1. Ecoulement laminaire stationnaire dans un convergent
axisymetrique : modelisation et experimentation
Y. Fujiso, B. Wu, X. Grandchamp, A. Van Hirtum
Gipsa-Lab, UMR CNRS 5216, Grenoble University, France

2. INTRODUCTION: FLOW DEVELOPMENT
Importance of flow inlet conditions and need for nozzle design
• Well established in aeronautics (wind tunnels, Re > 104) and
technological jet applications where efforts are done with respect
to nozzle design in order to generate and control well defined inlet
conditions [Metha1979, Mikhail1979, Morel1975, Fang2001]
- nozzle contraction ratio, length, exit diameter, etc.
• Not or marginally considered in bio fluid mechanics and bio
aero acoustics: respiration, whizzle, human speech sound
production, etc.
- with respect to upper airways:
most research deals with voiced sound
production
(e.g. vowels) for which inlet conditions are
neglected so far
Sataloff (1992)
2

3. INTRODUCTION
Importance of flow inlet conditions and need for nozzle design
BREATHING
LUNGS Flow through vocal tract
ZONE
Sataloff (1992) from in-vivo
observations
with Re < #104
➡ Suitable nozzle design for experimental studies on
simplified in-vitro replicas of upper airway portions providing
low turbulence intensity and uniform mean inlet flow
3

4. OBJECTIVES
• Nozzle design for experimental study of upper airway flow
- Re < 104, Ma < 0.2, geometrical dimensions
• Experimental characterisation of flow at the nozzle exit
- Reynolds number, mean velocity profile, turbulence intensity
• Modelling of flow as function of Reynolds number
➡ Study of flow through a small nozzle at low to
moderate Reynolds numbers
4

5. OUTLINE
• Introduction: inlet conditions & nozzle design
• Nozzle design
• Experimental characterisation of nozzle exit flow
• Modelling of nozzle flow
• Conclusion
5

6. NOZZLE DESIGN
‣ Sought inlet conditions:
- uniform mean flow
- reduced turbulence level
‣ Parameterised (L, D1, D2, xm) contraction nozzle R(x) of two
matched cubics inspired on wind tunnel design [Metha1979,
Mikhail1979, Morel1975, Fang2001].
D1 D2 (x/L)3 D2
R(x) = − 1− + , x ≤ xm
2 2 (xm /L)2 2
D1 D2 (1 − x/L)3 D2
R(x) = − + , x > xm
2 2 (1 − xm /L)2 2
inlet x=0 exit x=L
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7. NOZZLE DESIGN
Parameterised (L, D1, D2, xm) contraction nozzle:
100mm
20 - 25mm
60mm
52mm
• low and moderate Reexit (300 < Reexit < 20200)
=> small D2 (20 < D2 < 25mm)
• tolerance for flow irregularities such as flow separation
=> large area contraction ratio with D1 = 100mm (15 < CR < 22)
• total length short compared with upper airway length of 18cm
=> L = 6cm or 0.6 times the inlet diameter D1
• short outlet length to avoid boundary layer thickening at the exit
=> xm = 52mm or xm = 0.86L so that the outlet length is less than
5mm
7

8. OUTLINE
• Introduction: inlet conditions & nozzle design
• Nozzle design
• Experimental characterisation of nozzle exit flow
• Modelling of nozzle flow
• Conclusion
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9. EXPERIMENTAL VALIDATION OF NOZZLE EXIT FLOW
• Parameterised (mm) contraction nozzle: L=60, D1=100, D2=21.4, xm=52
• Single hot film anemometry at nozzle exit: transverse velocity profiles
- wall to wall (spatial accuracy 0.1mm)
- wall neighborhood (spatial accuracy 0.01mm)
D1/2
U (x=L,!D /2 ! y ! D /2)
e 2 2
5 < Q < 305 l/min U (x=0)
1
U (x=L)
2 " y = 0.1mm
flow conditioning U (0 ! x ! L,y=0)
" y = 0.01mm
c
(x!L)/D < 0.04
2
!D1/2
300 < Reexit < 20200 160000 samples at each measurement station
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10. EXPERIMENTAL VALIDATION OF NOZZLE EXIT FLOW
Comparison of transverse mean profiles near the wall for
different Re
- from wall to wall profiles with 0.1 mm steps (symbols)
- from wall neighboring profiles with 0.01 mm steps (dots)
D1/2
Ue(x=L,!D2/2 ! y ! D2/2)
U1(x=0) U (x=L) " y = 0.1mm
2
" y = 0.01mm
U (0 ! x ! L,y=0)
c
(x!L)/D < 0.04
2
!D1/2
good match between both profiles
=> no error due to positioning of the hot film at (x-L)/D2 < 0.04
10

11. EXPERIMENTAL VALIDATION OF NOZZLE EXIT FLOW
Normalised transverse mean velocity profiles:
• measured wall to wall for different Reexit (symbols)
• mean velocity profiles (lines)
1
661
0.8 1322 mean velocity profiles:
3966 - developed pipe flow (parabolic)
7932 - turbulent flow (1/7 power law)
U (y)/U [!]
0.6
11898 - top hat with without boundary layer (uniform)
2
15898 - top hat with boundary layer 2 = 0.004D2 (top
0.4 hat) δ
e
19830
parabolic
0.2 power law
uniform
top hat
0
!0.5 !0.4 !0.3 !0.2 !0.1 0 0.1 0.2 0.3 0.4 0.5
y/D2 [!]
✓uniform core flow (variation < 0.5%):
•|y|/D2 < 0.4 for Reexit > 3000
•|y|/D2 < 0.25 for Reexit < 3000
✓small velocity overshoot due to vena contracta
•same order of magnitude as observed by [Mi et al.
2001]
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12. EXPERIMENTAL VALIDATION OF NOZZLE EXIT FLOW
Turbulence intensities TU for different exit Reynolds numbers:
•wall to wall transverse profiles (symbols)
•initial centerline turbulence intensity levels (zoom)
< 1%
✓uniform mean core flow associated with uniform turbulence
intensities:
•|y|/D2 < 0.4 for Reexit > 3000
•|y|/D2 < 0.25 for Reexit < 3000
✓low turbulence intensities for all exit Reynolds numbers
•0.3 < TU < 1% along centerline (TU < 0.6% for Reexit > 3000)
•TU < 2% in boundary layer => laminar boundary layer ?
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13. EXPERIMENTAL VALIDATION OF NOZZLE EXIT FLOW
Determine nature of the boundary layer:
• compare with theoretical laminar Blasius profile
• compare with laminar law of the wall
!
✓laminar boundary
layer
13

14. OUTLINE
• Introduction: inlet conditions & nozzle design
• Nozzle design
• Experimental characterisation of nozzle exit flow
• Modelling of nozzle flow
• Conclusion
14

15. MODELLING OF NOZZLE FLOW
Model choice
- assume laminar flow and boundary layer
development
- model previously used with respect to upper
airway flow
Thwaites laminar axisymmetrical boundary layer solution
100mm
20 - 25mm
60m
52m
• flow properties throughout the nozzle
• validation of the model on measured transverse flow profiles at the exit
• evaluate influence of small variations of D2 and therefore contraction
ratio CR
in the range 300 < Reexit <
15

16. MODELLING OF NOZZLE FLOW
Thwaites laminar axisymmetrical boundary layer solution
implemented in an iterative algorithm
calculate laminar boundary layer momentum thickness as a function of x
∞ ∞
u(x, y) u(x, y) u(x, y)
δ2 = 1− dy δ1 = 1− dy
0 U (x) U (x) 0 U (x)
momentum thickness displacement thickness
with Thwaites equation using quasi-similarity assumptions
x
δ2 (x)R2 (x)U 6 (x)
2
− δ2 (0)R2 (0)U 6 (0)
2
= 0.45ν R2 (x)U 5 (x)dx
inlet conditions 0
2
δ2 ∂U (x) δ2 ∂U (x) δ1 (x) ∂U τ δ2
λ=− S(λ) = H(λ) = τ = ρν ⇒ S(λ) =
ν ∂x U (x) ∂y δ2 (x) ∂y ρνU
zero at
Thwaites parameter skin friction parameter shape parameter wall shear stress separation
•from experimental data as tabulated values or universal functions
•both can be put in same form by adding constants cS,H in universal functions:
0.018λ
S(λ)|0≤λ≤0.1 = 1.8λ2 + 1.57λ + 0.22 + cS S(λ)|−0.1≤λ≤0 = + 1.402λ + 0.22 + cS
0.107 + λ
H(λ)|0≤λ≤0.1 = 5.24λ2 − 3.75λ + 2.61 + cH 0.0731
H(λ)|−0.1≤λ≤0 = + 2.088 + cH
0.14 + λ
wit 0 ≤ cH ≤ 0.35 an −0.02 ≤ cS ≤ 0
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17. MODELLING OF NOZZLE FLOW
Thwaites laminar axisymmetrical boundary layer solution
choice two values for constants:
•tabulated values cS = 0.35 and cH = -0.02
•universal functions cS,H = 0
"
"
! !
non zero instead of zero constants
•H increases between 10 and 25%
•S decreases between 5 and 10%
➡ facilitates flow separation
validation
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18. MODELLING OF NOZZLE FLOW
• Parameterised (mm) contraction nozzle: L=60, D1=100, D2=21.4,
xm=52
• Flow through the nozzle for zero constants cH,S=0
normalised centerline velocity normalised flow acceleration
xm xm
maximum
!
CR=21.8
ideal fluid
•Uc increases with x until maximum of Uc at nozzle exit
•common flow acceleration and hence Uc for Reexit > 3000
•increased acceleration and so Uc with Reynolds number for Reexit < 3000
•maximum acceleration downstream xm independent from Reexit
•reduced flow acceleration associated with boundary layer development
• flow deceleration result in Uc > ideal fluid due to boundary layer development
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19. MODELLING OF NOZZLE FLOW
• Parameterised (mm) contraction nozzle: L=60, D1=100, D2=21.4,
xm=52
• Flow through the nozzle for zero constants cH,S=0
normalised displacement thickness normalised wall shear stress
maximum xm
minimum
!
! "
xm
• boundary layer develops in inlet and final section associated with flow
deceleration
• severe increase of boundary layer development for Reexit < 3000
• minimum corresponds to maximum flow acceleration
• shape factor reduced gradually with x: H ~ 2.95
• skin friction increases gradually with x: S ~ 0.2
• no flow separation along the nozzle wall
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20. MODELLING OF NOZZLE FLOW
• Parameterised (mm) contraction nozzle: L=60, D1=100, D2=21.4,
xm=52
• Modelled and measured exit centerline velocity
!
CR=21.8
ideal fluid
Reexit > 3000
• good match between model and measurements for Reexit > 3000 (<
2%)
• no influence of cH,S values for Reexit > 3000
• influence of cH,S for 300 < Reexit < 3000
• non zero cH,S improve model outcome for 1000 < Reexit < 3000 (<
4%)
• for Reexit < 1000 error increases to 20% or more 20

21. MODELLING OF NOZZLE FLOW
• Parameterised (mm) contraction nozzle: L=60, D1=100, D2=21.4,
xm=52
• Modelled and measured exit boundary layer characteristics
normalised displacement thickness normalised momentum thickness
" "
!
!
• match between model and spatial accurate boundary layer
measurements
• no influence of cH,S for momentum thickness
• no influence of cH,S values for Reexit > 3000
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22. OUTLINE
• Introduction: inlet conditions & nozzle design
• Nozzle design
• Experimental characterisation of nozzle exit flow
• Modelling of nozzle flow
• Conclusion
22

23. CONCLUSION
in the range 300 < Reexit <
• parameterised axisymmetrical nozzle suitable for study of upper airway
flow
• uniform mean flow
• low turbulence intensities
• laminar boundary layer
100mm
21.4mm
60mm
52mm
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24. CONCLUSION
Thwaites laminar axisymmetrical boundary layer solution
applied to flow through the nozzle
• no flow separation along the nozzle walls
• need for spatial accurate boundary layer measurements
• quantitatively good model outcome for Reexit > 1000
• for Reexit > 3000 no influence of cH,S
• 1000 < Reexit < 3000 influence of cH,S (non zero improves outcome)
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25. Thank you for your attention.
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