Multidimensional Plasma Sheath Modeling Using The Three ... · Aerospace & Energetics Research...

Plasma Dynamics GroupAerospace & Energetics Research Program

Multidimensional Plasma Sheath Modeling Using The Three Fluid Plasma Model

By R. Lilly and U. Shumlak

University of Washington16 November 2011


Abstract

There has been renewed interest in the use of plasma actuators for

high speed flow control applications. In the plasma actuator, current is driven through the surrounding weakly ionized plasma to impart control moments on the hypersonic vehicle. This expanded study employs the three-fluid (electrons, ions, neutrals) plasma model as it allows the capture of electron inertial effects, as well as energy and momentum transfer between the charged and neutral species. Previous investigations have typically assumed an electrostatic electric field. This work includes the full electrodynamics. Past work was conducted in 1- and 2-D. In this work, the problem is expanded to 3-D with the fluid equations extended from euler to Braginskii.


Outline

Accurate modeling of plasma sheath physics is of particular importance for weakly ionized plasmas (such as glow discharges). The two fluid model is useful because it allows the capture of electron inertial effects without the unbounded whistler wave that accompanies the Hall MHD approach.

The research presented here makes use of a “purely hyperbolic” form of Maxwell equations. The resulting boundary scheme that captures the sheath, in conjunction with the use of the purely hyperbolic Maxwell’s equation set, are reviewed and the results are discussed.

The addition of a third fluid allows the modeling of ionization and recombination physics.

This research includes the results of adding boundary conditions that will allow for the self-consistent modeling of a sheath in 1, 2, and 3 dimensions.

Extension of the approach to 3-D is in progress.


WARPX Capabilities

This research employs the code WARPX (Washington Approximate Riemann Problem) developed at the University of Washington.

WARPX solves non-linear, time-dependent equation systems with advanced algorithms for hyperbolic fluxes, parabolic fluxes, and source terms.

Two numerical algorithms are implemented in WARPX:– Finite Volume method using the wave propagation method, and– Finite Element method using the discontinuous Galerkin method.

Computational domain is discretized into a block-structured grid.

General geometries are fully supported in two and three dimensions for finite element, and in two dimensions for finite volume. Three-dimensional general geometry support for finite volume is in development.


The Three Fluid Plasma Model Captures Many Aerospace Plasma Phenomena

Plasma actuators have been proposed that would influence the neutral flow surrounding a hypersonic vehicle by controlling the accompanying plasma, either by direct energy addition, or shock location modification.

Influencing the bulk neutral flow would potentially provide an alternative flight control surface, and/or a means to reduce heat loading during high speed portions of flight/reentry.

A plasma actuator used as a flight control surface would have electrical (as opposed to mechanical) response times, which is critical given the high speed flight regime. Further, the ability to control shock location offers the prospect of controlling heat transfer.

As the electrode is the control surface for influencing the flow, modeling the plasma wall problem is key. A glow discharge is used to produce a plasma. Modeling the plasma formation requires an accurate representation of the sheath.

Journal of Spacecraft and Rockets, Vol. 46, No. 3, May-June 2009, by Bisek, N. and Boyd, N.


Multi-Fluid Plasma Model Captures the Relevant

Each fluid is represented by a set of Euler equations. The variable eα is the sum of the thermal and kinetic energy of the fluid. S, M, and A are collision terms connecting the charged species to the neutral fluid.

To capture the electrodynamics, Maxwell’s equations are modified to be purely hyperbolic. The variables φ and ψ are error correction scalar potentials. The variables χ and γ set the error correction wave speeds.


Fluid Boundary Conditions at the Electrode

The ion density and temperature are set as:

– dni/dx =0, and – dTi/dx =0.

It is assumed that the wall cannot source ions, but can appear as a sink where the ions recombine with the electrons. Ion normal momentum is therefore set to zero on inflow, and d(ρiui)/dx =0 on outflow.


Perpendicular Fluid Boundary Conditions at Electrode

Fluid dynamics are driven by electron/ion recombination at the electrode, which is assumed to occur at a rate equal to the particle momentum at the electron thermal speed kr. Secondary electron emission is specified by the incident ion flux and emission coefficient γ.

For the completely hyperbolic case, Gauss’s Law is implemented in 1-D as the BC for Ampere’s equation.


Implementation of Fixed-Voltage Boundary Conditions

Realistic applications apply a voltage to generate the desired sheath width.

One way to achieve this control uses feedback, which is introduced into the 1-D Poisson equation.

An observed voltage, Vcalc is calculated from the electric field and then compared to the applied voltage,VΔ, which is the potential between the applied left and right electrodes.


Applied Voltage Boundary Modifications for 2-D and 3-D However, the tangential electric

field components on the dielectric, the boundary condition is

For the perpendicular electric field components on both the electrode and dielectric

At the transition however, a relaxation function is required for the perpendicular electric field, and is of the form

– where “a” is a linear scaling factor representing the position on the transition region.

Dielectric

Electrode Conductor

Transition Region

In higher dimensions, the appropriate tangential electric field boundary conditions must be enforced.

For the electrode, the tangential electric field boundary condition is


Adding a Neutral Fluid and Collisions

In a weakly ionized plasma such as for a hypersonic aerospace application, the neutral fluid strongly affects the physics.

A third set of Euler equations represents the neutral species

The collisional source terms S, M, and A for each species must now be defined.


Collisional Source Terms for the Three Fluid Plasma Model-I


Collisional Source Terms for the Three Fluid Plasma Model-II

The following definitions are used for the source terms

where


Three-Fluid Simulation of Applied Voltage Plasma Sheath

Initial conditions– Ionization fraction of 10-6 for

a background neutral gas density of 1023 m-3

– All species, electrons, ions, and neutrals, have a temperature of 2 eV.

A higher neutral density (that is, a lower ionization fraction) is required to reduce the mean free path significantly below the domain size. This allows a three fluid equilibrium to be reached.

Ion and electron acceleration is limited by collisions with neutrals.


Finite Volume Method Overview The Finite Volume (Wave Propagation [1, 2]) Method employs the

Divergence theorem to create a readily discretizable formulation.

The fluxes normal to the interface, known as Godunov updates, are given by

The fluxes transverse to the interface are required to reach a CFL (a normalized timestep) of 1, and are required for stability in 3-D .


Transverse Fluxes

a

B+A+ ΔQi-1/2,j -1

A+ ΔQi-1/2,j -1

B-A- ΔQi+1/2,j +1

A+ΔQi-1/2,j+1 A- ΔQi+1/2,j +1

A- ΔQi+1/2,j -1

B-A+ ΔQi-1/2,j +1

B+A- ΔQi+1/2,j -1

A+ ΔQi-1/2,j A- ΔQi+1/2,j

Considering the figure at right for the 2-D case.

Godunov Fluxes are calculated for the x direction interfaces (Solid lines).

To account for diagonal motion, these fluxes are split to up and down going fluxes (dashed lines). These are the “transverse” fluxes, and are repeated for each interface direction.

For the 3-D case, a further split is required, and the fluxes are referred to as “double transverse” fluxes.

Transverse splits are also required for the second order corrections.

Qi,j


General Geometry (GG)

To enable the finite volume method to function on general hexahedral meshes, one must

Calculate the fluxes normal to each cell interface– This requires rotating into the local coordinate frame prior to flux

calculation

Rotating back to the global coordinate frame prior to updating the conserved quantities.

Considering next– Godunov updates– Transverse and second order corrections


General Geometry (GG) - Godunov Updates

Accounting for the rotation back to the global frame

The local frame fluctuations are given by

The split fluctuations must also be rotated into the local frame, ie


Mach 3 Inlet Ramp Validation Case

Mesh Details– 200x100x50 gridpoints– Ramp angle at 20 degrees– Ramp transition fillet provided @ 3<x<4, ie (x-3)2.

Normalized pressure and density initialized to 1.

Solid wall along bottom boundary

Open boundary at the right

Flow linearly increased from zero to Mach 3 over three normalized time units at the left and top boundaries.


Inlet Ramp Geometry Transient Results

Result at t=4 shows early bow shock interacting with pressure wave from ramp surface (lower right)

Pressure ratio of 11.17 illustrates capability to capture strong shocks


Shock Capturing Steady State Result

Ramp 20 degrees for Mach 3 flow, simulations shows convergence to shock angle of ~37 degrees, in accordance with oblique shock relations [4].

Close inspection reveals of the fillet the ability to capture shock compression in the parabolic transition region (3<x<4). Note how the weaker shockwaves converge into the main shockwave [4].


Three Dimensional Electrode Sheath Simulation

Domain 103 λD3.

Resolution 1003 gridpoints. Electrode radius of 2 λD

– With transition ring of 1 λD annular radius.

Bottom boundary consists of electrode, surrounded by dielectric.

All other boundaries are open, ie all quantities are extrapolated.

Simulation shows launch of Langmuir wave.

10λD

2λD

1λD


Three Dimensional Electrode Sheath Simulation - Result


Summary and Conclusions

Sheaths and plasma formation can be captured using the multi-fluid plasma model in 1-, 2-, and 3-D.

Higher dimensionality introduces the need for a transition region between the dielectric and the electrode conductor.

Multi-fluid simulations can capture physics across multiple time and spatial scales. This is particularly important when dealing with tightly coupled problems.

This work demonstrates an application of WARPX to 3-D finite volume simulations. This is an important step toward the solution of practical aerospace plasma problems.


Future Work

Complete general geometry validation for Maxwell’s equations, and, subsequently, the multi-fluid, five-moment model.

Adapt (and validate) already completed Braginskii heat flux and viscosity calculations from finite volume Cartesian implementation to finite volume general geometry.

Adapt the electrode BC from finite volume to discontinuous Galerkin finite element method and general geometries.

Complete a realistic hypersonic inlet ramp simulation with Braginskii and general geometry.


Endnotes

[1] Finite Volume Methods for Hyperbolic Problems, Randal J. LeVeque,2002.

[2] “A Wave Propagation Method for Three-Dimensional Hyperbolic Conservation Laws”, Jan Olav Langseth, and Randall J. LeVeque, Journal of Computational Physics 165, 126-166, 2000.

[3] “Simulation of a three-moment fluid model of a two-dimensional radio frequency discharge”, Mark H. Wilcoxson, and Vasilios I. Manousiouthakis, Chemical Engineering Science, Vol. 51, No. 7, pp 1089-1106, 1996.

[4] Elements of Gasdynamics, H. W. Liepmann and A. Roshko, 1957.

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