The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions

Salwen, Harold; Grosch, C. E.

doi:10.1017/s0022112081002991

Cited by 167 publications

(96 citation statements)

References 7 publications

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“…If an eigenfunction expansion of the disturbance field is sought, the definition of the adjoint problem is necessary in order to introduce a biorthogonality condition between the different eigenmodes (Morse & Feshbach 1953;Salwen & Grosch 1981;Tumin 2003). Applying the usual inner product for complex functions…”

Section: Direct and Adjoint Biglobal Instability Analysesmentioning

confidence: 99%

“…In the case l = m, the term in parenthesis vanishes and the inner product is non-zero. It can be shown (Salwen & Grosch 1981;Hill 1995) that if the eigenspectrum is complete, an arbitrary disturbance can be expressed as a linear combination of the direct eigenfunctions. The contribution of the disturbance to a discrete eigenmode / is obtained as the projection of the disturbance on the adjoint eigenfunction q t using the condition (3.16), with an adequate normalization.…”

Section: Direct and Adjoint Biglobal Instability Analysesmentioning

confidence: 99%

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Structural changes of laminar separation bubbles induced by global linear instability

Rodríguez

Theofilis

2010

J. Fluid Mech.

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The topology of the composite flow fields reconstructed by linear superposition of a two-dimensional boundary layer flow with an embedded laminar separation bubble and its leading three-dimensional global eigenmodes has been studied. According to critical point theory, the basic flow is structurally unstable; it is shown that in the presence of three-dimensional disturbances the degenerate basic flow topology is replaced by a fully three-dimensional pattern, regardless of the amplitude of the superposed linear perturbations. Attention has been focused on the leading stationary eigenmode of the laminar separation bubble discovered by Theofilis et al. (Phil. Trans. R. Soc. Lond. A, vol. 358, 2000, pp. 3229-3324); the composite flow fields have been fully characterized with respect to the generation and evolution of their critical points. The stationary global mode is shown to give rise to a three-dimensional flow field which is equivalent to the classical U-shaped separation, defined by Hornung & Perry (Z. Flugwiss. Weltraumforsch., vol. 8, 1984, pp. 77-87), and induces topologies on the surface streamlines that are resemblant to the characteristic stall cells observed experimentally. IntroductionThe instability of a laminar separation bubble (LSB), embedded inside a boundary layer and arising as a consequence of an adverse free-stream pressure gradient on a flat surface, or induced by changes of the surface curvature, constitutes a problem of prime technological significance: in an aeronautical context related applications include lifting surfaces, low-pressure turbines and wind-turbine blades. The presence of a separated shear layer as an integral part of the LSB suggests that the instability properties of the shear layer should play a decisive role in the reattachment of the separated boundary layer. The one-dimensionality of the model shear-layer profile and its resulting amenability to classic, ordinary-differential-equation-based analysis (Michalke 1964; Blumen 1970 and subsequent work in this 'local' vein) has generated a wealth of knowledge on the problem of shear-layer instability perse and as applied to the LSB flow. On the other hand, Gaster postulated (H. Fasel, private communication November 2004) that, besides paying attention to the amplification of incoming disturbances by solution of the local instability problem, analysis of LSB flows 'in the true sense' should also be performed, by which a global instability analysis of the entire separated boundary layer flow is implied.

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Section: Direct and Adjoint Biglobal Instability Analysesmentioning

confidence: 99%

Section: Direct and Adjoint Biglobal Instability Analysesmentioning

confidence: 99%

Structural changes of laminar separation bubbles induced by global linear instability

Rodríguez

Theofilis

2010

J. Fluid Mech.

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“…The adjoint linearised Navier-Stokes (ALNS) route is naturally suited for this purpose. Adjoint equations require that solutions be expanded as a bi-orthogonal set of eigensolutions 21 , and are advantageous over alternative receptivity schemes (i.e direct LNS) as they can be used to instantaneously predict the initial size of a disturbance to many environmental mechanisms. Hence, adjoint formulations are economical with both computational and time resources, and can be employed to conduct rapid and extensive Monte-Carlo type analysis.…”

Section: Introductionmentioning

confidence: 99%

On predicting receptivity to surface roughness in a compressible infinite swept wing boundary layer

2017

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The receptivity of crossflow disturbances on an infinite swept wing is investigated using solutions of the adjoint linearised Navier-Stokes equations. The adjoint based method for predicting the magnitude of stationary disturbances generated by randomly distributed surface roughness is described, with the analysis extended to include both surface curvature and compressible flow effects. Receptivity is predicted for a broad spectrum of spanwise wavenumbers, variable freestream Reynolds numbers and subsonic Mach numbers. Curvature is found to play a significant role in the receptivity calculations, while compressible flow effects are only found to marginally affect the initial size of the crossflow instability. A Monte-Carlo type analysis is undertaken to establish the mean amplitude and variance of crossflow disturbances generated by the randomly distributed surface roughness. Mean amplitudes are determined for a range of flow parameters that are maximised for roughness distributions containing a broad spectrum of roughness wavelengths, including those that are most effective in generating stationary crossflow disturbances. A control mechanism is then developed where the short scale roughness wavelengths are damped, leading to significant reductions in the receptivity amplitude.

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“…The same type of conditions are also imposed at the inflow boundary of open domains, when the analysis aims at excluding perturbations from entering the computational domain [65]. Vanishing of linear perturbations is also imposed at far-field boundaries, if the latter are taken far away from solid surfaces, although it should be stressed that, much like the situation in classic linear stability theory, this choice misrepresents or altogether eliminates the continuous branch of perturbations oscillatory to infinity [49,56]. At open outflow boundaries, it is in principle unclear what form the perturbation velocity may assume, although linear extrapolation from the interior has been found to not only work well, but also to have little effect on the form of linear perturbations in the interior of the domain, when the latter is analytically known [64].…”

Section: Modal Linear Biglobal Analysismentioning

confidence: 99%

The linearized pressure Poisson equation for global instability analysis of incompressible flows

Theofilis

2017

Theor. Comput. Fluid Dyn.

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The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.Keywords Global linear instability · Matrix-forming · Collocated grids · Pressure boundary conditions IntroductionGlobal linear instability theory is enjoying increasing acceptance in the fluid mechanics community, as also witnessed by the contents of this volume. The theory has permitted revealing previously unknown mechanisms responsible for laminar-turbulent flow transition in complex three-dimensional geometries with maximally one homogeneous spatial direction. Two main approaches are followed for the numerical work associated with modal and non-modal global linear instability analysis, namely matrix-forming vs. time-stepping, both of which have been discussed in a recent review [61]. The matrix-forming approach is interesting when dealing Communicated by Ati

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The continuous spectrum of the Orr-Sommerfeld equation. Part 2. Eigenfunction expansions

Cited by 167 publications

References 7 publications

Structural changes of laminar separation bubbles induced by global linear instability

Structural changes of laminar separation bubbles induced by global linear instability

On predicting receptivity to surface roughness in a compressible infinite swept wing boundary layer

The linearized pressure Poisson equation for global instability analysis of incompressible flows

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