Acoustic receptivity and transition modeling of Tollmien-Schlichting disturbances induced by distributed surface roughness

Abstract

Acoustic receptivity to Tollmien-Schlichting waves in the presence of surface roughness is investigated for a flat plate boundary layer using the time-harmonic incompressible linearized Navier-Stokes equations. It is shown to be an accurate and efficient means of predicting receptivity amplitudes, and therefore to be more suitable for parametric investigations than other approaches with DNS-like accuracy. Comparison with literature provides strong evidence of the correctness of the approach, including the ability to quantify non-parallel flow effects. These effects are found to be small for the efficiency function over a wide range of frequencies and local Reynolds numbers. In the presence of a two-dimensional wavy-wall, non-parallel flow effects are quite significant, producing both wavenumber detuning and an increase in maximum amplitude. However, a smaller influence is observed when considering an oblique Tollmien-Schlichting wave. This is explained by considering the non-parallel effects on receptivity and on linear growth which may, under certain conditions, cancel each other out. Ultimately, we undertake a Monte-Carlo type uncertainty quantification analysis with two-dimensional distributed random roughness. Its power spectral density (PSD) is assumed to follow a power law with an associated uncertainty following a probabilistic Gaussian distribution. The effects of the acoustic frequency over the mean amplitude of the generated two-dimensional Tollmien-Schlichting waves are studied. A strong dependence on the mean PSD shape is observed and discussed according to the basic resonance mechanisms leading to receptivity. The growth of Tollmien-Schlichting waves is predicted with non-linear parabolized stability equations computations to assess the effects of stochasticity in transition location.

An adjoint compressible linearised Navier–Stokes approach to model generation of Tollmien–Schlichting waves by sound

“…The problems can essentially be divided in two categories: the steady base flow and the acoustic signature are governed by parabolic equations which can be solved through an efficient streamwise marching procedure given an initial solution; the steady mean-flow distortion and the T-S wave, as well as their adjoint counterparts, are fully elliptic problems which upon numerical discretisation ultimately require solving very large systems of linear equations. The numerical schemes and procedure are described in detail in Mughal & Ashworth (2013) and Raposo et al (2018a). Only new aspects will be discussed hereafter.…”

Acoustic Receptivity of Compressible Tollmien-Schlichting Waves with an Efficient Time-Harmonic Linearized Navier-Stokes Method

“…We briefly describe the numerical methods to solve the four sub-problems leading to the determination of the T-S wave amplitude. Detailed accounts are found in [23,24].…”

“…Each perturbation is governed by a different set of equations that can be obtained by substitution of Eq. 1into the Navier-Stokes (N-S) equations (see [7,24] for a more detailed discussion). Flow and material quantities are made nondimensional with their respective far-field value.…”

“…The work reported here builds on previous work on time-harmonic LNS and applications to acoustic receptivity [23][24][25], but addresses additional challenges such as the modeling of the compressible Stokes layer. Ultimately, the problem of interest in the aeronautics sector is that of an airfoil operating in transonic flow conditions with acoustic waves incident at a generalized angle.…”

Acoustic excitation of Tollmien–Schlichting waves due to localised surface roughness

“…A total of 18 different roughness heights were used from 30 µm < |h| < 750 µm (0.025 < |h|/δ * B < 0.630). These were chosen to cover both the linear and the nonlinear amplitude forcing (Saric 1994;Raposo et al 2018). The minimum height needed to generate reliable and measurable disturbances was found to be approximately 30 µm.…”