Über das Spektrum bei Eigenschwingungen ebener Laminarströmungen

Grohne, D.

doi:10.1002/zamm.19540340851

Cited by 42 publications

(18 citation statements)

References 8 publications

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“…In this alternate approach we first note that the solutions of equation ( The solutions have been normalized, following Grohne (1954), so that [cf. equation (3.16)]…”

Section: The Characteristic Equationmentioning

confidence: 99%

“…As Synge (1938) has remarked, "the deduction of general results from this equation is well-neigh impossible" and some further approximations must obviously be made. In his early work on this problem, Hopf (1914) [see also Grohne (1954)J considered three approximations which lead to an enormous simplification of the characteristic equation. First, he effectively approximated the Airy functions A k «( + C( 2 S2) by Ak(O; this is clearly permissible since, from equation (2.31), Ak(O is simply the first term in the uniform expansions of A k «( + C( 2 S2) for bounded values of 1' 11.…”

Section: The Characteristic Equationmentioning

confidence: 99%

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Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

Reid

1974

Stud Appl Math

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This paper is concerned with the derivation of asymptotic approximations to the solutions of the Orr‐Sommerfeld equation and its adjoint which are uniformly valid in a full neighborhood of the critical point. In the case of plane Couette flow, it is shown that the expansions can be expressed to all orders in terms of a restricted class of generalized Airy functions and that the slowly varying coefficients in these expansions can be determined from simple recurrence relations. The expansions can easily be related to the Laplace integral representation of the solutions but this is not essential in their derivation. The structure of the theory also suggests how it can be immediately generalized to provide “first approximations” for a more general class of velocity profiles. Detailed results are given in the special case when the wave number α → 0. In this limit, the characteristic equation can be reduced to an especially simple form which permits a convenient global description of the modal structure.

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“…In this alternate approach we first note that the solutions of equation ( The solutions have been normalized, following Grohne (1954), so that [cf. equation (3.16)]…”

Section: The Characteristic Equationmentioning

confidence: 99%

Section: The Characteristic Equationmentioning

confidence: 99%

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

Reid

1974

Stud Appl Math

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“…The most famous and thoroughly studied approach, the spectral/modal approach (spectral expansion of disturbances in time and space, followed by the eigenfunction analysis), was derived by Orr. 1 A large collection of literature exists on the application of this approach to homogeneous incompressible [2][3][4][5][6] and compressible shear flows, [7][8][9] as well as inhomogeneous stratified or magnetized shear flows. [10][11][12][13][14] A comprehensive introduction to this field can be found, for instance, in the book by Schmid and Henningson.…”

Section: Introductionmentioning

confidence: 99%

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

Hau

Oberlack

Chagelishvili

2017

Journal of Mathematical Physics

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to deceptive conclusions (see, e.g., Ref. 16 and references herein), which do not coincide with experimental observations. 16,17 In the 1990s, the limiting nature of the classical modal approach was recognized and the nonnormal nature of non-uniform/shear flows was finally revealed and rigorously proven (see, e.g., Refs. 15,16,[18][19][20] and references herein)-a major breakthrough in the understanding of linear and nonlinear shear flow dynamics. In fact, the operators involved in the modal analysis of plane shear flows are non-normal, resulting in the non-orthogonality of the corresponding eigenfunctions, and hence strong interference phenomena among the eigenmodes. 19 The so-called non-modal approach, shifting the emphasis from the asymptotic to the short-time dynamics, became a well established alternative, taking into account the shortcomings of the modal approach. This resulted in the understanding and precise description of linear transient phenomena. In one branch of non-modal stability theory, the system's response to initial conditions is investigated, which is central to hydrodynamic stability theory. Here, the Kelvin mode approach, 21 stemming from the 1887 paper by Lord Kelvin, with a time-dependent shearwise wavenumber, has been extensively used for constant shear flows since the 1990s (see, e.g., . This mode represents the "simplest element" of constant shear flow physics 29 and has also been referred to as "flowing eigenfunctions" in expanding fluctuations. 30 In particular, there exist various terminologies for the Fourier mode with a time-dependent wave vector in the different areas of fluid mechanics, e.g., "Kelvin mode," "shear wave," "SH wave," "flowing eigenfunctions." Salhi and Cambon 31 gave a comprehensive survey of the method in the different communities (see Secs. I and II C).We make use of the Lie point symmetry analysis, as founded by the Norwegian mathematician Sophus Lie (1842-1899) and applied successfully in the area of fluid mechanics 32-48 to systematically derive invariant solutions in the context of linear stability of a two-dimensional (2-D) linearly sheared unbounded flow. The notion of invariant solutions was introduced by Sophus Lie, 49 and it denotes solutions whose representations are obtained with the help of any combinations of the admitted groups. This method identifies variable transformation by which new solutions can be generated from existing ones through the use of a differential operator, i.e., the generator of the group. The similarity analysis of systems of partial differential equations (PDEs) leading to group-invariant solution is systemically and well described in several textbooks, for instance, by Refs. 50-54. Compared to a simple dimensional analysis, this method is somehow superior. This is due to the fact that there is no need to analyze the equations and boundary/initial conditions at the same time in order to identify the desired similarity variables-non-dimensional combinations of variables. Further, it is possible to uncover types of similarity th...

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“…Numerical studies of the linear stability problem have been carried out by Grohne (1954), Gallagher andMercer (1962, 1964), Deardorff (1963), Davey (1973), and Gallagher (1974); however, all these studies found no evidence of instability. A number of analytic studies have also been carried out on this problem.…”

Section: Introductionmentioning

confidence: 99%

On the linear stability of compressible plane Couette flow

1994

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91-11267 ABSTRACTThe linear stability of compressible plane Couette flow is investigated. The correct and proper basic velocity and temperature distributions are perturbed by a small amplitude normal mode disturbance. The full small amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in sene detail. It is found that instability can occur, although the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wavespeed of the disturbances approaches the velocity of either of the walls, and these regimes are also analyzed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

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Über das Spektrum bei Eigenschwingungen ebener Laminarströmungen

Cited by 42 publications

References 8 publications

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

On the optimal systems of subalgebras for the equations of hydrodynamic stability analysis of smooth shear flows and their group-invariant solutions

On the linear stability of compressible plane Couette flow

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