Linear Stability of Flow in a Concentric Annulus

1977 ◽  
Vol 44 (2) ◽  
pp. 338-340
Author(s):  
A. K. Agarwal ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Annular Flow ◽  
Parallel Flow ◽  
Concentric Annulus ◽  
The Stability ◽  
Axisymmetric Disturbances

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.

Stability of Horizontal Boundary Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Mathematical Proof ◽  
Vertical Component ◽  
Ekman Layer ◽  
Numerical Results ◽  
Energy Estimates ◽  
Navier Stokes ◽  
The 1960S ◽  
The Stability

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

Stability of surfactant-laden core–annular flow and rod–annular flow to non-axisymmetric modes

2013 ◽  
Vol 716 ◽  
Author(s):  
M. G. Blyth ◽  
Andrew P. Bassom
Keyword(s):  
Linear Stability ◽  
Annular Flow ◽  
Convective Motion ◽  
Straight Pipe ◽  
Pipe Axis ◽  
Insoluble Surfactant ◽  
Fluid Layers ◽  
Axisymmetric Disturbances ◽  
Quiescent Fluid ◽  
Fluid Configuration

AbstractThe linear stability of core–annular fluid arrangements are considered in which two concentric viscous fluid layers occupy the annular region within a straight pipe with a solid rod mounted on its axis when the interface between the fluids is coated with an insoluble surfactant. The linear stability of this arrangement is studied in two scenarios: one for core–annular flow in the absence of the rod and the second for rod–annular flow when the rod moves parallel to itself along the pipe axis at a prescribed velocity. In the latter case the effect of convective motion on a quiescent fluid configuration is also considered. For both flows the emphasis is placed on non-axisymmetric modes; in particular their impact on the recent stabilization to axisymmetric modes at zero Reynolds number discovered by Bassom, Blyth and Papageorgiou (J. Fluid. Mech., vol. 704, 2012, pp. 333–359) is assessed. It is found that in general non-axisymmetric disturbances do not undermine this stabilization, but under certain conditions the flow may be linearly stable to axisymmetric disturbances but linearly unstable to non-axisymmetric disturbances.

Linear stability theory of oscillatory Stokes layers

1974 ◽  
Vol 62 (4) ◽  
pp. 753-773 ◽  
Author(s):  
Christian Von Kerczek ◽  
Stephen H. Davis
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Stability Theory ◽  
Absolute Stability ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Full Time ◽  
Full Theory ◽  
The Stability

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to bemore stablethan that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

scholarly journals The stability of developing pipe flow at high Reynolds number and the existence of nonlinear neutral centre modes

2011 ◽  
Vol 684 ◽  
pp. 284-315 ◽  
Author(s):  
Andrew G. Walton
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
High Reynolds Number ◽  
Pipe Wall ◽  
Fluid Velocity ◽  
Nonlinear Effects ◽  
Neutral Curve ◽  
High Reynolds ◽  
The Stability ◽  
Axisymmetric Disturbances

AbstractThe high-Reynolds-number stability of unsteady pipe flow to axisymmetric disturbances is studied using asymptotic analysis. It is shown that as the disturbance amplitude is increased, nonlinear effects first become significant within the critical layer, which moves away from the pipe wall as a result. It is found that the flow stabilizes once the basic profile has become sufficiently fully developed. By tracing the nonlinear neutral curve back to earlier times, it is found that in addition to the wall mode, which arises from a classical upper branch linear stability analysis, there also exists a nonlinear neutral centre mode, governed primarily by inviscid dynamics. The centre mode problem is solved numerically and the results show the existence of a concentrated region of vorticity centred on or close to the pipe axis and propagating downstream at almost the maximum fluid velocity. The connection between this structure and the puffs and slugs of vorticity observed in experiments is discussed.

Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids

2007 ◽  
Vol 592 ◽  
pp. 177-194 ◽  
Author(s):  
C. NOUAR ◽  
A. BOTTARO ◽  
J. P. BRANCHER
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Channel Flow ◽  
Shear Thinning ◽  
Transition To Turbulence ◽  
Shear Thinning Fluids ◽  
Viable Approach ◽  
The Stability ◽  
Definition Of ◽  
Do So

A viscosity stratification is considered as a possible mean to postpone the onset of transition to turbulence in channel flow. As a prototype problem, we focus on the linear stability of shear-thinning fluids modelled by the Carreau rheological law. To assess whether there is stabilization and by how much, it is important both to account for a viscosity disturbance in the perturbation equations, and to employ an appropriate viscosity scale in the definition of the Reynolds number. Failure to do so can yield qualitatively and quantitatively incorrect conclusions. Results are obtained for both exponentially and algebraically growing disturbances, demonstrating that a viscous stratification is a viable approach to maintain laminarity.

On the linear stability of spiral flow between rotating cylinders

1982 ◽  
Vol 382 (1782) ◽  
pp. 83-102 ◽  
Keyword(s):  
Reynolds Number ◽  
Numerical Study ◽  
Stability Boundary ◽  
Rotating Cylinders ◽  
Spiral Flow ◽  
Compound Matrix Method ◽  
Compound Matrix ◽  
High Reynolds ◽  
The Stability ◽  
Axisymmetric Disturbances

A numerical study is made of the effects of both axisymmetric and non-axisymmetric disturbances on the stability of spiral flow between rotating cylinders. If we let Ω 1 and Ω 2 be the angular speeds of the inner and outer cylinders, and R 1 and R 2 be their respective radii, then for fixed values of η = R 1 / R 2 and μ = Ω 2 / Ω 1 , the onset of instability depends on both the Taylor number T and the axial Reynolds number R . Here R is based on the gap width between the cylinders and the average axial velocity of the basic flow, while T is based on the average angular speeds of the cylinders. Using the compound matrix method, we have computed the complete stability boundary in the R , T -plane for axisymmetric disturbances with η = 0.95 and μ = 0. We find that, for sufficiently high Reynolds numbers, there are two distinct axisymmetric modes corresponding to the usual shear and rotational instabilities. We have also obtained the stability boundaries for non-axisymmetric disturbances for R ≼ 6000 for η = 0.95 and 0.77 with μ = 0. These last results are found to be in substantial agreement with the experimental observations of Snyder (1962, 1965), Nagib (1972) and Mavec (1973) in the low and moderate axial Reynolds number régimes.

Non-parallel flow effects on the stability of film flow down a right circular cone

1980 ◽  
Vol 96 (3) ◽  
pp. 585-601 ◽  
Author(s):  
Richard L. Zollars ◽  
William B. Krantz
Keyword(s):  
Asymptotic Solution ◽  
Wave Speed ◽  
Film Flow ◽  
Relative Increase ◽  
Parallel Flow ◽  
Circular Cone ◽  
Viscous Forces ◽  
Flow Effects ◽  
The Stability ◽  
Axisymmetric Disturbances

A global asymptotic solution for the linear stability of this flow with respect to axisymmetric disturbances is developed without invoking the usual quasi-parallel flow assumption. This solution predicts the occurrence of quasi-periodic spatially amplified disturbances relatively near the apex, which ultimately are stabilized further down the cone by the relative increase in viscous forces associated with the progressive thinning of the film. The wave speed and wavelength of these disturbances are found to decrease with increasing distance from the apex.

Stability of fluid flow in a flexible tube to non-axisymmetric disturbances

2000 ◽  
Vol 407 ◽  
pp. 291-314 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Phase Velocity ◽  
Poiseuille Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Flexible Tube ◽  
Moderate Reynolds Number ◽  
The Stability ◽  
Axisymmetric Disturbances

The stability of fluid flow in a flexible tube to non-axisymmetric perturbations is analysed in this paper. In the first part of the paper, the equivalents of classical theorems of hydrodynamic stability are derived for inviscid flow in a flexible tube subjected to arbitrary non-axisymmetric disturbances. Perturbations of the form vi = v˜i exp [ik(x − ct) + inθ] are imposed on a steady axisymmetric mean flow U(r) in a flexible tube, and the stability of mean flow velocity profiles and bounds for the phase velocity of the unstable modes are determined for arbitrary values of azimuthal wavenumber n. Here r, θ and x are respectively the radial, azimuthal and axial coordinates, and k and c are the axial wavenumber and phase velocity of disturbances. The flexible wall is represented by a standard constitutive relation which contains inertial, elastic and dissipative terms. The general results indicate that the fluid flow in a flexible tube is stable in the inviscid limit if the quantity Ud[Gscr ]/dr [ges ] 0, and could be unstable for Ud[Gscr ]/dr < 0, where [Gscr ] ≡ rU′/(n2 + k2r2). For the case of Hagen–Poiseuille flow, the general result implies that the flow is stable to axisymmetric disturbances (n = 0), but could be unstable to non-axisymmetric disturbances with any non-zero azimuthal wavenumber (n ≠ 0). This is in marked contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones (Squire theorem). Some new bounds are derived which place restrictions on the real and imaginary parts of the phase velocity for arbitrary non-axisymmetric disturbances.In the second part of this paper, the stability of the Hagen–Poiseuille flow in a flexible tube to non-axisymmetric disturbances is analysed in the high Reynolds number regime. An asymptotic analysis reveals that the Hagen–Poiseuille flow in a flexible tube is unstable to non-axisymmetric disturbances even in the inviscid limit, and this agrees with the general results derived in this paper. The asymptotic results are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the critical Reynolds number obtained for inviscid instability to non-axisymmetric disturbances is much lower than the critical Reynolds numbers obtained in the previous studies for viscous instability to axisymmetric disturbances when the dimensionless parameter Σ = ρGR2/η2 is large. Here G is the shear modulus of the elastic medium, ρ is the density of the fluid, R is the radius of the tube and η is the viscosity of the fluid. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

On the stability of flow in an elliptic pipe which is nearly circular

1978 ◽  
Vol 87 (2) ◽  
pp. 233-241 ◽  
Author(s):  
A. Davey
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Critical Reynolds Number ◽  
Major Axis ◽  
Reynolds Numbers ◽  
Minor Axis ◽  
Critical Reynolds Numbers ◽  
The Stability

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

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