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

2. 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.

3. 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

4. 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)

5. Plasma welding

6. Diamond deposition using a plasma arcjet

7. Diamond growth on silicon using an oxy-acetylene flame

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

9. 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

10. 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

11. 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.

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

13. 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.

14. 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

15. 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

16. 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.

17. 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

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

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

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

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

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

23. 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

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

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