A solver for free-surface flow in heterogeneous porous media

Transcription

1 A solver for free-surface flow n heterogeneous porous meda Olver Oxtoby Johan Heyns Aeronautc Systems, Councl for Scentfc and Industral Research Pretora, South Afrca

2 Free-surface flow: Sloshng Smple small-ampltude slosh for valdaton Sngle baffle confguraton nterfoam suffers sgnfcant spurous dsspaton Reducton n spurous dsspaton due to pecewse lnear pressure dscretsaton Slde 2 Emprcal data from Dodge, The new dynamc behavour of lquds n movng contaners", 2000

3 Free-surface flow: Sloshng Pecewse-lnear pressure nterpolaton reduces parastc currents on nonorthogonal grds nterfoam Hydro

4 Free-surface flow: Sloshng Improvement n capture of volent sloshng 2 nd order Crank-Ncolson, HRAC nterface capturng scheme used here Popular commercal codes suffer nterface smearng

5 Weak compressblty Accounts for temporal varaton of gas densty Sutable for low mach number, hgh densty rato flows No computatonal penalty Varatons n ρ g Absolute pressures Slde 5

6 Objectves Porous modellng n OpenFOAM : Implemented as porous drag sourceterm and pressure-jump BCs Regons of constant porosty + thn porous baffles Regons hard-wred n mesh generaton process Am: Develop a solver for multphase porous flow Arbtrary varyng porosty feld + dscontnutes Easy to specfy: (funky)setfelds Flexble: slow tme-dependence, etc.

7 Governng equatons Volume averagng: porous + multphase Contnuty: Momentum: Volume fracton: u: Intrnsc velocty F: Porosty a: Volumefracton F: Body force porous drag (Ergun) + gravty: ) ( 2 ) ( ) ( 0 ) ( j j j j j j u x t F S x x p u u x u t u x F F F F F F F a a p p g u D u D F F F F F ) ( ) ( u j j j x u x u S 2 1

8 Equaton dscretsaton Densty dscontnuty Consstent dscontnutes n convectve & temporal terms Pressure gradent dscontnuty Porosty dscontnuty Dscontnuty n velocty Consstent dscontnutes n pressure & convectve terms Volume fracton t ( Fu ) ( Fu x j j p u ) F x Velocty

9 Equaton dscretsaton Handlng dscontnutes by correct and consstent nterpolaton of porosty, densty, pressure & velocty to cell faces Consstent treatment n Rhe-Chow pressure-projecton equaton: Φ ave ρ ave p = Φu Δt Φu u + Save ρ ave Consderng a 1D cell par gves us necessary condtons whch the nterpolatons must meet F 1 F 2 F ave =?

10 Condtons for consstency 1. Steady sngle-phase 1D flow 1 Φ ave = 1 w 1 Φ 1 + w 1 Φ 2 and 1 Φ ave Sf = 1 Φ 1 S Φ 2 S 2 (w = convectve weghtng) 2. Steady u, non-unform ρ ave = ρ f 3. Unsteady u, non-unform F 1 F 2 p f = Φ 1ρ 2 p 1 + Φ 2 ρ 1 p 2 ρ f Φu u 2 u 1 1 w ρ 2 wρ Δx(ρ 2 S 1 ρ 2 S 2 ) Φ 1 ρ 2 + Φ 2 ρ 1

11 1D Channel Two-flud 1D flow F 1 F 0.5 F 1 Strong lnear pressure profle Need least-squares gradent on non-orthogonal grds

12 Porous baffle benchmark Experment: Porous dam-break (P. Ln, 1998) Mesh ndependence Valdaton 10k elt structured vs 40k elt structured vs 10k elt unstructured Open top Water Crushed stone / glass beads Stone: F = 0.49, D p = 1.59 cm Glass: F = 0.39, D p = 3 mm

13 Porous baffle benchmark Free surface plot Crushed stone baffle Mesh ndependence Valdaton: No calbraton of coeffcents Coarse (1x0.5 cm) Fne (0.5x0.25 cm) Unstructured Experment

14 Harbour breakwater Packed bed of breakwater armour unts Packed stone F = 0.1 D p = 0.25 m Gravel F = 0.2 D p = 1 cm Armour unts F = 0.65 D p = 0.4 m

15 Harbour breakwater Packed bed of breakwater armour unts

16 Tappng of melt from FeS furnace Rough approxmaton to a typcal ferroslcon smelter Actve zone F = 0.2 Inactve zone F = 0.1; Dp = 1cm Input: Gas producton ~ 10 m 3 /s (below electrodes) Estmated porostes/partcle szes Taphole Crust F = 0.1 D p = 0.5cm Electrode crater Melt zone F = 1 F = 0.5; 0.2; 0.1

17 Tappng of melt from FeS furnace Taphole gassng metal heght? Input: Gas producton ~ 10 m 3 /s Estmated porostes/partcle szes Valdaton : Crater pressure ~ 10 kpa Mass outflow rate ~ 10 kg/s Metal heght ~2 cm above taphole

18 Surface Tenson Smoothng used to amelorate spurous currents n surface-tenson domnated flows Addtonal smooth volume fracton * * a a a a 0 x x Normalsed smoothng parameter (relatve to mesh) no tunng requred nterfoam Hydro Water droplet transported n mneral ol n 100 mcron channel

19 Flud-structure nteracton Free-surface + 6DOF sold floatngblock tutoral Acceleraton relaxaton (nterfoam) Atken acceleraton (Hydro) Slde 19

20 Flud-structure nteracton Statonary bobbng boat Non-orthogonalty causes nterfoam blowup

21 Thank you Acknowledgements: Johan Heyns Development n collaboraton wth Engys Contact: Olver Oxtoby ooxtoby@csr.co.za