nozzle study - princeton university

Nozzle Study

Yan Zhan, Foluso Ladeinde

April 21, 2011

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

Region of interests

Hg delivery system at CERN

Simulation Regime

Interests: Influence of nozzle

& nozzle upstream on the jet exit

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

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)

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)

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

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)

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

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

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

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%

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)

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

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

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

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)

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)

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

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

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

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%

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)

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

Discussions (1)

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

(a) (b)

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.

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

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

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 =

∂∂

+∂∂

+=

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

Appendix(3)

• Turbulent Viscosity (SA)

vvC

fv

v /~ and function damping visouswhere31~

3

3

1~ ≡+

= χχχ

1~~

vt fνρµ =

. viscositykinematicmolecular theis v

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

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

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 ==+

=ε

ηηη

RTkMMY

xTgG

xu

uuG ttM

it

tib

i

jjik γ

ρµβρ ==∂∂

=∂∂′′−= ,2,

Pr, 2

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

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

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