Energy Growth Of Disturbances Research Articles

Nonmodal stability analysis of Poiseuille flow through a porous medium

We unravel the nonmodal stability of a three-dimensional nonstratified Poiseuille flow in a saturated hyperporous medium constrained by impermeable rigid parallel plates. The primary objective is to broaden the scope of previous studies that conducted modal stability analysis for two-dimensional disturbances. Here, we explore both temporal and spatial transient disturbance energy growths for three-dimensional disturbances when the Reynolds number and porosity of the material are high, based on evolution equations with respect to time and space, respectively. Modal stability analysis reveals that the critical Reynolds number for the onset of shear mode instability increases as porosity increases. Moreover, the Darcy viscous drag term stabilizes shear mode instability, resulting in a delay in the transition from laminar flow to turbulence. In addition, it demonstrates the suppression of three-dimensional shear mode instability as the spanwise wavenumber increases, thereby confirming the statement of Squire’s theorem. By contrast, nonmodal stability analysis discloses that both temporal and spatial transient disturbance energy growths curtail as the effect of the Darcy viscous drag force intensifies. But their maximum values behave like O(Re2) for a fixed porous material, where Re is the Reynolds number. However, for different porous materials, the scalings for both temporal and spatial transient disturbance energy growths are different. Furthermore, increasing porosity also suppresses both temporal and spatial disturbance energy growths. Finally, we observe that temporal transient disturbance energy growth becomes larger for a spanwise perturbation, while spatial transient disturbance energy growth becomes larger for a steady perturbation when angular frequency vanishes. The initial disturbance that excites the largest temporal energy amplification generates two sets of alternating high-speed and low-speed elongated streaks in the streamwise direction.

Stability of low-pressure turbine boundary layers under variable Reynolds number and pressure gradient

The free-stream turbulence induced transition occurring under typical low-pressure turbine flow conditions is investigated by comparing linear stability theory with wind tunnel measurements acquired over a flat plate subjected to high turbulence intensity. The analysis was carried out, accounting for three different Reynolds numbers and four different adverse pressure gradients. First, a non-similarity-based boundary layer (BL) solver was used to compute base flows and validated against pressure taps and particle image velocimetry (PIV) measurements. Successively, the optimal disturbances and their spatial transient growth were calculated by coupling classical linear stability theory and a direct-adjoint optimization procedure on all flow conditions considered. Linear stability results were compared with experimental particle image velocimetry measurements on both wall-normal and wall-parallel planes. Finally, the sensitivity of the disturbance spatial transient growth to the spanwise wavenumber of perturbations, the receptivity position, and the location where disturbance energy is maximized were investigated via the built numerical model. Overall, the optimal perturbations computed by linear stability theory show good agreement with the streaky structures surveyed in experiments. Interestingly, the energy growth of disturbances was found to be maximum for all the flow conditions examined, when perturbations entered the boundary layer close to the position where minimum pressure occurs.

Nonmodal and modal analyses of a flow through inhomogeneous and anisotropic porous channel

A study of nonmodal and modal stability analyses of a three-dimensional Poiseuille flow through a channel is performed where the bounding walls are partially covered by inhomogeneous and anisotropic porous layers. The evolution equations corresponding to normal velocity and normal vorticity are derived for fluid and porous layers to decipher the transient disturbance energy growth. The modal stability analysis reveals that the anisotropy parameter has a stabilizing impact, while the coefficient of inhomogeneity shows a peculiar behaviour on the most unstable shear mode. The nonmodal stability analysis predicts that the transient disturbance energy growth exists and attenuates with the increase in the value of the anisotropy parameter and larger values of the coefficient of inhomogeneity; however, it intensifies for smaller values of the coefficient of inhomogeneity.

Sensitivity and downstream influence of the impinging leading-edge vortex instability in a bileaflet mechanical heart valve

Bileaflet mechanical heart valves (BMHV) create unphysiological turbulent flow. Such turbulent flow involves multiple instability mechanisms interacting with one another in a confined geometry. For instance, an impinging leading-edge vortex (ILEV) instability creates disturbances at the leading edge of the valve leaflets, while potentially promoting turbulence downstream of the BMHV (Zolfaghari and Obrist, Phys. Rev. Fluids, vol. 4, 2019). In this article, we use adjoint-based methods to study the structural sensitivity of the ILEV instability in the BMHV, and to quantify the role of this instability in the maximum disturbance energy growth in the wake of the BMHV. We first present a direct numerical simulation to show the effect of the ILEV instability on the turbulent flow in the wake of the valve. Second, we perform a modal linear stability analysis on a two-dimensional subdomain attached to the leading edge of one leaflet. We investigate the sensitivity of the global modes associated with this flow using their adjoints, and then show a passive control scenario using a local feedback source. This results in a partial improvement in the flow oscillations downstream of the leaflets. We finally present a non-modal approach to identify the optimal initial conditions for achieving maximum energy growth at arbitrary locations. We show that, for sufficiently large times, the optimal initial condition for highest energy growth in the wake points at the leading edge, which includes the ILEV instability. Our study illustrates that an improved leading-edge shape can effectively reduce turbulence in the wake of the BMHV.

Disturbance energy budget of turbulent swirling premixed flame in a cuboid combustor

In order to clarify combustion oscillation based on the exact disturbance energy budget, three-dimensional direct numerical simulations (DNS) of hydrogen–air turbulent swirling premixed flames are conducted in this study, considering a detailed kinetic mechanism and temperature dependencies of transport and thermal properties. Stoichiometric and lean conditions under two swirl numbers (S=0.6 and 1.2) are investigated. The exact disturbance energy budget is closed to more than 92% for all computational conditions. Investigation of source term contributions reveals that the swirl number affects the order of contribution levels of some sources, while the equivalence ratio influences the fluctuation amplitude of the time derivative value of disturbance energy. It is shown that source terms related to fluctuations of entropy and heat sources are major contributors to disturbance energy growth and decay, respectively. In addition to these major terms, source terms related to non-equilibrium combustion chemistry are also determined to be influential to the time evolution of disturbance energy. The relation between disturbance energy generation and flame characteristics is discussed, based on the time–averaged distributions of some of the abovementioned important sources.

Osborne Reynolds pipe flow: Direct simulation from laminar through gradual transition to fully developed turbulence

The precise dynamics of breakdown in pipe transition is a century-old unresolved problem in fluid mechanics. We demonstrate that the abruptness and mysteriousness attributed to the Osborne Reynolds pipe transition can be partially resolved with a spatially developing direct simulation that carries weakly but finitely perturbed laminar inflow through gradual rather than abrupt transition arriving at the fully developed turbulent state. Our results with this approach show during transition the energy norms of such inlet perturbations grow exponentially rather than algebraically with axial distance. When inlet disturbance is located in the core region, helical vortex filaments evolve into large-scale reverse hairpin vortices. The interaction of these reverse hairpins among themselves or with the near-wall flow when they descend to the surface from the core produces small-scale hairpin packets, which leads to breakdown. When inlet disturbance is near the wall, certain quasi-spanwise structure is stretched into a Lambda vortex, and develops into a large-scale hairpin vortex. Small-scale hairpin packets emerge near the tip region of the large-scale hairpin vortex, and subsequently grow into a turbulent spot, which is itself a local concentration of small-scale hairpin vortices. This vortex dynamics is broadly analogous to that in the boundary layer bypass transition and in the secondary instability and breakdown stage of natural transition, suggesting the possibility of a partial unification. Under parabolic base flow the friction factor overshoots Moody's correlation. Plug base flow requires stronger inlet disturbance for transition. Accuracy of the results is demonstrated by comparing with analytical solutions before breakdown, and with fully developed turbulence measurements after the completion of transition.

Disturbance energy growth in core–annular flow

AbstractThe linear stability of the horizontal pipe flow of an equal density oil–water mixture, arranged as acore–annular flow(CAF), is here reconsidered from the point of view of non-modal analysis in order to assess the effects of non-normality of the linearized Navier–Stokes operator on the transient evolution of small disturbances. The aim of this investigation is to give insight into physical situations in which poor agreement occurs between the predictions of linear modal theory and classical experiments. The results exhibit high transient amplifications of the energy of three-dimensional perturbations and, in analogy with single-fluid pipe flow, the largest amplifications arise for non-axisymmetric disturbances of vanishing axial wavenumber. Energy analysis shows that the mechanisms leading to these transient phenomena mostly occur in the annulus, occupied by the less viscous fluid. Consequently, higher values of energy amplifications are obtained by increasing the gap between the core and the pipe wall and the annular Reynolds number. It is argued that these linear transient mechanisms of disturbance amplification play a key role in explaining the transition to turbulence of CAF.

Energy growth of disturbances in a non-Fourier fluid

Natural convection of non-Fourier fluid of the single-phase-lagging (SPL) type between two horizontal walls (Rayleigh–Benard) has been investigated. These fluids possess a relaxation time, reflecting the delay in the response of the heat flux and the temperature gradient with respect to one another. By invoking the role of the eigenvectors to detect and quantify short-time behavior, transient growth of energy of disturbances has been illustrated. The energy of the perturbations is introduced in terms of the primary variables as a disturbance measure in order to quantify the size of the disturbance. In contrast to linear stability analysis, one does not assume exponential time dependence, but monitor the evolution of initial conditions in the pre- and post-critical ranges of Rayleigh numbers. Different growth functions for different levels of non-Fourier effects have been found, which should be thought of as the envelope of the energy evolution of individual initial conditions. Also, it is found that nonlinearities are not required for the energy growth. Energy growth can occur if non-orthogonal eigenfunctions are available. A fundamental implication of the non-normality is that there can be significant energy growth in the energy of perturbations even if the flow is stable.

Non-normal dynamics of time-evolving co-rotating vortex pairs

AbstractTransient energy growth of disturbances to co-rotating pairs of vortices with axial core flows is investigated in an analysis where vortex core expansion and vortex merging are included by adopting a time-evolving base flow. The dynamics of pairs are compared with those of individual vortices in order to highlight the effect of vortex interaction. Three typical vortex pair cases are studied, with the pairs comprised respectively of individually inviscidly unstable vortices at the streamwise wavenumber that maximizes the individual instabilities, viscously unstable vortices also at the streamwise wavenumber maximizing the individual instabilities and asymptotically stable vortices at streamwise wavenumber zero. For the inviscidly unstable case, the optimal perturbation takes the form of a superposition of two individual helical unstable modes and the optimal energy growth is similar to that predicted for an individual inviscid unstable vortex, while where the individual vortices are viscously unstable, the optimal disturbances within each core have similar spatial distributions to the individually stable case. For both of these cases, time horizons considered are much lower than those required for the merger of the undisturbed vortices. However, for the asymptotically stable case, large linear transient energy growth of optimal perturbations occurs for time horizons corresponding to vortex merging. Linear transient disturbance energy growth exhibited by pairs in this stable case is two to three orders of magnitude larger than that for a corresponding individual vortex. The superposition of the perturbation and the base flow shows that the perturbation has a displacement effect on the vortices in the base flow. Direct numerical simulations of stable pairs seeded by optimal initial perturbations have been carried out and acceleration/delay of vortex merging associated with a dual vortex meandering and vortex breakup related to axially periodic acceleration and delay of vortex merging are observed. For axially invariant cases, the sign of perturbation has an effect, as well as magnitude; the sign dependence relates to whether or not the perturbation adds to or subtracts from the swirl of the base flow. For a two-dimensional perturbation that adds to the swirl of the base flow, seeding with the linear optimal disturbance at a relative energy level $1\ensuremath{\times} 1{0}^{\ensuremath{-} 4} $ induces the pair to move towards each other and approximately halves the time required for merger. Direct numerical simulation shows that the optimal three-dimensional perturbation can induce the vortex system to break up before merging occurs, since the two-dimensional nature of vortex merging is broken by the development of axially periodic perturbations.

Disturbance energy norms: A critical analysis

The question of which norm should be used to characterize the fluctuating disturbance energy in studies on thermoacoustic instabilities remains a topic of continued debate. In this paper we formulate a strategy for developing mathematically consistent norms that do not support spurious growth of disturbance energy in the absence of ‘physical’ sources of energy. A critical analysis of various disturbance energy norms existing in literature is conducted and important conclusions regarding positive definitiveness and susceptibility to exhibiting unphysical growth are drawn. It is shown that the energy norm proposed by Cantrell and Hart [Interaction between sound and flow in acoustic cavities: mass, momentum and energy considerations, Journal of Acoustical Society of America 36 (1964) 697–706] and Myers' first order disturbance energy norm [Transport of energy by disturbances in arbitrary steady flow, Journal of Fluid Mechanics 226 (1991) 383–400] are positive definite if M0<1 and M0<1/γ, respectively, where M0 is the magnitude of the local mean flow Mach number, γ is the ratio of specific heats at constant pressure and volume and the stipulated conditions should be met at every point in the flow domain. Our analysis shows that the disturbance energy norm proposed by Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215–234] does not exhibit unphysical growth. It is also shown that this property is not unique to Chu's disturbance energy norm and there exists a family of norms which satisfy this requirement. The analysis also leads to an interesting interpretation of the acoustic energy conservation principle of Cantrell and Hart. The potential held by various disturbance energy norms for exhibiting fictitious growth is quantified using tools from nonmodal stability theory. It is concluded that if the mean flow Mach number is small, Myers' norm is a suitable measure of the disturbance energy.

Generalized Reynolds-Orr Energy Equation with Wall Slip

The Reynolds-Orr energy equation is generalized to include the velocity slip effect at the walls, for investigating the energy gain or loss in the disturbed slip flow. Taken the Poiseuille flow, typical prototype of parallel shear flows as an example, it is found that for the very weak effect of wall slip, the disturbance energy being transferred from the basic flow overcomes the viscous dissipation, resulting in the growth of disturbance energy and the destabilizing role of wall slip. Otherwise, the viscous dissipation overcomes the energy production resulting in the decay of the disturbance energy and the stabilizing role of wall slip.

Effect of thermally induced perturbation in supersonic boundary layers

This paper investigates the mechanism of steady and unsteady thermal perturbation (also denoted as thermal bump) in a Mach 1.5 flat plate boundary layer. A high-fidelity upwind-biased third-order Roe scheme is used with the compressive van Leer harmonic limiter on a suitably refined mesh. The study consists of two parts. In the first part, the effects of the steady and pulsed thermal bumps are explored. It is shown that the finite-span thermal bumps generate streamwise vortices. With steady heating, the disturbance decays downstream. However, when the thermal bump is pulsed, vortex shedding is observed and the streamwise vortical disturbance grows with downstream distance, consistent with linear stability analysis. The integrated disturbance energy indicates that streamwise kinetic disturbance energy growth dominates over those associated with other two velocity and thermodynamic components. The second part of this paper explores the physical consequences of the nonlinear dynamics between the vortices produced by the pulsed bump and the compressible boundary layer. The resulting three-dimensional flow distortion generates hairpin structures which are aligned in the streamwise direction, suggesting that the transition process bears some similarity to K-type breakdown. The arrangement of these vortices is connected to the low-speed streaks observed in the evolving boundary layer. The shape factor, velocity, and Reynolds stress profiles suggest that the perturbed flow shows initiation of transition to turbulence, but remains transitional at the end of the plate.

Transient growth and rotor–stator interaction noise in mean swirling duct flow

The phenomenon of spatial transient growth in swirling duct flow with application to aeroengine duct acoustics is investigated. In a typical aeroengine a region of swirling flow exists between the rotor and the downstream stator. The possibility for optimal transient growth of disturbance energy, and of a norm related to the acoustic power radiated back upstream from the stator is shown. Significant transient growth of both energy and noise norms can be observed over length scales relevant to aeroengine size, and over a wide range of parameter values such as the degree of mean swirl, azimuthal wavenumber and frequency. The optimal disturbances are found to have significant three dimensionality. The growth in energy results from increases in the azimuthal and radial components of the disturbance velocity field as it propagates downstream, whilst increases in the azimuthal velocity bring about non-modal growth of the noise measure.

A singular value analysis of boundary layer control

Several approaches for boundary-layer control are analyzed from a linear system point of view. The singular value decomposition (SVD) is applied to the linearized Navier–Stokes system in the presence of control. The performance of control is examined in terms of the largest singular values, which represent the maximum disturbance energy growth ratio attainable in the linear system under control. It is shown that the maximum growth ratio is less in controlled systems than in the uncontrolled system only when control parameters are within a certain range of values. With opposition control, for example, when the detection plane is located too far away from the wall, the maximum energy growth ratio is larger, consistent with the results observed in direct numerical simulations. The SVD analysis of other controls also shows a similarity between the trend observed in the SVD analysis (linear) and that observed in direct numerical simulations (nonlinear), thus reaffirming the importance of linear mechanisms in the near-wall dynamics of turbulent boundary layers. The present study illustrates that the SVD analysis can be used as a guideline for designing controllers for drag reduction in turbulent boundary layers.

Vortex-induced instability of an incompressible wall-bounded shear layer

The unsteady separated flow produced by a finite-core vortex on a plane shear layer is studied here as a vortex-induced instability. The mechanism of such an interaction, where the distance between the wall and the vortex is many times the local boundary layer thickness, is shown here by flow visualization and the solution of the unsteady Navier–Stokes equation. A new theory is proposed here, which is generic to the Navier–Stokes equation without any assumptions, that is based on growth of disturbance energy in time. A dynamical systems approach based on the proper orthogonal decomposition technique is used to provide a quantitative measure.

A numerical study of wave propagation in a confined mixing layer by eigenfunction expansions

It is well known that the growth rate of instability waves of a two-dimensional free shear layer is reduced greatly at supersonic convective Mach numbers. In previous works, it has been shown that new wave modes exist when the shear layers are bounded by a channel due to the coupling effect between the acoustic wave modes and the motion of the mixing layer. The present work studies the simultaneous propagation of multiple stability waves using numerical simulation. It is shown here that the coexistence of two wave modes in the flow field can lead to an oscillatory growth of disturbance energy with each individual wave mode propagating linearly. This is particularly important when the growth rates of the unstable waves are small. It is also shown here that the propagation of two neutrally stable wave modes can lead to a stationary periodic structure of rms fluctuations. In the numerical simulations presented here the forced wave modes are propagating at same frequency, but with different phase velocities. In order to track the growth of each wave mode as it propagates downstream, a numerical method that can effectively detect and separate the contribution of the individual wave is given. It is demonstrated that by a least square fitting of the disturbance field with eigenfunctions the amplitude of each wave mode can be found. Satisfactory results as compared to linear theory are obtained.

On the growth of disturbances to forced and dissipated barotropic flows

Abstract We consider the growth of disturbances to large-scale zonally-asymmetric steady states in a truncated spectral model for forced and dissipated barotropic flow. A variant of the energy method is developed to optimize the instantaneous disturbance energy growth rate. The method involves solving a matrix eigenvalue problem amenable to standard numerical techniques. Two applications are discussed. (1) The global stability of a family of steady states is assessed in terms of the Ekman damping coefficient r. It is shown that monotonic global stability (i.e., every disturbances energy monotonically decays to zero) prevails when r≥rc . (2) Initially fastest-growing disturbances are constructed in the r<rc regime. Particular attention is paid to a subregion of the r<rc regime where initially-growing disturbances exist despite stability with respect to normal modes. Nonlinear time-dependent simulations are performed in order to appraise the time evolution of various disturbances.