Th The Kelvin-Hel Helmholtz In Instability in SPH Terrence Tricco CITA ttricco@cita.utoronto.ca http://cita.utoronto.ca/~ttricco
Accuracy and Correctness of SPH and SPMHD divB errors (Tricco & Price 2012) Spectra of magnetic energy in turbulence Propagation of magnetic waves (Tricco & Price 2013) (Tricco, Price & Federrath 2016) 0.1 10 0 0.08 10 -1 0.06 0.04 10 -2 0.02 P(B) B ε 0 10 -3 -0.02 Flash 128 3 10 -4 Flash 256 3 -0.04 Flash 512 3 -0.06 Phantom 128 3 10 -5 Phantom 256 3 -0.08 Phantom 512 3 -0.1 10 -6 0 0.5 1 1.5 2 0 0.5 1 1.5 2 1 10 100 x ξ x ξ k
The KH Instability in SPH Agertz et al (2007)
The KH Instability in SPH Agertz et al (2007) Hopkins (2015) Hayward et al (2014)
All the Mixing (Moving mesh / meshless finite volume) Springel (2010) Hopkins (2015)
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)
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 McNally et al (2012)
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
x velocity Initial Conditions 1 0.5 v x 0 -0.5 -1 0 0.5 1 1.5 2 y y velocity 0.005 0.004 0.003 0.002 0.001 v y 0 -0.001 -0.002 -0.003 -0.004 -0.005 0 0.5 1 1.5 2 y colour 1 0.8 colour 0.6 u flow = 1 z 1 = 0.5 a = 0.05 0.4 P 0 = 10 z 2 = 1.5 σ = 0.2 0.2 A = 0.01 0 0 0.5 1 1.5 2 y
x velocity Initial Conditions 1 0.5 v x 0 -0.5 I am using Δ 𝜍 / 𝜍 = 0 (uniform density) -1 0 0.5 1 1.5 2 y y velocity 0.005 0.004 0.003 0.002 0.001 v y 0 -0.001 -0.002 -0.003 -0.004 -0.005 0 0.5 1 1.5 2 y colour 1 0.8 colour 0.6 u flow = 1 z 1 = 0.5 a = 0.05 0.4 P 0 = 10 z 2 = 1.5 σ = 0.2 0.2 A = 0.01 0 0 0.5 1 1.5 2 y
Results of Lecoanet et al (2016) t = 0 t = 6 t = 2 t = 4
Stratified KH Test of Lecoanet et al (2016) converged here! converged here! converged here! (½ billion grid cells!)
SPH Simulations • I am using the Re=10 5 unstratified (uniform density) KH test ( ) • Comparison to n x = 2048 Dedalus calculation (spectral code) • Goal: obtain convergence of SPH results towards reference solution • Resolution: n x = 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 results (n x = 1024 particles) t = 0 t = 6 t = 2 t = 4
SPH results (n x = 1024 particles) t = 2 t = 4 SPH Dedalus t = 6
Colour Entropy 0.35 • Define entropy for colour 0.3 0.25 • and total colour entropy S 0.2 0.15 256 512 1024 0.1 2048 2048 Dedalus 0.05 0 2 4 6 8 10 t • Results are converging towards reference solution • Numerical dissipation (artificial viscosity) still relevant up till n x = 1024 or 2048, so don’t except convergence yet
L2 error Convergence (t = 2) 1 t = 2 L2 Error 0.1 ∝ n x -1 0.01 256 512 1024 2048 n x
L2 error Convergence (t = 4) 1 t = 2 1 t = 4 L2 Error 0.1 ∝ n x -1 L2 Error ∝ n x -1 0.01 256 512 1024 2048 0.1 n x 0.01 256 512 1024 2048 n x
L2 error Convergence (t = 6) 1 t = 2 1 t = 6 L2 Error 0.1 ∝ n x -1 L2 Error 0.01 ∝ n x 0 256 512 1024 2048 0.1 n x 1 t = 4 L2 Error ∝ n x -1 0.1 0.01 256 512 1024 2048 n x 0.01 256 512 1024 2048 n x
Kinetic Energy 0.92 0.9 0.88 kinetic energy 0.86 0.84 0.82 256 512 0.8 1024 2048 0.78 0 2 4 6 8 10 t • Dissipation rate of kinetic energy not yet converged! • Expected from analytic translation of artificial viscosity to physical dissipation
Quality of Smoothing Kernel Matters t = 4 Quartic Quintic Cubic Spline Sextic Heptic
Colour Entropy for Quintic Spline 0.35 0.3 0.25 S 0.2 0.15 256 512 0.1 1024 2048 Dedalus 0.05 0 2 4 6 8 10 t
Conclusions • 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 n x = 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.
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