Introduction to high order numerical schemes for compressibleNavier-Stokes equationsIntroduction to high order numerical schemes for compressibleNavier-Stokes equations

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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 iswell known and have been studied both experimentally and numerically for decades, the relatedmechanisms are still notwell understood.

Dynamics of the shock wave boundary layer interaction

We here describe the dynamic features of strong SWBLIs that are characterized by several unsteady phenomena whose characteristic scales spread over a large broadband spectrumrange. For SWTBLI, the incoming boundary layer is turbulent with themost energetic fluctuations at high frequencies characterized by a Strouhal number S± Æ f ± Ue » 1 (where f , ± and Ue are respectively 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 [2008]) 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 [2011], 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 cales excite the reflected shock wave.

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 behavior 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 [2001]. After this date, a significant improvement in the comprehension of the SWBLI unsteadiness have been obtained using experimental means andmodern 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 [2009]).
he first mechanism consists in a perturbation of the SWBLI by the large scales within the incoming turbulent boundary layer. Indeed, an experimental study (Ganapathisubramani et al. [2006]) of a separated compression ramp interaction at M Æ 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 Æ 0.025L ± 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 upstream 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] Ganapathisubramani et al. [2007b]).Nevertheless, the proposed mechanismwas challenged by other experimental and numerical results. For instance, an experimental study (Dupont et al. [2005]) of the interac- tion between an oblique shock wave and a turbulent boundary layer developing on a flat plate at M Æ 2.3 presented ratios L/± ‘ 5¡7 for the recirculation bubble and a SWBLI unsteadiness at SL ‘ 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 [2009]) 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. [2005], several numerical works advocate for this second mechanism. The DNS of Pirozzoli and Grasso [2006] and the LES of Aubard et al. [2013] that studied the interaction between an incident oblique shock wave and a turbulent boundary layer on a flat plate at MÆ 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 Aubard et al. [2013], 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 [2006], authors claim that the interaction between the vortical 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 selfsustained oscillations. In Piponniau et al. [2009], a simple model was proposed fromexperimental observations, based on the entrainment characteristics of the shear layer. The DynamicMode Decomposition (DMD) analysis performed by Priebe et al. [2016] on a previously DNS of aMach 2.9, 24± compression ramp SWBLI (Priebe andMartín [2012]) allowed authors to infere that the SWBLI unsteadiness would be related to the presence of Görtler-like vortices in the downstream separated flow due to an underliying centrifugal instability.

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Shock capturing procedures

Whatever the approach used, the high order numerical scheme needs a shock capturing procedure
to avoid the spurious oscillations arising fromthe high order discretization of discontinuities. Different shock capturing procedures have been introduced in the litterature. These methods introduce numerical diffusion near discontinuisties in amore or less implicit way.

Approaches used for high fidelity SWBLI simulations

Most of the high fidelity simulations of the SWBLI available in the literature have been performed using the method of lines approach with a Runge-Kutta time integration and finite differences for the spatial discretization.We here summarize the numerical approaches used in studies already cited in chapter 1 (section 1.3.2) that are representative from the classical numerical approaches used in the last decade to performhigh fidelity simulations of the SWBLI.
— In Pirozzoli and Grasso [2006], the authors use a 4th order Runge-Kutta time integration scheme. A 7th order WENO reconstruction of the characteristic inviscid fluxes that uses a local Lax–Friedrichs flux difference approach is used. A 4th order compact FD scheme (Lele [1992]) is used for the the viscous fluxes approximation.
— In Touber and Sandham [2009] the time integration is performed using a 3rd order Runge- Kutta scheme. The spatial discretization is performed by a 4th order central FD scheme both for the convective and the viscous fluxes. The so called « entropy splitting » method imposes the entropy conservation in order to stabilize the discretization of the convective fluxes. In order to enhance the stability of the viscous fluxes approximation, the laplacian formis used. The shock capturing procedure relies on a variant of the TVD constraint coupled with the Ducros sensor (Ducros et al. [1999]).
— In Aubard et al. [2013] a 6 steps 4th order Runge-Kutta scheme is used for the time integration. The convective fluxes are discretized by mean of a 4th order low dissipative and low dispertive FD scheme. An artificial viscosity method is used as a shock capturing procedure.
— In Priebe et al. [2016] the time integration is performed using a 3rd Runge-Kutta scheme. A 4th orderWENO FD scheme is used for the approximation of the convective fluxes whereas a 4th order standard FD scheme is used for the viscous fluxes.

Table of contents :

List of figures
List of tables
0.1 References
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
1.5 References
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
2.5 Conclusions
2.6 References
3 Validation of the numerical approach 
3.1 Taylor Green vortex at Re=1600
3.2 Shock-wave laminar boundary layer interaction.
3.3 Conclusions
3.4 References
4 Shock wave laminar boundary layer 
4.1 Physical parameters, computational domain and mesh
4.2 Mean flow
4.3 Flow Dynamics
4.4 Conclusion
4.5 References
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
5.4 Conclusion
5.5 References
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
6.5 Conclusions
6.6 References
Conclusions and perspectives 


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