Probability Density Function of Inter-

Particle Separation in Homogeneous

Isotropic Turbulence

Tianshu Liu

Department of Mechanical and Aerospace Engineering

Western Michigan University, Kalamazoo, MI 49008

Objectives

To derive the probability density function of the

separation between two inertial particles in

homogeneous isotropic turbulence.

To understand the physical mechanism of particle

collisions in turbulence.

Why do small inertial particles tend to collide

each other to form larger ones in small-scale

turbulence?

A problem related to experimental observations

Power-Law Distribution & Two-Particle Kinematics

(1)(2) XX r

Two-Particle Separation:

r

Eddy

A random differential equation for inter-particle separation:

where

S(t) is a Lagrangian particle strain rate

)t(nr)t(Sdt/dr

Equation in Two-Particle Kinematics

)1(n)1(mmn2 XVrrr)t(S )3,2,1n,m(

n(t) is a background noise.

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:

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

The linear term could not 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

Turbulent impinging jet

as an example

Conclusions

The power-law PDF of the inter-particle separation is

derived by considering the kinematics and dynamics

of two particles in homogeneous isotropic

turbulence.

(a) The power-law exponent is independent of Reynolds

number .

(b) The power-law exponent is a function of Stokes

number, where the linear term could not be

neglected even though the main term is quadratic.