12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

KOLMOGOROV CONSTANT FOR PARTICLE-LADEN TURBULENTCHANNEL FLOW

Naveen Rohilla
Department of Chemical Engineering
Indian Institute of Technology Bombay
Powai, Mumbai, Maharashtra, India-400076
naveenr621@gmail.com

Partha S. Goswami
Department of Chemical Engineering
Indian Institute of Technology Bombay
Powai, Mumbai, Maharashtra, India-400076
psg@iitb.ac.in

ABSTRACTsecond and third-order velocity structure-function and mentioned that small-scale isotropy should be satisfied for the existence of a universal inertial range. Heinz (2002) discussed
the variations of Kolmogorov constant for equilibrium turbulent boundary layer and homogeneous isotropic stationary turbulence, and mentioned that the value is near two if anisotropy
and acceleration fluctuations dominate in the energy budget.
And, the value is near six if these contributions disappear. Yeung & Zhou (1997) mentioned that for the existence of inertial range, isotropy should also be present along with −5/3
scaling. If isotropy is not satisfied, the Kolmogorov constant
may achieve a different value. The above studies point out
the deviation of Kolmogorov constant from universality due toanisotropy at low and moderate Reynolds numbers for unladencases. In turbulent flows, the local isotropy can be affected by
the presence of particles as well. It is evident that particles
can affect the turbulence intensity of the gas phase. In the
study by Gualtieri et al. (2013) for particle-laden shear flow,
they concluded that care should be taken while applying Kolmogorov theory as anisotropy is increased for particle-ladencases. Thus, in the present study, we want to explore whether
particles can induce anisotropy in the gas phase. And, what isthe effect of increased anisotropy on the Kolmogorov theory?
To answer the above question, simulations are performed
for turbulent channel flows with direct numerical simulations(DNS) at Reynolds number of 5600 based on average gas velocity and channel width. It is observed that the turbulence intensity of the gas phase decreases with an increase in particle
loading, along with the increase in anisotropy across the channel width. The increase in anisotropy affects the Kolmogorov
constant, which is discussed in the results section.

The Kolmogorov constant is a prefactor used to correlate
the spectral distribution of turbulent kinetic energy with the
turbulent dissipation rate. The Kolmogorov constant has beenused in stochastic turbulence models and turbulence models
based on large-scale turbulent structures, like the Smagorinsky
model and other eddy viscosity-based models used in largeeddy simulations (LES). Many research works have been performed to estimate the Kolmogorov constant for a wide rangeof turbulent flows like homogeneous isotropic turbulence, turbulent boundary layers, etc. In earlier work, Sreenivasan(1995) has consolidated and analyzed a large number of experimental data and concluded that at high Reynolds numbers,
the prefactor is a universal constant.
In the present work, we have estimated the Kolmogorov
constant for particle-laden turbulent channel flow using
second-order velocity structure-function based analysis. Our
study reveals that, with an increase in particle volume fraction
Kolmogorov constant decreases. The present analysis on thevariation of Kolmogorov constant with particle volume loading will be helpful to model the two-phase turbulent flows.

INTRODUCTION
The Kolmogorov constant is the proportionality prefactor
in Kolmogorov theory which states that the spectral energy inthe inertial subrange is E(k) = Cε 2/3 k−5/3 where ε is the meanviscous dissipation rate of turbulence kinetic energy and k thewavenumber. The Kolmogorov constant is obtained based
upon the Kolmogorov hypothesis for different flows such as
boundary layers, channel flows, etc. In the seminal work,
Sreenivasan (1995) has summarized that the Kolmogorov constant is universal and, independent of the flow configuration
and Reynolds number. At sufficiently high Reynolds number where isotropy is satisfied at dissipation scale and inertial
range, and will lead to a constant Kolmogorov prefactor. However, at moderate and low Reynolds numbers, the Kolmogorov
constant may differ from a universal value. Antonia et al.
(1997) performed experiments for channel flow and observeda lower value of Kolmogorov constant. They analyzed the

SIMULATION PARAMETERS
The fluid phase has been considered to be incompressible
and described by Navier-Stokes equation. The discrete smooth
point particles are simulated using Newton’s second law of
the motion. The simulations have been performed in a vertical channel domain with 8πδ ∗ 2δ ∗ (4/3)πδ in streamwise(x), wall-normal (y) and spanwise (z) directions, respectively.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(1941)) defined as ,Where δ is half channel width. No-slip boundary conditions
are applied on the walls in the y-direction. The bulk Reynolds
numbers (Reb = ρ f ∗ ū ∗ 2δ /µ f ) is fixed at 5600 based on thechannel width (2δ ) and average fluid velocity (u) which corresponds to Reτ = 180 based on the unladen frictional velocity and half-channel width. The detail simulation procedure
for calculation of feedback force, near wall corrections in liftand drag, corrections for undisturbed velocity at particle locations are discussed in our earlier work (Muramulla et al. (2020)
Goswami & Kumaran (2011)). The particles with Stokes numbers (St = τ p /τ f ) of 105.47 and 210.93 are simulated whereτ p = ρ p d 2p /18µ f , τ f = 2δ /u, ρ p is the particle density, d p is
the particle diameter and µ f is the fluid dynamic viscosity.

h(δ u)2 i = C2 (εr)2/3 ,(2)where r ( in the inertial range, η << r << L) being the
distance between the two points, η is the Kolmogorov lengthscale, L is the integral length scale, ε is the mean viscous dissipation rate and, δ u = u(x + r) − u(x), with u being the longitudnal fluctuations. The Kolmogorov constant is defined by
C2 and angular brackets denote the time averaging.

RESULTS
The anisotropy of small scales across the channel width
can be accounted from the ratio of Kolmogorov time scale tomean shear time scale (Antonia & Kim (1992); Saddoughi &Veeravalli (1994))<(δ u*)2> (r*)−2/3

Sc∗ = S(ν/ε)1/2

0(1)

where S is the mean shear, dU/dy. Antonia & Kim (1992)
did the DNS for channel flow and found a value of Sc∗ = 2.5
at the wall and reduction to low value for y+ > 60 (y+ is thewall-normal distance normalized with viscous units). The Sc∗
is plotted along the wall-normal direction for St = 210.47 fordifferent volume fractions (φ ), Fig. 1. It is found that the Sc∗ is
2.42 at the wall for the unladen case. A decrease in the Sc∗ is observed away from the wall. The Sc∗ increases across the channel as the particle volume loading is increased, suggesting an
increase in anisotropy of small scales for particle-laden cases.
The Sc∗ at the wall for φ = 0.0025 becomes almost 50% higher
than the unladen case. The turbulence collapse is observedat φ = 0.0028 (Rohilla et al. (2022); Kumaran et al. (2020)).
This increased anisotropy can affect the Kolmogorov constantfor particle-laden cases which is analyzed via a second-order
velocity structure-function.

+

60

0

2

10

10

r*
Figure 2: The second-order velocity structure-functions

multiplied with r∗−2/3 for unladen cases. The dashedline is plotted at 1.2.

The second-order velocity structure-function for unladencase is plotted in Fig. 2 using the Eqn. 2. The (∗ ) denotes thenormalization with Kolmogorov scales. The profiles are plotted for two locations, one in the near-wall region (y+ ∼ 15)
and the other at the channel center (y+ ∼ 180), and validated
against the experimental data of Antonia et al. (1997). Thereis a good agreement between the experimental data of Antonia
et al. (1997) and the present DNS results for both the channel
locations. The majority of the reported values of C2 in the literature are around 2 or more (Heinz (2002); Choi et al. (2004);
Sawford & Yeung (2011); Saddoughi & Veeravalli (1994);
Sreenivasan (1995)). However, a value of C2 can be lower than
two due to lower Reynolds number or if there is a deviation of
isotropy in dissipation and inertial-range (Sreenivasan (1995);
Antonia et al. (1997); Yeung & Zhou (1997)).
In Fig. 3, the second-order velocity structure-function isshown for different particle volume loadings for Reynolds
number of 5600 in the channel center location. There isa decrease observed in the value of second-order velocity
structure-function at all the r∗ locations with an increase involume fraction. The peak value decreases nearly to 0.4 for
φ = 0.0025 from 1.2 for unladen flow. The decrease observedin the value of second-order velocity structure-function, Fig. 3,
is due to an increase in anisotropy across the channel, as shownin Fig. 1.
In Fig. 4, the peak value of the second-order velocity
structure-function is plotted for two channel locations, one in
the near-wall and the other at the channel center. It is seenthat the Kolmogorov constant (C2 ) decreases significantly with

S*c

40

Unladen, Antonia et. al. 1997, y = 20

−3

10

1

20

y+ = 15, Reb = 5600
+

2

0
0

+

y = 180, Reb = 5600

−2

10

Unladen, Antonia et. al. 1997, y+ = 40

φ=0
φ = 0.0010
φ = 0.0015
φ = 0.0020
φ = 0.0025
φ = 0.0028

3

−1

10

Unladen, Antonia et. al. 1997, y = 180

5

4

10

80

y+
Figure 1: The ratio of Kolmogorov time scale to mean

shear time scale for different volume fractions at St =210.93.

The Kolmogorov constant (C2 ) can be calculated using the second-order velocity structure-function (Kolmogorov

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

5
0

4

−1

10

u′x /uk<(δ u*)2> (r*)−2/3

10

φ=0
φ = 0.0010
φ = 0.0015
φ = 0.0020
φ = 0.0025
φ = 0.0028

3

2

1

φ=0
φ = 0.0010
φ = 0.0015
φ = 0.0020
φ = 0.0025
φ = 0.0027

0
0

0.2

Unladen, Antonia et. al. 1997, y+ = 180

−2

10

1

2

10

10

0.4

0.6

0.8

1

y/δ

r*

Figure 5: The ratio of streamwise root mean square fluc-

Figure 3: The second-order velocity structure-functions

tuations (u0x ) and Kolmogorov velocity scale (η) plottedin the wall-normal direction for a range of volume fractions for St = 210.93.

multiplied with r∗−2/3 for different volume fractions atchannel center location (y+ = 180) with St = 210.93.

1.4

decrease of the Kolmogorov constant in the channel center region.

1.2
Turbulence
Collapse

1

SUMMARY
0.8

C2

We report that the Kolmogorov constant decreases withan increase in volume loading at moderate Reynolds number
due to an increase in anisotropy. In the case of unladen wallbounded flows, the Kolmogorov constant increases away fromthe wall. However, it is observed in the present study that theKolmogorov constant in channel center location is less than
the near-wall region for higher volume loadings. The demonstrated variation in the Kolmogorov constant will be helpful todevelop better turbulence models in stochastic modeling and
LES for particle-laden flows.

0.6
y+ = 180, St = 210.93+
y = 15, St = 210.93+
y = 180, St = 105.47
y+ = 15, St = 105.47+
Unladen, Antonia et. al. 1997, y = 180
Unladen, Antonia et. al. 1997, y+ = 20

0.4
0.2
0

0

0.5

1

1.5

φav

2

2.5

3

3.5
−3

x 10

Figure 4: The peak of second-order velocity structure-function plotted for different volume fractions. The
dashed lines are the fitting curves.

REFERENCES
Antonia, RA & Kim, J 1992 Isotropy of small-scale turbulence. In Proc. Summer Program of the Center for Turbulence Research, Stanford.
Antonia, RA, Zhou, Tongming & Romano, GP 1997 Secondand third-order longitudinal velocity structure functions ina fully developed turbulent channel flow. Physics of Fluids
9 (11), 3465–3471.
Choi, Jung-Il, Yeo, Kyongmin & Lee, Changhoon 2004 Lagrangian statistics in turbulent channel flow. Physics of fluids 16 (3), 779–793.
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in the channel flow of a turbulent particle gas suspension
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Gualtieri, Paolo, Picano, Francesco, Sardina, Gaetano & Casciola, Carlo Massimo 2013 Clustering and turbulence modulation in particle-laden shear flows. Journal of Fluid Mechanics 715, 134–162.
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Kumaran, Viswanathan, Muramulla, NSP, Tyagi, Ankit &Goswami, PS 2020 Turbulence collapses at a threshold par-increase in volume loading. The C2 value is not relevant atvolume loading of φ = 0.0028 and 0.003 for St = 210.93 and105.47 respectively as turbulence has collapsed at these volume loadings (Rohilla et al. (2022); Kumaran et al. (2020)).
Also, it is observed that the C2 increases from near-wall region
to the channel center for unldaden case (Antonia et al. (1997);
Choi et al. (2004)). However, the decrease in the peak value of
C2 is more for the channel center location than the near-wallregion with an increase in volume loading. It is interesting tonote that a crossover occurs as the volume loading is increased
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than the near-wall region for both the Stokes numbers. It isalso observed that this crossover occurs at low volume loading
for low Stokes number, Fig. 4.
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for St = 210.93. It is observed that the ratio decreases faster inthe channel center than in the near-wall region with increased
particle volume loading. This shows that the separation of velocity scales (large to small-scales) decreases more in the channel center than in the near-wall region. This explains the larger

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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