Studies of boundary-layer receptivity with parabolized stability equations

Herbert, Thorwald; Lin, Nay

doi:10.2514/6.1993-3053

Cited by 33 publications

(27 citation statements)

References 15 publications

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“…The efficiency function is then computed with Eq. (33). Results were confirmed to be geometry independent to the desired precision by studying convergence of the results with decreasing width, while numerical independence was ensured with 56 Chebyshev polynomials in the wall-normal direction and 10000 points in the streamwise direction.…”

Section: B Localized Receptivitymentioning

confidence: 87%

“…In particular, the multiple-scales method 30 was recently applied to the linearized Navier-Stokes (LNS) equations to model receptivity 31 . The parabolized stability equations 22,32,33 also partially capture non-parallel flow effects. 41 took on the acoustic receptivity problem by solving the compressible unsteady Navier-Stokes equations to compute the steady basic flows, and the LNS equations to obtain the unsteady perturbations.…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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.

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Section: B Localized Receptivitymentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

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

2018

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“…This normalization condition minimizes the streamwise change ∂φ/∂ξ in a weighted sense over the N −domain, which also keeps ∂φ/∂ξ small in accordance with our initial assumption. Other normalization conditions could be implemented (Herbert 1993;Andersson et al 1998), however we find this one to be most desirable because it gives results which are in excellent agreement with those for a boundary layer on a flat plate (Turner & Hammerton 2006). Although the formulation does not prove that a solution satisfying (2.11) exists, the agreement of the PSE results with those of Goldstein (1983) for a flat plate justifies this choice (Turner & Hammerton 2006;Turner 2005).…”

Section: Formulation Of the Parabolized Stability Equationmentioning

confidence: 85%

Analysis of the unstable Tollmien–Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

Turner

Hammerton

2009

J. Fluid Mech.

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The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation (PSE) in the Orr-Sommerfeld region along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalised to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien-Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

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“…This equation is known as the linearised unsteady boundary layer equation (LUBLE). Far downstream in this region, the solution for ψ 0 consists of a combination of a Stokes solution, and a sum of eigensolutions satisfying homogeneous boundary conditions (Lam & Rott 1960& 1993Brown & Stewartson 1973). The two sets of eigensolutions differ fundamentally and their precise relationship is unclear (Hammerton 1999).…”

Section: Leading Edge Receptivity Analysismentioning

confidence: 99%

“…Since the equations have been parabolized, an upstream boundary condition, sometimes referred to as an initial condition, is required. Previous papers which consider the PSE (Bertolotti et al 1992 andHerbert 1993) use approximations such as parallel Orr-Sommerfeld theory, or a local solution to the PSE as initial upstream conditions. However, such an approach does not take account of the amplitude of the unsteady disturbance at this point forced by the free-stream disturbances.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic receptivity analysis and the parabolized stability equation: a combined approach to boundary layer transition

Turner

Hammerton

2006

J. Fluid Mech.

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We consider the interaction of free-stream disturbances with the leading edge of a body and its effect on the transition point. We present a method which combines an asymptotic receptivity approach, and a numerical method which marches through the Orr-Sommerfeld region. The asymptotic receptivity analysis produces a three deck eigensolution which in its far downstream limiting form, produces an upstream boundary condition for our numerical Parabolized Stability Equation (PSE). We discuss the advantages of this method against existing numerical and asymptotic analysis and present results which justifies this method for the case of a semi-infinite flat plate, where asymptotic results exist in the Orr-Sommerfeld region. We also discuss the limitations of the PSE and comment on the validity of the upstream boundary conditions. Good agreement is found between the present results and the numerical results of Haddad & Corke (1998).

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Studies of boundary-layer receptivity with parabolized stability equations

Cited by 33 publications

References 15 publications

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

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

Analysis of the unstable Tollmien–Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

Asymptotic receptivity analysis and the parabolized stability equation: a combined approach to boundary layer transition

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