Transient Growth Of Disturbances Research Articles

Effect of Aspect Ratio on Optimal Disturbances of Duct Flows

The linear temporal stability of the Poiseuille flow through a rectangular duct is considered. The effect of the duct aspect ratio on the transient growth of disturbances, which causes the so-called subcritical laminar–turbulent transition, is studied numerically. In particular, it is shown that an increase in the aspect ratio promotes the subcritical transition in almost the entire considered range of the duct aspect ratios except a relatively narrow range, where the increase suppresses the transient growth of disturbances. Such peculiarity is qualitatively explained by considering the nonmodal stability of more simplified plane channel flow.

Non-modal transient growth of disturbances in pulsatile and oscillatory pipe flows

Abstract

Linear proportional–integral control for skin-friction reduction in a turbulent channel flow

In the present study, we apply a proportional (P)–integral (I) feedback control to a turbulent channel flow for skin-friction reduction. The instantaneous wall-normal velocity at a sensing plane above the wall is measured as a sensing parameter, and blowing/suction is provided at the wall based on the PI control. The performance of PI controls is estimated by the change in the skin friction while varying the sensing plane location $y_{s}$ and the proportional and integral feedback gains ($\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ respectively). The opposition control proposed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110) corresponds to a P control with $\unicode[STIX]{x1D6FC}=1$. When the sensing plane is located close to the wall ($y_{s}^{+}\lesssim 10$), PI controls result in greater skin-friction reductions than corresponding P controls. The root-mean-square (r.m.s.) sensing velocity fluctuations, considered as the control error, approach zero with successful PI controls, but do not with P controls. Successful PI controls reduce the strength of near-wall coherent structures and the r.m.s. velocity fluctuations above the wall apart from those near the wall due to the control input. The frequency spectra of the sensing velocity show that the I component of PI controls significantly reduces the energy at low frequencies, much more than P controls do. Proportional–integral controls are also applied to a linearized flow model having transient growth of disturbances. The performance of PI controls for a linearized flow model is very similar to that for a turbulent channel flow, i.e. the low-frequency components of disturbances are significantly reduced by the I component of PI controls, and the transient energy growth is suppressed more than by P controls.

Disturbance amplification in boundary layers over thin wall films

In single-fluid boundary layers, streaks can amplify at sub-critical Reynolds numbers and initiate early transition to turbulence. Introducing a wall film of different viscosities can appreciably alter the stability of the base flow and, in particular, the transient growth of the perturbation streaks. The formalism of seminorms is used to identify optimal disturbances which maximize the kinetic energy in the two-fluid flow. An examination of optimal growth over a range of viscosity ratios of the film relative to the outer flow reveals three distinct regimes of amplification, each associated with a particular combination of the eigenfunctions. In order to elucidate the underlying amplification mechanisms, a model problem is formulated: An initial value problem is solved using an eigenfunction expansion and is used to compute the evolution of pairs of eigenfunctions. By appropriately selecting the pair, the initial value problem qualitatively reproduces the temporal evolution of the optimal disturbance, and provides an unambiguous explanation of the dynamics. Two regimes of transient growth are attributed to the evolution of the interface mode along with free-stream vortical modes; the third regime is due to the evolution of the interface and a discrete mode. The results demonstrate that a lower-viscosity film can effectively reduce the efficacy of the lift-up mechanism and, as a result, transient growth of disturbances. However, another mechanism of amplification of wall-normal vorticity arises due to the deformation of the two-fluid interface and becomes dominant below a critical viscosity ratio.

Global instabilities and transient growth in Blasius boundary-layer flow over a compliant panel

We develop a hybrid of computational and theoretical approaches suited to study the fluid–structure interaction (FSI) of a compliant panel, flush between rigid upstream and downstream wall sections, with a Blasius boundary-layer flow. The ensuing linear-stability analysis is focused upon global instability and transient growth of disturbances. The flow solution is developed using a combination of vortex and source boundary-element sheets on a computational grid while the dynamics of a plate-spring compliant wall are couched in finite-difference form. The fully coupled FSI system is then written as an eigenvalue problem and the eigenvalues of the various flow- and wall-based instabilities are analysed. It is shown that coalescence or resonance of a structural eigenmode with either a flow-based Tollmien–Schlichting Wave (TSW) or wall-based travelling-wave flutter (TWF) modes can occur. This can render the nature of these well-known convective instabilities to become global for a finite compliant wall giving temporal growth of system disturbances. Finally, a non-modal analysis based on the linear superposition of the extracted temporal modes is presented. This reveals a high level of transient growth when the flow interacts with a compliant panel that has structural properties which render the FSI system prone to global instability. Thus, to design stable finite compliant panels for applications such as boundary-layer transition postponement, both global instabilities and transient growth must be taken into account.

Cross-correlation-aided transport in stochastically driven accretion flows.

The origin of linear instability resulting in rotating sheared accretion flows has remained a controversial subject for a long time. While some explanations of such non-normal transient growth of disturbances in the Rayleigh stable limit were available for magnetized accretion flows, similar instabilities in the absence of magnetic perturbations remained unexplained. This dichotomy was resolved in two recent publications by Chattopadhyay and co-workers [Mukhopadhyay and Chattopadhyay, J. Phys. A 46, 035501 (2013); Nath et al., Phys. Rev. E 88, 013010 (2013)] where it was shown that such instabilities, especially for nonmagnetized accretion flows, were introduced through interaction of the inherent stochastic noise in the system (even a "cold" accretion flow at 3000 K is too "hot" in the statistical parlance and is capable of inducing strong thermal modes) with the underlying Taylor-Couette flow profiles. Both studies, however, excluded the additional energy influx (or efflux) that could result from nonzero cross correlation of a noise perturbing the velocity flow, say, with the noise that is driving the vorticity flow (or equivalently the magnetic field and magnetic vorticity flow dynamics). Through the introduction of such a time symmetry violating effect, in this article we show that nonzero noise cross correlations essentially renormalize the strength of temporal correlations. Apart from an overall boost in the energy rate (both for spatial and temporal correlations, and hence in the ensemble averaged energy spectra), this results in mutual competition in growth rates of affected variables often resulting in suppression of oscillating Alfven waves at small times while leading to faster saturations at relatively longer time scales. The effects are seen to be more pronounced with magnetic field fluxes where the noise cross correlation magnifies the strength of the field concerned. Another remarkable feature noted specifically for the autocorrelation functions is the removal of energy degeneracy in the temporal profiles of fast growing non-normal modes leading to faster saturation with minimum oscillations. These results, including those presented in the previous two publications, now convincingly explain subcritical transition to turbulence in the linear limit for all possible situations that could now serve as the benchmark for nonlinear stability studies in Keplerian accretion disks.

Effect of heat source on transient energy growth analysis of a thermoacoustic system

Transient growth of disturbances in a non-normal combustion system could trigger thermoacoustic instability. In this work, the effect of heat source on triggering thermoacoustic instability in a system with Dirichlet or Neumann boundary conditions is investigated. For this a thermoacoustic model of a premixed laminar flame is developed. It is formulated in state-space by expanding flow disturbances via Galerkin series and coupling with a flame model, thus providing a platform on which to gain insight on the system stability behaviors. Transient energy growth analysis is then performed by discretizing the flame model into distributed monopole-like sound sources. It is shown that the thermoacoustic system is non-normal and associated with transient energy growth. Parametric studies are then conducted to study the effects of (1) the flame-acoustics interaction index I, (2) the eigenmode number N, (3) the mean temperature ratio T between pre- and after-combustion regions on the system stability behavior and non-normality. It is found that the maximum transient energy growth Gmax is increased with increased I. Furthermore, the thermoacoustic system with more eigenmodes is found to be associated with a larger Gmax.

Non-modal disturbances growth in a viscous mixing layer flow

The non-modal transient growth of disturbances in an isothermal viscous mixing layer flow is studied for Reynolds numbers varying from 100 up to 5000 at different streamwise and spanwise wavenumbers. It is found that the largest non-modal growth takes place at the wavenumbers for which the mixing layer flow is stable. In linearly unstable configurations, the non-modal growth can only slightly exceed the exponential growth at short times. Contrarily to the fastest exponential growth, which is two-dimensional, the most profound non-modal growth is attained by oblique three–dimensional waves propagating at an angle with respect to the base flow. By comparing the results of several mathematical approaches, it is concluded that within the considered mixing layer model with a tanh base velocity profile, the non-modal optimal disturbances growth is governed only by eigenvectors that are decaying far from the mixing zone. Finally, a full three-dimensional DNS with optimally perturbed base flow confirms the presence of the structures determined by the transient growth analysis. The time evolution of optimal perturbations is presented. It exhibits a growth and a decay of flow structures that sometimes become similar to those observed at the late stages of the time evolution of the Kelvin–Helmholtz billows. It is shown that non-modal optimal disturbances yield a strong mixing without a transition to turbulence.

Transient growth in linearly stable Taylor–Couette flows

AbstractNon-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor–Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers$\mathit{Re}$, the optimal transient energy growth always follows a$\mathit{Re}^{2/3}$scaling, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows, the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations, we show that linear stability and transient growth are independent of the cylinder rotation ratio and we derive a universal$\mathit{Re}^{2/3}$scaling of optimal energy growth using Wentzel–Kramers–Brillouin theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.

Scaling of streamwise boundary layer streaks and their ability to reduce skin-friction drag

AbstractSpanwise arrays of miniature vortex generators (MVGs) are used to generate energetic transient disturbance growth, which is able to modulate the boundary layer flow with steady and stable streak amplitudes up to 32 % of the free-stream velocity. This type of modulation has previously been shown to act in a stabilizing manner on modal disturbance growth described by classical instability theory. In an attempt to reproduce a more realistic flow configuration, in the present experimental set-up, Tollmien–Schlichting (TS) waves are generated upstream of the MVG array, allowing for a complete interaction of the incoming wave with the array. Fifteen new MVG configurations are investigated and the stabilizing effect on the TS waves is quantified. We show that the streak amplitude definition is very important when trying to relate it to the stabilization, since it may completely bypass information on the mean streamwise velocity gradient in the spanwise direction, which is an essential ingredient of the observed stabilization. Here, we use an integral-based streak amplitude definition along with a streak amplitude scaling relation based on empiricism, which takes the spanwise periodicity of the streaks into account. The results show that, applying the integral definition, the optimal streak amplitude for attenuating TS wave disturbance growth is around 30 % of the free-stream velocity, which corresponds to ${\sim }20\hspace{0.167em} \% $ in the conventional definition when keeping the spanwise wavelength constant. The experiments also show that the disturbance energy level, based on the full velocity signal, is significantly reduced in the controlled case, and that the onset of transition may be inhibited altogether throughout the measured region in the presence of an MVG array.

Stability analysis of unbounded uniform shear flows of dense, slightly inelastic spheres based on a frictional–kinetic theory

Asymptotic and transient stability analyses of unbounded uniform shear flows of dense, slightly inelastic, spherical particles were carried out using a frictional–kinetic theory. This model proposed for describing dense flows is based on a critical state plasticity theory and a simplified kinetic theory. In this model, the bulk and shear viscosities, the “ thermal” conductivity, and the energy dissipation rate are proportional to a “mean pressure” which is composed of a quasistatic-frictional-contribution pressure considered for slow, plasticity deformations and a granular-kinetic-theory collisional-contribution pressure. We studied two-dimensional stability analyses of layering disturbances (i.e., the perturbations whose wave number vectors are aligned only in the gradient) as well as nonlayering disturbances (the wave number vectors have nonzero streamwise components). Although this model has a simpler framework, it predicted similar results to those obtained using a more elaborate frictional–kinetic model. For instance, nonlayering disturbances are asymptotically stable at large time; the maximum transient growth of disturbances increases as the solids fraction or the friction coefficient is increased; and transient growths of disturbances can be significant due to the non-normality of the system. However, the prediction of the asymptotic stability of layering disturbances may be questionable because the collisional-contribution terms of the present model were over-simplified.

Transient growth in the asymptotic suction boundary layer

In the present experimental setup, the transient disturbance growth in a spatially invariant boundary layer flow, i.e., the asymptotic suction boundary layer (ASBL), has been investigated. The choi ...

Mechanisms of non-modal energy amplification in channel flow between compliant walls

The mechanisms leading to large transient growth of disturbances for the flow in a channel with compliant walls are investigated. The walls are modelled as thin spring-backed plates, and the flow dynamics is modelled using the Navier–Stokes equations linearized round the Poiseuille profile. Analysis for streamwise invariant perturbations show that this fluid-structure system can sustain oscillatory energy evolution of large amplitude, in the form of spanwise standing waves. Such waves are related to the travelling waves which a free wall can support, modified to account for an ‘added mass’ effect. Simple scaling arguments are found to provide results in excellent agreement with computations of optimal disturbances, for low-to-moderate values of the stiffness parameter characterizing the compliant surface.

The relative roles of divergence and velocity slip in the stability of plane channel flow

Wall slip and wall divergence are known to have large and opposing effects on the stability of flow in a two-dimensional channel. While divergence hugely destabilises, slip dramatically stabilizes the linear mode. In a non-parallel stability analysis, we study a combination of these two effects, since both will coexist in small-scale flows with wall roughness. Our main results are (i) that the stabilising effect of slip is reversed at higher angles of divergence, (ii) transient growth of disturbances is unaffected by either wall-divergence, or by slip at any divergence. Moreover, at the Reynolds numbers relevant here, transient growth is too low to be a significant player in transition to turbulence, which is more likely to be driven by linear instability.

The effect of wall heating on instability of channel flow

A comprehensive study of the effect of wall heating or cooling on the linear, transient and secondary growth of instability in channel flow is conducted. The effect of viscosity stratification, heat diffusivity and of buoyancy are estimated separately, with some unexpected results. From linear stability results, it has been accepted that heat diffusivity does not affect stability. However, we show that realistic Prandtl numbers cause a transient growth of disturbances that is an order of magnitude higher than at zero Prandtl number. Buoyancy, even at fairly low levels, gives rise to high levels of subcritical energy growth. Unusually for transient growth, both of these are spanwise-independent and not in the form of streamwise vortices. At moderate Grashof numbers, exponential growth dominates, with distinct Poiseuille–Rayleigh–Bénard and Tollmien–Schlichting modes for Grashof numbers up to ∼ 25 000, which merge thereafter. Wall heating has a converse effect on the secondary instability compared to the primary instability, destabilizing significantly when viscosity decreases towards the wall. It is hoped that the work will motivate experimental and numerical efforts to understand the role of wall heating in the control of channel and pipe flows.

Transient disturbance growth in a corrugated channel

Transient growth of small disturbances may lead to the initiation of the laminar-turbulent transition process. Such growth in a two-dimensional laminar flow in a channel with a corrugated wall is analysed. The corrugation has a wavy form that is completely characterized by its wavenumber and amplitude. The maximum possible growth and the form of the initial disturbance that leads to such growth have been identified for each form of the corrugation. The form that leads to the largest growth for a given corrugation amplitude, i.e. the optimal corrugation, has been found. It is shown that the corrugation acts as an amplifier for disturbances that are approximately optimal in the smooth channel case but has little effect in the other cases. The interplay between the modal (asymptotic) instability and the transient growth, and the use of the variable corrugation for modulation of the growth are discussed.

Receptivity of stationary transient disturbances to surface roughness

Recent experiments on transient disturbance growth in boundary layers indicate that disturbances generated by surface roughness undergo suboptimal growth. The implication is that the receptivity mechanism governing the distribution of disturbance energy among the continuous spectrum of damped Orr–Sommerfeld/Squire modes produces energy distributions that are significantly different from the theoretical optimum. Experiments presented here are intended to investigate how the amplitude and diameter of cylindrical roughness elements arranged in a spanwise array affect various features of transient growth. The objective is to infer how or to what extent the roughness features affect the continuous spectrum and to use this information as a foundation for future receptivity models. The results show that the energy of stationary disturbances varies as Rek2 and that the streamwise distance over which the disturbances grow increases slightly with increasing Rek. As the roughness diameter is varied, dramatic changes in the qualitative nature of the resulting transient growth occur. Both the variation of the growth length with Rek and the behavioral changes with roughness diameter indicate that the energy distribution among the continuous modes is a strong function of roughness features and that an accurate and sophisticated receptivity model will be necessary to accurately predict transient growth.

Spatial theory of optimal disturbances in boundary layers

A spatial theory is proposed for the linear transient growth of disturbances in a parallel boundary layer. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes and Tollmien–Schlichting (TS) like instability modes. Therefore, the problem of optimal disturbances may be considered as an initial value problem on the subset of the decaying eigenmodes and a TS wave, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. The results indicate that the most significant transient growth is associated with stationary streamwise vortices. Numerical examples illustrate that favorable pressure gradient decreases the overall amplification. Effects of compressibility and the wall cooling are investigated as well.

Transient growth in parallel two-phase flow: Analogies and differences with single-phase flow

Transient disturbance growth in parallel two-phase flow is studied. When the disturbance growth is measured in terms of the kinetic energy norm, which is commonly used for single-phase flow, the disturbance growth function does not converge as the number of eigenmodes used in the computation increases. A solution to this problem is presented in the form of a norm that also includes the potential energy of the disturbed interface. This solution is used to examine the two-phase flow experiment by Kao and Park.

Evidence for transient disturbance growth in a 1961 pipe-flow experiment

Evidence is presented for what appears to have been nonmodal disturbance growth in a pipe-flow experiment performed by Kaskel [Technical Report No. 32-138, Jet Propulsion Laboratory, California Institute of Technology, 1961], based on a comparison of the experimental traces of disturbance amplitude with predictions of transient amplitude growth calculated from the underlying linear stability operator. Owing to the geometry of the experimental disturbance generator, modes having azimuthal wavenumbers n=0, 6 and 12 are expected to contribute the bulk of the disturbance energy, and numerically predicted amplitude growth versus downstream distance is in good agreement with the experiment.