Linear Stability Of Flow Research Articles

The linear stability of slip channel flows

A linear stability investigation is undertaken on the two-dimensional flow that develops in a channel whose walls are coated with a superhydrophobic material. The surfaces are modeled as classical slip surfaces, exploiting a linear Navier slip condition imposed on the channel walls. Both symmetric and asymmetric slip walls are considered, whereby the linearized stability of the flow is determined via an Orr–Sommerfeld normal-mode approach. In both instances, the application of slip establishes a significant stabilizing effect and increases the critical Reynolds number associated with the onset of linearly unstable behavior. Indeed, for sufficiently large slip lengths, the upper and lower branches of the neutral stability curve coalesce. Consequently, the flow becomes linearly stable to all disturbances for all wavelengths and Reynolds numbers.

Linear stability of flow in a 90° bend

This paper considers two-dimensional flow in a channel that consists of straight inlet and outlet branches and a circular 90° curved bend. An incompressible viscous fluid flows through the elbow under the action of a constant pressure gradient between the inlet and outlet. Navier–Stokes equations were solved numerically using a high-fidelity spectral/hp element method. In a range of Reynolds numbers, an adaptive selective frequency damping method was used to obtain steady-state flow. It was found that three separation bubbles and vortex shedding can exist in the bend. The modal stability of two- and three-dimensional perturbations was investigated. The critical Reynolds number of two-dimensional disturbances was found by extrapolation from lower Reynolds number results. It is much greater than the three-dimensional one, but the two-dimensional flow could be subcritically unstable with respect to the externally imposed small-amplitude white noise. For three-dimensional perturbations, the dependence of critical Reynolds numbers on the bending radius was obtained. For the case of a moderate bending radius, a neutral curve is provided and eigenfunctions are studied in detail. Three-dimensional instability can be caused by a periodic or monotonically growing mode, and these unstable modes relate to recirculation bubbles that occur after the bend.

Stability of Slip Flow in Microbearings

Abstract The linear stability of flows with wall slip in microbearings is examined. The analytic expression of the basic flow for a narrow gap and the linearized differential equation governing flow instability for the case of axisymmetric disturbances are derived by using the Navier slip model. The generalized matrix eigenvalue problem of the flow stability is solved based on a Chebyshev collocation method. Neutral stability curves with wall slip are obtained. It is shown that, when the slip coefficient representing the degree of slip increases, the stable region expands in the instability diagram, showing the stabilizing effect of slip on the flow stability.

Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of As=3.19 and Re=2.47×105, where the saddle–node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio As is caused by the saddle–node bifurcation.

Instability of Slip Flow between Two Concentric Cylinders

The linear stability of flows confined between two concentric rotating cylinders with wall slip is examined. The curves of marginal stability in the plane of the angular velocities at the walls are obtained as a function of slip coefficient, showing the stabilizing effect of slip on the flow stability. The eigenfunction for the first mode at the onset of instability and the corresponding cell pattern are obtained for different slip coefficients. It is found that the number of cell pattern in the radial direction increases with increasing slip coefficients.

Linear instability of annular Poiseuille flow

The linear stability of flow along an annular pipe formed by two coaxial circular cylinders is considered. We find that the flow is unstable above a critical Reynolds number for all 0 < η ≤ 1, where η is the ratio between the radii of the inner and outer cylinders. This contradicts a recent claim that the flow is stable at all Reynolds numbers for radius ratio η less than a finite critical value. We find that non-axisymmetric disturbances become stable at all Reynolds numbers for η < 0.11686215, and we are able to study this ‘bifurcation from infinity’ asymptotically. However, axisymmetric disturbances remain unstable, with critical Reynolds number tending to infinity as η → 0. A second asymptotic analysis is performed to show that the critical Reynolds number Rec ∝ η−1 log(η−1) as η → 0, with the form of the mean flow profile causing the appearance of the logarithm. The stability of Hagen–Poiseuille flow (η = 0) at all Reynolds numbers is therefore interpreted as a limit result, and there are no annular pipe flows which share this stability.

Effect of slip on the linear stability of flow through a tube

A theoretical investigation of the linear stability of the flow of a Newtonian fluid through a tube is presented using an alternative boundary condition to the standard no-slip condition. The linear stability analysis is based on the classical method of infinitesimal axially symmetric harmonic perturbations super-imposed on the steady state solution. In this analysis the standard no-slip boundary condition is replaced with the Navier boundary condition, independently proposed over a hundred years ago by both Navier and Maxwell. This boundary condition contains an extra parameter, which may be regarded as a positive slip length. The aim of this analysis is to assess the effect of a nonzero slip length on the behavior of infinitesimal disturbances in the flow through a tube. It is demonstrated that for positive slip the rate of decay of the least damped disturbance is reduced, although the flow still remains stable to all infinitesimal disturbances of the type considered, as it does for the no-slip boundary condition.

Instabilities of the flow in a curved channel with compliant walls

The linear stability of flows in curved compliant channels is examined. The walls bounding the fluid are modelled as thin cylindrical shells supported by a rigid outer frame through arrays of springs and dampers; this is often referred to as Kramer-type coating. Sufficiently soft compliant walls have an influence on the large-scale streamwise vortices produced in the channel by the centrifugal force, although the effect is limited to modes of large enough spanwise wavelengths. For even longer wavelengths, a spanwise-periodic surface-based wave precedes the onset of the streamwise vortex instability. Longitudinal or oblique travelling-wave flutter modes may appear and dominate the transition scenario, for sufficiently compliant walls, depending on the receptivity conditions.

Stability of flow between two corotating disks in an enclosure

Linear stability of flow between two corotating disks in an enclosure is investigated numerically, and the result is confirmed experimentally with a visualization method. The outer cylindrical boundary of the flow is assumed to be stationary, whereas the inner cylinder rotates together with the two disks. Polygonal flow patterns have been observed in previous experiments and numerical simulations performed for large Reynolds numbers of O(103)–O(105). The flow pattern was classified into two groups, one of which is asymmetric with respect to the interdisk midplane and the other has the so-called shift-and-reflect symmetry. We found that the first instability of the axisymmetric flow that is a unique solution at small Reynolds numbers is caused by a disturbance with an unexpectedly large wavenumber such as from 20 to 30 for narrow disk spacings. The resultant flow from the instability has aligned vortices along the outer boundary and possesses the shift-and-reflect symmetry always at the criticality, which is expected to develop into polygonal flows at much larger Reynolds numbers than the critical value.

On the initial configurations of collapsible channel flow

This paper studies the effect of the initial configurations of the governing equations on flows in a collapsible channel where the upper elastic wall is replaced by a pre-stretched beam. The aim is to check the existence of a “tongue” shaped neutral stability curve in the Reynolds number–tension space from a fluid-beam model [Luo XY, Cai ZX. Effects of wall stiffness on the linear stability of flow in an elastic channel. In: de Langre E, Axisa F, editors. Proceedings of the eighth international conference on flow-induced vibrations, FIV2004, vol. II. Paris, France: 2004. p. 167–70], in a properly formulated initial strain configuration. It was found that, for a given Reynolds number, as the tension is lowered to a critical value, the system becomes unstable, which is to be expected. However, a further decrease of the tension re-stabilizes the system before it becomes unstable again. It was possible that this puzzling finding was an artefact since the elastic equations used in the model were not properly derived from the zero initial stress configuration (Ogden, private communication). To check this, in this paper, a range of steady solutions are studied with both zero and non-zero initial wall tension. These are compared with the results using the finite element package Adina 8.3 using both the initial strain and initial stress configurations. As expected, the fluid-beam model agrees with Adina when using the initial stress configuration, but not when using the initial strain configuration. For cases with a small initial tension or small deformation (very large initial tension), both initial stress and initial strain configurations lead to very similar results, however, when the initial tension is comparable with the stretching induced tension, there are obvious differences in these two configurations. The “tongue” stability curve is then re-calculated with a zero initial tension, and re-plotted in the Reynolds number–effective tension space. It is interesting to see that though slightly different in shape, the “tongue” stable zone appears again when the zero initial tension is used. Thus it is highly likely that the puzzling “tongue” in the neutral stability curve is not due to the modelling approximation, but indicating a real, interesting physical phenomenon.

Linear stability of flow in an internally heated rectangular duct

The linear stability of flow in a vertical rectangular duct subject to homogeneous internal heating, constant-temperature no-slip walls and a driving pressure gradient is investigated numerically in the framework of the Boussinesq hypothesis. A Galerkin method based upon modified Chebyshev polynomials is used to discretize the linearized Navier-Stokes equations in their most compact form, i.e. eliminating the pressure and the streamwise velocity and making use of the intrinsic symmetry properties. A classification of the basic flow in Grashof and Reynolds space in terms of inflectional properties is proposed. It is found that the flow loses stability at all aspect ratios for a combination of finite thermal buoyancy and pressure forces with opposed signs. In the square duct, the unstable region lies inside the range where the basic velocity profile exhibits additional inflection lines. Unstable eigenfunctions are obtained for all basic symmetry modes, consisting of rapidly propagating large-scale structures located in the vicinity of the inflection lines near the centre of the duct.

Linear Stability of Flows in a Squeeze Film

We study linear stability of viscous flows in a squeezelubrication film, in which the flow varies slowly in space and time,between two parallel plates moving normal to each other with a slowconstant speed, generalizing the inviscid results of Aristov and Gitman[J. Fluid Mech. 464 (2002) 209]. The temporalevolution of two-dimensional disturbances for this physical situation,including the asymptotic behaviour of a long term or the transientbehaviour of some time interval, is obtained by the construction of alow-dimensional Galerkin method. It is found that the wall boundariestypically play dual roles of stabilizer and destabilizer. Theyconstrain the development of disturbances and have stabilizinginfluences. However, they give rise to velocity shear, which isdiffused by viscosity and thereby tends to destabilize the flow.

The linear and nonlinear stability of thread-annular flow

The surgical technique of thread injection of medical implants is modelled by the axial pressure-gradient-driven flow between concentric cylinders with a moving core. The linear stability of the flow to both axisymmetric and asymmetric perturbations is analysed asymptotically at large Reynolds number, and computationally at finite Reynolds number. The existence of multiple regions of instability is predicted and their dependence upon radius ratio and thread velocity is determined. A discrepancy in critical Reynolds numbers and cut-off velocity is found to exist between experimental results and the predictions of the linear theory. In order to account for this discrepancy, the high Reynolds number, nonlinear stability properties of the flow are analysed and a nonlinear, equilibrium critical layer structure is found, which leads to an enhanced correction to the basic flow. The predictions of the nonlinear theory are found to be in good agreement with the experimental data.

Linear stability of flow in a differentially heated cavity via large‐scale eigenvalue calculations

Locates the onset of oscillatory instability in the fluid flow inside a differentially heated cavity with aspect ratio 2 by computing a steady‐state and analyzing the stability of the system via eigenvalue approximation. Discusses the choice of parameters for the Cayley transformation so that the calculation of selected eigenvalues of the transformed system will reliably answer the question of stability. Also presents an argument that due to the symmetry of the problem, the first two unstable modes will have eigenvalues that are nearly identical, and the numerical experiments confirm this. Finally, locates a co‐dimension 2 bifurcation signifying where there is a switch in the mode of initial instability. The results were obtained using a parallel finite element CFD code (MPSalsa) along with an Arnoldi‐based eigensolver (ARPACK), a preconditioned Krylov method code for the necessary linear solves (Aztec), and a stability analysis library (LOCA).

Vortex instability in a divergingconverging channel

Linear stability of flow in a diverging–converging channel is considered. The flow may develop under either the fixed mass or the fixed pressure gradient constraint. Both cases are considered. It is shown that under certain conditions the divergence–convergence of the channel leads to the formation of a secondary flow in the form of streamwise vortices. It is argued that the instability is driven by centrifugal effect. The instability has two modes and conditions leading to their onset have been identified. These conditions depend on the amplitude and the length of the channel diverging–converging section and can be expressed in terms of a critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required to create the instability for the specified amplitude of the variations of the channel opening are also given. It is shown that the flow developed under the fixed mass constraint is slightly more unstable than the flow developed under the fixed pressure constraint. This difference increases with an increase of the amplitude of the channel divergence–convergence.

Convective versus absolute instability in flow between counterrotating cylinders.

The linear stability of flow in a Couette-Taylor system with counterrotating cylinders is posed as an initial-value problem. The time-asymptotic solution is found by the method of steepest descent. We identify the saddle points of the complex dispersion relation and, from the growth rates evaluated at the saddle points, determine a Reynolds number range ${\mathit{R}}_{\mathit{c}}$\ensuremath{\le}R\ensuremath{\le}${\mathit{R}}_{\mathit{a}}$ within which the instability is convective and beyond which (Rg${\mathit{R}}_{\mathit{a}}$) it is absolute. The amount by which ${\mathit{R}}_{\mathit{a}}$ exceeds ${\mathit{R}}_{\mathit{c}}$ can be of order 1%, but loci in parameter space of zero group velocity are found where the instability is absolute at onset (${\mathit{R}}_{\mathit{a}}$=${\mathit{R}}_{\mathit{c}}$). Bicritical curves separating regions with different azimuthal wave numbers are compared for convective and absolute instabilities. The analysis reveals a new type of bicriticality where two different axial wave numbers simultaneously become unstable.

On the stability of high-Reynolds-number flows with closed streamlines

In steady, two-dimensional, inertia-dominated flows it is well known that the vorticity is constant along the streamlines, which, in a bounded domain, are necessarily closed. For inviscid flows, the variation of vorticity across the streamlines is arbitrary, while for forced, weakly dissipitative flows, it is determined by the balance between viscous diffusion and the forcing. This paper discusses the linear stability of flows of this type to two-dimensional disturbances. Arnol'd's stability theorems are discussed. An alternative functional to Arnol'd's is found, which gives the same stability criteria and which permits a representation of the problem in terms of a Schrödinger equation. Conditions for stability are derived from this functional. In particular it is shown that total flow reversals are potentially unstable. The results are illustrated with respect to the geometrically simple case when the streamlines are circular and the forcing is due to a rotating magnetic field, for which case the stability regions are calculated as a function of two parameters. It is shown that the entire theory, including Arnol'd's theorems, applies also to poloidal axisymmetric flows.

Linear Stability of Flow in a Concentric Annulus

The stability of fully developed, parallel flow in a rigid, concentric annulus to infinitesimal disturbances is analyzed. It is found that the annular flow is spatially stable to axisymmetric disturbances upto a modified Reynolds number of 10,000.

Linear stability of flow over a rotating cylinder

Linear stability of flow over a rotating cylinder

Stability of linear flow

It is shown that a finite disturbance independent of the streamwise coordinate may lead to instability of linear flow, even though the basic velocity does not possess any inflection point.