Magneto-hydrodynamic stability of plane Poiseuille flow with flexible boundaries

MHD stability of plane Poiseuille flow between parallel flexible walls with coplaner magnetic field is analysed. The study is restricted to sufficiently low values of magnetic Reynolds number. The eigen-value problem so posed is then solved graphically and neutral curves are obtained for various sets of magnetic parameter and flexible wall parameters. The nature of influence on flow stability depends on the values of these parameters.     

Hydromagnetic linear instability analysis of Giesekus fluids in plane Poiseuille flow

The effects of a fluid’s elasticity are investigated on the instability of plane Poiseuille flow on the presence of a transverse magnetic field. To determine the critical Reynolds number as a function of the Weissenberg number, a two-dimensional linear temporal stability analysis will be used assuming that the viscoelastic fluid obeys Giesekus model as its constitutive equation. Neglecting terms nonlinear in the perturbation quantities, an eigenvalue problem is obtained which is solved numerically by using the Chebyshev collocation method. Based on the results obtained in this work, fluid’s elasticity is predicted to have a stabilizing or destabilizing effect depending on the Weissenberg number being smaller or larger than one. Similarly, solvent viscosity and also the mobility factor are both found to have a stabilizing or destabilizing effect depending on their magnitude being smaller or larger than a critical value. In contrast, the effect of the magnetic field is predicted to be always stabilizing.     

Linear Stability of Pipe Poiseuille Flow at High Reynolds Number Regime

In this paper, we prove the linear stability of the pipe Poiseuille flow for general perturbations at high Reynolds number regime. This has been a long-standing problem since the experiments of Reynolds in 1883. Our work lays a foundation for the theoretical analysis of hydrodynamic stability of pipe flow, which is one of the oldest yet unsolved problems in fundamental fluid dynamics. © 2022 Wiley Periodicals LLC.     

Fluid flows around cascades

Flows induced by the small-amplitude and high frequency harmonic oscillations of a cascade of bodies in an unbounded fluid which is otherwise at rest are investigated theoretically. In the theoretical study we separate the flow into inner and outer regions. The flow in the inner region is governed by the Stokes boundary-layer equation. The first-order outer flow is governed by the potential solution which is found by using a conformai mapping technique. The second-order outer flow is governed by the full Navier-Stokes equation and the steady streaming flow has been obtained using a modified central-difference scheme for cascades with square cylinders and flat plates for values of the streaming Reynolds number,R _s , up to 70. These results show a complicated flow structure.     

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value ReE depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, ReE is shown to be at least 174.     

On the Nonlinear Development of Small Three-Dimensional Perturbations in Plane Poiseuille Flow: Single-Mode Approach

Nonlinear aspects of developing three-dimensional perturbations in plane Poiseuille flow have been elucidated at the primary, instead of the conventional secondary, level. Three-dimensional perturbation velocities generate normal vorticity by stretching and tilting the basic-flow vorticity. The amplitude of the induced normal vorticity, and hence that of the streamwise perturbation velocity, can grow temporally to significant peak values before the exponential decay predicted by the linear theory sets in. These growths, according to the linear theory, do not influence the amplitudes of the normal perturbation velocity that are monotonically decaying with time. It is shown in this study that the normal velocity continues to be oblivious to the development of induced normal vorticity, even in the nonlinear regime, if the perturbation velocities are described by waves traveling in a single oblique direction. Also, the Reynolds number dependence of the amplitude of the normal vorticity is discussed.     

Compressible Modes of the Rotating-Disk Boundary-Layer Flow Leading to Absolute Instability

This work is devoted to the clarification of the viscous compressible modes particularly leading to absolute instability of the three-dimensional generalized Von Karman's boundary-layer flow due to a rotating disk. The infinitesimally small perturbations are superimposed onto the basic Von Karman's flow to achieve linearized viscous compressible stability equations. A numerical treatment of these equations is then undertaken to search for the modes causing absolute instability within the principle of Briggs–Bers pinching. Having verified the earlier incompressible and inviscid compressible results of [ 1–3 ], and also confirming the correct match of the viscous modes onto the inviscid ones in the large Reynolds number limit, the influences of the compressibility on the subject matter are investigated taking into consideration both the wall insulation and heat transfer. Results clearly demonstrate that compressibility, as the Mach number increases, acts in favor of stabilizing the boundary-layer flow, especially in the inviscid limit, as far as the absolute instability is concerned, although wall heating and insulation greatly enhances the viscous absolutely unstable modes (even more dramatic in the case of wall insulation) by lowering down the critical Reynolds number for the onset of instability, unlike the wall cooling.     

高雷诺数下平面Poiseuille流稳定性渐近分析

本文主要讨论高雷诺数下平面Poiseuille流稳定性问题.应用多层结构分析理论,求出了描述该流体稳定性的Orr-Sommerfeld方程特征值的较为合理的逼近式及相应的特征函数的渐近解析解.     

Solutions of the Ginzburg-Landau Equation of Interest in Shear Flow Transition

The Ginzburg-Landau equation may be used to describe the weakly nonlinear 2-dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave- like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier-Stokes equations, describing pulses and fronts of instability in the flow.     

经过修正的平面Couette流的非线性稳定性研究

本文讨论了经过修正的平面Couette流在二维扰动下的非线性稳定性性质,并同经过修正的平面Poiseuille流的非线性稳定性性质进行了比较.计算结果表明,对于有限振幅的扰动,平面Couette流比平面Poiseulle流更不稳定.     

经过修正的层流流动的流动稳定性问题(Ⅲ)——经过修正的平行剪切流的非线性稳定性研究

本文根据文[1]给出的经过修正的层流流动的流动稳定性理论及平行剪切流中平均速度的一类修正剖面,研究了平行剪切流的非线性稳定性性质,并在本文的假设下,把背景湍流噪声的干扰引入了流动稳定性计算,对于平面Poiseuille流动和圆管Poiseuille流动,得到了与实验趋势相一致的结果.     

Nonlinear instability of flow in a rectangular pipe with large aspect ratio

Summary The stability of flow in a rectangular pipe with large aspect ratio can be studied by regarding the flow as a perturbation of plane Poiseuille flow. In the central region of the pipe, the disturbance has the form of a slowly modulated wave. The amplitude equation for this wave at slightly supercritical Reynolds numbers is identical with that for the unbounded flow. The regions in the vicinity of the side walls are studied and the boundary conditions for the amplitude equation at the walls are obtained. It is shown that the critical Reynolds number is only increased by a term proportional to (depth/width)², so that even moderate aspect ratios should be suitable for studying unbounded plane Poiseuille flow instability.

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Résumé Une perturbation de l'écoulement plan de Poiseuille permet d'étudier la stabilité de l'écoulement dans une conduite rectangulaire à grand rapport d'allongement. Dans la région centrale de la conduite, la perturbation se comporte comme une onde lentement modulée. L'équation donnant les amplitudes de cette onde est exactement celle de l'écoulement libre pour des nombres de Reynold légèrement au-dessus de la valeur critique. On étudie les régions voisines des parois et on obtient ainsi les conditions aux limites pour l'équation des amplitudes. On démontre que le nombre de Reynold critique n'est augmenté que d'un terme proportionnel au rapport (profondeur/largeur)². Ainsi, même de faibles rapports d'allongement conviennent à l'étude de l'instabilité de l'écoulement libre de Poiseuille dans un plan.

     

On the Effects of Nonparallelism,Three-Dimensionality,and Mode Interaction in Nonlinear Boundary-Layer Stability

A self-consistent theoretical investigation is described for the nonlinear stability, and spatial development, of disturbances in a plane boundary layer subject to a number of three-dimensional modes, their nonlinear interactions, and the effects of nonparallelism of the basic flow. For the largest weakly nonlinear disturbances considered, nonparallel-flow effects appear to be negligible at first sight, and primary, secondary, and/or tertiary bifurcations, usually supercritical but not always so, can occur when two fundamental modes are present. As a result the flow downstream then always has three ultimate possibilities: a unique stable disturbed state, two or more possible stable states, or no stable state possible. It is here that the nonparallel-flow effects exert their crucial influence. For nonparallelism comes into play significantly during the initial growth or decay of a disturbance, and that initial spatial development, from given initial conditions upstream, controls what happens subsequently as the disturbance increases. Thus in the first possibility above, the stable state is achieved through a smooth bifurcation, due to nonparallelism; in the second possibility the nonparallelism decides which stable state is attained (smoothly) from the initial conditions; and in the third possibility the nonparallel flow effects force the disturbance to terminate in a singular fashion. This singularity then leads to a fully nonlinear effect, locally on the boundary-layer flow. More complicated interactions can arise if more than two three-dimensional modes are present. The novel effect of the nonparallelism has a connection with related Navier-Stokes calculations even at near-critical Reynolds numbers.     

Stability of the Poiseuille Flow in a Channel with Comb Grooves

The stability of the Poiseuille flow in a channel with longitudinal comb grooves on the lower wall is studied numerically. Dependences of the linear and energy critical Reynolds numbers on the groove spacing and height are obtained and analyzed. The results are compared with data available for wavy grooves, which tend to comb grooves as one of the groove parameters approaches infinity.     

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [ 1 ], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [ 2 ] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter     , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [ 1 ] for zero-suction compressible modes.
The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.     

Optimal Disturbance Growth in Watertable Flow

The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter (S) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re² if k2 S Re² is kept constant, where k is the absolute value of the wavenumber vector. For small values of S exponential growth of free-surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flow.     

Degeneracies and Direct Resonances in Water-Table Flow

Degeneracies of the Orr-Sommerfeld eigenmodes and direct resonances between the Orr-Sommerfeld eigenmodes and vorticity eigenmodes are studied in water-table flow. The sensitivity of the characteristics of these algebraic mechanisms to flow parameters, such as the Reynolds number (R), the slope of the table (θ), and a material parameter (γ), are investigated. It is found that the mechanisms become operative at subtransitional R, and their damping rates decrease with increasing R. When the mean flow profile is slightly distorted from the ultimate parabolic profile, the characteristics of the direct resonances show remarkable variations. Also, some of the algebraic mechanisms in watertable flow are shown to have the same characteristics and modal structures as some of those in plane Poiseuille flow.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Viscous potential flow analysis of magnetohydrodynamic interfacial stability through porous media

In the view of viscous potential flow theory, the hydromagnetic stability of the interface between two infinitely conducting, incompressible plasmas, streaming parallel to the interface and subjected to a constant magnetic field parallel to the streaming direction will be considered. The plasmas are flowing through porous media between two rigid planes and surface tension is taken into account. A general dispersion relation is obtained analytically and solved numerically. For Kelvin-Helmholtz instability problem, the stability criterion is given by a critical value of the relative velocity. On the other hand, a comparison between inviscid and viscous potential flow solutions has been made and it has noticed that viscosity plays a dual role, destabilizing for Rayleigh-Taylor problem and stabilizing for Kelvin-Helmholtz. For Rayleigh-Taylor instability, a new dispersion relation has been obtained in terms of a critical wave number. It has been found that magnetic field, surface tension, and rigid planes have stabilizing effects, whereas critical wave number and porous media have destabilizing effects.     

平面Poiseuille流中弱非线性理论和二次失稳理论的关系*

文献[1]提出了平面Poiseuille流的二次失稳理论,本文则用弱非线性理论研究了同一问题.所得结果和二次失稳理论的结果是一致的,说明在平面Poiseuille流中弱非线性理论和二次失稳理论有内在联系.