essential features of moving particle with pressure mesh€¦ · mppm moving particle method with...

1

Essential Features of Moving Particle with Pressure Mesh

黃耀新 (Yao-Hsin Hwang) 國立高雄海洋科技大學輪機工程系 National Kaohsiung Marine University Department of Marine Engineering

2

Merits of particle method

Proceeds without topological connection among computational nodes (meshless)

Lagrangean treatment on convection term Easy particle manipulations (addition and/or

deletion) Versatility in engineering problems

3

MPPM

Moving particle method with embedded pressure mesh (MPPM) –

Incompressible Flow Computations

4

Features of existing schemes

SPH, MPS, FVPM…. Material particles (mass point, global mass

conservation) Operators realized with randomly particle

cloud Artificial compressibility or projection

method for continuity constraint

5

Disadvantages of existing schemes

Inaccurate (inconsistent) operator realization (particle smoothing)

Incommensurate particle distribution (invariable length scale)

Incapable particle management Assignment of boundary condition Computationally inefficient (Pressure

equation)

6

Motivation: role of pressure

Continuity constraints – Langragian or Eulerian?

Governing equation without convection term Two types of computational particles:

Langragian velocity/Eulerian pressure Velocity particle liberated from mass

constraint

7

Essentials of present proposition (1)

Inserted pressure mesh to realize pressure-related operators

Background mesh and length scale Projection method for continuity constraint Local mass conservation Particle as observation point

8

Essentials of present proposition (2)

No particle management constraint Feasible non-uniform distribution No special boundary treatment Constant and diagonally dominant

coefficient matrix of pressure equation

9

Governing equations

Continuity

Momentum

0∇ ⋅ =u

2D pDt

∇= − + ν∇

ρu u

10

Projection method

1. Intermediate step

2. Pressure step

3. Final step

* n 2 np p pt ,= + ∆ ν∇u u u * n *

p p pt= + ∆r r u

2 n 1 *pt

+ ρ∇ = ∇ ⋅

∆u

n 1pn 1 *

p p

pt ,

++ ∇

= − ∆ρ

u u 2

n 1 * n 1p p p

t p+ +∆= − ∇

ρr r

11

Pressure mesh Continuity

Facial velocity splitting

n 1 n 1 n 1 n 1e w n s(u u ) y (v v ) x 0+ + + +− ∆ + − ∆ =

n 1 n 1n 1 * E Pe e

t p pu ux

+ ++ ∆ −

= −ρ ∆

12

Pressure equation

Facial velocity interpolation

n 1 n 1 n 1 n 1P E N W

n 1 * * * *S e w n s

x y y y x2( )p p p py x x x yx p [(u u ) y (v v ) x]y t

+ + + +

+

∆ ∆ ∆ ∆ ∆+ = + +

∆ ∆ ∆ ∆ ∆∆ ρ

+ − − ∆ + − ∆∆ ∆

* *e p pe e pe e

p pu u (r , r ) / (r , r )= ω ω∑ ∑

13

Pressure gradient

Shape function

P E

N NE

p( , ) p (1 )(1 ) p (1 )p (1 ) p

ξ η = − ξ − η + ξ − η+ − ξ η + ξη

P E P(x x ) /(x x )ξ = − −

P N P(y y ) /(y y )η = − −

14

Velocity Diffusion

Particle smoothing

j i ji22i

j iji jij i

( )4r≠

≠

φ − φ ω∇ φ =

ω ∑∑

ee

e

r r r1(r, r ) r

otherwise0

≤−ω =

15

Pure convection

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

φ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1UDSKUD2UDQUICKparticle method

θ=15o

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

φ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


θ=30o

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

φ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


θ=45o

16

Pure convection

x

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

φ

-0.5

0.0

0.5

1.0

1.5

2.0

2.5


17

Lid-driven cavity

Particle velocity Particle distribution

Streamlines at various computation times Reference streamlines

18

Lid-driven cavity

Re=1000

19

Lid-driven cavity

Recirculation flow rate

20 Vertical velocity

Horizontal velocity

u

y

0.0

0.2

0.4

0.6

0.8

1.0

nC=40nC=80Ghia et al.Erturk et al.

0.0 0.0 0.0 0.0 0.0 0.0-0.5 0.5 1.0

Re=10

0

Re=4

00

Re=1

000

Re=2

500

Re=5

000

Re=1

0000

x0.0 0.2 0.4 0.6 0.8 1.0

v

nC=40nC=80Ghia et al.Erturk et al.

Re=100

Re=5000

Re=2500

Re=1000

Re=400

Re=10000

-1.0

0.0

0.0

0.0

0.0

0.0

0.5

-0.5

0.0

Lid-driven cavity

21

Weird consequences

Pronounced solution variations in low-Reynolds number flows

Inaccurate results with denser particle distribution

Inaccurate diffusion operators?

22

Analysis of Laplacian operator

j i ji j i jix2 2

j i j iji ji ji jij i j i

2j i ji j i ji

y xx2 2j i j iji ji

2j i j i ji j i ji

xy yy2 2j i j iji ji

( ) (x x )4 4 [r r

(y y ) (x x )1r 2 r

(x x )(y y ) (y y )1 ]r 2 r

≠ ≠≠ ≠

≠ ≠

≠ ≠

φ − φ ω − ω≈ φ

ω ω

− ω − ω+φ + φ

− − ω − ω+φ + φ

∑ ∑∑ ∑

∑ ∑

∑ ∑

Inconsistency: O(1/ )δ

23

Smoothing difference

gradient model

2 3 2q p q p q p qp q p q p qp q p qp q p q p qp

qp x y xx xy2 2 2 2q p q p q p q p q pqp qp qp qp qp qp

2q p q p qp

yy 2q p qp

x x (x x ) (x x )(y y ) (x x ) (x x ) (y y )1( )r r r r 2 r r

(x x )(y y )1 HOT2 r

≠ ≠ ≠ ≠ ≠

≠

φ − φ − − ω − − ω − ω − − ωω = φ + φ + φ + φ

− − ω+ φ +

∑ ∑ ∑ ∑ ∑

∑

2 2q p q p q p q p qp q p qp q p q p qp

qp x y xx2 2 2q p q p q p q pqp qp qp qp qp

2 3q p q p qp q p qp

xy yy2 2q p q pqp qp

y y (x x )(y y ) (y y ) (x x ) (y y )1( )r r r r 2 r

(x x )(y y ) (y y )1 HOTr 2 r

≠ ≠ ≠ ≠

≠ ≠

φ − φ − − − ω − ω − − ωω = φ + φ + φ

− − ω − ω+φ + φ +

∑ ∑ ∑ ∑

∑ ∑

24


Laplacian model 3 2 4 3

q p q p q p qp q p q p qp q p qp q p q p qp2qp x y xx xy2 4 4 4 4

q p q p q p q p q pqp qp qp qp qp qp

2 2q p q p qp

yy 4q p qp

x x (x x ) (x x ) (y y ) (x x ) (x x ) (y y )1( )r r r r 2 r r

(x x ) (y y )1 HOT2 r

≠ ≠ ≠ ≠ ≠

≠

φ − φ − − ω − − ω − ω − − ωω = φ + φ + φ + φ

− − ω+ φ +

∑ ∑ ∑ ∑ ∑

∑

2 2q p q p q p q p q p qp q p q p qp

qp x y2 4 4q p q p q pqp qp qp qp qp

3 2 2q p q p qp q p q p qp

xx xy4 4q p q pqp qp

3q p q p qp

yy 4q p qp

x x y y (x x ) (y y ) (x x )(y y )( )( )

r r r r r

(x x ) (y y ) (x x ) (y y )12 r r

(x x )(y y )1 HOT2 r

≠ ≠ ≠

≠ ≠

≠

φ − φ − − − − ω − − ωω = φ + φ

− − ω − − ω+ φ + φ

− − ω+ φ +

∑ ∑ ∑

∑ ∑

∑

2 3 2 2q p q p q p q p qp q p qp q p q p qp2

qp x y xx2 4 4 4q p q p q p q pqp qp qp qp qp

3 4q p q p qp q p qp

xy yy4 4q p q pqp qp

y y (x x )(y y ) (y y ) (x x ) (y y )1( )r r r r 2 r

(x x )(y y ) (y y )1 HOTr 2 r

≠ ≠ ≠ ≠

≠ ≠

φ − φ − − − ω − ω − − ωω = φ + φ + φ

− − ω − ω+φ + φ +

∑ ∑ ∑ ∑

∑ ∑

25


difference equation:

=Cq b

Tx y xx xy yy( , , , , )= φ φ φ φ φq

O( )δ

26

Three-point stencil

(0) 0,Φ = (1) 1,Φ = *( )Φ δ = φ

δ

φx

−60

−40

−20

0

20

PSSD and exact

δ

δ

-1.0 -0.8 -0.6 -0.4 -0.2

φx

−20

0

20

40

60

δ

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

φ*=2.

φ*=-0.5

φ*=0.5

φ*=-2. δ

φxx

−20

0

20

40

60

PSSD and exact

δ

δ

-1.0 -0.8 -0.6 -0.4 -0.2

φxx

−60

−40

−20

0

20

δ

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

φ*=2.

φ*=-0.5

φ*=0.5

φ*=-2.

Gradient Laplacian

27


One-dimensional case i

C

(i 1) ( 0.5)xn

− + α ξ −=

a=0.3: (a) nC=8; (b) nC=16.

28


29


Two-dimensional case

(a)particle; (b)exact (c)PS; (d)SD.

30


31

Pure conduction

32

Pure conduction

33

Cavity flow

time0 20 40 60 80 100 120

ψmin

PS (nC=40)SD (nC=40)Ghia et al.

Re=100

Re=400

Re=1000

-0.05

-0.10

-0.10

-0.10

-0.15

34

Cavity flow

35

Cavity flow

PS (t 100)= SD (t 100)= FV

36

Cavity flow

PS (t 100 105)= − SD (t 100 105)= − FV

37

Cavity flow

38

Backward facing step flow

39


40


41


42


43


44

CPU time

45

Axisymmetric: pure convection

r1.3 1.4 1.5 1.6 1.7

φ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactnC=8nC=16nC=32nC=64

46

Axisymmetric: developing pipe flow

47

Axisymmetric: developing pipe flow

u/um

r/r0

0.0

0.2

0.4

0.6

0.8

1.0

Re=200Re=300Re=400Sparrow et al.

0 0.5 1

x*=0.001

x*=0.005

x*=0.02

x*=0.03

x*=0.05

x*=0.08

x*=0.12

48

Axisymmetric flow: confined impinging jet

x0.0 0.2 0.4 0.6 0.8 1.0

u

0.0

0.5

1.0

1.5

2.0

Chatterjee and Deviprasathpresent

49

Axisymmetric flow: sudden expansion

time0 20 40 60 80 100

Lr

0

2

4

6

8

10

Re=20.6

55.477.6

109

156.1

211.1

Re0 50 100 150 200 250

Lr

0

2

4

6

8

10

12

Hammad et al. (exp.)present (∆r=0.05)

50

Axisymmetric flow: sudden expansion

x0 2 4 6 8 10 12 14 16

uc

0.0

0.5

1.0

1.5

2.0

2.5

Hammad et al. (exp.)present (∆r=0.05)present (∆r=0.025)

Re=20.6 156.1

211.1

55.4 77.6109

time0 20 40 60 80 100

Lr

7

8

9

10

11

∆r=0.05

∆r=0.025

Hammad et al. (upper, exp.)

Hammad et al. (lower, exp.)

51

CPM

Characteristic particle method (CPM) – Hyperbolic Equation Systems

52

緣起 Water hammer project Godunov-type finite-volume

method Particle method

53

Governing Equations (1)

2H H a VV 0X g Xτ

∂ ∂ ∂+ + =

∂ ∂ ∂

V V HV g J 0X Xτ

∂ ∂ ∂+ + + =

∂ ∂ ∂

K /a1 ( K / E )( D / e )

=+

ρ = =f V

J V V2D

Λ

54


h h vv 0t x x

∂ ∂ ∂+ + =

∂ ∂ ∂

v v hv vt x x

λ∂ ∂ ∂+ + = −

∂ ∂ ∂

55


t x∂ ∂

+ =∂ ∂V VA S

hv

=

V

v 11 v

=

A

0vλ

= −

S

56

Characteristic Analyses(1)

−= 1A RΛR

= −

1 11 1

R+

= −

v 1 00 v 1

Λ

− − −∂ ∂+ =

∂ ∂1 1 1

t xV VR ΛR R S

+−

−

+ = = = −

1 ( h v ) / 2w( h v ) / 2w

W R V

h w wv w w

+ −

+ −

+ = = = −

V RW

57


−∂ ∂+ =

∂ ∂1

t xW WΛ R S

w w( v 1) ( w w ) / 2t x

λ+ +

+ −∂ ∂+ + = − −

∂ ∂

w w( v 1) ( w w ) / 2t x

λ− −

+ −∂ ∂+ − = −

∂ ∂

1 v / 2 ( w w ) / 2v / 2 ( w w ) / 2

λ λλ λ

+ −−

+ −

− − − = = −

R S

58


= +dx v 1dt

dw w wdt 2 2

λ λ++ −+ =

= −dx v 1dt

dw w wdt 2 2

λ λ−− ++ =

(i) Along the characteristic line of

,

(ii) Along the characteristic line of ,

59

特徵線法MOC

60

Particle Method Moving along characteristic line No convection term error Adaptive Dual particles Particle interactions

61

Regular Particles (Transmitted) 1. For the right-running characteristic,

2. For the left-running characteristic,

n 1 n np p p

x x ( v 1) t+ + ++ = + + ∆

,n 1 ,n ,n t / 2 ,np p p p

w ( w w )e wλ∆+ + + +

+ + + − − −= − +

n 1 n np p p

x x ( v 1) t− − −+ = + − ∆

,n 1 ,n ,n t / 2 ,np p p p

w ( w w )e wλ∆− − − −

− + − + − += − +

62

Regular Particles (Interpolated) (i) For the right-running characteristic,

(ii) For the left-running characteristic,

n 1 n 1 n 1 n 1R p p L,n 1 ,n 1 ,n 1n 1 n 1 n 1 n 1p L RR L R L

x x x xw w w

x x x x− + + −

+ − −

− − − −

+ + + +

− + − + − ++ + + +

− −= +

− −

n 1 n 1 n 1 n 1R p p L,n 1 ,n 1 ,n 1n 1 n 1 n 1 n 1p L RR L R L

x x x xw w w

x x x x+ − − +

− + +

+ + + +

+ + + +

+ + + + + ++ + + +

− −= +

− −

63

Regular Particles

64

Particle Management (i) Addition,

(ii) Merge,

65

Shock Particle (1) (i) Conservation form

(ii) Rankine-Hugoniot relation

het x

∂ ∂+ =

∂ ∂U F S

h

h

ee v

=

U

h

h 2 h

e ve v e

=

+ F

hL R R L( ) x [ ( ) ( )] t e x tδ δ δ δ− + − =U U F U F U S

L R L R( ) ( ) ( )σ − = −U U F U F U

x / tσ δ δ=

66

Removal of a particle

67

Boundary Condition

+ + + + ++

+

− + −=

−2 1 1 2

1 2

n 1 n 1 * n 1B L L L B L

B * n 1L L

( x x )w ( x x )ww

( x x )

68

Particle Trajectory

69

Linear Problem

x0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

u

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactMOCparticle

∂ ∂+ =

∂ ∂u ua 0t x

( , ) ( , )= − ∆ − ∆u x t u x a t t t

70

Burger's Equation (1)

x-1.0 -0.5 0.0 0.5 1.0

u

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

exactt=0t=0.2t=0.6t=1.0

x-1.0 -0.5 0.0 0.5 1.0

u

0.25

0.50

0.75

1.00

1.25

exactt=0t=0.2t=0.6t=1.0

( , )u x 0 x= − . .( , )

. .1 0 x 0 5

u x 00 5 x 0 5

< −= > −

71

Burger's Equation (2)

x-1.0 -0.5 0.0 0.5 1.0

u

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5exactt=0t=0.1t=0.3t=0.5

time0.0 0.2 0.4 0.6 0.8 1.0

u

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactparticle method

u( x,0 ) sin( 2 x )= π

72

Sudden Valve Closure (1)

0h( x,0 ) h (1 x )= − 0v( x,0 ) v=IC:

0h(0,t ) h= 00

0

t tvv(1,t )

t t0<

= ≥BC:

73


−= 30v 10

74


x

h/h0

-202468

1012

presentanalytical (200 terms)

t=0.6t=1.0 t=0.8

x0.0 0.2 0.4 0.6 0.8 1.0

v/v0

-0.20.00.20.40.60.81.01.2

presentanalytical (200 terms)

t=0.6t=1.0 t=0.8

t=0.2

t=0.2

t

h/h0

-10

-5

0

5

10

t0 1 2 3 4 5 6

v/v0

-1.0

-0.5

0.0

0.5

1.0

x=0.1x=0.3x=0.5x=0.7analytical (200 terms)

75

Valve Closure Over Finite Time (1)

IC:

BC:

= Lh( x,0 ) h x 0v( x,0 ) v=

≤ −= + < < +

≥ +

00

0 00 0 C

C0 C

v for t tv ( t t )v(0,t ) [1 cos ] for t t t t2 t

for t t t0

π

76

Valve Closure Over Finite Time (2)

t

h/hL

-10

-5

0

5

10

t0 1 2 3 4 5 6

v/v0

-1.5

-1.0

-0.5

0.0

0.5

1.0

presentx=0.1 (analytical, 200 terms)x=0.3 (analytical, 200 terms)x=0.7 (analytical, 200 terms)x=0.9 (analytical, 200 terms)

t

h/hL

-5

0

5

10

t0 1 2 3 4 5 6

v/v0

-1.5

-1.0

-0.5

0.0

0.5


t

h/hL

0

2

4

6

t0 1 2 3 4 5 6

v/v0

-1.0

-0.5

0.0


tC=1.0

tC=5.0

tC=3.0

77

Start-up and Evolution to Steady State(1)

IC:

BC:

0h( x,0 ) h= v( x,0 ) 0=

0h(0,t ) h= 00

0

t thh(1,t )

t t0<

= ≥

78

Start-up and Evolution to Steady State(2)

t

h/h0

0.00.20.40.60.81.01.2

t0 2 4 6 8 10

v/v0

-0.2

0.0

0.2

0.4

0.6

0.8presentx=0.3 (analytical, 200 terms)x=0.7 (analytical, 200 terms)

79

Periodic Forcing (1)

0h( x,0 ) h (1 x )= − 0v( x,0 ) v=IC:

BC: 0 0h(0,t ) h a sin( t )ω= + h(1,t ) 0=

80

Periodic Forcing (2)

t

h/h0

0.2

0.4

0.6

0.8

1.0

1.2

t0 4 8 12 16 20 24

v/v0

0.98

1.00

1.02

1.04


81

Sound speed variations (1)

x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

u/u0

-0.20.00.20.40.60.81.01.2

h/h0

02468

101214

present, ∆x=1/400MOC, ∆x=1/800MOC, ∆x=1/3200

t=0.6t=1.0 t=0.75

t=0.6t=1.0 t=0.9

t=0.9t=1.1

t=1.1 t=0.75

t=1.1

t=1.1

82


83


x0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

u/u0

-0.20.00.20.40.60.81.01.2

h/h0

02468

101214

a*=0.8a*=1.0a*=1.2

t=0.85t=1.2

t=1.0

t=0.85

t=1.2

t=1.0

t=1.2

t=1.2

84

Branch

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

h/h0

t=0.5t=0.8, a*=1.0t=1.0, a*=1.0t=1.3, a*=1.0t=0.8, a*=1.2t=1.0, a*=1.2t=1.3, a*=1.2t=0.8, a*=0.8t=1.0, a*=0.8t=1.3, a*=0.8

t=0.8

t=1.3

t=1.0

Branch B

Branch A

t=1.3

t=1.3

t=1.0

t=1.3

t=1.3

t=1.0

t=0.8

t=0.8

0

5

10

5

10

15

15

Branch LBranch R

RJIL

85


0h( x,0 ) h (1 x )= − 0v( x,0 ) v=IC:

0h(0,t ) h= 00

0

t tvv(1,t )

t t0<

= ≥BC:

86


x

h

-0.1

0.0

0.1t=0.5t=1.0t=1.5 t=2.0

t=4.0 t=2.5t=3.0t=3.5

x0.0 0.2 0.4 0.6 0.8 1.0

v

-0.1

0.0

0.1

t=0.5t=1.0

t=1.5 t=2.0 t=2.5t=3.0

t=3.5 t=4.0

−= 10v 10

87

Shock Tube Problem (1)

x0.0 0.2 0.4 0.6 0.8 1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

exacth (t=0.2)h (t=0.5)v (t=0.2)v (t=0.5)

=L L( h , v ) ( 2,0 ) =R R( h , v ) (1,0 )

88


=L L( h , v ) (1,0 ) = −R R( h , v ) (1, 1)

x0.0 0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

exacth (t=0.2)h (t=0.5)v (t=0.2)v (t=0.5)

89


= −L L( h , v ) (1, 1) =R R( h , v ) (1,0 )

x0.0 0.2 0.4 0.6 0.8 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

exacth (t=0.2)h (t=0.5)v (t=0.2)v (t=0.5)

90

Valve Action (1)

p

1.00

1.01

1.02

x

p

1.0

1.1

1.2

x0 2 4 6 8 10

p

0.5

1.0

1.5

2.0finite volume (UD)finite volume (van Leer)particle method

∆u=∆p=0.01

∆u=∆p=0.1

∆u=∆p=0.5

91

Valve Action (2)

x0 2 4 6 8 10

p

0.5

1.0

1.5

2.0

x

p

1.0

1.5

2.0

p

1.0

1.5

2.0finite volume (UD)finite volume (van Leer)particle method

t=14

t=8

t=2

92

Valve Action (3)

x

time

0 2 4 6 8 100

2

4

6

8

10

12

14

x

time

0 2 4 6 8 100

2

4

6

8

10

12

14

x

time

0 2 4 6 8 100

2

4

6

8

10

12

14

UD Van Leer Present

93

Open Channel

x

0.2 0.3 0.4 0.5 0.6 0.7 0.8

v

0.0

0.5

1.0

1.5

2.0 exactSPH (n=1000), Chang et al.SPH (n=5000), Chang et al.present (n=64)

x

0.2 0.4 0.6 0.8 1.0

v

0.0

0.5

1.0

1.5

2.0 exactFV (n=120), Ying et al.present (n=64)

94

Tidal bore

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

b

-5

-4

-3

-2

-1

0

1

2

3

4

5

α+β tanh[γ(x-0.7)]

x0.0 0.2 0.4 0.6 0.8 1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

t=0.2t=0.4t=0.6t=0.8t=1.0

η

v

x

t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.05

1.45

1.075

1.1

1.025

1.5

1.35

1.5

1.45

1.4

1.3

1.25

1.2

1.15

95

Two-phase flow

x0.0 0.2 0.4 0.6 0.8 1.0

p/pref

0.0

0.2

0.4

0.6

0.8

1.0

1.2

v

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

exactp/pref

v

/ =L Rp p 100

96

Traffic flow (LWR model)

x-1.0 -0.5 0.0 0.5 1.0

ρ

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactt=0.5

1.02.0

3.0

t=0.

0.25

= − −L R L( x,0 ) sin[( x x ) /( x x )]ρ π

t0.0 0.5 1.0 1.5 2.0 2.5 3.0

xsh

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

exactpresent

0.3 0.4 0.5 0.6 0.7-0.64

-0.63

-0.62

-0.61exactδx=0.05δx=0.025

97

Traffic flow (Payne model): Branch

98

Traffic flow (Payne model): Red-Green

t

0.15

0.20

0.25ρu

t0 1 2 3 4 5 6 7 8 9 10

0.10

0.15

0.20

0.25

x=0.3

x=0.7

0.15

0.16

0.16

0.16

0.16

0.16

0.17

0.17

0.17

0.17

0.18

0.18

0.18

0.18

0.18

0.18

0.18

0.19

0.19

0.19

0.19

0.19

0.19

0.19

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.21

0.21

0.21

0.21

0.21

021

0.21

0.21

0.21 0.22

0.22 0.22

0.22

0.22

0.22

0.22

0.23

0.230.23

0.23

0.23

0.23

0.23

0.24

0.24

0.24

0.240.24

0.24

0.25

0.25

0.25

0.25

0.26

0.26

0.260.27

0.27

0.270.27

0.27

0.280.28

0.29

x

t

0 0.2 0.4 0.6 0.8 19

9.2

9.4

9.6

9.8

10

99

Gas dynamics

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactUDMinModESSGpresent

density

pressure

velocity

100

Gas dynamics

x0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ρ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

exactUDMinModESSG

0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

0.5present

essential features of moving particle with pressure mesh€¦ · mppm moving particle method with...

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