DIRECT NUMERICAL SIMULATION OF HOT AND HIGHLY PULSATED TURBULENT JET FLOWS

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HOT AND HIGHLY PULSATED TURBULENT JET FLOWS

V. Clauzon and T. Dubois

Laboratoire de Mathématiques, Université Blaise Pascal and CNRS (UMR 6620), 63177 Aubière, France

Summary. In this paper, we present numerical simulations of highly pulsated jet flows at 12 000 K developing in acolder 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. ForT∞= 3 000 K, the use of a high-order explicit filter was necessary in order to remove aliasing oscillations. The effect ofT^∞ on the flow properties is discussed.

Key words: DNS, plasma jet, turbulence, compact scheme, non-reflecting boundary 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

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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 acold environment with ambiant temperature 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 temperaturesT^∞ decreasing from 12 000 K to 6 000 K, are presented and discussed. Finally, a simulation was performed with the ambiant temperature set toT^∞= 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 temperatureT0, that isc0= 2 466 m.s⁻¹.

0 4000 8000 12000

0 1×10^-4 2×10^-4 3×10^-4

0 4000 8000 12000

0 1 2 3

Fig. 1. Tabulated values of the kinematic viscosity µ^⋆(T^⋆) (left) and the thermal conductivityκ^⋆(T^⋆).

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2.2 Governing equations

We consider that the flow variables, that is the density ρ, the velocity fieldu and the total energyE, satisfy the compressible Navier-Stokes equations

∂ρ

∂t +∂ρuj

xj

= 0,

∂ρui

∂t +∂ρuiuj

∂xj

+ ∂p

∂xi

= ∂τij

∂xj

,

∂E

∂t +∂(E+p)uj

∂xj

=∂τijui

∂xj

−∂qj

∂xj

,

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where

τij = µ(T) Re

“∂ui

∂xj

+∂uj

∂xi

−2 3δij

∂uk

∂xk

”

is the viscous stress tensor, and

qj = κ(T) Re P r

cp(T0) γ R

∂T

∂xj

is the thermal flux. The kinematic viscosity µ(T) =µ^⋆(T^⋆)/µ^⋆(T0) and the thermal conductivityκ(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

µ^⋆(T0) = 840, Pr =c^⋆p(T0)µ^⋆(T0)

κ^⋆(T0) = 0.5 and Ma =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

µ^⋆(T∞) . Renolds numbers Re∞and Re are related by

Re∞ =

„T0

T∞

«¹₂

µ^⋆(T0) µ^⋆(T∞)Re.

For plasma torches operating at T^∞= 300K, 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 ofNx×Ny×Nz mesh points. The grid is stretched in the lateral directions close the jet axis. In the streamwise direction, the mesh size

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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 pressurep^∞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

„1 2−1

2tanh“r−1 δ

”«

, (2)

vjet(0, y, z) =wjet(0, y, z) = 0, (3)

Tjet(0, y, z) =T^∞+` 1−T^∞´

„1 2−1

2tanh“r−1 δ

”«

, (4)

wherer=p

y²+z². The parameterδcontrolling the shear layer thickness was set toδ= 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 +εasin (Stat) +εhsin (Stht+θ)ri

whereθ= arccos (y/r). The parametersεaandεhdefining 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 6 000 K (Re∞= 1 441). Nevertheless, the resolution is sufficient to

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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. ForT∞= 3 000 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.

ForT∞>6 000K, the use of a sponge zone at the outflow boundary is necessary.

For smaller values, flow fluctuations are rapidly damped due to the difference between the jet and ambiant density. Indeed, the mean axial velocity forT∞≤6 000K (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 withT∞= 3 000K, 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 forx≥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 decay, 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 caseT∞= 3 000 K, most of the turbulence activity takes place in the region corresponding tox≥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 à la Modélisation Instationnaire et Tridimen- sionnelle du Comportement Dynamique de l’Arc Dans une Torche de Projection Plasma, Ph.D. thesis, Université de Limoges, France.

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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 axisy=z= 0 (left) and atx= 10 (right).

T^∞ = 12 000 K (dashed line), T^∞ = 9 000 K (dotted-dashed line), T^∞ = 6 000 K (double dotted-dashed line),T∞= 3 000 K (solid line).

X/R0

Y/R0

0 5 10 15 20

-10 -5 0 5 10

X

Y

0 5 10 15 20

-10 -5 0 5 10

Fig. 3. Instantaneous temperature (left) and magnitude of the instantaneous vorticity (right) for the simulation withT^∞= 3 000 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.

5. Danaila I, Boersma BJ (2000), Direct numerical simulation of bifurcating jets, Phys. Fluids 12:1255-1257.

6. Bogey C, Bailly C (2006), Computation of a high Reynolds number jet and its radiated noise using large eddy simulation based on explicit filtering, Computers and Fluids 35(10):1344-1358.