a selfconsistent unstructured solver for weakly ionized gasesoldr. pepe1, g. colonna2, a....

Digitare l'equazione qui.

• Chemical species

• Chemical reactions 𝑨𝑨𝟎 + 𝒆− ↔ 𝑨𝑨+ + 𝒆− + 𝒆− 𝑨𝑨𝟎 + 𝒆− ↔ 𝑨𝑨∗ + 𝒆− 𝑨𝑨∗ + 𝒆− ↔ 𝑨𝑨+ + 𝒆− + 𝒆− 𝑨𝑨𝟎 + 𝑨𝑨𝟎 ↔ 𝑨𝑨+ + 𝒆− + 𝒆− 𝑨𝑨𝟎 + 𝑨𝑨𝟎 ↔ 𝑨𝑨∗ + 𝒆− 𝑨𝑨∗ + 𝑨𝑨𝟎 ↔ 𝑨𝑨+ + 𝒆− + 𝒆−

A selfconsistent unstructured solver for weakly ionized gases R. Pepe1, G. Colonna2, A. Bonfiglioli1, A. D’Angola1, R. Paciorri3

1 Scuola di Ingegneria SI, Università degli Studi della Basilicata, via dell’Ateneo Lucano 10 - 85100, Potenza, Italy

2 Istituto di Metodologie Inorganiche dei Plasmi, CNR, via Amendola 122/D – 70126, Bari, Italy 3 Dipartimento di Meccanica e Aeronautica, Università di Roma La Sapienza,

via Eudossiana 18 – 00184, Roma, Italy

• Conservation equations Multispecies inviscid flow with finite rate chemistry are considered. The contribution of the magnetic field is neglected and a simple model for the internal ohmic source is considered[2].

𝜕𝜌𝑖𝜕𝑡

+ 𝛻 ∙ 𝜌𝑖𝐮 = 𝑆𝑖 (1.1) 𝜕𝜌𝐮

𝜕𝑡+ 𝛻 ∙ 𝜌𝐮𝐮 + 𝑝𝐈 = 𝟎 (1.2)

𝜕𝜌𝐸𝜕𝑡

+ 𝛻 ∙ [𝜌𝐮(𝐸 + 𝑝)] = 𝐣 ∙ 𝐄 (1.3)

• Laplace equation The electric field (𝐄 = −𝛻𝛻) is computed solving the Laplace equation for the electrical potential.

𝛻2𝛻 = 0 (2) • Ohm law The current density is computed using the Ohm law. For an argon mixture , the electrical conductivity is computed using the semi-analytic model of Chapman and Cowling[3-4].

𝐣 = 𝜎 E (3) • Circuit Ohm law: The electric field is generated by two electrodes connected in series to a power supply by an external circuit with a resistence 𝑅𝑐 [5].

𝑉𝑑𝑖𝑑 𝑡 = 𝑉𝑔(𝑡) − 𝑅𝑐𝐼(𝑡) (4) The electric current flowing in the circuit is computed integrating the current density on the electrodes surface.

𝐼 = ∫ 𝐣 ∙ 𝑑𝐀𝑒𝑙𝑒𝑐𝑡𝑙𝑙𝑑𝑒𝑑 (5)

A selfconsistent unstructured solver for weakly ionized gases modeling inviscid multi-species conservation equations coupled with a reduced state-to-state kinetics model is presented. Mathematical model have been implemented in EulFS [1], a CFD unstructured solver using Residual Distribution Schemes, to discretize convective fluxes and chemical source terms. A simple model which couples the weakly ionized gas with an electric field controlled by considering a power supply and an external circuit resistence Rc is considered. Preliminary results will be shown for an argon plasma considering a 2D geometrical configuration.

References: [1] Bonfiglioli Int. J. Comput. Fluid Dyn. vol. 14, pp. 21-39, 2000. [2] Giordano. AIAA Paper 2002-2165, 2002) [3] Bisiek, Boyd and Poggie AIAA Paper 2010-4487, 2010. [4]Cambel Plasma Physics and Magnetofluid-Mechanics, McGraw–Hill, New York, 1963. [5] Colonna and Capitelli J. Thermophys Heat Transfer. vol. 22, pp. 414-423, 2008. [6] Bacri and Gomes J. Appl. Phys. vol. 11, pp. 2185-2197, 1978. [7] Vlcek J. Appl. Phys. vol. 22, pp. 623-631, 1989.

Abstract Governing equations

Chemical model [5-7]

Preliminary results

• Unstructured mesh: The computational domain is tessellated into triangles (2D) or tetrahedra (3D).

• Fluctuation splitting approach: Numerical schemes used is 2° order accurate in space.

• Dual time stepping: Unsteady calculations are achieved using a dual time stepping scheme, which is 2° order accurate in time.

• Coupling between the fluid code and Laplace solver: At each time step, the boundary conditions for the Laplace equation are updated according to the circuit Ohm law. The electric conductivity depends upon the flow conditions.

Numerical method

Electron-atom

Atom-atom

𝑨𝑨𝟎 Neutral atoms (ground)

𝑨𝑨∗ Neutral atoms (metastable 3P2)

𝑨𝑨+ Positive ions

𝒆− Electrons

Neumann

Dirichlet (𝝋 = 𝝋𝑨𝒆𝒓 + Vdis)

Dirichlet (𝝋=𝝋𝑨𝒆𝒓)

Electric potential

𝐴 = 𝜋𝑟2

Circuit parameters 𝑽𝒈 4000 V

𝑹𝒄 1000 Ω

Initial flow conditions 𝑻 3000 K

𝒑 0.1bar

u 102 m/s

Boundary conditions (Laplace equation) Norm of the Electric field

Electric current (𝑰𝒄) Discharge potential (𝑽𝒅𝒅𝒅)

Density (𝑽𝒈≠𝟎) Flow speed (𝒖𝒙) (𝑽𝒈≠𝟎)

Temperature (𝑽𝒈 = 𝟎) Temperature (𝑽𝒈≠𝟎)