scholarly journals Instabilities of the flow around a cylinder and emission of vortex dipoles

2013 ◽  
Vol 730 ◽  
pp. 419-441 ◽  
Author(s):  
Ziv Kizner ◽  
Viacheslav Makarov ◽  
Leon Kamp ◽  
GertJan van Heijst
Keyword(s):  
Linear Stability ◽  
Linear Instability ◽  
Instability Region ◽  
Base Flow ◽  
Two Dimensional ◽  
Azimuthal Mode ◽  
Dimensional Parameter ◽  
Contour Dynamics ◽  
Instability Regions ◽  
Dynamics Simulations

AbstractInstabilities and long-term evolution of two-dimensional circular flows around a rigid circular cylinder (island) are studied analytically and numerically. For that we consider a base flow consisting of two concentric neighbouring rings of uniform but different vorticity, with the inner ring touching the cylinder. We first study the inviscid linear stability of such flows to perturbations of the free edges of the rings. For a given ratio of the vorticity in the rings, the governing parameters of the problem are the radii of the inner and outer rings scaled on the cylinder radius. In this two-dimensional parameter space, we determine analytically the regions of linear stability/instability of each azimuthal mode $m= 1, 2, \ldots . $ In the physically most meaningful case of zero net circulation, for each mode $m\gt 1$, two regions are identified: a regular instability region where mode $m$ is unstable along with some other modes, and a unique instability region where only mode $m$ is unstable. After the conditions of linear instability are established, inviscid contour-dynamics and high-Reynolds-number finite-element simulations are conducted. In the regular instability regions, simulations of both kinds typically result in the formation of vortical dipoles or multipoles. In the unique instability regions, where the inner vorticity ring is much thinner than the outer ring, the inviscid contour-dynamics simulations do not reveal dipole emission. In the viscous simulation, because viscosity has time to widen the inner ring, the instability develops in the same manner as in the regular instability regions.

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.

scholarly journals On the stability of plane Couette–Poiseuille flow with uniform crossflow

2010 ◽  
Vol 656 ◽  
pp. 417-447 ◽  
Author(s):  
ANIRBAN GUHA ◽  
IAN A. FRIGAARD
Keyword(s):  
Linear Stability ◽  
Poiseuille Flow ◽  
Energy Production ◽  
Base Flow ◽  
Two Dimensional ◽  
Wall Velocity ◽  
Resonant Interactions ◽  
Flow Stabilization ◽  
Tollmien Schlichting ◽  
The Stability

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

scholarly journals Parabolized stability analysis of jets from serrated nozzles

2016 ◽  
Vol 789 ◽  
pp. 36-63 ◽  
Author(s):  
Aniruddha Sinha ◽  
Kristján Gudmundsson ◽  
Hao Xia ◽  
Tim Colonius
Keyword(s):  
Linear Stability ◽  
Near Field ◽  
Pressure Fluctuations ◽  
Three Dimensional ◽  
Hydrodynamic Pressure ◽  
Linear Instability ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Instability Wave ◽  
Mean Flow

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

Stability of the flow past a sphere

1990 ◽  
Vol 211 ◽  
pp. 73-93 ◽  
Author(s):  
Inchul Kim ◽  
Arne J. Pearlstein
Keyword(s):  
Linear Stability ◽  
Axisymmetric Flow ◽  
Base Flow ◽  
Dimensionless Frequency ◽  
Azimuthal Mode ◽  
Spectral Technique ◽  
Flow Past ◽  
Axisymmetric Vortex ◽  
Axisymmetric Disturbances ◽  
Flow Past A Sphere

Experiment shows that the steady axisymmetric flow past a sphere becomes unstable in the range 120 < Re < 300. The resulting time-dependent non-axi-symmetric flow gives rise to non-axisymmetric vortex shedding at higher Reynolds numbers. The present work reports a computational investigation of the linear stability of the steady axisymmetric base flow. We use a spectral technique to represent the base flow. We then perform a linear stability analysis with respect to axisymmetric and non-axisymmetric disturbances. A spectral technique similar to that employed in the base-flow calculation is used to solve the linear-disturbance equations in stream-function form (for axisymmetric disturbances), and in a modified primitive variables form (for non-axisymmetric disturbances). The analysis shows that the axisymmetric base flow undergoes a Hopf bifurcation at Re = 175.1, with the critical disturbance having azimuthal wavenumber m = 1, and dimensionless frequency (non-dimensionalized as a Strouhal number, St) 0.0955. The critical Re, azimuthal mode number, and St are favourably compared to previous experimental work.

Stability of a temporally evolving natural convection boundary layer on an isothermal wall

2019 ◽  
Vol 877 ◽  
pp. 1163-1185 ◽  
Author(s):  
Junhao Ke ◽  
N. Williamson ◽  
S. W. Armfield ◽  
G. D. McBain ◽  
S. E. Norris
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Linear Stability ◽  
Transition Point ◽  
Three Dimensional ◽  
Base Flow ◽  
Convection Boundary Layer ◽  
Two Dimensional ◽  
Integral Boundary ◽  
Convection Boundary

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.

Compressibility effects on the structural evolution of transitional high-speed planar wakes

2016 ◽  
Vol 796 ◽  
pp. 5-39 ◽  
Author(s):  
Jean-Pierre Hickey ◽  
Fazle Hussain ◽  
Xiaohua Wu
Keyword(s):  
Mach Number ◽  
Linear Stability ◽  
High Speed ◽  
Three Dimensional ◽  
Structural Evolution ◽  
Base Flow ◽  
Length Scale ◽  
Two Dimensional ◽  
Linear Mode ◽  
Compressibility Effects

The compressibility effects on the structural evolution of the transitional high-speed planar wake are studied. The relative Mach number ($Ma_{r}$) of the laminar base flow modifies two fundamental features of planar wake transition: (i) the characteristic length scale defined by the most unstable linear mode; and (ii) the domain of influence of the structures within the staggered two-dimensional vortex array. Linear stability results reveal a reduced growth (approximately 30 % reduction up to $Ma_{r}=2.0$) and a quasilinear increase of the wavelength of the most unstable, two-dimensional instability mode (approximately 20 % longer over the same $Ma_{r}$ range) with increasing $Ma$. The primary wavelength defines the length scale imposed on the emerging transitional structures; naturally, a longer wavelength results in rollers with a greater streamwise separation and hence also larger circulation. A reduction of the growth rate and an increase of the principal wavelength results in a greater ellipticity of the emerging rollers. Compressibility effects also modify the domain of influence of the transitional structures through an increased cross-wake and inhibited streamwise communication as characteristic paths between rollers are deflected due to local $Ma$ gradients. The reduced streamwise domain of influence impedes roller pairing and, for a sufficiently large relative $Ma$, pairing is completely suppressed. Thus, we observe an increased two-dimensionality with increasing Mach number: directly contrasting the increasing three-dimensional effects in high-speed mixing layers. Temporally evolving direct numerical simulations conducted at $Ma=0.8$ and 2.0, for Reynolds numbers up to 3000, support the physical insight gained from linear stability and vortex dynamics studies.

scholarly journals A Discrete Linear Stability Analysis Framework for Two-Dimensional Laminar Flows

2021 ◽  
Vol 16 ◽  
pp. 34-42
Author(s):  
Nitin Kumar ◽  
Sunil Chamoli ◽  
Sachin Tejyan ◽  
Pawan Kumar Pant
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Laminar Flows ◽  
Base Flow ◽  
Two Dimensional ◽  
Analysis Framework ◽  
Square Cylinder

A discrete linear stability analysis framework for two-dimensional laminar flows is presented. Using two case studies involving analysis of thermal and laminar flows, the stability of flows in the discrete numerical sense is addressed. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finitedifference method in two-dimensions and are subsequently written in the form of perturbed equations with twodimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 and the critical Rayleigh number is 6 1.88×10 for the square cylinder and double-glazing problem, respectively, The results are consistent with the previous numerical and experimental observations.

scholarly journals A Discrete Linear Stability Analysis of Two-dimensional Laminar Flow Past a Square Cylinder

2021 ◽  
Vol 16 ◽  
pp. 109-119
Author(s):  
Nitin Kumar ◽  
Sachin Tejyan ◽  
Sunil Chamoli ◽  
Pawan Kumar Pant
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Two Dimensions ◽  
Base Flow ◽  
Mode Analysis ◽  
Two Dimensional ◽  
Square Cylinder ◽  
Flow Past

The present study focuses on the development of a numerical framework for predicting the onset of vortex sheading due to flow past a square cylinder. For this a discrete linear stability analysis framework for two-dimensional laminar flows have used. Initially the frame work is validating by using the analysis of thermal stability of flows in the discrete numerical sense. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finite-difference method in two-dimensions and are subsequently written in the form of perturbed equations with two-dimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 for the flow past a square cylinder. The results are consistent with the previous numerical and experimental observations.

scholarly journals Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid–liquid flows

2014 ◽  
Vol 750 ◽  
pp. 464-506 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Prashant Valluri ◽  
David M. Scott ◽  
Iain Bethune ◽  
Peter D. M. Spelt
Keyword(s):  
Linear Theory ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Base Flow ◽  
Navier Stokes ◽  
Late Time ◽  
Two Dimensional ◽  
Two Phase ◽  
Weakly Nonlinear

AbstractWe consider the linear and nonlinear stability of two-phase density-matched but viscosity-contrasted fluids subject to laminar Poiseuille flow in a channel, paying particular attention to the formation of three-dimensional waves. A combination of Orr–Sommerfeld–Squire analysis (both modal and non-modal) with direct numerical simulation of the three-dimensional two-phase Navier–Stokes equations is used. For the parameter regimes under consideration, under linear theory, the most unstable waves are two-dimensional. Nevertheless, we demonstrate several mechanisms whereby three-dimensional waves enter the system, and dominate at late time. There exists a direct route, whereby three-dimensional waves are amplified by the standard linear mechanism; for certain parameter classes, such waves grow at a rate less than but comparable to that of the most dangerous two-dimensional mode. Additionally, there is a weakly nonlinear route, whereby a purely spanwise wave grows according to transient linear theory and subsequently couples to a streamwise mode in weakly nonlinear fashion. Consideration is also given to the ultimate state of these waves: persistent three-dimensional nonlinear waves are stretched and distorted by the base flow, thereby producing regimes of ligaments, ‘sheets’ or ‘interfacial turbulence’. Depending on the parameter regime, these regimes are observed either in isolation, or acting together.

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