Non-modal Approach Research Articles

Linear stability analysis of the viscoelastic Navier–Stokes–Voigt fluid model through Brinkman porous media: Modal and non-modal approaches

The linear stability analysis of a viscoelastic Navier-Stokes-Voigt fluid flow, or the Kelvin-Voigt fluid of zero order in a Brinkman porous medium, is investigated using both modal and non-modal analysis. The numerical solution is obtained using the Chebyshev collocation method. The combined effects of the medium's porosity, represented by the porous parameter, the fluid viscosity, represented by the ratio of effective viscosity to fluid viscosity, and the fluid elasticity, represented by the Kelvin-Voigt parameter are investigated using both modal and non-modal analysis. The modal analysis describes the long-term behavior of the system, obtained through plotting the eigenspectrum, eigenfunctions, growth rate curves, neutral stability curves, and streamline plots, along with accurate values of critical triplets. In non-modal analysis, the pseudospectrum of the Orr-Sommerfeld operator, transient energy growth curves, and regions of stability, instability, and potential instability are depicted. The results obtained from modal analysis indicate that the porous parameter, Kelvin-Voigt parameter, and the ratio of effective viscosity to fluid viscosity act as stabilizing agents. However, using non-modal analysis, it is observed that while the porous parameter and the ratio of effective viscosity to fluid viscosity act as stabilizing agents, the Kelvin-Voigt parameter acts as a destabilizing agent over shorter periods.

Joint-mode diffusion analysis of discontinuous Galerkin methods: Towards superior dissipation estimates for nonlinear problems and implicit LES

We present a new linear eigensolution analysis technique that provides superior estimates of dissipation distribution in wavenumber space for the discontinuous Galerkin (DG) method. The technique builds upon traditional dispersion-diffusion analyses that have been applied to spectral/hp element methods, but in particular is an improvement upon the non-modal eigenanalysis approach proposed by Fernandez et al. in [1]. The present technique takes into account the indirect effects that dispersion may have on dissipation, as recently discussed by Moura et al. in [2], in order to better represent dissipation itself. Also, a concept often used with dynamic mode decomposition (DMD) techniques is invoked to weight the relative contribution of the multiple diffusion curves that stem from temporal eigenanalysis. This allows for obtaining a single dissipation profile in wavenumber space, so that the proposed technique is named joint-mode analysis. Although the non-modal approach also provides a single diffusion curve, the joint-mode dissipation curve is shown to correlate significantly better with the energy spectrum of Burgers' turbulence at large and intermediate scales, which is particularly relevant for implicit large-eddy simulation (LES). The proposed technique is readily extensible to other spectral/hp element methods.

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.

Linear stability of electrocapillary convection in an infinite liquid layer

Electrocapillary convection is the flow driven by the interfacial shear stress induced by a surface-charge density and electric fields. It has become an important topic in electrohydrodynamics recently. The present work examine the convective instability for electrocapillary liquid layers by the modal and non-modal approaches. The modal analysis finds two kinds of modes, namely purely hydrodynamic and electrohydrodynamic modes, while both of them are stable. In the non-modal analysis, the amplifications of initial and external disturbances are measured by the transient growth and the response function, respectively. Both the perturbation velocity and surface-charge density can be amplified significantly.

Stability analysis of a zero-dimensional model of PWR core using non-modal stability theory

In this work, non-modal stability theory is used to study the non-linear dynamics of the Pressurised Water Reactor (PWR), using a zero-dimensional model. Usually, stability analysis for nuclear reactors is performed using modal methods. Instead, the non-modal approach used in the present work offers a different perspective. This approach considers the so-called short-term behaviour following a disturbance instead of the asymptotic (long-term) one. In the model, point-reactor kinetics with six groups delayed neutrons is used for neutronics, whereas thermal-hydraulics is treated using a two-region model (fuel and coolant). The non-orthogonality of the dynamic matrix of the obtained system shows the possibility of transient energy growth following a disturbance, and the proposed stability analysis offers proof of this. A sensitivity analysis of some parameters is then carried out to study how their variation influences the short-term response of the system. The results of this analysis allow for a ranking of the parameters, allowing the identification of the most significant ones regarding system stability.

Effect of Couette component on the stability of Poiseuille flow of a Bingham fluid–porous system: Modal and non-modal approaches

In this study, both modal and non-modal stability analyses are attempted in case of Couette–Poiseuille flow of a Bingham fluid overlying a porous layer. Such a flow configuration is widely encountered in the geophysical context in case of oil drilling. The solution of the modal problem yields no unstable eigenvalue, similar to the flow of a viscoplastic fluid in a non-porous channel configuration. Thus, non-modal analysis is performed to throw light on the short-time characteristics. The primary goal is to unveil the complex interplay between the upper plate velocity (Couette component) and the parameters characterizing the porous layer in dictating the flow transition characteristics. The current study is possibly the first attempt at investigating the effect of the Couette flow on the stability of a fluid–porous system for any kind of non-Newtonian fluid and reveals marked departure from the results reported in the literature for a similar flow configuration involving Newtonian rheology. The reason for the deviation is attributed to the role of yield stress, quantified by the Bingham number, and its complex interaction with the Couette number and porous layer parameters (depth, permeability, anisotropy, inhomogeneity, etc.). The relative interaction between fluid and porous modes in an environment of non-linear viscosity variation (owing to the rheology of the viscoplastic fluid), coupled with enhanced shearing (imparted by the Couette component), is found to demonstrate unique, non-monotonic flow transition characteristics. The possible physical mechanism governing short-time (non-modal) amplifications via interaction between the mean shear flow and the perturbation waves is also explored in detail.

Linear stability of plane Poiseuille flow of a Bingham fluid in a channel with the presence of wall slip

In the present paper, the effects of a channel symmetric wall slip are investigated on the stability of plane Poiseuille flow of a Bingham fluid, based on the modal and non-modal linear stability analysis approaches. Both streamwise and spanwise slip conditions are considered and their results are compared with those of the no-slip condition, in terms of the flow stability. The results are obtained for different dimensionless groups that govern the stability picture, for example the Bingham number, the Reynolds number and the streamwise/spanwise slip numbers. The linearized perturbation equations are obtained and the corresponding eigenvalue problem is solved based on the Chebyshev collocation method at the Gauss-Lobatto grid points, using the QZ algorithm. In the modal analysis, the focus is on the changes made by the channel wall slip on the eigenvalue spectra. The flow is shown to be linearly stable for no-slip and streamwise slip conditions. However, for the spanwise slip condition, an unstable eigenvalue is found for large values of the spanwise slip number. The non-modal stability analysis is based on the transient energy growth of the perturbations, which is calculated using the eigenfunctions of the eigenvalue problem representing the amplitude of the perturbations. Increasing the streamwise slip decreases the maximum energy growth, eventually leading to a flow for which the transient energy only decays in time. Thus, two distinct flows showing the energy growth and energy decay regimes can be identified, for a range of Bingham, Reynolds and streamwise slip numbers.

The nonmodal kinetic theory for the electrostatic instabilities of a plasma with a sheared Hall current

The kinetic theory for the instabilities driven by the Hall current with a sheared current velocity, which has the method of the shearing modes or the so-called nonmodal approach as its foundation, is developed. The developed theory predicts that in the Hall plasma with the inhomogeneous electric field, the separate spatial Fourier mode of the perturbations is determined in the frame convected with one of the plasma components. Because of the different shearing of the ion and electron flows in the Hall plasma, this mode is perceived by the second component as the Doppler-shifted continuously sheared mode with time-dependent wave numbers. Due to this effect, the interaction of the plasma components forms the nonmodal time-dependent process, which should be investigated as the initial value problem. The developed approach is applied to the solutions of the linear initial value problems for the hydrodynamic modified two-stream instability and the kinetic ion-sound instability of the plasma with a sheared Hall current with a uniform velocity shear. These solutions reveal that the uniform part of the current velocity is responsible for the modal evolution of the instability, whereas the current velocity shear is the source of the development of the nonmodal instability with exponent growing with time as ∼(t−t0)3.

Modelling the radiation efficiency of orthotropic cross-laminated timber plates with simply-supported boundaries

In this paper two prediction models to evaluate the radiation efficiency of orthotropic plates, developed with different approaches, are presented. A sound radiation model, based on an analytical/modal approach, is developed for a thin orthotropic plate, with the principal directions aligned with the edges. The model allows to consider the contribution of each mode, either resonant or non-resonant, as well as the influence of fluid loading on the plate dynamic response and on sound radiation. Moreover, a statistical model to evaluate the average radiation efficiency, based on a non-modal approach, which only considers the contribution of resonant modes, is presented. These two models have been used in order to predict the radiation efficiency of orthotropic cross-laminated timber (CLT) plates. CLT is an engineered wood material constituted by an odd number of lumber beams glued together, which have become very popular in the last twenty years in the building construction market. Due to their layered structure, CLT plates might exhibit a highly orthotropic behaviour. Both prediction models are validated by comparing the simulated results with the experimental radiation efficiency, obtained by means of vibro-acoustic measurements on three CLT plates. Finally, the influence of fluid loading on sound power radiated by CLT plates is investigated.

Nonmodal modified Simon-Hoh instability of a plasma with a shearing Hall current

A new analytical nonmodal approach to investigate the plasma instabilities driven by sheared current is presented and applied to the analysis of the linear evolution of modified Simon-Hoh (MSH) instability of a plasma in an inhomogeneous electric field. We found analytically a strong nonmodal growth for this instability which is missed completely in the normal mode analysis. It dominates the normal mode growth when the shearing rate of the current velocity is above the growth rate of the MSH instability with a uniform current. It also includes the nonmodal growth for subcritical perturbations, which do not develop in plasmas with uniform Hall current.

Continuum-Based Geometry/Analysis Approach for Flexible and Soft Robotic Systems

Control and stability of flexible and soft robotic systems (FSRS), which have complex geometry and experience desirable and undesirable deformations, are of major concern, particularly when lightweight soft materials are used. Nonetheless, there is no unified continuum-based geometry/analysis approach that can be used for the efficient FSRS virtual prototyping and design. The goal of this article is to propose a new FSRS geometric modeling and analysis methodology by addressing fundamental virtual prototyping challenges that include: (1) integration of the robot geometry and analysis; (2) implementation of general and unconventional material models and actuation forces; (3) use of new concepts for modeling FSRS joints; and (4) development of efficient and robust algorithms for the FSRS virtual prototyping. To address these challenges, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system computational algorithms are used. ANCF FEs allow for modeling arbitrarily large and coupled displacements, correctly capture complex geometries, allow for implementing general and nonconventional material models, provide accurate definitions of conventional and nonconventional actuation forces, lead to a constant inertia matrix that defines optimally sparse matrix structure of the dynamic equations, and allow for exploiting new geometry concepts to define linear and more general joint constraints instead of the less general and nonlinear joint constraints currently used for robot systems. Using the general structure of the FSRS nonlinear dynamic equations of motion, a nonmodal continuum-based approach can be developed and used to correctly capture the FSRS complex geometry and large deformations.

Linear global stability of two incompressible coaxial jets

The linear stability of two incompressible coaxial jets, separated by a thick duct wall, is investigated by means of both a modal and a non-modal approach within a global framework. The attention is focused on the range of unitary velocity ratios for which an alternate vortex shedding from the duct wall is known to dominate the flow. In spite of the inherent convective nature of jet flow instabilities, such behaviour is shown to originate from an unstable global mode of the dynamics linearised around the axisymmetric base flow. The corresponding wavemaker is located in the recirculating-flow region formed behind the duct wall. At the same time, the transient-growth analysis reveals that huge amplifications (up to $20$ orders of magnitude) of small flow perturbations at the nozzle exit can occur in the subcritical regime, especially for high ratios between the outer and the inner velocities.

Obligation as Optimal Goal Satisfaction

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation Oq is a logical consequence of the obligations Op and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation Op is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.

Integration of geometry and analysis for the study of liquid sloshing in railroad vehicle dynamics

A new continuum-based liquid sloshing approach that accounts for the effect of complex fluid and tank-car geometry on railroad vehicle dynamics is developed in this investigation. A unified geometry/analysis mesh is used from the outset to examine the effect of liquid sloshing on railroad vehicle dynamics during curve negotiation and during the application of electronically controlled pneumatic (ECP) brakes that produce braking forces uniformly and simultaneously across all cars. Using a non-modal approach, the geometry of the tank-car and fluid is accurately defined, a continuum-based fluid constitutive model is employed, and a fluid-tank contact algorithm is developed. The liquid sloshing model is integrated with a three-dimensional multibody system (MBS) railroad vehicle algorithm which accounts for the nonlinear wheel/rail contact. The three-dimensional wheel/rail contact force formulation used in this study accounts for the longitudinal, lateral, and spin creep forces that influence the vehicle stability. In order to examine the effect of the liquid sloshing on the railroad vehicle dynamics during curve negotiation, a general and precise definition of the outward inertia force is defined, and in order to correctly capture the fluid and tank-car geometry, the absolute nodal coordinate formulation (ANCF) is used. The balance speed and centrifugal effects in the case of tank-car partially filled with liquid are studied and compared with the equivalent rigid body model in curve negotiation and braking scenarios. In particular, the results obtained in the case of the ECP brake application of two freight car model are compared with the results obtained when using conventional braking. The traction analysis shows that liquid sloshing has a significant effect on the load distribution between the front and rear trucks. A larger coupler force develops when using conventional braking compared with ECP braking, and the liquid sloshing contributes to amplifying the coupler force in the ECP braking case compared to the equivalent rigid body model which does not capture the fluid nonlinear inertia effects. Furthermore, the results obtained in this study show that liquid sloshing can exacerbate the unbalance effects when the rail vehicle negotiates a curve at a velocity higher than the balance speed.

Energy growth in Hagen–Poiseuille flow of Herschel–Bulkley fluid

Linear stability of Poiseuille flow of Herschel–Bulkley fluid in a cylindrical pipe is studied using modal and non-modal approaches. The first part of the present study thus deals with the classical normal mode approach in which the resulting eigenvalue problem is solved using a Chebyshev collocation method. Within the considered range of parameters, the modal-linear theory predicts that perturbations are dumped exponentially. In the second part, the effect of the rheological behavior of the fluid on the pseudospectra and the most amplified perturbations is investigated. At very low Herschel–Bulkley number (Hb < <1), the optimal perturbation consists of almost streamwise vortices, and the amplification of the kinetic energy is provided by the lift-up mechanism. In contrast, for sufficiently large values of Hb, the optimal perturbation is axisymmetric and the growth of the kinetic energy is provided by the Orr-mechanism. For intermediate values of Hb, the optimal perturbation is oblique. The amplification of such perturbation is due to a synergy between Orr and lift-up mechanisms. In the last part of the study, the maximal value of the Reynolds number, RecE, below which the perturbation energy decreases monotonically with time is computed for a large range of Hb. Asymptotic behaviors of RecE for Hb < <1 and Hb > >1 are established. The influence of the terms arising from the viscosity perturbation is highlighted throughout this study.

Singular spectrum analysis combined with ARMAX model for structural damage detection

Time series analysis is being used popularly in structural health monitoring mainly because of its output-only and non-modal approach. Generally, the damage features are extracted either from the coefficients or the prediction errors of the time series models. However, when the incipient damage is small like minor cracks, the damage features of popularly used time series models, constructed using only the coefficients/prediction errors, are not sensitive. Therefore, identifying the presence or exact spatial damage location becomes difficult. In view of this, in this paper, we present an approach to enhance the sensitivity of the damage features by augmenting Singular Spectrum Analysis (SSA) to ARMAX model, enabling it to locate the smaller damage like cracks. The damage index is obtained from the Cepstral distance between any two ARMAX models. Numerical simulation studies have been carried out by considering an example of a simply supported beam girder with single and multiple cracks. Experimental studies on a simply supported RCC beam is conducted to demonstrate the effectiveness of the proposed algorithm. A benchmark problem associated with the bookshelf frame structure, proposed by Engineering Institute –Los Alamos National Laboratory, is used as another example for experimental verification of the proposed technique. SSA is found to improve the sensitivity of the damage features devised from the ARMAX models for detection of minor damage and damage localization on the structures.

Optimal perturbations in time-dependent variable-density Kelvin–Helmholtz billows

We analyse the influence of the specific features of time-dependent variable-density Kelvin–Helmholtz (VDKH) roll-ups on the development of three-dimensional secondary instabilities. Due to inertial (high Froude number) baroclinic sources of spanwise vorticity at high Atwood number (up to 0.5 here), temporally evolving mixing layers exhibit a layered structure associated with a strain field radically different from their homogeneous counterpart. We use a direct-adjoint non-modal linear approach to determine the fastest growing perturbations over a single period of the time-evolving two-dimensional base flow during a given time interval $[t_{0},T]$. When perturbations are seeded at the initial time of the primary KH mode growth, i.e. $t_{0}=0$, it is found that additional mechanisms of energy growth are onset around a bifurcation time $t_{b}$, a little before the saturation of the primary two-dimensional instability. The evolution of optimal perturbations is thus observed to develop in two distinct stages. Whatever the Atwood number, the first period $[t_{0},t_{b}]$ is characterised by a unique route for optimal energy growth resulting from a combination of the Orr and lift-up transient mechanisms. In the second period $[t_{b},T]$, the growing influence of mass inhomogeneities raises the energy gain over the whole range of spanwise wavenumbers. As the Atwood number increases, the short spanwise wavelength perturbations tend to benefit more from the onset of variable-density effects than large wavelength ones. The extra energy gain due to increasing Atwood numbers relies on contributions from spanwise baroclinic sources. The resulting vorticity field is structured into two elongated dipoles localised along the braid on either side of the saddle point. In return they yield two longitudinal velocity streaks of opposite sign which account for most of the energy growth. This transition towards three-dimensional motions is in marked contrast with the classic streamwise rib vortices, so far accepted as the paradigm for the transition of free shear flows, either homogeneous or not. It is argued that the emergence of these longitudinal velocity streaks is generic of the transition in variable-density shear flows. Among them, the light round jet is known to display striking side ejections as a result of the loss of axisymmetry. The present analysis helps to renew the question of the underlying flow structure behind side jets, otherwise based on radial induction between pairs of counter-rotating longitudinal vortices (Monkewitz &amp; Pfizenmaier, Phys. Fluids A, vol. 3 (5), 1991, pp. 1356–1361). Instead, it is more likely that side ejections would result from the convergence of the longitudinal velocity streaks near the braid saddle point. When the injection time is delayed so as to suppress the initial stage of energy growth, a new class of perturbations arises at low wavenumber with energy gains far larger than those observed so far. They correspond to the two-dimensional Kelvin–Helmholtz secondary instability of the baroclinically enhanced vorticity braid discovered by Reinaud et al. (Phys. Fluids, vol. 12 (10), pp. 2489–2505), leading potentially to another route to turbulence through a two-dimensional fractal cascade.

A comparative numerical analysis of linear and nonlinear aerodynamic sound generation by vortex disturbances in homentropic constant shear flows

Aerodynamic sound generation in shear flows is investigated in the light of the breakthrough in hydrodynamics stability theory in the 1990s, where generic phenomena of non-normal shear flow systems were understood. By applying the thereby emerged short-time/non-modal approach, the sole linear mechanism of wave generation by vortices in shear flows was captured [G. D. Chagelishvili, A. Tevzadze, G. Bodo, and S. S. Moiseev, “Linear mechanism of wave emergence from vortices in smooth shear flows,” Phys. Rev. Lett. 79, 3178-3181 (1997); B. F. Farrell and P. J. Ioannou, “Transient and asymptotic growth of two-dimensional perturbations in viscous compressible shear flow,” Phys. Fluids 12, 3021-3028 (2000); N. A. Bakas, “Mechanism underlying transient growth of planar perturbations in unbounded compressible shear flow,” J. Fluid Mech. 639, 479-507 (2009); and G. Favraud and V. Pagneux, “Superadiabatic evolution of acoustic and vorticity perturbations in Couette flow,” Phys. Rev. E 89, 033012 (2014)]. Its source is the non-normality induced linear mode-coupling, which becomes efficient at moderate Mach numbers that is defined for each perturbation harmonic as the ratio of the shear rate to its characteristic frequency. Based on the results by the non-modal approach, we investigate a two-dimensional homentropic constant shear flow and focus on the dynamical characteristics in the wavenumber plane. This allows to separate from each other the participants of the dynamical processes — vortex and wave modes — and to estimate the efficacy of the process of linear wave-generation. This process is analyzed and visualized on the example of a packet of vortex modes, localized in both, spectral and physical, planes. Further, by employing direct numerical simulations, the wave generation by chaotically distributed vortex modes is analyzed and the involved linear and nonlinear processes are identified. The generated acoustic field is anisotropic in the wavenumber plane, which results in highly directional linear sound radiation, whereas the nonlinearly generated waves are almost omni-directional. As part of this analysis, we compare the effectiveness of the linear and nonlinear mechanisms of wave generation within the range of validity of the rapid distortion theory and show the dominance of the linear aerodynamic sound generation. Finally, topological differences between the linear source term of the acoustic analogy equation and of the anisotropic non-normality induced linear mechanism of wave generation are found.

Linear stability of Taylor–Couette flow of shear-thinning fluids: modal and non-modal approaches

In this paper, the response of circular Couette flow of shear-thinning fluids between two infinitely long coaxial cylinders to weak disturbances is addressed. It is highlighted by transient growth analysis. Both power-law and Carreau models are used to describe the rheological behaviour of the fluid. The first part of the paper deals with the asymptotic long-time behaviour of three-dimensional infinitesimal perturbations. Using the normal-mode approach, an eigenvalue problem is derived and solved by means of the spectral collocation method. An extensive description and the classification of eigenspectra are presented. The influence of shear-thinning effects on the critical Reynolds numbers as well as on the critical azimuthal and axial wavenumbers is analysed. It is shown that with a reference viscosity defined with the characteristic scales$\hat{{\it\mu}}_{ref}=\hat{K}(\hat{R}_{1}\hat{{\it\Omega}}_{1}/\hat{d})^{(n-1)}$for a power-law fluid and$\hat{{\it\mu}}_{ref}=\hat{{\it\mu}}_{0}$for a Carreau fluid, the shear-thinning character is destabilizing for counter-rotating cylinders. Moreover, the axial wavenumber increases with$\mathit{Re}_{2}$and with shear-thinning effects. The second part investigates the short-time behaviour of the disturbance using the non-modal approach. For the same inner and outer Reynolds numbers, the amplification of the kinetic energy perturbation becomes much more important with increasing shear-thinning effects. Two different mechanisms are used to explain the transient growth, depending on whether or not there is a stratification of the angular momentum. On the Rayleigh line and for Newtonian fluids, the optimal perturbation is in the form of azimuthal streaks, which transform into Taylor vortices through the anti-lift-up mechanism. In the other cases, the optimal perturbation is initially oriented against the base flow, then it tilts to align with the base flow at optimal time. The scaling laws for the optimal energy amplification proposed in the literature for Newtonian fluids are extended to shear-thinning fluids.

Global dynamics analysis of nappe oscillation

The unsteady global dynamics of a gravitational liquid sheet interacting with a one-sided adjacent air enclosure, typically referred to as nappe oscillation, is addressed, under the assumptions of potential flow and absence of surface tension effects. To the purpose of shedding physical insights, the investigation examines both the dynamics and the energy aspects. An interesting re-formulation of the problem consists of recasting the nappe global behavior as a driven damped spring-mass oscillator, where the inertial effects are linked to the liquid sheet mass and the spring is represented by the equivalent stiffness of the air enclosure acting on the average displacement of the compliant nappe centerline. The investigation is carried out through a modal (i.e., time asymptotic) and a non-modal (i.e., short-time transient) linear approach, which are corroborated by direct numerical simulations of the governing equation. The modal analysis shows that the flow system is characterized by low-frequency and high-frequency oscillations, the former related to the crossing time of the perturbations over the whole domain and the latter related to the spring-mass oscillator. The low-frequency oscillations, observed in real life systems, are produced by the (linear) combination of multiple modes. The non-normality of the operator is responsible for short-time energy amplifications even in asymptotically stable configurations, which are confirmed by numerical simulations and justified by energy budget considerations. Strong analogies with the edge-tone problem are encountered; in particular, the integer-plus-one-quarter resonance criterion is uncovered, where the basic frequency to be multiplied by n+14 is just the one related to the spacing among the imaginary parts of the eigenvalues.