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Nozzle Study Yan Zhan, Foluso Ladeinde April 21, 2011

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Page 1: Nozzle Study - Princeton University

Nozzle Study

Yan Zhan, Foluso Ladeinde

April 21, 2011

Page 2: Nozzle Study - Princeton University

Outlines

• Region of interests• Bend Combinations without nozzle

– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions

• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions

• Appendix

Page 3: Nozzle Study - Princeton University

Region of interests

Hg delivery system at CERN

Simulation Regime

Interests: Influence of nozzle

& nozzle upstream on the jet exit

Page 4: Nozzle Study - Princeton University

Outlines

• Region of interests• Bend Combinations without nozzle

– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions

• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions

• Appendix

Page 5: Nozzle Study - Princeton University

Bend Combinations Without Nozzle (1)

Geometry of bends with varying anglesNondimensionalized by pipe diameter (p.d.);φ range: 0⁰, 30⁰, 60⁰, 90⁰.

Mesh at the cross-sectionnr × nθ =152×64 (the 1st grid center at y+≈1)

Page 6: Nozzle Study - Princeton University

Geometries without nozzle: (a) 0⁰/0⁰ ; (b) 30⁰/30⁰; (c) 60⁰/60⁰; (d) 90⁰/90⁰ nx =40 +nφ+20+ nφ +100where nφ depends on the bend angle φ

Bend Combinations Without Nozzle (2)(a)

(b) (c)

(d)

Page 7: Nozzle Study - Princeton University

Turbulence Model Comparisons (2)

Sudo’s Experiment (1998) for 90⁰ bend*

•K. Sudo, M. Sumida, H. Hibara, 1998. Experimental investigation on turbulent flow in a circular-sectioned 90-degrees bend, Experiments in Fluids. 25, 42-49.

uave = 8.7 m/s; Re=6×104; ρair = 1.2647; μ air = 1.983×10-5; Pr= 0.712

Page 8: Nozzle Study - Princeton University

Inlet

Outlet

Turbulence Model Comparisons (3)

Mesh at the cross-sectionnr × nθ =80×50 (the 1st grid center at y+≈1)

Mesh for the test pipe

nφ =75 (dφ=1.2⁰)

Page 9: Nozzle Study - Princeton University

Turbulence Model Comparisons (4)

Longitudinal distribution of wall static pressureCp pressure coefficientpref reference value of p at z’/d=-17.6

RKε is the best of the three in simulating curved pipe flow

Page 10: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (1)

• Steady incompressible turbulent flow

• Boundary Conditions– Inlet: Fully developed velocity;

– Outlet: Outflow;

– Wall: non-slip

• Schemes– 3rd order MUSCL for momentum and turbulence equations

– SIMPLE schemes for pressure linked equations

• Convergence Criterion– 10-5

Page 11: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (2)

• Turbulence Characteristics– Turbulence Level (Flutuating Velocity)

– Momentum Thickness (Mean Velocity)

Measure of the momentum loss within the boundary layer due to viscosity.

mm uk

uuI 3/2

=′

∫∫ −=⇒−= R

t

R

t drUu

UudyuUuUU

00)1( )( θρρθ

Small I

Big θt

Page 12: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (3)

• Turbulence Level (1)

Fig. Two shown planes at the pipe outlet

Pipes Imean at outlet

90⁰/90⁰ 5.204039%

60⁰/60⁰ 5.323943%

30⁰/30⁰ 5.368109%

0⁰/0⁰ 4.649036%

Table. Mean turbulence level at pipe outlet Mean Iin=4.600041%

Page 13: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (4)• Turbulence Level (2)

Turbulence intensity distribution at the pipe exit(a) Horizontal plane (from inner side to outer side )

(b) Vertical (from bottom to top )

(a) (b)

Page 14: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (4)

• Momentum Thickness

Momentum thickness distributionalong the wall at the pipe exit(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bend

Page 15: Nozzle Study - Princeton University

Discussions

• Bend Effects– Bend and turbulence level

• Bend enhances the turbulence level but not too much;– The 0⁰/0⁰ bend has the lowest turbulence level;

• Symmetry I for the 90⁰/90⁰ bend;

– Bend and θt

• Bend effects θt not linearly with the increasing bends– The 60⁰/60⁰ bend seems like a turning point

• Less uniform θt distribution in larger bend– θt is bigger near the inner side and smaller near the outer side

Page 16: Nozzle Study - Princeton University

Outlines

• Region of interests• Bend Combinations without nozzle

– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions

• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions

• Appendix

Page 17: Nozzle Study - Princeton University

Bend Combinations With Nozzle

Geometry of bends with varying anglesNondimensionalized by pipe diameter (p.d.);φ range: 0⁰, 30⁰, 60⁰, 90⁰.

Mesh at the cross-section (half model)nr × nθ =152×64 (the 1st grid center at y+≈1)

Page 18: Nozzle Study - Princeton University

Mesh for bends(a) 0⁰/0⁰ ; (b) 30⁰/30⁰; (c) 60⁰/60⁰; (d) 90⁰/90⁰ nx =40 +nφ+20+ nφ +60+70where nφ depends on the bend angle φ

(a)

(b) (c)

(d)

Page 19: Nozzle Study - Princeton University

Mercury Flow in Curved Nozzle Pipes (1)

Longitudinal distribution of static pressure for 90⁰/90⁰ CombinationVelocity inlet condition

Pin 30bar;uin fully developed velocity profile;

Outlet flow condition

Page 20: Nozzle Study - Princeton University

Mercury Flow in Curved Nozzle Pipes (2)

• Main Loss

Assume smooth pipe, thus Pmain=0

• Minor Loss– Elbow Loss

– Contraction Loss

• Total Loss

2/)/(2/)/( 22ininoutoutmain uDLudlP ρλρλ +=

mguh inelbow 2618.028.9136.43.02/ 22 =÷÷×== ξ

mgKuh outcontr 9694.028.9200475.02/ 22 =÷÷×==

Pa 198196944 )2( =++= contrelbowmainloss hhgPP ρ

Page 21: Nozzle Study - Princeton University

Mercury Flow in Curved Nozzle Pipes (3)

• Bernoulli’s Law

• Comparison

Pa 3534.208465944.1981965.013546205.013546136.410 3

2/5.02/5.0226

12

12

=⇒+××+=××+×

+++=++

out

out

lossoutoutinin

PP

PghuPghuP ρρρρ

%0898.9%1003534.208465

3534.2084653.189516

%100

=×−

=

×−

=ana

ananum

PPPError

Page 22: Nozzle Study - Princeton University

Mercury Flow in Curved Nozzle Pipes (4)

• Turbulence Level (1)

Pipes Imean(Nozzle)

Imean(No nozzle)

90⁰/90⁰ 5.143341% 5.204039%

60⁰/60⁰ 2.858747% 5.323943%

30⁰/30⁰ 2.701366% 5.368109%

0⁰/0⁰ 2.182027% 4.649036%

Fig. Two shown planes at the pipe outlet Table. Mean turbulence level at pipe outlet Mean Iin=4.599195%

Page 23: Nozzle Study - Princeton University

Mercury Flow in Curved Nozzle Pipes (5)• Turbulence Level (2)

Turbulence intensity distribution at the pipe exit(a) Horizontal plane ; (b) Vertical plane

(a) (b)

Page 24: Nozzle Study - Princeton University

Mercury Flow in Curved Pipes (6)

• Momentum Thickness

Momentum thickness distributionalong the wall at the pipe exit(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bend

Page 25: Nozzle Study - Princeton University

Discussions (1)

Turbulence intensity distribution at the pipe exit(a) Horizontal plane (b)Vertical plane

(a) (b)

Page 26: Nozzle Study - Princeton University

Discussions (2)

The comparison of momentum thickness distribution at the pipe exit between pipe with/without nozzle(a) 0⁰/0 ⁰ bend (b)30⁰/30 ⁰ bend (c) 60⁰/60 ⁰ bend (d) 90⁰/90 ⁰ bendRed: pipe without nozzle;Blue: pipe with nozzle.

Page 27: Nozzle Study - Princeton University

Discussions (3)

• Nozzle Effects– Nozzle and turbulence level

• Nozzle reduces the turbulence level;– The 90⁰/90⁰ bend doesn’t change too much;

• I of the 90⁰/90⁰ bend is far different from other bends

– Nozzle and θt

• Nozzle decreases θt

• Very uniform and similar θt distribution at the nozzle exit for all bend pipes

Page 28: Nozzle Study - Princeton University

Outlines

• Region of interests• Bend Combinations without nozzle

– Bend Combinations Without Nozzle– Turbulence Model Comparisons– Mercury Flow in Curved Pipes – Discussions

• Bend Combinations with Nozzle– Bend Combinations With Nozzle– Mercury Flow in Curved Nozzle Pipes – Discussions

• Appendix

Page 29: Nozzle Study - Princeton University

Appendix(1)

• Continuity EquationThe two-phase model considers mixture comprising of liquid, vapor and non-condensable gas (NCG). Gas is compressible, the liquid and vapor are impressible. The mixture is modeled as incompressible.

• Momentum Equations

0=∂∂

j

j

xu

j

ji

ij

ji

xxP

xuu

∂∂

+∂∂

−=∂∂ τ

ρ

)( and )( where2

t ερµµµτ µ

kCxu

xu

i

j

j

itij =

∂∂

+∂∂

+=

Page 30: Nozzle Study - Princeton University

Appendix(2)

• Spalart-Allmaras Model

νν

ν

ν

νρννρµσ

νρ~

22

~

])~

(~

)~([1~SY

xC

xxG

xu

j

b

jji

i +−∂∂

+∂∂

+∂∂

+=∂

)/(~~ and ~~ termproduction The 222~1 dfvSSvSCG vb κρν +≡=

21 )/~( n termdestructio The dvfCY ww ρν =

3.0,1.7 2/3, ,4187.0,622.0,1355.0 21~21 ====== wvbb CCCC νσκ

2/)(,2 and )1/(1 where ,,1~2~ ijjiijijijvv uuSff −=ΩΩΩ≡+−= χχ

)~/(~ ),( ,)]/()1[( where 2262

6/163

663 dSvrrrCrgCgCgf wwww κ≡−+=++=

vbbww CCCC ~22

113 /)1(/2.0, σκ ++==

stresses Reynolds theestimating ignored , termsource defined-User vS

Page 31: Nozzle Study - Princeton University

Appendix(3)

• Turbulent Viscosity (SA)

vvC

fv

v /~ and function damping visouswhere31~

3

3

1~ ≡+

= χχχ

1~~

vt fνρµ =

. viscositykinematicmolecular theis v

Page 32: Nozzle Study - Princeton University

Appendix (4)

• Standard K-ε model

εεεε

ε

ερεεσµµ

ερρε Sk

CGCGk

Cxxx

ut bk

j

t

jj

j +−++∂∂

+∂∂

=∂∂

+∂∂ 2

231 )(])[(

kMbk

jk

t

jj

j SYGGxk

xxku

tk

+−−++∂∂

+∂∂

=∂∂

+∂∂ ρε

σµµ

ρρ ])[(

RTkMMY

xTgG

xu

uuG ttM

it

tib

i

jjik γ

ρµβρ ==∂∂

=∂∂′′−= ,2,

Pr, 2

3.1,0.1,09.0,92.1,44.1 21 ===== εµεε σσ kCCC

Page 33: Nozzle Study - Princeton University

Appendix (5)

• Turbulent Viscosity (SKε)

ερµ µ

2kCt =

constant. a is µC

gradients;ity mean veloc toduek of generation :kG

buoyance; toduek of generation :bG

rate;n dissipatio overall the toturbulence lecompressibin dilatation gfluctuatin theofon contributi :MY

; and for numbers Prandtl turbulent:, εσσ ε kk

terms;source defined-user :, εSSk

Page 34: Nozzle Study - Princeton University

Appendix (6)

• Realizable K-ε model

kMbk

jk

t

jj

j SYGGxk

xxku

tk

+−−++∂∂

+∂∂

=∂∂

+∂∂ ρε

σµµ

ρρ ])[(

εεε

ε

εε

ερερεσµµ

ερρε SCCk

Cvk

CSCxxx

ut b

j

t

jj

j +++

−+∂∂

+∂∂

=∂∂

+∂∂

31

2

21])[(

ijij SSSkSC 2,,]5

,43.0max[1 ==+

ηηη

RTkMMY

xTgG

xu

uuG ttM

it

tib

i

jjik γ

ρµβρ ==∂∂

=∂∂′′−= ,2,

Pr, 2

Page 35: Nozzle Study - Princeton University

Appendix (7)

• Turbulent Viscosity (RKε)

ερµ µ

2kCt =

kijkijijkijkijijijijijij

s

SSUkUAAC ωεωε

ε

µ −Ω=Ω−Ω=ΩΩΩ+≡+

= ,2~,~~,1 **

0

jiijkijkij

s SSSS

SSSWWAA =====

− ~,~,3

)6(cos,cos6,04.43

1

0 φφ

2.1,0.1,9.1,44.1,2 21 ====+

= εε σσ kjiij

ij CCuu

S

Page 36: Nozzle Study - Princeton University

Appendix(8)Models Pros Cons

Spalart-Allmaras(SA)

Low-cost;Aerospace applications involving wall-bounded flows and with mild separation;

Can not predict the decay of homogeneous, isotropic turbulence; No claim is made regarding its applicability to all types of complex engineering flows;

Standard K-ε(Skε)

Robust and reasonably accurate;Most widely-used engineering model industrial applications;Contains submodels for buoyancy, compressibility, combustion, etc;

Must use wall function for near wall calculation;Performs poorly for flows with strong separation, large streamline curvature, and large pressure gradient;

Realizable K-ε(Rkε)

superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation

Page 37: Nozzle Study - Princeton University

Appendix (9)

• Properties of fluid

• Steady incompressible turbulent flow

Variable ValuesPipe ID 0.884 inch

Nozzle exit ID 0.402 inch

Driving pressure 30 bar

Frequency for Hg supply 12 s

Maximum Cycles 100

Jet Velocity 20 m/s

Jet Diameter 1 cm

Jet Flow rate 1.6 L/sec

Environment 1 atm air/vacuum

Mercury Properties (25 ⁰C )

Density 13.546 kg/L

Sound Speed 1451 m/s

Bulk Modulus 2.67×1010Pa

Dynamic Viscosity 1.128×10-7 m2/s

Thermal Conductivity

8.69 W/m⋅K

Electrical Conductivity

106 Siemens/m

Specific Heat 0.139 J/kg⋅KPrandtl Number 0.025

Surface Tension 465 dyne/cm