Three-dimensional Disturbances Research Articles

A note on the linear stability of a two-dimensional Stokes layer

The fluid motion generated adjacent to an infinite flat plate undergoing orbital motion in its own plane is a generalization of the classical Stokes-layer profile. We show that the stabilities of these flows can be related to each other via two transformations directly analogous to the well-known Squire transformation. The main result obtained is that, in general, the two-dimensional Stokes layer is more stable than the corresponding unidirectional Stokes layer. A further by-product of the analysis is the construction of a shear flow having identical neutral stability conditions when subject to either two- or three-dimensional disturbances.

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.

Global MHD Instabilities in a Three‐dimensional Thin‐Shell Model of Solar Tachocline

We generalize the linear analysis of the global instability of coexisting differential rotation and toroidal magnetic fields in the solar tachocline to include continuous radial stratification, thermodynamics, and finite tachocline thickness, as perturbed by three-dimensional disturbances of longitudinal wavenumbers m = 1, 2. For radiative tachocline stratification, the instability for both banded and broad toroidal field profiles is similar to the two-dimensional and shallow water cases studied previously, even though the unstable modes have substantial vertical structure. For overshoot tachocline stratification, instability for banded toroidal fields with peaks 20 kG is similar to the corresponding shallow water case, but for substantially higher (perhaps unrealistic) toroidal fields there is no low-subadiabaticity cutoff, and modes appear with much higher growth rate and increasingly negative phase velocities in longitude, analogous to those found earlier by Cally. All of these results are only modestly sensitive to the tachocline thickness chosen. For broad toroidal field profiles, the instability results are similar to the two-dimensional and shallow water cases for toroidal field peaks up to at least 80 kG, unless the shell has a thickness that is a substantial fraction of a pressure scale height. For thinner shells, only above about 94 kG do the high growth rate, low phase velocity modes appear, with structure similar to the instability Cally found. But even in this case, we find that the slower growing clam-shell instability eventually replaces the faster growing polar kink modes. We conclude that for conditions most likely to occur in the solar tachocline, such as peak toroidal fields limited by the dynamo to 20 kG or less, the two-dimensional and shallow water type unstable modes are likely to predominate even when the tachocline has finite thickness and the modes can have radial structure.

The stability of low-amplitude three-dimensional disturbances in a nonequilibrium compressible boundary layer

The stability of Tollmien-Schlichting waves propagating at an angle to the main flow in a nonequilibrium compressible supersonic boundary layer is investigated within the linear theory of hydrodynamic stability. The dependences of the critical Reynolds number on the degree of disequilibrium and on the Mach number of undisturbed flow are found at different angles of wave propagation. It is demonstrated that the critical Reynolds number in a nonequilibrium medium may decrease appreciably with increasing degree of disequilibrium, which results in the reduction of the characteristic length of the linear region of transition to turbulence.

A conceptual model to characterize disturbance severity in forest harvests

I present a three-dimensional disturbance severity model and illustrate the potential usefulness of the model for characterizing disturbance conditions caused by forest harvesting. Disturbance is most often described in terms of disturbance type, size and frequency, with less emphasis on severity. In addition, studies of disturbance impacts have emphasized the dominant tree and woody understory layers, overlooking the species-rich herbaceous layer. For a wide range of harvesting treatments (clearcutting to selection systems with and without site preparation), I characterized disturbance severity based on hypothesized patterns of canopy removal, understory vegetation removal and forest floor and soil modification. To illustrate the application of the model, I plotted the data from three studies in North American coniferous and mixed coniferous–deciduous forests in the three-dimensional diagram of disturbance severity, using actual measures of the degree of disturbance on each axis or estimates inferred from other related variables. The plotted points showed general agreement with the hypothesized ranges of disturbance severity for the treatments represented in these studies. The three axes of disturbance severity can be affected differently depending on disturbance type, emphasizing the importance of characterizing the degree of disturbance on each of these three axes. Short-term responses of the herbaceous layer are strongly influenced by disturbance severity on each of these three axes. Disturbance conditions must be quantified before vegetation indicators of these conditions can be identified. The conceptual model provides a framework for comparing forest harvesting treatments to natural disturbance events. These comparisons are critical to designing ecologically sustainable forest management strategies.

DNS of a plane mixing layer for the investigation of sound generation mechanisms

Direct numerical simulations in two and three dimensions have been performed to investigate the sound generation by vortex pairing in a compressible plane mixing layer with Ma 1 = 0.5 being the upper and Ma 2 = 0.25 being the lower stream Mach number. The Reynolds number based on the vorticity thickness at the inflow and the velocity of the upper stream is Re = 500 . The flow is forced at the inflow with eigenfunctions obtained from viscous linear stability theory including three-dimensional disturbances. The results are verified with linear stability theory and the two-dimensional simulations performed by Colonius et al. [Colonius T, Lele SK, Moin P. Sound generation in a mixing layer. J Fluid Mech 1997;330:375–409]. The excitation of a steady longitudinal vortex mode leads to an early three-dimensional deformation of the travelling spanwise vortices and reduced sound emission to the slower fluid stream side.

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman’s boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary cross-flow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic frameworks introduced in [Proc. Roy. Soc. London Ser. A406(1986), 93–106] and [Proc. Roy. Soc. London Ser. A413(1987), 497–513] for the stationary linear and non-linear modes, it is revealed here that the non-stationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A406(1986), 93–106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.

Amplification of three-dimensional perturbations during parallel vortex–cylinder interaction

A computational study has been performed to examine the amplification of three-dimensional flow features as a vortex with small-amplitude helical perturbations impinges on a circular cylinder whose axis is parallel to the nominal vortex axis. For sufficiently weak vortices with sufficiently small core radius in an inviscid flow, three-dimensional perturbations on the vortex core are indefinitely amplified as the vortex wraps around the cylinder front surface. The paper focuses on the effect of viscosity in regulating amplification of three-dimensional disturbances and on assessing the ability of two-dimensional computations to accurately model parallel vortex–cylinder interaction problems. The computations are performed using a multi-block structured finite-volume method for an incompressible flow, with periodic boundary conditions along the cylinder axis. Growth of three-dimensional flow features is examined using a proper-orthogonal decomposition of the Fourier-transformed vorticity field in the azimuthal and axial directions. The interaction is examined for different axial wavelengths and amplitudes of the initial helical vortex waves and for three different Reynolds numbers.

Growing Hydrodynamic Modes in Keplerian Accretion Disks during Secondary Perturbations: Elliptical Vortex Effects

The origin of hydrodynamic turbulence, and in particular of an anomalously enhanced angular momentum transport, in accretion disks is still an unsolved problem. This is especially important for cold disk systems that are practically neutral in charge, since the turbulence in those systems cannot be of magnetohydrodynamic origin. While the flow must exhibit some instability and then turbulence in support of the transfer of mass inward and angular momentum outward, according to the linear perturbation theory, in the absence of magnetohydrodynamic effects, it should always be stable. We demonstrate that the three-dimensional secondary disturbance to the primarily perturbed disk, consisting of elliptical vortices, gives significantly large hydrodynamic growth in such a system and hence may suggest a transition to an ultimately turbulent state. This result is essentially applicable to accretion disks around quiescent cataclysmic variables, in protoplanetary and star-forming disks, and the outer region of disks in active galactic nuclei, where the gas is significantly colder, and thus the magnetic Reynolds number is smaller than 104.

Three-dimensional linear stability of natural convection in horizontal concentric annuli

We have performed three-dimensional linear stability analyses of natural convection of air ( Pr = 0.71) in horizontal annuli between concentric cylinders by using a spectral element method. A linear stability theory is applied to a basic two-dimensional steady solution which is called a crescent-shaped convection, and an eigenvalue problem of matrix form is constituted. We evaluate a critical Rayleigh number where the basic flow loses its stability with respect to three-dimensional disturbances over a wide range of aspect ratio (1 ⩽ A ⩽ 20). It is found that the instability mode which destabilizes the basic convection exchanges among three different modes, and the new transition lines are proposed.

Two- and three-dimensional instabilities in the film blowing process

We examine the film blowing process (FBP), which is widely used for manufacturing biaxially stretched films of polymeric materials. The viscoelastic property of the material is taken into account by employing the Upper Convected Maxwell, the Oldroyd-B or the Phan-Thien and Tanner constitutive model. In contrast to all previous theoretical works, which followed the now classical method developed by Pearson and Petrie [J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part 1. Formal mathematical representation, J. Fluid Mech. 40 (1) (1970) 1–19; J.R.A. Pearson, C.J.S. Petrie, The flow of a tubular film. Part 2. Interpretation of the model and discussion of solutions, J. Fluid Mech. 42 (3) (1970) 609–625], we analyze the process by starting with the general three-dimensional mass and momentum balances and by formally and systematically applying the thin-film approximation. This procedure results in two-dimensional dynamic balances in both the axial and azimuthal directions. Although these balances are highly non-linear and more complicated than the original momentum balance, they are reduced by one spatial dimension and, more importantly, they are more general than the classical ones, whereas they are developed in a rigorous and straightforward manner. When we assume axial symmetry and steady state, we recover the earlier model equations. However, this new methodology allows us to examine not only axisymmetric, but also non-axisymmetric disturbances to this base flow and to retain the time derivatives in all the governing equations. This procedure is an extension of our earlier one used to study transient annular extrusion [K. Housiadas, J. Tsamopoulos, Unsteady flow of an axisymmetric annular film under gravity, Phys. Fluids 10 (10) (1998) 2500–2516; K. Housiadas, J. Tsamopoulos, Unsteady extrusion of a viscoelastic annular film: I. General model and its numerical solution, J. Non-Newton. Fluid Mech. 88 (3) (2000) 229–259], which also involved the thin-film approximation and three moving interfaces, but under the assumption of axial symmetry. Viscous, elastic, inertial, gravitational and capillary forces are included in our model. The base state is computed using finite differences to simultaneously predict bubble shape, film thickness, velocity, pressure and polymer extra-stress profiles. Subsequently, its linear stability is examined to two- and three-dimensional disturbances by solving the full eigenvalue problem to determine the stability regions of the process. It is shown that under typical operating conditions the bubble becomes unstable first to non-axisymmetric disturbances, although two-dimensional instability is also predicted by our model, in agreement with recent experiments.

Control of Laminar Separation Bubbles Using Instability Waves

This paper presents detailed investigations related to active transition control in laminar separation bubbles. The investigations rely on direct numerical simulations based on the complete Navier-Stokes equations for a flat-plate boundary layer. A laminar separation bubble is created by imposing a streamwise adverse pressure gradient at the freestream boundary of the integration domain. Different steady and unsteady boundary layer disturbances are then introduced at a disturbance strip upstream of separation and their effects on the separation bubble are studied. It is shown that the size of the separated region can be controlled most efficiently by very small periodic oscillations, which lead to traveling instability waves that grow to large levels by the hydrodynamic instability of the flow. Indications for the preferred frequency of these waves can be obtained from linear stability theory, but since the problem is nonlinear, only direct numerical simulations can really qualify or disqualify the predictions. Overall, it turns out that unsteady two- or three-dimensional disturbances have a stronger impact on the size of the bubble than steady disturbances, because they directly provide initial amplitudes for the laminar-turbulent transition mechanism.

Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid

In this paper, we investigate the three-dimensional stability of the Moore–Saffman elliptical vortex in a rotating stratified fluid. By means of an asymptotic analysis for long vertical wavelength perturbations and small Froude number, we study the effects of Rossby number, external strain, and ellipticity of the vortex on the stability of azimuthal modes m = 1 (corresponding to a bending instability) and m = 2 (corresponding to a twisting instability). In the case of a quasi-geostrophic fluid (small Rossby number), the asymptotic results are in striking agreement with previous numerical stability analyses even for vertical wavelengths of order one. For arbitrary Rossby number, the key finding is that the Rossby number has no effect on the domains of long-wavelength instability of these two modes: the two-dimensional or three-dimensional nature of the instabilities is controlled only by the background strain rate γ and by the rotation rate Ω of the principal axes of the elliptical vortex relative to the rotating frame of reference. For the m = 1 mode, it is shown that when Ω< −γ , the vortex is stable to any long-wavelength disturbances, when − γ< Ω 0, two-dimensional perturbations are most unstable, when 0 Ω<γ , long-wavelength three-dimensional disturbances are the most unstable, and finally when γ< Ω, short-wavelength three-dimensional perturbations are the most unstable. Similarly, the m = 2 instability is two-dimensional or three-dimensional depending only on γ and Ω, independent of the Rossby number. This means that if a long-wavelength three-dimensional instability exists for a given elliptical vortex in a quasi-geostrophic fluid, a similar instability should be observed for any other Rossby number, in particular for infinite Rossby number (strongly stratified fluids). This implies that the planetary rotation plays a minor role in the nature of the instabilities observed in rotating strongly stratified fluids. The present results for the azimuthal mode m = 1 suggest that the vortex-bending instabilities observed previously in quasi-geostrophic fluids (tall-column instability) and in strongly stratified fluids (zigzag instability) are fundamentally related.

Linear analysis of three-dimensional instability of non-Newtonian liquid jets

The instability behaviour of non-Newtonian liquid jets moving in an inviscid gaseous environment is investigated theoretically for three-dimensional disturbances. The corresponding dispersion relation between the wave growth rate and the wavenumber is derived. Results for axisymmetrical non-Newtonian jets, the Newtonian jets, and the inviscid jets are recovered, and it is shown that two-dimensional disturbances are the most dangerous for the considered set of parameters.

The evolution of a finite-amplitude localized disturbance in an irrotational unbounded plane stagnation flow

The evolution of a finite-amplitude three-dimensional localized disturbance, having an initial dipole Gaussian vorticity distribution, embedded in an external unbounded irrotational plane stagnation flow (U = (Ay,Ax,0)) is investigated. Using the fluid impulse integral as a characteristic of such a disturbance, the viscous vorticity equation is integrated analytically. Accordingly, the fluid impulse associated with such disturbances decays and grows exponentially along the principal axes x=y and x=-y, respectively. Numerical simulations, carried out for both linear and nonlinear disturbances at a Reynolds number of 40, confirm the above predictions. The simulations have also been compared with the solution of the linear viscous vorticity disturbance equation. While the solution predicts the vorticity distribution for the linear case, it fails to predict the essential characteristics of a nonlinear disturbance associated with its self-induced movement. Finally, it is shown that the fluid impulse and the disturbance kinetic energy follow the same trend, i.e. when the fluid impulse increases with time so does the kinetic energy and vice versa. The correspondence between them suggests the use of the fluid impulse to predict the stability of a localized disturbance. © 2006 Cambridge University Press.

Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Nonmodal transient growth studies and an estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The steady mean flow is characterized by a nonuniform shear rate and a varying temperature across the wall-normal direction for an appropriate perfect gas model. The maximum amplification of perturbation energy over time, Gmax, is found to increase with increasing Reynolds number Re, but decreases with increasing Mach number M. More specifically, the optimal energy amplification Gopt (the supremum of Gmax over both the streamwise and spanwise wave numbers) is maximum in the incompressible limit and decreases monotonically as M increases. The corresponding optimal streamwise wave number, αopt, is nonzero at M=0, increases with increasing M, reaching a maximum for some value of M and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers (in contrast to incompressible Couette flow), the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law G(t∕Re)∼Re2, the reasons for which are shown to be tied to the dominance of some terms (related to density and temperature fluctuations) in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid algebraic instability. The decrease of transient growth with an increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow (Ė1) in the same limit. The sharp decay of the viscous eigenfunctions with increasing Mach number is responsible for the decrease of Ė1 for the present mean flow.

Effect of Permeability on Steady Flow in a Dendrite Layer

We consider the problem of nonlinear steady convective flow in a horizontal dendrite layer during alloy solidification. We analyze the effect of permeability of the layer on the stationary modes of convection in the form of two-dimensional rolls and threedimensional patterns. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the twoand three-dimensional solutions to the weakly nonlinear problem by using a perturbation technique, and the stability of these solutions are investigated with respect to arbitrary three-dimensional disturbances. An inverse form of the permeability function introduces two non-negative non-dimensional parameters K1 and K2 that are significant in the present problem. The results of the analyses in particular range of values of the magnitude |e| of the amplitude of convection indicate, in particular, that the effects of K1 and K2 on the flow pattern are destabilizing and stabilizing, respectively, and different types of flow pattern can be stable for particular values of these parameters. For sufficiently small and non-zero values of |e| and K1, the steady flow pattern in the form of subcritical hexagons can stable. For |e| beyond some value and depending on the values of the parameters of the problem, supercritical rolls, squares or rectangles can be stable. For K1=0.0, the only stable flow pattern is that due to steady rolls.

Three-dimensional study of multiple transitions for natural convection in horizontal annuli

Natural convection in horizontal differentially heated annuli is numerically investigated both by linear stability analysis and by solving the three-dimensional flow equations for a radius ratio R = 1.7. The governing equations were solved numerically by using a finite volume method. The stability of the basic crescent-shaped flows to three-dimensional disturbances is investigated. The existence of a stability region corresponding to a reversal from 3D- to 2D-flows is predicted for the first time. Three-dimensional computations show that multiple solutions are possible in the new stability region according to the initial conditions. Computations performed at supercritical Rayleigh numbers for the oscillatory régime elucidated the influence of the annulus length. Conflicting results reported in the archival literature are clarified.

Transformation of Long Nonlinear Waves in a Two-Layer Viscous Fluid Between a Gently Sloping Lid and Bottom

Dynamics of three-dimensional disturbances of the interface between two fluid layers of different densities is considered analytically and numerically. An evolutionary integrodifferential equation is derived, which takes into account long-wave contributions of inertia of the layers and surface tension, small but finite amplitude of disturbances of the interface between two incompressible immiscible fluids, gentle slopes of the lid and bottom, and nonstationary shear stresses at all boundaries. Numerical solutions of this model equation for several (most typical) nonlinear problems of transformation of two- and three-dimensional waves are obtained.

On upstream influence in supersonic flows

The general problem of propagation of three-dimensional disturbances in viscous supersonic flows is considered in the framework of characteristic analysis. Unlike previous results for linear disturbances we deduce a condition determining nonlinear characteristic surfaces which is exact and therefore allows both qualitative and quantitative studies of the speed of propagation as a function of various physical phenomena. These include negative and adverse pressure gradients, and effects of wall cooling and suction–blowing, which are studied in this work as an illustration of the general theory.