A solver for free-surface flow in

heterogeneous porous media

Oliver Oxtoby Johan Heyns

Aeronautic Systems,

Council for Scientific and Industrial Research

Pretoria, South Africa

© CSIR 2014 Slide 2

Free-surface flow: Sloshing

• Simple small-amplitude slosh for validation

• Single baffle configuration

• interFoam suffers significant spurious dissipation

• Reduction in spurious dissipation due to piecewise

linear pressure discretisation

Empirical data from Dodge, “The new dynamic behaviour of

liquids in moving containers", 2000

© CSIR 2014

Free-surface flow: Sloshing

• Piecewise-linear pressure

interpolation reduces

parasitic currents on non-

orthogonal grids

interFoam Hydro

© CSIR 2014

Free-surface flow: Sloshing

• Improvement in capture of

violent sloshing

• 2nd order Crank-Nicolson,

HiRAC interface capturing

scheme used here

• Popular commercial codes

suffer interface smearing

© CSIR 2014 Slide 5

Weak compressibility

• Accounts for temporal variation of gas density

• Suitable for low mach number, high density ratio flows

• No computational penalty

Variations in ρg Absolute pressures

© CSIR 2014

Objectives

Porous modelling in OpenFOAM®:

• Implemented as porous drag sourceterm and

pressure-jump BCs

• Regions of constant porosity + thin porous

baffles

• Regions ‘hard-wired’ in mesh generation

process

Aim: Develop a solver for multiphase porous flow

• Arbitrary varying porosity field +

discontinuities

• Easy to specify: (funky)SetFields

• Flexible: slow time-dependence, etc.

© CSIR 2014

Governing equations

Volume averaging: porous + multiphase

• Continuity:

• Momentum:

• Volume fraction:

u: Intrinsic velocity F: Porosity a: Volume-

fraction

F: Body force – porous

drag (Ergun) + gravity:

)(

2)()(

0)(

j

j

iij

ji

ij

j

i

i

i

uxt

FSxx

puu

xu

t

ux

F

FF

FF

F

F

aa

ii

p

i

p

i guD

uD

F F

F

F

F ||

)1(75.1

)1(150

23

2

u

i

j

j

iij

x

u

x

uS

2

1

© CSIR 2014

Equation discretisation

Density discontinuity

• Consistent discontinuities in

convective & temporal terms

• Pressure gradient discontinuity

Porosity discontinuity

• Discontinuity in velocity

• Consistent discontinuities in

pressure & convective terms

Velocity

Volume fraction

FF

F

i

ij

j

ix

puu

xu

t)()(

© CSIR 2014

Equation discretisation

• Handling discontinuities by correct and

consistent interpolation of porosity,

density, pressure & velocity to cell faces

• Consistent treatment in Rhie-Chow

pressure-projection equation:

• Considering a 1D cell pair gives us

necessary conditions which the

interpolations must meet

F1 F2

Fave = ?

𝛻 ∙Φ𝑎𝑣𝑒

𝜌𝑎𝑣𝑒𝛻𝑝 = 𝛻 ∙

Φ𝒖𝒊

Δ𝑡− Φ𝒖 ∙ 𝛻𝒖 +

𝑺𝑎𝑣𝑒

𝜌𝒂𝒗𝒆

© CSIR 2014

Conditions for consistency

1. Steady single-phase 1D flow

2. Steady u, non-uniform

3. Unsteady u, non-uniform

F1 F2 ⟹ 𝜌𝑎𝑣𝑒 = 𝜌𝑓

⟹ 1

Φ𝑎𝑣𝑒 = 1 − 𝑤1

Φ1+ 𝑤

1

Φ2 and

1

Φ𝑎𝑣𝑒 𝑆𝑓 =1

Φ1𝑆1 +

1

Φ2𝑆2

⟹ 𝑝𝑓= Φ1𝜌2𝑝1 + Φ2𝜌1𝑝2 − 𝜌𝑓Φ𝑢 𝑢2 − 𝑢1 1 − 𝑤 𝜌2 − 𝑤𝜌1 + 0.5Δ𝑥(𝜌2𝑆1 − 𝜌2𝑆2)

Φ1𝜌2 + Φ2𝜌1

(w = convective

weighting)

© CSIR 2014

Two-fluid 1D flow

Strong linear pressure profile

→ Need least-squares gradient on non-orthogonal grids

1D Channel

F 1 F 0.5 F 1

© CSIR 2014

Porous baffle benchmark

Experiment: Porous dam-break

(P. Lin, 1998)

• Mesh independence

• Validation

Water

Crushed stone / glass beads

Open top

Stone: F = 0.49, Dp = 1.59 cm

Glass: F = 0.39, Dp = 3 mm

10k elt structured vs

40k elt structured vs

10k elt unstructured

© CSIR 2014

Porous baffle benchmark

• Mesh independence

• Validation: No

calibration of

coefficients

Free surface plot –

Crushed stone baffle

Coarse (1x0.5 cm)

Fine (0.5x0.25 cm)

Unstructured

Experiment

© CSIR 2014

Harbour breakwater

Packed bed of breakwater armour units

Armour units

F = 0.65

Dp = 0.4 m

Packed stone

F = 0.1

Dp = 0.25 m

Gravel

F = 0.2

Dp = 1 cm

© CSIR 2014

Harbour breakwater

Packed bed of breakwater armour units

© CSIR 2014

Tapping of melt from FeSi furnace

Inactive zone

F = 0.1; Dp = 1cm Active zone

F = 0.2

Crust

F = 0.1

Dp = 0.5cm

Electrode crater

F = 1 Melt zone

F = 0.5; 0.2; 0.1

Taphole

Input:

• Gas production

~ 10 m3/s

(below electrodes)

• Estimated

porosities/particle

sizes

Rough approximation to a

typical ferrosilicon smelter

© CSIR 2014

Tapping of melt from FeSi furnace

Input:

• Gas production

~ 10 m3/s

• Estimated

porosities/particle

sizes

‘Validation’:

• Crater pressure

~ 10 kPa

• Mass outflow rate

~ 10 kg/s

Taphole gassing – metal height?

Metal height ~2 cm above taphole

© CSIR 2014

Surface Tension

Water droplet transported in mineral oil in 100 micron channel

• Smoothing used to ameliorate spurious

currents in surface-tension dominated flows

• Additional smooth volume fraction

• Normalised smoothing parameter (relative to

mesh) – no tuning required

0**

ii xx

a

aaa

interFoam Hydro

© CSIR 2014 Slide 19

Fluid-structure interaction

• Free-surface + 6DOF solid

• floatingBlock tutorial

Aitken acceleration (Hydro) Acceleration relaxation (interFoam)

© CSIR 2014

Fluid-structure interaction

• Stationary bobbing boat

• Non-orthogonality causes

interFoam blowup

Acknowledgements:

• Johan Heyns

• Development in collaboration with Engys

Contact:

Oliver Oxtoby

ooxtoby@csir.co.za

Thank you