Boussinesq global modes and stability sensitivity, with applications to stratified wakes

2017 ◽  
Vol 812 ◽  
pp. 1146-1188
Author(s):  
Kevin K. Chen ◽  
Geoffrey R. Spedding
Keyword(s):  
Bluff Body ◽  
Boussinesq Equations ◽  
Base Flow ◽  
Mode Analysis ◽  
Flow Density ◽  
Global Mode ◽  
Multiple Production ◽  
Small Density ◽  
Global Modes ◽  
The Stability

For the Boussinesq equations, we present a theory of linear stability sensitivity to base flow density and velocity modifications. Given a steady-state flow with small density variations, the sensitivity of the stability eigenvalues is computed from the direct and adjoint global modes of the linearised Boussinesq equations, similarly to Marquetet al.(J. Fluid Mech., vol. 615, 2008, pp. 221–252). Combinations of the density and velocity components of these modes reveal multiple production and transport mechanisms that contribute to the eigenvalue sensitivity. We present an application of the sensitivity theory to a stably linearly density-stratified flow around a thin plate at a$90^{\circ }$angle of attack, a Reynolds number of 30 and Froude numbers of 1, 8 and$\infty$. The global mode analysis reveals lightly damped undulations pervading through the entire domain, which are predicted by both inviscid uniform base flow theory and Orr–Sommerfeld theory. The sensitivity to base flow velocity modifications is primarily concentrated just downstream of the bluff body. On the other hand, the sensitivity to base flow density modifications is concentrated in regions both immediately upstream and immediately downstream of the plate. Both sensitivities have a greater upstream presence for lower Froude numbers.

scholarly journals Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities

2009 ◽  
Vol 622 ◽  
pp. 1-21 ◽  
Author(s):  
OLIVIER MARQUET ◽  
MATTEO LOMBARDI ◽  
JEAN-MARC CHOMAZ ◽  
DENIS SIPP ◽  
LAURENT JACQUIN
Keyword(s):  
High Speed ◽  
Passive Control ◽  
Three Dimensional ◽  
Adjoint Problem ◽  
Base Flow ◽  
Mode Analysis ◽  
Global Mode ◽  
Recirculation Bubble ◽  
Global Modes ◽  
The Stability

The stability of the recirculation bubble behind a smoothed backward-facing step is numerically computed. Destabilization occurs first through a stationary three-dimensional mode. Analysis of the direct global mode shows that the instability corresponds to a deformation of the recirculation bubble in which streamwise vortices induce low- and high-speed streaks as in the classical lift-up mechanism. Formulation of the adjoint problem and computation of the adjoint global mode show that both the lift-up mechanism associated with the transport of the base flow by the perturbation and the convective non-normality associated with the transport of the perturbation by the base flow explain the properties of the flow. The lift-up non-normality differentiates the direct and adjoint modes by their component: the direct is dominated by the streamwise component and the adjoint by the cross-stream component. The convective non-normality results in a different localization of the direct and adjoint global modes, respectively downstream and upstream. The implications of these properties for the control problem are considered. Passive control, to be most efficient, should modify the flow inside the recirculation bubble where direct and adjoint global modes overlap, whereas active control, by for example blowing and suction at the wall, should be placed just upstream of the separation point where the pressure of the adjoint global mode is maximum.

scholarly journals Global linear stability of the non-parallel Batchelor vortex

2009 ◽  
Vol 629 ◽  
pp. 139-160 ◽  
Author(s):  
C. J. HEATON ◽  
J. W. NICHOLS ◽  
P. J. SCHMID
Keyword(s):  
Numerical Simulation ◽  
Linear Stability ◽  
Stokes Equations ◽  
General Structure ◽  
Base Flow ◽  
Illustrative Case ◽  
Arnoldi Method ◽  
Unstable Flow ◽  
Global Mode ◽  
Global Modes

Linear stability of the non-parallel Batchelor vortex is studied using global modes. This family of swirling wakes and jets has been extensively studied under the parallel-flow approximation, and in this paper we extend to more realistic non-parallel base flows. Our base flow is obtained as an exact steady solution of the Navier–Stokes equations by direct numerical simulation (with imposed axisymmetry to damp all instabilities). Global stability modes are computed by numerical simulation of the linearized equations, using the implicitly restarted Arnoldi method, and we discuss fully the numerical and convergence issues encountered. Emphasis is placed on exploring the general structure of the global spectrum, and in particular the correspondence between global modes and local absolute modes which is anticipated by weakly non-parallel asymptotic theory. We believe that our computed global modes for a weakly non-parallel vortex are the first to display this correspondence with local absolute modes. Superpositions of global modes are also studied, allowing an investigation of the amplifier dynamics of this unstable flow. For an illustrative case we find global non-modal transient growth via a convective mechanism. Generally amplifier dynamics, via convective growth, are prevalent over short time intervals, and resonator dynamics, via global mode growth, become prevalent at later times.

scholarly journals Local stability analysis and eigenvalue sensitivity of reacting bluff-body wakes

2016 ◽  
Vol 788 ◽  
pp. 549-575 ◽  
Author(s):  
Benjamin Emerson ◽  
Tim Lieuwen ◽  
Matthew P. Juniper
Keyword(s):  
Stability Analysis ◽  
Acoustic Waves ◽  
Local Stability ◽  
Bluff Body ◽  
Density Ratio ◽  
Global Mode ◽  
Eigenvalue Sensitivity ◽  
Local Stability Analysis ◽  
Measured Time ◽  
The Stability

This paper presents an experimental and theoretical investigation of high-Reynolds-number low-density reacting wakes near a hydrodynamic Hopf bifurcation. This configuration is applicable to the wake flows that are commonly used to stabilize flames in high-velocity flows. First, an experimental study is conducted to measure the limit-cycle oscillation of this reacting bluff-body wake. The experiment is repeated while independently varying the bluff-body lip velocity and the density ratio across the flame. In all cases, the wake exhibits a sinuous oscillation. Linear stability analysis is performed on the measured time-averaged velocity and density fields. In the first stage of this analysis, a local spatiotemporal stability analysis is performed on the measured time-averaged velocity and density fields. The stability analysis results are compared to the experimental measurement and demonstrate that the local stability analysis correctly captures the influence of the lip-velocity and density-ratio parameters on the sinuous mode. In the second stage of the analysis, the linear direct and adjoint global modes are estimated by combining the local results. The sensitivity of the eigenvalue to changes in intrinsic feedback mechanisms is found by combining the direct and adjoint global modes. This is referred to as the eigenvalue sensitivity throughout the paper for reasons of brevity. The predicted global mode frequency is consistently within 10 % of the measured value, and the linear global mode shape closely resembles the measured nonlinear oscillations. The adjoint global mode reveals that the oscillation is strongly sensitive to open-loop forcing in the shear layers. The eigenvalue sensitivity identifies a wavemaker in the recirculation zone of the wake. A parametric study shows that these regions change little when the density ratio and lip velocity change. In the third stage of the analysis, the stability analysis is repeated for the varicose hydrodynamic mode. Although not physically observed in this unforced flow, the varicose mode can lock into longitudinal acoustic waves and cause thermoacoustic oscillations to occur. The paper shows that the local stability analysis successfully predicts the global hydrodynamic stability characteristics of this flow and shows that experimental data can be post-processed with this method in order to identify the wavemaker regions and the regions that are most sensitive to external forcing, for example from acoustic waves.

scholarly journals Global mode analysis of axisymmetric bluff-body wakes: Stabilization by base bleed

2009 ◽  
Vol 21 (11) ◽  
pp. 114102 ◽  
Author(s):  
E. Sanmiguel-Rojas ◽  
A. Sevilla ◽  
C. Martínez-Bazán ◽  
J.-M. Chomaz
Keyword(s):  
Bluff Body ◽  
Mode Analysis ◽  
Global Mode ◽  
Base Bleed

Nonlinear global modes in inhomogeneous mixed convection flows in porous media

2008 ◽  
Vol 595 ◽  
pp. 367-377 ◽  
Author(s):  
M. N. OUARZAZI ◽  
F. MEJNI ◽  
A. DELACHE ◽  
G. LABROSSE
Keyword(s):  
Porous Media ◽  
Mixed Convection ◽  
Scaling Laws ◽  
Axial Flow ◽  
Base Flow ◽  
Global Mode ◽  
First Case ◽  
Inhomogeneous Temperature ◽  
Global Modes ◽  
Convection In Porous Media

The aim of this work is to investigate the fully nonlinear dynamics of mixed convection in porous media heated non-uniformly from below and through which an axial flow is maintained. Depending on the choice of the imposed inhomogeneous temperature profile, two cases prove to be of interest: the base flow displays an absolute instability region either detached from the inlet or attached to it. Results from a combined direct numerical simulations and linear stability approach have revealed that in the first case, the nonlinear solution is a steep nonlinear global mode, with a sharp stationary front located at a marginally absolutely unstable station. In the second configuration, the scaling laws for the establishment of a nonlinear global mode quenched by the inlet are found to agree perfectly with the theory. It is also found that in both configurations, the global frequency of synchronized oscillations corresponds to the local absolute frequency determined by linear criterion, even far from the threshold of global instability.

scholarly journals Transition to turbulence through steep global-modes cascade in an open rotating cavity

2011 ◽  
Vol 688 ◽  
pp. 493-506 ◽  
Author(s):  
Bertrand Viaud ◽  
Eric Serre ◽  
Jean-Marc Chomaz
Keyword(s):  
Threshold Value ◽  
Secondary Instability ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Small Scale ◽  
Global Mode ◽  
Rotating Boundary ◽  
The Stability ◽  
Real Flow ◽  
Secondary Instabilities

AbstractThe transition to turbulence in a rotating boundary layer is analysed via direct numerical simulation (DNS) in an annular cavity made of two parallel corotating discs of finite radial extent, with a forced inflow at the hub and free outflow at the rim. In a former numerical investigation (Viaud, Serre & Chomaz J. Fluid Mech., vol. 598, 2008, pp. 451–464) realized in a sectorial cavity of azimuthal extent $2\lrm{\pi} / 68$, we have established the existence of a primary bifurcation to nonlinear global mode with angular phase velocity and radial envelope coherent with the so-called elephant mode theory. The former study has demonstrated the subcritical nature of this primary bifurcation with a base flow that keeps being linearly stable for all Reynolds numbers studied. The present work investigates the stability of this elephant mode by extending the cavity both in the radial and azimuthal direction. When the Reynolds number based on the forced throughflow is increased above a threshold value for the existence of the nonlinear global mode, a large-amplitude impulsive perturbation gives rise to a self-sustained saturated wave with characteristics identical to the 68-fold global elephant mode obtained in the smaller cavity. This saturated wave is itself globally unstable and a second front appears in the lee of the primary where small-scale instability develops. These secondary instabilities are identical for the $2\lrm{\pi} / 68$ and the $2\lrm{\pi} / 4$ long sectorial cavities, indicating that transition involves a Floquet mode of zero azimuthal wavenumber. This secondary instability leads to a very disorganized state, defining the transition to turbulence. The observed transition to turbulence linked to the secondary instability of a global mode confirms, for the first time on a real flow, the possibility of a direct transition to turbulence through an elephant mode cascade, a scenario that was up to now only observed on the Ginzburg–Landau model.

Global modes and transient response of a cold supersonic jet

2011 ◽  
Vol 669 ◽  
pp. 225-241 ◽  
Author(s):  
JOSEPH W. NICHOLS ◽  
SANJIVA K. LELE
Keyword(s):  
Acoustic Radiation ◽  
Near Field ◽  
Supersonic Jet ◽  
Mode Analysis ◽  
Wave Radiation ◽  
Far Field ◽  
Transient Growth ◽  
Global Mode ◽  
Laminar Jet ◽  
Global Modes

Global-mode analysis is applied to a cold, M = 2.5 laminar jet. Global modes of the non-parallel jet capture directly both near-field dynamics and far-field acoustics which, in this case, are coupled by Mach wave radiation. In addition to type (a) modes corresponding to Kelvin–Helmholtz instability, it is found that the jet also supports upstream-propagating type (b) modes which could not be resolved by previous analyses of the parabolized stability equations. The locally neutrally propagating part of a type (a) mode consists of the growth and decay of an aerodynamic wavepacket attached to the jet, coupled with a beam of acoustic radiation at a low angle to the jet downstream axis. Type (b) modes are shown to be related to the subsonic family of modes predicted by Tam & Hu (1989). Finally, significant transient growth is recovered by superposing damped, but non-normal, global modes, leading to a novel interpretation of jet noise production. The mechanism of optimal transient growth is identified with a propagating aerodynamic wavepacket which emits an acoustic wavepacket to the far field at an axial location consistent with the peaks of the locally neutrally propagating parts of type (a) modes.

Sensitivity analysis and passive control of cylinder flow

2008 ◽  
Vol 615 ◽  
pp. 221-252 ◽  
Author(s):  
OLIVIER MARQUET ◽  
DENIS SIPP ◽  
LAURENT JACQUIN
Keyword(s):  
Sensitivity Analysis ◽  
Vortex Shedding ◽  
Passive Control ◽  
Control Device ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Sensitivity Analyses ◽  
Global Modes ◽  
The Stability ◽  
Steady Force

A general theoretical formalism is developed to assess how base-flow modifications may alter the stability properties of flows studied in a global approach of linear stability theory. It also comprises a systematic approach to the passive control of globally unstable flows by the use of small control devices. This formalism is based on a sensitivity analysis of any global eigenvalue to base-flow modifications. The base-flow modifications investigated are either arbitrary or specific ones induced by a steady force. This leads to a definition of the so-called sensitivity to base-flow modifications and sensitivity to a steady force. These sensitivity analyses are applied to the unstable global modes responsible for the onset of vortex shedding in the wake of a cylinder for Reynolds numbers in the range 47≤Re≤80. First, it is demonstrated how the sensitivity to arbitrary base-flow modifications may be used to identify regions and properties of the base flow that contribute to the onset of vortex shedding. Secondly, the sensitivity to a steady force determines the regions of the flow where a steady force acting on the base flow stabilizes the unstable global modes. Upon modelling the presence of a control device by a steady force acting on the base flow, these predictions are then extensively compared with the experimental results of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, p. 71). A physical interpretation of the suppression of vortex shedding by use of a control cylinder is proposed in the light of the sensitivity analysis.

Self-destabilising loop of a low-speed water jet emanating from an orifice in microgravity

2016 ◽  
Vol 797 ◽  
pp. 146-180 ◽  
Author(s):  
Akira Umemura
Keyword(s):  
Mode Analysis ◽  
Unstable Wave ◽  
Complex Conjugate ◽  
Transfer Energy ◽  
Global Mode ◽  
Weakly Nonlinear ◽  
Low Speed ◽  
Nonlinear Growth ◽  
Breakup Length ◽  
The Stability

A one-dimensional global mode analysis is conducted for low-speed water jets emanating from a circular orifice in microgravity, in which the observed spontaneous convective instability causes almost periodic jet disintegrations at a fixed location for each jet-issue speed that exceeds a certain threshold. The inviscid spatial linear stability analysis identifies four wave modes excitable at the frequency: the Plateau–Rayleigh (PR) unstable wave, its complex conjugate and two neutral waves which may transfer energy upstream. Their linear combination satisfying the orifice exit condition may describe the synchronised reproduction of a PR unstable wave from each neutral wave at the orifice exit. On the other hand, a weakly nonlinear analysis shows that the growth of the nonlinear PR unstable wave produces the two neutral waves near the orifice. Thus, the same PR unstable wave can be reproduced on a newly issued liquid surface owing to the neutral waves produced by its own nonlinear growth. This self-destabilising loop, dominantly operating for the most unstable PR wave, determines the initial PR unstable wave amplitude and, consequently, the breakup length as a function of jet-issue speed. The predicted initial amplitude of the PR unstable wave is in reasonably good agreement with the value calculated from the average breakup length measured in our microgravity experiments. It is found that that the loop consists mainly of the downstream- and upstream-moving neutral waves at relatively high and low jet speeds, respectively. The stability of the self-destabilising loop is also discussed.

Stability of Viscoelastic Channel Flow

2007 ◽  
Author(s):  
Nariman Ashrafi
Keyword(s):  
Reynolds Number ◽  
Viscosity Ratio ◽  
Weissenberg Number ◽  
Base Flow ◽  
Galerkin Projection ◽  
Large Reynolds Number ◽  
One Dimensional ◽  
Stress Effects ◽  
The Stability ◽  
The One

The nonlinear stability and bifurcation of the one-dimensional channel (Poiseuille) flow is examined for a Johnson-Segalman fluid. The velocity and stress are represented by orthonormal functions in the transverse direction to the flow. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. The stability picture is dramatically influenced by the viscosity ratio. The range of shear rate or Weissenberg number for which the base flow is unstable increases from zero as the fluid deviates from the Newtonian limit as decreases. Typically, two turning points are observed near the critical Weissenberg numbers. The transient response is heavily influenced by the level of inertia. It is found that the flow responds oscillatorily. When the Reynolds number is small, and monotonically at large Reynolds number when elastic effects are dominated by inertia.

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