AIAA JOURNAL

Length Scales and Time Scales of a Heated
Shock-Wave/Boundary-Layer Interaction
V. Jaunet∗
Centre National d'Etudes Spatiales, Direction des Lanceurs, Paris, France
andJ. F. Debiève† and P. Dupont†
Aix-Marseille Université, 13453 Marseille, France

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DOI: 10.2514/1.J052869
The wall-temperature effects on the behavior of a shock-induced separation at Mach number 2.3 are experimentallyinvestigated. Schlieren visualizations, velocity measurements with particle image velocimetry, and time-resolved hotwire measurements are used to describe how the wall temperature influences the length and timescales of the interaction.
It is shown that heating has a large influence on the size and on the low-frequency unsteadiness of the interaction, but
influences only slightly the onset of the separation. Models, initially proposed to describe adiabatic shock-wave/
boundary-layer interaction, are shown to be effective to obtain similarities for the length and timescales between theheated and adiabatic interactions considered at the same external conditions, as long as the heating effects on the friction
coefficient are taken into account. Comparisons with cooled interactions at similar external conditions are also given.

NomenclatureaCfgr; sg3 θ; β
L
Mc
qe
R
r
Se
SrL
SrL
s
T
Tr
Tw
U0
U1
X0
β
γ
δ
δ0
θ
ΦMc 

=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=

sound velocity
friction coefficientcorrective function for the Strouhal number
normalizing function for L
interaction length
convective Mach number
dynamic pressure
specific gas constantvelocity ratio across the mixing layer
separation criterion based on external parameters
Strouhal number
weighted Strouhal number
density ratio across the mixing layer
static temperature
recovery temperaturewall temperatureexternal velocity
flow velocity downstream of the incident shock
mean position of the reflected shock
shock angle
ratio of specific heatboundary-layer displacement thickness
upstream boundary-layer thickness
flow deviation
normalized spreading rate of the compressible
mixing layers

I.

I

Introduction

N MANY aeronautical applications, unsteady events can createstrong and local loads that can affect the behavior of the system

Presented as Paper 2012-1095 at the 50th AIAA Aerospace Sciences
Meeting Including the New Horizons Forum and Aerospace Exposition,
Nashville, TN, 9–12 January 2012; received 4 July 2013; revision received 2
February 2014; accepted for publication 11 February 2014; published online
27 May 2014. Copyright © 2014 by the American Institute of Aeronautics andAstronautics, Inc. All rights reserved. Copies of this paper may be made for
personal or internal use, on condition that the copier pay the $10.00 per-copy
fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,
MA 01923; include the code 1533-385X/14 and $10.00 in correspondencewith the CCC.
*Postgraduate; currently PIV Specialist, Dantec Dynamics A/S,
Skovlunde, Denmark.
†Chargé de Recherche, IUSTI, UMR, CNRS 7343.

and even be harmful for the mechanical structure. In supersonic flow,such unsteadiness is found, for example, when a shock wave
impinges on a boundary layer and makes it separate. This interaction
produces strong low-frequency aerodynamic loads that are felt along
the whole flow downstream of the shock wave. The interaction withan incident shock and the compression ramp has been the two mostdocumented cases (see Délery and Marvin [1]). A recent review of themain properties of these flows can be found in Dolling [2]. The originof the low frequencies is not totally understood, and several models
have been suggested to explain their development.
On the one hand, some authors propose to relate the low-frequency
movements to instantaneous behavior of the upstream turbulent
boundary layer [3–6]. They conjectured that some motions of theshock in a M  5 compression corner could be related to the lowfrequency events of the upstream boundary layer. On the other hand,strong links have been observed between the low-frequency shock
oscillations and the dynamics of the separated region that develops
downstream [7,8]. Finally, some authors propose to describe theinteraction as an equivalent low-pass filter applied on the large band
fluctuations of the upstream boundary layer [9].
Dupont et al. [10] underlined the existence of a mixing layer within
the separation region of a M  2.3 shock-induced interaction. They
also showed that Kelvin/Helmholtz-like structures are created in thismixing layer. Recently, Piponniau et al. [11] proposed a model to link
the unsteady behavior of the shock motion to these structures
evolving in the mixing layer of the separated region. By taking into
account compressibility effects, they obtained a collapse of nondimensional frequencies, or Strouhal number, of the separated-region
motions from subsonic cases up to M  5. Finally, Souverein [12]and Souverein et al. [13] obtained recently a convincing collapse ofinteraction length using a displacement-thickness normalization anda separation criterion based on the upstream inviscid properties of theflow, which is consistent with the free-interaction concept.
When nonadiabatic wall conditions are considered, these different
results on the time- and length scales of the interaction have to bereconsidered. Indeed, the timescales of velocity and temperaturefluctuations in the upstream boundary layer may depend on the wall
heating flux as well as the size of the interaction.
The aim of this study was to provide experimental information on
the effects of wall heating on both the length and timescales of theinteraction. It is shown in this paper that the normalization of thelength of the interaction, as proposed by Souverein et al. [13], is still
relevant to explain the heating effects on the interaction length, butthat a slightly modified separation criterion has to be considered.
Moreover, Piponniau et al. [11] argued that compressibility and

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density effects have a direct influence on the interaction timescales.
Such effects can be controlled by heating the wall, and we propose
to verify that the observed changes in the low-frequency timescalesof the interaction are in agreement with the model of Piponniau
et al. [11].
The experimental setup is described in Sec. II. In Sec. III.A, we
examine the heating effects on the interaction length, which is used in
the Strouhal number used to qualify the frequency unsteadiness.
Then, the low-frequency shock unsteadiness of the adiabatic andheated cases is described in Sec. III.D, and the parametric dependencies given by [11] are discussed for the heated case.

Experimental Setup

The experiment was performed in S8, the Mach 2.3 supersonicwind tunnel at the Institut Universitaire des Systèmes Thermiques
Industriels. This is a continuous facility with a closed-loop circuit; its
test section is 120 mm high and 170 mm wide. The nominal
conditions are presented in the following. The incoming boundary
layer is turbulent, fully developed, and has a thickness ofδ0 0.99 × U0  ≃ 11 mm. In the external flow, the stagnationpressure is set to 0.5 × 105 Pa, and the total temperature is 300 K.
An oblique shock wave is produced using a shock generator madeof a sharp-edged plate. The shock generator is located in thefreestream on the top of the test section, and spans the entire width of
the test section. Its angle with respect to the external flow θ can be set
from 3 up to 10 deg. These flow deviations correspond, respectively,
to attached and massively separated cases.
The origin of the longitudinal coordinate x is chosen at the mean
position of the reflected shock at the wall X0 . The length of interaction L is defined as the distance between X0 and the extrapolation to
the wall of the incident shock. The floor of the S8 nozzle is equippedwith a heated section, which allows the ratio of wall to recovery
temperature T w ∕T r to be modified from adiabatic conditions(T w ∕T r  1) to T w ∕T r  1.9. This model is the same used inprevious studies [14–16]. The heated section starts downstream of theFig. 1

Schematic diagram of the heating apparatus.

2.2

2.2

z=0
z = -40z = -20z = 20z = 40

2.1

1.9

2
1.9

P/P1

Tp/Tf

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II.

contoured nozzle with a length of 500 mm ≃ 50 × δ0 . The strong
thermal stresses forced to design a block heater that does not span thewhole test section. Therefore, the heating part is 130 mm wide and iscentered with respect to the wind-tunnel axis. A sketch of the heating
apparatus is given in Fig. 1. A proportional–integral–derivative controller is used to control the wall temperature during the measurements.
This provides variations of wall temperature of less 2% of T w ∕T r .
The heating block is equipped with type-K thermocouples
mounted flush with the floor along the centerline of the nozzle tocontrol the wall-temperature distribution in the streamwise direction.
The spanwise temperature distribution is measured by sensors
symmetrically placed with respect to the longitudinal axis at two
different stations. A typical wall-temperature distribution is reported
in Fig. 2a for T w ∕T r  1.9. As the reader can notice, the wall
temperature is reasonably uniform in streamwise and transverse
directions within variations of less 2% of T w ∕T r .
In previous adiabatic experiments [8,11], unsteady wall-pressure
measurements were used to document the different timescales along
the interaction. In the present experiments, it was not possible toimplement such sensors in the heated wall. Therefore, hot-wire measurements are achieved with a StreamLine system from Dantec
Dynamics. The anemometer is used in a symmetric bridge configuration, and is operated at an overheat ratio of 0.6. The wire has adiameter of 5 μm with an aspect ratio of about 200. The bandwidth of
the system is about 100 kHz. Signals are digitized with a PXI-6133
National Instruments fast digitizer at a sampling frequency of
200 kHz with respect to the Nyquist criterion. Spectral analysis wasderived from 2 · 106 point samples corresponding to 10 s ofacquisition.
A particle-image-velocimetry (PIV) investigation is carried out
using a Dantec Dynamics system equipped with a 14 bits and 11
megapixels Hisense camera. The acquired field of view is 100 mm
long and 60 mm high, providing measurements in the wholeinteraction, including a portion of the incoming boundary layer. This
leads to a 38 pixels∕mm resolution minimizing the peak-locking
phenomenon. The light sheets are generated by a double-pulse Nd:
YAG laser New Wave Solo II, which delivers 30 mJ∕pulse, separated
in time by 1 μs. The measurements are made in a vertical plane, along
the longitudinal axis of the test section.
The boundary layer of the flow is seeded with incense smoke
particles. The particles are injected from the wall, upstream of thesonic section, using three holes placed symmetrically with respect tothe wind-tunnel axis.
The velocity vectors are obtained from an adaptive correlation
algorithm. Attention has been paid to the choice of the interrogationwindow size. Indeed, such windows have to be large enough to makepossible the calculation of correlations, and sufficiently small todiscretize the boundary layer. That is the reason why the image
correlations are calculated using an initial rectangular interrogationof 64 × 32 pixels during the first pass, and a smallest one of 32 × 16

1.8

1.6

θ=8.0, Tp/Tf=1.9
θ=8.0, Tp/Tf=1.9, Laurent [15]

1.3

1.7
1.6

1
1.5
0

100

200

300

400

500

220

x [mm]

240

260

280

x [mm]a) Temperature
Fig. 2b) Pressure
Wall quantities.

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pixels for the last pass. It leads to a physical cell size of about
0.8 × 0.4 mm representing 0.07δ0 × 0.035δ0 , which gives a sufficient
discretization of the domain. The interrogation cells are weighted by
Gaussian function to avoid Fourier-transform windowing effects, aswell as effects of particles entering or going out of the interrogation
area. This allows the use of an overlap of 75% between two interrogation areas. The data are then filtered using a 6 rms limiter.
Figures 3a and 3b present, respectively, the mean and second-ordermoments of velocity fluctuations in the upstream boundary layer.
In previous studies, the velocity and density profiles were derived
from pitot tube and thermocouple or hot-wire measurements (seeDebiève et al. [17] for more details). The velocity profiles arecompared with the present results in Fig. 3a. As it can be observed, avery good agreement is obtained for both experiments, showing thatthe experiment is highly repeatable. As for the mean velocity, thesecond-order moments of velocity fluctuations (see Fig. 3b) do notshow a significant difference between the adiabatic and heated
1.1

cases. From the velocity and density profiles, the boundary-layer
displacement thickness δ was derived in the upstream section of theinteraction. An increase of about 15% was obtained for the heated casewith respect to the adiabatic thickness (respectively, 3.5 and 4.1 mm).
In previous works [14,16], the friction coefficients were estimated for
adiabatic and heated interactions, and a similar reduction of about 15%
was observed (respectively, 2 × 10−3 and 1.7 × 10−3 ). Thus, theheating effects on the incoming boundary layer may be fully accounted
through changes in the temperature (i.e., density) profile.

III.

Effects of Heating on the Interaction

A. Characteristic Length of the Interaction

The overall organization of the shock-wave/boundary-layerinteraction (SWBLI) obtained by schlieren visualization is presented
in Fig. 4, in the case θ  8 deg, for the adiabatic and heated cases.
0.14

1u’/Ue, Tw/Tr=1v’/Ue, Tw/Tr=1

0.12u’/U , T /T =1.9ew

rv’/Ue, Tw/Tr=1.9

U/Ue

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0.9

0.1

0.8

0.08

0.7

0.06

0.6

0.04
Tw/Tr=1 (present)

0.5
0.4

Debiève et al. [16]T /T =1.9 (present)w r
Debiève et al. [16]

0

2

4

6

8

10

y(mm)

a) Streamwise velocity
Fig. 3Fig. 4

12

14

16

0.02
0

0

2

4

6

8

10

12

y(mm)

b) Second-order moments of velocity fluctuations
Mean profiles for the adiabatic and heated incoming boundary layer.

Schlieren visualization of the interaction for various wall temperatures, flow deviations of θ  8 deg.

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boundary layer separate. Note that this behavior is preserved when
the wall is heated, but with a slight shift, toward smaller flow
deviations, of the point where the change in slope occurs. This point
will be addressed further. For all angles, L increases significantly
when the wall temperature is set to T w ∕T r  1.9.
Using mass-conservation properties along the interaction,
Souverein [12] and Souverein et al. [13] proposed to normalize thelength of the interaction L by the displacement thickness of the
upstream boundary layer δ and a function g3 , taking into account theimposed flow deviation and the Mach number. From this model, thedimensionless length of interaction is given byLδ g3 β; θ(1)sinβ − θsinβ sinθ

(2)

with
g3 β; θ 

in which θ and β represent the deviation and shock angles,
respectively.
Using the normalized length of interaction L and a separation
criterion Se , it was possible to represent a large compilation of the
SWBLI’s interaction length on a single curve, with limited dispersion. In Souverein [12] and Souverein et al. [13], the separation
criterion was defined as Se  ΔP∕ΔPsep , in which ΔP is theimposed pressure jump, and ΔPsep is the necessary pressure jump

3

0

00

L 

6

0

0

20

3

For this flow deviation, the adverse pressure gradient is strong
enough to make the boundary layer separate.
PIV measurements have also been achieved for the 8 deg case forthe adiabatic and heated cases. We present the longitudinal-velocity
standard-deviation maps in Fig. 5. This quantity highlights the development of a highly turbulent shear layer downstream the reflected
shock (see [11]). Although the spatial extent of the interaction isstrongly increased when the wall is heated, it is clear that the spatialorganization, as well as the turbulence intensities, is similar between
the adiabatic and heated cases.
An evaluation of the length of the interaction has been derived fromschlieren pictures by extrapolation of the inviscid shock waves to thewall (see Fig. 4). The absolute uncertainty on the interaction length,
determined with this method, has been evaluated to be less than
2 mm. Therefore, the relative uncertainties on the interaction length
range between 7%, for the smallest interaction, and 2% for the largest
one. It is relevant to emphasize that those uncertainties are small
compared to the variations of the phenomenon being studied, giving agreat confidence in the results.
For both the adiabatic and heated cases, a wide range of flow
deviations has been documented: from 3.5 to 9.5 deg. These flow
deviations correspond to interactions ranging from attached to largely
separated cases. A comparison of the interaction length L of theadiabatic and heated cases is given in Fig. 6a. Both curves exhibit
linear behavior, with a slope change when increasing θ. This can beinterpreted as the change in flow regime when the pressure risethrough the incident shock wave becomes strong enough to make the
30

30

0

90
90
250
260

240

60

60

90

30

30

30

5

30

3060

10

90

y[mm]

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90 60 90
270
280

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290

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320

90
310

300

330

340

a) Tω /Tr = 1
9

20

0

60
250

60

260

90

90

90

30 60
290
X[mm]

60 90
30
270
280

300

60

60

6060300
0 90
9
240

60

90

30

90

60
30

60

30

30

y[mm]

10
5

30

30

15

90

60
310

30
320

330

340

b) Tω /Tr = 1.9
Fig. 5

Isocontours of the longitudinal-velocity standard deviations for the adiabatic and heated cases.
6

110

T /T =1.0

T /T =1.0 (M=2.3, reflection)

T /T =1.9

T /T =1.9

w

100

w

r

w

r

w

r
r

5
90
4

L/(g3 δ )

80
70

*

L[mm]

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60

−7.5%

3

2

50
40

1
30
20

3

4

5

a)
Fig. 6

6

θ

7

8

9

10

0

0

1

2

3

4

5

θ

6

7

b)
a) Interaction length vs deviation angle; b) normalized interaction length vs deviation angle.

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3ΔP
qe

(3)

As the external conditions remain constant for the adiabatic andheated cases, the separation criterion Se depends only on the imposedpressure step; otherwise, on the imposed flow deviation θ. This is
illustrated in Fig. 7, in which the flow deviation θ is plotted vs Se , inwhich ΔP is evaluated from shock relations. The best-fit power law(pSne , with p  7 and n  0.85) is also reported in the figure for the
Mach 2.3 case. From the figure, it can be observed that, up to Mach 3,
the exponent of the power law (n  0.85) depends weakly on the
Mach number. On the other hand, the coefficient p depends on the
Mach number considered.
From Fig. 7, we can writeΔP nθ  pSne  p 3
qe(4)

As n  0.85 ≃ 1, Eq. (4) indicates that the angles of deviation are
valid representations for the separation criterion Se , at least for agiven Mach number, as the constant p depends on this parameter. The
normalized interaction length L is plotted against the deviation
angle θ in Fig. 6b. The displacement thickness δ , used in thecomputation of L , is an integral quantity, which can therefore be
calculated accurately. In our case, δ has been determined with anuncertainty of less than 3%; and therefore, those uncertainties are
smaller than the variations the phenomenon studied in this paper.
Here, the function g3 is not essential, as the external parameters areconstant for the adiabatic and heated cases. It is clear that most of theheating effects on the length of interaction are taken into account by
the changes in incoming conditions, through the mixing-layer
displacement thickness δ . Nevertheless, a horizontal sift (about one-half degree) can still be observed between the adiabatic and heatedinteractions, whatever the imposed flow deviation.
These results show that this representation is not accurate enoughin the heated case, to correctly describe the interaction-length scale.
In the following, we will try to explain the shift observed in Fig. 6b by
a difference between the separation states for the adiabatic andheated cases.
B. Separation State of the Interaction

For the adiabatic case, the change in the slope of the length ofinteraction occurs in the vicinity of a flow deviation of 7 deg. Fromprevious adiabatic experiments [11,18], this angle corresponds to thedevelopment of a separated region, and is in accordance with Se  1,
the estimation of the separation threshold given by Eq. (3). Moreover,the change in the slope of the dimensionless length of interaction
is observed near L∕δ g3  ≃ 1.5 − 2. This value was shown tocorrespond to the onset of the separations for various SWBLIs [13].
From Fig. 6b, one can argue that the heated case will separate for aslightly smaller flow deviation than the adiabatic case (typically 1 degor less). This is illustrated in Fig. 8, in which the mean longitudinalvelocity profiles, for the same flow deviation (θ  8.0 deg), arereported for the adiabatic and heated configurations. In both cases,
profiles with maximum reverse flow have been selected. It is clearthat the heated case presents the significantly larger reverse flow(−50 ms−1 compared to 20 ms−1 for the adiabatic case).
Moreover, as the separation criterion derived by Souverein [12]and Souverein et al. [13] solely depends on external flow parameters,
the adiabatic and heated interactions having similar length should
show similar separation state. In Fig. 8 are also reported the velocity
profiles for the 9.5 deg adiabatic case, which has a similar separation
length than the 8 deg heated case (cf. Fig. 6a). It is obvious that the
9.5 deg adiabatic interaction shows a stronger reverse flow than the
8 deg adiabatic one. Consequently, a comparison of the different
interactions (adiabatic or heated wall) at constant length of interaction does not lead to compare flows in similar separation states.
From the results shown in Fig. 8, it seems that the separation
criterion defined by 3 has to be refined to take into account the heating
effects. It is necessary to evaluate more precisely the minimumpressure jump through the interaction needed to make the flow
separate ΔPsep . From Fig. 6b, the correction to apply has to be of theorder of a half-degree of deviation; it means that the onset ofseparation will be about 7 deg for the adiabatic case and about 6.5 deg
for the heated case. If the angle for which separation is observed isnoted θsep , using relation 4, we have
ΔPsep n(5)
θsep  p k
qe
and from Eqs. (4) and (5), it is possible to derive the following relation
between the separation criterion and the deviation angles:
20

power law pxn (p=7, n=0.85)
M=2.3

10

18

1

θ=8°, Tw/Tr=1.0
T /T =1.9
w

r

w

r

θ=9.5°, T /T =1.0

16
14

y(mm)

12

θ[°]

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to make the boundary layer separate. In other words, Se < 1corresponds to the attached cases, Se ≃ 1 corresponds to the incipientinteractions, and Se > 1 represents the separated interactions.
The estimation of the pressure ΔPsep was based on the inviscid
properties of the upstream boundary layer, and was estimated as
ΔPsep ≃ 1∕kqe , in which qe  1∕2ρe U2e is the dynamic pressureand k a constant to adjust. This constant has been shown to be
independent on the Mach number, and to depend only slightly on the
Reynolds number: no significant dependency was observed for large
Reynolds numbers (3 × 105 > Reθ > 1 × 104 ), and a value of 2.5
can be used for this range of flows. In low-Reynolds-number flows(Reθ ≃ 5 × 103 ), a larger value has been derived: k  3.0. This value
has been used in the present study.
This criterion was found to collapse correctly the separation stateof several SWBLIs in the adiabatic cases. For the Reynolds number
considered in our case, the separation criterion was finally given bySe 

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JAUNET, DEBIÈVE, AND DUPONT

M=3

10
8
6
4

M=1.4
10

2

0

10

−1

3∆ /q

10

0

0

P e

Fig. 7 Flow deviation angle and separation criterion Se correspondence.

0

100

200

300

400

500

600

U(m/s)

Fig. 8Mean longitudinal velocity in the section of maximum reverse flow.

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θ

ΔP
ΔPsep

JAUNET, DEBIÈVE, AND DUPONTn(6)

in which the exponent n  0.85, for Mach number up to 3.
Previous results show that, if the separation onset is qualitatively
predicted from the inviscid criterion Se based on the externalparameters of the flow, some limited corrections between the heated
and adiabatic configurations have to be applied. A dimensionlessanalysis, as proposed by Morgan et al. [19], suggests that thefollowing parameters have to be considered: Reynolds number Re,
T w ∕T r , Cf , and Cp . Nevertheless, some well-known aerodynamic
relations exist between these parameters, allowing the reduction of
the number of variables to be considered.
This suggests that some wall scales have also to be considered. As
the heating modifies mainly the near-wall properties of the flow
through density variations due to the large temperature increase near
the wall, it can be expected that the reduction of the separationthreshold, in respect with the adiabatic case, is mainly due to themodifications of the near-wall properties of the boundary layer. We
have shown that the onset of separation is reduced by about 7%, when
the friction coefficient is reduced by 15% (see Sec. II): this suggests asquare-root dependence on the friction coefficient.
In the classical free-interaction concept [20,21], Chapman et al. [20],using classical aerodynamic properties, proposed a normalized
pressure distribution depending on parameters as the Mach number
and the upstream skin friction Cf0 . From this dependency, somecharacteristic points, such as pressure at separation Ppsep , pressure in
the plateau region Ppl , and so on, can be expressed as
Ppsep − P0 Ppl − P0
∝
∝
qe
qe

s
2Cf0
M2e − 11∕2(7)

The proportionality coefficient depends on the point considered(separation or plateau), as well as on the state of the incoming boundary
layer (laminar or turbulent boundary layer). To make the flow separate,
the pressure jump through the whole interaction should reach at least
the plateau pressure. From these considerations, we assume that thepressure jump through the interaction necessary to make the boundary
layer separate (ΔPsep ), is also proportional to these parameters, itmeanss
2Cf0(8)
ΔPsep ∝ qe
M2e − 11∕2Here, we are considering flows with constant external parameters, andqe and Me are considered as constant. From the previous relation, we
can write
1∂log ΔPsep   ∂ log Cf0
2

with n  0.85. Therefore, from Eq. (9), it is possible to predictthe change in separation angle θsep from the variation in Cf0 . To
apply Eq. (9), Cf0 has to be estimated. Several different techniques
can be used to determine this quantity, all of them leading to an
important uncertainty in the computed value. We retained thetechnique used in Debiève et al. [16]), leading to an uncertainty of
about 10% in Cf0 . Nevertheless, because we are analyzing variations
in ratios of Cf0 , the uncertainties in Cf0 have a low impact in the
results presented here.
Applied to the present case, relation 9 predicts a reduction of theseparation angle of 0.4 deg for the heated case compared to theadiabatic case. This is coherent with the previous estimation based on
experimental observations (a reduction of 0.5 deg).
The influence of heating on the separation threshold of theboundary layer seems to be correctly estimated from Eq. (9).
Therefore, it is of interest to consider the opposite case, in which thewall is cooled. Unfortunately, to the best of our knowledge, there is noshock-wave reflection with cooled wall, in which the onset ofseparation and the length of interaction are discussed. Nevertheless,as the dimensionless length-scale model given by Eq. (1) does notdepend on the geometry of the interaction [13], Spaid and Frishett’s
experiments have been considered [23].
The authors considered a Mach 2.9 compression corner at similar
Reynolds number (3.9 × 104 ) with adiabatic conditions and ratio ofwall temperature to recovery temperature of 0.47. They obtained anincrease of the friction coefficient of 46% (from 1.94 × 10−3 to2.85 × 10−3 ). The authors evaluated the length of interaction of theircompression corner from schlieren visualizations, as well as fromstatic-pressure data. In both cases, as in the present results, thevariation of the slope of the length of interaction when plotted vs theimposed flow deviation is observed (see fig. 9 of their paper). These
slope variations occur at 13 and 16 deg, respectively, for the adiabaticand cooled conditions. The flow deviation of 13 deg was foundto correspond to the onset of separation for the adiabatic case.
Applying relation 9 to the Spaid and Frishett’s [23] results would leadto a variation of 20% of θsep ; it means a separation occurring for anangle of 15.6 deg, very similar to the angle derived from the slopechange of the length of interaction curve. These results are
summarized in Fig. 9, in which the different normalized lengths ofinteraction are represented vs the imposed flow deviation θ. (Theflow deviation was divided by 2 for the compression corner, tocompare with the reflection case.) The variations of the square-root
friction coefficient between, respectively, the adiabatic andnonadiabatic conditions are reported by horizontal arrows in the
figure. In both cases (Mach 2.3 heated shock reflection and Mach 2.9
cooled compression corner), the influence of the heat transfer on theseparation threshold of the boundary layer seems to be correctly
estimated from Eq. (9).

6
Tw/Tr=1.0 (M=2.3, reflection)

Developing the previous relations and using relation 4, we have


n
(9)
∂log θsep   ∂ log Cf0
2

T /T =1.9w

5r

Tw/Tr=1.0 (M=2.9, corner)
T /T =0.47
w

r

4
*

with log as the logarithmic function and ∂  being a differential
operator for constant external parameters, it means at the same external
Mach and Reynolds number. The logarithmic derivative ∂log Cf0 
represents the relative variation of the friction coefficient to theadiabatic case (Cf0 ∕Cf0a − 1), in which Cf0a is the friction coefficientof the adiabatic flow with the same external parameters. Therefore, thisderivative depends only on the heating parameter [22], and no more on
the Reynolds number.
From Eq. (6), we obtain




θ
ΔP
 n∂ log
∂ log
θsep
ΔPsep

L/(g3 δ )Downloaded by Centre d'Etudes Aerodynamiques et Thermiques on June 3, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.J052869

θsep




/

−7.5%

3
+25%

2

1

0

0

1

2

3

4

5

θ or θ/2

6

7

8

9

10

Fig. 9 Interaction length similarity between adiabatic, cooled, andheated cases using adabatic Se .

AIAA Early Edition/

The results presented in the previous section suggest that thenormalized length of interaction defined by relation 1 effectively
compares various SWBLIs, including heating effects, as long as theseparation criterion defined by Se  ΔP∕ΔPsep is correctly evaluated. The estimate Se derived from adiabatic compilations [seeEq. (3)] is not accurate enough when heating effects are considered,despite similar Reynolds numbers with the adiabatic cases.
The variation of the flow deviation necessary to make the flow
separate for the heated cases, in comparison with the adiabatic
configurations, has been found described accurately by the classicalrelation proposed by Chapman et al. to evaluate the pressure at the
separation point or the plateau region [20].
This relation suggests that the pressure step necessary to obtainseparation (ΔPsep ) should be compared to the dynamic pressure q0 ,as an estimate for Se, but with a weighting function depending on thesquare root of the friction coefficient and the Mach number. Suchcorrection cannot be expected to be relevant for the various turbulent
adiabatic SWBLI cases considered in Souverein et al. [13]. Indeed,
only a limited dependency on the Reynolds number was observed,
despite the wide range of flow considered (3 × 103 < Reθ < 3 × 105 ),in accordance with other classical separation criteria [24–26]. Thissuggests that the constant of proportionality in Eq. (8) may depend
itself on the Reynolds number.
As the present and Spaid and Frishett’s [23] experiments involve
similar Reynolds number for their adiabatic configuration, we tested
relation 8 to evaluate the quantity ΔPsep. The constant of proportionality has been derived from our adiabatic experiment, in which
the separation was obtained for an imposed flow deviation of 7 deg. A
constant of 7.14 has been obtained and used for all the other cases(heated reflection and adiabatic or cooled compression corner).
We represent in Fig. 10 the normalized interaction length L vsΔP∕ΔP0 , with ΔP0 estimated from Eq. (8) with a constant of 7.14. It
is clear that this representation leads to a very good collapse of theadiabatic and heated interactions for the whole range of flow
deviations: from the attached to largely separated cases. Moreover,
the Mach 2.9 compression corner, adiabatic and cooled, collapse wellwith the present experiments, despite the geometry difference and theinitial incertitude on the length of interaction derived from schlieren
visualizations or from static-pressure measurements.
In conclusion, the heat-flux effects on the mean properties of theinteractions can be summarized as 1) an increase of the boundarylayer displacement thickness of the upstream boundary layer due tothe lower density in the near-wall region when the wall is heated, and2) a dependence on the heating flux of the flow-deviation angle
necessary to make the flow separate. This reduction has been related
to variations of the skin-friction coefficient with the wall heating flux.
6
T /T =1.0 (M=2.3, reflection)r

Tw/Tr=1.9

5

T /T =1.0 (M=2.9, corner)wr

T /T =0.47
w

r

4

3

*

L/(g δ )Downloaded by Centre d'Etudes Aerodynamiques et Thermiques on June 3, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.J052869

C. Similarity for the Length of Interaction Including Heat Transfer
Effects

w

3

2

1

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

7

JAUNET, DEBIÈVE, AND DUPONT

2∆ P /∆ P0

Fig. 10 Similarity for the length of interaction between adiabatic andheated cases.

Therefore, it seems that the evolution proposed by relation 8 iscorrect when flows with adiabatic conditions correspond to similar
Reynolds numbers. This confirms that the constant of proportionality
may depend on the Reynolds number, even for the turbulent cases.
Relation 8 also applies a direct Mach-number correction through theterm M2 − 1−1∕4 . Unfortunately, this term varies only by 13%
between the reflection and the compression-corner cases. Therefore,
to generalize the present similarity, it would be necessary to consider
the SWBLI at larger Mach numbers, keeping the Reynolds number inthe range here under consideration.
In the following, the heating effects on the unsteadiness inside
the interaction will be considered. Only the θ  8 deg case will be
discussed: this angle of flow deviation is chosen to be sure that theboundary layer separates in both the adiabatic and heated interactions, and because it is small enough to avoid any blockage issue.
D. Unsteadiness of the Heated Interaction

As mentioned in the Introduction, the SWBLI is well known to havean unsteady behavior, leading to low-frequency pressure fluctuations
inside the recirculation area. These low-frequency pressure fluctuations are related to the unsteady behavior of the reflected shock. For the
adiabatic SWBLI at Mach number larger than 2, a typical Strouhal
number SrL associated with these low frequencies was proposed [27].
It was defined as SrL  fL∕U1 , in which U1 is the velocity
downstream of the incident shock. For a wide range of configurations,
SrL ≃ 0.03. Thus, assuming that this Strouhal number is also valid forthe heated interaction, one can expect a change in frequency of order ofabout 50%, as well as the interaction length.
Therefore, the unsteadiness of the reflected shock is investigatedby means of hot-wire measurements in the external flow. The spectraof hot-wire signals taken at the mean reflected shock position are
given in Fig. 11a. The fluctuations are produced by the unsteadyshock moving back and forth around the probe. As expected, thepremultiplied spectra are dominated by low frequencies for both the
adiabatic and heated interactions. It is obvious that lower frequencies
are present in the heated interaction than in the adiabatic one.
Considering the equilibrium between the mass of fluid in theseparated bubble and the mass-entrainment rate through the shear
layer, which develops downstream the separation point, Piponniau
et al. [11] propose to take into account the compressibility and density
effects on the spreading rate of the shear layer to weigh the Strouhal
number SrL. They propose the weighted Strouhal number defined as
SL 

SL
ΦMc  × gr; s

with
SrL 

(10)

f×L
U

in which Mc is the convective Mach number, r is the velocity ratio,
and s is the density ratio across the mixing layer, and ΦMc  is thenormalized spreading rate for the mixing layers (see [28]).
Equation (10) suggests that the wall heating can influence thecharacteristic low-frequency unsteadiness of the interaction throughthree effects: 1) the variation of the length of interaction L (seeFig. 6a), 2) the velocity and density ratio across the shear layer, and3) the compressibility effects, depending on the variation of theconvective Mach number.
These different effects are now evaluated. The premultiplied power
spectral densities (PSDs) for the θ  8.0 deg case, are plottedagainst the Strouhal number SrL in Fig. 11b. It is clear that thisrepresentation overpredicts the influence of heating on thecharacteristic frequency. Now, we take into account the effect ofheating on the functions ΦMc  and gr; s. As shown in the previous
section, the velocity ratio between the adiabatic and heatedinteractions will change only slightly (see Fig. 8). Conversely,
heating will change dramatically the density ratio: typically reduced
by a factor 2. Nevertheless, as shown in [11], the function gr; s does
not significantly change with the parameter s. This is illustrated inFig. 12a: a reduction of only 15% of gr; s can be expected for the

8

AIAA Early Edition

JAUNET, DEBIÈVE, AND DUPONT

/

14

14

Tw/Tr = 1

12

Tw/Tr = 1

12

Tw/Tr = 1.9

10

10

8

8

6

6

4

4

2

2

0

0

102

103

10-2

104

10-1

f [Hz]a) Energy vs frequency

100

SL

b) Energy vs strouhal number
Fig. 11 Normalized premultiplied spectra of the hot-wire signals.

0.03
1
0.025
0.8

Φ(Mc)

0.02

g(r,s)

0.015

0.6

0.4

0.01

0.2

0.005

s=1
s=0.56(adiabatic.)s=0.28 (T /T =2)wr0−0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

0

0

0

0.2

0.4

0.6

0.8

r

a)

1

1.2

1.4

1.6

1.8

Mc

b)
Fig. 12 Strouhal number weighting funtions : a) gr;s, b) ΦMc ].

heated case in respect with the adiabatic conditions. Moreover, thisreduction will increase the difference observed in Fig. 11b.
On the other hand, the density variation will also influence theconvective Mach number through the change in sound velocity near
the wall. Indeed, the convective Mach number can be expressed byMc ΔUa1  a 2p
with ai  γRT i , in which T is the static temperature of the fluid.
Therefore, the convective Mach number will be reduced through thewall heating and the increase of the sound velocity on the lowvelocity side of the mixing layer. Reducing the convective Mach
number, the function ΦMc  will increase (see Fig. 12b adapted from
Smits and Dussauge [29]) and the normalized Strouhal number SrL
will decrease.
In Fig. 13, we report the normalized spectra for both the heated andadiabatic interactions. An excellent agreement is obtained for thelocation of the low-frequency peak, indicating that the normalizationof the Strouhal number of Eq. (10) is correct for the present results.
This confirms that the low-frequency unsteadiness of the interaction
has to be related with the development of the mixing layer, whichdevelops downstream the reflected shock. The reader may notice thatthe spectra of the adiabatic interaction show a secondary peak at
SrL  200, which is not present for the heated case. This is due to thefact that the measurements were carried out at the same distance fromthe wall for both interactions. Therefore, because the size of theinteraction increases with heating, the adiabatic-case measurement
was carried out at a higher distance from the interaction. As shown by
Agostini et al. [30], at higher distances from the wall, the shock
motion is not only altered by the low-frequency unsteadiness of theseparation bubble, but also by the medium-frequency perturbations
coming from the underneath developing mixing layer.
14

Tw/Tr = 1

12

Tw/Tr = 1.9
10

f.S(f)*

Downloaded by Centre d'Etudes Aerodynamiques et Thermiques on June 3, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.J052869

f.S(f)*

f.S(f)*

Tw/Tr = 1.9

8
6
4
2
0
100

101

S*L

102

103

Fig. 13 Premultiplied and normalized spectra vs weighted Strouhal
number.

AIAA Early Edition

Downloaded by Centre d'Etudes Aerodynamiques et Thermiques on June 3, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.J052869

IV.

/

JAUNET, DEBIÈVE, AND DUPONT

Conclusions

An investigation of the impact of wall temperature on a M  2.3
shock-induced boundary-layer separation has been carried out
experimentally. This study highlighted that the interaction length
significantly increases when the wall temperature is elevated. The
results obtained seem to imply that this increase is mainly aconsequence of changes in the wall incoming conditions, which canbe taken into account through the changes on the displacement
thickness and friction coefficient of the upstream boundary layer. Theinteraction-length increase can then be accurately described using amodel based on mass conservation across the interaction. Anothereffect of heating is a slight modification of the separation state.
Classical dependencies, inferred from the free-interaction theory,
seem relevant to describe this influence of the wall conditions, forconstant external parameters (Mach and Reynolds numbers).
Combining both models, a full similarity between the adiabatic andheated interactions has been obtained, from the attached to separated
cases. Moreover, the same model has also been applied with successto cooled-wall interaction.
From the present results, it seems that low frequencies of thereflected shock motion can also be related to the same definition ofStrouhal number as in the adiabatic case. The proposal of Piponniau
et al. [11] for normalizing the shock beating frequency has been
evaluated and discussed for nonadiabatic cases, and the method
succeeds in collapsing the characteristic low-frequency unsteadinessof the reflected shock for the heated interaction.
These results confirm that the overall spatiotemporal organizationof the heated-wall interaction can be effectively compared to aprevious analysis achieved in the adiabatic cases, and suggest similar
results for the cooled-wall cases.

Acknowledgments
Part of this work was carried out with support from the Research
Pole Centre National d'Etudes Spatiales Aérodynamique des Tuyèreset Arrière-Corps and a grant of the Agence Nationale de la Recherche
through the DÉcollements COMpressibles et OScilations autoinduites (DECOMOS) program. Their support is gratefully
acknowledged.

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G. Elliott
Associate Editor