Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow

Layered flows that are commonly observed in stratified turbulence are susceptible to the Taylor–Caulfield Instability. While the modal stability properties of layered shear flows have been examined, the non-modal growth of perturbations has not been investigated. In this work, the tools of Generalized Stability Theory are utilized to study linear transient growth within a finite time interval of two-dimensional perturbations in an inviscid, three-layer constant shear flow under the Boussinesq approximation. It is found that, for low optimization times, small-scale perturbations utilize the Orr mechanism and achieve growth equal to that in the case of an unstratified flow. For larger optimization times, transient growth is much larger compared to growth for an unstratified flow as the Kelvin–Orr waves comprising the continuous spectrum of the dynamical operator and the gravity edge-waves comprising the discrete spectrum interact synergistically. Maximum growth is obtained for perturbations with scales within the region of instability, but significant growth is maintained for modally stable perturbations as well. For perturbations with scales within the unstable region, the unstable normal modes are excited at high amplitude by their bi-orthogonals. For perturbations with modally stable scales, the Orr mechanism is utilized to excite at high amplitude neutral propagating waves resembling the neutral Taylor–Caulfield modes.

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

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.

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $$\mathbb {T}\times \mathbb {R}$$ T × R . In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $$L^2$$ L 2 norm grows as $$t^{1/2}$$ t 1 / 2 and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $$\nu ^{-1/6}$$ ν - 1 / 6 (with $$\nu ^{-1}$$ ν - 1 being proportional to the Reynolds number) on a time-scale $$\nu ^{-1/3}$$ ν - 1 / 3 , after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Optimal energy growth in pulsatile channel and pipe flows

Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.

Transient Thermoacoustic Responses of Methane/Hydrogen Flames in a Pressurized Annular Combustor

The present article experimentally investigates the triggering and transient growth of azimuthal instabilities in a pressurized laboratory-scale annular combustor featuring twelve methane/hydrogen flames, as the equivalence ratio is ramped up and down. The ramping rate of equivalence ratio is varied to examine its effect on the transient thermoacoustic response and the driving mechanisms, highlighting a number of previously unseen features. As the equivalence ratio is dynamically increased, all cases were observed to feature a distinct modal trajectory, during the onset of high amplitude instabilities. Strongly spinning counter-clockwise modes are first excited before a dynamic transition to strongly spinning clockwise modes occurs. Furthermore, the strength of the spinning mode (quantified through the spin ratio or nature angle) was shown to feature a local minima before the spinning mode stabilized in the system, which corresponds to an almost pure spinning state. Hysteresis behaviour was observed in both the amplitude and nature of the mode, resulting in different thresholds for the onset and decay of the instability, depending on the time history of the combustor. Increasing the ramping rate was found to reduce the amount of hysteresis in the system. Furthermore, the high amplitude of the instability resulted in significant harmonic components. The behaviour of the harmonics generally resembles the fundamental component, albeit with some notable exceptions.

Transient Thermoacoustic Responses of Methane/Hydrogen Flames in a Pressurized Annular Combustor

The present article experimentally investigates the triggering and transient growth of azimuthal instabilities in a pressurized laboratory-scale annular combustor featuring twelve methane/hydrogen flames, as the equivalence ratio is ramped up and down. The ramping rate of equivalence ratio is varied to examine its effect on the transient thermoacoustic response and the driving mechanisms, highlighting a number of previously unseen features. As the equivalence ratio is dynamically increased, all cases were observed to feature a distinct modal trajectory, during the onset of high amplitude instabilities. Strongly spinning counter-clockwise modes are first excited before a dynamic transition to strongly spinning clockwise modes occurs. Furthermore, the strength of the spinning mode (quantified through the spin ratio or nature angle) was shown to feature a local minima before the spinning mode stabilized in the system, which corresponds to an almost pure spinning state. Hysteresis behaviour was observed in both the amplitude and nature of the mode, resulting in different thresholds for the onset and decay of the instability, depending on the time history of the combustor. Increasing the ramping rate was found to reduce the amount of hysteresis in the system. Furthermore, the high amplitude of the instability resulted in significant harmonic components. The behaviour of the harmonics generally resembles the fundamental component, albeit with some notable exceptions.