Prediction of Fluid Dynamics in The Inertial Confinement Fusion Chamber

by Godunov Solver With Adaptive Grid Refinement

Zoran Dragojlovic, Farrokh Najmabadi, Marcus Day

Accomplishments• IFE Chamber dynamics code SPARTAN is developed: Simulation of Physics by Algorithms based on Robust

Turbulent Approximation of Navier-Stokes Equations.• SPARTAN current features:

– 2-D Navier Stokes equations, viscosity and thermal conductivity included

– arbitrary geometry– adaptive mesh refinement

• SPARTAN tests:– Acoustic wave propagation.– Viscous channel flow.– Mach reflection.– Analysis of discretization errors to find code accuracy.

• Initial conditions from BUCKY code are used for simulations.• Two Journal articles on SPARTAN are in preparation.

• IFE Chamber dynamics code SPARTAN is developed: Simulation of Physics by Algorithms based on Robust

Turbulent Approximation of Navier-Stokes Equations.• SPARTAN current features:

– 2-D Navier Stokes equations, viscosity and thermal conductivity included

– arbitrary geometry– adaptive mesh refinement

• SPARTAN tests:– Acoustic wave propagation.– Viscous channel flow.– Mach reflection.– Analysis of discretization errors to find code accuracy.

• Initial conditions from BUCKY code are used for simulations.• Two Journal articles on SPARTAN are in preparation.

Governing Equations of Fluid Flow

2-D Navier-Stokes equations in conservative form:

Solution vector:

Governing Equations of Fluid Flow

Flux terms in x and y direction:

Viscosity

In general, depends on (T, Z)

T – temperature, Z – average ionization stage

2 extreme cases: neutral gas and fully ionized gas on a temperature range (10,000-60,000)K

neutral gas:

2.21 1016– T

2.5Ai

0.5

Z4 ln

---------------------kgms-------=

fully ionized gas: := (4.9 x 10-11 – 4.4 x 10-9) Pas

:= (2.6 x 10-4 – 6.55 x 10-4) Pas

Thermal Conductivity

Thermal conductivity depends on (T, Z), as well as viscosity.

2 extreme cases: neutral gas and fully ionized gas on a temperature range (10,000-60,000)K

neutral gas:

fully ionized gas:

:= (6.2 x 10-2 – 0.156) W/(mK)

k 5.86 1010– T

2.5

Z ln------------

WmK---------= := (0.022 – 1.94) W/m-K

Godunov Method

Introduced in 1959, as a finite volume method with a special method of upwinding.

Uses solution to a 1-D Riemann problem in order to estimate fluxes at the interface between cells.

Formulation of Riemann problem:

1-D governing equation

initial condition

U(x,0) does not necessarily satisfy the conservation laws, breaks into

fans, shocks and contact discontinuities.

Godunov Method

Solution procedure for Riemann problem uses Hugoniot jump relations and second law or thermodynamics to estimate pressure at the discontinuity, speed of propagation of discontinuities and the corresponding values of state variables.

Numerical application:

x

u,t

j j+1j-1

wave diagram for Riemann problem

uj+1uj

Uj-1

Adaptive Mesh Refinement• Motivation: efficient grid distribution results in reasonable CPU

time.• Grid organized into levels from coarse to fine.• Solution tagging based on density and energy gradients.• Grid is refined at every time step.• Solution interpolated in space and time between the grid levels.• Referenced in: Almgren et al., 1993.

• Motivation: efficient grid distribution results in reasonable CPU time.

• Grid organized into levels from coarse to fine.• Solution tagging based on density and energy gradients.• Grid is refined at every time step.• Solution interpolated in space and time between the grid levels.• Referenced in: Almgren et al., 1993.

chamber beam channel

wall

Example of Adaptive Mesh Refinement

maxmin

geometry

density contour plot

“forbidden” domain

fluid domain

embedded boundary

Embedded Boundary Algorithm

Arbitrary geometry imposed onto regular grid.

This results in formation of cells irregular in shapes and sizes.

Conservative update of irregular cells needs to be consistent and stable.

Embedded Boundary Algorithm

Conservative update of irregular cells, consistent with split formulation of governing equations:

FwestFeast

Fsouth

FnorthFB

Embedded Boundary Algorithm

irregular cell

cell for nonconservative update

fluid domain irregular cell’s

neighborhood

non-conservative update

preliminary update

Error Analysis

0.00E+00

2.00E-03

0.0000 8.0000

Reflection of shock waves from the walls of cylindrical chamber was studied.

Initial condition imposed by rotation of 1-D BUCKY solution about the center of the chamber.

The wave was propagated for 0.001s, until it reflected from the wall and started converging back towards the center.

Error Analysis

Two cases considered:non-diffusive flow (m, k=0.0) and diffusive flow (m, k given by Sutherland law).

Error analysis was done to determine the influence of the viscous and thermal diffusion on the accuracy of the solution.

The errors were estimated by 4th order Richardson extrapolation.

Error Analysis

density

no diffusion

min=3.68e-5 kg/m3

max=1.19e-3 kg/m3

density

diffusive flow

min=3.69e-5 kg/m3

max=2.09e-3 kg/m3

error

no diffusion

avg. = 3%

520x520 grid

error

diffusive flow

avg. = 3.1%

520x520 grid

Error Analysis

X-momentum

no diffusion

min= -1.67kg/(m2s)

max= 1.63kg/(m2s)

X-momentum

diffusive flow

min=-1.07kg/(m2s)

max=0.96kg/(m2s)

error

no diffusion

avg. = 3.6%

520x520 grid

error

diffusive flow

avg. = 3.2%

520x520 grid

Error Analysis

pressure

no diffusion

min=50.01 Pa

max=1.262e3 Pa

pressure

diffusive flow

min=50.42 Pa

max=1.24e3 Pa

error

no diffusion

avg. = 1.57%

520x520 grid

error

diffusive flow

avg. = 1.52%

520x520 grid

Error Analysis

energy

no diffusion

min=130.59J

max=4.492e3J

energy

diffusive flow

min=131.19J

max=3.66e3 J

error

no diffusion

avg. = 1.86%

520x520 grid

error

diffusive flow

avg. = 1.7%

520x520 grid

Validation Problemscylindrical shock reflection from rectangular geometry

10.1 11.2 12.1 12.9 r[m]

initial disturbance

Mach reflection

Viscous Flow Through the Channel

IFE Chamber Dynamics Simulations

Objectives• Determine the influence of the following factors on the chamber

state at 100 ms:- viscosity- blast position in the chamber- heat conduction from gas to the wall.

• Chamber density, pressure, temperature, and velocity distribution prior to insertion of next target are calculated.

Objectives• Determine the influence of the following factors on the chamber

state at 100 ms:- viscosity- blast position in the chamber- heat conduction from gas to the wall.

• Chamber density, pressure, temperature, and velocity distribution prior to insertion of next target are calculated.

Numerical Simulations

IFE Chamber Simulation• 2-D cylindrical chamber with a laser beam channel on the side.• 160 MJ NRL target• Boundary conditions:

– Zero particle flux, Reflective velocity– Zero energy flux or determined by heat conduction.

• Physical time: 500 s (BUCKY initial conditions) to 100 ms.

IFE Chamber Simulation• 2-D cylindrical chamber with a laser beam channel on the side.• 160 MJ NRL target• Boundary conditions:

– Zero particle flux, Reflective velocity– Zero energy flux or determined by heat conduction.

• Physical time: 500 s (BUCKY initial conditions) to 100 ms.

Numerical Simulations

Initial Conditions• 1-D BUCKY solution for density, velocity and temperature at

500 s imposed by rotation and interpolation.• Target blast has arbitrary location near the center of the

chamber.• Solution was advanced by SPARTAN code until 100 ms were

reached.

Initial Conditions• 1-D BUCKY solution for density, velocity and temperature at

500 s imposed by rotation and interpolation.• Target blast has arbitrary location near the center of the

chamber.• Solution was advanced by SPARTAN code until 100 ms were

reached.

0.00E+00

2.00E-03

0.0000 8.0000

Effect of Viscosity on Chamber State at 100 ms

inviscid flow at 100 ms

pressure, pmean = 569.69 Pa

temperature, Tmean = 5.08 104 K

(CvT)mean = 1.412 103 J/m3

pressure, pmean = 564.87 Pa

temperature, Tmean = 4.7 104 K

(CvT)mean = 1.424 103 J/m3

viscous flow at 100 ms

Viscosity makes a difference due to it’s strong dependence on temperature.

Viscosity makes a difference due to it’s strong dependence on temperature.

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Effect of Blast Position on Chamber State at 100ms

pressure, pmean = 564.87 Pa

temperature, Tmean = 4.7 104 K

centered blast at 100 mspressure, pmean = 564.43 Pa

temperature, Tmean = 4.74 104 K

eccentric blast at 100 ms

Large disturbance due to eccentricity of blast and small numerical disturbances have the same effect after 100 ms.

Large disturbance due to eccentricity of blast and small numerical disturbances have the same effect after 100 ms.

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Effect of Blast Position on Chamber State at 100 ms

0

1000

2000

3000

4000

5000

6000

0 0.02 0.04 0.06 0.08 0.1

time [s]

pre

ss

ure

[P

a]

centered blasteccentric blast

pressure at the wall pressure at the mirror

•Mirror is normal to the beam tube.

•Pressure is conservative by an order of magnitude.

•Pressure on the mirror is so small that the mechanical response is negligible.

•Mirror is normal to the beam tube.

•Pressure is conservative by an order of magnitude.

•Pressure on the mirror is so small that the mechanical response is negligible.

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

time [s]

pre

ss

ure

[P

a]

centered blasteccentric blast

Effect of Wall Heat Conduction on Chamber State at 100 ms

pressure, pmean = 564.431 Pa

temperature, tmean = 4.736 104 K

insulated wall

pressure, pmean = 402.073 Pa

temperature, tmean = 2.537 104 K

wall conduction

Ionized gas or plasma makes a difference by the means of heat conduction.

Ionized gas or plasma makes a difference by the means of heat conduction.

maxmin

pressure

temperature

density

Chamber Gas Dynamicsmaxmin

Prediction of chamber condition at long time scale is the goal of chamber simulation research.

Chamber dynamics simulation program is on schedule. Program is based on: Staged development of Spartan simulation code. Periodic release of the code and extensive simulations while development

of next-stage code is in progress.

Chamber dynamics simulation program is on schedule. Program is based on: Staged development of Spartan simulation code. Periodic release of the code and extensive simulations while development

of next-stage code is in progress.

Documentation and Release of Spartan (v1.0) Two papers are under preparation

Exercise Spartan (v1.x) Code Use hybrid models for viscosity and thermal conduction. Parametric survey of chamber conditions for different initial conditions

(gas constituent, pressure, temperature, etc.) Need a series of Bucky runs as initial conditions for these cases. We should run Bucky using Spartan results to model the following

shot and see real “equilibrium” condition. Investigate scaling effects to define simulation experiments.

Documentation and Release of Spartan (v1.0) Two papers are under preparation

Exercise Spartan (v1.x) Code Use hybrid models for viscosity and thermal conduction. Parametric survey of chamber conditions for different initial conditions

(gas constituent, pressure, temperature, etc.) Need a series of Bucky runs as initial conditions for these cases. We should run Bucky using Spartan results to model the following

shot and see real “equilibrium” condition. Investigate scaling effects to define simulation experiments.

Several upgrades are planned for Spartan (v2.0)

Numeric:

Implementation of multi-species capability: Neutral gases, ions, and electrons to account for different thermal

conductivity, viscosity, and radiative losses.

Physics:

Evaluation of long-term transport of various species in the chamber (e.g., material deposition on the wall, beam tubes, mirrors) Atomics and particulate release from the wall; Particulates and aerosol formation and transport in the chamber.

Improved modeling of temperature/pressure evolution in the chamber: Radiation heat transport; Equation of state; Turbulence models.

Numeric:

Implementation of multi-species capability: Neutral gases, ions, and electrons to account for different thermal

conductivity, viscosity, and radiative losses.

Physics:

Evaluation of long-term transport of various species in the chamber (e.g., material deposition on the wall, beam tubes, mirrors) Atomics and particulate release from the wall; Particulates and aerosol formation and transport in the chamber.

Improved modeling of temperature/pressure evolution in the chamber: Radiation heat transport; Equation of state; Turbulence models.