Orr-Sommerfeld Research Articles

A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials

The article discusses two spectral methods, namely the Galerkin and the Petrov–Galerkin methods, for linear stability analysis of magneto-hydrodynamic (MHD) equations describing the flow of an electrically conducting fluid in the presence of a tangential magnetic field. The stability and spectral accuracy of both methods have been compared by examining the most unstable eigensolution of the Orr–Sommerfeld (OS) and induction equations. The Petrov–Galerkin spectral method (PGSM) used in this work has been developed by choosing function spaces and basis functions that always lead to banded coefficient matrices. The Galerkin spectral method (GSM), on the contrary, leads to dense matrices when Chebyshev polynomials are utilized in a weighted inner product space. We have found that both the GSM and the PGSM can produce results with minimal round-off errors, as confirmed by computing the most unstable eigenvalue of the OS equations (Re =104) to 14 decimal places of accuracy in double precision. We show that with properly scaled basis functions the GSM leads to coefficient matrices with bounded condition numbers, both for the OS equation and for the coupled OS and induction equations. This allows to achieve accurate results with double precision for any number of N for both the GSM and the PGSM. The analysis of the different behavior of the condition numbers suggests that the proposed two methods, based on the Chebyshev polynomials, can become a useful computer-based tool that is capable of finding a numerical solution to both the hydrodynamic and the MHD equations at very high Reynolds numbers.

Shear-imposed falling film on a vertical moving plate with disrupted time-reversal

We propose a mathematical model to study the stability and dynamics of a shear-imposed thin film flow on a vertical moving plate, incorporating the influence of odd viscosity. This odd viscosity effect is vital in conventional fluids when there is a disruption in time-reversal symmetry. Our motivation to study the dynamics with odd viscosity arises from recent studies (Kirkinis & Andreev, vol. 878, 2019, pp. 169–189; Chattopadhyay & Ji, vol. 455, 2023, pp. 133883) where the odd viscosity effectively reduces flow instabilities under different scenarios. Utilizing a long wave perturbation method, we derive a nonlinear evolution equation at the liquid–air interface, which is influenced by the motion of the vertical plate, imposed shear, odd viscosity, and inertia. We first perform a linear stability analysis of the model to get firsthand information on various flow parameters. Three distinct conditions for the vertical plate, quiescent, upward-moving, and downward-moving, are considered, accounting the imposed shear and odd viscosity. Additionally, employing the method of multiple scales, we conduct a weakly nonlinear stability analysis for the traveling wave solution of the evolution equation and explore its bifurcation analysis. The bifurcation analysis reveals the existence of subcritical unstable and supercritical stable zones for crucial flow parameters: odd viscosity, imposed shear, and motion of the vertical plate. Both linear and weakly nonlinear stability analyses demonstrate that the destabilizing effect induced by the upward motion of the vertical plate can be alleviated by applying uphill shear, while the destabilizing effect of downhill shear can be mitigated when the vertical plate is in a downward motion. Moreover, we define an eigenvalue problem that mirrors the Orr–Sommerfeld (OS) model for analyzing normal modes and identifying the critical Reynolds number. We investigate the dynamics of surface waves through numerical solutions of the OS eigenvalue problem using the Chebyshev spectral collocation method. We observe the consistent enhancement of stabilization in the presence of odd viscosity. In the low to moderate Reynolds number range, vertical plate motion and odd viscosity show similar behavior in OS analysis, while imposed shear exhibits distinct changes. The Benney-type model does not agree with the OS problem when the Reynolds number is moderate with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity. However, when the Reynolds number is low with or without the three key parameters: vertical plate motion, imposed shear, and odd viscosity, the Benney-type model agrees with the OS. Further, numerical simulations of the evolution equation corroborate the results obtained from linear stability, weakly nonlinear stability, and OS analyses. Finally, the Hopf bifurcation analysis of the fixed point reveals that the wave speed is influenced by both the motion of the plate and the imposed shear while it remains independent of odd viscosity.

Algebraic growth of 2D optimal perturbation of a plane Poiseuille flow in a Brinkman porous medium

The impact of various factors on the stability of a plane porous Poiseuille flow is investigated in detail. Both modal and non-modal linear stability analyses are used to study the effects of the porosity of the porous media and the ratio of effective viscosity to the fluid viscosity. The present analysis includes solving the linearized Navier-Stokes equation in the form of the Orr-Sommerfeld (O-S) type equation by applying the 2D perturbation to the basic mean flow. The Chebyshev collocation method is used to solve the Orr-Sommerfeld equation numerically. Through modal analysis, the accurate values of the critical triplets (αc,Rc,cc), the eigen-spectrum, the growth rate curves, and the marginal stability curves are studied. Then, by using non-modal analysis, the transient energy growth G(t) of two-dimensional optimal perturbations, the ε-pseudospectrum of the non-normal O-S operator (L), and the regions of stability, instability, and potential instability of the fluid flow system are investigated in detail. The collective results of both modal and non-modal analysis show that the porous parameter and the ratio of effective viscosity to the fluid viscosity have stabilizing effects on the fluid system due to the postponement of the onset of stability.

Shear imposed falling film with odd viscosity effects

This study examines the behavior of a thin liquid film flowing on an inclined plane subject to imposed shear stress. The liquid’s time-reversal symmetry is broken, and we consider the odd part of the viscosity to describe the Navier–Stokes equation. To investigate the interplay between the imposed shear and odd viscosity on the flow dynamics, we develop a weighted residual model (WRM). We determine the model’s critical Reynolds number (Rec) through linear stability analysis. Our findings indicate that uphill shear tends to stabilize the flow, while downhill shear enhances the instability, albeit reduced by the presence of odd viscosity. We also construct an Orr–Sommerfeld (OS) type eigenvalue problem for normal mode analysis and derive Rec. We discover that in the longwave regime, RecWRM=RecOS. Finally, our numerical simulations of the model align well with our linear stability analysis.

Improved assessment of the statistical stability of turbulent flows using extended Orr–Sommerfeld stability analysis

The concept of statistical stability is central to Malkus's 1956 attempt to predict the mean profile in shear flow turbulence. Here we discuss how his original attempt to assess this – an Orr–Sommerfeld (OS) analysis on the mean profile – can be improved by considering a cumulant expansion of the Navier–Stokes equations. Focusing on the simplest non-trivial closure (commonly referred to as CE2) that corresponds to the quasilinearized Navier–Stokes equations, we develop an extended OS analysis that also incorporates information about the fluctuation field. A more practical version of this – minimally extended OS analysis – is identified and tested on a number of statistically steady and, therefore, statistically stable turbulent channel flows. Beyond the concept of statistical stability, this extended stability analysis should also improve the popular approach of mean flow linear analysis in time-dependent shear flows by including more information about the underlying flow in its predictions as well as for other flows with additional physics such as convection.

Linear instability of a surfactant-laden shear imposed falling film over an inclined porous bed

The influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory. The hydrodynamic instability of the flow system corresponding to infinitesimal disturbances is examined in the framework of the Orr–Sommerfeld (OS) boundary value problem. Furthermore, the generalized OS model is obtained by including the Marangoni stress and external shear on the flow dynamics. The formulated stability problem is solved as an eigenvalue problem by the Chebyshev spectral collocation technique. The analysis encounters the existence of different classes of unstable modes, namely, the surface, surfactant, and shear modes. The surface mode instability occurs in the low range of Reynolds number and is the dominant mode of instability in particular parameter ranges. The imposed shear at the top surface along and opposite to the flow direction induces possible destabilization and stabilization of the flow, respectively. The permeability and porosity of the porous medium have a mixed impact on the surface mode instability. The temporal growth rate of the surface mode enhances for a thicker porous medium. The surface mode of the flow contaminated by an insoluble surfactant is less unstable than that of the clean free surface flow. This is due to the co-existence of the damped surfactant mode together with the unstable surface mode. On the other hand, the shear mode instability is identified at higher Reynolds numbers for a very small inclination angle, and the shear mode propagates faster for stronger imposed shear in the downstream direction. This trend is reversed for the upstream imposed shear. Moreover, the Marangoni effects exhibit the stabilizing influence on the shear mode. Conclusively, the external shear force would be helpful in regulating the instability of the surfactant-laden film flow down a porous medium.

Surface wave and thermocapillary instabilities on flowing film under the sway of Hall viscosity

The present study investigates the influence of Hall/odd-viscosity on the surface wave and thermocapillary instabilities in a thin viscous film flowing down an inclined plane. A detailed analysis considering a generalized scaling for the velocity field to understand the effect of odd-viscosity in the hydrodynamic and thermocapillary modes of instabilities has been performed, and later the results are presented for a specific velocity scale. Starting with Orr–Sommerfeld (OS) linear stability analysis for all ranges of wavenumber and then constructing two low-dimensional models; namely, the Long wave expansion model (LWE) and the Weighted residual integral boundary layer model (WRIBL) under the assumption of small wavenumber are analysed for linear and nonlinear behaviour of the instabilities analytically and numerically. Two energy transferring mechanisms reinforce each other to the disturbance: hydrodynamic H-mode and thermocapillary S-mode are focused and found that the odd viscosity plays a stabilizing role for both the modes of instabilities. For small Reynolds number (Re) values, both LWE and WRIBL models give consistent results as that of the OS. However, for a large value of Re, WRIBL gives way better match with OS than LWE. Increasing odd viscosity flattens the streamlines in a moving frame of reference, confirming the odd viscosity’s stabilizing effect. However, being non-dissipative, odd viscosity has no significant influence on the temperature field. In a static frame, the intensity of capillary separation and flow reversal phenomena decreases with the increase of odd viscosity, confirming its stabilizing effect on the flow.

Electrohydrodynamic instability of viscous liquid films: Electrostatically modified second-order Benney and Kuramoto–Sivashinsky models

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter (ε) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O(ε2) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number (Re). Furthermore, at subcritical Re with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

Weakly viscoelastic film on a slippery slope

We study the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane. The focus is to investigate the interaction of the bottom slip with the viscoelastic parameter as well as the influence of the other flow parameters on the stability of the flow. For the slippery substrate, we use the Navier-slip boundary condition at the solid–liquid interface. The dimensionless slip length β is first assumed to be small and its order is considered same as the order of the film aspect ratio ϵ=H/L, where H is the mean film thickness and L is a typical wavelength. To discuss the coupled effect of slip length β and viscoelastic parameter γ, we have used the classical Benney equation model (BEM) as well as the weighted residual method (WRM). For linear stability, the normal mode analysis is carried out to capture the instability threshold. The critical Reynolds numbers (Rec) are obtained for BEM and WRM separately for the system. We found that the first-order WRM is a better choice to capture the instability threshold in comparison with a first-order BEM when β is small. Another noteworthy result we obtain is that, in the absence of β, WRM and BEM yield the same expression for the critical Reynolds number. Further, we have extended the study for moderate values of β, that is, β of order unity and it is found that both BEM and WRM are able to capture the effects of β and γ. We derive the Orr–Sommerfeld (OS) type eigenvalue problem and an asymptotic analysis is performed for small perturbation wavenumbers, which gives an expression for the critical Reynolds number for the instability of very long perturbations. The critical Reynolds number obtained by the OS eigenvalue problem exactly matches with that obtained by BEM. Finally, we validate our analytical predictions by performing a direct numerical simulation of the WRM and good agreement between the results of the linear stability analysis, weighted residual model, and the numerical simulations is found.

Mittag‐Leffler stabilization for coupled fractional reaction‐diffusion neural networks subject to boundary matched disturbance

In this paper, we study the boundary stabilization for a class of neural networks with non‐identical nodes described by the fractional Orr‐Sommerfeld (OS) equation cascaded with the fractional Squire equation and fractional ordinary differential equations (ODEs), in which the disturbances flow to the control end of both the OS equation and the Squire equation. By applying the Backstepping transform, the diffusion subsystems in the considered neural networks are decoupled, while only the ODEs are still cascaded with the diffusion terms through the Neumann interconnection. By the active disturbance rejection control (ADRC) approach, two auxiliary systems are constructed to compensate each disturbance by an estimator without high gain. Finally, a boundary feedback control law is designed to achieve the Mittag‐Leffler stability of the neural networks in the closed‐loop.

Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers

In the dynamical systems approach to describing turbulent or otherwise chaotic flows, an important quantity is the Lyapunov exponents and vectors that characterize the strange attractor of the flow. In particular, knowledge of the Lyapunov exponents and vectors will help identify perturbations that the system is most sensitive to, and quantify the dimension of the attractor. However, reliably computing these quantities requires robust numerical algorithms. While several perturbation-based techniques are available in literature, their application to commonly used turbulent flow solvers as well the numerical convergence properties have not been studied in detail. The goal of this work is to examine the convergence properties of the Lyapunov exponents when computed using a low Mach number solver, for a series of progressively complex flow problems. In particular, a manufactured solutions approach is devised using the Orr-Sommerfeld (OS) perturbation theory to extract Lyapunov exponents and vectors from OS eigen solutions. Overall the spatial convergence rates of Lyapunov exponents follow the truncation order for the discretization scheme. However, temporal convergence rates were found to be only a weak function of timestep used. Additionally, for some configurations, the convergence properties are found to depend on the Lyapunov exponent itself. These results indicate that convergence properties do not follow universal rates, and require careful analysis for specific configurations considered.

Simulation of Surface Instability at the Interface of Two Fluids

In this theoretical and numerical study, the physical mechanisms leading to the production of surface waves generated at the interface of two fluids (liquid/gas or liquid/liquid) are investigated. Particular attention is devoted to the Kelvin-Helmholtz (KH) type instability, which appears in the area of high shear located at the fluid-fluid interface. The subsequent disturbances in velocity and pressure associated with the wave motion are assumed to be 2D and sufficiently small to justify the linearization of the equations of the motion. The Navier-Stokes (NS), Orr-Sommerfeld (OS) and KH equations are the primary ones used to investigate of surface instability. During the study, the main characteristics of a surface instability such as wavelength, wave number, frequency, amplitude, wave speed and growth rate are investigated according to fluid viscosity. The effect of gravity is investigated by 2D simulations without these external forces or with them in one of the following configurations:

On the sensitivity of planar jets

Planar jets are widely researched flows. Many aspects of their instability have been well-known for a long time. One of them is the observation that the jet is more sensitive to disturbances near the orifice than elsewhere. This paper aims at giving an explanation to this hitherto unexplained phenomenon. To this end, linear stability investigations on various velocity profiles were carried out using the Orr–Sommerfeld (OS) equation. The velocity profiles were provided by analytical approximations and numerical computational fluid dynamics (CFD) simulations. A special method called compound matrix method (CMM) was used for solving the OS equation to obtain sufficiently accurate results. The adaptation of the method for symmetric jets was briefly derived. The stability of various velocity profiles was compared based on the local spatial growth rate. The results of the comparison clearly show that velocity profiles near the orifice are more unstable than the downstream ones. The outcomes also show that the velocity profile close to the orifice and the wall thickness of the nozzle have impact on the stability properties of the flow there.

Boundary-layer disturbances subjected to free-stream turbulence and simulation on bypass transition

The phenomena associated with the entrainment of free-stream turbulence (FST) into boundary-layer flows are relevant for a number of subjects. It has been believed that the continuous spectra of the Orr-Sommerfeld (O-S)/Squire equations describe the entrainment process, and thus they are used to specify the inlet condition in simulation of bypass transition. However, Dong and Wu (Dong, M. and Wu, X. On continuous spectra of the Orr-Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances. Journal of Fluid Mechanics, 732, 616–659 (2013)) pointed out that continuous spectra exhibit several non-physical features due to neglecting the non-parallelism. They further proposed a large-Reynolds-number asymptotic approach, and showed that the non-parallelism is a leading-order effect even for the short-wavelength disturbance, for which the response concentrates in the edge layer. In this paper, the asymptotic solution is verified numerically by studying its evolution in incompressible boundary layers. It is found that the numerical results can be accurately predicted by the asymptotic solution, implying that the latter is adequate for moderate Reynolds numbers. By introducing a series of such solutions as the inflow perturbations, the bypass transition is investigated via the direct numerical simulation (DNS). The transition processes, including the evolution of streaks, the amplification of secondary-instability modes, and the emergence of turbulent spots, agree with the experimental observations.

A computational study of an atomizing liquid sheet

Linear instability predictions of liquid sheets injected into a gas medium are well established in the literature. These analyses are often used in Lagrangian-Eulerian spray simulations, a prominent simulation method, to model the dynamics occurring in the near-nozzle region. In the present work, these instability predictions are re-examined by first generalizing the treatment of interfacial conditions and related assumptions with a two-phase Orr-Sommerfeld (OS) system, and second, by employing highly resolved-Volume-of-Fluid (VoF) simulations. After presenting some validation exercises for both the VoF and OS solvers, the OS predictions are compared to earlier studies from the literature leading to reasonable agreement in the limit as the boundary layer thickness tends to zero. Results from VoF simulations of liquid sheet injection are used to characterize the range of scales of the liquid structures immediately before atomization. The mean value in this range is found to be approximately two to three orders of magnitude larger than the corresponding predictions from previous studies. A two-phase mixing layer under the same physical conditions is used to examine this disparity, revealing that within the linear regime, relatively good agreement exists between the VoF and OS predicted instability mechanisms. However, the most unstable mode in the linear regime is too small to cause a fracture or atomization of the liquid sheet and hence cannot be directly responsible for the atomization. The generation of a much larger mode, which emerges well beyond the linear regime, is the one causing breakup.

On continuous spectra of the Orr–Sommerfeld/Squire equations and entrainment of free-stream vortical disturbances

AbstractSmall-amplitude perturbations are governed by the linearized Navier–Stokes equations, which are, for a parallel or nearly parallel shear flow, customarily reduced to the Orr–Sommerfeld (O-S) and Squire equations. In this paper, we consider continuous spectra (CS) of the O-S and Squire operators for the Blasius and asymptotic suction boundary layers, and address the issue of whether and when continuous modes can represent free-stream vortical disturbances and their entrainment into the shear layer. For the Blasius boundary layer, we highlight two particular properties of the CS: (i) the eigenfunction of a continuous mode simultaneously consists of two components with wall-normal wavenumbers $\pm {k}_{2} $, a phenomenon which we refer to as ‘entanglement of Fourier components’; and (ii) for low-frequency disturbances the presence of the boundary layer forces the streamwise velocity in the free stream to take a much larger amplitude than those of the transverse velocities. Both features appear to be non-physical, and cast some doubt about the appropriateness of using CS to characterize free-stream vortical disturbances and their entrainment into the boundary layer, a practice that has been adopted in some recent studies of bypass transition. A high-Reynolds-number asymptotic description of continuous modes and entrainment is present, and it shows that the entanglement is a result of neglecting non-parallelism, which has a leading-order effect on the entrainment. When this effect is included, entanglement disappears, and moreover the streamwise velocity is significantly amplified in the edge layer when ${R}^{- 1} \ll \omega \ll 1$, where $R$ is the Reynolds number based on the local boundary-layer thickness. For the asymptotic suction boundary layer, which is an exactly parallel flow, both temporal and spatial CS may be defined mathematically. However, at a finite $R$ neither of them represents the physical process of free-stream vortical disturbances penetrating into the boundary layer. The latter must instead be characterized by a peculiar type of continuous modes whose eigenfunctions increase exponentially with the distance from the wall. In the limit $R\gg 1$, all three types of CS are identical at leading order, and hence can be used to represent free-stream vortical disturbances and their entrainment. Low-frequency disturbances are found to generate a large-amplitude streamwise velocity in the boundary layer, which is reminiscent of longitudinal streaks.

Hydromagnetic thin film flow: Linear stability

This paper deals with the long wave instability of an electroconductor fluid film, flowing down an inclined plane at small to moderate Reynolds numbers, under the action of electromagnetic fields. A coherent second order long wave model and two simplified versions of it, referred to as first and second reduced models (FRM and SRM), are proposed to describe the nonlinear behavior of the flow. The modeling procedure consists of a combination of the lubrication theory and the weighted residual approach using an appropriate projection basis. A suitable choice of weighting functions allows a significant reduction of the dimension of the problem. The full model is naturally unique, i.e., independent of the particular form of the trial functions. The linear stability of the problem is investigated, and the influence of electromagnetic field on the flow stability is analyzed. Two cases are considered: the applied magnetic field is either normal or parallel to the fluid flow direction, while the electric field is transversal. The numerical solution of the Orr-Sommerfeld (OS) eigenvalue problem and those of the depth averaging model are used to assess the accuracy of the reduced models. It is found that the current models have the advantage of the Benney-like model, which is known to asymptote the exact solution near criticality. Moreover, far from the instability threshold, the current reduced models continue to follow the OS solution up to moderate Reynolds numbers, while the averaging model diverges rapidly. The model SRM gives better results than FRM beyond sufficiently high Reynolds numbers.

Surface instability of a thin electrolyte film undergoing coupled electroosmotic and electrophoretic flows in a microfluidic channel

We consider the stability of a thin liquid film with a free charged surface resting on a solid charged substrate by performing a general Orr-Sommerfeld (O-S) analysis complemented by a long-wave (LW) analysis. An externally applied field generates an electroosmotic flow (EOF) near the solid substrate and an electrophoretic flow (EPF) at the free surface. The EPF retards the EOF when both the surfaces have the same sign of the potential and can even lead to the flow reversal in a part of the film. In conjunction with the hydrodynamic stress, the Maxwell stress is also considered in the problem formulation. The electrokinetic potential at the liquid-air and solid-liquid interfaces is modelled by the Poisson-Boltzmann equation with the Debye-Hückel approximation. The O-S analysis shows a finite-wavenumber shear mode of instability when the inertial forces are strong and an LW interfacial mode of instability in the regime where the viscous force dominates. Interestingly, both the modes are found to form beyond a critical flow rate. The shear (interfacial) mode is found to be dominant when the film is thick (thin), the electric field applied is strong (weak), and the zeta-potentials on the liquid-air and solid-liquid interfaces are high (small). The LW analysis predicts the presence of the interfacial mode, but fails to capture the shear mode. The change in the propagation direction of the interfacial mode with the zeta-potential is predicted by both O-S and LW analyses. The parametric range in which the LW analysis is valid is thus demonstrated.

State-Space Approximations of the Orr-Sommerfeld System with Boundary Inputs and Outputs

Finite-dimensional state-space approximations of the Orr–Sommerfeld equation for plane Poiseuille flow with boundary input and output are discretized to state-space models using two spectral techniques. The models are compared for accuracy and discussed in the context of metrics important for a nonnormal dynamical system. Both state-space models capture the sensitivity behavior of the pole-zero perturbations and the pseudospectrum properties. The effects of balanced truncation ordered by Hankel singular values are discussed. For practical purposes, both methods are effective, and yet certain basic variations in the state-space properties exist. The merits and penalties of the two techniques are also provided. I. Introduction D ISTRIBUTED-PARAMETERS dynamical systems are described by partial differential equations (PDE) with associated boundary conditions. These systems are solved by numerical discretization to yield sets of ordinary differential equations similar to those in lumped-parameters systems [1,2]. This paper considers the state-space modeling of the two-dimensional (2-D) perturbations of plane Poiseuille flow described by the well-known Orr– Sommerfeld (OS) equation with the incorporation of an input/output structure. Two different spectral techniques are evaluated here. Of interest is existence of some basic variation in the state-space properties consisting of certain nontruncatable superdamped eigenvaluesandaninputfeedthroughtermintheoutputequation.Astrong motivation exists todemonstrate the effectiveness of using the finitedimensional state-space approximations in representing an infinitedimensional system. The state-space representation is increasingly popular,becausenumeroustoolboxeshavebeendevelopedbasedon the state-space form. MATLAB®, for example, offers toolboxes for balancing, model reduction, control design, data analysis, Simulink, and system theory [3]. All require standard state-space representation.Foraplanegeometry,spectraltechniques[4,5]arepowerfulin constructing state-space models. These are the original models that represent the infinite-dimensional systems. For certain flows, excitable by the input and observable through the output, the original models can be further balanced and order reduced. The balanced reduced-order models (BROMs) are commonly used in designing estimators and controllers. The OS system is of interest in flow controls, in association with thefundamentalproblemoftransitiontoturbulencein fluid flows ata much lower Reynolds number than the ones predicted by linear stability theory [6]. For the plane Poiseuille flow, the OS differential operators are highly nonnormal [7,8]. It is well known that, for nonnormal systems, the eigenvalues alone are inadequate to predict the stability of the flow. It can be demonstrated that, even when all eigenmodes are damped, large bursts of transient responses (orders of magnitude greater than the excitations) can be generated for certain transfer functions of a nonnormal system. Although exerting no exponential growth, these large transient bursts are believed as capable of triggering a transition to turbulence via a subcritical route to instability [6–8]. The aim of this paper is to compare and validate two spectral techniques applied to the OS system and to point out the basic differences between the eigenvalue properties and the state-space representations. A second aim is to demonstrate the sensitivity of the nonnormality of the OS operator [7] and of the transfer function, as well as to show the effects of balanced model truncation.

On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers

Studies of vortical interactions in boundary layers have often invoked the continuous spectrum of the Orr–Sommerfeld (O-S) equation. These vortical eigenmodes provide a link between free-stream disturbances and the boundary-layer shear – a link which is absent in the inviscid limit due to shear sheltering. In the presence of viscosity, however, a shift in the dominant balance in the operator determines the structure of these eigenfunctions inside the mean shear. In order to explain the mechanics of shear sheltering and the structure of the continuous modes, both numerical and asymptotic solutions of the linear perturbation equation are presented in single- and two-fluid boundary layers. The asymptotic analysis identifies three limits: a convective shear-sheltering regime, a convective–diffusive regime and a diffusive regime. In the shear-dominated limit, the vorticity eigenfunction possesses a three-layer structure, the topmost being a region of exponential decay. The role of viscosity is most pronounced in the diffusive regime, where the boundary layer becomes ‘transparent’ to the oscillatory eigenfunctions. Finally, the convective–diffusive regime demonstrates the interplay between the the accumulative effect of the shear and the role of viscosity. The analyses are complemented by a physical interpretation of shear-sheltering mechanism. The influence of a wallfilm, in particular viscosity and density stratification, and surface tension are also evaluated. It is shown that a modified wavenumber emerges across the interface and influences the penetration of vortical disturbances into the two-fluid shear flow.