Application of Transient Growth Theory to Bypass Transition

The early nonlinear transition process initiated by a small-amplitude pair of oblique waves is studied using both temporal numerical simulation and theoretical considerations. This investigation is performed under the flow conditions of the experiments by Corke et al. (AIAA J., vol. 40, 2002, pp. 1015–1018) who investigated a sharp $7^{\circ }$ cone in the NASA Mach 3.5 Quiet Tunnel. In particular, both the linear and the nonlinear mechanisms prior to transition onset are investigated in great detail as the physics of this regime predetermine the flow topology of the nonlinear transition zone. The objective of this study is (i) to advance the understanding of the underlying physical mechanisms relevant for the early nonlinear transition regime of oblique breakdown and (ii) to make the connection to oblique transition, the incompressible scenario for bypass transition investigated by Schmid & Henningson (Phys. Fluids A, vol. 4, 1992, pp. 1986–1989). The dominance of the longitudinal vortex mode in oblique breakdown is shown to be a consequence of a constantly forced transient growth instability. In particular, the primary pair of oblique waves serves as an ‘actuator’ that is permanently introducing disturbances into the longitudinal mode where these disturbances exhibit transient growth. The effect of the transient growth instability on the longitudinal mode is to raise its amplitude rather than change the growth rate of this mode.

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Experiments in Transient Growth and Roughness-Induced Bypass Transition

10.21236/ada433231 ◽

2005 ◽

Cited By ~ 1

Author(s):

Edward B. White

Keyword(s):

Bypass Transition ◽

Transient Growth

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Convective instability in steady stenotic flow: optimal transient growth and experimental observation

Journal of Fluid Mechanics ◽

10.1017/s0022112010001229 ◽

2010 ◽

Vol 655 ◽

pp. 504-514 ◽

Cited By ~ 15

Author(s):

M. D. GRIFFITH ◽

M. C. THOMPSON ◽

T. LEWEKE ◽

K. HOURIGAN

Keyword(s):

Experimental Observation ◽

Convective Instability ◽

Growth Analysis ◽

Optimal Growth ◽

Reynolds Numbers ◽

Bypass Transition ◽

Transient Growth ◽

Shear Layer Instability ◽

Chaotic Region ◽

Flow Through

An optimal transient growth analysis is compared with experimental observation for the steady flow through an abrupt, axisymmetric stenosis of varying stenosis degree. Across the stenosis range, a localized sinuous convective shear-layer instability type is predicted to dominate. A comparison of the shape and development of the optimal modes is made with experimental dye visualizations. The presence of the same sinuous-type disturbance immediately upstream of the highly chaotic region observed in the experimental flow is consistent with the optimal growth predictions. This, together with the fact that the flow is unstable globally only at much higher Reynolds numbers, suggests bypass transition.

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On hydrodynamic shear turbulence in Keplerian disks: Via transient growth to bypass transition

Astronomy and Astrophysics ◽

10.1051/0004-6361:20030269 ◽

2003 ◽

Vol 402 (2) ◽

pp. 401-407 ◽

Cited By ~ 70

Author(s):

G. D. Chagelishvili ◽

J.-P. Zahn ◽

A. G. Tevzadze ◽

J. G. Lominadze

Keyword(s):

Bypass Transition ◽

Transient Growth ◽

Hydrodynamic Shear ◽

Shear Turbulence

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Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition

Journal of Fluid Mechanics ◽

10.1017/s0022112010004453 ◽

2010 ◽

Vol 667 ◽

pp. 272-308 ◽

Cited By ~ 122

Author(s):

MATTHEW P. JUNIPER

Keyword(s):

Periodic Solution ◽

Simple Model ◽

Initial Energy ◽

Bypass Transition ◽

Transient Growth ◽

Governing Equations ◽

Initial State ◽

Rijke Tube ◽

Sustained Oscillations ◽

Sequence Of Events

With a sufficiently large impulse, a thermoacoustic system can reach self-sustained oscillations even when it is linearly stable, a process known as triggering. In this paper, a procedure is developed to find the lowest initial energy that can trigger self-sustained oscillations, as well as the corresponding initial state. This is known as the ‘most dangerous’ initial state. The procedure is based on adjoint looping of the nonlinear governing equations, combined with an optimization routine. It is developed for a simple model of a thermoacoustic system, the horizontal Rijke tube, and can be extended to more sophisticated thermoacoustic models. It is observed that the most dangerous initial state grows transiently towards an unstable periodic solution before growing to a stable periodic solution. The initial energy required to trigger these self-sustained oscillations is much lower than the energy of the oscillations themselves and slightly lower than the lowest energy on the unstable periodic solution. It is shown that this transient growth arises due to non-normality of the governing equations. This is analogous to the sequence of events observed in bypass transition to turbulence in fluid mechanical systems and has the same underlying cause. The most dangerous initial state is calculated as a function of the heat-release parameter. It is found that self-sustained oscillations can be reached over approximately half the linearly stable domain. Transient growth in real thermoacoustic systems is 105–106 times greater than that in this simple model. One practical conclusion is that, even in the linearly stable regime, it may take very little initial energy for a real thermoacoustic system to trigger to high-amplitude self-sustained oscillations through the mechanism described in this paper.

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