dles7_clauzon_dubois.dviLaboratoire de Mathematiques, Universite
Blaise Pascal and CNRS (UMR 6620), 63177 Aubiere, France
[email protected]
[email protected]
Summary. In this paper, we present numerical simulations of highly
pulsated jet flows at 12 000 K developing in a colder environment.
Such flows are used to model plasma jets generated by direct
current plasma torch. Plasma spraying is used to deposit thick
coatings on a substrate. We focus here on situations where plasma
jets are randomly forced by an electric arc resulting in a highly
turbulent flow at large Reynolds number. DNS were performed for
different ambiant temperatures T∞ ≥ 6 000 K. For T∞ = 3000 K, the
use of a high-order explicit filter was necessary in order to
remove aliasing oscillations. The effect of T∞ on the flow
properties is discussed.
Key words: DNS, plasma jet, turbulence, compact scheme,
non-reflecting bound- ary conditions
1 Introduction
Plasma spraying is a materials processing technique for producing
coatings using a plasma jet. Deposits can be produced from metals
or ceramics which are introduced into the plasma jet. The jet
temperature is typically of the order of 10 000 K so that the
material is melted and propelled towards a substrate. The resulting
materials are used for engineering applications including
automotive, biomedical or aerospace areas.
In direct durrent (dc) plasma torch, the plasma forming gas, here
Ar−H2, enters the torch with a speed of 30 m.s−1 and a temperature
of 1 000 K. While passing between the two concentric electrodes,
the gas is heated by Joule effects reaching 12 000 K. The mean
axial velocity of the plasma jet measured at the torch exit is 1
600 m.s−1.
Flows induced by plasma jets develop in a chamber filled with air
at 300 K. When plasma torches are operated in restrike mode (see
[1]), plasma jets are forced by the chaotic movement of the arc.
The induced flow is highly turbulent. Due
2 V. Clauzon and T. Dubois
to the large difference between the plasma and the ambiant gas
temperature, the Reynolds number based on environment
characteristics is of the order of 55 000. DNS at this Reynolds
number is not accessible with the computational ressources which
are available. To overcome this difficulty, we focus in this paper
on hot and highly pulsated jet flows developping in a cold
environment with ambiant temper- ature above 3 000 K, which
corresponds to Reynolds numbers smaller than 3 500. In order to
model the chaotic behavior of the arc, perturbations up to 30% of
the signal magnitude are imposed to the jet inflow. DNS, for
ambiant temperatures T∞
decreasing from 12 000 K to 6 000 K, are presented and discussed.
Finally, a simula- tion was performed with the ambiant temperature
set to T∞ = 3 000 K. In this case, an explicit filtering is used to
remove aliasing oscillations.
2 Formulation of the problem
2.1 Modeling assumptions
We assume that plasma and ambiant gas are the same, that is Ar −
H2. Also, as in [2] and [1], the flow is assumed to be in local
thermodynamic equilibrium (LTE). Therefore, the plasma is
considered as a compressible, perfect gas. Thermodynamic and
transport properties are evaluated by using tabulated values for
the viscosity and the thermal conductivity (see Figures 1). This is
essential in order to capture changes due to gas ionization and
dissociation occuring at 3 500 K.
As mathematical model, we use the compressible Navier-Stokes
equations sup- plemented with the equation of state for ideal gas.
The flow variables are made non-dimensional by using as reference
values the torch radius R0 = 3.5 mm, the temperature of the plasma
jet at the torch exit, T0 = 12 000 K, and the speed of sound at
temperature T0, that is c0 = 2466 m.s−1.
0 4000 8000 12000 0
1×10 -4
2×10 -4
3×10 -4
1
2
3
Fig. 1. Tabulated values of the kinematic viscosity µ(T ) (left)
and the thermal conductivity κ(T ).
DNS of hot and highly pulsated jet flows 3
2.2 Governing equations
We consider that the flow variables, that is the density ρ, the
velocity field u and the total energy E, satisfy the compressible
Navier-Stokes equations
∂ρ
∂t +
∂ρuj
xj
= 0,
∂ρui
∂t +
∂ρuiuj
∂xj
+ ∂p
∂xi
= ∂τij
∂xj
qj = κ(T )
∂T
∂xj
is the thermal flux. The kinematic viscosity µ(T ) = µ(T )/µ(T0)
and the ther- mal conductivity κ(T ) = κ(T )/κ(T0) strongly depend
on the temperature (see Figures 1). The constants R and γ, computed
from tabulated values of the speed of sound for Ar − H2, are set to
R = 187 J.kg−1.K−1 and γ = 1.2. The Reynolds, Prandtl and Mach
numbers are respectively defined by
Re = ρ0c0R0
p(T0) µ(T0)
u0
c0
= 0.7.
The ambiant gas temperature T∞ is assumed to be smaller than T0.
Due to the large variations of the viscosity with respect to the
gas temperature (see Figures 1), the viscosity of the ambiant gas
is much smaller than the jet inflow one. In order to characterize
the jet flow in its expansion region, we introduce a Reynolds
number Re∞ based on the environment characteristics and defined
by
Re∞ = ρ∞c∞R0
Re∞ =
2 µ(T0)
µ(T∞) Re.
For plasma torches operating at T∞ = 300 K, we have Re∞ = 54 600.
DNS at this Reynolds number can not be performed with the available
computational ressources.
2.3 Numerical procedure
The computational domain is a rectangular box of size (0, Lx) × (0,
Ly) × (0, Lz) discretized by a cartesian grid of Nx × Ny × Nz mesh
points. The grid is stretched in the lateral directions close the
jet axis. In the streamwise direction, the mesh size
4 V. Clauzon and T. Dubois
is constant for x ≤ 20R0 and increases linearly until the outflow
boundaries are reached.
The spatial derivatives are approximated by using a sixth-order
compact finite difference scheme (see [3]). The time discretization
is achieved with a third-order explicit Runge-Kutta scheme written
in compact form.
Due to the large variation of temperature and density, the velocity
fluctuations are rapidly damped in the transversal directions.
Therefore, the use of a large enough computational domain combined
with a sponge zone ensure that artificial oscillations due to wave
reflections at the boundaries will have minor effects on the jet
flow.
The non-reflecting conditions of Poinsot and Lele [4] are applied
at the inflow and outflow in order to approximate the streamwise
partial derivatives. A sponge zone acts on 20% of the domain length
in the streamwise direction. A weak condition smoothly forces the
pressure to adjust to the freestream pressure p∞ at the
outflow.
The jet temperature and velocity are prescribed at inflow.
Classical hyperbolic tangent profiles are used for the mean
streamwise velocity and temperature, namely
ujet(0, y, z) = Ma
Tjet(0, y, z) = T∞ + `
where r = p
y2 + z2. The parameter δ controlling the shear layer thickness was
set to δ = 0.2 in our simulations. Meshes in the transverse
directions are choosed so that approximately 10 grid points are
located in the jet shear layer. In order to model the effects of
the chaotic movement of the arc inside the plasma torch on the jet
flow, axial and helical fluctuations are imposed on the inlet
streamwise velocity, that is
u(0, r, θ, t) = ujet
h
1 + εa sin (Sta t) + εh sin (Sth t + θ) r i
where θ = arccos (y/r). The parameters εa and εh defining the
amplitude of the axial and helical perturbations were set to 0.3;
note that Boersma and Danaila in [5] used two times weaker
perturbations in their study of bifurcating incompressible jet
flows. The Strouhal numbers were set to Sta = 0.36 and Sth = 0.45
which respectively correspond to frequencies of arc fluctuations of
40 khz and 50 khz. These values are above the critical Strouhal
number for jet flows so that turbulence rapidly develops. Note that
experimental values for dc plasma torch (see [1]) are a factor of
2-4 smaller.
3 Numerical results
Numerical simulations for ambiant gas temperature T∞ in the range 3
000 K to 12 000 K were performed in the computational domain (0,
60R0) × (0, 40R0) × (0, 40R0). The computational parameters are
reported in Table 1.
For T∞ ≥ 9 000 K (Re∞ < 900), the 320 × 161 × 161 grid is fine
enough to provide an efficient resolution of the flow variables as
well as their derivatives. Grid oscillations appear in the shear
layer and close to the jet inlet (x ≤ 5) when T∞
is decreased to 6 000K (Re∞ = 1 441). Nevertheless, the resolution
is sufficient to
DNS of hot and highly pulsated jet flows 5
Table 1. Computational parameters
T∞ Nx Ny = Nz Re∞ Outflow sponge Spatial filtering
12 000 320 161 840 Yes No 9 000 320 161 882 Yes No 6 000 320 161 1
441 Yes No 3 000 500 241 3 442 No Sixth-order
avoid accumulations of aliasing oscillations. For T∞ = 3000 K, the
grid was refined to 500 × 241 × 241 points and the use of a
high-order compact filter, as in [6], was necessary in order to
remove grid oscillations. In all cases, equations were integrated
on 200 000 time iterations. Statistics are accumulated on 100 000
time iterations so that at least first order statistics can be
considered as converged values.
For T∞ > 6 000 K, the use of a sponge zone at the outflow
boundary is necessary. For smaller values, flow fluctuations are
rapidly damped due to the difference be- tween the jet and ambiant
density. Indeed, the mean axial velocity for T∞ ≤ 6 000 K (see
Figure 2) is about 30% smaller than the one obtained with larger
temperature : this results in weaker advection of flow
perturbutions in the axial direction. In the case with T∞ = 3000 K,
the non-reflecting BCs adjust values of the flow variables at the
outflow without requiring the use of a sponge zone. The mean axial
velocity (see Figures 2) has a rapid decay in the lateral
directions. Also, the expansion of the jet is such that only small
flow fluctuations reach the lateral sponge zone for x ≥ 30R0.
Reflections on the lateral boundaries are weak so that their effect
are negligible. On Figures 2, we observe that the virtual origins
of these heated jet flows are of the order of 5 which smaller than
the values observed for classical jet flows. A faster de- cay, with
respect to the jet distance, is also observed when the ambiant
temperature is decresased. Finally, Figures 3 representing the
temperature and the magnitude of the vorticity shows that, in the
case T∞ = 3 000 K, most of the turbulence activity takes place in
the region corresponding to x ≥ 7R0 and −5R0 ≤ y ≤ 5R0. Further in
the domain, that is for x ≥ 20R0 and |y| > 5R0, flow
fluctuations are rapdily damped indicating that turbulence
decays.
Acknowledgments
The numerical simulations presented in this paper were performed on
the cluster of 10 vectorial supercomputers NEC-SX8 of the
Supercomputing Center IDRIS of CNRS (Orsay, France,
http://www.idris.fr).
References
1. Trelles P, Heberlein JVR (2006), Simulation Results of Arc
Behavior in Different Plasma Spray Torches, J. Thermal Spray Tech.
15:563-569.
2. Baudry C (2003), Contribution a la Modelisation Instationnaire
et Tridimen- sionnelle du Comportement Dynamique de l’Arc Dans une
Torche de Projection Plasma, Ph.D. thesis, Universite de Limoges,
France.
6 V. Clauzon and T. Dubois
0 10 20 30 40 50 60 0
0.2
0.4
0.6
0.8
-4 -3 -2 -1 0 1 2 3 4 0
0.1
0.2
0.3
0.4
0.5
x/R0 y/R0
Fig. 2. Mean axial velocity along the jet axis y = z = 0 (left) and
at x = 10 (right). T∞ = 12 000 K (dashed line), T∞ = 9 000 K
(dotted-dashed line), T∞ = 6000 K (double dotted-dashed line), T∞ =
3000 K (solid line).
X/R0
-5
0
5
10
X
Y
-5
0
5
10
Fig. 3. Instantaneous temperature (left) and magnitude of the
instantaneous vor- ticity (right) for the simulation with T∞ = 3000
K on the grid with 500× 241× 241 mesh points.
3. Lele SK (1992), Compact finite difference schemes with
spectral-like resolution, J. Comput. Phys. 103:16-42.
4. Poinsot TJ, Lele SK (1991), Boundary Conditions for direct
Simulations of Compressible Viscous Flows, J. Comput. Phys.
101:104-129.
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bifurcating jets, Phys. Fluids 12:1255-1257.