Results Of Linear Stability Analysis Research Articles

Dynamics of a thin film of viscoelastic fluid flowing down an inclined or vertical plane

The stability characteristics of a thin film of viscoelastic (Walters’ B′ model) fluid flowing down an inclined or vertical plane are analyzed under the combined influence of gravity and surface tension. A nonlinear free surface evolution equation is obtained by using the momentum-integral method. Normal mode technique and multiple scales method are used to obtain the results of linear and nonlinear stability analysis of this problem. The linear stability analysis gives the critical condition and critical wave number kc which include the viscoelastic parameter Γ, angle of inclination of the plane θ, Reynolds number Re and Weber number We. The weakly nonlinear stability analysis that is based on the second Landau constant J2, reveals the condition for the existence of explosive unstable and supercritical stable zone along with the other two (unconditional stable and subcritical unstable) flow zones of this problem which is 3(1+3Γ)Re−3cotθ−4ReWek2 = 0. It is found that all the four distinct flow zones of this problem exist in Re-k-, θ-k- and Γ-k-plane after the critical value of Rec,θc and Γc, respectively. A novel result of this analysis is that the film flow is stable (unstable) for a negative (positive) value of Γ irrespective of the values of Re and θ, as for example, a solution of polyisobutylene in cetane, compared with the viscous (Γ=0) film flow case. Finally, we scrutinize the effect of Γ on the threshold amplitude and nonlinear wave speed by depicting some numerical examples in supercritical stable as well as subcritical unstable zone of this problem.

Accuracy Assessment of Numerical Morphological Models Based on Reduced Saint‐Venant Equations

AbstractSustainable river management often requires long‐term morphological simulations. As the future is unknown, uncertainty needs to be accounted for, which may require probabilistic simulations covering a large parameter domain. Even for one‐dimensional models, simulation times can be long. One of the acceleration strategies is simplification of models by neglecting terms in the governing hydrodynamic equations. Examples are the quasi‐steady model and the diffusive wave model, both widely used by scientists and practitioners. Here, we establish under which conditions these simplified models are accurate. Based on results of linear stability analyses of the St. Venant‐Exner equations, we assess migration celerities and damping of infinitesimal, but long riverbed perturbations. We did this for the full dynamic model, that is, no terms neglected, as well as for the simplified models. The accuracy of the simplified models was obtained from comparison between the characteristics of the riverbed perturbations for simplified models and the full dynamic model. We executed a spatial‐mode and a temporal‐mode linear analysis and compared the results with numerical modeling results for the full dynamic and simplified models, for very small and large bed waves. The numerical results match best with the temporal‐mode linear analysis. We show that the quasi‐steady model is highly accurate for Froude numbers up to 0.7, probably even for long river reaches with large flood wave damping. Although the diffusive wave model accurately predicts flood wave migration and damping, key morphological metrics deviate more than 5% (10%) from the full dynamic model when Froude numbers exceed 0.2 (0.3).

Hemline breakup of gel drops subjected to a continuous air flow

To explore the effect of yield stress on the secondary breakup of gel drops, experimental and theoretical investigations are carried out by employing a high-speed camera. A unique hemline-type breakup, as a modified behaviour of sheet-thinning breakup, occurs when the air velocity increases to a high region. The edges of the drops constantly deform into thin membranes when the high-velocity air skims over the gel drops. These membranes vibrate vertically, and breaking points occur at high amplitudes, causing the formation of reticular fragments. The results of linear stability analysis indicated that the yield stress of the gel drops has an influence on the formation and breakup of the gel membranes. The breakup regime map and breakup times are also studied.

Predicting the effect of inertia, rotation, and magnetic field on the onset of convection in a bidispersive porous medium using machine learning techniques

Effects of the magnetic field and inertia on the onset of thermal convection in a horizontal bidispersive porous layer, rotating about a vertical axis, are analyzed. The Darcy equation with same temperature in the micro- and macrophases is used to characterize the fluid motion. The Vadasz number is taken into account in a generalized Darcy equation for the macrophase. The eigenvalue problem obtained from the linear stability analysis is solved analytically for free–free boundaries. Moving one step further from the traditional linear stability analysis, machine learning tools are introduced in this paper to include the effect of multiple parameters on the marginal state of the system. Machine learning techniques have been implemented to identify the mode of instability with respect to different parameters. In particular, classification algorithms, namely, Artificial Neural Networks (ANN) and Support vector machine, are used to examine the onset of oscillatory convection and stationary convection. The required data for training of the algorithms are generated from the results of linear stability analysis. It is found that ANN with the sufficient number of hidden layers along with good choice of training dataset can predict the mode of instability even on the small variation in a given parameter. The combined effect of rotation, magnetic field, and inertia is to reduce the oscillatory mode of instability; hence, the system exhibits the steady mode of instability for a significant region in the three dimensional space comprising the Taylor number, the Hartman number, and the Vadasz number.

INTERFACIAL INSTABILITY ANALYSIS OF VISCOUS-VISCOELASTIC FLUID INTERFACE IN AN ANNULAR POROUS MEDIUM

The transport of mass and heat from one fluid to another fluid in a porous medium has numerous applications such as film boiling and heat exchangers. This paper reports the results of linear stability analysis of the capillary instability of viscoelastic liquid-viscous gas interface with heat and mass transfer. The viscoelastic liquid and viscous gas lie in an annular region enclosed by two rigid cylinders. The annular region is a saturated porous medium with constant porosity and permeability. The analysis is performed using the potential flow theory for viscoelastic liquid satisfying the Oldroyd-B model. A third-order polynomial in terms of growth rate is achieved and analyzed numerically. It is observed that porous media slows the growth of perturbations, but an increase in porosity makes the interface unstable. The interface goes towards stability when the transfer of heat-mass increases.

On a hybrid continuum-kinetic model for complex fluids

In the present work, we first introduce a general framework for modelling complex multiscale fluids and then focus on the derivation and analysis of a new hybrid continuum-kinetic model. In particular, we combine conservation of mass and momentum for an isentropic macroscopic model with a kinetic representation of the microscopic behavior. After introducing a small scale of interest, we compute the complex stress tensor by means of the Irving-Kirkwood formula. The latter requires an expansion of the kinetic distribution around an equilibrium state and a successive homogenization over the fast in time and small in space scale dynamics. For a new hybrid continuum-kinetic model the results of linear stability analysis indicate a conditional stability in the relevant low speed regimes and linear instability for high speed regimes for higher modes. Extensive numerical experiments confirm that the proposed multiscale model can reflect new phenomena of complex fluids not being present in standard Newtonian fluids. Consequently, the proposed general technique can be successfully used to derive new interesting systems combining the macro and micro structure of a given physical problem.

Cold‐Formed Steel Profiles with Multiple Stiffeners Effective Section Calculation Using Alternative Methodology of the EN1993‐1‐3 Standard

SummaryIn this article, the use of paragraph 5.5.1 (7) is applied, only for stiffeners, and the use of the numerical results of linear stability analyses (LSA) obtained by the GBT (Generalized Beam Theory) methods is tested for the section, namely the distortional modes, relating them to the formulas currently present in EC3‐1‐3. For comparison and analysis of results, studies present in the bibliography were used that show experimental and numerical results, relevant and representative, of sections with stiffeners. In the calculation of effective sections, the Class4Section application, developed by the authors, is used to run several tests combining EC3‐1‐3 specifications and LSA results.

Stability analysis for laminar separation flutter of an airfoil in the transitional flow regime

Laminar separation flutter (LSF) is a type of aeroelastic instability phenomenon characterized by small-amplitude low-frequency pitching oscillations of the airfoil. The present study aims to gain insight into the intrinsic dynamics of LSF via data-driven stability analysis. The proposed data-driven approach relies on the autoregressive with exogenous input (ARX) technique to design reduced-order models (ROMs) of unsteady aerodynamics in a state-space format. First, high-fidelity full-order numerical simulations of the LSF phenomenon are performed using the incompressible Unsteady Reynolds-Averaged Navier–Stokes equations and the Shear-Stress Transport k−ω turbulence model with Low-Reynolds-number correction. The calculated LSF responses show good agreement with previous experimental data in the literature. Then, linear stability analysis (LSA) of the aeroelastic system is carried out to reveal the underlying fluid-structure interaction mechanism. The LSA model is developed by coupling the ROM with the structure motion equation. LSA results indicate that the LSF phenomenon is primarily caused by the instability of the structure mode (SM), which is induced by the mutual repulsion effect between one static fluid mode (FM) and the SM. The presence of laminar separation near the trailing-edge of the airfoil can significantly reduce the stability of the static FM, which ultimately strengthens the fluid-structure coupling effect and leads to LSF. We would like to emphasize that LSF is essentially different from other flow-induced vibration phenomena, such as transonic buffeting of an airfoil and vortex-induced vibration of bluff bodies, for which the instabilities are triggered by the coupling between one dynamic FM and the SM. Finally, the effects of the mass ratio, structural damping ratio, and freestream turbulence intensity on the aeroelastic system are also investigated.

Transient growth analysis of hypersonic flow over an elliptic cone

Non-modal linear stability analysis results are presented for hypersonic flow over an elliptic cone with an aspect ratio of two at zero angle of attack, completing earlier modal instability analysis of flow around the same geometry. The theoretical framework to perform transient growth analysis of compressible flows on a generalized two-dimensional frame of reference is developed for the first time and is then applied to solve the initial-value problem governing non-modal linear instability on planes perpendicular to the cone axis, taken at successive streamwise locations along the elliptic cone. Parameter ranges examined here are chosen so as to model flight of the Hypersonic International Flight Research Experimentation 5 (HIFiRE-5) test geometry at altitudes of 21 km and 33 km, corresponding to Mach numbers 7.45 and 8.05 and unit Reynolds numbers $Re' = 1.07\times 10^7$ and $1.89\times 10^6$ , respectively. Results obtained indicate that the significance of the non-modal growth for laminar–turbulent transition increases with increasing flight altitude (decreasing Reynolds number). At a given set of flow parameters, transient growth is stronger in the vicinity of the tip of the cone and in azimuthal locations away from both of the minor (centreline) and major (attachment line) axes of the cone. Linear optimal disturbances calculated at conditions of maximal transient growth are found to peak in the crossflow region of the elliptic cone. These structures are elongated along the streamwise spatial direction, while being periodic along the spanwise direction with periodicity lengths of the same order of magnitude as the well-known structures identified as crossflow vortices in both experiments and simulations.

Linear stability of exact solutions for the generalized Kaup-Boussinesq equation and their dynamical evolutions

<p style='text-indent:20px;'>The integrability, classification of traveling wave solutions and stability of exact solutions for the generalized Kaup-Boussinesq equation are studied by prolongation structure technique and linear stability analysis. Firstly, it is proved that the generalized Kaup-Boussinesq equation is completely integrable in sense of having Lax pair. Secondly, the complete classification of exact traveling wave solutions of the generalized Kaup-Boussinesq equation are given and a family of exact solutions are proposed. Finally, the stability of these exact solutions are investigated by linear stability analysis and dynamical evolutions, and some stable traveling wave solutions are found. It is shown that the results of linear stability analysis are in excellent agreement with the results from dynamical evolutions.</p>

Onset of vortex shedding around a short cylinder

This paper presents results of three-dimensional direct numerical simulations (DNS) and global linear stability analyses of a viscous incompressible flow past a finite-length cylinder with two free flat ends. The cylindrical axis is normal to the streamwise direction. The work focuses on the effects of aspect ratios (in the range of $0.5\leq {\small \text{AR}} \leq 2$ , cylinder length over diameter) and Reynolds numbers ( $Re\leq 1000$ based on cylinder diameter and uniform incoming velocity) on the onset of vortex shedding in this flow. All important flow patterns have been identified and studied, especially as ${\small \text{AR}}$ changes. The appearance of a steady wake pattern when ${\small \text{AR}} \leq 1.75$ has not been discussed earlier in the literature for this flow. Linear stability analyses based on the time-mean flow has been applied to understand the Hopf bifurcation past which vortex shedding happens. The nonlinear DNS results indicate that there are two vortex shedding patterns at different $Re$ , one is transient and the other is nonlinearly saturated. The vortex-shedding frequencies of these two flow patterns correspond to the eigenfrequencies of the two global modes in the stability analysis of the time-mean flow. Wherever possible, we compare the results of our analyses to those of the flows past other short- ${\small \text{AR}}$ bluff bodies in order that our discussions bear more general meanings.

Rayleigh–Taylor instability of classical diffusive density profiles for miscible fluids in porous media: a linear stability analysis

A Rayleigh–Taylor instability typically develops when a denser layer overlies a less dense one in the gravity field. In that case, the initial base state density profile is a step function for which linear stability analysis results are well known. We investigate here analytically the linear stability analysis of other classical diffusive density profiles for porous media flows. We find that, for a species A initially distributed in the upper half of the domain with an initial concentration profile of \((-X)^m\) for \(0<m<1\) where X is the vertical coordinate, and absent from the bottom half of the domain, for large times the eigenfunctions grow like \(\exp \left( \omega _0 T^{(m+1)/2} + \omega _1 \ln (T) \right) \) where \(\omega _0\) and \(\omega _1\) are constants and T is time. Thus, the growth rate defined by \((1/A) (\mathrm{{d}}A/\mathrm{{d}}T)\) decays like \(c_1 T^{(m-1)/2} +c_2 T^{(m-2)/3}\) whilst the maximum growing wavenumber scales with \(T^{(m-2)/6}\). These results are compared to the growth rates obtained using numerical linear stability analysis. Our analytical predictions provide a set of generalised results that pave the way to the analysis of Rayleigh–Taylor instabilities of nontrivial density profiles.

Multi-scale study of the transitional shock-wave boundary layer interaction in hypersonic flow

A high-fidelity simulation of the massively separated shock/transitional boundary layer interaction caused by a 15-degrees axisymmetrical compression ramp is performed at a free stream Mach number of 6 and a transitional Reynolds number. The chosen configuration yields a strongly multiscale dynamics of the flow as the separated region oscillates at low-frequency, and high-frequency transitional instabilities are triggered by the injection of a generic noise at the inlet of the simulation. The simulation is post-processed using Proper Orthogonal Decomposition to extract the large scale low-frequency dynamics of the recirculation region. The bubble dynamics from the simulation is then compared to the results of a global linear stability analysis about the mean flow. A critical interpretation of the eigenspectrum of the linearized Navier–Stokes operator is presented. The recirculation region dynamics is found to be dominated by two coexisting modes, a quasi-steady one that expresses itself mainly in the reattachment region and that is caused by the interaction of two self-sustained instabilities, and an unsteady one linked with the separation shock-wave and the mixing layer. The unsteady mode is driven by a feedback loop in the recirculation region, which may also be relevant for other unsteady shock-motion already documented for shock-wave/turbulent boundary layer interaction. The impact of the large-scale dynamics on the transitional one is then assessed through the numerical filtering of those low wavenumber modes; they are found to have no impact on the transitional dynamics.

An Improved Lattice Hydrodynamic Model by considering the Effect of “Backward-Looking” and Anticipation Behavior

In order to prevent the occurrence of traffic accidents, drivers always focus on the running conditions of the preceding and rear vehicles to change their driving behavior. By taking into the “backward-looking” effect and the driver’s anticipation effect of flux difference consideration at the same time, a novel two-lane lattice hydrodynamic model is proposed to reveal driving characteristics. The corresponding stability conditions are derived through a linear stability analysis. Then, the nonlinear theory is also applied to derive the mKdV equation describing traffic congestion near the critical point. Linear and nonlinear analyses of the proposed model show that how the “backward-looking” effect and the driver’s anticipation behavior comprehensively affect the traffic flow stability. The results show that the positive constant γ , the driver’s anticipation time τ , and the sensitivity coefficient p play significant roles in the improvement of traffic flow stability and the alleviation of the traffic congestion. Furthermore, the effectiveness of linear stability analysis and nonlinear analysis results is demonstrated by numerical simulations.

An extended two-lane car-following model considering the influence of heterogeneous speed information on drivers with different characteristics under honk environment

As the driver is the decision-maker and executor of driving behavior, the driver’s subjective awareness plays a considerable role in the traffic environment. In the real two-lane traffic environment with no isolation zone, the speed of surrounding vehicles will affect the driver’s car-following behavior on the main road. Drivers with different characteristics are likely to react differently to the same information. Therefore, providing heterogeneous information to the driver may improve the stability of the traffic flow. Considering the influence of the heterogeneous information of the preceding vehicle speed on drivers with different characteristics, an extended car-following model that also considers the speed of adjacent lanes under the honk effect is proposed in this paper. The results of linear stability analysis show that timid drivers and skilled drivers are more beneficial to improve the stability of traffic flow. In addition, providing heterogeneous information for drivers of different personalities has a positive effect on maintaining the stability of the traffic flow. The simulation results are consistent with the theoretical results, which proves that the proposed model can effectively characterize the car-following behavior and has good stability.

Thin-film evolution and fingering instability of self-rewetting films flowing down an inclined plane

This paper examines the evolution patterns and essential mechanisms of flow instability of a self-rewetting fluid (SRF) coating on an inclined plane. Considering that the self-rewetting liquid has an anomalous surface tension with temperature change, some interesting phenomena will be found and should be explained. Using the thin-film model, the evolution equation of the air–liquid interface is derived, and the thickness of the liquid film is determined by a fourth-order partial differential equation. Taking T0 (temperature corresponding to the minimum of surface tension) as a cutoff point, two representative cases of the nonlinear flow are comprehensively discussed. One is the case of Ti &amp;gt; T0, and the other is Ti &amp;lt; T0 (interfacial temperature Ti). Based on traveling wave solutions, linear stability analysis (LSA) of the small perturbation applied to the initial condition is given, and the results of LSA are confirmed and explained by the numerical simulations. Results show that the inclined angle and the Weber number always stabilize the free surface, while the Marangoni effect and the Biot number play different roles for the two cases. As Ti − T0 varies from a negative value to a positive value, the Marangoni effect switches to the reverse Marangoni effect. With Ti − T0 &amp;lt; 0, the Marangoni effect enhances the fingering instability, while the Marangoni effect makes the flow more stable if Ti − T0 &amp;gt; 0. The Biot number Bi = 1 corresponds to the most unstable state for Ti &amp;lt; T0 and to the most stable state for Ti &amp;gt; T0.

An unconditionally stable second-order accurate method for systems of Cahn–Hilliard equations

In this paper, we develop an unconditionally stable linear numerical scheme for the N-component Cahn–Hilliard system with second-order accuracy in time and space. The proposed scheme is modified from the Crank–Nicolson finite difference scheme and adopts the idea of a stabilized method. Nonlinear multigird algorithm with Gauss–Seidel-type iteration is used to solve the resulting discrete system. We theoretically prove that the proposed scheme is unconditionally stable for the whole system. The numerical solutions show that the larger time steps can be used and the second-order accuracy is obtained in time and space; and they are consistent with the results of linear stability analysis. We investigate the evolutions of triple junction and spinodal decomposition in a quaternary mixture. Moreover, the proposed scheme can be modified to solve the binary spinodal decomposition in complex domains and multi-component fluid flows.

The effect of tension-compression asymmetry on the formation of dynamic necking instabilities under plane strain stretching

This paper brings to light the effect of tension-compression asymmetry in flow stresses on the formation of dynamic necking instabilities in isotropic metallic plates subjected to plane strain stretching. For that purpose, a two-pronged approach which includes finite element calculations and linear stability analysis has been used. In both approaches, for the description of the plastic behavior the isotropic form of Cazacu et al. (2006) criterion and isotropic hardening was assumed. It is shown that although this criterion involves dependence of the third-invariant of the stress deviator, it is possible to develop a linear stability analysis and obtain the value of the growth rate of the perturbation at different loading times, and track the history of the growth rate of all the growing modes during the post-critical deformation process. Furthermore, an original procedure for calibration of linear stability analysis from finite element calculations was developed. Both linear stability analysis and finite element results indicate an important effect of the tension-compression asymmetry on the necking behavior and the same overall trends. In particular, the results show that while the necking time, and thus the specimen elongation when necking occurs, is roughly the same irrespective of the tension-compression asymmetry ratio, the necking strain and the necking energy are significantly greater for a material that displays a larger flow stress in uniaxial compression than in uniaxial tension. A key outcome of this investigation is to demonstrate that such behavior is due to the larger plastic dissipation undergone, under plane strain stretching, by such a material as compared to a von Mises material and a material with larger flow stress in uniaxial tension than in uniaxial compression.

Characterization of Viscous Fingering and Channeling for the Assessment of Polymer-Based Heavy Oil Displacements

The ability to determine a predictive stability criterion is of great practical importance for designing stable polymer-based displacements. Where one usually resorts to a limited number of core-scale experiments or coarse-scale reservoir simulations, the first ones are potentially impacted by length scale issues, while the second ones possibly smooth out sharp displacing fronts and physical instability due to numerical diffusion. This paper proposes a new hydrodynamical stability criterion based on the previous linear stability analysis results. This criterion is tested for 2D polymer oil displacement by performing high viscosity contrasts, high-resolution numerical experiments at pilot scale. We investigate mesh resolution issues and several perturbation ideas. Different factors are considered such as mobility ratios, polymer adsorption and degradation and heterogeneities. The analysis is based on a combination of reservoir simulation and image processing techniques. We show the development of viscous fingering in homogeneous porous media is driven by the shock mobility ratio defined as the ratio of the total fluids upstream mobility over the total fluids downstream mobility. This stability criterion proves to predict both the polymer upstream and polymer-free downstream saturation fronts stability, typical of a polymer displacement, whether polymer adsorbs on the rock or degradates, or not. The observed fingers dynamical behavior is in line with previous works addressing single-phase miscible flow or immiscible oil displacement in porous media: fingers transversally merge while growing in the flow direction. Time evolution of fingers spreading and number is linear. Investigation on porous media of variable heterogeneity distributions shows how viscous fingering couples with heterogeneity and leads to even more marked, distorted and unstable flow patterns. In those cases, flow patterns are not solely driven by the porous medium heterogeneity. The more unstable the flow is, the more sensitive it is to heterogeneity. In-depth fingers analysis shows a very specific time evolution behavior, quite different from viscous fingering in homogeneous media. Such a flow pattern is related to production data such as water and polymer breakthrough times and/or oil recovery profiles as a function of time, which can be used in turn to interpret displacement stability and porous medium heterogeneity features.

Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion

Auto-wave solutions in nonlinear time-fractional reaction–diffusion systems are investigated. It is shown that stability of steady-state solutions and their subsequent evolution are mainly determined by the eigenvalue spectrum of a linearized system and level of anomalous diffusion (orders of fractional derivatives). The results of linear stability analysis are confirmed by computer simulations. To illustrate the influence of anomalous diffusion on stability properties and possible dynamics in fractional reaction–diffusion systems, we generalized two classical activator–inhibitor nonlinear models: FitzHugh–Nagumo and Brusselator. Based on them a common picture of typical nonlinear solutions in nonlinear incommensurate time-fractional activator–inhibitor systems is presented.