Th The Kelvin-Hel Helmholtz In Instability in SPH Terrence - - PowerPoint PPT Presentation

Th The Kelvin-Hel Helmholtz In Instability in SPH

Terrence Tricco CITA ttricco@cita.utoronto.ca http://cita.utoronto.ca/~ttricco

• 0.1
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• 0.06
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• 0.02

0.02 0.04 0.06 0.08 0.1 0.5 1 1.5 2 Bε xξ 0.5 1 1.5 2 xξ

Accuracy and Correctness of SPH and SPMHD

Propagation of magnetic waves (Tricco & Price 2013)

10-6 10-5 10-4 10-3 10-2 10-1 100 1 10 100 P(B) k Flash 1283 Flash 2563 Flash 5123 Phantom 1283 Phantom 2563 Phantom 5123

Spectra of magnetic energy in turbulence (Tricco, Price & Federrath 2016)

divB errors (Tricco & Price 2012)

Agertz et al (2007)

The KH Instability in SPH

Hopkins (2015) Hayward et al (2014) Agertz et al (2007)

The KH Instability in SPH

Hopkins (2015) Springel (2010)

All the Mixing (Moving mesh / meshless finite volume)

Mo’ mixing, Mo’ problems

• Robertson et al (2010), McNally et al (2012), Lecoanet et

al (2016) introduce KH tests with well-posed initial conditions that demonstrate convergence

Lecoanet et al (2016)

McNally et al (2012)

Mo’ mixing, Mo’ problems

• Robertson et al (2010), McNally et al (2012), Lecoanet et

al (2016) introduce KH tests with well-posed initial conditions that demonstrate convergence

The KH Tests of Lecoanet et al (2016)

• Two-dimensional tests with well-posed initial conditions
• Introduce a scalar “colour” field to measure degree of mixing
• Include physical dissipation, that is Navier-Stokes viscosity and

thermal conductivity (also colour diffusion!) – dissipation is numerically independent!

• Lecoanet et al (2016) show converged solutions between grid

(Athena) and spectral methods (Dedalus) in the non-linear regime

• 1
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0.5 1 0.5 1 1.5 2 vx y

• 0.005
• 0.004
• 0.003
• 0.002
• 0.001

0.001 0.002 0.003 0.004 0.005 0.5 1 1.5 2 vy y

x velocity y velocity colour

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 colour y

a = 0.05 σ = 0.2 A = 0.01 uflow = 1 P0 = 10 z1 = 0.5 z2 = 1.5

Initial Conditions

a = 0.05 σ = 0.2 A = 0.01

I am using Δ𝜍/𝜍 = 0 (uniform density)

uflow = 1 P0 = 10

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• 0.5

0.5 1 0.5 1 1.5 2 vx y

• 0.005
• 0.004
• 0.003
• 0.002
• 0.001

0.001 0.002 0.003 0.004 0.005 0.5 1 1.5 2 vy y

x velocity y velocity colour

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 colour y

z1 = 0.5 z2 = 1.5

Initial Conditions

t = 0 t = 4 t = 6 t = 2

Results of Lecoanet et al (2016)

converged here! converged here! converged here! (½ billion grid cells!)

Stratified KH Test of Lecoanet et al (2016)

• I am using the Re=105 unstratified (uniform density) KH test ( )
• Comparison to nx = 2048 Dedalus calculation (spectral code)
• Goal: obtain convergence of SPH results towards reference solution
• Resolution: nx = 256, 512, 1024, 2048 particles (~8 million)
• Dissipation Implementation: direct second derivative style for Navier-

Stokes viscosity, thermal conduction, and colour diffusion (efficiency, consistency)

SPH Simulations

t = 0 t = 4 t = 6 t = 2

SPH results (nx = 1024 particles)

t = 4 t = 2

SPH results (nx = 1024 particles)

t = 6 SPH Dedalus

• Define entropy for colour
• and total colour entropy

0.05 0.1 0.15 0.2 0.25 0.3 0.35 2 4 6 8 10 S t 256 512 1024 2048 2048 Dedalus

• Results are converging towards reference solution
• Numerical dissipation (artificial viscosity) still relevant up till nx = 1024
• r 2048, so don’t except convergence yet

Colour Entropy

0.01 0.1 1 256 512 1024 2048

∝ nx

• 1

t = 2 L2 Error nx

L2 error Convergence (t = 2)

0.01 0.1 1 256 512 1024 2048

∝ nx

• 1

t = 4 L2 Error nx

0.01 0.1 1 256 512 1024 2048

∝ nx

• 1

t = 2 L2 Error nx

L2 error Convergence (t = 4)

0.01 0.1 1 256 512 1024 2048

∝ nx

t = 6 L2 Error nx

0.01 0.1 1 256 512 1024 2048

∝ nx

• 1

t = 4 L2 Error nx 0.01 0.1 1 256 512 1024 2048

∝ nx

• 1

t = 2 L2 Error nx

L2 error Convergence (t = 6)

Kinetic Energy

0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 2 4 6 8 10 kinetic energy t 256 512 1024 2048

• Dissipation rate of kinetic energy not yet converged!
• Expected from analytic translation of artificial viscosity to physical dissipation

Cubic Spline Quartic Quintic Sextic Heptic t = 4

Quality of Smoothing Kernel Matters

Colour Entropy for Quintic Spline

0.05 0.1 0.15 0.2 0.25 0.3 0.35 2 4 6 8 10 S t 256 512 1024 2048 Dedalus

• SPH can activate the Kelvin-Helmholtz instability!

(that is, SPH can do hydrodynamics – not a surprise to anyone in this room)

• May need to use nx = 4096 to achieve formal convergence

(32 million particles – I hope not!)

• Currently running octic and nonic splines (R = 5h!) to check kernel bias

convergence.

• It may not be as difficult (resolution requirement, kernel bias) to activate KH

as found here for other conditions (i.e., Reynolds number).

• Not shown, but Wendland family of kernels demonstrate same behaviour.
• My belief is that SPH will converge to the agreed solution.

Conclusions