The
features of the motion of a dispersed impurity in the form of particles in
turbulent gas flows and its inverse effect on the turbulence characteristics of
the carrier phase are key problems of the theory of two-phase flows [1-4]. The inverse
problem is to study the effect of particles on the characteristics of the gas
stream carrying them. The solution to this problem involves determining the
characteristics of a gas in the presence of particles: velocity and temperature
fields, friction and heat transfer coefficients, etc. [5-7]. The flow with
large particles is characterized by the fact that the relaxation time of
particles significantly exceeds the characteristic time of large-scale
turbulent vortices, i.e.
.
Such particles will not react to the turbulent pulsations of the velocity of
the carrier phase, and the distributions of their averaged velocities will be
almost uniform along the section of the channel (pipe). The data [8] can serve
as a clear confirmation of this.
The
purpose of this work is to visualize the flow formed in the wake of large
particles moving in an ascending turbulent air flow in a channel based on
numerical modeling.
A
characteristic feature of turbulent flows is the presence of random
fluctuations in all flow parameters. Due to the variability of parameters not
only in space, but also in time, various methods of averaging and smoothing are
used in the study of turbulent flows, allowing to move from probabilistic fields
of characteristics to their regular average values.
A
complete system of Reynolds averaged equations (RANS) describing the motion of
a viscous fluid in tensor form:
|
|
(1)
|
where
xj – Cartesian coordinates;
ui, uj – components of the velocity vector of the averaged flow;
E – specific total energy;
‑ specific total enthalpy of a gas;
Ò – temperature;
– gas density;
ð – pressure;
R – gas constant.
Components
of the Reynolds stress tensor
and the Reynolds heat flux vector
They
appear when averaging nonlinear convective terms of the initial Navier-Stokes
equations and energy transfer, and their relationship with the parameters of
the averaged flow is unknown.
Since
the RANS equations are not closed due to the Reynolds stress tensor and the
turbulent heat flow vector, it is necessary to use additional relations
(turbulence models) linking these values with the characteristics of the
averaged flow.
In this paper, the SST k-
ω
turbulence model, the Menter model, is used to close the Navier-Stokes
equations averaged by Reynolds [9]. This turbulence model was chosen because
there are no additional (source) terms in the transfer equations of the
two-parameter Menter model, which take into account the generation of
turbulence by large particles.Actually, the additional generation of turbulence
energy does not occur due to an additional source term in the equation, but due
to the fact that the flow around large particles is calculated directly as
objects with solid walls.
The Mentor model is implemented in the
following form:
|
|
(2)
|
where
Gk
‑ generation of turbulence kinetic energy
k,
Gω
– generation of turbulence kinetic energy
dissipation ω per unit
k,
Yk
– dissipation
of turbulence kinetic energy,
Yω
– dissipation
ω,
,
‑ the cross-diffusion term.
To solve the problem, a computational grid
with 1.11 million cells was built. The study of grid convergence was carried
out in terms of the sufficiency of the thickening of the grid in particles. The
value of y+ in the flow of particles does not exceed 3, which suggests that the
first cell is located in a viscous sublayer, wall functions are not used.
The
authors of the work chose the Ansys Fluent software to visualize the flow in
the wake of large particles, since this software is a powerful tool for
modeling complex problems in the field of hydrogas dynamics. Ansys Fluent
allows you to model the behavior of liquids and gases in various conditions,
including laminar and turbulent flows, taking into account viscosity, density
and other properties, and also has many tools for visualizing calculation
results [10].
Modeling
of large particles is implemented using the module Dynamic Mesh in Ansys Fluent.
Smoothing was carried out using the method Spring/Laplace/Boundary Layer – this
is a smoothing function when rebuilding the calculated grid. In the method
used, the edges of the cells are represented as springs and a coefficient is
set, which is actually the proportionality coefficient in Hooke's law, thereby
it is possible to regulate the thickening of the grid during rebuilding. The
dynamic grid tracks the movement of particles and is rebuilt, smoothing
functions are used to ensure that the quality of the grid remains close to the
initial one. The grid was rebuilt using the local cell method with a minimum
scale of 0.0005 m and a maximum scale of 0.1 m. The movement of particles was
set by one degree of freedom using the Six DOF (degree of freedom) module,
which allows calculating forces and moments acting on an object that lead to a
change in its position in space with a given time step. The paper assumes that
particles have one degree of freedom – they can only move along the Y axis
without rotation. Accordingly, in addition to the boundary and initial
conditions for the particles, a mass is set to take into account gravity (in
the work, the mass of each particle is 0.5 g).
Fig.1
The scheme of particle motion in the carrier gas
Figure
1 shows a diagram of the movement of particles at a speed of
in
the carrier gas
.
The center of the rectangular coordinate system (x – y) it is located on the
axis of symmetry of the channel with a diameter of
.
Large particles are spheres with a diameter of
,
which are placed vertically one after the other along the length of the channel
L
at different distances
from each other in order to exclude the mutual influence of vortex traces
behind the spheres. The spheres are located with an offset relative to the axis
of symmetry of the channel by 5 mm.
Table
1. Main characteristics of the studied flow
|
Air velocity
 , m/s
|
Particle velocity
 , m/s
|
Particle diameter
 , m
|
Pipe diameter
 , m
|
Pipe length
 , m
|
Reynolds number
Rep
|
Reynolds number
Rec
|
|
14.9
|
5.7
|
0.003
|
0.0305
|
0.2
|
1120
|
3
|
For a detailed analysis of
the process of additional generation of air turbulence energy in a sample
section of the pipe, an original methodological technique was proposed,
consisting in the arrangement of particles on the same line (Fig. 1).
Using
this technique, it was possible to obtain visual distributions of turbulence
energy along the length of the channel and identify qualitatively different
areas (Fig. 2): 1) the initial section characterized by increased values of
turbulence energy of "clean" air (single-phase flow); 2) areas of
growth of turbulence energy of the gas phase behind individual particles; 3)
the area of "quasi-stationary" single-phase flow.
Fig.
2 Distribution of turbulence energy
according to the length of the channel for
The
numbers are indicated by: 1 – the initial section with an increased
2
– growth
in the case of individual particles; 3 – "quasi-stationary"
single-phase flow
The data in Fig. 2 indicate that developed turbulent
traces are formed behind large particles, which are characterized by the
presence of non-stationary three-dimensional vortex structures.
Figures 3-5 show typical turbulence energy
distributions
in
the carrier gas in the channel and in the traces of the particles. The
visualization of the gas and particle flow is performed in the Ansys Fluent
software. It should be noted that the equations of motion of the particles
were not integrated, and the particles themselves moved at the same speeds. Thus,
a simplified version of the approach, called "two–way coupling" in
English literature, was implemented, i.e. taking into account the inverse
effect of particles on gas characteristics. Let's call it a quasi–approach or,
using the English abbreviation, "quasi – two-way coupling", TWC(Q).
In this setting, when the calculation of the flow of gas around each single
particle is performed and the interphase boundary is resolved (such
calculations are called "particle – resolved" (PR)). Thus, the
approach used in this work can be classified as PR – TWC(Q) – RANS.
Fig.
3 Distribution of the turbulence kinetic energy behind a single particle,
=
0.1, 1/ cm3
Fig.
4 Distribution of the turbulence kinetic energy behind a group of particles,
=
0.5, 1/cm3
Fig.
5 Distribution of the turbulence kinetic energy behind a group of particles,
=
1, 1/
ñm3
Fig.
6 Growth in turbulence energy generation depending on the calculated particle
concentration
Figure
6 shows data on the increase in turbulence energy of the carrier gas from the
value of the calculated particle concentration (the number of particles per
unit volume).
The
paper visualizes the flow behind large particles moving in an ascending
turbulent air flow in the channel. Numerical simulation is performed using a
simplified version of TVC(Q). Some results of numerical modeling of the
characteristics of turbulent traces behind large moving particles based on
Reynolds averaged Navier-Stokes equations (RANS) are presented. The dependence
of the turbulence energy value on the calculated particle concentration is
revealed.
The
work is supported by the Russian Scientific Foundation (Project ¹ 23-19-00734).
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