Transient growth associated with continuous spectra of the Batchelor vortex

2012 ◽  
Vol 697 ◽  
pp. 35-59 ◽  
Author(s):  
X. Mao ◽  
S. J. Sherwin
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Vortex Core ◽  
Azimuthal Angle ◽  
Original Position ◽  
Local Analysis ◽  
Transient Growth ◽  
Transient Effects ◽  
Potential Region ◽  
Optimal Perturbation

AbstractThe spectrum of the Batchelor vortex can be broadly split into a discrete spectrum, a potential spectrum and a free-stream spectrum where, since the last two spectra are both continuous, they can also be considered as one continuous spectrum. The discrete spectrum has been extensively studied but the continuous spectrum has received limited attention in the context of vortex flow. A local transient growth study is conducted and the contribution of the discrete spectrum and the continuous spectrum to the transient growth is separated by constructing optimal perturbations on the discrete or continuous sub-eigenspaces separately. It is found that the significant transient growth is mainly due to the non-normality of the continuous eigenmodes/spectrum whilst the discrete eigenmodes/spectrum have little contribution to the transient energy growth. A matrix-free method, which reduces to the local analysis when appropriate periodic boundary conditions are imposed, is also applied to investigate the transient growth in both a plane of constant azimuthal angle and a plane constant axial location. Previously studies by other authors have demonstrated that at zero azimuthal wavenumber the transient growth reaches infinitely large values over infinite time intervals while the optimal perturbations are located far from the vortex core. Therefore we limited our scope to small values of the time horizon so as to obtain reasonably strong transient effects stemming from physically relevant optimal perturbations. Two mechanisms of transient growth are observed: namely a redistribution of the azimuthal velocity to the azimuthal vorticity and interaction between out-of-vortex-core structures with those within the vortex core. A direct numerical simulation (DNS) of the vortex perturbed by optimal perturbations is conducted to investigate the nonlinear development of the optimal perturbations. In the azimuthally constant decomposed case, it is found that the optimal perturbation induces a string of bubble structures to be generated as a consequence of the non-orthogonality of continuous eigenmodes and the breakdown bubble is induced by viscous diffusion, while in the axially constant decomposition transient growth analysis, it is observed that the optimal perturbations associated with the continuous eigenmodes drive the vortex to vibrate around the initial vortex centre before eventually returning to its original position at larger times. This transient effect provides a mechanism for the ‘vortex meandering’ observed in previous experimental and numerical studies. These optimal perturbations associated with the continuous spectrum with out-of-vortex-core structures are observed to be activated by anisotropic inflow perturbations in the potential region.

Transient growth in strongly stratified shear layers

2014 ◽  
Vol 758 ◽  
Author(s):  
A. K. Kaminski ◽  
C. P. Caulfield ◽  
J. R. Taylor
Keyword(s):  
Shear Layer ◽  
Normal Mode ◽  
Richardson Number ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Buoyancy Frequency ◽  
Transient Growth ◽  
Time Intervals ◽  
Optimal Perturbation ◽  
Target Time

AbstractWe investigate numerically transient linear growth of three-dimensional perturbations in a stratified shear layer to determine which perturbations optimize the growth of the total kinetic and potential energy over a range of finite target time intervals. The stratified shear layer has an initial parallel hyperbolic tangent velocity distribution with Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=U_0 h/\nu =1000$ and Prandtl number $\nu /\kappa =1$, where $\nu $ is the kinematic viscosity of the fluid and $\kappa $ is the diffusivity of the density. The initial stable buoyancy distribution has constant buoyancy frequency $N_0$, and we consider a range of flows with different bulk Richardson number ${\mathit{Ri}}_b=N_0^2h^2/U_0^2$, which also corresponds to the minimum gradient Richardson number ${\mathit{Ri}}_g(z)=N_0^2/(\mathrm{d}U/\mathrm{d} z)^2$ at the midpoint of the shear layer. For short target times, the optimal perturbations are inherently three-dimensional, while for sufficiently long target times and small ${\mathit{Ri}}_b$ the optimal perturbations are closely related to the normal-mode ‘Kelvin–Helmholtz’ (KH) instability, consistent with analogous calculations in an unstratified mixing layer recently reported by Arratia et al. (J. Fluid Mech., vol. 717, 2013, pp. 90–133). However, we demonstrate that non-trivial transient growth occurs even when the Richardson number is sufficiently high to stabilize all normal-mode instabilities, with the optimal perturbation exciting internal waves at some distance from the midpoint of the shear layer.

Continuous spectrum analysis of roughness-induced transient growth

2009 ◽  
Vol 21 (11) ◽  
pp. 114105 ◽  
Author(s):  
N. A. Denissen ◽  
E. B. White
Keyword(s):  
Continuous Spectrum ◽  
Spectrum Analysis ◽  
Transient Growth

The Effect of Non-Normality of Navier-Stokes Operator Mechanism on Dynamics of an Axisymmetric Swirling Flow

2013 ◽  
Vol 681 ◽  
pp. 72-78
Author(s):  
Cheng Chen ◽  
Ya Yong Shi
Keyword(s):  
Swirling Flow ◽  
Initial Perturbation ◽  
Fluid Motion ◽  
Nonlinear Behavior ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Transient Growth ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Oseen Vortex

The effect of non-normality of the Navier-Stokes operator on the dynamics of an axisymmetric swirling flow, namely, the Oseen vortex, has been investigated. The eigenvalue analysis and transient growth analysis have been employed in order to obtain the least stable eigenmode and the global optimal perturbation, which are both considered as the initial perturbation. Three stages of dynamic process have been put into evidence for the evolution of the optimal perturbation. The early (linear) stage is characterized by the amplification of radial perturbation, consistent with the prediction of transient growth theory. Having come into the nonlinear stage, the perturbation energy growth is suppressed by the interaction between the vortex ring and the Oseen vortex core. Finally, the phenomena of secondary energy growth are also observed. Compared with the results obtained by applying the least stable eigenmode as the initial disturbance, the nonlinear behavior of the optimal perturbation features radial fluid motion and the rapid production of small eddies, which are both thought to be beneficial to fluid entrainment or mixing. The effect of perturbation amplitude on the nonlinear evolution of flows is also studied.

scholarly journals Spectral analysis of singular Sturm-Liouville operators on time scales

2018 ◽  
Vol 72 (1) ◽  
pp. 1 ◽  
Author(s):  
Bilender P. Allahverdiev ◽  
Huseyin Tuna
Keyword(s):  
Spectral Analysis ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Time Scales ◽  
Liouville Operator ◽  
Adjoint Operator ◽  
The Self ◽  
Sturm Liouville ◽  
Liouville Operators

In this paper, we consider properties of the spectrum of a Sturm-Liouville<br />operator on time scales. We will prove that the regular symmetric<br />Sturm-Liouville operator is semi-bounded from below. We will also give some<br />conditions for the self-adjoint operator associated with the singular<br />Sturm-Liouville expression to have a discrete spectrum. Finally, we will<br />investigate the continuous spectrum of this operator.

Transient Growth of Three-Dimensional Vortex Perturbations During Near-Parallel Collision With a Circular Cylinder

2006 ◽  
Author(s):  
X. Liu ◽  
J. S. Marshall
Keyword(s):  
Circular Cylinder ◽  
Vortex Core ◽  
Computational Study ◽  
Three Dimensional ◽  
The Body ◽  
Transient Growth ◽  
Two Dimensional ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Flow Features

A computational study is reported that examines the transient growth of three-dimensional flow features for nominally parallel vortex-cylinder interaction problems. We consider a helical vortex with small-amplitude perturbations that is advected onto a circular cylinder whose axis is parallel to the nominal vortex axis. The study assesses the applicability of the two-dimensional flow assumption for parallel vortex-body interaction problems in which the body impinges on the vortex core. The computations are performed using an unstructured finite-volume method for an incompressible flow, with periodic boundary conditions along the cylinder axis. Growth of three-dimensional flow features is quantified by use of a proper-orthogonal decomposition of the Fourier-transformed velocity and vorticity fields in the cylinder azimuthal and axial directions. The interaction is examined for different axial wavelengths and amplitudes of the initial helical waves on the vortex core, and the results for cylinder force are compared to the two-dimensional results. The degree of perturbation amplification as the vortex approaches the cylinder is quantified and shown to be mostly dependent on the dominant axial wavenumber of the perturbation. The perturbation amplification is observed to be greatest for perturbations with axial wavelength of about 1.5 times the cylinder diameter.

On the transition of transient growth mechanism in Taylor–Dean flow

2021 ◽  
pp. 2150185
Author(s):  
Cheng Chen ◽  
Liu Zhang ◽  
Wei Zhang
Keyword(s):  
Pressure Gradient ◽  
Outer Cylinder ◽  
Azimuthal Velocity ◽  
Transient Growth ◽  
Narrow Gap ◽  
Optimal Perturbation ◽  
Dean Flow ◽  
Energy Growth ◽  
Axisymmetric Perturbation ◽  
Wide Gap

We investigate optimal perturbation and its transient growth characteristics in Taylor–Dean flow theoretically. The parameter [Formula: see text], accounting for the ratio of average pumping velocity induced by azimuthal pressure gradient to rotating velocity by rotating cylinders, is varied from −5 to 5. The results show that for the rigid rotation case, the energy growth of optimal perturbation is increased with increasing magnitude of azimuthal pressure gradient. Further, both the main and secondary peak of the amplitude of azimuthal velocity are seen to be shifted towards the outer cylinder for wide gap case, and both are shifted oppositely towards the inner cylinder for narrow gap case. Viewing the time evolution of the energies in the three velocity components for wide gap case, anti-lift-up mechanism replaces lift-up mechanism as the dominant mechanism for energy growth, when [Formula: see text] changes from −5 to 5. While for narrow gap case, lift-up mechanism is always responsible for transient growth of axisymmetric perturbation, no matter how strong azimuthal pressure gradient is considered.

On the competition of anti-lift-up mechanism and Reynolds stress mechanism in Spiral Poiseuille flow

2021 ◽  
Author(s):  
Cheng Chen ◽  
Cheng-Jun He ◽  
Li-Hua Gao
Keyword(s):  
Reynolds Stress ◽  
Poiseuille Flow ◽  
Outer Cylinder ◽  
Radial Velocities ◽  
Wave Number ◽  
Transient Growth ◽  
Optimal Perturbation ◽  
Axisymmetric Case ◽  
Azimuthal Wave Number ◽  
Number Formula

This work is devoted to the studies of optimal perturbation and its transient growth characteristics in Spiral Poiseuille flow (SPF). The Poiseuille number [Formula: see text], representing the dimensionless axial pressure gradient, is varied from 0 to 20,000. The results show that for the axisymmetric case, with the increase of axial shear, the peaks of the amplitudes of azimuthal and radial velocities are both shifted towards the inner cylinder, and a second peak appears near the outer cylinder for both velocity components. Viewing the time evolution of azimuthal shear contribution [Formula: see text] and axial shear contribution [Formula: see text] to the kinetic energy growth of the optimal perturbation, while [Formula: see text] is large enough ([Formula: see text], 20,000), the Reynolds stress mechanism in the meridional plane [Formula: see text] is dominant for the transient growth behavior in SPF relative to anti-lift-up mechanism, which is dominant in the absence of axial flow for co-rotating Taylor–Couette flow with wide gap. For the oblique mode with azimuthal wave number [Formula: see text], which becomes the optimal azimuthal mode over a wide range of azimuthal wave number ([Formula: see text]–10) when [Formula: see text] is large enough, the peaks of the amplitudes of azimuthal and radial velocities are both shifted towards the outer cylinder, opposite to the axisymmetric case.

scholarly journals The Discrete and Continuous Retardation and Relaxation Spectrum Method for Viscoelastic Characterization of Warm Mix Crumb Rubber-Modified Asphalt Mixtures

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3723 ◽  
Author(s):  
Fei Zhang ◽  
Lan Wang ◽  
Chao Li ◽  
Yongming Xing
Keyword(s):  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Creep Compliance ◽  
Relaxation Modulus ◽  
Crumb Rubber ◽  
Asphalt Mixtures ◽  
Time Range ◽  
Modified Asphalt ◽  
Dynamic Modulus Test ◽  
Discrete And Continuous Spectrum

To study the linear viscoelastic (LVE) of crumb rubber-modified asphalt mixtures before and after the warm mix additive was added methods of obtaining the discrete and continuous spectrum are presented. Besides, the relaxation modulus and creep compliance are constructed from the discrete and continuous spectrum, respectively. The discrete spectrum of asphalt mixtures can be obtained from dynamic modulus test results according to the generalized Maxwell model (GMM) and the generalized Kelvin model (GKM). Similarly, the continuous spectrum of asphalt mixtures can be obtained from the dynamic modulus test data via the inverse integral transformation. In this paper, the test procedure for all specimens was ensured to be completed in the LVE range. The results show that the discrete spectrum and the continuous spectrum have similar shapes, but the magnitude and position of the spectrum peaks is different. The continuous spectrum can be considered as the limiting case of the discrete spectrum. The relaxation modulus and creep compliance constructed by the discrete and continuous spectrum are almost indistinguishable in the reduced time range of 10−5 s–103 s. However, there are more significant errors outside the time range, and the maximum error is up to 55%.

Transient growth in vortices with axial flow

2007 ◽  
Vol 587 ◽  
pp. 271-301 ◽  
Author(s):  
C. J. HEATON ◽  
N. PEAKE
Keyword(s):  
Vortex Breakdown ◽  
Resonance Effect ◽  
Axial Flow ◽  
Practical Interest ◽  
Transient Growth ◽  
Transient Effects ◽  
Long Wavelength ◽  
Instability Modes ◽  
Large Growth ◽  
High Reynolds

We investigate transient growth in high-Reynolds-number vortices with axial flow. Manycases of vortex instability are not fully explained by strong exponential instability modes, and transient growth could offer an alternative route to breakdown in such cases. Strong transient growth is found, in agreement with previous studies. We first discuss the problem by reference to ducted vortices which model aeroengine flow. The transient growth is inviscid in character, and in this paper we specifically interpret it as an effect of the inviscid continuous spectrum. The relevant inviscid theory explains new scalings which we find for the transient growth, which are generalizations of the quadratic scaling seen previously in two-dimensional flows and non-swirling pipe flows. We then turn to a second case, of interest for vortex breakdown, the Batchelor vortex, and present calculations of the transient growth. Large growth is possible, especially for the helical modes (with azimuthal wavenumber |m| = 1). The general trends are complicated by a number ofissues, including a long-wavelength effect and a resonance effect, both of which were recently discovered for a vortex without axial flow and are found here to be present in the Batchelor vortex also. Overall, the results suggest that strong transient effects are present in the moderate- to high-swirl regime of practical interest (swirl number q ≳ 2). Foraxisymmetric (m = 0) and higher (|m| > 1) modes, however, transient effects are not found to be significant.

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