Tamer A. Zaki

Estimating past events in wall turbulence based solely on surface measurements and first principles is an ill-posed problem that is complicated by chaos. The sensitivity of a measurement to the earlier flow state is described by the adjoint Navier–Stokes equations, which are solved in reverse time starting from the measurement kernel at the sensing position and time. The resulting adjoint field is the spatio-temporal domain of dependence (DOD) of the sensor, which is a dual to the concept of the domain of influence (DOI) of an actuator in the linearised forward equations. In channel turbulence, the energy of each adjoint realisation grows exponentially in backward time according to the Lyapunov exponent, even though the energy of the ensemble average should decay. We introduce a linear eddy-viscosity closure model in the ensemble-averaged adjoint equations, and directly compute the mean DOD and compare our prediction with the ensemble average. Furthermore, we demonstrate that the DOD of a wall-stress measurement and the DOI resulting from a wall-stress perturbation exhibit respective universal behaviours across Reynolds numbers. However, their spatio-temporal structures differ qualitatively, due to the time asymmetry of the governing equations. The DOD field has a two-part structure: one component is associated with the Orr mechanism, characterised by rapid reorientation under mean shear, and the other is related to self-similar expanding streaky structures. These two components jointly define the sensitivity of the wall-stress measurement to past flow events.

Ensemble-variational (EnVar) assimilation of wall-pressure measurements in direct numerical simulations of Mach 6 flow over a cone–flare is performed. The experimental data include pressure spectra and intensities from seven wall-mounted PCB sensors positioned upstream, within and downstream of the separation region induced by the compression corner. Assimilation of the first two sensors only, all upstream of separation, is insufficient to accurately predict the downstream flow. Assimilating all the sensor data is shown to be essential to correctly predict separation onset and the downstream wall-pressure data. Similar to the experiments, the assimilated flow features intense rope-like structures in the attached region. The simulations additionally predict a localised amplification of disturbances beneath the separation shock, where experimental data are not available. This amplification results from the interaction of the boundary-layer instability modes with the compression shock. The simulations also capture the sharp decrease in wall-pressure intensity across separation, and the amplification of low-frequency three-dimensional disturbances within the recirculation bubble. Additionally, the computations highlight the uncertainty in the post-separation predictions due to the low-frequency unsteadiness of the separation shock. Oscillations of the streamwise velocity modulate the boundary-layer thickness, which in turn introduces variability in disturbance amplification.

Interactions of turbulent boundary layers with a compliant surface are investigated experimentally at Re _τ = 3300–8900. Integrating tomographic particle tracking with Mach–Zehnder interferometry enables simultaneous mapping of the compliant wall deformation and the three-dimensional velocity and pressure fields. Our initial study (J. Fluid. Mech. vol. 980, R2) shows that the flow–deformation correlations decrease with increasing Re _τ , despite an order of magnitude increase in deformation amplitude. To elucidate the mechanisms involved, the same velocity, pressure and kinetic energy fields are decomposed to ‘wave-coherent’ and ‘stochastic’ parts using a Hilbert projection method. The phase dependent coherent variables, especially the pressure, are highly correlated with the wave, but decrease with increasing Re _τ . While the coherent energy is 6 %–10 % of the stochastic level, the pressure root mean square is comparable near the wall. The energy flux between the coherent and stochastic parts and the pressure diffusion reverse sign at the critical layer. To explain the Re _τ dependence, the characteristic deformation wavelength (three times the thickness) is compared with the scales of the energy-containing eddies in the boundary layer represented by the k⁻¹ range in the energy spectrum. When the deformation wavelength is matched with the k _x E _uu peak at the present lowest Re _τ , the flow–deformation correlations and coherent pressure become strong, even for submicron deformations. In this case, the flow and wall motion become phase locked, suggesting resonant behaviours. As Re _τ increases, the wall wavelengths and spectral range of attached eddies are no longer matched, resulting in reduced correlations and lower coherent energy and pressure, despite larger deformation.