A Case Study in Computational Science

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A Case Study in Computational Science
EngineeringSupersonic flow of ionized gas
through a nozzle
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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.

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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

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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)
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Plasma welding
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Diamond deposition using a plasma arcjet
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Diamond growth on silicon using an oxy-acetylene
flame
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• There are numerous examples of reacting flows in
  industrial applications
• Energy generation conversion combustion
  processes
• Automotive engines
• Gas Turbine engines

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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

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Existing modeling tools

• Canned programs exist
• Fluent, Fidap, StarCD, Chemkin, etc.
• Why write ones own code?
• greater flexibility
• speed (canned codes trade off speed for
  user-friendliness), and most importantly
• ability to model additional phenomena

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• This quarters 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.

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

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Case Study Problem
Adiabatic walls, i.e. no heat flow
Po1 atm To300 K
L 1 m
Argon gas flow
CONVERGING-DIVERGING OR CD NOZZLE
Zone of heat addition
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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

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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.

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Governing equations for quasi 1-D flow
r(x) ni(x) P(x) T(x) u(x) A(x)
r(xdx) ni(xdx) P(xdx) T(xdx) u(xdx) A(xdx)
x
dx
xdx
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Conservation of mass
(1)
At steady state, we have
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Conservation of momentum
(2)
Conservation of energy
(3)
Species conservation (electrons)
(4)
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Equation of State
(5)
Definition of Density
(6)
Unknowns r, u, P, T, ne, and nA
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Governing equations for quasi 1-D steady flow
(1)
(2)
(3)
(4)
(5)
(6)
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Governing equations for quasi 1-D, steady,
adiabatic, frictionless, compositionally frozen
flow
(1)
(2)
(3)
(4)
(5)
(6)
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Choking condition

• The foregoing equations can be combined to yield
• where Mu/(gRT)1/2 is the Mach number based on
  the isentropic speed of sound
• Note that when M1, dA/dx must be zero in order
  for there to be smooth acceleration through M1.
• Further, for Mlt1, du/dxgt0 for dA/dxlt0
• Similarly, for Mgt1, du/dxgt0 for dA/dxgt0

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Implications of the choking condition

• Subsonic nozzles are supersonic diffusers
• Subsonic diffusers are supersonic nozzles

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Two-dimensional flow

• Conservation of mass
• Conservation of momentum
• Conservation of energy
• Species conservation

Navier-Stokes Equations
y
x