Local Linear Stability Analysis Research Articles

Characterizing disturbance growth and transition of laminar separation bubbles subjected to varying freestream turbulence

The spatial manifestation of a boundary layer subjected to an adverse pressure gradient for varying freestream turbulence (FST) is discussed here through experiments and a wall-resolved large eddy simulation (LES). Three levels of FST, i.e., 1.02%, 2.1%, and 3.5%, are considered at a Reynolds number of 0.2 × 106 based on the inlet velocity and plate length. A laminar separation bubble (LSB) appears for FST levels of 1.02%, where the shear layer becomes unstable via the Kelvin-Helmholtz (K-H) mechanism. Bubble suppression is apparent, where transition occurs via the Klebanoff mode (K-mode), bypassing the K-H instability as the FST level is increased to 3.5%. The streamwise evolution of fluctuations reveals an exponential growth at a low FST of 1.02%, whereas it changes to algebraic at a higher FST level of 3.5%. Notably, a dual growth of velocity fluctuations, i.e., an algebraic growth in the initial part of LSB followed by an exponential growth, illustrates the coexistence of K-H instability and K-mode at an FST level of 2.1%: a finding corroborated by LES. Additionally, local linear stability analysis and the evolution of intermittency provide valuable insights into the growth of modal and non-modal instabilities and their interactions.

The boundary layer instability beneath internal solitary waves and its sensitivity to vortex wakes

We investigated the stability of the bottom boundary layer (BBL) beneath periodic internal solitary waves (ISWs) of depression over a flat bottom through two-dimensional direct numerical simulations. We explored the convective versus absolute/global nature of the BBL instability in response to changes in Reynolds number, and the sensitivity of the instability to seeding noise in the front of the ISW – spanning laboratory to geophysical scales. The BBL was laminar at $Re_{ISW}=90$ and convectively unstable at $Re_{ISW}=300$ . At laboratory-scale $Re_{ISW}=300$ , the convective wave packet was periodically amplified by each successive ISW, until vortex shedding occurred. The associated noise-amplification behaviour potentially explains the discrepancies of the critical $Re_{ISW}$ between the lock–release laboratory experiments and our Dubreil–Jacotin–Long-initialized numerical simulations as the result of the difference in background noise. Instability energy decreased under the front shoulder of the ISW, analogous to flow relaminarization under a favourable pressure gradient. At geophysical-scale $Re_{ISW}=900$ , the BBL was initially convectively unstable, and then the instability tracked with the ISW, appearing phenomenologically similar to a global instability. The simulated initial convective instability at both $Re_{ISW}=300$ and $Re_{ISW}=900$ is in agreement with local linear stability analysis which predicts that the instability group speed is always lower than the ISW celerity. Increased free stream perturbations in front of the ISW and larger $Re_{ISW}$ shift the location of vortex shedding (and enhanced bed shear stress) beneath the wave, closer to the ISW trough, thereby potentially changing the location of maximum sediment resuspension, in agreement with field observations at higher $Re_{ISW}$ .

The seizure classification of focal epilepsy based on the network motif analysis

Due to the complexity of focal epilepsy and its risk for transiting to the generalized epilepsy, the development of reliable classification methods to accurately predict and classify focal and generalized seizures is critical for the clinical management of patients with epilepsy. In order to holistically understand the seizure propagation behavior of focal epilepsy, we propose a three-node motif reduced network by respectively simplifying the focal region, surrounding healthy region and their critical regions as the single node. Because three-node motif can richly characterize information evolutions, the motif analysis method could comprehensively investigate the seizure behavior of focal epilepsy. Firstly, we define a new seizure propagation marker value to capture the seizure onsets and intensity. Based on the three-node motif analysis, it is shown that the focal seizure and spreading can be categorized as inhibitory seizure, focal seizure, focal-critical seizure and generalized seizures, respectively. The four types of seizures correspond to specific modal types respectively, reflecting the strong correlation between seizure behavior and information flow evolution. In addition, it is found that the intensity difference of outflow and inflow information from the critical node (connection heterogeneity) and the excitability of the critical node significantly affected the distribution and transition of the four seizure types. In particular, the method of local linear stability analysis also verifies the effectiveness of four types of seizures classification. In sum, this paper computationally confirms the complex dynamic behavior of focal seizures, and the study of criticality is helpful to propose novel seizure control strategies.

Stability analysis of a streaky boundary layer generated by miniature vortex generators

Delaying laminar-turbulent transition is an attractive method to reduce friction drag on streamlined bodies. In the case of natural transition, in which turbulence is triggered by the growth of two-dimensional Tollmien–Schlichting (TS) instabilities, this growth of TS instabilities can be attenuated by introducing steady streamwise streaks into the boundary layer. These streamwise streaks are generated by streamwise vortices, which rearrange the flow via the lift-up mechanism. Such streamwise vortices can be induced by so-called miniature vortex generators (MVGs). Although recently considerable attention has been devoted to the investigation of MVGs, and multiple studies investigated MVGs with different parameters, the previously investigated MVG configurations are likely suboptimal. This study aims at (partially) filling this knowledge gap by complementing previous studies by conducting a parametric study of rectangular MVGs and assessing them with local linear stability analysis. Three parameters are varied simultaneously: the height, the spanwise distance of the MVG pairs, and the distance between MVGs in each pair. First, the modelling approach is presented, which consist of an efficient base flow computation, and BiGlobal stability analysis. The base flow calculation is validated by comparing our results with the experiments of Shattarzadeh and Fransson (2015). Then, the streak amplitude, the growth factors of the TS-waves and the secondary shear-layer instabilities are analysed in a parametric study. It is shown that with the variation of the spanwise distances, the maximum amplitude and the streamwise extent of the streaks in the boundary layer can be controlled, reminiscent of the behaviour of the optimal vortices. The findings and the test of the methodology provide valuable insight, which may be useful for the optimisation of MVGs.

Identification of the precessing vortex core in a hydro turbine model using local stability analysis and stochastic modeling

Stochastic modeling and local linear stability analysis (LSA) is employed to predict the onset of the precessing vortex core (PVC) in the hydro turbine model. The method of the stochastic modeling based on the pressure fluctuation signals correctly predicts the instability of the azimuthal mode m = 1 at flow rates below 0.7Q c . This is in line with local LSA that shows that the azimuthal modes m = 1 and m = 2 are absolutely unstable below the flow rate of 0.7Q c . The absolute instability of mode m = 2 is a new observation in the part load regimes of hydro turbines and plays a significant role in the dynamics of the PVC. As demonstrated in this paper, local LSA and stochastic modelling are both methods to uncover the driver of the PVC using sparse experimental data stemming from either spatially resolved but non-timeresolved PIV snapshots or single-point time-resolved wall pressure recordings, respectively. This makes these methods suitable to be applied to configurations of industrial relevance.

Development of a Shock-Stable and Contact-Preserving Scheme for Multidimensional Euler Equations

The shock instability, notably the disastrous carbuncle phenomenon, has plagued many low-dissipation Godunov-type numerical schemes in calculations of high Mach flows. In the current work, two popular stability analysis tools and corresponding numerical experiments are used to investigate the root cause of shock instability of the low-dissipation Harten, Lax, and van Leer with contact (HLLC) scheme. The local linear stability analysis reveals that the HLLC flux could attenuate all perturbations in the streamwise direction but not perturbations of density and shear velocity in the transverse direction. Numerical results also indicate that the shock instability of the HLLC scheme is only related to the transverse flux. The viscosity terms of entropy wave and shear wave are introduced into the transverse flux, and the stability analysis shows that with the help of additional viscosities the shock instability of the HLLC scheme is cured. To preserve the capability of resolving contact discontinuities and shear waves, a pressure-based switching function is incorporated into the expressions of viscosity terms so that the additional viscosities are activated only when calculating the transverse flux in the shock layer. A series of numerical experiments give evidence of the robustness and accuracy of the proposed scheme, and the strategy adopted here can be easily applied to cure shock instabilities of other low-dissipation numerical schemes, e.g., the Roe scheme and the AUSM+ scheme.

Instability of a planar fluid interface under a tangential electric field in a stagnation point flow

The interface between two immiscible fluids can become unstable under the effect of an imposed tangential electric field along with a stagnation point flow. This canonical situation, which arises in a wide range of electrohydrodynamic systems including at the equator of electrified droplets, can result in unstable interface deflections where the perturbed interface gets drawn along the extensional axis of the flow while experiencing strong charge build-up. Here, we present analytical and numerical analyses of the stability of a planar interface separating two immiscible fluid layers subject to a tangential electric field and a stagnation point flow. The interfacial charge dynamics is captured by a conservation equation accounting for Ohmic conduction, advection by the flow and finite charge relaxation. Using this model, we perform a local linear stability analysis in the vicinity of the stagnation point to study the behaviour of the system in terms of the relevant dimensionless groups of the problem. The local theory is complemented with a numerical normal-mode linear stability analysis based on the full system of equations and boundary conditions using the boundary element method. Our analysis demonstrates the subtle interplay of charge convection and conduction in the dynamics of the system, which oppose one another in the dominant unstable eigenmode. Finally, numerical simulations of the full nonlinear problem demonstrate how the coupling of flow and interfacial charge dynamics can give rise to nonlinear phenomena such as tip formation and the growth of charge density shocks.

Scaling Analysis on the Dynamic and Instability Characteristics of Isolated Wingtip Vortex

The isolated wingtip vortex is generated by an M6 wing under different angles of attack and Reynolds numbers , and is measured within 16 chord length downstream regions. Then, the spatiotemporal local linear stability analysis (LSA) is performed at the azimuthal wavenumber of to analyze the instability characteristics. Experimental results show that the radial circulation of wingtip vortex satisfies a scaling formula. Moreover, the LSA results demonstrate that the instability characteristics, including the stability curve, growth rate eigenvalue spectrum, and perturbation mode, are uniformed in time and space, laying the basis for scaling the wingtip vortex by considering them. In this situation, by plotting the circulation-based Reynolds number and dimensionless wandering amplitude in log scale, it is found that they are in the form of , where and . Particularly, along with the streamwise development, the swirl number of wingtip vortex asymptotically closes to the unstable region in the (, ) plane defined by Fabre and Jacquin (Journal of Fluid Mechanics, Vol. 500, Jan. 2004, pp. 239–262). By considering the role of growth rate in the vortex wandering, an alternative scaling formula , where and , is suggested to describe the wandering development.

Absolute instability of double annular jets: local stability analysis

The paper presents the local linear stability analysis of the double annular jets. The calculations show that the first absolutely unstable helical mode can be generated in the non-swirling annular jets by the back-flow in the central recirculation zone or sufficiently strong back-flow in the external recirculation zone. The influence of the back-flow magnitude on the frequency, growth rate and eigenfunctions of the first helical mode is discussed. The calculations are completed with an analysis of the influence of the swirl intensity in the internal and external jets on the characteristics of the first absolutely unstable helical mode.

Experimental detection of Hopf bifurcation in two-dimensional dynamical systems

In this work, we propose a strategy based on an analog active network to detect Hopf bifurcations in two–dimensional dynamical systems described by ordinary differential equations. With the proposed strategy, two parameters of the nonlinear system are established in the analog active network by using external controllable voltage levels in order to explore the dynamical evolution of the system in a fast, easy, and accessible way, making our approach a powerful tool to detect Hopf bifurcations in a two-dimensional space. To demonstrate the proposed strategy’s potential and functionality, we electronically implement the kinetics of a reaction-diffusion model proposed by Barrio et al. in 1999 [1], called BVAM model. Hopf bifurcations are detected by following the changes in the system, going from stationary to periodic solutions when varying the control parameters. Local linear stability analysis is performed to show the quantitative agreement between analytical and experimental bifurcations. We additionally found that global effects are detected by the experimental approach, which cannot be predicted from a local analysis. The proposed strategy opens the way to use analog active networks to detect bifurcations in dynamical systems experimentally.

Numerical study of total temperature effect on hypersonic boundary layer transition

The total temperature in the hypersonic wind tunnel test can be significantly different from that in real flight conditions, and this leads to a large discrepancy in the measurement of the boundary layer transition between ground experiment measurements and flight tests. Even at the same Mach number and Reynolds number, different wind tunnels may yield different transition data for the same model due to the total temperature effect. In this paper, the boundary layer transition on a 7° half-angle sharp cone (at a 0° angle of attack) with four freestream total temperatures is investigated using both the simulations of a local correlation-based transition model and linear stability analysis. The results show that as the freestream total temperature increases, the starting point of the transition on the sharp cone gradually moves backward and the length of the transition region decreases. The N factor of the unstable wave gradually decreases with increasing freestream total temperature, causing the transition onset to move backward. The total temperature effects on boundary transition as determined by both methods of analysis were in good agreement.

Effect of Pressure Gradient on the Development of Görtler Vortices

Boundary layers over concave surfaces may become unstable due to centrifugal instability that manifests itself as stationary streamwise counter rotating vortices. The centrifugal instability mechanism in boundary layers has been extensively studied and there is a large number of publications addressing different aspects of this problem. The results on the effect of pressure gradient show that favorable pressure gradients are stabilizing and adverse pressure gradient enhances the instability. The objective of the present investigation is to complement those works, looking particularly at the effect of pressure gradient on the stability diagram and on the determination of the spanwise wave number corresponding to the fastest growth. This study is based on the classic linear stability theory, where the parallel boundary layer approximation is assumed. Therefore, results are valid for Görtler numbers above 7, the lower limit where local mode linear stability analysis was identified in the literature as valid. For the base flow given by the Falkner-Skan solution, the linear stability equations are solved by a shooting method where the eigenvalues are the Görtler number, the spanwise wavenumber and the growth rate. The results show stabilization due to favorable pressure gradient as the constant amplification rate curves are displaced to higher Görtler numbers, with the opposite effect for adverse pressure gradient. Results previously unavailable in the literature identifying the fastest growing mode spanwise wavelength for a range of Falkner-Skan acceleration parameters are presented.

Nonaxisymmetric magnetorotational instability in spherical Couette flow.

We investigate numerically the flow of an electrically conducting fluid in a rapidly rotating spherical shell where the inner boundary spins slightly faster than the outer one. The magnetic field evolves self-consistently from an initial dipolar configuration of weak amplitude, and a toroidal field is produced by winding this poloidal field through the internal differential rotation. First, we characterize the axisymmetric field solutions obtained at long times when the Lorentz force is negligible and the flow follows the steady, purely hydrodynamical solution. We then examine the stability of these solutions, focusing on the regime of large magnetic Reynolds numbers where the field is dominantly toroidal. When the ratio of the azimuthal Alfvén frequency to the rotation frequency exceeds a certain value, a nonaxisymmetric instability develops. We show that the instability properties are compatible with those expected for the magnetorotational instability. Finally, we compare the instability properties with predictions obtained from a local linear stability analysis. The linear analysis agrees well with the numerical simulation results, except in a number of cases where the discrepancies are attributed to shearing effects on the unstable modes.

On the Effect of Rotational Forces on Rotor Blade Boundary-Layer Transition

Laminar-turbulent boundary-layer transition is investigated on the suction side of Mach-scaled helicopter rotor blades in climb and analyzed in view of the effect of rotational forces. Transition positions are detected with temperature-sensitive paint and complemented by surface pressure measurements at two radial blade sections. The effect of rotational forces is investigated by systematic variation of the Rossby number from to at and . The findings do not show a significant effect on boundary-layer transition in the investigated parameter range and suggest predominantly two-dimensional flow behavior. The result is supported by subsequent validation with two-dimensional numerical tools. Based on quantitative agreement between measured and calculated surface pressures, measured transition positions are predicted to within chord if a critical amplification factor of is used in the coupled Euler/boundary equation solver MSES for transition prediction in the rotor test facility of the DLR, German Aerospace Center in Göttingen, Germany. The measured transition onset positions are also correlated with integral growth rates, obtained from separate two-dimensional compressible boundary-layer computations and subsequent local linear stability analysis to get transition factors of . Differences in factors are discussed in view of the different approaches used. Overall, transition -factor correlations are independent of relative chord Reynolds number and incompressible shape factor at the detected transition onset. The findings underline the capability of two-dimensional numerical techniques based on linear stability theory to model boundary-layer transition in the investigated parameter range.

Coexistence of magneto-rotational and Jeans instabilities in an axisymmetric nebula

Aims. We analyze the magneto-rotational instability (MRI) effects on gravitational collapse and its influence on the instability critical scale. Methods. In particular, we study an axisymmetric nonstratified differentially rotating cloud, embedded in a small magnetic field, and we perform a local linear stability analysis, including the self gravity of the system. Results. We demonstrate that the linear evolution of the perturbations is characterized by the emergence of an anisotropy degree of the perturbed mass densities. Starting with spherical growing overdensities, we see that they naturally acquire an anisotropy of order unity in their shape. Despite the linear character of our analysis, we infer that such a seed of anisotropy can rapidly grow in a nonlinear regime, leading to the formation of filament-like structures. However, we show how such an anisotropy is essentially an intrinsic feature of the Jean instability, and how MRI only plays a significant role in fixing the critical scale of the mode spectrum. We then provide a characterization of the present analysis in terms of the cosmological setting, in order to provide an outlook of how the present results could concern the formation of large-scale structures across the Universe.

Local Spatial Linear Stability Analysis of Low-Pressure Turbine Endwall Boundary Layer

An implicit large-eddy simulation of a low-pressure turbine cascade for an axial chord–based Reynolds number of 100,000 revealed steady longitudinal structures near the endwall that are reminiscent of crossflow vortices. A local spatial linear stability theory analysis with Newtonian eddy model of a profile extracted from the strongly curved section did not indicate any spatially growing steady modes. An unstable steady mode could be identified when the quasi-laminar assumption was made. The mode shape roughly agreed with the observed steady flow structures. Further analysis revealed that this mode was most strongly amplified when the frequency was nonzero. Based on a comparison of the Reynolds stress profiles upstream and inside the cascade, it is argued that the unstable mode may affect the mean flow through a modification of the Reynolds shear stresses.

Stability and Sensitivity Analysis of Hydrodynamic Instabilities in Industrial Swirled Injection Systems

The hydrodynamic instability in an industrial, two-staged, counter-rotative, swirled injector of highly complex geometry is under investigation. Large eddy simulations (LES) show that the complicated and strongly nonparallel flow field in the injector is superimposed by a strong precessing vortex core (PVC). Mean flow fields of LES, validated by experimental particle image velocimetry (PIV) measurements, are used as input for both local and global linear stability analysis (LSA). It is shown that the origin of the instability is located at the exit plane of the primary injector. Mode shapes of both global and local LSA are compared to dynamic mode decomposition (DMD) based on LES snapshots, showing good agreement. The estimated frequencies for the instability are in good agreement with both the experiment and the simulation. Furthermore, the adjoint mode shapes retrieved by the global approach are used to find the best location for periodic forcing in order to control the PVC.

Experimental study of double-cavity flow

The flow through two facing, identical cavities (double-cavity) is characterized experimentally, as the inflow velocity and the distance between the cavities is varied. Standard 2D2C particle image velocimetry measurements in the spanwise mid-plane provide information on the instantaneous and mean velocity flow fields. Laser Doppler velocimetry measurements at several points in the double-cavity domain reveal the global character of the streamwise fluctuating velocity spectra. The flow is characterized based on time series, recorded in the shear layer of one of the cavities, for a wide range of inflow velocities and intercavity distances. In a detailed spectral study, we show how the shear layer spectra get affected when the two cavities are brought closer together. Based on the experimental data, a temporal local linear stability analysis was carried out, which was able to explain why the frequency peaks for close intercavity distances broaden and move to higher Strouhal numbers.

Mode coupling instability mitigation in friction systems by means of nonlinear energy sinks: Numerical highlighting and local stability analysis

In this paper, we study the problem of passive control of friction-induced vibrations due to mode coupling instability in braking systems. To achieve that, the well-known two degrees of freedom Hultén’s model, which reproduces the typical dynamic behavior of friction systems, is coupled to two ungrounded nonlinear energy sinks (NES). The NES involves an essential cubic restoring force and a linear damping force. First, using numerical simulations it is shown that the suppression or the mitigation of the instability is possible and four steady-state responses are highlighted: complete suppression, mitigation through periodic response, mitigation through strongly modulated response, and no suppression of the mode coupling instability. Then the system is analyzed applying a complexification-averaging method and the resulting slow-flow is finally analyzed using geometric singular perturbation theory. This analysis allows us to explain the observed steady-state response regimes and predict some of them. The boundary values of the friction coefficient for some of the transitions between these regimes are predicted. However, the appearance of a three-dimensional super-slow flow subsystem highlights the limitation of the local linear stability analysis of the slow-flow to predict all these boundaries.

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

A five‐dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self‐diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans‐critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease‐free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.