Different shock wave boundary layer interactions in 2D flows
In supersonic flows, the solutions of the Navier-Stokes equations can be discontinuous. These discontinuities can be contact discontinuities or shock waves. Moreover, at solid body walls, vis-cous flows satisfy the non slip condition. The effects of the viscosity are significant in a thin boun-dary layer in the vicinity of the solid wall, where the velocity of the fluid varies from the null velocity at the wall to the free stream velocity. SWBLI occurs when a shock wave interacts with a boundary layer. During the past decades, the fundamental physics of SWBLI have been extensively studied in 2D canonical situations. Five canonical flows are described in Babinsky and Harvey : Each canonical interaction corresponds to a supersonic viscous flow in which a shock wave provoked by a geometrical modification interacts with a boundary layer:
1. impinging oblique-shock wave reflection (figure 1.1):
FIGURE 1.1 – Schlieren visualization of an impinging oblique-shock reflection on a flat plate from Babinsky and Harvey .
This interaction occurs when a supersonic boundary layer developing on a solid wall is im-pinged by an oblique shock wave that is reflected at the wall.
2. ramp flow (figure 1.2):
FIGURE 1.2 – Schlieren visualization of a ramp induced shock wave from Babinsky and Harvey .
When a supersonic boundary layer develops on a solid wall that presents a sudden deviation with a positive slope, a shock wave is created at the wall due to the deflection of the flow.
3. oblique shock wave induced by a forward-facing step (figure 1.3)
FIGURE 1.3 – Schlieren visualization of an oblique shock wave induced by a forward-facing step from Na-rayan and Govardhan .
When a boundary layer developing on a solid wall encounters an obstacle (for example, a forward-facing step), the boundary layers separates upstream of the obstacle and a recircu-lation zone is created. When the boundary layer is supersonic, a shock wave is created at the separation point.
4. normal shock wave (figure 1.4)
FIGURE 1.4 – Schlieren visualization of a normal shock wave from Babinsky and Harvey .
A normal shock wave can be produced in a supersonic flows such as supersonic chanel flows or transonic airfoils. The supersonic flow becomes subsonic by passing through the shock wave. A SWBLI occurs at the shock foot where a boundary layer develops either on the cha-nel boundaries or around a transonic airfoil.
5. imposed pressure jump (figure 1.5)
FIGURE 1.5 – Schlieren visualization of an adaptation shock wave at an overexpanded nozzle exit from Ba-binsky and Harvey .
When an internal supersonic flow ends up at an atmosphere at higher pressure (for example at the end of an over expanded nozzle), an oblique adaptation shock wave is created whose foot is located at the extremity of the nozzle.
For each canonical flow introduce above, the boundary layer can be either laminar (SWLBLI) or turbulent (SWTBLI). SWBLIs have important repercutions on the flows that will be introduced in the following sections tackling the mean properties and the dynamics features of the SWBLI.
We introduce the mean flow organisation of a SWBLI. As in Délery and Dussauge , for conciseness we concentrate our attention on the first canonical interaction, namely the interac-tion induced by the impact of an oblique shock wave on a boundary layer developing on a surface. Indeed, this flow corresponds to the direct numerical simulations of the SWBLI performed in this thesis. Moreover, as claimed in Délery and Dussauge , most of the conclusions could be ap-plied mutatis mutandis to other kinds of interactions.
SWBLIs can be splitted into two categories: weak SWBLIs for which the boundary layer does not separates and strong SWBLIs for which separation occurs.
A sketch of a weak interaction between an oblique shock wave and a laminar boundary layer is shown in figure 1.6 coming from Délery and Dussauge . By weak interaction we mean that no separation of the boundary layer occurs. The free stream flow undergoes a deviation of angle ¢’1 through the incident shock wave C1. The flow undergoes a deviation of angle ¢’2 ˘ ¡¢’1 through the reflected shock wave C2 so that the downstream flow is parallel to the wall. In an inviscid flow where no boundary layer would exist, the regular reflection would be located at a point on the wall (inviscid shock wave reflection on a solid surface). The viscous character of the fluid implies the existence of a boundary layer in the vicinity of the wall which modifies the organization of the flow. Indeed, when the incident shock wave C1 penetrates into the boundary layer, it progressively bends because of the local Mach number decrease. Moreover, the strength of the shock wave decreases until it vanishes when it reaches the sonic line within the boundary layer. The adverse pressure gradient felt by the boundary layer due to the incident shock wave C1 implies a thickening of the boundary layer and consequently a deviation of the streamlines near the wall. The subsonic layer is then also thickened and the pressure rise due to the shock wave C1 is transmitted upstream through this subsonic region. The deviation of the flow implies the creation of compression waves (•) that coalesce and form the reflected shock wave C2. The consequence of the presence of the boundary layer is then to spread the interaction zone that would be reduced to a point in an inviscid shock wave reflection.
The effect of the viscosity on the solution is highlighted by considering the pressure distribu-tion at the wall for a viscous weak SWBLI compared with an inviscid shock wave regular reflection as shown in figure 1.7.
FIGURE 1.7 – Pressure distribution at the wall for an inviscid shock wave regular reflection and for a viscous weak SWBLI from Délery and Dussauge .
The weak interaction differs from the strong interaction where the boundary layer separates in the sense that accounting for the viscous effects is a mere correction to the inviscid solution that is already close to the reality. When the incident shock wave is strong enough, the adverse pressure gradient felt by the boundary layer is likely to provoke its separation so that a separation bubble is created. Depending on the strength of the shock wave, the separation can be incipient or well developed. The incipient case corresponds to interactions where isolated and intermittent spots exist containing fluid with negative velocity but producing no average separation. The well developed separations correspond to flows where an entire specific zone experience reverse ve-locity during periods long enough to produce an average separated bubble. In the following, we focus on SWBLI with well developed separations. Indeed the low frequency unsteadiness of the SWBLI that is the subject of interest in this work (see section 1.3) occures only for this kind of in-teraction. A strong SWBLI with well developed separation is sketched in figure 1.8 (coming from Délery and Dussauge ) where the separation point is noted S and the reattachement point is noted R. A shear layer develops, that bounds the upper part of the separation bubble. The flow being subsonic under the sonic line (S), the pressure rise due to the incident shock wave (C1) is sensed upstream of the location where the incident shock wave would impact the wall because of slow acoustic waves, explaining the location of the separation point upstream of the impact loca-tion. The presence of the recirculation bubble induces compression waves that converge to form the reflected shock wave (C2). The incident shock wave (C1) is transmitted as (C4) through the separation shock wave (C2) and is reflected as expansion waves. At the reattachement point R, the deviation of the supersonic flow due to the presence of the wall leads to compression waves that also coalesce to form the so called reattachment shock wave. In such a strong SWBLI, the viscosity at play in the boundary layer leads to a complete restructuring of the flow even in the outer region where a different system of shock waves is created, with respect to the inviscid Mach shock wave reflexions on a wall.
FIGURE 1.8 – Sketch of a strong SWBLI from Délery and Dussauge .
Such a strong interaction is characterized by a typical wall pressure distribution as shown in figure 1.9 where the wall pressure distribution for a strong shock wave boundary layer interaction is compared with the distribution for an inviscid Mach shock wave regular reflection. The first part of the interaction consists in a steep rise of the pressure associated with the separation followed by a plateau like of pressure characteristic of separated flows. The second part of the interaction consists in a second wall pressure rise associated with the reattachment process, leading to the same pressure downstream the reattachment as in the inviscid case. The wall pressure distribution is then an important quantity to assess the accuracy of the simulations.
FIGURE 1.9 – Pressure distribution along the wall for a strong shock wave boundary layer interaction from Délery and Dussauge 
The extent of the recirculation bubble is driven by the intensity of the incident shock wave, let say the pressure ratio from each side of the shock wave, and by the incoming boundary layer velocity profile. Indeed, the stronger the shock wave, the stronger the adverse pressure gradient leading to the separation of the boundary layer. Furthermore, the larger the normal to the wall velocity gradient within the boundary layer, the better the boundary layer is able to resist to the separation caused by the adverse pressure gradient. For instance, a laminar boundary layer is more prone to separation than a turbulent one when subjected to a steep adverse pressure gradient. The separation bubble extent then characterizes the interaction studied. It is largely bigger when the incoming boundary layer is laminar (Sansica et al. ).
Strong shock wave boundary layer unsteadiness
For strong SWBLIs, it is well established that the separation bubble and the system of shock waves (included the reflected shock wave whose foot is located upstream from the separation point of the boundary layer) are subjected to low frequency longitudinal oscillations called “the unsteadiness” of the SWBLI. This oscillatory motion can spread over a large extent with respect to the boundary layer thickeness. Even if this phenomenon is well known and have been studied both experimentally and numerically for decades, the related mechanisms are still not well understood.
Dynamics of the shock wave boundary layer interaction
We here describe the dynamic features of strong SWBLIs that are characterized by several uns-teady phenomena whose characteristic scales spread over a large broadband spectrum range.
High frequency features
For SWTBLI, the incoming boundary layer is turbulent with the most energetic fluctuations at f – high frequencies characterized by a Strouhal number S– ˘ Ue » 1 (where f , – and Ue are respec-tively the characteristic frequency of the fluctuations, the boundary layer thickeness before the interaction and the free stream velocity).
Low amplitude oscillations of the reflected shock wave have been observed in several numerical simulations of strong SWBLIs (for example Touber and Sandham ) at the same frequency scale than the most energetic fluctuations of the incoming boundary layer (i.e. at high frequency). As explained in Babinsky and Harvey , these small unsteady ripples of the reflected shock waves were also observed in simulations of weak interactions. These high frequency oscillations of the reflected shock waves are then linked to the incoming turbulence whose most energetic scales excite the reflected shock wave.
Medium frequency features
The dynamics of subsonic separated and reattaching flows have been extensively studied and characteristic frequency scales have been characterized ( Cherry et al.  Kiya and Sasaki ). In particular these study highlighted that the shear layer bounding the upper part of the separa-tion bubble, is subjected to two instabilities of medium characteristic frequencies.
The shear layer is submitted to a convective instability (Kelvin-Helmholtz waves). The non linear evolution of the Kelvin-Helmholtz waves leads to a vortex shedding at a Strouhal number around f L SL ˘ Ue ’ 0.6 ¡0.8 based on the length (L) of the recirculation bubble .
The shear layer is also submitted to an absolute instability called « flapping » of the shear layer that f L has the characteristic frequency SL ˘ Ue ’ 0.12 ¡0.15. This flapping consists in successive enlarg-ment and shrinkage of the recirculation bubble. The shrinkage is associated to a vortex shedding downstream of the recirculation bubble.
Low frequency features
For supersonic flows, a low frequency flapping mode of the shear layer have also been observed in addition to the medium frequency flapping at a Strouhal number of SL ˘ Ue ’ 0.03 ¡ 0.04 (Dupont et al.  Piponniau et al. ). This low frequency flapping mode is also called the « breathing » of the separation bubble.
A low frequency oscillation of the whole SWBLI system (the recirculation bubble in phase with the system of shock waves) is also observed in simulations and experiments (Délery and Dus-sauge ). This instability, called the « unsteadiness » of the SWBLI, consists in an oscillation of the recirculation bubble coupled to the shock wave system. This low frequency phenomenon has the same characteristic Strouhal number as the breathing of the separation bubble, namely SL ˘ Ue ’ 0.03 ¡ 0.04. It is shown in figure 1.10 from Dussauge et al.  where the dominant Strouhal mumber SL is plotted against the Mach number (ranging from 0 to 5) for several data for separated flows available in the literature. The SWBLI unsteadiness is further discussed in the followings.
FIGURE 1.10 – Dimensionless frequency (SL) based on the mean separation length (L) of the shock wave oscillation in various configurations versus the Mach number: ( ) subsonic separation from Kiya and Sasaki  ; ( ) compression ramp cases ; (*) IUSTI reflection cases ; (+) overexpanded nozzle (restricted shock wave separation) ; ( F) blunt fin ; (†) Touber and Sandham  ; (–) estimated superstructures upstream influence for the 8– IUSTI case. Figure adapted from Dussauge et al. 
Recent works and mechanisms proposed in the literature
Until the 2000’s, researches about SWBLI unsteadiness have essentially been experimental. These works allowed to characterize the mean properties of the flow as well as to detect the low frequency oscillations of the SWBLI. Nevertheless, the prediction of unsteady pressure loads, the precise characterization of the unsteadiness and the causal explanation of this low frequency be-havior of the interaction remained open research fields. A overview of the knowledge available in the literature about SWBLIs in 2001 can be found in Dolling . After this date, a significant improvement in the comprehension of the SWBLI unsteadiness have been obtained using experi-mental means and modern simulation techniques. In particular, successive increases of computer capabilities allowed high fidelity simulations over times long enough to capture and characterize the low frequency unsteadiness. These works mainly focused on impinging oblique-shock wave reflections as well as ramp flows, that are configurations that share the same dynamic features. These research efforts leaded to new explanations of physical mechanisms responsible for the SWBLI unsteadiness. These mechanisms can be coarsly classified in two main categories (Délery and Dussauge ).
The first mechanism consists in a perturbation of the SWBLI by the large scales within the inco-ming turbulent boundary layer. Indeed, an experimental study (Ganapathisubramani et al. ) of a separated compression ramp interaction at M ˘ 2 has evidenced very long (’ 30–) coherent structures in the incoming turbulent boundary layer convected at a speed of 0.75U1. The resulting frequency is therefore SL ˘ 0.025 L– expressed in terms of a Strouhal number based on the length of the separated zone L (– being the boundary layer thickness). If L is of the order of the boundary layer thickness (L » –), the resulting frequency is of the order of the characteristic frequency of the SWBLI unsteadiness (SL ’ 0.03¡0.04). Consequently, the excitation of the reflected shock wave by these long structures were suspected to trigger the SWBLI unsteadiness. Strong links between ups-tream large scales and unsteadiness in the interaction in the case of a Mach 2 compression ramp flow have been evidenced by same authors (Ganapathisubramani et al. [2007a] Ganapathisubra-mani et al. [2007b]). Nevertheless, the proposed mechanism was challenged by other experimental and numerical results. For instance, an experimental study (Dupont et al. ) of the interaction between an oblique shock wave and a turbulent boundary layer developing on a flat plate at M ˘ 2.3 presented ratios L/– ’ 5 ¡ 7 for the recirculation bubble and a SWBLI unsteadiness at S L ’ 0.03¡0.04. The authors claimed that links between upstream large scales and unsteadiness in the interaction were not significant in their experimental flow. Moreover, the SWBLI unsteadiness have been observed in a large eddy simulation (LES) (Touber and Sandham ) in which no elongated coherent structure was identified in the incoming turbulent boundary layer.
A second kind of mechanism has also been introduced in the literature, for which the dynamics of the recirculation bubble must be related to the unsteadiness of the whole SWBLI system and and in particular to the reflected shock wave oscillations. In addition to the work of Dupont et al. , several numerical works advocate for this second mechanism. The DNS of Pirozzoli and Grasso  and the LES of Aubard et al.  that studied the interaction between an inci-dent oblique shock wave and a turbulent boundary layer on a flat plate at M ˘ 2.25 proposed two slightly distinct mechanisms to explain the oscillations of the reflected shock wave based on the coupling between this shock wave and the dynamics of the boundary layer recirculation. In Au-bard et al. , the reflected shock wave have been suspected to behave as a low pass filter. The foot of the shock wave being excited by the recirculation dynamics, the characteristic frequency of the oscillatory movement is suspected to be prescribed by the low frequency breathing of the separation zone. The separation point was then observed to oscillate with the foot of the reflected shock wave. In Pirozzoli and Grasso , authors claim that the interaction between the vorti-cal structures of the shear layer and the incident shock wave generates feedback pressure waves that excite the separation point region at the frequency of the SWBLI unsteadiness, producing self-sustained oscillations. In Piponniau et al. , a simple model was proposed from experimental observations, based on the entrainment characteristics of the shear layer. The Dynamic Mode De-composition (DMD) analysis performed by Priebe et al.  on a previously DNS of a Mach 2.9, 24– compression ramp SWBLI (Priebe and Martín ) allowed authors to infere that the SWBLI unsteadiness would be related to the presence of Görtler-like vortices in the downstream separa-ted flow due to an underliying centrifugal instability.
Despite arguments in favor of each mechanism, no definitive explanation have yet been provided. Some authors argue that both mechanisms are at play in the onset of the SWBLI unsteadiness but the influence of the first mechanism decreases as far as the strength of the interaction decreases (Souverein et al.  Clemens and Narayanaswamy ).
In additions to these two main families of mechanims, a third kind of model have been introdu-ced in the literature. It accounts for the incoming turbulence without requiring the existence of superstructures. Touber and Sandham  proposed a quantitative model involving the inco-ming boundary layer disturbrances in which the shock wave / separation bubble system is seen as a black-box filter/amplifier converting incoherent background disturbances into the observed spectra. In mathematical terms the model is a first order ordinary differential equation with sto-chastic forcing. This was first proposed empirically by J. Plotkin . This model only requires very low amplitude background disturbances in the flow and these do not need to be in the form of coherent structures. Nevertheless, this model have the major drawback of not explaining the causes of the SWBLI unsteadiness. The relevance of this model have been strengthened by San-sica et al.  who reproduced numerically the SWBLI unsteadiness (in particular the low fre-quency oscillations of the separation point) by forcing a 2D SWLBLI with white noise. This white noise perturbation of the SWLBLI (involving no incoming turbulent fluctuations), mimicking the broadband background distrurbances due to a turbulent incoming boundary layer, were found sufficient to produce the low frequency unsteadiness wherease simulations of a non forced 2D SWLBLI did not exhibit these low frequency oscillations of the separation point and the reflected shock wave (Fournier et al. ).
In a recent work (Adler and Gaitonde ) the authors performed a statistically stationary linear response analysis of the SWTBLI using the synchronized large-eddy simulation method. Their re-sults demonstrated that the SWTBLI fosters a global absolute linear instability corresponding to a time-mean linear tendency of the reflected shock for restoration to more moderate displacements when experiencing an extrem (upstream or downstream) displacement. They interpreted the low frequency unsteadiness of the SWTBLI as a non linear forcing of the reflected shock wave by the medium frequency flapping of the separation bubble. The competition between the linear resto-ring tendency of the reflected shock wave and the non linear flapping of the separation bubble have been said responsible for the low frequency oscillations of the bubble. This mecanism ga-thers some aspects of several mecanisms cited hereinabove. Indeed, this model confirms the li-near restoring tendency of the reflected shock wave which is the premise of the model proposed by J. Plotkin . Moreover, this linear behavior of the reflected shock wave is coupled with a non linear mass-depletion mechanism wich corresponds to the mechanism proposed by Pipon-niau et al. .
Conclusions and outlook of the work
In this chapter, an overview of the knowledge available in the literature about the SWBLI has been presented. In particular we focused on the the low frequency SWBLI unsteadiness pheno-menon. As pointed out in section 1.3.2, non consensus has been reached in the explanation of the physical mechanism leading to these low frequency oscillations of the whole SWBLI system. Some explanations link the SWBLI unsteadiness to the large coherent structures from the incoming tur-bulent boundary layer whereas other proposed mechanisms link the SWBLI unsteadiness to the low frequency dynamics of the recirculations bubble.
In order to better understand the mechanisms leading to the unsteadiness of the SWBLI, we have first chosen to perform a SWBLI simulation suppressing one of the two suspected mechanisms leading to the unsteadiness. By simulating the interaction between a laminar boundary layer and a incident shock wave, we have suppressed the suspected influence of the large scale turbulent structures within the boundary layer on the SWBLI unsteadiness. The only remaining suspected cause of unsteadiness would be the dynamics of the separation bubble. The results are shown in chapter 4. The numerical approach used shows its ability to capture the dynamics of the recir-culation zone. Mainly, the detachment of the boundary layer recovers a steady location while the reattachment location is sensitive to the instabilities of the recirculation bubble, namely the low frequency breathing, the flapping and the Kelvin-Helmholtz frequencies. The unsteadiness of the whole SWBLI system have however not been recovered for this interaction with a laminar boun-dary layer. Nevertheless, the oscillatory motion of the reattachment shock wave have been recor-ded obviously in phase with the motion of the reattachment point. These results point out that the dynamics of the recirculation bubble is not the only phenomenon responsible to the SWBLI uns-teadiness and the turbulent structures of the incoming boundary layer might play an important role in triggering the whole unsteadiness.
The rest of this manuscript is then devoted to the simulation and analysis of the SWTBLI.
Table of contents :
1 Shock wave boundary layer interactions
1.1 Different shock wave boundary layer interactions in 2D flows
1.2 Mean flow
1.3 Strong shock wave boundary layer unsteadiness
1.4 Conclusions and outlook of the work
2 Equations and numerical approach
2.1 The governing equations
2.2 Introduction to high order numerical schemes for compressibleNavier-Stokes equations
2.3 Numerical approach applied
2.4 Domain and boundary conditions
3 Validation of the numerical approach
3.1 Taylor Green vortex at Re=1600
3.2 Shock-wave laminar boundary layer interaction
4 Shock wave laminar boundary layer
4.1 Physical parameters, computational domain and mesh
4.2 Mean flow
4.3 Flow Dynamics
5 Initiating a turbulent compressible boundary layer
5.1 Turbulent inflow boundary conditions
5.2 Implemented Synthetic EddyMethod
5.3 Simulation of a turbulent compressible boundary layer over a flat plate
6 Shock wave turbulent boundary layer interaction
6.1 Physical parameters, computational domain and mesh
6.2 Mean flow organization
6.3 Dynamics of the flow
6.4 POD analysis
Conclusions and perspectives