The continuous spectrum of the Orr-Sommerfeld equation: note on a paper of Grosch & Salwen

1991 ◽  
Vol 226 ◽  
pp. 565-571 ◽  
Author(s):  
A. D. D. Craik

Grosch & Salwen (1978) discuss the continuous-spectrum contribution in both temporal and spatial stability problems that are governed by the linear Orr–Sommerfeld equation. Their computed temporal continuum eigenfunctions for the Blasius boundary layer and for a laminar jet profile show surprising differences. This note provides an improved physical understanding of the results through a simple model, and shows that these differences are more apparent than real.

1978 ◽  
Vol 87 (1) ◽  
pp. 33-54 ◽  
Author(s):  
Chester E. Grosch ◽  
Harold Salwen

It is shown that the Orr-Sommerfeld equation, which governs the stability of any mean shear flow in an unbounded domain which approaches a constant velocity in the far field, has a continuous spectrum. This result applies to both the temporal and the spatial stability problem. Formulae for the location of this continuum in the complex wave-speed plane are given. The temporal continuum eigenfunctions are calculated for two sample problems: the Blasius boundary layer and the two-dimensional laminar jet. The nature of the eigenfunctions, which are very different from the Tollmien-Schlichting waves, is discussed. Three mechanisms are proposed by which these continuum modes could cause transition in a shear flow while bypassing the usual linear Tollmien-Schlichting stage.


1976 ◽  
Vol 77 (1) ◽  
pp. 81-104 ◽  
Author(s):  
D. Corner ◽  
D. J. R. Houston ◽  
M. A. S. Ross

Using the Orr-Sommerfeld equation with the wavenumber as the eigenvalue, a search for higher eigenstates in the stability theory of the Blasius boundary layer has revealed the existence of a number of viscous states in addition to the long established fundamental state. The viscous states are discrete, belong to two series, and are all heavily damped in space. Within the limits of the investigation the number of viscous states existing in the layer increases as the Reynolds number and the angular frequency of the perturbation increase. It is suggested that the viscous eigenstates may be responsible for the excitation of some boundary-layer disturbances by disturbances in the free stream.


Author(s):  
Anatoly I. Ruban

Chapter 1 discusses the flows that can be described in the framework of Prandtl’s 1904 classical boundary-layer theory, including the Blasius boundary layer on a flat plate and the Falkner–Skan solutions for the boundary layer on a wedge surface. It presents Schlichting’s solution for the laminar jet and Tollmien’s solution for the viscous wake. These are followed by analysis of Chapman’s shear layer performed with the help of Prandtl’s transposition theorem. It also considers the boundary layer on the surface of a fast rotating cylinder with the purpose of linking the circulation around the cylinder with the speed of its rotation. It concludes discussion of the classical boundary-layer theory with analysis of compressible boundary layers, including the interactive boundary layers in hypersonic flows.


1976 ◽  
Vol 73 (3) ◽  
pp. 497-520 ◽  
Author(s):  
Leslie M. Mack

A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity (c = cr + ici) plane on which the real and imaginary parts of the secular determinant are zero. This method gives only a finite and small number of discrete eigenvalues for a wide range of Reynolds numbers and wavenumbers. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards cr = 1 and the spacing between eigenvalues goes to zero. It is concluded, on the basis of this result and the contour method, that the original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow over a wide portion of the c plane for cr < 1 and cr > 1. It is suggested that the discrete spectrum is supplemented by a continuous spectrum which lies along the cr = 1 axis for ci < −α/R.


1981 ◽  
Vol 104 ◽  
pp. 445-465 ◽  
Author(s):  
Harold Salwen ◽  
Chester E. Grosch

The expansion of an arbitrary two-dimensional solution of the linearized stream-function equation in terms of the discrete and continuum eigenfunctions of the Orr-Sommerfeld equation is discussed for flows in the half-space, y ε [0, ∞). A recent result of Salwen is used to derive a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem.For the case of temporal stability, the orthogonality relation obtained is equivalent to that of Schensted for bounded flows. This relationship is used to carry out the formal solution of the initial-value problem for temporal stability. It is found that the vorticity of the disturbance at t = 0 is the proper initial condition for the temporal stability problem. Finally, it is shown that the set consisting of the discrete eigen-modes and continuum eigenfunctions is complete.For the spatial stability problem, it is shown that the continuous spectrum of the Orr-Sommerfeld equation contains four branches. The biorthogonality relation is used to derive the formal solution to the boundary-value problem of spatial stability. It is shown that the boundary-value problem of spatial stability requires the stream function and its first three partial derivatives with respect to x to be specified at x = 0 for all t. To be applicable to practical problems, this solution will require modification to eliminate disturbances originating at x = ∞ and travelling upstream to x = 0.For the special case of a constant base flow, the method is used to calculate the evolution in time of a particular initial disturbance.


Author(s):  
S. G. Rajeev

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.


Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3745
Author(s):  
Tristan Revaz ◽  
Fernando Porté-Agel

Large-eddy simulation (LES) with actuator models has become the state-of-the-art numerical tool to study the complex interaction between the atmospheric boundary layer (ABL) and wind turbines. In this paper, a new evaluation of actuator disk models (ADMs) for LES of wind turbine flows is presented. Several details of the implementation of such models are evaluated based on a test case studied experimentally. In contrast to other test cases used in previous similar studies, the present test case consists of a wind turbine immersed in a realistic turbulent boundary-layer flow, for which accurate data for the turbine, the flow, the thrust and the power are available. It is found that the projection of the forces generated by the turbine into the flow solver grid is crucial for rotor predictions, especially for the power, and less important for the wake flow prediction. In this context, the projection of the forces into the flow solver grid should be as accurate as possible, in order to conserve the consistency between the computed axial velocity and the projected axial force. Also, the projection of the force is found to be much more important in the rotor plane directions than in the streamwise direction. It is found that for the case of a wind turbine immersed in a realistic turbulent boundary-layer flow, the potential spurious numerical oscillations originating from sharp force projections are not harmful to the results. By comparing an advanced model which computes the non-uniform distribution of the turbine forces over the rotor with a simple model which assumes uniform effects of the turbine forces, it is found that both can lead to accurate results for the far wake flow and the thrust and power predictions. However, the comparison shows that the advanced model leads to better results for the near wake flow. In addition, it is found that the simple model overestimates the rotor velocity prediction in comparison to the advanced model. These elements are explained by the lack of local feedback between the axial velocity and the axial force in the simple model. By comparing simulations with and without including the effects of the nacelle and tower, it is found that the consideration of the nacelle and tower is relatively important both for the near wake and the power prediction, due to the shadow effects. The grid resolution is not found to be critical once a reasonable resolution is used, i.e. in the order of 10 grid points along each direction across the rotor. The comparison with the experimental data shows that an accurate prediction of the flow, thrust, and power is possible with a very reasonable computational cost. Overall, the results give important guidelines for the implementation of ADMs for LES.


2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.


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