072611 15B.9 droplet #2, droplet radius = 685 m, 

airfoil chord = 0.710 m, airfoil velocity = 90 m/sec. 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 0704-0188 

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the 

data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this 

burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. 

Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB 

control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

01-09-2012 

2. REPORT TYPE

Technical Memorandum 

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Droplet Deformation Prediction With the Droplet Deformation and Breakup Model (DDB) 

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Vargas, Mario 

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

WBS 648987.02.02.03.10 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

John H. Glenn Research Center at Lewis Field 

Cleveland, Ohio 44135-3191 

8. PERFORMING ORGANIZATION

 REPORT NUMBER

E-18392 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

10. SPONSORING/MONITOR'S

 ACRONYM(S)

NASA 

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2012-217705 

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited 

Subject Category: 03 

Available electronically at http://www.sti.nasa.gov 

This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

13. SUPPLEMENTARY NOTES

14. ABSTRACT

The Droplet Deformation and Breakup Model was used to predict deformation of droplets approaching the leading edge stagnation line of 

an airfoil. The quasi-steady model was solved for each position along the droplet path. A program was developed to solve the non-linear, 

second order, ordinary differential equation that governs the model. A fourth order Runge-Kutta method was used to solve the equation. 

Experimental slip velocities from droplet breakup studies were used as input to the model which required slip velocity along the particle 

path. The center of mass displacement predictions were compared to the experimental measurements from the droplet breakup studies for 

droplets with radii in the range of 200 to 700 mm approaching the airfoil at 50 and 90 m/sec. The model predictions were good for the 

displacement of the center of mass for small and medium sized droplets. For larger droplets the model predictions did not agree with the 

experimental results. 

15. SUBJECT TERMS

Icing; Engine icing 

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF

 ABSTRACT

UU 

18. NUMBER

 OF 

 PAGES

24 

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email:help@sti.nasa.gov) 

a. REPORT

U 

b. ABSTRACT

U 

c. THIS 

PAGE

U 

19b. TELEPHONE NUMBER (include area code)

443-757-5802 

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. Z39-18