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# Numerical Investigation of High-Speed Wall-Bounded Turbulence Subject to Complex Wall Impedance

Laminar or turbulent flows over porous surfaces have received extensive attention in the past few decades, due to their potential to achieve passive flow controls. These surfaces either in natural exhibit roughness or are engineered in purpose, and usually entail special features such as increasing/reducing surface drags. An increasing interest has arisen in the interaction between these surfaces and high-speed compressible flows, which could inform the next-level flow control studies at supersonic and hypersonic speeds for the designs of high-speed vehicles. In this dissertation, the interaction between high-speed compressible turbulent flows and acoustically permeable surface is investigated. The surface property is modeled via the Time-Domain Impedance Boundary Condition (TDIBC), which avoids the inclusion of the geometric details in the numerical simulations.

We first perform Large-Eddy Simulations of compressible turbulent channel flows over one impedance wall for three bulk Mach numbers:Mb = 1.5, 3.5 and 6.0. The bulk Reynolds number Reb is tuned to achieve similar viscous Reynolds number Re∗τ ≈ 220 across all Mb to ensure a nearly common state of near-wall turbulence structures over impermeable walls. The TDIBC based on the auxiliary differential equations (ADE) method is applied to bottom wall of the channel. A three-parameter complex impedance model with a resonating frequency tuned to the large-eddy turn-over frequency of the flow is adopted. With a sufficiently high permeability, a streamwise traveling instability wave that is confined in nature and that increases the surface drag, is observed in the near-wall region and changes the local turbulent events. As a result, the first and second order mean flow statistics are found to deviate from that of a flow over impermeable walls. We then perform a linear stability analysis using a turbulent background base flow and confirm that the instability wave is triggered by a sufficiently high permeability and manifests a confined nature. The critical resistance Rcr (interpreted as the inverse of the permeability), above which the instability is suppressed, is found to be sub-linearly proportional to the bulk Mach number Mb, indicating less permeability required to trigger the instability in high Mach number flows.

Due to the extremely high computational cost in high Mach number wall-bounded flow calculations, the next-phase optimization/flow control design using the porous surface becomes unaffordable. An ’economical’ flow setup that can server the purpose of rapid flow generation would greatly benefit the planned research. For such reason, we carry out a study about the effect of the domain size on the near-wall turbulence structures in compressible turbulent channel flows, to identify such type of flow setup. Apart from the concept of minimal flow units (MFU, as in the literature) entailing a minimal domain size required for near-wall turbulence to be sustained, efforts have also been made to identify a range of the domain size that can sustain both the inner and outer layer turbulence, and lead to only small deviations in mean flow statistics from the baseline data, which herein defined as minimal turbulent channel (MTC). The motivation of proposing the concept of MTC is to provide a computationally efficient setup for the rapid generation of near-wall turbulence with minimal compromise on the fidelity of the simulated field for investigations requiring numerous simulations, such as machine learning, flow control/optimization designs. It is found that the mean flow statistics from a computational domain spanning 700 − 1100 and 230 − 280 local viscous units in streamwise and spanwise directions, respectively, agree reasonably well with the reference calculations of all three Mach numbers under investigation, and are thus identified as the range in which the MTC stays. The large scale near-wall turbulence structures observed in full scale DNS simulations, and their spatially coherent connections, are roughly preserved in MTC, indicated by the existence of the grouped streamwise aligned hairpin vortices of various sizes and the resulted patterns of uniform momentum zones and thermal zones in the instantaneous flow field. In an MTC, the energy transfer paths among the kinetic energy of the mean field, turbulent kinetic energy and mean internal energy are slightly modified, with the most significant change observed in the viscous dissipation. The mean wall-shear stress and mean wall heat flux see less than 5% error as compared to the full scale simulations. Such reduced-order flow setup requires less than 3% of the computational resource as compared to the full scale simulations.

## Funding

### Boundary Layer Turbulence Control via Acoustically Resonating Porous Surfaces

Directorate for Engineering

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