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The behaviour of the normal fluctuation terms in the case of attached and detached turbulent boundary-layers
G.A. Gerolymos, Y.N. Kallas, K.D. Papailiou
To cite this version:
G.A. Gerolymos, Y.N. Kallas, K.D. Papailiou. The behaviour of the normal fluctuation terms in
the case of attached and detached turbulent boundary-layers. Revue de Physique Appliquée, Société
française de physique / EDP, 1989, 24 (3), pp.375-387. �10.1051/rphysap:01989002403037500�. �jpa-
00246060�
The behaviour of the normal fluctuation terms in the case of attached and detached turbulent boundary-layers
G. A. Gerolymos, Y. N. Kallas and K. D. Papailiou
National Technical University, Athens, Greece
(Reçu le 2 mai 1988, révisé le 30 août 1988, accepté le 27 septembre 1988)
Résumé.
2014Il a été établi par des mesures expérimentales que quand une couche limite turbulente s’approche
du décollement, décolle ou interagit avec une onde de choc, les termes de fluctuation turbulente augmentent
en valeur et exercent une influence importante sur le bilan de quantité de mouvement du fluide. En
conséquence les intégrales des termes normaux de fluctuation devraient être prises en compte dans des formulations intégrales pour le calcul des couches limites turbulentes. Dans le présent travail des données
expérimentales sont examinées, dans le but de déterminer des corrélations entre ces intégrales et des quantités
de l’écoulement moyen. Les corrélations qui sont élaborées sont valables pour des couches limites attachées et décollées, compressibles et incompressibles. Elles sont aussi valables pour des écoulements d’interaction onde de choc/couche limite turbulente. En plus, des corrélations sont élaborées pour les maxima des fluctuations normales. Finalement, la structure de la turbulence dans des couches limites décollées est discutée.
Abstract.
2014Experimental measurements have established that when a turbulent boundary-layer approaches separation, separates or interacts with a shock-wave, the turbulence normal fluctuation terms increase in
magnitude and exert an appreciable influence upon fluid-momentum balance. Consequently, the integrals of
the normal fluctuation terms should be included in integral formulations for the calculation of turbulent
boundary-layers. In the present work, available experimental data are examined in view of establishing
correlations between these integrals and mean-flow quantities. The correlations that are elaborated are valid both for attached and detached, incompressible and compressible turbulent boundary-layers. They are also
valid for shock-wave/turbulent boundary-layer interaction flows. Furthermore correlations for the maxima of the normal fluctuation terms are presented. Finally the turbulence structure in separated boundary-layers is
discussed.
Classification Physics Abstracts
47.25F
-47.40N
Nomenclature :
a
=anisotropy (Eq. (13))
A
=normal fluctuations integral in the mean
momentum equation (Eq. (3))
B
=normal fluctuations integral in the mean
energy equation (Eq. (4))
Cd = dissipation-factor j { (~/03C1e u3e) dy}
Cf
=skin-friction coefficient {2 Tw/Pe u2e}
H12 = boundary-layer shape-factor {03B41/03B42}
H13 = boundary-layer shape-factor {03B41/03B43}
J12
=boundary-layer shape-parameter
Pk
=boundary-layer length-scale (Eq. (7))
M
=Mach-number p
=static pressure
r
=wall-temperature recovery-factor
u
=velocity-component parallel to the wall
u,
=friction-velocity {sign (w) ~|w|/03C1w}
v
=velocity-component normal to the wall
x
=coordinate parallel to the wall
y
=coordinate normal to the wall
w
=transverse velocity-component
y
=isentropic exponent
à
=boundary-layer thickness 03B41
=displacement thickness
8 2
=momentum thickness
5 3
=energy thickness
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01989002403037500
/3
=equilibrium-parameter (Eq. (11)) e
=dimensionless length (Eq. (16))
p
=density
T
=shear-stress
03A6
=dimensionless velocity (Eq. (15))
~
=turbulent dissipation
Subscripts
e = external flow k
=kinematic value
w
=wall-value
N
=value at the maximum backflow-velocity lo-
cation
max
=maximum value in the boundary-layer Superscripts
(* ) = mean quantity (. )’ = fluctuating quantity
Introduction.
Early measurements in turbulent boundary-layers
had established that the turbulence normal fluctu- ation terms u’2, V ’2, w’2 which appear in the en-
semble-averaged Navier-Stokes equations [1] in-
crease when the boundary-layer approaches separ- ation [2, 3]. Detailed measurements in flows with
large separation [4, 5, 6] confirmed this and showed that the turbulence normal fluctuation terms increase
rapidly in the separated flow region, so that their
effect on mean fluid momentum balance becomes substantial. The same phenomenon is observed in shock-wave/turbulent boundary-layer interactions both when the boundary-layer separates [7] or not [8, 9]. Furthermore their value remains important in reattaching turbulent boundary-layers [10, 11].
Computer power growth has favorized the devel-
opment of field methods for viscous flow compu- tations [12, 13]. However, integral methods for boundary-layer calculations are of current use in
applications where computing speed is essential,
such as aerodynamic design. Most integral bound- ary-layer computation methods are based on the
solution of 2 integral equations for the evolution of the momentum and energy thickness, respectively.
If the normal fluctuation terms are not neglected,
the integral mean momentum equation takes the following form [15]
By replacing the turbulence production term which
appears in the mean kinetic energy equation by the
appropriate balance indicated by the turbulence
kinetic energy equation, the integral mean kinetic
energy equation takes the following form [15]
In the following attention will be focused on the terms
appearing in equation (1) and
appearing in equation (2) (a combination of the 2nd and 3rd right-hand-side terms).
Most prediction methods neglect the terms A and
B. Experience with a particular integral method [16- 18] has indicated that separation could not be well predicted unless thèse terms were included in the formulation. A first model for these terms [18, 19]
which proved to work well for attached boundary- layers was found to be inadequate for separated
ones. Furthermore, recent measurements provide
the possibility to validate turbulence models in the
separated flow region where few reliable and de- tailed measurements existed in the past.
The purpose of the present work is to analyse
available experimental data in order to formulate models for the terms A and B, defined in equations (3) and (4) respectively. These models should be applicable both to attached and detached compressible turbulent boundary-layers including
the case of shock-wave/turbulent boundary-layer
interaction. It will be attempted, as in reference [18],
to correlate these terms with mean-flow quantities.
In this way, turbulence closure in the mathematical formulation of integral turbulent boundary-layer computation methods may be obtained without
solving any additional integral turbulence transport
equation. Furthermore, the turbulence structure in
nonequilibrium turbulent boundary-layers is investi- gated, with special emphasis on flows with large separation. It will be shown in the following that it is possible to elaborate correlations for the maximum
profile values of the normal fluctuation terms, which
are valid in the entire range of boundary-layer flows
that were examined.
Expérimental Data.
The data used in this work are taken from 5 well documented experiments, the main features of
which are summarized in table I. They cover a wide
range of external-flow Mach-numbers and bound-
ary-layer shape-factors, including 2 cases of transonic shock-wave/turbulent boundary-layer interaction.
Accurate and detailed measurements of mean-flow
quantities and Reynolds-stresses are available in the references cited in table I.
The transverse velocity-fluctuation was not
measured in experiments 3 and 4 (Tab. I). Its value
is necessary for computing B (Eq. (4)). It was
assumed that (w’2
=v’2) everywhere w’2 was not measured. This assumption is based on the data of
Simpson, Chew and Shivaprasad in experiment 2,
who concluded [6] that (03C9’2
=v’2) in the outer 90 %
of the layer, both upstream and downstream of
separation. It should however be noted that
Schubauer and Klebanoff [24] found that (2 w’ 2
u’2+ v’2) in a zero pressure-gradient turbulent boundary-layer. Furthermore, Acharya [25] studying
the effects of compressibility on boundary-layer
turbulence found that in the Mach-number range from 0.2 to 0.6, w’2 increases with respect to the
value established by Klebanoff as the extemal-flow Mach-number increases. However, Acharya men-
tions the presence of secondary-flows in the measur- ing section which could account for the elevated
values of w’2. No data on flow 2-dimensionality are
available for the experiment of Schubauer and Klebanoff. On the contrary, the mean transverse velocity was found to be zero within the measure- ment uncertainties in the experiment of Simpson,
Chew and Shivaprasad. It is believed that the
assumption (w’2
=v’2 ) does not greatly affect the results of the present work.
In order to establish the accuracy which should be
expected from correlations established on the basis of the afore-mentioned data (Tab. I), a note on the
uncertainties of measurement is appropriate, es- pecially since data from different experimental set-
ups and instrumentation are used. Simpson, Chew
and Shivaprasad [4] estimated that using laser
anemometry the values of u’2 and v’2 were measured
to ± 4 % of the maximum profile value. Considering
the form of the u’2 and v’2 profiles this corresponds
to an average ± 8 % uncertainty. When using cross- hot-wire anemometry [4] the values of u’2 and
V,2 were measured with an estimated uncertainty of
± 10 %. This accuracy is about the best achieved in the experiments used in the present work (Tab. I).
Furthermore when considering quantities involving
sums of u’2 and v’2 (Eqs. (3) and (4)) a ± 20 %
uncertainty is to be expected if u’2 and v’2 are known
individually to ± 10 %. Furthermore the assumption
(w’2 = v’2) used in the present work introduces further uncertainties in the determination of quan- tities dependent on w’2 (Eq. (4)).
Table I.
-Summary of the experiments used in establishing the correlations
Modeling approach.
It is well established that the turbulent boundary- layer mean-velocity profile is adequately described by a 2-parameter family of profiles [26, 27]. From
REVUE DE
PHYSIQUE APPLIQUÉE. -
T.24,
?3,
MARS 1989this experimentally verified assumption span various
integral methods for the computation of turbulent
boundary-layer development. These methods were
initially developed and applied to incompressible
flows, but were successfully extended to compress-
26