Procedures of extracting Boundary-I data for NS solver

Subroutine Extract_Boundary-I_data_for_NS_solver

#

for each cell interface k such that FBIk = 1 do iR = the right-hand side cell of cell interface k iL = the left-hand side cell of cell interface k if ( D_iR = 0 ) then

set flow data of cell interface k with flow properties of cell iR from DSMC domain

else

set flow data of cell interface k with flow properties of cell iL from DSMC domain

end if end for

Table 3.1 Free-stream conditions in supersonic flow over quasi-2-D 25o wedge.

Gas ρ_∞ U_∞ T_∞ M_∞

N2 6.545E-4kg/m³ 1111m/s 185.6K 4

Table 3.2 Four simulation sets with various parameters in supersonic flow over quasi-2-D 25^o wedge.

Set No. 1 2 3 4

Cell layer No. of ΩB 4 2 4 4

Cell layer No. of ΩC 2 1 2 2

. Thr

Knmax 0.02 0.02 0.04 0.02

. Thr

PTne 0.03 0.03 0.03 0.06

Final DSMC cells ~85,000 ~71,000 ~64,000 ~84,000

Table 3.3 Total computational time (hours) with pure NS solver, pure DSMC and coupled method in supersonic flow over quasi-2-D 25^o wedge.

Pure NS Pure DSMC Coupled DSMC-NS method^*

One-shot NS DSMC NS

2.8 12.2 9.2

Total time 2.8 16.3 24.2

*10 coupling iterations are used in the coupled method for 2-D 25^o wedge flow.

Table 3.4 Sonic conditions at the orifice exiting plane in two parallel near-continuum orifice free jets flow.

Gas ρthroat Uthroat Tthroat Rethroat

N2 6.52E-3 kg/m³ 314 m/s 237.5 K 401

Table 3.5 Simulational conditions of two parallel near-continuum orifice free jets flow.

Method DSMC for coupled method

Cell No. ~470,000

Sim. Particle No. ~6,200,000

Reference ∆t (sec) 3.01E-09

Sampling time steps 13,500

Cell layer No. of Ω^B 1

Cell layer No. of ΩC 0

Thr.

Knmax 0.05

. Thr

PTne 0.1

* Total cell No. of computational domain for coupled DSMC-NS method is 520,830

Table 3.6 Total computational time (hours) with pure NS solver and coupled method in two parallel near-continuum orifice free jets flow.

Pure NS Coupled DSMC-NS method^*

One-shot NS DSMC NS

2.5 22.2 4.1

Total time 2.5 30.8

*3 coupling iterations are used for a converged solution in the coupled method.

Table 3.7 Flow conditions of the plume simulation issuing from RCS thruster

Gas Po To Tw Dthroat Rethroat Area ratio

N2 0.1 bar 300K 300K 4.36mm 6,256 60

Table 3.8 Simulation conditions of the plume simulation issuing from RCS thruster.

Method DSMC for coupled method

Cell No. 210,000~250,000

Sim. Particle No. 2,200,000~2,500,000

Reference ∆t (sec) 6.9E-08

Sampling time steps 8,000

Cell layer No. of Ω^B 2

Cell layer No. of ΩC 1

Thr.

Knmax 0.03

. Thr

PTne 0.03

* Total number of the hexahedral cells for coupled DSMC-NS method is 380,100.

Fig. 1.1 Sketch of expanding reaction control system plumes.

Continuum Domain

Breakdown Domain

Turbulent flow

&

Complicated geometry

Fig. 1.2 Sketch of expanding flying object plume at high altitude.

Breakdown Domain

Continuum Domain

Radiation effects

&

Reversed flow

Fig. 1.3 Sketch of a sketch of spiral-grooved turbo booster pump.

High compression

Outlet Continuum Domain

Breakdown Domain

.

Fig. 1.4 Configuration of a jet-type CVD reactor.

Continuum Domain

Breakdown Domain

Chemical reactions precursor

pumping

Fig. 1.5 Schematic sketch of solution method applicability in a dilute gas.

move all molecules

enter new molecules

print out the data NO

NO reset

sampling data YES

YES sort (index) molecules

reach steady flow?

sufficient sampling?

start

set initial state and read system data

collide molecules

sample flow field

stop

Fig. 2.1 Flow chart of the DSMC method.

n=(a,b,c) (x_f, y_f)

(x_i, y_i)

(x_o, y_o)

: the initial position

: the intersection position

(x_i, y_i)

: the final position

(x_f, y_f) (x_o, y_o)

∆ 't

∆ 't

: intersecting time Fig. 2.2 Sketch of the particle movement in three-dimensional unstructured mesh.

Ω

_A

Ω ∪ : Domain with flow properties updated from DSMC calculation

D

C Ω

Ω ∪ : Domain with flow properties updated from NS calculation

B

A Ω

Ω

= ∪

S(I) : Boundary-I (Dirichlet B.C.) for NS simulations

C

B Ω

Ω

= ∪

S(II) : Boundary-II for updated solution using different solvers

D

C Ω

Ω

= ∪

S(III) : Boundary-III (Dirchlet B.C.) for DSMC simulations

Fig. 2.3 Sketch of domain distribution of the present coupled DSMC-NS method with overlapping regions and boundaries

Fig. 2.4 A typical example of the overlapping regions between the particle and the

Fig. 3.1 Sketch of a hypersonic flow over 25^o 2-D wedge N_2, Ma=4

X

Y

X

Fig. 3.2a Grid sensitivity test of HYB3D on quasi-2-D 25^o wedge flow (Density).

Fig. 3.2b Grid sensitivity test of HYB3D on quasi-2-D 25^o wedge flow (Total temperature).

0 1 2 3 4

Breakdown Parameter Knmax or PTne

SYM DATA SOLVER

Fig. 3.3a Breakdown parameters along the line normal to the wedge at x=0.5mm.

0 1 2 3 4 5

Breakdown Parameter Knmax or PTne

SYM DATA SOLVER

Fig. 3.3b Breakdown parameters along the line normal to the wedge at x=5mm.

0 5 10 15 20

δn (mm)

0 0.1 0.2 0.3 0.4 0.5

Breakdown Parameter Knmax or PTne

SYM DATA SOLVER

x δn

Knmax HYB3D (Initial) Knmax Coupled (15^th iteration) P_Tne Coupled (15^th iteration)

K n_max =0.02 P_{T ne} ^Thr.=0.03

T hr.

Fig. 3.3c Breakdown parameters along the line normal to the wedge at x=50mm.

Fig. 3.4a Initial continuum breakdown (NS) domain distribution in quasi-2-D 25^o wedge flow.

X Y

Z

: Ω

A

= Ω

Knmax

Fig. 3.4b Initial DSMC domain including the overlapping regions in quasi-2-D 25^o wedge flow.

X Y

Z

: _Ω

_A

_{∪ Ω}

_B

_{∪ Ω}

_C

Fig. 3.5a Breakdown domain distribution at 15^th coupling iteration in quasi-2-D 25^o wedge flow.

:

X Y

Z

⎭ ⎬

⎫

⎩ ⎨

⎧ Ω Ω

=

Ω

A P_Tne

∪

Kn_max

Fig. 3.5a DSMC domain including the overlapping regions at 15^th coupling iteration in quasi-2-D 25^o wedge flow.

:

X Y

Z

C B

A

Ω Ω

Ω ∪ ∪

X (mm)

Fig. 3.6a Density comparison between the DSMC method and the present coupled DSMC-NS method in quasi-2-D 25^o wedge flow.

X (mm)

Fig. 3.6b Translation temperature comparison between the DSMC method and the present coupled DSMC-NS method in quasi-2-D 25^o wedge flow.

0 1 2 3 4 5

ρ

/

ρ

_∞

0 0.5 1 1.5 2 2.5

δn (mm)

SYM DATA Pure DSMC Pure NSS Coupled Method

HYB3D

PDSC

x δn

*NSS=Navier-Stokes Solver

Fig. 3.7a Density profile along a line normal to the wedge at x=0.5mm.

1 1.5 2 2.5 3 T_Tr/T_∞

0 0.5 1 1.5 2 2.5

δn (mm)

SYM DATA Pure DSMC Pure NSS Coupled Method

HYB3D

PDSC

x δn

*NSS=Navier-Stokes Solver

Fig. 3.7b Translational temperature profile along a line normal to the wedge at x=0.5mm.

0 0.2 0.4 0.6 0.8 1 1.2

|V|/V_∞ 0

0.5 1 1.5 2 2.5

δn (mm) SYM DATA

Pure DSMC Pure NSS Coupled Method

HYB3D

PDSC

x δn

*NSS=Navier-Stokes Solver

Fig. 3.7c Velocity profile along a line normal to the wedge at x=0.5mm.

Fig. 3.8a Density profile along a line normal to the wedge at x=5mm.

1 1.5 2 2.5 3

Fig. 3.8b Translational temperature profile along a line normal to the wedge at x=5mm.

0 0.2 0.4 0.6 0.8 1 1.2

Fig. 3.8c Velocity profile along a line normal to the wedge at x=5mm.

0 1 2 3 4 5

ρ

^/

ρ

∞

0 5 10 15 20

δn (mm)

SYM DATA Pure DSMC Pure NSS Coupled Method

x δn

*NSS=Navier-Stokes Solver

III: PDSC IV: HYB3D

I: PDSC

II: HYB3D

Fig. 3.9a Density profile along a line normal to the wedge at x=50mm.

0.8 1.2 1.6 2 2.4 2.8 T_Tr/T_∞

0 5 10 15 20

δn (mm)

SYM DATA Pure DSMC Pure NSS Coupled Method

x δn

*NSS=Navier-Stokes Solver

III: PDSC IV: HYB3D

I: PDSC

II: HYB3D

Fig. 3.9b Translational temperature profile along a line normal to the wedge at x=50mm.

0 0.2 0.4 0.6 0.8 1

|V|/V_∞ 0

5 10 15 20

δn (mm) SYM DATA

Pure DSMC Pure NSS Coupled Method

x δn

*NSS=Navier-Stokes Solver

III: PDSC IV: HYB3D

I: PDSC II: HYB3D

Fig. 3.9c Velocity profile along a line normal to the wedge at x=50mm.

0 2 4 6 8 10 12 14 16 Iteration Numbers

1.00E-005 1.00E-004 1.00E-003

L2-Norm Deviation of Density (kg/m3)

SYM TEST_SET Set 1

Set 2 Set 3 Set 4

*ρ_∞= 6.545e-004 (kg/m³)

Fig. 3.10 Convergence history of L2-norm deviation of density among the four simulation sets in quasi-2-D 25^o wedge flow.

0 2 4 6 8 10 12 14 16 Iteration Numbers

1 10 100

L2-Norm Deviation of Total Temperature (K)

SYM TEST_SET Set 1

Set 2 Set 3 Set 4

* T_∞= 185.6 K

Fig. 3.11 Convergence history of L2-norm deviation of total temperature among the four simulation sets in quasi-2-D 25^o wedge flow.

0 1 2 3 4 5

Fig. 3.12a Comparison of density along a line normal to the wedge at x=0.5mm among the four simulation sets.

1 1.5 2 2.5 3

T

_Tr/

T

_∞

0 0.5 1 1.5 2 2.5

δn (mm)

SYM DATA Pure DSMC Set 1 (Coupled) Set 2 (Coupled) Set 3 (Coupled) Set 4 (Coupled)

HYB3D

PDSC

Fig. 3.12b Comparison of translational temperature along a line normal to the wedge at x=0.5mm among the four simulation sets.

Fig. 3.13 Sketch of two parallel near-continuum orifice free jets flow.

Fig. 3.14 Mesh distribution of two parallel near-continuum orifice free jets flow simulation.

Fig. 3.15 Surface mesh distribution of breakdown domain (DSMC domain) of two parallel near-continuum orifice free jets flow simulation with an exploded view (after 2 coupled iterations).

Fig. 3.16 Density contours of two parallel near-continuum orifice free jets flow.

Fig. 3.17a Thermal non-equilibrium contours of two parallel near-continuum orifice free jets

flow.

Fig. 3.17b Thermal non-equilibrium contours near the orifice.

0 5 10 15 20 X/D

0 0.5 1

Normalized Density

SYM DATA SOURCE

Experiment Soga et al. [1994]

DSMC-NS Present Pure NS Present

* Each datum is normalized to its own peak value.

Fig. 3.18a Normalized density distribution along the symmetric line of two parallel near-continuum orifice free jets flow.

0 5 10 15 20 X/D

0 0.0001 0.0002 0.0003 0.0004 0.0005

Density (kg/s)

SYM DATA SOURCE DSMC-NS Present

Pure NS Present

Fig. 3.18b Density distribution along the symmetric line of two parallel near-continuum orifice free jets flow.

0 5 10 15 20 X/D

50 100 150 200 250 300

Rotational Temperature (K)

SYM DATA SOURCE

Experiment Soga et al. [1984]

DSMC-NS Present Pure NS* Present

*Total Temp. is used to represent Rot. Temp. in pure NS

Fig. 3.19 Rotational temperature distribution along the symmetric line of two parallel near-continuum orifice free jets flow.

0 2 4 6 8 10 12 Iteration Numbers

1.00E-005 1.00E-004

L2-Norm Deviation of Density (kg/m3)

*ρ_throat= 6.52e-3 kg/m³

Convergence History

Fig. 3.20 Convergence history of density for two parallel near-continuum orifice free jets flow.

0 2 4 6 8 10 12 Iteration Numbers

1.00E+000 1.00E+001 1.00E+002

L2-Norm Deviation of Total Temperature (K)

*T_throat= 237.5K

Convergence History

Fig. 3.21 Convergence history of total temperature for two parallel near-continuum orifice free jets flow.

Fig. 3.22a Mach number distribution of quasi-2-D supersonic wedge flow (gray areas: DSMC domain; others: NS domain).

Fig. 3.22b Mach number distribution of two parallel near-continuum free jets near orifice (white region: breakdown interface).

Fig. 3.23 Sketch of the plume simulation issuing from RCS thruster.

AR=60

Vacuum Boundary conditions (For DSMC) Supersonic Boundary conditions (For NS solver)

Isothermal wall (300K) Po= 0.1bar

T_o =300K

N

₂

Fig. 3.24 Mesh distribution of the plume simulation issuing from RCS thruster.

Fig. 3.25 Spatial computational domain distributions.

(a)

(b) (c)

Fig. 3.26 Domain decomposition for 6 processors (a) NS CPU domain; (b) Initial DSMC CPU domain; (c) Final DSMC CPU domain (6^th iteration).

Fig. 3.27 Continuum breakdown distribution of the plume simulation issuing from RCS thruster.

Fig. 3.28 Distribution of non-equilibrium ratio in the RCS plume simulation.

Fig. 3.29 Particle per cell in the RCS plume simulation.

(a)

(b)

Fig. 3.30 Density distribution of the plume simulation issuing from RCS thruster (a) One-shot NS method; (b) Coupled DSMC-NS method (6^th iteration).

(a)

(b)

Fig. 3.31 Temperature distribution of the plume simulation issuing from RCS thruster (a) One-shot NS method; (b) Coupled DSMC-NS method (6^th iteration).

(a)

(b)

Fig. 3.32 Stream lines of the plume simulation issuing from RCS thruster (a) One-shot NS method; (b) Coupled DSMC-NS method (6^th iteration).

(a)

(b)

Fig. 3.33 Mach number distribution of the plume simulation issuing from RCS thruster (a) One-shot NS method; (b) Coupled DSMC-NS method (6^th iteration).

0 2 4 6 8 Iteration Numbers

1.00E-005 1.00E-004 1.00E-003 1.00E-002 1.00E-001

L2 Norm Deviation of Density (kg/m3)

*ρ_throat= 0.107 kg/m³

Convergence History

Fig. 3.34 Convergence history of density for the plume simulation issuing from RCS thruster.

0 2 4 6 8 Iteration Numbers

1E+001 1E+002

L2 Norm Deviation of Total Temperature (K)

*Tthroat= 250K

Convergence History

Fig. 3.35 Convergence history of total temperature for the plume simulation issuing from RCS thruster.

Fig. 4.1 Surface mesh distribution of 2-D wedge flow for the kinetic study of velocity sampling.

Fig. 4.2 Locations of velocity sampling and continuum breakdown distribution of 2-D wedge flow for the kinetic study of velocity sampling.

Fig. 4.3 Domain distribution of dominating breakdown parameter of 2-D wedge flow for the kinetic study of velocity sampling.

Leading Edge

3 7

Fig. 4.4a Locations of velocity sampling Point 3-7 (Leading edge region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.4b Random velocity distributions to each direction at Point 3 (Leading edge region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.4c Random velocity distributions to each direction at Point 4 (Leading edge region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.4d Random velocity distributions to each direction at Point 5 (Leading edge region).

-4 -3 -2 -1 0 1 2 3 4 V_i/(2KT_i)^1/2

0 0.2 0.4 0.6 0.8 1 1.2

SYM DATA

V_x V_y V_z

MB Distribution

T_x:T_y:T_z:T_Rot:T_Tot= 1.87 : 0.88 : 0.93 : 0.66 : 1.00

Point 6

Fig. 4.4e Random velocity distributions to each direction at Point 6 (Leading edge region).

-4 -3 -2 -1 0 1 2 3 4 V_i/(2KT_i)^1/2

0 0.2 0.4 0.6 0.8 1 1.2

SYM DATA

V_x V_y V_z

MB Distribution

T_x:T_y:T_z:T_Rot:T_Tot= 1.38 : 0.85 : 1.01 : 0.88 : 1.00

Point 7

Fig. 4.4f Random velocity distributions to each direction at Point 7 (Leading edge region).

Shock Region

14

19

Fig. 4.5a Locations of velocity sampling Point 14-19 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.5b Random velocity distributions to each direction at Point 14 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.5c Random velocity distributions to each direction at Point 15 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4 V_i/(2KT_i)^1/2

0 0.2 0.4 0.6 0.8

1 SYM DATA

V_x V_y V_z

MB Distribution

T_x:T_y:T_z:T_Rot:T_Tot= 1.07 : 1.11 : 0.97 : 0.92 : 1.00

Point 16

Fig. 4.5d Random velocity distributions to each direction at Point 16 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4 V_i/(2KT_i)^1/2

0 0.2 0.4 0.6 0.8 1 1.2

SYM DATA

V_x V_y V_z

MB Distribution

T_x:T_y:T_z:T_Rot:T_Tot= 1.22 : 1.29 : 0.96 : 0.77 : 1.00

Point 17

Fig. 4.5e Random velocity distributions to each direction at Point 17 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.5f Random velocity distributions to each direction at Point 18 (Oblique shock region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.5g Random velocity distributions to each direction at Point 19 (Oblique shock region).

Boundary Layer

26

30

Fig. 4.6b Locations of velocity sampling Point 26-30 (Boundary layer region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.6b Random velocity distributions to each direction at Point 26 (Boundary layer region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.6c Random velocity distributions to each direction at Point 27 (Boundary layer region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.6d Random velocity distributions to each direction at Point 28 (Boundary layer region).

-4 -3 -2 -1 0 1 2 3 4

Fig. 4.6e Random velocity distributions to each direction at Point 29 (Boundary layer region).

在文檔中 平行化三維DSMC-NS偶合數值方法之發展及驗證 (頁 111-200)