Emile Touber

Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available lowfrequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

A parametrized non-equilibrium wall-model for large-eddy simulations

Wall-models are essential for enabling large-eddy simulations (LESs) of realistic problems at high Reynolds numbers. The present study is focused on approaches that directly model the wall shear stress, specifically on filling the gap between models based on wall-normal ordinary differential equations (ODEs) that assume equilibrium and models based on full partial differential equations (PDEs) that do not. We develop ideas for how to incorporate non-equilibrium effects (most importantly, strong pressure-gradient effects) in the wall-model while still solving only wall-normal ODEs. We test these ideas using two reference databases: an adverse pressure-gradient turbulent boundary-layer and a shock/boundary-layer interaction problem, both of which lead to separation and re-attachment of the turbulent boundary layer.

Shock-induced energy transfers in dense gases

Dense gases are characterised by molecules featuring large numbers of active degrees of freedom (quantified by the cv/R ratio). The isentropes in such gases have the distinct property of following rather closely the isotherms (the two become identical in the limit of cv/R going to infinity). Near the liquid-vapour critical point, this makes the isentropes very shallow and possibly concave (in the pressure-specific volume diagram). Whilst shallow isentropes are desirable when designing expanders (i.e. a large specific-volume increase may be achieved for virtually no pressure drop), could such extreme compressibility effects modify turbulence in a profound manner? This paper discusses two particularly interesting aspects: (i) shock-refraction properties (i.e. the way a shock can redistribute the energy of incoming perturbations), (ii) enstrophy production in homogeneous turbulence. A linear interaction analysis (LIA) is conducted on the shock configuration for which the incoming perturbation is decomposed into linear modes of the compressible Euler equations. The transmission coefficients relative to each eigen modes are solved analytically and results are compared against fully non-linear compressible direct numerical simulation reproducing the weak perturbation of an isolated two-dimensional compression shock wave. The linear analysis is found to be capable of predicting the shock-induced redistribution of the energy of the incoming perturbation between the different eigen modes. Non-ideal gas effects are observed both analytically and numerically with especially an unusual selective response for some particular choice of incoming Mach number. A two-dimensional isotropic turbulence configuration is then numerically investigated for the case of an inviscid compressible dense-gas flow close to the liquid-vapour critical point. Strong non-ideal-gas effects on enstrophy production are observed with the formation of eddy shocklets. In both cases non-convex isentropes close to the liquid-vapour critical point are extremely influential in letting both the shock and the turbulence redistribute any supply of turbulence kinetic energy in ways which are simply not observable in ideal gases. This will hopefully spark enthusiasm amongst turbulence modellers (and their end users?).

Large-Eddy Simulations of an Oblique Shock Impinging on a Turbulent Boundary Layer: Effect of the Spanwise Confinement on the Low-Frequency Oscillations

The low-frequency motions found in two large-eddy simulations of the same oblique-shock/turbulent-boundary-layer interaction with significantly different domain widths are investigated. The narrow domain artificially confines the shock-induced separation bubble, which is seen to grow significantly. In addition, the low-frequency/large-amplitude shock oscillations are found to be enhanced, therefore suggesting that they originate from an intrinsic two-dimensional mechanism. By reducing the spanwise confinement, large coherent structures as wide as one separation-bubble length are found to develop inside the interaction. Those structures can move sideways and survive for extended periods of time. Their proper resolution is therefore challenging in terms of computational cost and their meandering motions can significantly bias the interpretation of a spectral analysis performed at a fixed point.

Numerical simulations of shock-wave/boundary-layer interaction phenomena

We review recent simulations of shock-induced separation of boundary layers and then consider in detail some properties of the detached shear layer that forms after separation of a turbulent boundary layer at Mach 2.3. Whilst still challenging in terms of numerical methods, due to the simultaneous presence of shock waves and turbulence, both direct numerical simulation and large eddy simulations (LES) are useful tools to investigate fundamental issues of shock-wave/boundary-layer interaction. In particular with LES it is feasible to calculate the long run times and wide domains necessary to study low frequency motions that occur under the reflected shock foot. We show here that a simplified model based on growth rate of the separated shear layer leads to predictions of frequency that are significantly higher than the low-frequency peak seen in LES and experiment.