Orr-Sommerfeld Equation Research Articles

Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid

The linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. A pair of coupled Orr-Sommerfeld eigenvalue equations are derived and solved using an efficient spectral collocation method for cases in which unyielded regions are absent. An asymptotic analysis is also carried out in the long-wave limit, the results of which are in excellent agreement with the numerical predictions. Our analytical and numerical results indicate that increasing the dimensionless yield stress, prior to the formation of unyielded plugs below the interface, is destabilizing. Increasing the shear-thinning tendency of the lower fluid is stabilizing.

Experimental investigation into the routes to bypass transition and the shear-sheltering phenomenon

The objective of this investigation is to give experimental support to recent direct numerical simulation (DNS) results which demonstrated that in bypass transition the flow first breaks down to turbulence on the low-speed streaks (or so-called negative jets) that are lifted up towards the boundary-layer edge region. In order to do this, wall-normal profiles of the streamwise fluctuation velocity are presented in terms of maximum positive and negative values over a range of turbulence intensities (1.3–6%) and Reynolds numbers for zero pressure gradient flow upstream of, and including, transition onset. For all turbulence intensities considered, it was found that the peak negative fluctuation velocity increased in magnitude above the peak positive fluctuations and their positions relative to the wall shifted as transition onset approached; the peak negative value moved towards the boundary-layer edge and the peak positive value moved toward the wall. An experimental measure of the penetration depth (PD) of free-stream disturbances into the boundary layer has been gained through the use of the skewness function. The penetration depth (measured from the boundary-layer edge) scales as PD ∝ (ω Rexτw)−0.3), where ω is the frequency of the largest eddies in the free stream, Rex is the Reynolds number of the flow based on the streamwise distance from the leading edge and τw is the wall shear stress. The parameter dependence demonstrated by this scaling compares favourably with recent solutions to the Orr–Sommerfeld equation on the penetration depth of disturbances into the boundary layer. The findings illustrate the importance of negative fluctuation velocities in the transition process, giving experimental support to suggestions from recent DNS predictions that the breakdown to turbulence is initiated on the low-speed regions of the flow in the upper portion of the boundary layer. The representation of pre-transitional disturbances in time-averaged form is shown to be inadequate in elucidating which disturbances grow to cause the breakdown to turbulence.

Linear stability of a two-layer film flow down an inclined channel: A second-order weighted residual approach

Our interest is with the long-wavelength instability in a two-dimensional gravity-driven flow of two superposed, incompressible, immiscible, and viscous fluids bounded by an upper and a lower rigid plate. Approximate boundary-layer-like equations valid up to second order in the channel width-to-wavelength ratio are first derived. An extension of the weighted residual approach first suggested by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] to model the single-layer flow is then presented. It is shown that a suitable choice of trial and weighted functions allows us to lower appreciably the dimensionality of the problem. Marginal stability results clearly indicate that unlike the Shkadov approach, which gives erroneous critical conditions, the proposed second-order model, its simplified version corresponding to parabolic velocity profiles and referred to as the one-mode Galerkin approach, and the lubrication theory (long wave expansion procedure) are all in good agreement with the Orr-Sommerfeld equations near criticality where the viscous effects are dominant. The one-mode Galerkin model is therefore sufficient to correctly predict the near-critical behavior of the long wave interfacial mode. As inertia increases, the lubrication theory provides results that diverge. The Shkadov approach gives bad results as well while the second-order weighted residual model and its simplified version continue to follow the numerical results up to moderate Reynolds numbers, the approximation being better for the former than for the latter.

Three-dimensional disturbances in channel flows

In this paper we conduct a linear stability analysis of three-dimensional two-fluid flows and use an energy method to comment on its stability. The governing equations are solved using a Chebyshev-tau D2 method that reduces the order of the coupled governing Orr-Sommerfeld and Squire equation and hence achieves more accurate results. A new norm, called the M-norm, is defined to overcome the problem of nonconvergence of the disturbance energy. The maximum amplification of O(103) is achieved for streamwise independent disturbances due to the “lift-up effect,” as is the case of three-dimensional single-fluid flow. In contrast to two-dimensional flows, where the adjoint of the leading mode influences the growth, the three-dimensional single-fluid flow growth is influenced by the adjoint of the second mode. Although most growth in three-dimensional two-fluid flow is due to the contribution of the adjoint of the second mode, at large time the interfacial mode contributes to most growth.

Organised structures in wall turbulence as deduced from stability theory-based methods

In earlier work, we have explored the relevance of hydrodynamic stability theory to fully developed turbulent wall flows. Using an extended Orr-Sommerfeld Equation, based on an anisotropic eddy-viscosity model, it was shown that there exists a wide range of unstable wave numbers (wall modes), which mimic some of the key features of turbulent wall flows. Here we present experimental confirmation for the same. There is good qualitative and quantitative agreement between theory and experiment. Once the dominant coherent structure is obtained from stability theory, control of turbulence would be the next logical step. As shown, the use of a compliant wall shows considerable promise.

Instability and transition to turbulence in a free shear layer affected by a parallel magnetic field

Instability and transition to turbulence in a temporally evolving free shear layer of an electrically conducting fluid affected by an imposed parallel magnetic field is investigated numerically. The case of low magnetic Reynolds number is considered. It has long been known that the neutral disturbances of the linear problem are three-dimensional at sufficiently strong magnetic fields. We analyse the details of this instability solving the generalized Orr–Sommerfeld equation to determine the wavenumbers, growth rates and spatial shapes of the eigenmodes. The three-dimensional perturbations are identified as oblique waves and their properties are described. In particular, we find that at high hydrodynamic Reynolds number, the effect of the strength of the magnetic field on the fastest growing perturbations is limited to an increase of their oblique angle. The dimensions and spatial shape of the waves remain unchanged. The transition to turbulence triggered by the growing oblique waves is investigated in direct numerical simulations. It is shown that initial perturbations in the form of superposition of two symmetric waves are particularly effective in inducing three-dimensionality and turbulence in the flow.

Hydromagnetic stability of plane Couette flow of an upper convected Maxwell fluid

Hydrodynamic stability of plane Couette flow of an upper convected Maxwell fluid is investigated in presence of a transverse magnetic field assuming that the magnetic Prandtl number is sufficiently small. The resulting equation is a modified Orr-Sommerfeld equation. The equations of stability are solved numerically using Chebyshev collocation method with QZ algorithm. The critical values of Reynolds number, wave number and wave speed are computed and the results are shown through the neutral curves. By increasing the amount of elasticity to a certain value, it is shown that, as the Hartmann number increases, the minimum critical Reynolds number decreases and it does not increase again in contrast to the Newtonian case.

Temporal stability of small disturbances in MHD Jeffery–Hamel flows

In this paper, the temporal development of small disturbances in magnetohydrodynamic (MHD) Jeffery–Hamel flows is investigated, in order to understand the stability of hydromagnetic steady flows in convergent/divergent channels at very small magnetic Reynolds number R m . A modified form of normal modes that satisfy the linearized governing equations for small disturbance development asymptotically far downstream is employed [A. McAlpine, P.G. Drazin, On the spatio-development of small perturbations of Jeffery–Hamel flows, Fluid Dyn. Res. 22 (1998) 123–138]. The resulting fourth-order eigenvalue problem which reduces to the well known Orr–Sommerfeld equation in some limiting cases is solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The results indicate that a small divergence of the walls is destabilizing for plane Poiseuille flow while a small convergence has a stabilizing effect. However, an increase in the magnetic field intensity has a strong stabilizing effect on both diverging and converging channel geometry.

Stability of density stratified flow in the boundary layer

The stability of the laminar flat plate boundary layer is investigated numerically by solving the linear Orr–Sommerfeld equations for the disturbulence amplitude function. These equations include the terms of viscosity, density stratification, and diffusion. Neutral stability curve and the critical Re numbers are computed for various Richardson ( Ri) numbers and Schmidt ( Sc) numbers. The results show that the larger the Ri, the larger the critical Re for Sc<10. The flow is stable for Ri<0, when Sc is very small or the mass diffusion coefficient is very large. But for Ri>0, the effects of diffusion are reversed for Sc<10. For Sc>10, the critical Re rapidly decreases to zero as the Sc increases for a given Ri number. The critical Re rapidly decreases as the Ri increases.

Inertialess temporal and spatio-temporal stability analysis of the two-layer film flow with density stratification

This paper presents a temporal, spatial, and spatio-temporal linear stability analysis of the two-layer film flow down a plate tilted at an angle θ. It is based on a zero Reynolds number approximation to the Orr-Sommerfeld equations and a zero surface tension approximation to both surface boundary conditions. The combined effects of density and viscosity stratifications are systematically investigated. The subtle influence of density stratification is first put into light by a temporal analysis for θ=0.2; when increasing/decreasing the density ratio (upper fluid/lower fluid), the two-layer film flow becomes much more unstable/stable with respect to the finite wavelength instability. Moreover, below a critical density ratio this finite wavelength instability even disappears, whatever the viscous ratio. Concerning the long wave instability, it becomes dominant when decreasing the density ratio below 1 and is even triggered in a region which was stable for equal density layers. The spatio-temporal analysis shows that the instability is convective for incline angles that are not too small as θ=0.2. The study of the local growth rates of the spatio-temporal instability as a function of the ray velocity V shows that there is a transition between long wave and short wave instabilities which has been determined by using the Briggs-Bers collision criterion. Accordingly, there exists a jump for the local oscillatory frequency, spatial amplification rate, and spatial wave number due to this transition. Due to the existence of the absolute Rayleigh-Taylor instability for θ=0, the transition from convective to absolute instability can be detected for values of θ smaller than 0.2, and absolute/convective instability boundary curves have been obtained for varying characteristic parameters.

The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory

According to parallel-flow theory based on the Orr–Sommerfeld equation, a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there exist a non-zero Reynolds number below which disturbances cannot extract net energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism using the minimal composite theory and the properties of adjoints, following Govindarajan & Narasimha (Theor. Comput. Fluid Dyn. vol. 19, 2005, p. 229) resolves the issue by yielding a non-zero critical Reynolds number for co-flowing streams of any velocity ratio. The critical Reynolds number for the total disturbance kinetic energy is found to be close to 30 for all velocity ratios in the range from zero to unity.

Numerical research on the hydrodynamic stability of Blasius flow with spectral method

Abstract Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials. The numerical results show a good agreement with Howarth's solution, with relatively low computational cost. This method is then applied to the stability of flat plate boundary layer flow compared with the finite difference method; our study shows that the expansions in Chebyshev polynomials are more suitable for the solution of hydrodynamic stability problems than the expansions in finite difference method.

Convective instability in wake intermediate asymptotics

The paper presents a multiscale analysis of the intermediate region of the two-dimensional convectively unstable wake past a bluff body. A recent asymptotic expansion solution was used as basic flow (Tordella &amp; Belan, Phys. Fluids, vol. 15, 2003, 1897). This solution was obtained by matching an inner to an outer flow, both of which are Navier–Stokes solutions. By introducing a spatio-temporal multiscaling into the instability problem, an inhomogeneous Orr–Sommerfeld equation and an associated modulation equation are obtained. The streamwise variation of the instability characteristics can then be deduced from the wave modulation, by considering the system to be perturbed by waves with a complex wavenumber that corresponds of the dominant saddle point of the local dispersion relation, taken at different positions downstream of the wake, and at different Reynolds numbers. The corrections of no parallelism are remarkable in the intermediate wake. When the disturbance is related to an early intermediate station, the corrections lead to absolute instability in the upstream portion of the intermediate wake, where, in addition, the spatial growth rate decreases. When the disturbance is related to a section in the far field, conditions of minimal temporal stability are reached about 20 body scales downstream. In the far field the temporal damping increases with the Reynolds number.

Leading-edge receptivity by adjoint methods

The properties of adjoint operators and the method of composite expansion are used to study the generation of Tollmien–Schlichting (TS) waves in the leading-edge region of an incompressible, flat-plate boundary layer. Following the classical asymptotic approach, the flow field is divided into an initial receptivity region, where the unsteady motion is governed by the linearized unsteady boundary-layer equation (LUBLE), and a downstream linear amplification area, where the evolution of the unstable mode is described by the classical Orr–Sommerfeld equation (OSE). The large behaviour of the LUBLE is analysed using a multiple-scale expansion which leads to a set of composite differential equations uniformly valid in the wall-normal direction. These are solved numerically as an eigenvalue problem to determine the local properties of the Lam and Rott eigensolutions. The receptivity coefficient for an impinging acoustic wave is extracted by projecting the numerical solution of the LUBLE onto the adjoint of the Lam and Rott eigenfunction which, further downstream, turns into an unstable TS wave. In the linear amplification region, the main characteristics of the instability are recovered by using a multiple-scale expansion of the Navier–Stokes equations and solving numerically the derived eigenvalue problems. A new matching procedure, based on the properties of the adjoint Orr–Sommerfeld operator, is then used to check the existence and the extent of an overlapping domain between the two asymptotic regions. Results for different frequencies are discussed.

Investigation of stability of a laminar flow in a parallel-plate channel filled with a fluid saturated porous medium

Instability of a laminar flow in a parallel-plate channel filled with a fluid saturated porous medium is investigated on the basis of a modified Orr-Sommerfeld equation. This equation takes into account three drag terms: the Darcy term that describes friction between the fluid and the porous matrix, the Forchheimer quadratic drag term that describes a form drag due to the solid obstacles, and the Brinkman term, which is a viscous term similar to the Laplacian term in the Navier-Stokes equations. Numerical analysis is carried out using the collocation method. The dependence of the critical Reynolds number on porosity and permeability of the porous medium is analyzed numerically.

On the spatial linear growth of gravity-capillary water waves sheared by a laminar air flow

The initial growth of mechanically generated small amplitude water waves below a laminar air stream was examined numerically and experimentally in order to explore the primary growth mechanism, that is, the interfacial instability of coupled laminar air and water flows. Measurements of the laminar velocity profile in the air over the water surface were found to be consistent with Lock’s [Q. J. Mech. Appl. Math. 4, 42 (1951)] theory. This profile was then used to calculate the spatial growth rates by solving the Orr-Sommerfeld equations. The simulation shows that the growth of the boundary layer affects the exponential growth of water waves along the fetch. The sensitivity of the growth rate is observed to vary by a factor of 2 for changes in the laminar velocity profile as small as 2% at the water surface. This indicates that the interfacial instability is strongly influenced by the wind-induced surface current. A laminar airflow was produced in the wind tunnel over mechanically generated monochromatic gravity-capillary water waves with the ka value in the order of 10−3. The novel experiment was designed to measure the minute changes in the wave slope and phase velocity simultaneously using a highly sensitive reflected twin laser beam technique. Agreement between linear theory and experiments for the spatial development of wave height and phase velocity suggests that the linear instability mechanism determines the initial stages of development of small-scale water waves.

Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases

The equations of the (2+1)-dimensional boundary-layer perturbation split into eigenmodes: a vortex wave and two acoustic waves. We assume that the equations of state (Taylor series approximation) are arbitrary. We realize a mode definition via local-relation equations extracted from the linearization of the general system over the boundary-layer flow. Each such link determines an invariant subspace and the corresponding projector. We examine the nonlinear equation for a vortex wave using a special orthogonal coordinate system based on streamlines. The equations for the orthogonal curves are linked to the Laplace equations via Laplace and Moutard transformations. The nonlinearity determines the proper form of the interaction between vortical and acoustic boundary-layer perturbation fields fixed by projecting to a subspace of the Orr-Sommerfeld equation solutions for the Tollmienn-Schlichting (linear vortical) wave and by the corresponding procedure for the acoustic wave. We suggest a new mechanism for controlling the nonlinear resonance of the Tollmienn-Schlichting wave by sound via a four-wave interaction.

The stability of laminar symmetric separated wakes

Time-dependent computations of the two-dimensional incompressible uniform-velocity laminar flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the well-known anti-symmetric (von Karman-type) vortex shedding is suppressed by the imposition of a symmetry plane on the downstream plate centreline. With a further symmetry plane at the channel's upper boundary, the only two governing parameters in the problem are the channel half-width, H, and the Reynolds number, Re (based on the body half-width and the upstream velocity, U). The former is restricted to the range 3?H?30 and the interest lies in determining the nature of the initial instability which occurs in the separated wake as Re is gradually increased. It is found that for sufficiently large H and at a critical Re, a long-time-scale global (supercritical) instability is initiated, which in its saturated (limit) state takes the form of ‘lumps’ of vorticity being periodically shed from the tail end of the separated bubble. Stability calculations of corresponding mean flow profiles (typical of those found in the separated wake) are undertaken by examining the impulse response of particular profiles via appropriate solution of the Orr–Sommerfeld equation. The results of this analysis extend those available from related published work and are consistent with the behaviour found from the numerical computations. Taken together, all the results suggest that this type of global instability may be generic to many kinds of separated wakes and, indeed, may provide the fundamental explanation for the very low-frequency oscillations often noticed in fully turbulent wake bubbles.

Stability and eigenvalue bounds of the flow of a dipolar fluid between two parallel plates

In this article, we derive the Orr–Sommerfeld equation for the stability of parallel flows of a dipolar fluid. The classical results found by Squire, for viscous Newtonian fluids, are generalized to the case of dipolar fluids. A sufficient condition for stability is obtained for dipolar fluids and eigenvalue bounds for the Orr–Sommerfeld equation are found.

Long-wavelength equation for vertically falling films

An equation is derived for describing wave evolution on the surface of a vertically falling viscous film. The traditional long-wavelength scaling is replaced by a new scaling to reduce the (We) must be used instead of the Reynolds number (Re) to distinguish between viscous and inertia dominated regimes for vertically falling films. This equation includes viscous dissipation and pressure correction terms that are missing in the existing single evolution equations at the same order. Comparison of the neutral stability curves and growth rates predicted by different models to that of the Orr-Sommerfeld (OS) equation shows that our equation matches with the OS results better than the existing single evolution equations. However, our equation is not free from finite time blowup. Selective regularization leads to a two mode model in flow rate and film thickness. The regularized equation is free from finite time blowup and predicts two families of solitary waves. Numerical simulations of the derived equation and its regularized version in the traveling wave coordinate show the transition of wave structure from regular (periodic) to chaotic profiles. Model predictions on maximum wave amplitude on the low celerity branch show good agreement with experimental data.