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

ρ

mf

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

v2vvv

δy

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

δyvρ2μu

δy

μ2

δyvρ2μ

δy 2u

y

WWxxWWyWxWWtW

b2b

bWWWW2

b

W

bWWW

2b

b'

F)(vμ)(p)(vu)(vρ

vy2

δyvρ2μv

δy

μ2

δyvρ2μ

δy 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

Fx

σ

ρ

1

Dt

Du,

x

uρ

Dt

Dρ

αβαβαβ τpδσ

The equations are ( = 1,2)

where

αβ

2αβαβ

1αβ dλdητλτ

,α

β

β

ααβ

x

u

x

ud

β

ααβ

x

uκ

Further Reduction

αβαβ

1

2αβ Sdλ

λητ

αβ

1

2

1

αβ

1

αγβγγβαγαβ

dλ

λ1

λ

ηS

λ

1-SκSκ

Dt

DS

Using

giving

At the Wall

y

WWW

22yW

21xWyWyxyy21

xWWW

12yW

11xWxWxyyy21

FρSSpu2vηλ

FρSSpvuηλ

Wxxxytx

WWxy

WxyxyW

WyyWyx

uρρvρρ

1v

uρρuρ

1vu

,

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

vλ1λ

2ηS

λ

1Sv2SuS

vλ1λ

ηS

λ

1SvSuSuS

Sλ

1Su2SuS

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

SSS

1nαβ,W

nαβ,W

Wαβt

,δy

SSS

b

αβb

αβW

W

αβy

b

αβb

αβb'

W

αβy δy 2

SSS

Density Update Equation

1n

Wy

1n

WxnW

1nWn

W

1n

WynW

1n

WxnW

1nW

nW

WyWWxWWt

vΔt1

ρuΔtρρ

vρρuΔtρρ

vρρuρ

Polymeric Stress Update Equations

1n

Wy211

n12,W

1

n12,W

1n

Wyn22,

W1n

Wy

1n

W12xW

1-n12,W

n12,W

n11,W

1

n12,W

1n

Wy

1n

W11xW

1-n11,W

n11,W

1n

Wy211

n22,W

1

n22,W

1n

Wy

1n

W22xW

1-n22,W

n22,W

vλ1λ

η

Sλ

1SvSuSu-

ΔtSS

Sλ

1Su2Su-ΔtSS

vλ1λ

2η

Sλ

1Sv2Su-

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-vλ1λ

ηΔtS

Su-vλ1λ

2ηΔtS

S

S

S

λ

1Δt1u Δt 2-0

0vλ

1Δt1uΔt-

00v2λ

1Δt1

n12,W

n21,W

1

1n

W11xW

n12,W

1n

Wy1-n11,

Wn11,W

1n

Wy1

n22,W

1n

Wy

1n

W12xW

1n

Wy211

1-n12,W

n12,W

1n

Wy1

1n

W22xW

1n

Wy211

1-n22,W

n22,W

SS

λΔt

1

Su-Su2ΔtSS

vλ1

Δt1

SuSu-vλ1λη

ΔtS

S

v2λ1

Δt1

Su-vλ1λ2η

ΔtS

S

Velocity Update Equations

yWWW

22yW

21xWy

21Wxy

WWyy

xWWW

12yW

11xWx

21Wxxxytx

WWyy

FρSSpηλ

ρuρ

1v

FρSSpηλ

uρρvρρ

1u

1

1

yWW

b

22b

22W

W21x

21x

b

bb'

W2b

bWb'

xWW

b

12b

12W

W11xWx

21xxx

b

bb'tx

W2b

bWb'

Fρδy

SSS

ηλ

1ρ

δy 2

uu

ρ

1

δy

v2vv

Fρδy

SSSp

ηλ

1uρρ

δy 2

vvρ

ρ

1

δy

u2uu

Solution for ub’ and vb’

n

W

12y

12x

3nb

n

WxnW

n

W

12y

11x

2nb

2nW

2nb

n

WtxnW21

ny,W

3nb

n

Wx

2nW21

nb

nb

n

WxnW21

nx,W

nW

n

Wx

2nb

2nW

nb

2nb

2

Wnx

2nW21

nW

2nb

n

WxxnW21

nW

2nb

n

W2x

2nW21

nb'

SSδyρ2ρSSδyρ4

δyρρλ 4ηFδyρρλ 2η

vδyρρλ 4ηFρpδyρ4

uδyρρ4ηλ

uδyρ2ρηλ4ρ

δyρρ4ηλ

1u

n

W

12y

11x

3nb

n

WxnW

n

W

12y

12x

2nb

2

Wnx

3nb

n

Wtxx21

nx,W

nW

n

Wx3b

n

WxnW

nb

nb

n

WxnW21

nW

2nb

n

WxxnW

nb

n

Wx21ny,

W

2nb

3nW

nb

2nb

2

Wnx

2nW21

2nb

n

W2x

2nW21

nb'

SSδyρ2ρ

SSδyρ4δyρρλ 2η

Fρpδyρ2ρuδyρρλ 4η

uδyρ2ρδyρλ 2ηFδyρ4

vδyρρ4ηλ

δyρρ4ηλ

1v

If x = 0, 21 = 1, = S

yWW

2b

bb'

xWW

2b

bWb'

Fρμ

δyvv

Fρμ

δyu2uu

Equivalent Newtonian Update Equations

yWWWyWxy

WWyy

xWWWxWxxxytx

WWyy

Fρpμ

1ρu

ρ

1v

Fρpμ

1uρρvρ

ρ

1u

Giving

2bWtxW

yW

3bWx

2W

bbWxWx

WWWx2b

2W

b2bW

2x

2W

W2bWxxWW

2bW

2x

2W

b'

δyρρ 4μFδyρρ 2μ

vδyρρ 4μFρpδy4ρ

uδyρ4ρμ

uδyρ2ρμ4ρ

δyρ4ρμ

1u

3bWtxx

xWWWx

3bWxWbbWxW

W2bWxxWbWx

yW

2b

3W

b2bW

2x

2W

2bW

2x

2W

b'

δyρρ2μ

Fρ-pδyρ2ρuδyρρ 4μ

uδyρ2ρδyρ2μFδy4ρ

vδyρ4ρμ

δyρ4ρμ

1v