**LES MODELING OF HEAT AND MASS TRANSFER IN TURBULENT
RECIRCULATED FLOWS**

BAAKE E., NACKE B.

*Institute
for Electrothermal Processes, **University** of **Hanover*

*baake@ewh.uni-hannover.de,
nacke@ewh.uni-hannover.de*

UMBRASHKO A., JAKOVICS A.

*Laboratory
for Mathematical Modeling of Environmental and Technological Processes, **University** of **Latvia**, Zellu str. 8, LV-1002 **Riga**, **LATVIA*

*andrey@modlab.lv, ajakov@latnet.lv*

**Abstract:** Experimental results show that heat and mass transfer processes in
the turbulent melt flow of induction furnaces are significantly influenced by
low-frequency large scale oscillations of the main flow eddies. Large Eddy
Simulation (LES) of the turbulent melt flow in induction crucible furnace
carries out with good conformity the transient three-dimensional oscillations
of the dominating toroidal flow eddies. This 3D transient model offers new
possibilities for simulation of heat and mass transfer processes in induction
furnaces.

1. Introduction

Fluid flow in industrial metallurgical installations has become a subject of numerical modeling many years ago. The variety of geometries, boundary conditions, fluid forcing factors and a whole range of flow Reynolds numbers provide a good challenge for numerical investigations. The absence of an universal and always reliable modeling approach together with a wide choice of available non-universal turbulence schemes turns it into a non-trivial problem, at least while up to now direct numerical simulation of the turbulent melt flow remains practically inapplicable. As a wide spread example can be mentioned the melting of alloys in induction furnaces. The flow pattern in these installations is formed by the influence of electromagnetic forces and usually takes form of several dominating vortices. Flow patterns obtained with two-dimensional solvers based on Reynolds Averaged Navier-Stokes (RANS) equations usually are in good agreement with estimated and measured time-averaged values [1-3]. Resulting spatial distribution of the temperature and alloys compound concentration depends strongly on the heat and mass exchange between these vortices. Numerical investigations show that two-dimensional turbulence models fail to describe correctly the heat and mass transfer processes between the main vortices. There was developed an engineering approach for this problem described in [1], but for more generic and therefore more flexible solution it is necessary to investigate advanced simulation methods. Today it is possible to run un-stationary three-dimensional numerical calculations of fluid dynamic problems using advanced turbulent models with higher grid resolution requirements and get reliable results in reasonable time. Concluding all these preconditions the calculations presented in this paper were based on Large Eddy Simulation (LES) method, which can be described as a compromise between the solving of RANS equations and Direct Numerical Simulation (DNS). The main idea is that main flow structure is resolved directly, while only small eddies, which size is comparable with grid size, are modeled additionally. Finer meshing is required and, consequently, more computational resources than for two parameter turbulence models, e.g. k-ε turbulence model, but less than it is necessary for the DNS.

**2.
Low-frequency oscillations**

Most of the experimental and numerical
investigations were carried out with a model induction crucible furnace (ICF),
which has a radius of 158 mm and a height of 756 mm, where the inductor height
is 570 mm (Fig.1). Wood's metal, which has a melting point of 72°C, and a
dynamic viscosity of 4.2∙10^{-3} kg/m∙s, a density of 9,700
kg/m^{3} and a conductivity of 1∙10^{6 }S/m
was used as a model melt. Experimental velocity measurements carried out using
a potential probe show presence of low-frequency flow oscillations in this
model furnace (Fig.2). Most intensive of them have a characteristic period
about 8-12 seconds depending on inductor current I. These oscillations can be
described very simple as periodical up and down movements of the toroidal
eddies of the mean melt flow, where the axis symmetry of the melt flow does not longer exist. The main oscillation frequency f
increases in dependence on the time-averaged velocity: f ~ v_{ch} ~ I_{ind}

*Figure 2*.
Measured axial velocity oscillations at half-height of the inductor near the crucible wall (left) and at half radius (right)

These flow instabilities have a large
characteristic time and space scale and therefore they can not be considered as
turbulence. So it is doubtfully, that stationary calculations with help of any
existing turbulence modeling approach could correctly simulate scalars
transport processes in such flow formation. This statement was verified with
several widespread turbulence models. All of them do not predict the high
intensity of heat exchange in the discussed region. As expected, unstationary
2D calculations using k-ε turbulence model also didn't lead to acceptable
results. The absence of azimuthal velocity in 2D analysis limits the dynamics
of the large eddies in incompressible fluid. Also 3D calculations using
k-ε turbulence model lead to not acceptable results because the
low-frequency oscillations are dumped by the high turbulent viscosity.
Transient 3D calculations appeared to be the only possible way to resolve the
given problem.

3. Numerical modeling

LES and
two-equation RANS approach are using different turbulence modeling techniques,
therefore, the calculated subgrid turbulent viscosity distribution has
principally different character. The highest values of the LES subgrid
viscosity are in the zone of eddies interaction in the near-wall region. Another
unsimilarity is that turbulent viscosity in case of LES is one order of
magnitude less than it is predicted by k-ε model. This difference
increases if we improve the spatial resolution of the numerical grid. The
smaller become cells, the less energy contain eddies, which are modeled with
subgrid turbulent viscosity (Fig. 3). Therefore, further mesh refining will
lead to the situation when eddies of all considerable scales are resolved
directly. In LES this tendency is obvious when examine the expression for
subgrid turbulent viscosity m_{sub} in Smagorinsky-Lilly scheme used in FLUENT [4-6]:

_{}, _{} _{},

where *d *is the distance from the closest wall, *k *= 0.42, C_{s} = 0.1 is Smagorinsky constant and *V _{c}* is volume of the
computational cell. The averaged radius or cell size directly influences this
expression. Kinetic energy of turbulence predicted with k-ε model doesn't
depend on spatial resolution, and averaged velocity field doesn't change
significantly with mesh refinement. Hence, velocity gradients, which are
responsible for modeling of turbulent parameters, also remain the same.
Certainly, if we use smaller cell sizes, then additional flow details may
appear and influence the resulting turbulence field. But this dependence is not
as obvious as in LES. For example, increasing number of mesh elements from
¼ of million to 3 ½ millions the subgrid viscosity calculated
with LES becomes one order of magnitude less, but turbulent viscosity of k-ε
model remains almost the same (Fig.3).

In k-ε simulation, due to high predicted turbulent
viscosity, the viscous forces are comparable with inertial ones and, therefore,
all possible large-scale pulsations are damped. Calculating with LES the
subgrid viscosity is at least ten times smaller even on relatively coarse mesh.
As result, oscillations, which arise from any disturbing factors, develop in
time and reach significant amplitude.

Transient
calculations started with uninitialized flow field **v**(**r**)≡0.
After several seconds flow pattern comes to the symmetric state, which
resembles results of 2D stationary modeling (Fig.4B), and then begins to
oscillate (Fig.4C). There were chosen three control points for velocity along
the crucible radius at half-height of the inductor, where the time-averaged
velocity is approximately zero. It is possible to analyze the oscillations and
compare the time-depended behavior of the velocity components with the
experimental results. Velocity oscillations obtained with transient numerical
simulation using LES turbulence model are shown on the Fig.5. The
Smagorinsky-Lilly subgrid-scale model on tetrahedral mesh with 3.5**∙**10^{6} elements and with
time step 10 ms is taken as a base variant. Other cases differ in only one
parameter. In case of 0.4**∙**10^{6}
elements mesh were calculated 130 seconds, in other cases only about 60
seconds. Simulated oscillations amplitudes are in good agreement with those
from measurements and don’t depend on number of elements. In good agreement
with the experiment the maximum of the oscillation intensity is near the wall
and decreases in the center of the furnace (Fig.2,3).
Table 1 lists the parameters of different numerical investigations and
resulting characteristic values of low-frequency oscillations. The oscillations
intensity in the given point is estimated by the time average of axial velocity
square.

*Figure
4.* Flow pattern after 2, 6 and 10 seconds of
calculations

Number of grid
elements, x10 |
Subgrid turbulence
scheme |
Total calculated
flow time, s; [time step], ms |
Oscillations
intensity <V |
Low-frequency
oscillations period, seconds |

0.4 |
S-L |
130 [10] |
88 |
14 |

3.5 |
S-L |
60 [10] |
110 |
12 |

3.5 |
S-L |
60 [5] |
95 |
8.5 |

3.5 |
RNG |
60 [10] |
60 |
10.5 |

Experimental data [7] |
56 (measured) |
122 |
9 |

*Table **1*. Comparison between calculated
and measured low-frequency oscillations intensity and characteristic period
between the main flow vortices

Apparently we deal with a nonlinear system of Navier-Stokes equations, which has a solution tending to the limiting cycle. This means, that it is possible to estimate, with a good probability, the flow direction in a given point after relatively long time period. The numerical investigations show, that characteristic period of this main cycle doesn't change significantly when we improve the spatial or temporal resolution. Its length is determined by electromagnetic forces intensity and domain geometry; therefore, the initial conditions have no strong influence.

The discrete
particle tracing approach has been carried out to investigate convective scalar
transport mechanism in considered flow. In the very beginning virtual particles
are placed on the top of computational domain. These particles are assumed to
have the same density as fluid and this leads to the expectation that their
path will coincide with the streamlines of the flow. When the flow in closed
domain without inlets and outlets is stationary, the streamlines also are
closed and particle trajectories should be looped. Then it is improbable that
particle will penetrate into the neighboring flow region if the turbulent
transfer is neglected. Therefore, transport processes between the main flow
eddies generally would have diffusive character in steady-state flow. In this
case scalar exchange intensity will strongly depend on the semi-empiric
turbulent parameters like turbulent viscosity and turbulent Prandtl (δ_{t}=c_{p}μ_{t}/λ_{t})
or Schmidt (Sc_{t}=μ_{t}/ρD_{t}) number.
Latter parameters magnitude often depends both on the type of fluid and on the
type of the flow and has to be determined experimentally. The flexibility of
using such approach for different industrial installations is rather low.

The subgrid viscosity model seems to have more universal
character. Transient simulations, with a small time step and appropriate
meshing, allow resolving the wide range of flow formations involved in scalar
transport. Particle trajectories, traced in such unstationary simulations,
show, that estimated convective mass exchange between the main flow eddies is
quite intensive (Fig.6A,B).

Four particles were launched simultaneously at z = z_{max},
r = r_{max}/2, δφ =π/2 and
traced 20 seconds (about 5-6 eddy turnover times). The choice of starting
position was caused by conformity with industrial alloying process, when
additional components are added on the melt surface. The typical tracing result
states, that only one particle didn't change the host eddy, but others didn't
stay in one eddy longer than two or three turnover times. Therefore we can make
the conclusion, that convective transport mechanism plays significant role in
the heat and mass exchange between the main flow
eddies. The same tracing procedure was used with the averaged velocity field
from transient LES calculations. As expected, all particles rotated in the
initial eddy with relatively small azimuthal drift (Fig.6C). Probably, if we
would take longer time for averaging, the trajectories would tend to those in
steady-state flow pattern.

4. Conclusions

Large Eddy Simulation turbulence model was applied for three-dimensional transient calculations and it proved to be a very promising tool for numerical simulation of complex turbulent flows. Produced results for velocity field and flow oscillations are in good agreement with experimental data. An extensive numerical analysis was performed to study how spatial and temporal resolution together with subgrid turbulence model choice influence solution stability and reliability. Particle tracing in transient flow showed the high efficiency of convective transfer due to simulated low-frequency oscillations. These results state, that, due to constant increasing of computational power, Large Eddy Simulation could become an universal tool for practical engineering applications.

Further numerical research in this area may include calculations with deformed surface of the melt in order to study its influence on the flow regime. Also the numerical simulation and optimization of the melting processes in industrial installations could be a logical continuation of the presented work.

5. References

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Nacke
B., Baake E., Jakovics A., Umbrashko A.: Heat and mass
transfer in turbulent flows with several recirculated flow eddies. Fourth
International PAMIR Conference, Presquile de

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R.M., Harvey A.D., Acharya S.: Two-equation turbulence modeling for impeller
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3.
Bojarevics
V., Pericleous K., Cross M.: Modeling turbulent flow dynamics in AC magnetic
field. The 3^{rd} International Symposium on Electromagnetic Processing
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4.
Fluent
5 User’s guide. Fluent Inc. 1999

5.
Smagorinsky
J.S.: General circulation model of the atmosphere. Monthly Weather Review. p.91-99, 164, 1963.

6.
Lilly
D.K.: On the application of the eddy viscosity concept in the inertial subrange
of turbulence. NCAR Manuscript, 123, 1966.

7.
Baake, E.: Grenzleistungs- und Aufkohlungsverhalten von
Induktions-Tiegelöfen. VDI-Verlag Düsseldorf 1994.