Ahmad Shakibaeinia Assistant Professor Department of Civil, Geological & Mining Engineering Polytechnique Montreal

Transcription

1 Natonal Center for Atmospherc Research (NCAR) IMAGe TOY2017: Workshop on Multscale Geoscence Numercs Ahmad Shakbaena Assstant Professor Department of Cvl, Geologcal & Mnng Engneerng Polytechnque Montreal May 2017

2 Flud flow n nature Often volent & extreme Challengng characterstcs: Large deformatons/ fragmentatons Volent flow structure, hghly erosve Often nvolve nteracton of more than one phase (multphase flow) wave breakng Athabasca Falls Photo from: slverdoctors.com Photo from:

3 Numercal modellng of flud flow How to represent computatonal doman? Mesh-based vs Mesh-free Mesh representaton (adaptve mesh) Mesh adaptablty and connectvty Physcal Phenomena Mesh representaton (fxed mesh) complcated nterface trackng technques Sharp nterfaces

4 Numercal modellng of flud flow How to represent comput. doman? Mesh-based vs Mesh-free Mesh representaton (adaptve mesh) Mesh adaptablty and connectvty Physcal Phenomena Mesh representaton (fxed mesh) complcated nterface trackng technques Sharp nterfaces node/partcle Mesh-free (Partcle) representaton No connectvty Handles any deformaton & fragmentaton Smple, yet natural and powerful

5 Numercal modellng of flud flow How to represent comput. doman? Mesh-based vs Mesh-free Mesh representaton (adaptve mesh) Mesh adaptablty and connectvty Physcal Phenomena Mesh representaton (fxed mesh) complcated nterface trackng technques Sharp nterfaces node/partcle Mesh-free (Partcle) representaton No connectvty Handles any deformaton & fragmentaton

6 What are partcle methods How to solve the governng equaton on these partcles? Mesh-free partcle methods (Fully Lagrangan) Movng Partcle Sem-mplct method (MPS) Smoothed Partcle Hydrodynamcs (SPH) Molecular dynamcs (MD) Dscrete Element Method (DEM),

7 Smoothed partcle hydrodynamcs (SPH) Smoothed partcle hydrodynamcs (SPH) Lucy 1977, Gngold & Monaghan 1977 Drac delta functon 0 x / 2 ( x) lm 0 1 x / 2 Useful propertes of ( x) dx 1 delta functon f ( x ) d f x Partton of unty Kronecker delta propery Drac delta functon can not be used for nterpolaton because t s a Generalzed functon. (lacks some requred propertes for a well behaved functon such as contnuty and dfferentablty) To used the useful propertes of delta functon, the man ngredent of SPH s to choose a smooth kernel functon to mmc the valuable part of Drac delta functon.

8 Smoothed partcle hydrodynamcs (SPH) Interpolaton ' f r f r W r r ' d r ' Gradent N j1 f j1 rj Wr rj m f ( r) f ( r ') W r r ' dv N rj Wr rj f Laplacan 2 f ( r) 2 f ( r ') W r r ' dv N j1 j j m 2 rj Wr rj f j j m j j

9 MPS (Movng Partcles Sem-mplct) MPS (Movng Partcles Sem-mplct) method s based on a local weghted averagng of quanttes and dervatves. Interpolaton f f W r, r V j Laplacan j j e j 2 0 n j 2 0 n j j j Gradent & Dvergence formula d f f j f W r, r 0 ej j e n r d j j j f 0 n j rj f W r, r j j e W r, r 2d f f f W r r 2d f f f f f e W j j, e j W rj, re r, r j j e j e n j W r, r r e j e j Partcle number densty r e f j f j f j r e f f r j j r j r j f j 1 f 1 f 2 2 f 2 f f f. f r f 2r r f j 2 j j 2 j 2 j j 2 j 2 r r r r j r j rj rj r j rj f 2 j f j f f 2 j 2 rj

10 Sngle-phase flows Conservaton of mass and momentum n the Lagrangan frame D u 0 Dt Du Dt Dr u Dt 2 p u f MPS approxmaton of spatal dervatves u d u u r n r r 0 2 j j j 2d u u u 2 0 n j d p p r j j p W r, r j n r r 0 2 j j j j j 0 2 j j j W j W rj, re r, r d p pˆ r j j p W r, r j n r r j e e e

11 Sngle-phase flows Tme ntegraton: a 2-step predcton-correcton algorthm * k t 2 u u u f 1p ' where 1 k 1 u ' t p k 1 * u u u For an ncompressble flud, Dρ/Dt s equal to zero. Therefore, by keepng partcle number densty constant, the contnuty equaton s satsfed. k k1 * k1 * ' const. n n n' const. 1 ' 1 n' u' u' 0 0 t n t

12 Sngle-phase flows Pressure Calculaton Fully Incompressble method FI-MPS 1 n ' u ' 0 n t u ' c p 2 n1 1 k 1 t p 2 n1 0 c n n p * 1 0 t Weakly compressble MPS (WC-MPS) 0 0 n * p 2 Numercal value of 2 b sound speed c0 n n p (pa) Explct relaton: EOS max V Vb FL,, ; L 0 W( r ) p n n W r1 12 j 0 * 1 j 1 ( * 21) p W r W r j n n j W( r 2... ( N 1) N) td... 0 * W( rn( N1) ) WrNjp n n N N j Posson eq. : Solved usng one of the matrx teratve solvers Ma 0 2 WC-MPS FI-MPS Exact soluton t (s) Pressure gradent d p pˆ j p W r, r 0 e n r j j j j e 12

13 Sngle-phase flows Neghborng search strategy All-par search Lnked-cell search

14 Sngle-phase flows Test case: Poseulle flow n F ( 1) d F (2n 1) y (2n 1) u( y, t) ( y d ) cos exp t n0 2d 4d 2n 1 v( y, t) 0 14

15 0.018 m Sngle-phase flows Test case : 2D Dam-break 1.5 m/s Gate 0.15 m 0.38 m IJNMF, 2010

16 Sngle-phase flows Test case: 3D Dam-break wave httng structure H 2 H 1 Board 1.00 m Water y z 0.40 m x 1.22 m 1.25 m 0.75 m 0.55 m H1 H2

17 Sngle-phase free-surface flows Example: 3D Dam-break wave httng structure

18 Sngle-phase flows Test case: Hydraulc jump Reproduced from Chanson (2009) 18

19 Sngle-phase flows Test case: Hydraulc jump Reproduced from Chanson (2009) 19

20 Sngle-phase flows Test case: Vortex flow 5m/s d 5d

21 Multphase flows Multphase challenges Vscosty dfference Densty dfference Phase nteracton Flud 1: 1, μ 1 Reproduced from Flud 2: 2, μ 2 nterface Gas-lqud Multphase Large densty dfference Instablty Surface tenson Frequent topology change n dfferent scales (spray, bubble, slug, ) Water: ρ 1000 kg/m 3 Wave break Ar: ρ 1 kg/m 3 Reproduced from Petroff (1993)

22 Multphase flows A mult-densty mult-vscosty partcle model D Contnuty : u 0 Dt Du Momentum : p τ f Dt Equaton of state : p f( ) Dr Lagrangan moton : u D t Densty dscontnuty Large densty dscontnuty Sudden pressure change across the nterface resultng n nstablty Soluton: Usng the smoothed value of densty for pressure calculaton p 1 p ρ I ρ II d 0 n j Transton regon Densty dscontnuty s automatcally taken care of j j ej W rj, re r j r Phase I Transton regon Interface Phase II

23 Multphase flows A mult-densty mult-vscosty partcle model Vscosty dscontnuty Du p τ f Dt T 2 τ u u u u u 2 D Interface 2d τ 0 u u n j W r, r j j j e Phase I u k Phase II j = = j Interacton vscosty 1 2 j j j Mn(, k) < k= k < Max(, k) Two-layer Poseulle flow 23

24 Multphase flows A mult-densty mult-vscosty partcle model Surface tenson Du Dt 2 p s s u f f fs n Phase II n Phase I Contnuum Surface Force (CSF) model N W ( rj, re ) Normal vector: n ; N C jj N C ; J j : Phase I j I W ( rj, re ) Curvature: j n MPS approxmaton: d Cj C C j W rj, r e ρ = 100 n 0 j r j σ = 1 d n n n 0 j r j j N e n ej W rj, re Volume fracton ρ = 100 Analytcal perod 24

25 Multphase flows Test case: K-H nstablty Downtown Vctora, 28 Nov CMAME, 2012

26 Multphase flows Test case: rsng bubble r 0 =0.25 8r 0 A 2D crcular bubble, rsng n an ambent flud under ts densty dfference Densty rato: 1:10 Vscosty rato: 1:10 g=1 σ=25 partcles wth sze of r 0 2r 0 4r 0 g=

27 Multphase flows Example: Two-phase dam-break Water 1.5 m/s m m PEO soluton m CMAME, 2012

28 Multphase flows: Gas-lqud Test case: Hydraulc jump Reproduced from Chanson (2009) 28 Two-phase model Sngle-phase model IAHR, 2015

29 Multphase granular flows What s granular flow Flow of macroscopc small grans (e.g. sand) Crtcal role n geophyscal & hydro-envronmental processes landsldes, eroson, debrs flow, mnng slurres, sedment transport,... Submarne-landslde Landslde Debrs flow Mnng talng slurres Reservor flushng Photo from: thechnatmes.com Photo from: Rver bank falure Dam-break Photo from: slverdoctors.com

30 Multphase granular flows Complex mechancs Behavor: sold? lqud? gas? Non-everythng Multphase: even more complex (effect of ambent flud) sold regme Sand Water gaseous regme lqud regme

31 Multphase granular flows Numercal models for multphase granular flows Sngle-phase +Emprcal/ sememprcal relatons) Multphase granular models (sedments as a separate phase) Sedment phase as contnuum Sedment phase as dscrete Mesh-based Euleran methods (FEM, FVM, ) No scalablty ssue (computatonally affordable) Problem dealng the nterfacal deformaton and fragmentatons Mesh-free methods (e.g. MPS, SPH) No scalablty ssue Can deal wth any deformaton/fragmentaton Mesh-free Lagrangan methods (DEM, MD) Good for n-depth analyss Scalablty ssue (computatonally expensve

32 Multphase granular flows A mesh-free partcle model for multphase granular flows: flud partcles Water ambent flud nterface 0<ϕp <1 Sand ϕ g ϕ p grans pore flud granular partcles 0 ϕ 0 1 ϕ D Contnuty : u 0 Dt Du Momentum : p τ f Dt Equaton of state : p f( ) Dr Lagrangan moton : u D t Devatorc stress tensor: Water phase Newtonan τ E u 2 eff Granular phase Non-Newtonan flud Rheologcal model

33 Multphase granular flows A mesh-free partcle model for multphase granular flows: eff eff y 0 E 2 E 0 N 1 sp 2 s p E 2 E I / I 1 Herschel Bulkley (H B) 1 µ (I) Rheology D Contnuty : u 0 Dt Du Momentum : p τ f Dt Equaton of state : p f( ) Dr Lagrangan moton : u D t Devatorc stress tensor: Water phase Newtonan τ E u 2 eff Granular phase Non-Newtonan flud Rheologcal model

34 y/h Multphase granular flows Test case: Vscoplastc Poseulle flow p 0 h y 0 u 0 yeld lne y x p 0 + p h u(y) yeld lne l τy=1.0 (analytcal) τy=0.6 (analytcal) τy=0.0 (analytcal) τy=1.0 (numercal) τy=0.6 (numercal) τy=0.0 (numercal)

35 Multphase granular flows Test case: Sand jet

36 Multphase granular flows Test case: Sand jet AWR, 2012

37 Multphase granular flows Example applcaton: Landsldes JCP, 2016 falure lne

38 Multphase granular flows Test case: Submarne landsldes 0.65 m 0.10 m 0.10 m Sand 0.65 m 1.60 m y 45 x 4.00 m

39 Challenges & opportuntes Partcle methods: Very promsng methods, have the potental to become tomorrow's numercal tools Stll relatvely young Numercal stablty Unphyscal fluctuatons Conservaton Performance Stll relatvely computatonally expensve Boundary condtons how to nclude BCs wthout loosng conservaton propertes? How to nclude BCs for complex geometres? Applcablty to new problems Complex geometres? Many multphase flows

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