Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis

TLDR: In this paper, a transonic channel flow over a bump is investigated, where a shock wave causes the separation of the boundary layer in the form of a recirculating bubble downstream of the shock foot.

Abstract: A transonic interaction between a shock wave and a turbulent boundary layer is experimentally and theoretically investigated. The configuration is a transonic channel flow over a bump, where a shock wave causes the separation of the boundary layer in the form of a recirculating bubble downstream of the shock foot. Different experimental techniques allow for the identification of the main unsteadiness features. As recognised in similar shock-wave/boundary-layer interactions, the flow field exhibits two distinct characteristic frequencies, whose origins are still controversial: a low-frequency motion which primarily affects the shock wave; and medium-frequency perturbations localised in the shear layer. A Fourier analysis of a series of Schlieren snapshots is performed to precisely characterise the structure of the perturbations at low- and medium-frequencies. Then, the Reynolds-averaged Navier–Stokes (RANS) equations closed with a Spalart–Allmaras turbulence model are solved to obtain a mean flow, which favourably compares with the experimental results. A global stability analysis based on the linearization of the full RANS equations is then performed. The eigenvalues of the Jacobian operator are all damped, indicating that the interaction dynamic cannot be explained by the existence of unstable global modes. The input/output behaviour of the flow is then analysed by performing a singular-value decomposition of the Resolvent operator; pseudo-resonances of the flow may be identified and optimal forcings/responses determined as a function of frequency. It is found that the flow strongly amplifies both medium-frequency perturbations, generating fluctuations in the mixing layer, and low-frequency perturbations, affecting the shock wave. The structure of the optimal perturbations and the preferred frequencies agree with the experimental observations.

Figures

Figure 4.15: Optimal response for medium-frequency unsteadiness.

Figure 3.13: Eigenvalue decomposition for linearisation methods.

Figure 4.8: Stability analysis spectrum for the buffet configuration.

Figure 2.20: Horizontal density gradient obtained with a Schlieren apparatus, using a vertically oriented knife edge, and by numerical simulation from chapter 3.

Figure 2.17: Effect of the number of blocks on the spectra.

Figure A.4: Correlations pour un point de référence dans la couche de mélange.

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Instationnarités dans les interactions choc/couche-limite
en régime transsonique : étude expérimentale et analyse
de stabilité
F. Sartor
To cite this version:
F. Sartor. Instationnarités dans les interactions choc/couche-limite en régime transsonique : étude ex-
périmentale et analyse de stabilité. Mécanique des uides [physics.class-ph]. Aix-Marseille Université,
2014. Français. �tel-01018720�

AIX-MARSEILLE UNIVERSITÉ
THÈSE
présentée pour obtenir le titre de
Docteur de Aix-Marseille université
Ecole Doctorale 353 : Sciences pour l’Ingenieur
Spécialité : Mécanique et Physique des Fluides
par
Fulvio Sartor
Unsteadiness in transonic shock-wave/boundary-layer
interactions: experimental investigation and global
stability analysis
Soutenue le 17 mars 2014 devant le jury composé de
Jean-Christophe Robinet Professeur, ENSAM Paris Rapporteur
Neil D. Sandham Professeur, University of Southampton Rapporteur
Jean-Paul Dussauge Directeur de recherche au CNRS, IUSTI, Marseille Directeur de Thèse
Denis Sipp Maitre de recherche ONERA, Meudon Examinateur
Pierre Dupont Chargé de Recherche au CNRS, IUSTI, Marseille Examinateur
Uwe Ehrenstein Professeur, Aix-Marseille université Examinateur
Reynald Bur Maitre de recherche ONERA, Meudon Invité

Abstract
A transonic interaction between a shock wave and a turbulent boundary layer is ex-
perimentally and theoretically investigated. The configuration is a transonic channel
flow over a bump, where a shock wave causes the separation of the boundary layer
and a recirculating bubble is observed downstream of the shock foot.
First, the mean flow is experimentally investigated by means of PIV, then differ-
ent techniques allows to identify the main unsteadiness of this shock-wave/boundary-
layer interaction. As recognised in similar configurations, the flow presents two
distinct characteristic frequencies, whose origins are still unknown.
Numerical simulations are performed solving Reynolds-averaged Navier-Stokes
equations. Results are in good agreement with the experimental investigation on
the mean flow, but the approach fails to predict the unsteady behaviour of the
configuration. The solution of RANS equations is then considered as a base flow, and
a global stability analysis is performed. Eigenvalue decomposition of the linearised
Navier-Stokes operator indicates that the interaction is stable, and the dynamics
cannot be described by unstable global modes.
A linearised approach based on a singular-value decomposition of the global
Resolvent is then proposed: the noise-amplifier behaviour of the flow is highlighted
by the linearised approach. Medium-frequency perturbations are shown to be the
most amplified in the mixing layer, whilst the shock wave behaves as a low-pass filter.
Optimal forcing and optimal response are capable to reproduce the mechanisms that
are responsible for these two phenomena. A restriction on the location of the forcing
can give an insight on the origin on the unsteadiness.
The same approach is then applied to a transonic flow over the OAT15A profile,
where the flow can present, for a range of angles of attack, high-amplitude self-
sustained shock oscillations. Global stability analysis indicates that the shock buffet
onset is linked to a Hopf bifurcation, and the eigenvalue decomposition can describe
the phenomenon when an unstable global mode is present. Regardless of the angle
of attack, singular-value decomposition of the global Resolvent can describe the
convective instabilities responsible of medium-frequency unsteadiness.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Instationnarités dans les interactions choc/couche-limite en régime transsonique: étude expérimentale et analyse de stabilité" ?

A linearised approach based on a singular-value decomposition of the global Resolvent is then proposed: the noise-amplifier behaviour of the flow is highlighted by the linearised approach.