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Simulation of the Turbulent Flow from a Non-Transferred Arc Plasma Torch

S.M. Modirkhazeni1 and J.P. Trelles1

1 University of Massachusetts Lowell, Department of Mechanical Engineering, Lowell, MA, United States of America

Abstract: Non-transferred arc plasma torches are at the core of diverse applications such as plasma spray and waste treatment. The flow in these torches transitions from laminar inside the torch to turbulent in the emerging jet. There is no established approach for the modeling and simulation of turbulent plasma flows. The Variational Multiscale-n method is presented for the comprehensive modeling of general multiscale transport problems, such as turbulent plasma flows, and applied to the simulation of the flow in an arc plasma torch. Keywords: Plasma turbulence, subgrid scale modelling, thermodynamic non-equilibrium.

1. Introduction

Plasma torches are among the most established atmospheric-pressure plasma sources which generate a directed flow of plasma from a stream of working fluid. These torches are used in diverse applications such as plasma spraying, cutting, electric welding, metallurgy, and waste treatment. Direct-current (DC) arc plasma torches are typically differentiated between transferred and non-transferred configurations. In non-transferred torches, the electrodes are inside the torch, where a DC current between the electrodes produces an electric arc that generates the plasma. Figure 1 displays a schematic of a DC non-transferred arc plasma torch and the associated plasma flow.

Fig. 1. Schematic representation of the flow from a

non-transferred arc plasma torch. Turbulent flow behaviour, which enhances the

exchange of energy and momentum between heavy-species (molecules, atoms, ions) and electrons, is often observed in non-transferred arc plasma torches. The flow inside the torch can be laminar or in transition to become turbulent, while the generated jet is generally turbulent. The turbulent nature of the emerging plasma jet also depends on the dynamics of the arc inside the torch, which cause major fluctuations in the jet core. Moreover, deviations from Local Thermodynamic Equilibrium (LTE), manifested as deviations between the temperature of electrons and that of the heavy-species, is often a consequence of the interaction of the plasma with the processing media [1]. The turbulent nature of the flow can drastically affect the LTE state of the plasma. Hence, reliable plasma torch simulations need to be based on non-LTE (NLTE) models.

The direct simulation of turbulent plasma flows is extremely computationally expensive and often unfeasible due to the diversity of phenomena involved as well as the large range of scales that needs to be resolved. Large Eddy Simulation (LES), the de facto standard for the computational exploration of turbulent incompressible flows, largely relies on assumptions that are not valid for plasmas, such as isotropy and mild nonlinearity [2]. The adaptation of LES approaches to plasma flows often leads to methods that are not-consistent (e.g., the turbulent model equations are not valid for non-turbulent conditions) and/or are not-complete (e.g., employ empirical constants).

The Variational Multiscale-n (VMSn) method is presented as a consistent and complete approach for the simulation of turbulent plasma flows. VMSn is a superset of the VMS approach [1-3], one of the most versatile and robust methods used in multiphysics solvers based on the Finite Element Method (FEM). The method uses a variational decomposition of scales together with a residual-based approximation of the small scales without the need for empirical parameters. A major component of VMS methods is the handling of the nonlinearity dependence of the small scales, which VMSn addresses by a fixed-point procedure (i.e. n indicates the level of approximation used, i.e., from the classical VMS method for n = 0, to an exact description for n = ∞). The VMSn method is validated against established LES approaches with the simulation of an incompressible jet, and subsequently used, for the first time, for the comprehensive simulation of the flow a non-transferred arc plasma torch.

2. Nonequilibrium plasma flow model

The nonequilibrium plasma flow model in the present study is given by the set of conservation equations for total mass, mass-averaged momentum, internal energy of heavy-species, and internal energy of electrons (i.e. two-temperature NLTE plasma model); together with the equation of conservation of total charge and the magnetic induction equation, both expressed in terms of electromagnetic potentials. The set of coupled fluid

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– electromagnetic equations is shown in Error! Reference source not found.. The nomenclature and details of the mathematical model are described in [1]. The modeling approach treats the flow model as a general system of transient, advective, diffusive, and reactive (TADR) transport equations. These equations can be expressed in compact form as a coupled system for the vector of unknowns Y, as indicated in Eq. [1], where A0, Ai, Kij, S1 are matrices used to describe each transport process [1]: R (Y) = A0∂tY

transient!"#

+ (Ai∂i )Yadvective!"# $#

−

∂i (K ij∂ jY)diffusive

! "# $#− (S1Y+S0 )

reactive! "# $#

= 0,

(1)

The set of variables used is given by: Y = [ p uT Th Te φ p A

T ]T . (2)

3. Variational Multiscale-n (VMSn) Formulation

The so-called strong form of the NLTE plasma flow model for the array of unknowns Y is given by:

R (Y) = LY−S0 = 0, withL = A0∂t +Ai∂i −∂i (K ij∂ j )−S1,

(3)

where L is the TADR transport operator. Considering that the coefficient matrices are, in general, function of the vector of unknowns Y, )(YLL = , and hence Eq. [1] is nonlinear. Using the variational form of )(YR , decomposing the solution field Y into a large (coarse, grid-scale) component Y and a small (fine, sub-grid) scale component Yʹ (i.e., Y =Y+Y ' ), and applying the same decomposition to the basis function W (characteristic of the weak form of the problem), two

equations, one for the large scales and one for the unresolved scales, are obtained:

0SYYW =−+ Ω))'(,( 0L , and(W',L(Y+Y ' )−S0 )Ω = 0.

(4)

It is assumed that non-linearities existing in the above equations are non-separable, and therefore the dependence of the transport matrices on Y cannot be explicitly divided among the large and small scale terms (i.e., )( qpA +i ≠ )(pAi + )(qAi , hence: iA ≠iA + iAʹ ). Consequently, for the transport operator:

)'()(')( YYYYY +++= L-LLLL (5) Algebraic VMS formulations rely on approximating L with an algebraic operator τ , the intrinsic time scales matrix that approximates the inverse of the operator L and is the main modelling component [3]. The form of τ used here borrows ideas from the works of [1, 3].

Considering Eq. [5], Eq. [4] can be re-written in the form of Eqs. [6] and [7] for the large and small scales:

(W,R )Ω

Galerkin!"# $#

+ (L*W,Y)Ω

VMS! "# $#

+

(W,(L - L )Y)Ω+ ((L - L )*W,Y ')

Ω

non-linear correction ! "###### $######

= 0,,

(6)

Y ' = −τ(LY−S0 ), . (7)where Eq. (7) is a non-linear equation given that τ and L are function of 'Y .

Due to the depencency of Eq. [7] on both, the Y and Yʹ solution components, to capture the role of the small scales, a Picard iterative process (for n = 0 to n = ∞) is to be used, through which progressively more accurate approximations of the small scales is obtained.

Table 1: Set of equations of the nonequilibrium plasma flow model; for each equation: Transient + Advective – Diffusive – Reactive = 0.

Equation Transient Advective Diffusive Reactive

Conservation of total mass

ρ t∂ u u ⋅∇+∇⋅ ρρ 0 0

Conservation of linear

momentum u t∂ρ p∇+∇⋅ uu ρ

∇⋅µ(∇u+∇uT )−∇⋅ ( 23 µ(∇⋅u)δ)

BJ ×q

Thermal energy of

heavy species ρ ∂thh hh∇⋅u ρ )( hhr T∇⋅∇ κ u

u∇−−

+∇⋅+∂

:)( τheehhht

TTKpp

Thermal energy of electrons

ρ ∂the eh∇⋅u ρ )( ee T∇⋅∇ κ eqe

Bkq

rheeheet

T

TTKpp

∇⋅+×+⋅+

−−−∇⋅+∂

JBuEJ

u

25)(

4)( πε

Charge conservation 0 0

∇⋅ (σ∇φp )−∇⋅ (σu× (∇×A))

0

Magnetic induction

A t∂σµ0 µ0σ∇φp −

µ0σu× (∇×A) A2∇ 0

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4. Problem Set-Up and Simulation Results As a preliminary step to the plasma flow simulations,

the simulation of an incompressible turbulent free jet with air at a Reynolds number (Re) of 5000, a benchmark turbulent flow problem, is considered. The description of the problem, including the geometry and boundary conditions, are depicted in Fig. 2. In order to reduce the computational time to observe the evolution of the flow into a turbulent state, the inflow velocity profile is modified with uniform random perturbations of up to 5% the magnitude of the average velocity. Such approach, standard of LES, seeds to flow with perturbations that trigger excitable instabilities. These instabilities typically decay if the flow is laminar or if the numerical approach is not suitable for turbulent flow simulation (e.g. if the method produces excessive numerical dissipation). In the case of a turbulent flow, the resulting flow characteristics are independent of the type and magnitude of the seeded perturbations (as long as they are appropriately small).

Figure 2 depicts the capability of the VMSn method for turbulent flow simulation by comparing it against the VMS and the standard and dynamic Smagorinsky LES methods. The results using the VMSn method are more accurate than those from VMS, and are comparable to the dynamic Smagorinsky method, which is considered the de facto standard for incompressible turbulent flows. Both, the VMSn and dynamic Smagorinsky results, compare well with experimental observations.

Fig. 2. Incompressible jet flow: instantaneous

normalized velocity magnitude for different methods. As a preliminary step to the VMSn simulation of the

NLTE plasma flow in an arc plasma torch, the incompressible flow through a non-transferred arc plasma torch was simulated using the VMS and VMSn approaches. The domain geometry is shown in Fig. 3. The corresponding boundary conditions are similar to

those used for free jet problem. Representative simulation results are shown in Fig. 4. Figure 4 shows the instantaneous and time-averaged normalized velocity magnitudes. The white arrows indicates the locations where decay of the maximum velocity is dominated by turbulent dissipation. Figure 4 suggests that using the same geometry, computational domain, initial and boundary conditions, the VMSn formulation is more capable of revealing the small features in the flow than what is expected for VMS. Therefore, for same Re number, the incompressible VMSn simulation appears to be more turbulent than the one of VMS.

Fig. 3. Domain and boundary for the non-transferred

torch simulations.

Fig. 3. Incompressible flow in a non-transferred torch: instantaneous and time-averaged normalized velocity

magnitude using the VMS and VMSn approaches. The simulation of the NLTE plasma flow through a

non-transferred arc torch is subsequently carried out using the geometry displayed in Fig. 3 and boundary conditions in Table 2. The inflow velocity profile (the same as used in [1]) is modified with uniform random

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perturbations of up to 5% the magnitude of the average velocity. The simulation results depicted in Fig. 5 include the instantaneous temperature of heavy-species and normalized velocity magnitude using the VMS method. The instability and movement of the arc inside the torch, together with the large temperature and rapid acceleration of the flow near the cathode are well captured. The observed cathode jet is a result of the large heating an self-induced electromagnetic confinement of the plasma. It is observed that, although no explicit turbulence model is used for this simulation, the results resemble the experimental observations of the dynamics of plasma jets. Hence, the VMS method provides a reasonable coarse-grained representation of the flow, and therefore is a suitable platform for the development of more comprehensive turbulent plasma flow simulation aimed by the VMSn method.

Table 2: Set of boundary conditions for the non-

transferred arc plasma torch simulations.

Fig. 5. Non-transferred arch plasma torch simulation:

NLTE simulation using VMS method: instantaneous normalized velocity magnitude ||u||/uin and heavy-

species temperature Th. Ongoing efforts include performing NLTE turbulent

plasma flow simulations of the non-transferred arc torch using the new developed VMSn method. The results of the new approach will be validated with 3D time-dependent experimental data [4].

6. Summary and Conclusions

Non-transferred arc plasma torches are very versatile devices at the core of diverse applications, such as plasma spraying and waste treatment. Although the

flow inside these torches can be laminar or transitional, the emerged jet is inherently multiscale and turbulent. The flow characteristics associated to these torches are drastically affected by the dynamics of the arc inside the torch. The simulation of turbulent plasma flows is very challenging, especially because no method exists for their comprehensive coarse-grained modeling, which has prompted the use of turbulent models designed for incompressible flows. The Variational Multiscale-n (VMSn) method is presented as a comprehensive (i.e., consistent and complete) formulation for the modelling of general multiscale transport problems, particularly nonequilibrium turbulent plasmas flows. VMSn uses a variational decomposition of scales together with a non-linear residual-based approximation of the small scales without the need for empirical parameters. The effectiveness of the method is established through simulations using the VMS method and the dynamic Smagorinsky Large Eddy Simulation (LES) approach, which is so far the state of the art turbulent flow modeling strategy. Results of the incompressible turbulent flow in a free jet and on a non-transferred arc torch showed that the VMSn method is more capable of capturing the small scales in turbulent flows than the VMS, and that its accuracy is comparable to the dynamic Smagorinsky LES approach. Moreover, NLTE plasma flow simulations using the VMS method are able to reproduce the main experimentally-observed flow dynamics of the plasma jet (although no the small-scale features), which establishes prompts to the higher accuracy expected with the use of the VMSn approach. Ongoing efforts are centered on the validation of the VMSn approach with experimental turbulent flow data from a non-transferred arc torch.

Acknowledgements The authors gratefully acknowledge support from the U.S. National Science Foundation, Division of Physics, through award number PHY-1301935. References [1] J.P. Trelles, S.M. Modirkhazeni, Computer

Methods in Applied Mechanics and Engineering, 282 (2014).

[2] S. ModirKhazeni, J. Trelles, 22nd International Symposium on Plasma Chemistry, 2015.

[3] S.M. Modirkhazeni, J.P. Trelles, Computer Methods in Applied Mechanics and Engineering, 306 (2016).

[4] J. Hlína, J. Šonský, V. Něnička, A. Zachar, Journal of Physics D: Applied Physics, 38 (2005).