Fluid Mechanics

of

Deposition of Micron Liquid Droplets on

Wall in Impinging Turbulent Air Jet

Tianshu Liu, J. Nink, P. Merati

Department of Mechanical and Aerospace Engineering

Western Michigan University, Kalamazoo, MI 49008

Objectives

To understand the physical mechanisms of micron

oil droplet deposition in an impinging turbulent jet

based on measurements and theoretical analysis

● Flow structures and formation of droplet

deposition ring patterns

● Droplet collisions induced by small-scale

turbulence

Techniques

Experimental Techniques:

• UV Luminescence

• Microscopy

• Global Velocity Diagnostics

• Global Skin Friction Diagnostics

Theoretical Analysis:

Probability density function (PDF) of inter-particle

separation in turbulence

Splashing, Deposition and Rebound Domains

3m/kg850 , m/N102 2 , 22 m/N10 , 5.2rmax , and

deg70 .

Rebound

Deposition

Splash

~ 1-2 microns

Impinging Jet Testing Setup

• Jet

– Diameter: 9.5 mm

– Height: 10 mm

– Impingement Angle: 90 deg

– Reynolds Number: 21,076

– Mean Exit Velocity: 30 m/s

• Olive Oil Particles

– Mean Diameter: 1 micron

– UV Doping Agent: Dye

• Plate

– Material: Glass

(a)

IMPINGEMENT SURFACE

TSI MODEL 3407 OIL ATOMIZER

AEROSOL-AIR MIXING CHAMBER

REDUCTION CHAMBER

NOZZL

E

Z

Y

X

Free Jet Characteristics

Mean Velocity Distributions Turbulent Intensity Distributions

Deposited Droplet Distributions Visualized

Using Luminescence

Deposition Rings & Self-Similar Growth

Self-Similar Growth of Deposition Ring

Relative Oil Thickness

at the Stagnation Point

Normalized Oil Thickness

Distributions

Microscopic Images at 400x of Droplet Deposition

• Microscopic inspection

shows that the ring structure

is created by discrete

individual droplets, not the

creation of an oil film.

• Image analysis will

determine oil droplet size

distribution and density

Select Sequential Line Segment -

Microscopic Measurements and Deposited Oil

Droplet Statistics along a Horizontal Line

x/D =0.0 x/D =0.7 x/D =1.7 x/D = 1.3

Jet Center First Ring Dark Region Second Ring

Ra

w Im

ag

e P

rocessed

Ima

ge

Pa

rticle Size

Distrib

utio

n

Deposited Droplet Mean Diameter and

Deposited Oil Area

PIV/Optical Flow Diagnostics

Typical PIV image

Snap-shot velocity

field & vorticity

Mean velocity

field & vorticity

Turbulent kinetic

energy

Characteristics of Velocity Fields

max (U)

Half-

Width

<U>/max(U)

<V>/max(U)

Reynolds Stress

Flow Structures & Droplet Deposition

Intermittent large

downward velocity Sharply turning flow

x/D =0.7 x/D = 1.3 x/D =1.7

First Ring Second Ring

Probability of Normal Velocity Toward Wall

Second Ring

Skin friction

feature of

second ring ??

Second Ring

(a)

(b)

(c)

Turbulence & Droplet Deposition:

Jet Nozzle Height Effect

H/D = 0.35

H/D = 1.58

H/D = 2.63

Turbulence & Droplet Deposition:

Thin-Wire-Tripping Effect

D=0.65 mm

(a)

(b)

(c)

(d)

Histograms of Deposited Droplet Diameters

at the Stagnation Point

D=0.65 mm D=0.35 mm

D=0.13 mm No wire

Effect of Turbulent Kinetic Energy on Droplet

Deposition

Relative Deposited Oil Area Integral of Deposited Oil Distribution

● Inner Ring

– The inner ring is formed from sharp turning of the jet

at the wall. The larger droplets no longer follow the

airflow path due to their inertia in the turning, and

they deposit upon the wall forming the inner ring.

● Outer Ring

– The outer ring is created by high turbulence intensity

and intermittent downward velocity induced by large

vortical structures to entrain droplets to the wall.

Summary of Experimental Results

Small-scale turbulence induces droplet collision

to form larger droplets.

● Effect of Small-Scale Turbulence

Why ???

Why do small droplets tend to collide each other

to form larger ones in small-scale turbulence?

A Theoretical Problem Related to the

Experimental Observations

Probability Density Function (PDF) of

Inter-Particle Separation in Turbulence

A random differential equation for inter-particle separation:

where S(t) is a Lagrangian particle strain rate

& n(t) is a background noise.

Two-Particle Kinematics

)t(nr)t(Sdt/dr

r

Eddy

)1(n)1(mmn2 XVrrr)t(S

)3,2,1n,m(

(1)(2) XX r

Two-Particle Separation:

Lagrangian Strain Rate:

r

)t,r(p)r(N

r2

1)t,r(p)r(M

r

)r(N

4

1

rt

t)p(r,

Application of Stratonovich’s perturbation method gives

a Fokker-Planck equation for PDF

)r,t()r(M corr

'd)r,'t(),r,t(K2)r(N

02corr

)t(nr)t(S)r,t(

d)t(S),t(SK

02

corr

where

)t(S),t(SK2 212121 ),(K

The Steady-State Solution for PDF

dr

arara

nrSexp

arara

C)r(p

r

r 322

12/1

322

1 0

The Pearson System

d)t(S),t(SKa

0

1

d)t(S),t(nK)t(n),t(SKa

0

2

d)t(n),t(nKa

0

3

where

13.0)t(S f 4.1corr 18.0)t(Scorr Estimates:

What is a Physically Feasible Solution for PDF ??

322

1 arara

nrS

dr

)r(glnd

The Pearson System:

Gaussian Distribution: 0aa 21 0n

Excluded Solutions:

Gamma Distribution: 0a1 3122 aa4a 0a/S 3

0a1 since it represents the integral timescale of S(t)

0S 0a3 since material lines are always stretched

in turbulence in a mean sense

3122 aa4a violates the Schwartz inequality in mathematics

Reasons:

The Pearson-Type IV Distribution

as a Physically Feasible Solution:

230

3

230

32

2/)c1(

032 cc4

cr̂2tana

cc4

cc2exp

cr̂cr̂

C)r̂(p

1

)a/(ac 2130 11 a/Sc )a/(nc 12 )a/(ac 123

where

2231 aaa4 222

nSn,S

d)t(n),t(SKn,S

0

Condition for the Pearson-Type IV: The Schwartz Inequality

Scalar Product:

The Power-Law PDF for

where

2/)c1(

02 1

cr̂C)r̂(p

/rr̂

1r̂

)S,S(KSc corr1

1c1 1

0r̂ )r̂(pmax[ ]

Behavior:

Consequence:

Two particles tend to approach each other,

leading to collisions in small-scale turbulence.

Power-law region

Tail region

Power-law region

Tail region

Comparison with DNS

)S,S(KSc corr1 Power-Law Exponent:

Two-Particle Dynamics

)t(S)t(SS/dtd wp

The power-law exponent as a function of Stokes number (St):

1/St p

(1) The exponent is weakly dependent of Reynolds number

(2) The exponent is 2

122132111 St)q̂k̂q̂k̂k̂(St)k̂q̂k̂(c1

for

Main Theoretical Findings:

A generalized Langevin equation for the Lagrangian strain rate:

2

321

2321

1Stk̂Stk̂k̂

StStq̂Stq̂q̂1c

Cannot be neglected

Statistical Mechanical Modeling of N-Particle System

The Radial Distribution Function:

)Vn()r̂(p)1N(N)r̂(g 2

ppp

The Collision Rate per Unit Volume:

r0

2

p

2

0c w)d(gnd5.0N

~ 800 collisions in 1 mm cube

Conclusions

The formation of the outer ring is caused by high

turbulence activities through the following

mechanisms:

(a) Large-scale vortical structures entrain droplets

toward the wall.

(b) Small-scale turbulence induces droplet collisions to

form larger droplets.