Daniel Price Monash University Melbourne, Australia. Guillaume Laibe University of St Andrews, Scotland

Transcription

1 Dust+gas in SPH en.wikipedia.org Daniel Price Monash University Melourne, Australia Guillaume Laie University of St Andrews, Scotland Image: Gemini Oservatory/ AURA Artwork y Lynette Cook HST/ 10th SPHERIC meeting, 16th-18th June 2015, Parma, Italy,

2 Dust + Gas: A simple eample of a two-fluid miture n Two fluids coupled y a drag + r. ( gv g ) = + r. ( dv d ) = +(v g.r) v g +(v d.r) v d = rp g + K (v d v g )+f, g g K (v d v g )+f, d

3 Stopping time K=0.01 t stop d g K( d + g ) K=100

4 Two fluid dust+gas in SPH Monaghan & Kocharyan (1995), Monaghan (1997), Maddison et al. (2003), Laie & Price (2012a, MNRAS) a g = X i d = X j m W ( r a r,h a ), m j W ( r i r j,h i ), dv a g dt = X m " P a a 2 r a W a (h a )+ g,a P a 2 r a W a (h ) g, # X j m j v aj ˆr aj ( a + j )t s aj D aj (h a ), dv i d dt du dt = = X X j P a 2 g,a v i ˆr i m ( i + )t s D i (h ), i X m v a r a W a (h a ) (v aj ˆr aj ) 2 m j ( a + j )t s D aj (h a ), aj Doule hump kernel Two sets of particles coupled y drag terms

5 Two prolems with two fluids

6 1) Overdamping prolem No drag=no damping SPH=eact Intermediate drag = strong damping in oth SPH + eact Red=analytic solution for dust/gas waves derived y Laie & Price (2011) MNRAS 418, 1491 High drag = no damping ut SPH strongly damped

7 Overdamping prolem: Resolution Criterion Laie & Price, 2012, MNRAS 420, 2345 Temporal: t<t stop (can e fied with implicit timestepping methods) Spatial: h<t stop c s (much more difficult to fi) t stop! 0 (K!1) implies t! 0! 0 n Require infinite timesteps AND infinite resolution in the ovious limit of perfect coupling!

8 2) Dust trapping prolem y [AU] 20 0 t=3413 yrs gas dust 0 log column density [g/cm 2 ] [AU] [AU] Fig. 2. Gas (left) and dust (right) column density in a two-fluid simulation Dust particles feel no pressure, can ecome of material oriting in a protoplanetary disc. Once the dust smoothing length ecomes smaller than the typical gas smoothing length (solid redcircleshows a 2h for a representative gas particle) the dust particles ecome artificial trapped the in gas, highforming density rings, artificial due to the lack structures of mutual repulsion etween SPH dust particles. `trapped if they fall elow the resolution length of

9 ONE FLUID TO RULE THEM ALL LAIBE & PRICE (2014A,B,C)

10 TWO BECOME ONE A phoeni from the ashes n One Two miture fluids with coupled a differential y a drag velocity term d d / + r. ( gv g ) = 0, dt = (r v), v = dv d + g v d + r. ( dv d ) = 0, dt = 1 r [ (1 ) v], v = v d v +(v rp g g.r) v g = + K (v d v g )+f, dt = rp 1 r [ (1 ) v v]+f, g g d@v d +(v v K = d.r) v+ rp d = ( v (v d r)v v+ g )+f, 1 dt g d 2 r (2 1) v 2. t s No approimations! Laie & Price (2014) MNRAS = d + g

11 SPH one fluid method ρ a = m W a (h a ), (26) dϵ a dt = m [ ϵa (1 ϵ a ) Ω a ρ a v a a W a (h a ) + ϵ (1 ϵ ) Ω ρ v a W a (h ) ], (27) P ] Ω ρ 2 W a (h ) ONE set of particles representing the miture dv a dt d v a dt du a dt = [ Pa m Ω a ρ 2 W a (h a )+ a m [ ϵa (1 ϵ a ) v a Ω a ρ a v a. W a (h a ) + ϵ (1 ϵ ) v Ω ρ v W a (h ) + f a, (28) = v a t s,a ρ a ρ g,a [ Pa m Ω a ρ 2 W a (h a )+ a P Ω ρ m (v a v ) v a. W a (h a ) ρ a Ω a + 1 [ m (1 2ϵa ) va 2 2ρ a Ω a ] ] W a (h ), (1 2ϵ ) v] 2 Wa (h a ), (29) = P a m (v g,a v g, ) W a (h a ) Ω a ρ a ρ g,a ϵ a m (u a u ) v a. W a (h a ) Ω a ρ a + ϵ a v 2 a t s,a. (30) Fig. 4. Solution to a d solution with the one fl solution in the limit wh now correctly repr oth strong and we It is intuitive why t prolem with the tw was related to the length etween gas there is no physica dust particles e phases. (Very!) ca this, since to visua simply made two c one with the gas p and one with the d (18). Afurtherconfirm resolution issue is s of a shock in the m in this case should with the shock pro to the weight of t method gives resul solution (red line), particles in one dim solution with the t It is also intuitiv solves the dust tra of dust particles, resolution of the g is now tied to the separate for each ph

12 Tests of one fluid method Laie & Price (2014) no drag strong drag

13 The diffusion approimation for dust Laie & Price (2014) Price & Laie (2015) MNRAS 451, 5332

14 Terminal velocity approimation d = (r v) dt dv dt = rp + f d dt = 1 r ( t srp ) Use SPH second derivative Valid when t stop < t Laie & Price (2014)

15 Diffusion approimation: tests Fall of a layer of dust from Monaghan (1997), JCP t=0 t=1 t=2 t=3 t=4 t=0 t=1 t=2 t=3 t= y Diffusion method Two fluid ρ ρ Fig. 8. Fall of a layer of dust in a stratified atmosphere, comparing results with the diffusion method (left) to the two fluid approach(right). d dt = X m a ( a t s,a + t s, )(P a P )F a Figure 8 compares the results using the diffusion approimation (left) to the results otained with the standard two fluid method Eplicit (right). The onetimestepping! fluid method enefits from the regularisation of the miture particles y the gas pressure,

16 Settling of grains in a protoplanetary disc

17 Some issues Epsilon can go negative. Solution: if (eps < 0) eps = 0. Found prolem with harmonic mean: d dt = X m a 4Da D D a + D Weird discretisation in du/dt - is it correct? (P a P )F a du a dt = 1 X 2(1 a ) a m (u a u )(D a + D )(P a P )F a

18 APPENDIX A: PROOF THAT EQUATION 46 IS A DISCRETE FORM OF EQUATION 4 Here, we prove that the epression otained for the second term in Eq. 46 y enforcing the conservation o 1 m (u a u )(D a + D )(P a P ) F a 2(1 ϵ a )ρ a ρ r a, is indeed a discrete form of the corresponding term in Eq. 4, i.e. ϵt s ρ g P u. We proceed, following Price (2012), y identifying 2F a / r a as equivalent to the second derivative of a 2 Y a 2F a r a. It may e shown straightforwardly that this new kernel Y a indeed satisfies the normalisation conditions derivative (see Price 2012 for more details). We can then take thelaplacianofthestandardsphsummati A A a m Y a, ρ to give 2 A a m A ρ 2 Y a. By writing (A1) in the form 1 m (u 4ρ a a u )(D a + D )(P a P ) 2 Y a, g ρ

19 Summary New general method for dusty gas with SPH Small grains/strong drag = usual SPH equations + evolution equation for dust fraction Widely applicale Refs: Two fluid: Laie & Price (2012a,) MNRAS One fluid: Laie & Price (2014a,,c) MNRAS Diffusion method: Price & Laie (2015) MNRAS