15TH EUROPEAN TURBULENCE CONFERENCE, 25-28 AUGUST, DELFT,. THE NETHERLANDS
UNSTEADY PARTICLE ACCUMULATION IN WALL TURBULENCE
Dmitrii Ph. Sikovsky1,2
Institute of Thermophysics of Siberian Branch of Russian Academy of Science, Novosibirsk, Russian
Federation
2
Novosibirsk State University, Novosibirsk, Russian Federation
1
Abstract We propose the asymptotic theory of unsteady accumulation of inertial particles in the viscous sublayer of wall-bounded
particle-laden turbulent flow. We derive the diffusion equation for the particle concentration in the viscous sublayer and find the selfsimilar exact solution of this equation at large times. It is shown that near the wall the maximal concentration grows as t 1/ 2 , while
the distance from the wall to the concentration pike as well as its width decay as t results of stochastic Lagrangian simulations.
1/ 2
. The obtained solution is corroborated by the
In [1] it was shown that the steady-state solution for the kinetic equation for the probability density function of particle
position and velocity (PDF) exhibits the near-wall singularity. It was assumed that the infinite time is needed for the
point particles to reach such steady solution with the singularity of particle concentration. The aim of the present paper
is to study the unsteady accumulation of particles in the viscous sublayer of wall-bounded turbulent flow.
In the thin viscous sublayer the intensity of wall-normal fluid velocity decays to zero as v f¢ = qy 2 + O(y 3 ) , where y is
the distance to the wall and wall units are implicit throughout for all variables. Above the viscous sublayer, in the buffer
zone the particles interact with the near-wall coherent structures such as the low-speed streaks and energetic streamwise
vortices which radius is of order of viscous sublayer thickness. Based on the observations from DNS/Largangian
tracking numerical data [2] and the results of asymptotic analysis [1], we can divide the particles on two groups.
Particles of the first group are captured by the near-wall energetic eddies and slung toward the wall with the wallnormal velocities of order of one, which are much larger than local v f¢ . Thus, they are not influenced by the fluid
velocity fluctuations and can be referred to as ballistic, or free-flight particles. Particles of the second group are trapped
by the low-speed streamwise streaks and have velocities of the order of v f¢ and can be referred to as diffusional
particles. In [1] it was shown that the near-wall build-up of particles is created by the slow diffusional particles.
Consider the motion of diffusional particles in the limit y ® 0
dv v
(1)
+ = qy 2 + O(y 3 )
dt
t
where t is the particle relaxation time, q = q(x (t ), z (t ), t ) x (t ), z (t ) are the streamwise and spanwise positions of
particle.
During the time interval of order of decorrelation time of fluid turbulence near the wall – Lagrangian timescale – T L ,
the diffusional particle shifts in wall-normal direction on a distance D y ~ qy 2T L , which is much less than its initial
position y (t ) , the shifts in stream- and spanwise directions are small as well D x , z ~ y . Thus we can assume the
particle wall-normal position y (t ) in (1) to be a slow variable as compared to a fast variable q(x (t ), z (t ), t ) , in which in
turn we can assume the particle position to be almost fixed: q(x (t ), z (t ), t ) » q(t ) . Then from (1) it follows the
Kramers-Moyal expansion for particle PDF P (v, y, t ) = ád(v - v(t ))d(y - y (t ))ñ [1]
æ ¶ 2P
ö
¶ (vP )
¶ (vP )
¶P
¶ 2P ÷
÷
+
- t-1
= t - 1y 4 çççl 0
+ l1
+ O (y 5 )
÷
çè ¶ v 2
¶t
¶y
¶v
¶ v¶ y ÷
ø
where l 0 = t -
1
¥
¥
ò áq(0)q(t )ñexp(- t / t )dt , l 1 =
ò áq(0)q(t )ñ[1 -
0
(2)
exp(- t / t )]dt .
0
Multiplying (2) on v n and integrating over all v gives the transport equations for the particle concentration
F =
ò Pdv , flux
j =
ò vPdv , and energy áv 2 ñF
=
ò v 2Pdv
æ¶
ö
l 1y 4 ¶ F
¶F
¶j
¶ (áv 2 ñF )
1÷
+
= 0 , çç + ÷
j
+
=
çè ¶ t
÷
¶t
¶y
ø
t ÷
¶y
t ¶y
(3)
Figure 1. Time growth of maximal concentration of point particles in turbulent channel flow with t = 25 .
æ¶
ö 2
l 0y 4
l 1y 4 ¶ j
¶ (áv 3 ñF )
÷
çç + 2 ÷á
v
ñ
F
+
=
2
2
(4)
÷
çè ¶ t
ø
t ÷
¶y
t
t ¶y
At large times t > > t , the quasi-equilibrium approximation [1,3] of (3),(4) gives the following diffusion equation
¶F
¶ é 4æ
¶F
a F öù
÷ú,
êDy çç
=
+
÷
(5)
÷ú
ç
ê
è¶y
¶t
¶y ë
y ø÷
û
¥
where D =
ò áq(0)q(t )ñdt
= áq 2 ñT L , a =
0
4t l 0
tl 0 + l1
»
4t
(cf. [1]).
t + TL
We found the exact self-similar solution of (5) for the case of inertial particles with t > T L / 3
F = 2 Dt F (2y Dt ) , F ( h) = C h- a exp(- h- 2 )
(6)
¥
æ - 1÷
ö
÷
F dy .
where C = 2G ç
÷
çè 2 ÷
øò
0
- 1 ça
The solution (6) describes the unsteady accumulation of inertial particles near the wall at large times, when all particles
becomes diffusional. The latter is the consequence of the irreversibility of the transition from free-flight particles to
diffusional one expressed by (5), since it is almost impossible for the former free-flight particle retarded near the wall to
increase their velocity up to order of one values again. Particle concentration profile has the near-wall pike with
magnitude growing as t 1/ 2 and the thickness collapsing as t - 1/ 2 . According to (6) after the infinite time all particles
will deposit the wall, so that F ~ d(y ) at t = ¥ . The real particles have the finite radius r , such that the solution (6)
is valid at times t < < D - 1r 2 . Beyond these times t > > D - 1r 2 the regularized steady-state singularity solution [1] is
reached. At large distances y > > (Dt )- 1/ 2 the particle concentration obeys the power law F ~ y - a , which is similar
to the steady solution obtained in [1].
We perform the stochastic Lagrangian simulations using the model [3] of turbulent channel flow at Ret = 180 seeded
with the point particles. The results of simulation corroborate the expression (6) (Fig.1). As seen from Fig.1, at the
initial stage of accumulation the maximal concentration grows linearly with time, which can be explained with the help
of another self-similar solution of (5) having the form F ~ t F1(2y Dt ) .
References
[1] D.Ph.Sikovsky. Singularity of inertial particle concentration in the viscous sublayer of wall-bounded turbulent flows. Flow, Turb. and Comb.
92: 41-64 ,2014.
[2] A. Soldati, C. Marchioli. Physics and modelling of turbulent particle deposition and entrainment: Review of a systematic study. Int. J.
Multiphase Flow 35: 827-839, 2009.
[3] D.Ph.Sikovsky. Stochastic Lagrangian simulation of particle deposition in a turbulent channel flows. 10th International ERCOFTAC
Symposium on Engineering Turbulence Modeling and Measurements, Marbella, Spain, 17-19 September 2014. DOI: 10.13140/2.1.4204.6722.