Linear Stability Theory Research Articles

Non-linear analysis of the flexural-torsional stability of slender tropical glulam beams

The main objective of this manuscript is to model the flexural-torsional buckling of slender tropical glulam beams using the finite element method. The non-linear approach to quantify both the buckling strength and the lateral geometrical features of wooden beams is proposed. A very rigorous methodology is used which is based on the kinematic Vlasov’s theory to determine the displacement field of the beam section by considering an updated Lagrangian description. For the finite element discretization, the Author develop a single wood element characterized by 14 degrees of freedom, i.e. seven degrees for each nodal point, and derive non-linear stiffness matrices by using the virtual works principle. The model is implemented in MatLab and the numerical results are compared with results based on the classical linear stability theory of slender wood beams. The selected case-studies are a cantilever beam and a beam on two supports.

Impact of rotation on thermal instability of Darcy–Brinkman porous layer filled with a Jeffrey nanofluid

The novelty of this research lies in its investigation of the initiation of thermal instability within a Darcy–Brinkman porous layer containing a Jeffrey nanofluid under the influence of rotation. While non-Newtonian fluids complex flow patterns and heat transfer enhancements due to thermal gradients are known, this study addresses a specific and previously unexplored aspect, the impact of rotation on Darcy–Brinkman porous layer containing a Jeffrey nanofluid. When a fluid is subjected to rotation, it can induce additional forces and flow patterns within the fluid. The linear stability theory and normal mode analysis method are used to analyze the stability of the system analytically and are computed numerically using the software Mathematica version 11.3. The investigation focuses on understanding the influence of various factors, namely, the Taylor number, the Lewis number, the Jeffrey parameter, the modified diffusivity ratio, nanoparticles Rayleigh number, the Darcy–Brinkman number, and medium porosity on the onset of thermal instability in the physical system. This novel perspective can contribute to a deeper understanding of heat transfer processes in non-Newtonian fluids and has potential applications in improving the efficiency of industrial systems, such as cooling systems, heat exchangers, and microfluidic devices where efficient heat transfer is crucial for performance improvement.

Interactions between oblique second mode and oblique waves in a high-speed boundary layer

Interactions between oblique second mode and oblique waves at a high-speed boundary at Mach 4.5 are studied using linear stability theory, nonlinear parabolized stability equations (NPSE) and direct numerical simulation (DNS). Parametric analysis based on the NPSE suggests that the oblique second mode can amplify both the oblique first and second modes, with the former experiencing a higher amplification. Our analysis reveals that the mean-flow distortion and difference mode contribute to this enhancement, with the latter exerting a key influence through the parametric resonance process. Kinetic energy transfer analysis demonstrates that the oblique waves gain energy from the mean flow, rather than from the oblique second mode. Furthermore, we find that the mechanism underlying the interaction between a pair of second oblique waves and a single oblique wave is similar to that between an oblique second mode and an oblique wave, as the steady modes generated by the pair of oblique second modes have a limited impact on the oblique wave. Finally, DNS confirms the validity of two transition paths proposed in this study based on the NPSE results. The first path suggests that a pair of low-amplitude second oblique waves alone are insufficient to cause oblique breakdown, but the introduction of a pair of low-amplitude damping first oblique modes could lead to boundary layer breakdown. The second path involves the formation of a domino-like effect through the combination of different types of oblique waves with the appropriate parameters. These two nonlinear paths can lead to a fully developed turbulent boundary layer.

The equivalence between BE-LSE and NS-LSEs under continuum assumption

The linear stability theory, which is based on the Navier–Stokes equations, has been extensively studied in the field of hydrodynamic stability. However, due to the continuum assumption, there exist various limitations when attempting to solve problems that involve rarefaction effects. In contrast, based on kinetic theory, the Boltzmann equation is well-suited for analyzing linear stability throughout the entire regime, ranging from continuous to rarefied flows. In this paper, a linear stability equation derived from the Boltzmann–Bhatnagar–Gross–Krook (Boltzmann-BGK) equation is introduced, and the relationship between small perturbations of the velocity distribution function and macroscopic physical variables is established. Under the continuum assumption, the linear stability equation based on the Boltzmann equation (BE-LSE) and the linear stability equations based on the Navier–Stokes equations (NS-LSEs) have the same numerical solutions, which indicates that the microscopic BE-LSE can recover to macroscopic NS-LSEs theoretically. Nevertheless, there is still a lack of mathematical proof for this theoretical outcome. To address this issue, the research on the equivalence of BE-LSE and NS-LSEs under continuum assumption is carried out in this paper, which can establish the theoretical relationship between BE-LSE and NS-LSEs. These efforts lay a solid theoretical foundation for stability research based on the BE-LSE.

Input–output finite-time stabilisation for periodic piecewise polynomial systems with nonlinear actuator faults: an observer-based approach

This paper is concerned with the input–output finite-time (IO-FT) stabilisation problem for a class of continuous-time periodic piecewise polynomial systems (PPPSs) with immeasurable states and external disturbances via state estimation-based robust reliable controller. Firstly, to reflect the actuality, a parameter uncertainty that exhibits the random nature and the actuator faults with nonlinear character are considered in the addressed PPPSs and the control scheme, respectively. In detail, the randomness of uncertain parameters is portrayed by the stochastic variable and it is presumed to pursue the Bernoulli distributions. Secondly, to estimate the immeasurable states of PPPSs, a periodic piecewise polynomial observer is designed depending on the output of PPPSs. The main intent of this article is to devise a state estimation-based robust reliable controller to ascertain the IO-FT stabilisation of the PPPSs. Moreover, by bridging the Lyapunov stability theory, linear matrix inequality technique and IO-FT stability theory, the required IO-FT stabilisation conditions in the frame of linear matrix inequality are procured for the system under consideration. Eventually, simulation results of the addressed PPPSs are shown in line with the proposed analytical findings, revealing the competence, inherent capability and utility of the devised control protocol.

Effects of Small Leading-Edge Bluntness on High-Speed Boundary-Layer Instabilities on Flat Plates

A numerical investigation of the effect of small leading-edge bluntness on the linear stability regime in high-speed boundary layers for flat plates was carried out using linear stability theory. While it is known that increasing leading-edge bluntness leads to stabilization of the linear instability, it is not completely understood when leading-edge bluntness effects become relevant; in other words, for which leading-edge radii the primary instability regime is affected for flat plates at higher Mach numbers. For the so-called “sharp” geometries, it is assumed that the influence of the nose radius on stability and transition is negligible. For cones, it is generally accepted that nose radii resulting in Reynolds numbers (based on nose radius) smaller than 1000 can be considered sharp. However, no clear guidelines supported by numerical or experimental data exist to conclusively determine what constitutes a sharp leading edge for a flat plate. In this work, bluntness effects are assessed by considering the influence of the leading-edge radius on the linear stability characteristics for high-speed flat-plate boundary layers. It was found from linear stability calculations that surprisingly small leading-edge radii have significant effects on the critical Reynolds number and spatial amplification rates, thus resulting in substantially reduced maximum -factors.

Magneto-Rotatory Convection in Couple-Stress Fluid

Background: Thermal convection is the most convective instability when crystals are produced from a single element like silicon and the thermal instability of a fluid layer heated from below plays an important role in geophysics, oceanography, atmospheric physics, etc. The flow through porous media is of considerable interest for petroleum engineers, for geophysical fluid dynamicists and has importance in chemical technology and industry. Many of the flow problems in fluids with couple-stresses indicate some possible experiments, that could be used for determining the material constants, and the results are found to differ from those of Newtonian fluid. Keeping this in view, the present work was to study the effect of a uniform vertical magnetic field on the couple-stress fluid heated from below in the presence of a uniform vertical rotation through permeable media. Methodology: The present problem is studied using the linearized stability theory, Boussinesq approximation, normal mode analysis, and the dispersion relation is obtained. Results: The stationary convection, stability of the system, and oscillatory modes are discussed. In the case of stationary convection, the rotation postpones the onset of convection. The magnetic field and couple-stress may hasten the onset of convection in the presence of rotation while in the absence of rotation; they always postpone the onset of convection. The medium permeability hastens the onset of convection in the absence of rotation while in the presence of rotation, it may postpone the onset of convection. The rotation and magnetic field are found to introduce oscillatory modes in the system which was non-existent in their absence. A sufficient condition for the non-existence of overstability is also obtained.

Thermo-electric convection in a cylindrical annulus during a sounding rocket flight

An experiment on convective flows induced by the dielectrophoretic force was performed under the microgravity condition provided during a sounding rocket flight. The dielectrophoretic force possesses a non-conservative term that can be seen as resulting from an electric gravity. That gravity can be responsible for an electric Rayleigh–Bénard convection between a hot inner cylinder and a cold outer cylinder when an electric field is applied in the radial direction. Four cells with independent temperature and electric field controls allowed the investigation of eight different values of the electric Rayleigh number relatively close to the onset of the thermo-electric instability. A linear stability analysis is performed to predict the stability threshold and the evolution of the growth rate of the instability. The three-dimensional structure of the flow is captured by simultaneous particle image velocimetry and by shadowgraphy. The amplitude of the instability modes and the time evolution of the flow is analysed, and various methods are proposed to extrapolate the experimental critical value of the electric Rayleigh number for the onset of convection. The measured critical electric Rayleigh number is in agreement with the prediction of the linear stability theory. The comparison of the new experimental results with previous ones from parabolic flight campaigns highlights the importance of long-term microgravity for the achievement of thermal convection at low values of the control parameters.

Dynamical systems analysis of an interacting scalar field model in an anisotropic universe

In this paper, we investigate a non-canonical scalar field model in the background dynamics of anisotropic Locally Rotationally Symmetric (LRS) Bianchi type I universe where gravity is coupled minimally to scalar field which is taken as dark energy and pressureless dust as dark matter are the main matter content of the universe. We perform dynamical system analysis to characterize the cosmological evolution of the model with and without interaction in the dark sector separately. First, we convert the evolution equation into an autonomous system of ordinary differential equations by using a suitable choice of dimensionless variables, which are normalized over the Hubble scale. We choose scalar field coupling and potential in such a way that the autonomous system converted to a 2D system. Linear stability theory is employed to the extracted critical points to find the nature. From the analysis, we find some interesting cosmological scenarios, such as late-time scalar-field dominated solutions, which evolve in the quintessence era, cannot solve the coincidence problem. Accelerated scaling attractors are also obtained that correspond to the late phase evolution in agreement with present observational data, and these solutions also provide possible mechanisms to alleviate the coincidence problem. A complete cosmic evolution is obtained from early inflation to a late-time dark energy-dominated phase, connecting through a matter-dominated transient phase of the universe. Furthermore, we find that for different values of the interaction parameter α, the evolutionary trajectories of the Hubble parameter, and the distance modulus forecasted by the model are in quite well agreement with observational datasets.

The effect of cant angle on the sound production of mutually inclined rectangular supersonic jets

Rectangular nozzle geometries offer a number of benefits over their axisymmetric counterparts, particularly for military aviation. Among these benefits are altered thermal and noise signatures, ease of airframe integration, and simpler thrust vectoring. Many next-generation tactical aircraft concepts employ two rectangular jets in close proximity. These jets can couple together, producing high-amplitude acoustic tones that are deleterious to aircraft structure. In this work, we take a combined experimental and theoretical approach to examine the effect of cant angle (i.e., mutual inclination angle) on the jets' proclivity to couple. We employ modal decomposition of flow visualization data alongside resonance predictions based on linear stability theory to explore the degree to which mutual inclination can suppress resonance.

Aerodynamic drag reduction through a hybrid laminar flow control and variable camber coupled wing

This paper presents computational fluid dynamics (CFD) results of a transonic transport aircraft wing, featuring a hybrid laminar flow control (HLFC) and variable camber (VC) technology coupling. The goal of the paper is a quantitative assessment of synergistic effects for aerodynamic drag reduction, when combining the possibility of actively shaping the surface pressure distribution of the wing through VC with the passive natural laminar flow aspect of HLFC. Modeling approaches for both VC integration, achieved through deflections of an Adaptive Dropped Hinge Flap (ADHF), and boundary layer suction into a CFD workflow are presented. Transition prediction is accomplished with the γ−Reθ+Cross-flow(CF) transition turbulence model, and linear stability theory (LST) coupled to a two-N-factor transition prediction method. Both methods reflect synergy-driven effects. While the extent of laminar flow predicted by the γ−Reθ+CF model is limited, both favorable ADHF settings and boundary layer suction lead to an overall drag reduction. This is correspondingly reflected through the analyses via LST. Furthermore, the latter predicts a pronounced downstream shift in transition position with increasing suction strength, where the inherent differentiation of transition mechanisms in the two-N-factor method between Tollmien-Schlichting (TSI) and cross-flow instabilities (CFI) reflects a marked reduction in TSI N-factors with increasing ADHF deflection angles. Even though N-factors associated to CFI partially increase through ADHF deflections, the intended synergistic effects are represented throughout the investigated parameter range, leading to an increase in the extent of laminar flow for adequate ADHF deflection angles.

High-subsonic boundary-layer flows of an organic vapour

Boundary layers of Novec649, a low-global-warming potential fluid of interest for low-grade heat recovery, are investigated numerically by means of linear stability theory, direct numerical simulation (DNS) and large-eddy simulations (LES). This organic vapour is of interest in organic Rankine cycle (ORC) turbines and realistic thermodynamic conditions are selected. Under these conditions, the vapour behaves as a dense gas and, due to its high molecular complexity, real-gas effects occur. In addition, the fluid exhibits large and highly variable heat capacities and density- as well as temperature-dependent transport properties. More specifically we report the first direct and LES of transitional and turbulent boundary layers of Novec649 at high-subsonic conditions$M=0.9$. A controlled transition is performed by using oblique modes determined by linear stability theory extended to dense gases. An oblique-type transition is obtained as in low-speed air flows, where sinuous streaks develop by the lift-up mechanism and break down into turbulence. In the turbulent state, the profiles of dynamic flow properties (velocities, turbulent intensities, turbulent kinetic energy budgets) are little affected by the gas properties and remain very close to incompressible DNS, despite the high-subsonic flow speed. The fluctuations levels for thermodynamic properties have been quantified with respect to air flows. Notwithstanding a drastic reduction, genuine compressibility effects are present. For example, the fluctuating Mach number and the acoustic mode are characteristic of high-speed flows. The influence of forcing frequency and amplitude on the established turbulent state has been investigated using LES. An analysis of integral quantities shows a slow relaxation towards a canonical equilibrium turbulent state for all cases due to the high Reynolds numbers typical of dense gas flows. Overall the present DNS constitutes a valuable reference not only for forthcoming experiments but also for future studies of free-stream transition and loss mechanisms in ORC turbines.

Effect of cone rotation on the nonlinear evolution of Mack modes in supersonic boundary layers

In this paper, we present a systematic study of the nonlinear evolution of the travelling Mack modes in a Mach 3 supersonic boundary layer over a rotating cone with a $7^{\circ }$ half-apex angle using the nonlinear parabolic stability equation (NPSE). To quantify the effect of cone rotation, six cases with different rotation rates are considered, and from the same streamwise position, a pair of oblique Mack modes with the same frequency but opposite circumferential wavenumbers are introduced as the initial perturbations for NPSE calculations. As the angular rotation rate $\varOmega$ increases such that $\bar \varOmega$ (defined as the ratio of the rotation speed of the cone to the streamwise velocity at the boundary-layer edge) varies from 0 to $O(1)$ , three distinguished nonlinear regimes appear, namely the oblique-mode breakdown, the generalised fundamental resonance and the centrifugal-instability-induced transition. For each regime, the mechanisms for the amplifications of the streak mode and the harmonic travelling waves are explained in detail, and the dominant role of the streak mode in triggering the breakdown of the laminar flow is particularly highlighted. Additionally, from the linear stability theory, the dominant travelling mode undergoes the greatest amplification for a moderate $\varOmega$ , which, according to the $e^N$ transition-prediction method, indicates premature transition to turbulence. However, this is in contrast to the NPSE results, in which a delay of the transition onset is observed for a moderate $\varOmega$ . Such a disagreement is attributed to the different nonlinear regimes appearing for different rotation rates. Therefore, the traditional transition-prediction method based on the linear instability should be carefully employed if multiple nonlinear regimes may appear.

Influence of distributed hydrogen injection and combustion on supersonic boundary layer stability and transition

This theoretical study recorded the influence of the distributed injection of hydrogen through a permeable surface and its combustion on the stability and transition of a supersonic boundary layer (BL). The laminar base flow for a multicomponent flat-plate BL was computed using the computational fluid dynamics solver with finite-rate chemistry for free-stream Mach number (M) = 2. Linear stability theory (LST) equations for a reactive gas mixture were derived in the local parallel base flow approximation. Stability calculations based on the developed theory revealed the possibility of a decrease in the local spatial amplification rates of unstable perturbations. A double reversal in the maximal amplification rate magnitudes of perturbations was obtained, wherein at first, they increased, then decreased to zero, and then rose again. The influence of the distributed injection and combustion of hydrogen in the supersonic BL on the position of the laminar-turbulent transition was estimated using the LST-based eN-method. Calculated and experimentally obtained dependencies of the relative transition Reynolds number on the relative mass flow rate of injected hydrogen were compared. It was found that depending upon the flow conditions, the hydrogen diffusion flame could accelerate or decelerate the transition in the supersonic M = 2 BL, when compared to absence of hydrogen burning. Two counteracting effects compete: heat supply by combustion exerts stabilizing influence, while low-molecular-weight gas blowing from the surface exerts destabilizing influence. Depending on the interplay of these two factors, it would be possible to obtain acceleration or deceleration of transition.

Regular and Chaotic Oscillations in a Geostrophic Flow with Vertical Shear

In the framework of a two-level quasi-geostrophic model, the stability of flow with a constant vertical shear is investigated. Analytical expressions for the increment of perturbation growth in linear stability theory were obtained. The Galerkin method with three basic Fourier harmonics was used to describe the nonlinear dynamics of perturbations. A nonlinear system of ordinary differential equations is formulated for amplitudes of Fourier harmonics. It is shown that in the absence of bottom friction all solutions of the system describe periodic mode of nonlinear oscillations or vascillations. The situation changes fundamentally in the model with bottom friction. In this case, for a wide range of parameter values, the system solutions exhibit complex chaotic behavior. Thus, chaos or turbulence emerges for large-scale motions.

Reynolds number induced growth of the large-scale rolls in plane Couette flow using resolvent analysis

In direct numerical simulations (DNS) of turbulent Couette flow, the observation has been made that the long streamwise rolls increase in length with the Reynolds number (Lee & Moser, J. Fluid Mech., vol. 842, 2018, pp. 128–145). To understand this, we employ both linear stability theory and its extension to resolvent analysis. For this, we emphasise the high Reynolds number ( $Re \rightarrow \infty$ ) and small streamwise wavenumbers ( $\alpha \rightarrow 0$ ) limit, imposing the distinguished limit $Re_{\alpha }=Re \, \alpha = O(1)$ . We find that in case of linear stability theory, $Re_{\alpha }$ acts as a global invariant in the resulting eigenvalue problem, while in case of resolvent analysis, $Re_{\alpha }$ acts as a local invariant in the behaviour of the energy of the system characterised through the first singular value $\sigma _1$ of the resolvent operator within the investigated asymptotic limit. In order to obtain constant streamwise structures for increasing Reynolds numbers, the respective streamwise wavenumber has to decrease, which verifies the observations from DNS studies of an increasing length of the streamwise structures with the Reynolds number. In linear stability theory, a parameter reduction is achieved for the above asymptotic limit, resulting in the modified Orr–Sommerfeld and Squire equations being dependent only on $Re_{\alpha }$ . The behaviour of both the coherent structures obtained from linear stability theory and resolvent analysis are compared with each other and show similar behaviours over $Re_{\alpha }$ .

Amplification factor transport transition model for high-speed flows

A fully local amplification factor transport model is developed for high-speed transitional flows. On the basis of the approximate eN envelope method of linear stability theory, two transport equations are established to describe the evolution of amplification factors for first- and second-mode instabilities in high-speed boundary layers. An intermittency factor transport equation is then constructed based on the transported amplification factors and is coupled with Menter’s k−ω shear-stress transport eddy-viscosity turbulence model. The new model is validated through several test cases under different flow conditions, including flat plates, straight and flared cones, and a double ramp configuration. Comparisons with linear stability theory and experimental data demonstrate the ability of the model to predict the transition behavior induced by first- or second-mode instabilities. The model provides a reasonable reflection of the effects of different parameters that influence transitions, including the Mach number, temperature, nose bluntness, and pressure gradient.

Oscillatory Modes on the Onset of Electrohydrodynamic Instability in Oldroydian Nanofluid Saturated Anisotropic Porous Layer

This work deals with an analytical study on the initiation of oscillatory convection in a rheological nanofluid saturating anisotropic porous layer with inclusion of vertical AC electric field using modified boundary conditions with negligible flux of volume fraction of nanoparticles. The rheological properties of the nanofluid are described using Oldroyd model. The Darcy model extended by Brinkman model is deployed to characterize the solid matrix behavior. The used model for nanofluid with inclusion of electric field integrates the additional effect of electrophoresis with that of thermophoresis and Brownian motion in the conservation equations of motion. The partial differential equations are simplified to non-dimensional linear equations using infinitesimal perturbations, Boussinesq approximation, normal mode technique and linearized stability theory. The characteristic equation is solved analytically for stress-free boundary conditions and the expressions for Rayleigh number of non-oscillatory and oscillatory modes initiation are determined. The oscillatory modes are found to occur for both the cases of top-/bottom-heavy nanoparticles distributions. The electric Rayleigh number, thermal Prandtl number and stress relaxation parameter advances whereas the Brinkman-Darcy number are found to delay initiation of both stationary and oscillatory convection.

Numerical study of the hydrodynamic stability of a wind-turbine airfoil with a laminar separation bubble under free-stream turbulence

The interaction of several instabilities and the influence of free-stream turbulence on laminar-turbulent transition on a 20% thick wind-turbine blade section with a laminar separation bubble (LSB) are investigated with wall-resolved large-eddy simulations (LES). Turbulence intensities (TI) of 0%, 2.2%, 4.5%, 8.6%, and 15.6% at chord Reynolds number 105 are considered. Linear receptivity occurs for the most energetic disturbances; high-frequency perturbations are excited via non-linear mechanisms for TI≥8.6%. Unstable Tollmien–Schlichting (TS) waves appear in the inflectional flow region for TI≤4.5%, shifting to inviscid Kelvin–Helmholtz (KH) modes upon separation and forming spanwise rolls. Sub-harmonic secondary instability occurs for TI=0%, with rolls intertwining before transition. Streaks spanwise modulate the rolls and increase their growth rates with TI for TI≤4.5%, reducing separation and shifting transition upstream. The TI=4.5% case presents the highest perturbations, leading to the smallest LSB and most upstream transition. Earlier inception of TS/KH modes occurs on low-speed streaks, inducing premature transition. However, for TI=8.6%, the effect of the streaks is to stabilize the attached mean flow and front part of the LSB. This occurs due to the near-wall momentum deficit alleviation, leading to the transition delay and larger LSB than TI=4.5%. This also suppresses separation and completely stabilizes TS/KH modes for TI=15.6%. Linear stability theory predicts well the modal evolution for TI≤8.6%. Optimal perturbation analysis accurately computes the streak development upstream of the inflectional flow region but indicates higher amplification than LES downstream due to the capture of low-frequency, oblique modal instabilities from the LSB. Only low-amplitude [O(1%)] streaks displayed exponential growth in the LES since non-linearity precludes the appearance of these modes.

Convective instability of blood‐based Au‐Fe3O4 hybrid nanoliquid under the presence of magnetic field with internal heat source: Application to cancer treatment

AbstractTargeted drug delivery is one of the most promising aspects of cancer diagnosis and treatment. Gold nanoparticles are widely used for drug delivery and in the treatment of tumors, but due to the low absorption of infrared light, tumor cells get destroyed. However, iron‐oxide‐coated gold nanoparticles could be used for drug delivery to a targeted spot with the help of an external magnetic field. The present article aims to investigate the thermal instability of the blood flow transporting gold and iron‐oxide nanoparticles through the artery. The Casson fluid model is used to characterize the blood flow, and Maxwell equations, together with Navier–Stokes equations, are used to describe the flow behavior. Further, the linear stability theory and normal mode analysis are used to obtain the secular equation of the thermal Rayleigh number. The influence of pertinent flow governing parameters such as heat source parameter, Chandrashekhar number, the diameter of nanoparticles, and volume fraction of nanoparticles are discussed graphically on the convective instability of the system. The addition of gold nanoparticles makes the system unstable due to their large size and heat generation within the system. On the other hand, the magnetic field stabilizes the system by controlling the trajectory of the nanoparticles injected into the blood vessels.