THE OFFICE OF NONPROLIFERATION

1
Wichita State UniversityJanuary 28, 2008

• Multiphase computational fluid dynamics
• with applications
• Tianshi Lu
• Computational Science Center
• Brookhaven National Laboratory
• In collaboration with
• Roman Samulyak, Brookhaven National Laboratory
• James Glimm, Xiao-Lin Li, Stony Brook University
• Zhi-Liang Xu, Notre Dame University

2
Main Features

• CFD for compressible fluids
• sound wave, shock wave propagation
• hyperbolic system of conservation laws
• Heterogeneous approach to multiphase flows
• front tracking for free surface flows
• verification of homogeneous models

3
Talk outline

• Mathematical theory of conservation laws
• Numerical methods for conservation laws
• Numerical methods for multiphase CFD
• Applications of multiphase CFD

4
Mathematical theory of conservation laws
A general system of conservation laws
A good theory for conservation laws in higher
dimensions is unavailable.
1D system of conservation laws in
non-conservative form
Hyperbolic system
A(u) is diagonalizable with a real spectrum.
5
Linear system of conservation laws
If A(u) is a constant matrix, the system can be
solved by the method of characteristics.
l1
l2
Solution to the Riemann problem
x
0
vL
vR
6
Nonlinear system of conservation laws
Traffic flow
Burgers equation
Inviscid gas dynamics
Shallow water equations
7
Nonexistence of smooth solution in general
Rarefaction, F(u)u2/2 is convex
u(x,Tb)
Characteristics cross, the wave breaks.
u(x,0)
Shock formation in Burgers equation
8
Weak solution and entropy condition
Integral solution to the conservation laws
Rankine-Hugoniot Conditions
Lax entropy condition
Shock A discontinuity that satisfies the
Rankine-Hugoniot conditions and the entropy
condition.
9
Solution to the Riemann problem
Assuming strict hyperbolicity (distinct
eigenvalues), and for each eigenvalue either
genuine nonlinearity (lk strictly increasing
along the corresponding eigenvector rk) or linear
degeneracy (lk invariant along rk), the Riemann
problem has a scale-invariant local solution
consisting of single waves.
Wk
non-admissible shock
W1
Wm
Rk rarefaction curve du/ds rk lk(uR) gt lk(uL)
x
uL
Sk- shock curve su F s(uRuL)
lk lk(uR) lt lk(uL)
d lk/ds D lk rk gt 0 s(lk) defined u(x,t)
u(s(x/t))
contact discontinuity
SkRk for linear degeneracy
s
uL
uR
uL
uR
uL
uR
10
Riemann problem for gas dynamics
left wave (rarefaction or shock)
right wave (rarefaction or shock)
Rankine-Hugoniot Conditions
Contact
x
0
11
Talk outline

• Mathematical theory of conservation laws
• Numerical methods for conservation laws
• Numerical methods for multiphase CFD
• Applications of multiphase CFD

12
Conservative numerical scheme
The numerical scheme must be in conservative form
Weak solution depends on the conservative form of
the equations.
13
Shock capturing scheme

• 1st order method has strong diffusion
• 2nd order method has spurious oscillation
• high resolution shock capturing method

14
MUSCL a Godunov-type shock capturing method
Monotone Upstream-centered Schemes for
Conservation Laws
Fully discretized, 2nd order accuracy in smooth
region
uM
uL
uR
15
Talk outline

• Mathematical theory of conservation laws
• Numerical methods for conservation laws
• Numerical methods for multiphase CFD
• Applications of multiphase CFD

16
Main ideas of front tracking
Front Tracking A hybrid of Eulerian and
Lagrangian methods

• Two separate grids to describe the solution
• A volume filling rectangular mesh
• An unstructured codimension-1 Lagrangian mesh to
  represent interface

• Major components
• Front propagation and redistribution
• Wave (smooth region) solution

• Advantages of explicit interface tracking
• No numerical interfacial diffusion
• Real physics models for interface propagation
• Different physics / numerical approximations in
  domains separated by interfaces

17
Level-set vs. front tracking method
Explicit tracking of interfaces preserves
geometry and topology more accurately.
5th order level set (WENO)
4th order front tracking (Runge-Kutta)
18
Propagation of a tracked front (1)
Operator splitting to separate normal and
tangential propagation

• Normal propagation move the front position and
  couple the interior wave solution to front
  dynamics.
• Tangential propagation complete the front
  evolution.

19
Propagation of a tracked front (2)
Normal propagation solve a non-local Riemann
problem by method of characteristics
contact
new front position and states
20
Conservative hyperbolic solver
SR dx/dt sR
SL dx/dt sL
Space-time Volume
Flux determined by MUSCL
Tracked interface
Flux determined by interface states, which
enforces conservation across the interface by R-H
conditions
21
2D grid based conservative front tracking (1)
Grid based spatial interface reconstruction
The 14 isomorphically distinct cases for
space-time volumes
Grid free
Grid based
22
2D grid based conservative front tracking (2)

• Space-time interface construction and
  triangulation
• Space-time volume construction
• Volume merging for CFL condition

23
The FronTier code

• FronTier is a parallel 3D multi-physics code
  based on front tracking
• Physics models include
• Compressible fluid dynamics
• MHD
• Flow in porous media
• Elasto-plastic deformations
• Realistic EOS models
• Exact and approximate Riemann solvers
• Phase transition models
• Adaptive mesh refinement

Interface untangling by the grid based method
24
FronTier MHD numerical schemes
Elliptic step
Hyperbolic step
Point Shift (top) or Embedded Boundary (bottom)

• Propagate interface
• Untangle interface
• Update interface states

• Apply hyperbolic solvers
• Update interior hydro states

• Calculate electromagnetic fields
• Update front and interior states

• Generate finite element grid
• Perform mixed finite element discretization
• or
• Perform finite volume discretization
• Solve linear system using fast Poisson solvers

25
Embedded Boundary Elliptic Solver

• Main Ideas
• Based on the finite volume discretization
• Domain boundary is embedded in the rectangular
  Cartesian grid, and the solution is treated as a
  cell-centered quantity
• Using finite difference for full cell and linear
  interpolation for cut cell flux calculation
• Advantage robust, readily parallelizable,
  compatible with FronTier grid-based interface
  tracking algorithm.

26
Talk outline

• Mathematical theory of conservation laws
• Numerical methods for conservation laws
• Numerical methods for multiphase CFD
• Applications of multiphase CFD

27
Main FronTier applications

• Rayleigh-Taylor instability

Supernova explosion
Richtmyer-Meshkov instability
Targets for future accelerators
Tokamak refuelling through the ablation of frozen
D2 pellets
Liquid jet break-up and atomization
28
Rayleigh-Taylor instability
Single mode, bubble and spike
Multimode
Courtesy of J. Grove, LANL
The growth rate predicted by FronTier agrees with
experiments, while the prediction from untracked
(TVD) simulations was about half.
29
Two models for bubbly and cavitating fluids
30
Theory on bubbly flows (1)
Mass and momentum conservation
The Keller equation for bubble growth
31
Theory on bubbly flows (2)
Dispersion relation
Sound speed in bubbly fluid (m/s)
cf
cg
c low frequency sound speed cf sound speed in
pure fluid wB resonant frequency d damping
coefficient
Bubble volume fraction b
32
Dispersion relation measured from simulations
Phase velocity
Attenuation rate
V (cm/ms)
a (dB/cm)
33
Shock wave propagation (1)
Shock speeds measured from the simulations are
within 10 deviation from the theoretical steady
state values.
Shock Profile

• The oscillation amplitude is smaller for gas with
  larger g.
• The oscillation period is longer for larger
  bubble volume fraction.

34
Shock wave propagation (2)
Shock profile of SF6 gas bubbles
Experiment
Simulation
The oscillation period is shorter than the
experimental value by 28.
Courtesy of Beylich Gülhan
35
SNS and cavitation mitigation
Courtesy of Oak Ridge National Laboratory
36
DNS of pressure wave propagation in the SNS target
Pure Mercury
Bubbly Mercury ( R1.0mm, b2.5 )
Statistical average of collapsing bubble pressure
predicts the mitigation efficiency of 32 for
injected air bubbles of radius 1mm and void
fraction 2.5.
37
Liquid Jet Breakup and Spray Formation

• Breakup Regimes
• Rayleigh breakup
• first wind-induced breakup
• second wind-induced breakup
• atomization
• Proposed breakup mechanism
• cavitation, partial liquid fuel evaporation

Inlet pressure fluctuation
Courtesy of S.P.Lin and R,D. Reitz
Boundary rearrangement effect
38
Previous work on the simulation of phase
transitions
Rayleigh (17) Simplified equation of motion for
inertia controlled growth of a spherical vapor
bubble. Menikoff etal. (89) Compressible fluids
with transition zone treated as the macroscopic
mixture of the two phases at equilibrium. Welch
(95) Interface tracking with mass transfer, the
interface assumed to be in thermal and chemical
equilibrium. Juric etal. (98) Simulation of
boiling flows in incompressible fluids. Hao etal.
(99) Dynamics of vapor bubbles in acoustic
pressure fields assuming the vapor was
saturated. Matsumoto etal. (94) Numerical study
on the influence of internal phenomena on vapor
bubble motion with complete conservation laws for
compressible fluids and the interfacial dynamics
of phase transitions
39
Front tracking with phase transitions
Governing equations
Rankine-Hugoniot conditions
Phase boundary conditions

• Mev evaporation rate
• k thermal conductivity
• L latent heat
• a evaporation coefficient

40
Phase boundary propagation

• Characteristic equations are solved with the
  boundary conditions using an iterative solver.
• A subgrid model was developed to account for thin
  thermal layers next to the phase boundary.

41
Validation (1) convergence to analytical solution
b 0 temperature wave decoupled ( t 1 )
(G, k, rcp, c, m)l (1, 1, 1, 1, 1),
(p, u, T)l0 (0, 0, 1) (G, k, rcp, c, m)r (½,
½, ½, ½, 1), (p, u, T)r0 (0, 0, 0).
42
Validation (2) comparison with contact
discontinuity
Pressure jump due to the mass flux across the
phase boundary
43
Validation (3) condensation under outgoing heat
flux
Early Stage
T0 20 C p0 Psat(T0) Tx0 -2.5 C/mm
t 30 ms
Late Stage
Convergence to steady states
44
Dynamic insertion of cavitation bubbles
45
Simulation of diesel jet atomization through a
nozzle

• Micron-size vapor bubbles created and collapsed
  dynamically
• Adaptive Mesh Refinement implemented for the
  axisymmetric simulations
• Opening angle too large if jet were treated as
  gas, no opening if treated as pure liquid

46
Comparison with experiments
Mass flux through a window located 1 mm from
nozzle exit
Tip velocity of the diesel jet
47
Applications of FronTier - MHD code
Neutrino Factory / Muon Collider target has been
proposed as a free mercury jet interacting with
an intensive proton pulse in a 20Tesla magnetic
field

• Tokamak applications
• Pellet ablation
• Striation instabilities
• Laser driven pellet acceleration
• Gyrotron driven pellet acceleration
• Plasma disruption mitigation

Laser ablation plasma plume
Laser driven pellet acceleration
Injection of a high speed gaseous jet
48
Summary

• The mathematical theory and numerical methods for
  systems of conservation laws.
• Conservative front tracking technique for
  multiphase CFD, and the FronTier-MHD code.
• Direct numerical simulations of acoustic and
  shock wave propagation in bubbly flows, and
  application to cavitation mitigation.
• Front tracking with phase transitions, code
  validations, and simulations of the diesel jet
  atomization.