Hypersonic blunt bodies: modelling streaks and their effect on different transition mechanisms

Caillaud ^{1,2, a}, E. Martini^{1, b}, G. Lehnasch^1,c, and P. Jordan ^1,d

¹ Institut PPRIME UPR3346 CNRS, ISAE-ENSMA, Université Poitiers,86961 Futuroscope, France

² CEA-CESTA, 15 avenue des sablières, CS 60001, 33116 Le Barp Cedex, France

^a clement.caillaud@isae-ensma.fr

^b eduardo.martini@isae-ensma.fr

^c guillaume.lehnasch@isae-ensma.fr (Corresponding author)

^d peter.jordan@univ-poitiers.fr

Keywords : resolvent analysis, blunt body, Direct Numerical Simulation, streaks

Abstract

Linear stability of planar hypersonic boundary layers evolving over smooth walls has been extensively described, highlighting the specific role of Mack and entropy modes in hypersonic regime. The transition scenario may dramatically change when curvature and roughness come into play.

The boundary layer structure, altered by isolated or distributed roughness and/or pressure gradient effects, can lead to new unstable modes and non-modal and/or non-linear mechanisms whose exact nature remains unclear. The transient growth of streaks is in particular suspected to play a key role [1]. The present contribution aims at giving an overview of recent studies [7] carried out to better understand such mechanisms.

 

A set of numerical tools, suitable for considering both complex geometries and large datasets, has been developed. Their most salient features will be first discussed. They concern :

1. a Resolvent Analysis (RA) code, adapted from [2] and [3],
2. a Direct Numerical Simulation (DNS) solver CurviCREAMS, adapted from [4] with optimized high order schemes and adaptive shock capturing methods in generalized coordinates
3. a Linearized DNS (LDNS) solver, implemented through a discrete linearization procedure of the DNS code by following the approach of [5].

 

Selected results will highlight more particularly two key aspects:

1. receptivity of streaks perturbations on paraboloid geometries
2. their influence on growth of instabilities in hypersonic planar streaky boundary layer subject to upstream white noise.

RA is first used to characterize the non-modal mechanisms which are likely to trigger the emergence and growth of flow streaks in the leading-edge region of bodies showing a paraboloid geometry. The sensitivity of the receptivity process and growth rate of streaks perturbations are characterized for a large range of azimuthal mode, Reynolds number, Mach number and bluntness level.

 

An example of leading forcing and response modes at zero frequency are illustrated in Fig. 1(a,b). Sub-optimal modes show similar spatial extension, just yielding radial oscillations in the forcing components. The liftup mechanism, associated to growth of such disturbances of streamwise vorticity convected into the boundary layer, thus appears optimally excited upstream of the body, near the stagnation point, rather than directly within the boundary layer.

 

Increasing the Reynolds number leads to rather similar features of this forcing domain. Their growth rate is found to be significantly reduced by comparison with the flat plate case but increases with the body bluntness, more particularly for lower azimuthal wavenumbers.

 

The compressibility effects, illustrated in Fig. 1(c), also increase the amplification of perturbations and reduces the optimal azimuthal wavenumber when the flow approaches sonic speeds.

Following the (linear PSE) study of Paredes et al [6], the effect of steady streaks on receptivity, growth and dynamics of instabilities is also here studied through DNS and LDNS. Optimal steady disturbances of increasing amplitude are determined through 2D LST and used to build different base flows containing steady spatially growing streaks.

 

DNS and LDNS of these base flows with upstream white-noise forcing are then carried out (see Fig. 2). The local stability of baseflows modulated by streaks is first analysed with a 2D Local Stability Analysis. Then the unsteady solutions from DNS and LDNS are decomposed with the introduction of a Floquet-SPOD formulation.

 

A rich variety of unsteady waves is found and their properties are discussed by comparing 2D-LST and SPOD results. The usual first and second Mack modes are found along with specific streaks induced unstable waves. For these waves families, stabilising properties of streaks are found at low amplitude, but this is no longer verified at higher amplitude where the first mode is strongly destabilised. These results highlight the sensitivity of the transition process to small boundary layer modulations that are likely to appear in practical conditions.