A Case Study in Computational Science & Engineering A Case Study in Computational Science & Engineering: Supersonic flow of ionized gas through a nozzle

Course Objectives To provide insight and understanding of issues and difficulties in computational modeling through a quarter-long case-study. Evaluate relative merits of various methods using programs you have written and pre-developed software modules. To understand the differences in performance when computations are done serially versus in parallel.

Introduction to Case Study Many engineering applications and processes involve reacting and plasma flows. Some important examples: plasma processes in manufacturing of integrated circuits (etching, deposition) manufacturing processes (welding, coatings, synthesis of novel materials) space propulsion (positioning and station-keeping of satellites) Gas lasers, wind-tunnel test facilities, nozzles/shock tubes for studying chemistry

Manufacturing of Semiconductor Devices A trench 0.2 mm wide by 4 mm deep in single-crystal Si, produced by plasma etching (from Lieberman & Lichtenberg)

Plasma welding

Diamond deposition using a plasma arcjet

Diamond growth on silicon using an oxy-acetylene flame

There are numerous examples of reacting flows in industrial applications: Energy generation & conversion: combustion processes Automotive engines Gas Turbine engines

Motivation for modeling Gives detailed insight and basic understanding into the problem Helpful for design, control and optimization; can identify improved geometries for reactors, scale-up, etc. Availability of detailed experimental measurements enable in-depth understanding of cause-effect relationships (important for process control). Helpful in interpreting system or sensor response (e.g.. Ionization probe), and experimental data

Existing modeling tools “Canned” programs exist: Fluent, Fidap, StarCD, Chemkin, etc. Why write one’s own code? greater flexibility speed (canned codes trade off speed for user-friendliness), and most importantly ability to model additional phenomena

This quarter’s case study will focus on an illustrative example involving a supersonic flow in a nozzle with ionization & recombination processes. This case study is intended to help bring out issues related computational modeling of a prototypical engineering problem, using high-performance computing methods.

OSU supersonic afterglow wind-tunnel Supersonic afterglow of Nitrogen over a wedge Supersonic afterglow of Helium over a wedge

Case Study Problem Argon gas flows through a converging-diverging channel of known cross sectional area Given: upstream total pressure, total temperature, and channel geometry desire supersonic flow in the diverging portion of the channel Find: distributions of velocity, density, pressure, temperature, Mach number, electron density, and ionization fraction throughout the channel, at steady state/transient state.

CONVERGING-DIVERGING Case Study Problem Adiabatic walls, i.e. no heat flow Po=1 atm To=300 K L = 1 m Argon gas flow CONVERGING-DIVERGING OR CD NOZZLE Zone of heat addition

Background Flow of a gas at high speeds such as in CD nozzles, is characterized by changing density, r. the mass density, r, of a gas can change due to temperature changes or pressure changes. When r changes because of pressure changes, the flow is called a compressible flow. To illustrate some of the basic characteristics of such a flow through a varying area channel, we begin by with a quasi one-dimensional (quasi 1-D) model of steady flow

Quasi 1-D steady flow Quasi 1-D  that flow varies in the streamwise, i.e. flow direction only, and transverse variations are ignored. Steady  , i.e., no time variation. Governing equations, i.e. rules that govern such a flow are conservation of mass (or continuity), conservation of linear momentum, conservation of energy and species number density.

Governing equations for quasi 1-D flow r(x) ni(x) P(x) T(x) u(x) A(x) r(x+dx) ni(x+dx) P(x+dx) T(x+dx) u(x+dx) A(x+dx) x dx x+dx

Conservation of mass (1) At steady state, we have

Conservation of momentum (2) Conservation of energy (3) Species conservation (electrons) (4)

Equation of State Definition of Density (5) Definition of Density (6) Unknowns: r, u, P, T, ne, and nA

Governing equations for quasi 1-D steady flow (1) (2) (3) (4) (5) (6)

Governing equations for quasi 1-D, steady, adiabatic, frictionless, compositionally frozen flow (1) (2) (3) (4) (5) (6)

Choking condition The foregoing equations can be combined to yield: where M=u/(gRT)1/2 is the Mach number based on the isentropic speed of sound Note that when M=1, dA/dx must be zero in order for there to be smooth acceleration through M=1. Further, for M<1, du/dx>0 for dA/dx<0 Similarly, for M>1, du/dx>0 for dA/dx>0

Implications of the choking condition Subsonic nozzles are supersonic diffusers: Subsonic diffusers are supersonic nozzles:

Two-dimensional flow Conservation of mass Conservation of momentum Conservation of energy Species conservation Navier-Stokes Equations y x