Two-dimensional Disturbances Research Articles

Linear stability of a doubly periodic array of vortices to three-dimensional perturbations

The linear stability of a doubly periodic array of vortices to three-dimensional perturbations is studied. The instabilities are separated into symmetric and antisymmetric modes. For two-dimensional disturbances only the symmetric mode is found to be unstable. The antisymmetric mode shows a peak in growth rate at long wavelengths. This is attributed to the Crow instability. For short wavelengths both symmetric and antisymmetric modes are found to have similar growth rates. This is attributed to the elliptical instability and it is found to occur even when the vortex cells are square, a physical explanation for which is provided, but for elongated vortex cells the growth rates are higher. Viscosity is found to have a strong stabilizing influence on short wavelength perturbations.

Instabilities in a liquid film flow over an inclined heated porous substrate

Stability of a thin viscous Newtonian fluid draining down a uniformly heated porous inclined plane is examined. The long-wave linear stability analysis is performed within the generic Orr-Sommerfeld framework both theoretically and numerically. An evolution equation for the local film thickness for two-dimensional disturbances is derived to analyze the effect of long-wave instabilities. The parameters governing the film flow system and the porous substrate strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The long-time wave forms are either time-independent wave forms that propagate or time-dependent modes that oscillate slightly in the amplitude. The role of permeability and Marangoni number is to increase the amplitude of the disturbance leading to the destabilization state of the film flow system. The permeability of the porous medium promotes the oscillatory behavior.

NONLINEAR KELVIN-HELMHOLTZ INSTABILITY OF TWO SUPERPOSED DIELECTRIC FINITE FLUIDS IN POROUS MEDIUM UNDER VERTICAL ELECTRIC FIELDS

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field and in the absence of surface charges at their interface is investigated in three dimensions. The method of multiple scales is used to obtain a dispersion relation for the linear problem and a Ginzburg-Landau equation with complex coefficients for the nonlinear problem, describing the behavior of the system. The stability of the system is discussed both analytically and numerically in both cases, and the corresponding stability conditions are obtained. It is found, in the linear case, that the stability criterion is independent of the medium permeability and that the medium porosity, surface tension, and dimension all have stabilizing effects the fluid viscosities, velocities, and depths have destabilizing effects, and the electric field has a dual role in the stability of the system. In the nonlinear analysis, it is found that the electric field has a stabilizing effect in two-dimensional disturbances and destabilizing effect in three-dimensional disturbances cases. The surface tension, fluid depths, and medium porosity have stabilizing effects in both two- and three-dimensional disturbance cases and the fluid viscosities, velocities, and medium permeability have destabilizing effects in both cases, and this stability or instability occurs faster for three-dimensional disturbance cases. It is found also that the system is unstable in the absence of fluid velocities or for nonporous media. Finally, the dimension was found to have a dual role (stabilizing as well as destabilizing) in the considered system, while it has a destabilizing effect in the case of nonporous media.

Nonlinear disturbance evolution in a two-dimensional boundary-layer along an elastic plate and induced radiated sound

The interaction between a boundary-layer flow and an elastic plate is addressed by direct numerical simulation, taking into account the full coupling between the fluid flow and the flexible wall. The convectively unstable flow state is harmonically forced and two-dimensional nonlinearly saturated wavelike disturbances are computed along archetype-plates with respect to stiffness and natural frequencies. In the aim of determining the low-Mach-number radiated sound for the system, the simulation data are used to solve the Lighthill's equation in terms of a Green's function in the wavenumber-frequency space. Different degrees of fluid-structure coupling are implemented in the radiated sound model and the resulting acoustic pressure levels are compared. The sound radiation levels are shown to be increased in the presence of flexible walls with however significant differences in the radiated pressure levels for different coupling assumptions.

Transient growth in a two-fluid channel flow under normal electric field

A linear model of a layered channel flow of two perfectly dielectric viscous fluids in the presence of uniform normal electric field is built. The effect of the normal electric field on transient growth of small disturbances is studied at two values of Weber number. The numerical result shows that the electric field enhances the transient growth for both two-dimensional and three-dimensional disturbance cases. The contours of optimal energy growth are represented in the wave number plane. When the electric field is small, the optimal disturbance that corresponds to the peak value of optimal growth is two dimensional. It is governed by the lift-up mechanism and is little influenced by the electric field. However, when the electrical Euler number exceeds a critical value, the optimal disturbance is three dimensional with streamwise uniform wave number and is partially dominated by the electric field, and moreover, the spanwise wave number has a linear relationship with the electrical Euler number. The comparison of exponential growth and transient growth is performed. It is shown that exponential growth becomes profound and even predominant over transient growth when the electric field is sufficiently strong. In addition, the mechanism of transient growth is discussed and it is found that the existence of material interface may cause transient growth in the absence of shear in basic flow.

Nonlinear stage of instability development in a stratified shear flow with an inflection-free velocity profile

A plane-parallel shear flow with an inflection-free velocity profile Vx=U(y), having an embedded thin layer of stable density stratification, is known to be unstable in ideal incompressible fluid, for an arbitrary bulk Richardson number J>0 [S. M. Churilov, J. Fluid Mech. 539, 25 (2005); S. M. Churilov, J. Fluid Mech.617, 301 (2008)], and it is the three- rather than two-dimensional (z-independent) disturbances that are most unstable within a wide range of parameters. We examine the weakly nonlinear evolution of a pair of unstable oblique waves in such a flow, in the unsteady critical layer regime. For this purpose, we derive the evolution equation which has the form of a nonlinear integral equation and is valid for both thin and thick critical layers, including the case where the critical layer width exceeds the stratification layer thickness. The solutions of this equation are inspected both analytically and numerically, and it is shown that during the nonlinear stage, the disturbance evolves, as a rule, explosively.

Input–output analysis, model reduction and control of the flat-plate boundary layer

The dynamics and control of two-dimensional disturbances in the spatially evolving boundary layer on a flat plate are investigated from an input–output viewpoint. A set-up of spatially localized inputs (external disturbances and actuators) and outputs (objective functions and sensors) is introduced for the control design of convectively unstable flow configurations. From the linearized Navier–Stokes equations with the inputs and outputs, controllable, observable and balanced modes are extracted using the snapshot method. A balanced reduced-order model (ROM) is constructed and shown to capture the input–output behaviour of the linearized Navier–Stokes equations. This model is finally used to design a2-feedback controller to suppress the growth of two-dimensional perturbations inside the boundary layer.

Long-Wave Instabilities in a Non-Newtonian Film on a Nonuniformly Heated Inclined Plane

A thin liquid layer of a non-Newtonian film falling down an inclined plane that is subjected to nonuniform heating has been considered. The temperature of the inclined plane is assumed to be linearly distributed and the case when the temperature gradient is positive or negative is investigated. The film flow is influenced by gravity, mean surface tension, and thermocapillary forces acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. A nonlinear evolution equation is derived by applying the long-wave theory, and the equation governs the evolution of a power-law film flowing down a nonuniformly heated inclined plane. The linear stability analysis shows that the film flow system is stable when the plate temperature decreases in the downstream direction while it is less stable for increasing temperature along the plate. Weakly nonlinear stability analysis using the method of multiple scales has been investigated and this leads to a secular equation of the Ginzburg–Landau type. The analysis shows that both supercritical stability and subcritical instability are possible for the film flow system. The results indicate the existence of finite-amplitude waves, and the threshold amplitude and nonlinear speed of these waves are influenced by thermocapillarity. The nonlinear evolution equation for the film thickness is solved numerically in a periodic domain in the supercritical stable region, and the results show that the shape of the wave is influenced by the choice of wave number, non-Newtonian rheology, and nonuniform heating.

Stability of the plane shear flow of dilute polymeric solutions

Three variants of Oldroyd-B model are analyzed for stability of the base profile in plane Couette flow of dilute polymeric fluid at moderate Reynolds number. The stability to two-dimensional disturbances is analyzed for the linearized problem as well as the weakly nonlinear flow. We begin with the classical Oldroyd-B model with emphasis on the disturbances with axial wavenumber α∼Re1/2, where Re is the Reynolds number based on maximum velocity and channel width. For linearly stable flow, the finite amplitude stability is analyzed using the equilibrium flow method, wherein the nonlinear flow is assumed to be at the transition point. For the classical Oldroyd-B fluid, the threshold kinetic energy for the equilibrium wall mode disturbances is found to be higher for the viscoelastic fluid than for the Newtonian fluid. In the second variant, the Oldroyd-B model with additional artificial diffusivity is studied. In this model, the diffusion modes, not present in the classical Oldroyd-B model, are introduced. For large wavenumber disturbances, the diffusion modes become the slowest decaying modes in comparison to the wall modes. The threshold energy for the diffusive Oldroyd-B model is smaller than that for the Newtonian fluid. The third variant of the Oldroyd-B model accounts for the nonhomogeneous polymer concentration coupled with the polymeric stress field. While the base profile is linearly stable for the first two models, the nonhomogeneous Oldroyd-B fluid exhibits an instability in the linear analysis. The “concentration mode” becomes unstable when the fluid Weissenberg number exceeds a certain transition value. This mode of instability, driven by the stress-induced fluctuations in polymer number density, renders the uniform polymer concentration profile unstable leading to the well-known phenomenon of flow-induced demixing.

Study of stability and secondary regimes of thermocapillary flow in a liquid layer under localized heating

The problem of thermocapillary convection in a thin horizontal layer of viscous incompressible fluid with a deformable free boundary under the action of space-inhomogeneous localized temperature field is considered. The system of nonlinear differential equations for temperature, surface deformation and vorticity amplitudes in the long-wave approximation is solved numerically. The Galerkin method with polynomial basic functions is used to investigate steady states and their stability to infinitesimal two-dimensional disturbances in the cases of planar and axisymmetrical heat fluxes. The dependences of a disturbance decrement on the wave number are obtained for different steady states and parameter values. The forms of eigenfunctions for the most dangerous disturbances are presented. The nonlinear behavior of the disturbances of localized one-dimensional equilibrium states is investigated by the pseudo-spectral method in the two-dimensional formulation. It is shown that there is a domain of parameters in which these states are steady. Beyond its limits, diverse variants of the nonlinear development of disturbances can be realized, which, depending on the parameter values and disturbance magnitude, could lead to formation of the localized structures of another symmetry, global cellular structures of different symmetry and oscillatory regimes of thermocapillary convection.

Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation

We investigate stability with respect to two-dimensional (independent of z) disturbances of plane-parallel shear flows with a velocity profile Vx=u(y) of a rather general form, monotonically growing upwards from zero at the bottom (y=0) to U0 as y → ∞ and having no inflection points, in an ideal incompressible fluid stably stratified in density in a layer of thickness ℓ, small as compared to the scale L of velocity variation. In terms of the ‘wavenumber k – bulk Richardson number J’ variables, the upper and lower (in J) boundaries of instability domains are found for each oscillation mode. It is shown that the total instability domain has a lower boundary which is convex downwards and is separated from the abscissa (k) axis by a strip of stability 0 < J < J0(−)(k) with minimum width J*=O(ℓ2/L2) at kL=O(1). In other words, the instability domain configuration is such that three-dimensional (oblique) disturbances are first to lose their stability when the density difference across the layer increases. Hence, in the class of flows under consideration, it is a three- not two-dimensional turbulence that develops as a result of primary instability.

On Temporal Stability Analysis for Hydromagnetic Flow in a Channel Filled with a Saturated Porous Medium

In this paper, a linear stability analysis is presented to trace the time evolution of an infinitesimal, two-dimensional disturbance imposed on the base flow of an electrically conducting fluid in a channel filled with a saturated porous medium under the influence of a transversely imposed magnetic field. An eigenvalue problem is obtained and solved numerically using the Chebyshev collocation spectral method. The critical Reynolds number Rec, the critical wave number αc and the critical wave speed cc are obtained for a wide range of the porous medium shape factor parameter S and Hartmann number H. It is found that an increase in the magnetic field intensity and a decrease in porous medium permeability have a stabilizing effect on the fluid flow.

Interplay Between Inertia and Elasticity in Film Casting

The influence of inertia and boundary conditions on the steady state and stability of isothermal film casting of viscoelastic fluids is examined using a Phan-Thien–Tanner rheological model. The elongational flow between the die exit and the take-up point is investigated. In general, the steady-state film tends to contract for low-inertia flow; this contraction, however, is significantly diminished by inertia. The polymeric normal stresses and primary normal stress difference decrease in the most of the air gap as inertia increases. In contrast, the stress and stress difference increase considerably near the take-up point due to a dramatic increase in the elongation rate. The linear stability analysis for two-dimensional disturbances is carried out. For a polymer with no degradation, and in the absence of inertia (Re=0), the analysis predicts critical draw ratios that form an envelope to an unstable region. This region of unstable conditions reduces as inertia increases. Two branches of neutral stability curve are observed for higher-inertia flow as opposed to a single curve for Re=0. The unstable region expands as α increases, where α is a measure of polymer degradation. When α becomes sufficiently large, the elasticity tends to destabilize the flow. It is also found that boundary conditions have an important influence on the steady-state profiles and stability region, particularly for high-elasticity fluids.

선형 안정성 이론을 이용한 압축성 축 대칭 원뿔 경계층의 천이지점 예측

본 연구에서는 축 대칭 원뿔 형상 위의 압축성 경계층의 천이 지점을 선형 안정성 이론과 -method를 이용하여 예측하였다. 축 대칭 좌표계에서의 압축성 유동 지배 방정식으로부터 압축성 원뿔 경계층의 선형 안정성 방정식을 얻었으며 안정성 방정식을 2차 정확도의 유한 차분법을 이용하여 계산하는 수치 프로그램을 개발하였다. 개발 된 코드로 원뿔 경계층의 안정성 특성 및 2차원 교란의 증폭률을 계산하고 실험결과와의 비교를 통해 검증을 수행하였다. 얻어진 교란의 증폭률을 활용하여 -method를 통해 천이지점 예측을 수행하였다. 풍동 시험 및 비행 시험 결과와의 비교를 통해 비행 조건에 있는 마하수 4와 8사이의 원뿔 경계층에 대한 본 연구의 천이지점의 예측 능력을 확인하였다. 또한 벽면 냉각이 경계층 내부 교란의 안정성 및 천이 지점에 미치는 영향을 분석하였다. In this study, the transition Reynolds number of compressible axi-symmetric sharp cone boundary layer is predicted by using a linear stability theory and the -method. The compressible linear stability equation for sharp cone boundary layer was derived from the governing equations on the body-intrinsic axi-symmetric coordinate system. The numerical analysis code for the stability equation was developed based on a second-order accurate finite-difference method. Stability characteristics and amplification rate of two-dimensional second mode disturbance for the sharp cone boundary layer were calculated from the analysis code and the numerical code was validated by comparing the results with experimental data. Transition prediction was performed by application of the -method with N=10. From comparison with wind tunnel experiments and flight tests data, capability of the transition prediction of this study is confirmed for the sharp cone boundary layers which have an edge Mach number between 4 and 8. In addition, effect of wall cooling on the stability of disturbance in the boundary layer and transition position is investigated.

Approximation asymptotique uniformément valable d'un écoulement perturbé sur une plaque plane avec incidence

In the context of the laminar steady two-dimensional flow of an incompressible Newtonian fluid, we propose, with the matched asymptotic expansions method, a uniformly valid approximation in the whole domain for a disturbed flow over a flat plate with incidence. Following Libby and Fox's ideas, two-dimensional disturbances generated in the boundary layer, without disturbed pressure gradient, have been studied for a basic flow without incidence. The established theoretical perturbation is then validated from the numerical simulation of the Navier–Stokes equations linearized in the vicinity of the basic flow. To cite this article: S. Saintlos Brillac, K. Debbagh, C. R. Mecanique 336 (2008).

Two-dimensional instabilities of time-dependent zonal flows: linear shear

In this study, we investigate the stability of time-dependent zonal flows to two-dimensional (zonally symmetric) disturbances. While steady currents can only experience inertial instability (II) in this setting, unsteady ones may be destabilized in other ways. For example, time-periodic flows can be subject to parametric subharmonic instability (PSI). Motivated by observations of salinity interleaving patterns in the upper equatorial Pacific Ocean, our objective is to determine the basic properties of dominant instabilities (their generation mechanism, spatial and temporal characteristics, and finite-amplitude development) for background flows that are representative of those in the upper-equatorial ocean, yet still amenable to a computational sweep of parameter space. Our approach is to explore the stability of solutions to linear and nonlinear versions of a two-dimensional model for an idealized background flow with oscillating linear shear. To illustrate basic properties of the instabilities, the f-plane and equatorial β-plane scenarios are studied using a linear model. Stability regime diagrams show that on the f-plane there is a clear separation in dominant vertical scales between PSI- and II-dominated regimes, whereas on the equatorial β-plane the parameter space contains a region where dominant instability is a mixture of the two types. In general, PSI favours lower vertical modes than II. The finite-amplitude development of instabilities on the equatorial β-plane is explored using a nonlinear model, including cases illustrating the equilibration of pure II and the development of pure PSI and mixed instabilities. We find that unless the instabilities are weak enough to be equilibrated by viscosity at low amplitude, disturbances continue to grow until the vertical shear of their meridional velocity field becomes large enough to allow for Richardson numbers less than 1/4; as a consequence, PSI-favoured vertical modes are able to reach higher amplitudes than II-favoured modes before becoming susceptible to Kelvin–Helmholtz instability, and induce tracer intrusions of a considerably larger meridional extent.

The use of global modes to understand transition and perform flow control

The stability of nonparallel flows is considered using superposition of global modes. When perturbed by the worst case initial condition, these flows often exhibit a large transient growth associated with the development of wave packets. The global modes of the systems also provide a good starting point for the design of reduced order models used to control the growing disturbances. Three recent investigations are reviewed. The first example is the growth of a wave packet on a falling liquid sheet. The optimal perturbation analysis shows that the worst case initial condition is a localized disturbance that creates a propagating wave packet that hits the downstream end, regenerating a wave packet upstream through a global pressure pulse. Second, we consider two-dimensional disturbances in the Blasius boundary layer. It is found that a wave packet is optimally excited by an initial condition consisting of localized backward leaning Orr structures. Finally, the control of a globally unstable boundary-layer flow along a shallow cavity is considered. The disturbance propagation is associated with the development of a wave packet along the cavity shear layer, unstable to the Kelvin–Helmholtz mechanism, followed by a global cycle related to the two unstable global modes. Direct numerical simulations of this flow are coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and provides the actuation at the upstream lip. A reduced order model for the control is obtained by a projection on the least stable global eigenmodes. The linear-quadratic-Gaussian controller is run in parallel to the Navier–Stokes time integration and it is shown to damp out the global oscillations.

Effect of an electric field on the stability of contaminated film flow down an inclined plane

The stability of a liquid film flowing down an inclined plane is considered when the film is contaminated by an insoluble surfactant and subjected to a uniform normal electric field. The liquid is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. Previous studies have shown that, when acting in isolation, surfactant has a stabilizing influence on the flow while an electric field has a destabilizing influence. The competition between these two effects is the focus of the present study. The linear stability problem is formulated and solved at arbitrary parameter values. An extended form of Squire's theorem is presented to argue that attention may be confined to two-dimensional disturbances. The stability characteristics for Stokes flow are described exactly; the growth rates of the normal modes at finite Reynolds number are computed numerically. We plot the neutral curves dividing regions of stability and instability, and trace how the topology of the curves changes as the intensity of the electric field varies both for a clean and for a contaminated film. With a sufficiently strong electric field, the neutral curve for a clean film consists of a lower branch trapping an area of stable modes around the origin, and an upper branch above which the flow is stable. With surfactant present, a similar situation obtains, but with an additional island of stable modes disjoint from the upper and lower branches.

Stability of a binary mixture of a chemically reacting incompressible fluid in a plane couette flow

The stability of a binary mixture of a chemically reacting incompressible fluid in plane couette flow is considered with respect to a two-dimensional infinitesimal disturbance. The eighenvalues are found with the aid of microcomputer. Neutral stability curves for various Schmidt numbers at Reynolds numbers up to 150 are obtained by Gallagher method. It is found that the principle of the exchange of stabilities holds for this problem. Thus, using the basic orthogonal functions, the differential equations are reduced to a set of paired matrix equations with the conclusion that all fluids are unstable at different values of stability parameter C=1.40 x 10 to 2.0 x 10 for wave number (a) = 0.2 to 2 (based on half-channel width and half the relative velocity of the plates). Keywords: Incompressible fluid, binary mixture, wave number, couette flow and stability parameter. Nigerian Journal of Physics Vol. 18 (2) 2006: pp. 155-160

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.