Unstable Steady-state Solutions Research Articles

Including the parallel mass flow in calculating the steady-state solutions and stability of the momentum balance equations for a quasisymmetric stellarator

The Helically Symmetric Experiment (HSX) is a quasisymmetric stellarator with minimal parallel viscous damping in a helical direction. The parallel flow (Vǁ) along the magnetic field is similarly weakly damped by viscosity. In this paper, the self-consistent steady-state parallel and poloidal momentum balance equations are used to show that a large Vǁ on the order of the ion thermal velocity can increase the ion resonant radial electric field (Er) beyond the value calculated using the typical approximation that Vǁ is zero. By altering the damping of Vǁ, either by degrading the quasisymmetry or varying the neutral density, the ion resonant Er can shift in a controllable fashion. It is shown explicitly that there exist stable and unstable steady-state solutions in the two-dimensional space of Vǁ and Er. A stability analysis of each solution is performed by calculating the eigenvalues and eigenvectors of the Jacobian. The unstable solution corresponds to a saddle point in which the eigenvalues have opposite signs. The analysis leads to the conclusion that unstable solutions occur when the derivative of the total poloidal damping with respect to Er is positive. A hysteresis in Er and Vǁ is observed when the radial current density is linearly increased to a maximum and then decreased back to zero. Jumps in the radial electric field and the parallel flow are observed as the radial current density drives the evolution from one stable point to the next. This result is similar to experimental data observed on several devices.

Stability and bifurcation of dynamic contact lines in two dimensions

The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number,$Ca_{{crit}}$, above which no steady-state solution can be found. Below$Ca_{{crit}}$, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against$Ca$, a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when$Ca< Ca_{{crit}}$due to finite-amplitude perturbations. Furthermore, when$Ca>Ca_{{crit}}$, we show that the trajectories in phase space closely follow the unstable branch.

Convergence enhancement of SIMPLE-like steady-state RANS solvers applied to airfoil and cylinder flows

Iterative steady Reynolds-averaged Navier–Stokes (RANS) solvers are widely used to obtain polars representing the aerodynamic characteristics of airfoils. Unsteadiness is encountered in the flow for many of the simulated angles of attack due to phenomenon such as vortex shedding and separation bubbles. In such cases, the steady RANS solver may become numerically unstable. As a consequence it is unable to find a converged solution, with the solver residuals often entering limit cycle oscillations. The BoostConv method is an alternative to the classical Newton’s method to compute unstable steady-state solutions of dynamical systems. It has been previously applied for obtaining steady laminar solutions of the Navier–Stokes equations. However, it demonstrated a lack of robustness by failing to always find the steady-state turbulent solutions of the RANS equations. To overcome this problem, we propose a modification in the application of the BoostConv method to compute steady-state turbulent solutions of the RANS equations. The modified BoostConv method is coupled with an iterative steady RANS solver for computing converged two-dimensional steady-state flows over airfoils. The proposed modification in the application procedure introduces a relaxation parameter that improves the convergence and robustness of the BoostConv method. We show this by finding lift and drag curves with angles of attack spanning 360°for airfoils representative of those used in modern wind turbines. The flow solutions are converged to machine precision at all angles of attack using the novel application procedure of the BoostConv method. Additionally, for flow cases convergent with the original BoostConv method, the modified BoostConv method showed faster convergence rates. The proposed modification provides an enhancement of the BoostConv method that can be easily integrated into existing implementations. By improving the method’s robustness and convergence rates in finding steady-state turbulent solutions of the RANS equations, it opens up the modified BoostConv method to be used in a wide variety of industrial flow cases.

Compressible Gas Flow past a Plate with Upstream-Moving Surface

The numerical solution of the problem of compressible gas flow past a plate with upstream-moving surface is obtained. Unstable time-periodic and stable steady-state solutions of this problem are discussed. A new type of unsteady periodic flow in which the varying characteristics turn out to be integrally asymmetric at transonic velocities is obtained.

Dissipative time crystal in an asymmetric nonlinear photonic dimer

We investigate the behavior of two coupled non-linear photonic cavities, in presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly non-classical properties and its dynamics exhibits the long lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.

Critical Evolution of Finite Perturbations of a Water Evaporation Surface in Porous Media

It is shown that the approximate steady-state solutions, which satisfy the model dissipative equation that describes the process of water evaporation in the neighborhood of the instability threshold of a phase transition interface, determine localized damped finite-amplitude perturbations when a certain condition is fulfilled. These steady-state solutions can be used for forecasting the scenario of the development of a perturbation with sufficient accuracy if this perturbation has no common points with any steady-state solution. If the initial position of the phase transition front is located between the spectrally stable solution and any of the steady-state solutions, this front damps. If the initial position of the front is located above at least one of the spectrally unstable steady-state solutions, then the solution is catastrophically restructured.

Effectivity and efficiency of selective frequency damping for the computation of unstable steady-state solutions

Selective Frequency Damping (SFD) is a popular method for the computation of globally unstable steady-state solutions in fluid dynamics. The approach has two model parameters whose selection is generally unclear. In this article, a detailed analysis of the influence of these parameters is presented, answering several open questions with regard to the effectiveness, optimum efficiency and limitations of the method. In particular, we show that SFD is always capable of stabilising a globally unstable systems ruled by one unsteady unstable eigenmode and derive analytical formulas for optimum parameter values. We show that the numerical feasibility of the approach depends on the complex phase angle of the most unstable eigenvalue. A numerical technique for characterising the pertinent eigenmodes is presented. In combination with analytical expressions, this technique allows finding optimal parameters that minimise the spectral radius of a simulation, without having to perform an independent stability analysis. An extension to multiple unstable eigenmodes is derived. As computational example, a two-dimensional cylinder flow case is optimally stabilised using this method. We provide a physical interpretation of the stabilisation mechanism based on, but not limited to, this Navier–Stokes example.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACTThe unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

An adaptive selective frequency damping method

The selective frequency damping (SFD) method is an alternative to classical Newton’s method to obtain unstable steady-state solutions of dynamical systems. However, this method has two main limitations: it does not converge for arbitrary control parameters, and when it does converge, the time necessary to reach a steady-state solution may be very long. In this paper, we present an adaptive algorithm to address these two issues. We show that by evaluating the dominant eigenvalue of a “partially converged” steady flow, we can select a control coefficient and a filter width that ensure an optimum convergence of the SFD method. We apply this adaptive method to several classical test cases of computational fluid dynamics and we show that a steady-state solution can be obtained with a very limited (or without any) a priori knowledge of the flow stability properties.

Structure and stability of steady porous medium convection at large Rayleigh number

A systematic investigation of unstable steady-state solutions of the Darcy–Oberbeck–Boussinesq equations at large values of the Rayleigh number $\mathit{Ra}$ is performed to gain insight into two-dimensional porous medium convection in domains of varying aspect ratio $L$. The steady convective states are shown to transport less heat than the statistically steady ‘turbulent’ flow realised at the same parameter values: the Nusselt number $\mathit{Nu}\sim \mathit{Ra}$ for turbulent porous medium convection, while $\mathit{Nu}\sim \mathit{Ra}^{0.6}$ for the maximum heat-transporting steady solutions. A key finding is that the lateral scale of the heat-flux-maximising solutions shrinks roughly as $L\sim \mathit{Ra}^{-0.5}$, reminiscent of the decrease of the mean inter-plume spacing observed in turbulent porous medium convection as the thermal forcing is increased. A spatial Floquet analysis is performed to investigate the linear stability of the fully nonlinear steady convective states, extending a recent study by Hewitt et al. (J. Fluid Mech., vol. 737, 2013, pp. 205–231) by treating a base convective state, and secondary stability modes, that satisfy appropriate boundary conditions along plane parallel walls. As in that study, a bulk instability mode is found for sufficiently small-aspect-ratio base states. However, the growth rate of this bulk mode is shown to be significantly reduced by the presence of the walls. Beyond a certain critical $\mathit{Ra}$-dependent aspect ratio, the base state is most strongly unstable to a secondary mode that is localised near the heated and cooled walls. Direct numerical simulations, strategically initialised to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically steady porous medium convection results from a balance between the competing effects of these two types of instability.

Computational Analysis of Two Phase Flow Distribution in a Horizonatal Pipe

In this paper, a two phase flow distribution in a horizontal pipe is numerically analyzed by solving one dimensional steady state momentum equation for predicting the pressure drop (∆P), quality of steam at outlet of the pipe (X), void fraction (α). The heat absorbed by the pipe (Q) and the mass flow rate (W) of water are varied over a wide range to investigate the above parameters. The locations of the two phase mixture are discussed. Pressure drop along the pipe is inconsistent for different flow rate so, the stable and unstable steady state solution is also carried out using Linear Stability analysis. The present numerical results are compared with the reported data from the literature and found that they are in good agreement. This study is used to calculate the pressure, temperature, hold up and quality within the horizontal pipe.

Harmonic response of a class of finite extensibility nonlinear oscillators

Finite extensibility oscillators are widely used to simulate those systems that cannot be extended to infinity. For example, they are used when modelling the bonds between molecules in a polymer or DNA molecule or when simulating filaments of non-Newtonian liquids. In this paper, the dynamic behavior of a harmonically driven finite extensibility oscillator is presented and studied. To this end, the harmonic balance method is applied to determine the amplitude–frequency and amplitude–phase equations. The distinguishable feature in this case is the bending of the amplitude–frequency curve to the frequency axis, making it asymptotically approach the limit of maximum elongation of the oscillator, which physically represents the impossibility of the system reaching this limit. Also, the stability condition that defines stable and unstable steady-state solutions is derived. The study of the effect of the system parameters on the response reveals that a decreasing value of the damping coefficient or an increasing value of the excitation amplitude leads to the appearance of a multi-valued response and to the existence of a jump phenomenon. In this sense, the critical amplitude of the excitation, which means here a certain value of external excitation that results in the occurrence of jump phenomena, is also derived. Numerical experiments to observe the effects of system parameters on the frequency–amplitude response are performed and compared with analytical calculations. At a low value of the damping coefficient or at a high value of excitation amplitude, the agreement is poor for low frequencies but good for high frequencies. It is demonstrated that the disagreement is caused by the neglect of higher-order harmonics in the analytical formulation. These higher-order harmonics, which appear as distinguishable peaks at certain values in the frequency response curves, are possible to calculate considering not the linearized frequency of the oscillator but its actual frequency, which is strongly amplitude dependent. On the other hand, for a high value of the damping coefficient or a low value of excitation amplitude, the agreement between numerical and analytical calculations is excellent. For these cases, the system is prevented from exploring large amplitudes of vibration and therefore the nonlinearity is not much evidenced.

Transient solution for flow of evaporating fluid in parallel pipes using analysis based on flow patterns

Flow with evaporation in parallel lines with common inlet and outlet headers may result in an uneven flow distribution among the parallel pipes. The prediction of the flow rate distribution in steady state as well as under transient conditions was based on simplified models. In this paper a more accurate time dependent model based on the temporal-local flow pattern in the pipe is presented. The pipe is subdivided into numerical sections and the calculation of the pressure drop in each cell is based on mechanistic models that are specific for the flow pattern in the cell. The analysis is used to calculate the steady state solutions for the flow rate distribution among the pipes and the temperature, holdup, mass concentration and quality along each pipe. A linear stability analysis is performed to demarcate between stable and unstable steady state solutions. In addition transient simulations are carried out to predict the system response to finite disturbances and to calculate the system behavior due to variations in the operating conditions such as inlet flow rate and heating power input.

Global stability of a jet in crossflow

A linear stability analysis shows that the jet in crossflow is characterized by self-sustained global oscillations for a jet-to-crossflow velocity ratio of 3. A fully three-dimensional unstable steady-state solution and its associated global eigenmodes are computed by direct numerical simulations and iterative eigenvalue routines. The steady flow, obtained by means of selective frequency damping, consists mainly of a (steady) counter-rotating vortex pair (CVP) in the far field and horseshoe-shaped vortices close to the wall. High-frequency unstable global eigenmodes associated with shear-layer instabilities on the CVP and low-frequency modes associated with shedding vortices in the wake of the jet are identified. Furthermore, different spanwise symmetries of the global modes are discussed. This work constitutes the first simulation-based global stability analysis of a fully three-dimensional base flow.

Stability of the annular Poiseuille flow of a Newtonian liquid with slip along the walls

The annular Poiseuille flow a Newtonian fluid is studied assuming that slip occurs along the walls. Different slip models relating the wall shear stress to the slip velocity are employed. In the case of non-monotonic slip models with a maximum followed by a minimum, there exist linearly unstable steady-state solutions with one or both the slip velocities along the inner and outer cylinders of the annulus corresponding to the (unstable) negative-slope branch of the slip equation. The resulting flow curve is non-monotonic with one or even two narrow unstable branches corresponding to the stick–slip instability regime. The sizes of these two unstable regimes are reduced as the radii ratio is reduced. It is demonstrated that the second unstable branch may not be observed at all due to the presence of stable steady-states. These results provide a partial explanation for the absence of the stick–slip instability in annular extrusion experiments.

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.

Limit cycle theory of self-sustained current oscillations in sequential tunneling of superlattices

A unified theory of self-sustained current oscillations (SSCOs) is presented. We establish these oscillations as the manifestations of limit cycles, around unstable steady-state solutions caused by the negative differential conductance. This theory implies that both the generation and the motion of an electric field domain boundary are universal in the sense that they do not depend on the initial conditions. Under an extra, weak ac bias with a frequency ω a c , the frequency must be either ω a c or an integer fractional of ω a c if the tunneling current oscillates periodically in time, indicating the periodic doubling for this nonlinear dynamical system.

Interaction of thermal contact resistance and frictional heating in thermoelastic instability

Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V 0 above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V> V 0 there are some initial conditions from which the contact pressure grows without limit in time, whereas for V< V 0 the system will always tend to one of the stable steady states.

Multiple Equilibria and Low-Frequency Variability of the Wind-Driven Ocean Circulation

Abstract The link between low-frequency time-dependent variability and the existence of multiple unstable steady-state solutions in a reduced gravity quasigeostrophic ocean model for the midlatitude wind-driven circulation is investigated. It is shown that a sequence of successive symmetry-breaking pitchfork bifurcations lead to multiple equilibria that differ from each other primarily in the elongation of the recirculation cell, in the amount of meandering present in the intergyre jet, and in a north–south shift in the eastward jet. The elongation of the recirculation cells and the meandering of the jet play compensating roles in the establishment of the global energy and vorticity balance. The solutions also have distinct energy levels, but general agreement between them and the bumps in a histogram of the total energy obtained from a 1200-yr time-dependent simulation is not found. Nevertheless, a substantial fraction of the variance (30%) can be accounted for by four coherent structures that capture th...

On the crossing of intermediate unstable steady state solutions for thermal ignition in a sphere

AbstractSteady state solutions for spontaneous thermal ignition in a unit sphere are considered. The multiplicity of unstable, intermediate, steady state, temperature profiles is calculated and shown for selected parameter values. The crossing of the temperature profiles corresponding to the unstable, intermediate, steady states is exhibited in a particular case and is proven in general using elementary ideas from analysis. Estimates of the location of crossing points are given.