No-Slip Boundary Conditions in Smoothed Particle Hydrodynamics

by

Frank Bierbrauer

Updating Fluid Variables

• In SPH fluid variables f are updated through interpolation about a given point (xa,ya) using information from surrounding points (xb,yb) .

• Each surrounding point is given a weight Wab with respect to the distance between point a and b.

abb

N

1b b

ba Wf

ρmf

Particle Deficiency

• Near a no-slip boundary there is a particle deficiency• Any interpolation carried out in this region will

produce an incorrect sum

Three Ways to Resolve the Particle Deficiency Problem

1. Insert fixed image particles outside the boundary a distance dI away from the boundary c.f. nearest fluid particle at distance dF

2. Insert fixed virtual particles within the fluid and in a direct line to the fixed image particles

– Avoids creation of errors when fluid and image particles are not aligned

3. Co-moving image particles with dI = dF

1 2 3

Velocity Update Using Image Particles

1. Fixed image approach: uI = uF+(1+dI /dF)(uW - uF)

2. Virtual image approach: uI = uV+(1+dI /dV)(uW - uV)

– Virtual velocities uV are created through interpolation

Velocity Update Using the Navier-Stokes Equations

• Update the velocity using the Navier-Stokes equations and a second order finite difference approximation to the velocity derivatives

At the no-Slip Wall (W)

Wy

yyxxyWyxt

Wx

yyxxxWyxt

)F-)vμ(vp()ρvvρuvv (ρ

)F-)uμ(up()ρvuρuuu (ρ

2b

bWb'yy2

b

bWb'yy δy

v2vvvδy

u2uuu

,

Navier-Stokes Equations

Finite-Difference Approximation at the wall

Velocity Update

x

WWxxWWxWxWWtW

b2b

bWWWW2

b

W

bWWW

2b

b'

F)(uμ)(p)(uu)(uρ

uδy 2

δyvρ2μuδyμ2

δyvρ2μδy 2u

y

WWxxWWyWxWWtW

b2b

bWWWW2

b

W

bWWW

2b

b'

F)(vμ)(p)(vu)(vρ

vy2

δyvρ2μvδyμ2

δyvρ2μδy 2v

Much of this reduces down as, in general, a no-slip wall has condition uW=(U0,0). Therefore, at the wall, ut= ux= uxx= v= vx= vxx= 0

The Viscoelastic Case

αβ

αβα

β

β

Fxσ

ρ1

DtDu,

xuρ

DtDρ

αβαβαβ τpδσ

The equations are ( = 1,2)

where

αβ

2αβαβ

1αβ dλdητλτ

,α

β

β

ααβ

xu

xud

β

ααβ

xuκ

Further Reduction

αβαβ

1

2αβ Sdλλητ

αβ

1

2

1

αβ

1

αγβγγβαγαβ

dλλ1

ληS

λ1-SκSκ

DtDS

Using

giving

At the Wall y

WWW22yW

21xWyWyxyy21

xWWW

12yW

11xWxWxyyy21

FρSSpu2vηλ

FρSSpvuηλ

Wxxxytx

WWxy

WxyxyW

WyyWyx

uρρvρρ1v

uρρuρ1vu

,

As well as

Non-Newtonian (elastic) Stress

12W

21W

Wy211

22W

1

22WWyW

22xWW

22t

Wy211

12W

1

12WWy

22WWyW

12xWW

12t

11W

1

21WWyW

11xWW

11t

SS

vλ1λ2ηS

λ1Sv2SuS

vλ1ληS

λ1SvSuSuS

Sλ1Su2SuS

Only have the velocity condition uW = (U0,0) as well as y=0

Must Solve

• Need ub’ and vb’ and W

• Need as well as St and• e.g.

221211 ,, WWW SSS 22y

12y S,S

Δt

SSS1nαβ,

Wnαβ,

WW

αβt

,δy

SSSb

αβb

αβW

Wαβy

b

αβb

αβb'

Wαβy δy 2

SSS

Density Update Equation

1n

Wy

1nWx

nW

1nWn

W

1n

WynW

1nWx

nW

1nW

nW

WyWWxWWt

vΔt1ρuΔtρ

ρ

vρρuΔtρρ

vρρuρ

Polymeric Stress Update Equations

1n

Wy211

n12,W

1

n12,W

1n

Wyn22,

W1n

Wy1n

W12xW

1-n12,W

n12,W

n11,W

1

n12,W

1n

Wy1n

W11xW

1-n11,W

n11,W

1n

Wy211

n22,W

1

n22,W

1n

Wy1n

W22xW

1-n22,W

n22,W

vλ1λη

Sλ1SvSuSu-

ΔtSS

Sλ1Su2Su-ΔtSS

vλ1λ2η

Sλ1Sv2Su-

ΔtSS

1n

W11xW

1-n11,W

1n

W12xW

1n

Wy211

1-n12,W

1n

W22xW

1n

Wy211

1-n22,W

n11,W

n12,W

n22,W

1

1n

Wy

1n

Wy1

1n

Wy

1n

Wy1

Su ΔtS

Su-vλ1ληΔtS

Su-vλ1λ2ηΔtS

S

S

S

λ1Δt1u Δt 2-0

0vλ1Δt1uΔt-

00v2λ1Δt1

n12,W

n21,W

1

1n

W11xW

n12,W

1n

Wy1-n11,

Wn11,W

1n

Wy1

n22,W

1n

Wy1n

W12xW

1n

Wy211

1-n12,W

n12,W

1n

Wy1

1n

W22xW

1n

Wy211

1-n22,W

n22,W

SS

λΔt1

Su-Su2ΔtSS

vλ1Δt1

SuSu-vλ1ληΔtS

S

v2λ1Δt1

Su-vλ1λ2ηΔtS

S

Velocity Update Equations

yWWW

22yW

21xWy

21Wxy

WWyy

xWWW

12yW

11xWx

21Wxxxytx

WWyy

FρSSpηλ

ρuρ1v

FρSSpηλ

uρρvρρ1u

1

1

yWW

b

22b

22W

W21x

21x

b

bb'

W2b

bWb'

xWW

b

12b

12W

W11xWx

21xxx

b

bb'tx

W2b

bWb'

Fρδy

SSSηλ

1ρδy 2

uuρ1

δyv2vv

Fρδy

SSSpηλ

1uρρδy 2

vvρρ1

δyu2uu

Solution for ub’ and vb’

n

W12y

12x

3nb

nWx

nW

n

W12y

11x

2nb

2nW

2nb

nWtx

nW21

ny,W

3nb

nWx

2nW21

nb

nb

nWx

nW21

nx,W

nW

nWx

2nb

2nW

nb

2nb

2

Wnx

2nW21

nW

2nb

nWxx

nW21

nW

2nb

n

W2x

2nW21

nb'

SSδyρ2ρSSδyρ4

δyρρλ 4ηFδyρρλ 2η

vδyρρλ 4ηFρpδyρ4

uδyρρ4ηλ

uδyρ2ρηλ4ρ

δyρρ4ηλ1u

n

W12y

11x

3nb

nWx

nW

n

W12y

12x

2nb

2

Wnx

3nb

nWtxx21

nx,W

nW

nWx

3b

nWx

nW

nb

nb

nWx

nW21

nW

2nb

nWxx

nW

nb

nWx21

ny,W

2nb

3nW

nb

2nb

2

Wnx

2nW21

2nb

n

W2x

2nW21

nb'

SSδyρ2ρ

SSδyρ4δyρρλ 2η

Fρpδyρ2ρuδyρρλ 4η

uδyρ2ρδyρλ 2ηFδyρ4

vδyρρ4ηλ

δyρρ4ηλ1v

If x = 0, 21 = 1, = S

yWW

2b

bb'

xWW

2b

bWb'

Fρμ

δyvv

Fρμ

δyu2uu

Equivalent Newtonian Update Equations

yWWWyWxy

WWyy

xWWWxWxxxytx

WWyy

Fρpμ1ρu

ρ1v

Fρpμ1uρρvρ

ρ1u

Giving

2bWtxW

yW

3bWx

2W

bbWxWx

WWWx2b

2W

b2bW

2x

2W

W2bWxxWW

2bW

2x

2W

b'

δyρρ 4μFδyρρ 2μ

vδyρρ 4μFρpδy4ρ

uδyρ4ρμ

uδyρ2ρμ4ρ

δyρ4ρμ1u

3bWtxx

xWWWx

3bWxWbbWxW

W2bWxxWbWx

yW

2b

3W

b2bW

2x

2W

2bW

2x

2W

b'

δyρρ2μ

Fρ-pδyρ2ρuδyρρ 4μ

uδyρ2ρδyρ2μFδy4ρ

vδyρ4ρμ

δyρ4ρμ1v