12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

BOUNDARY LAYER TRANSITION IN SONIC CONVERGING-DIVERGINGNOZZLES

Bruno Zebrowski
Department of Fluid, Thermal and Combustion
Institut Pprime - CNRS, ENSMA
86962 Chasseneuil-du-Poitou, France
bruno.zebrowski@ensma.fr

Peter Jordan
Department of Fluid, Thermal and Combustion
Institut Pprime - CNRS, Université de Poitiers
86962 Chasseneuil-du-Poitou, France
peter.jordan@univ-poitiers.fr

Vincent Jaunet
Department of Fluid, Thermal and Combustion
Institut Pprime - CNRS, ENSMA
86962 Chasseneuil-du-Poitou, France
vincent.jaunet@ensma.fr

ABSTRACT
We study the laminar-to-turbulent boundary layer transition of converging-diverging sonic nozzles used as standard
flow meters. Stability characteristics are addressed using thelinear stability framework under the parallel flow assumption.
We show that the boundary layer is stable to modal disturbances but can experience transient-growth of energy in theform of streaks of high and low velocity. The peak of energy
associated to transient-growth is expected to happen within thethroat region of the nozzle.
Transition trends are investigated using N factors. A
higher acceleration associated with a small inlet radius tends
to stabilise the non-modal mechanism. Freestream turbulenceand surface roughness are discussed as possible receptivity
mechanisms for transient-growth to occur. It is shown thatroughness is unlikely to trigger the transition process.

R

4◦

D

Figure 1.

Sketch of the nozzle geometry under investigation

ifi et al. (1996), Tumin & Reshotko (2001)), where streamwise vortices result in the formation of streaks of high and low
streamwise velocity via the lift-up.
We investigate the stability characteristics of boundary
layers in nozzles with an arc-circle throat shown in Figure
1, similar to those used as references in high-pressure mass
flow rate applications. We first present the model used to obtain boundary layers for such flows that will be used as baseflow. We investigate the spatial modal stability using the linear stability analysis framework for a compressible boundary
layer. Then, we study non-modal mechanisms through transient growth analysis. N factors are computed and comparedwith literature data. The analysis is extended to other nozzle
geometries by varying the ratio R/D. Finally, the roughnessinduced transition is discussed based on the numerical simulation database of Bernardini et al. (2014) and through measurements of an analogous surface.

INTRODUCTION
This work is motivated by boundary layer transition that
occurs in converging-diverging sonic nozzles used in industrial
applications as flow meters. The boundary layer displacement
thickness at the nozzle throat influences the mass flow rateby reducing the effective sonic area. In standard dischargecoefficient models used to calibrate flow meters, laminar andturbulent boundary layer models were established for a set ofnozzle shapes by Stratford (1964) and Geropp (1971). However, there is little information about the transitional regime,
introducing uncertainty about the validity range of these models.
A common approach to study boundary layer transition isbased on modal growth of infinitesimal disturbances on a linearised base flow. The boundary layer in the converging section of the nozzle ranges from subsonic to transonic. For a flat
plate in a transonic regime, modal transition is characterized
by a Tollmiens-Schlichting (T.S.) wave that is most unstablefor plane 2D wave (Mack (1984) and Özgen & Kircali (2008)).
However, a favourable pressure gradient is known to havea stabilizing effect on this mode Masad & Zurigat (1994)).
Boundary layers are also subject to transient growth (Han-

BASE FLOWThe boundary layer is modelled using the analytic solution proposed by Geropp (1971). The model is based on thecompressible boundary layer equations with pressure gradientand an adiabatic wall. The equations are simplified by introducing the Prandtl number of unity and by using a linear relation between viscosity and temperature instead of the classicalSutherland’s formula. The equations are worked out by in-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
troducing self-similar velocity profiles that can be set for any
given nozzle geometry by evaluating the streamwise pressure
gradient at the nozzle throat. At the nozzle throat, the boundary layer equation is

D
y=



γ+1
2

 γ+1

4(γ−1)"r√
2 + u/uδ
−
2
atanh
 1
3
γ−1 2(γ−1)

√R0 m 1 − γ+1√
2 atanh

r

2
+
3

##
r
γ−1 "
r
u
2
2
u
γ+1
√ −
+2 ,
(1 − )3uδuδ
3
1 − γ−1γ+1(1)Figure 2. Boundary layer velocity profile in nozzle. Comparison between Geropp self-similar model and Blasius flat plate
solution.

where y is the wall normal position for a given value of velocity u/uδ , where the value of u/uδ ∈ [0, 1], R0 = a0 D/ν0
is a stagnation-based Reynolds number with a0 the stagnation
speed of sound, D the nozzle throat diameter, ν0 the kinematic
viscosity at rest, M the Mach number, γ is the isentropic exponent and the subscript δ denotes the value at the boundary
layer edge. The acceleration parameter m is here chosen sothat it has the same pressure gradient at the throat as a nozzle
with an arc-circle throat

The linearised N.-S. operator L is further simplified byconsidering a locally parallel baseflow where the wall-normal
direction is inhomogenousq̄(x, y, z,t) = q̄(y),The density and temperature are made non-dimensionalby their reference values at the boundary layer edge δ and thevelocities by the speed of sound outside the boundary layercδ . The reference length is the boundary layer displacement
thickness δ ∗ . The corresponding Reynolds number isr

4 D
γ +1 Rm= 
 1−2γ ,γ + 1 γ−12(2)Reδ ∗ =

where R is the radius of the nozzle profile at the throat. In
Equation 1, the boundary layer scales with the Reynolds number R0 and the acceleration parameter m. As a result, the velocity profile shown in Figure 2 is self-similar and holds forany ratio R/D. In order to facilitate comparison with available
data in the literature, we will use a Reynolds number based on
the nozzle throat diameterρ u D
Ret = δ δ ,µ0(5)ρ̄δ ūδ δ ∗
µ̄δ

(6)

For a nozzle with R/D = 2 at Ret = 106 , the computed
displacement thickness-based Reynolds number is Reδ ∗ =1057.

Spatial modal stability
The spatial modal stability of the boundary layer isadressed by considering the normal-mode ansatz(3)

where µ0 is the stagnation viscosity.
In the following, unless otherwise specified, the nozzle ischosen with a ratio R/D = 2, corresponding to the most commonly used nozzle geometry for mass flow measurement discussed in the introduction. The throat Reynolds number is set
at Ret = 106 , corresponding to conditions where the transition
is commonly observed.

q0 (x, y, z,t) = q̂(y)ei(αx+β z−ωt) ,(7)

where α and β are the streamwise and spanwise wavenumbersand ω the frequency. In matrix form, the perturbation equationishi−iω q̂ = A0 (β ) + Ax (β )α + Axx α 2 q̂.

(8)

STABILITY ANALYSIS
To perform the stability investigation, the compressible
Navier-Stokes (N.-S.) equations are linearised by introducing the decomposition q = q̄ + q0 , where the state vectorq = [ρ, u, v, w, T ]> contains the density, the 3 velocity components and the temperature, q0 is the state vector of fluctuating
quantities and q̄ the base state. Under the small perturbation
assumption, the linearised N.-S. equations are obtained∂ 0
q = Lq0 .
∂twith α ∈ C, β ∈ R and ω ∈ R. The matrices A0 , Ax and Axx
are the constant coefficients of the linearised N.-S. equations.
The polynomial eigenvalue problem in Equation 8 is solvedfor prescribed values of β and ω. Only the downstreampropagating modes are retained by applying the Briggs-Bers
criterion (Briggs (1964) and Bers (1983)). The stability of theresulting eigenvalues α is given by Equation 7.
The eigenvalue problem is solved for pairs of wavenumbers varying from β = [0, 3] and ω = [0, 3]. Figure 3 presents
a map of minimum imaginary part over all eigenvalues found(4)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
and the resulting perturbation equation
∂∂2−iω q̂ = L q̂ = A0 (β ) + A1 (β ) + A2 2 q̂.
∂x
∂x

(10)

The solution to Equation 10 is sought in the formq̂(x, y) = qˆ0 (y)eL x(11)where qˆ0 (y) = q̂(y, x = 0) is an initial small pertubation vector.
The spatial growth of an initial perturbation is given by

Figure 3.

Map of Imaginary part of α at Re∗δ = 1057.

G(x, β , ω) =for each computed pair of [β , ω], where acoustic modes havebeen filterd out. For each pair of wavenumbers we have thatmin(ℑ(α)) > 0. Since we have only kept the downstreamtravelling modes, it shows that the flow is modally stable.
In flows without pressure gradient, the Tollmien-Schlichting
wave is unstable. In the present case, it is stabilised by thefavorable pressure gradient, as illustrated in Figure 4.

kq̂(x)k2E.
kqˆ0 k2E(12)where G is the gain, defined as the energy norm of the perturbation at a given x-location q̂(x) produced by an initial disturbance q̂0 over the energy norm of that same inital disturbance.
Following the work of Chu (1965), the energy norm is chosen
for the inner product hq, qiE = qH We q
We = DiagT̄ρ̄, ρ̄, ρ̄, ρ̄,.
ρ̄γT̄ γ(γ − 1)

(13)

Equation 12 is solved using optimal perturbations by performing the singular value decomposition of the matrix eL x
after application of the energy norm. The detailed method to
obtain the gain can be found in Jordan et al. (2017).
Spatial non-modal growth is computed for ω ∈ [0, 0.5]
and β ∈ [0, 2.5] for a total of 2500 pairs. The highest gain
achieved over distance
Gmax (ω, β ) = max G(ω, β , x).
x(14)

is presented on Figure 5. It is shown that significant transient-Figure 4. Spatial eigenspectrums for β = 0, ω = 0.1 andReδ ∗ = 1057. Comparison of Blasius B.L. without pressure
gradient and the current nozzle B.L. with favorable pressure
gradient.

This has been repeated for higher Reynolds numbers (upto Ret = 107 ) and the same conclusions have been drawn. We
therefore conclude that the boundary layer in the choked region of the nozzle considered is modally stable, and transition
must therefore be underpinned by non-modal mechanisms.

Spatial non-modal growth
Despite a stable modal behavior, transient-growth occurs
due to the non-orthogonality of the linear operator L. If small
perturbations experience enough growth before decaying, they
may either affect the base flow q̄ in a new modally unstablestate or trigger nonlinear terms leading to turbulence.
The transient-growth analysis is performed by considering the normal-mode ansatz
q0 (x, y, z,t) = q̂(x, y)ei(β z−ωt) ,Figure 5. Gmax map at Re∗δ = 1057.

growth occurs for ω = 0, corresponding to an optimal perturbation made of streamwise vortices leading to steady streaks ofhigh and low velocity. The highest gain is acheived at β ∼ 0.61
and ω = 0. In general, high gains are confined at low frequencies ω and spanwise wavenumbers β ∈ [0.4 − 1].

(9)312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
TRANSITION CHARACTERISTIC
The gain is found to scale with the Reynolds number aswell as the spatial length of development, as mentionned in thecalculation of Tumin & Reshotko (2001). In the case of thenozzle boundary layer, it is found that the dimensional lengthof development x∗ scales with the throat diameter. The ratio
R/D also modifies the boundary layer displacement thickness
at the nozzle throat, therefore the length of streaks development is also affected by this ratio. Gains are presented in Figure 6 for various R/D. The highest gain are achieved at the∗ between 0.4D and 0.9D, depending on
optimal distance xopt
R/D, for ω = 0 and β = 0.61.

0.996

9

0.994

8

0.992

7

0.99

6

0.988

5

0.986
10 5

10 6

4
10 7

10 -4

7

Figure 7. Discharge coefficient (Cd): black curve (left axis)
and N factor: orange curve (right axis) as function of throat
based Reynolds number Ret . The discharge coefficient is a fitof experimental data for a nozzle of R/D = 2 (see Ishibashi(2015)).

6
5
4
3

From Ret = 105 to Ret = 106 , the boundary layer is laminar.
The drop in discharge coefficient, attributed to transition, is
progressive over the range Ret = [1 · 106 − 2 · 106 ]. It then increases again in the turbulent regime Ret > 2 · 106 . The N factor associated to this nozzle is continuoulsy increasing fromN = 4.3 at Ret = 105 to N = 9 at Ret = 107 . In this comparison, the transition has occurred at N ≈ 7.

2
1
0
0

0.5

1

1.5

2

Figure 6. Spatial Gain G(ω = 0, β = 0.61) scaled with
Reynolds number. The distance is expressed in throat diameter, x∗ /D and is independent of Reynolds number.

The N factor calculation is repeated for other R/D andpresented in Figure 8. At a given Reynolds number, the N
factor increases with R/D as a result of a thicker boundary
layer at the nozzle throat.

These optimal lengths indicates that the peak of energy
will occur in the throat region. This makes the non-modal spatial growth a plausible candidate to explain the transition inthese accelerated transonic flows.

109

N factors

8

The relations found between the gain, the Reynolds number and R/D allow to compute N factors

7
6

N = ln(Gmax )

(15)

5
4

to predict the trend of transition in a parametric study. The N
factor is a parameter used to correlate the turbulence intensityof the freestream flow to the transition. It is commonly applied to the exponential growth of unstable mode on flat plateboundary layers. Since the present model is based on nonmodal growth, the N factor cannot be directly compared withflat plate boundary layers. However, the N factors can be presented alongside experimental discharge coefficients obtained
for the same nozzle geometry. We base the reference pointon the experiments carried by Ishibashi (2015), in which thedischarge coefficient is measured for a nozzle of R/D = 2. In
Figure 7, it is compared the discharge coefficient obtained by
Ishibashi from experimental data and the N factor computedby the present method for the same nozzle. As the Reynolds
number increases, the discharge coefficient approaches unity
as a result of a thinner boundary layer at the nozzle throat.

3
10 5

10 6

10 7

Figure 8. N factor as a fonction of Ret for various nozzle
ratio R/D

In a scenario where the transition occurs at a fixed N factor, the lower R/D, the higher the transition Reynolds number
will be. The higher acceleration associated with smaller R/Dtends to stabilise the non-modal mechanism. This trend is inaccordance with the experiments of Ishibashi (2015), where
the higher R/D showed an earlier transition.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
THE DISTURBANCES ORIGINSThe spatial transient-growth analysis is based on the optimal disturbance theory, where we have selected the optimal
perturbation in order to maximize energy growth. An insight
into the possible origins of disturbances is necessary to complete the analysis. There are two main candidates: the first is
freestream forcing due to the turbulence that naturally occurs
in any flow. The second is wall roughness, where distributed
or isolated roughness elements can introduce disturbances intheir wake.

Freestream disturbances
Among the experiments conducted by Ishibashi (2015),
the discharge coefficients of two nozzles of the same geometry were compared using two different configurations. In thefirst, the standard shape of Figure 1 was used. In the second,an obstacle is introduced upstream of the nozzle, as shown in
Figure 9

Figure 10. Evolution of the roughness based Reynolds number Re2k as a function of roughness height k/D. The shaded
area represents the expected critical Re2k above which thewake behind the roughness element becomes turbulent. Different curves are for different flow conditions Ret .

Rρk evaluated in the undisturbed boundary layer at the height k
and the viscosity evaluated at wall µk
4◦DFigure 9.

Re2k =

ρk uk kµk

(16)Nozzle with obstacle
is convenient to describe the critical roughness height that willcause transition to turbulence. Depending on the roughness
shape, the wake introduced behind the obstacle would breakdown to turbulence if Re2k > [400 − 700]. Using the analytical formulation of the boundary layer profile we consider, it ispossible to calculate this Reynolds number as a function of theroughness height, and thus to determine the critical roughness
height as a function of Ret .
Figure 10 presents Re2k at the nozzle throat as a function of roughness height for five Ret . We note that at constant Re2k , as Ret is increased, the critical roughness heightdecreases as a result of a thinner boundary layer. In therange where Ishibashi (2015) observed transition, that is Ret =[1 · 106 − 2 · 106 ], the computed critical roughness height lies
somewhere in the range k/D ' [0.5 − 1] · 10−3 , which correspond to k ' [1 − 2] · 10−5 m when scaled by the nozzle diameter used in the experiments. These values of roughness heightare incompatible with the polished surface used. They correspond to the surface quality obtained via sanding with roughsandpaper. When standard polishing techniques are applied,the expected height of the peaks of roughness is of the orderof k ' 1 · 10−7 m, two orders of magnitude below the minimaltransition height estimated. Hence, in that case, it is thoughtthat roughness-induced transition is not the dominant mechanism.
Standard nozzles for mass flow rate measurement are required to have a maximum surface roughness proportional to
the throat diameter to reduce effects on the discharge coefficient. This requirement is set at Ra < 15 · 10−6 × D, where Ra
is the arithmetical mean deviation of the roughness profile z

The nozzles were mirror polished to eliminate roughness
effects on transition. The transition Reynolds number was
lower when the obstacle was introduced. Although no quantitative measurements were carried out, the earlier transition
was suggested to be related to upstream conditions, such as
geometry and disturbances. We can further add that the modifications made on the tested nozzles do not modify the laminar boundary layer profile, as the discharge coefficient in thelaminar regime remained unaffected by the changes in geometry. The present analysis suggests that the modified geometry introduces disturbances into the boundary layer. Since thebase flow is believed to be the same for both nozzles, the transition mechanism we discuss in this paper is consistent withthe hypothesis of Ishibashi. This also suggests that the turbulence intensity of the incoming flow may influence the transition Reynolds number, despite the strongly favorable pressure
gradient in the converging section of the nozzle. This is ofimportance because this is currently not considered when performing mass flow rate measurement on different facilities that
may have different incoming turbulence levels.

Disturbance from isolated roughness
Wall roughness is known to cause transition to turbulence
provided it is sufficiently large compared to the thickness ofthe boundary layer. In a numerical investigation, Bernardini
et al. (2014) performed a parametric study of boundary layertransition induced by an isolated roughness element. They varied the shape, the height, the Reynolds number, and the Mach
number and found that a Reynolds number Re2k based on theheight of the roughness elements k, the velocity uk and density

Ra =

5

1
l

Z l
0

|z|dx

(17)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
transition occurs at values of N = 7, which is lower than thevalue typically associated with a modal transition. The analysis shows that a smaller inlet in the convergent section will
result in delayed transition, indicating that stronger acceleration attenuates non-modal growth mechanisms.
Experimental investigations in the literature have shownthat the transition Reynolds number is sensitive to the disturbances injected upstream of the nozzle throat. The current
study suggests that the incoming disturbances are amplified via
the lift-up mechanism. We have provided an analysis of thepotential of isolated roughness-induced transition and shownthat it is unlikely to occur for nozzles with surface finish required in industrial configurations. Moreover, the analysis of asanded surface shows that the surface roughness height distribution is out of scale compared to optimal disturbances. Thisconfirms and strengthens the hypothesis that roughness is not
the underpinning factor for transition in these configurations.
Figure 11. Flow field of the optimal initial disturbance. The
dashed line is the boundary layer thickness δ , the black line is
the measured wall roughness. Ret = 1.75 · 106 .

REFERENCES
Bernardini, Matteo, Pirozzoli, Sergio, Orlandi, Paolo & Lele,
Sanjiva K. 2014 Parameterization of Boundary-Layer Transition Induced by Isolated Roughness Elements 52 (10),2261–2269.
Bers, A. 1983 Space-time evolution of plasma instabilitiesabsolute and convective. In Basic plasma physics., , vol. 1.
Amsterdam : North-Holland.
Briggs, R.J. 1964 Electron-stream interaction with plasmas.
Tech. Rep.. MIT Press.
Chu, Boa-Teh 1965 On the energy transfer to small disturbances in fluid flow (Part I) 1 (3), 215–234.
Geropp, D. 1971 Laminare grenzschichten in ebenen undrotationssymmetrischen lavaldsen. In Deutsche Luft- undRaumfahrt; 71,90.
Hanifi, A, Schmid, P.J. & Henningson, D.S. 1996 Transient
growth in compressible boundary layer flow. Physics of Fluids 8 (3), 826–837.
Ishibashi, M. 2015 Discharge coefficient equation for criticalflow toroidal-throat venturi nozzles covering the boundarylayer transition regime. Flow Measurement and Instrumentation 44, 107–121.
Jordan, Peter, Zhang, Mengqi, Lehnasch, Guillaume & Cavalieri, Andr V. 2017 Modal and non-modal linear wavepacket
dynamics in turbulent jets. In 23rd AIAA/CEAS Aeroacoustics Conference. American Institute of Aeronautics and Astronautics.
Mack, L. M. 1984 Boundary layer stability theory. In AGARD
Report No. 709: Special Course on Stability and Transitionof Laminar Flow. AGARD.
Masad, J. A. & Zurigat, Y. H. 1994 Effect of pressure gradienton first mode of instability in compressible boundary layers.
Physics of Fluids 6 (12), 3945–3953.
Özgen, S. & Kircali, S. A. 2008 Linear stability analysis incompressible, flat-plate boundary-layers. Theoretical andComputational Fluid Dynamics 22 (1), 1–20.
Stratford, B. S. 1964 The calculation of the discharge coefficient of profiled choked nozzles and the optimum profile
for absolute air flow measurement. The Journal of the Royal
Aeronautical Society 68 (640), 237245.
Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers 13 (7), 2097–2104.

over the length l considered. To ensure that the current analysis
is also valid in standard cases and not only applies to mirrorpolished nozzles, we perform a roughness study. A piece ofaluminum is sanded to meet the roughness required by thenorm. The surface roughness is characterized using a Talysurf 6000 CCI (Coherence correlation interferometry) with awall-normal resolution of 0.01nm.
The roughness obtained is Ra = 15 · 10−6 × D, effectively
meeting the requirement. A roughness profile is extracted andcompared to an optimal disturbance in Figure 11. It showsthat the roughness height is orders of magnitude smaller thanthe boundary layer thickness and the optimal disturbance. The
periodicity of the optimal disturbance is also much larger thanthat of the roughness profile. The scale separation further indicates that roughness is not the underpinning factor for transition in these configurations.

CONCLUSION
We have used an analytical boundary layer formulation toinvestigate the transition observed within the throat section ofa converging-diverging nozzle. In the locally parallel framework, the stability analysis reveals that transition is not related
to a modal mechanism. The T.S. mode is stabilised by thefavorable pressure gradient, and no unstable mode is foundfor nozzle baseflow. Transient-growth analysis shows that significant energy amplification is likely to occur via the lift-up
mechanism, which would result in streamwise streaks of low
and high speed.
The spatial transient-growth analysis is performed in aparabolized formulation in which upstream-traveling modes
are removed. While the gain is proportional to Re2δ ∗ , the spatial
length at which the maximum gain is achieved is found to scalewith the throat diameter. This finding is an important result as
it indicates that transient-growth is compatible with the geometry, the spatial length for maximum energy growth being of
the order of the nozzle diameter. The optimal gain formulation
was used to compute N factors for various nozzle geometries.
Comparison with experimental data suggests that non-modal

6