Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

A Suggested Mechanism for Nonlinear Wall Roughness Effects on High Reynolds Number Flow Stability

The interactions between an uneven wall and free stream unsteadiness and their resultant nonlinear influence on flow stability are considered by means of a related model problem concerning the nonlinear stability of streaming flow past a moving wavy wall. The particular streaming flows studied are plane Poiseuille flow and attached boundary-layer flow, and the theory is presented for the high Reynolds number regime in each case. That regime can permit inter alia much more analytical and physical understanding to be obtained than the finite Reynolds number regime; this may be at the expense of some loss of real application, but not necessarily so, as the present study shows. The fundamental differences found between the forced nonlinear stability properties of the two cases are influenced to a large extent by the surprising contrasts existing even in the unforced situations. For the high Reynolds number effects of nonlinearity alone are destabilizing for plane Poiseuille flow, in contrast with both the initial suggestion of earlier numerical work (our prediction is shown to be consistent with these results nevertheless) and the corresponding high Reynolds number effects in boundary-layer stability. A small amplitude of unevenness at the wall can still have a significant impact on the bifurcation of disturbances to finite-amplitude periodic solutions, however, producing a destabilizing influence on plane Poiseuille flow but a stabilizing influence on boundary-layer flow.     

Investigation of secondary flow instability on an infinite yawed cylinder with 15 per cent. Thick joukowski profile as section

This paper contains a theoretical investigation of the secondary flow instability in the incompressible boundary layer on an infinite yawed cylinder with chordwise section as Joukowski profile of 15 per cent. thickness at zero incidence and with homogeneous suction, the suction mass flow coefficient being equal to 0·2085. Values of the instability criterion are obtained at different points of the wing section and for various angles of sweepback. It is found that the values of the criterion increase with the increasing sweepback whether the pressure gradient is favourable or adverse. The effect of adverse pressure gradient on the variation of the criterion is more pronounced than that of a favourable pressure gradient. At some points in adverse pressure gradients, there are two values of the criterion for a given sweepback. It is also found that the flow is intermittently laminar and turbulent for low values of the chordwise free stream Reynolds number and consists of an irregular sequence of laminar and turbulent regions.     

Effect of wall alignment in a very short rotating annulus

This paper reports numerical results of the study of effects of cylinders wall alignment in a small aspect ratio Taylor–Couette system. The investigation concerns bifurcations of steady vortical structures when the cylindrical walls defining the gap are not perfectly parallel. The imperfection is introduced by opening the outer fixed cylinder with a certain angle with regard to the vertical to form a tapered very short liquid column and keeping the inner rotating cylinder wall vertical. The numerical results obtained for the velocity components have revealed that bifurcation from a particular mode to another one occurs at a range of specific values of the inclination angle of the outer cylinder. The band width of the angle at which bifurcation occurred depended on the Reynolds number Re and was found to become narrower as Re increased. It is shown that geometrically broken symmetry can yield flow symmetry for specific combinations of geometrical and dynamical parameters.     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.     

Stability of flow past cylinder cascade

The global stability analysis of the two–dimensional incompressible unsteady flow past a circular clinder cascade is studied using linear stability analysis. A new numerical technique is used to find the value of critical Reynolds number that causes the flow to undergo Hopf bifurcation. An attempt has also been made to study the blockage effect on critical Reynolds number and associated Strouhal number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experiments on the flow stability in a double-lid-driven cavity

The stability of the two-dimensional, steady, incompressible flow in a rectangular square cavity is investigated experimentally for the parallel motion of two facing walls. The critical Reynolds numbers for the onset of three-dimensional steady flow, its structure, and the bifurcation diagram of the velocity field, measured by LDV, agree with numerical predictions. It is observed that the wavelength of the selected pattern increases with the Reynolds number. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiple bifurcation from “simple” eigenvalues

Consider an operator equation G(u,λ) = 0 where λ is a real parameter. Suppose 0 is a “simple” eigenvalue of the Fréchet derivative G_u at (u₀, λ₀). We give a hierarchy of conditions which completely determines the solution structure of the operator equation. It will be shown that multiple bifurcation as well as simple bifurcation can occur. This extends the standard bifurcation theory from a simple eigenvalue in which only one branch bifurcates. We also discuss limit point bifurcations. Applications to semilinear elliptic equations and the homotopy method for the matrix eigenvalue problem are also given.     

On the stability and transition for the Navier-Stokes-alpha model

The main aim of this paper is to investigate the stability and transition of the Navier-Stokes-alpha model. By using the continued-fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non-dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio-temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid-axis of the channel.     

On the Effects of Nonparallelism,Three-Dimensionality,and Mode Interaction in Nonlinear Boundary-Layer Stability

A self-consistent theoretical investigation is described for the nonlinear stability, and spatial development, of disturbances in a plane boundary layer subject to a number of three-dimensional modes, their nonlinear interactions, and the effects of nonparallelism of the basic flow. For the largest weakly nonlinear disturbances considered, nonparallel-flow effects appear to be negligible at first sight, and primary, secondary, and/or tertiary bifurcations, usually supercritical but not always so, can occur when two fundamental modes are present. As a result the flow downstream then always has three ultimate possibilities: a unique stable disturbed state, two or more possible stable states, or no stable state possible. It is here that the nonparallel-flow effects exert their crucial influence. For nonparallelism comes into play significantly during the initial growth or decay of a disturbance, and that initial spatial development, from given initial conditions upstream, controls what happens subsequently as the disturbance increases. Thus in the first possibility above, the stable state is achieved through a smooth bifurcation, due to nonparallelism; in the second possibility the nonparallelism decides which stable state is attained (smoothly) from the initial conditions; and in the third possibility the nonparallel flow effects force the disturbance to terminate in a singular fashion. This singularity then leads to a fully nonlinear effect, locally on the boundary-layer flow. More complicated interactions can arise if more than two three-dimensional modes are present. The novel effect of the nonparallelism has a connection with related Navier-Stokes calculations even at near-critical Reynolds numbers.     

Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Iterative solvers and inflow boundary conditions for plane sudden expansion flows

Incompressible laminar flow in a symmetric plane sudden expansion is studied numerically. The flow is known to exhibit a stable symmetric solution up to a critical Reynolds number above which symmetry-breaking bifurcation occurs. The aim of the present study is to investigate the effect of using different iterative solvers on the calculation of the bifurcation point. For this purpose, the governing equations for steady two-dimensional incompressible flow are written in terms of a stream function-vorticity formulation. A second order finite volume discretization is applied. Explicit and implicit solvers are used to solve the resulting system of algebraic equations. It is shown that the explicit solver recovers the stable asymmetric solution, while the implicit solver can recover both the unstable symmetric solution or the stable asymmetric solution, depending on whether the initial guess is symmetric or not. It is also found that the type of inflow velocity profile, whether uniform or parabolic, has a significant effect on the onset of bifurcation as uniform inflows tend to stabilize the symmetric solution by delaying the onset of bifurcation to a higher Reynolds number as compared to parabolic inflows.     

Forced convective flow and heat transfer past an unconfined blunt headed cylinder at different angles of incidence

In the present paper, a numerical investigation has been carried out to study the forced convective flow and heat transfer characteristics past a blunt-headed cylinder in crossflow. Employing air as an operating fluid, calculations are carried out for a range of Reynolds number (Re) from 40 to 160. The angle of incidence is varied in the range of 0^∘ ≤ α ≤ 180 ^∘. The thermofluid features of flow and heat transport are analysed in detail for different angles of incidence. To analyse the aerodynamic characteristics, several parameters such as drag and lift coefficients, moment coefficient, Strouhal number, recirculation length, and local time-averaged vorticity flux have been calculated. Furthermore, a stability analysis has been undertaken by using the Stuart Landau equation to enumerate the critical Reynolds number at each angle of incidence. Heat transfer characteristics are studied by computing local and time-averaged values of Nusselt numbers. When compared to a rectangular cylinder, a blunt-headed cylinder exhibits an enhanced heat transfer rate. In the end, an entropy generation analysis has been carried out to study the effects of Re and angle of incidence on the efficiency of thermofluid transport characteristics.     

Conforming Chebyshev Spectral Collocation Methods for the Solution of Laminar flow in a Constricted Channel

The numerical simulation of steady planar two-dimensional laminarflow of an incompressible fluid through an abruptly contractingchannel using spectral domain decomposition methods is described.The key features of the method are the decomposition of theflow region into a number of rectangular subregions and spectralapproximations that are pointwise C¹ continuous across subregioninterfaces. Spectral approximations to the solution are obtainedfor Reynolds numbers in the range [0, 500]. The size of thesalient corner vortex decreases as the Reynolds number increasesfrom 0 to around 45. As the Reynolds number is increased further,the vortex grows slowly. A vortex is detected downstream ofthe contraction at a Reynolds number of around 175 that continuesto grow as the Reynolds number is increased further.     

Modeling unsteady flow characteristics using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method has been extensively used to simulate unsteady free surface flows. The works dedicated to simulation of unsteady internal flows have been generally performed to study the transient start up of steady flows under constant driving forces and for low Reynolds number regimes. However, most of the fluid flow phenomena are unsteady by nature and at moderate to high Reynolds numbers. In this study, first a benchmark case (transient Poiseuille flow) is simulated to evaluate the ability of SPH to simulate internal transient flows at low and moderate Reynolds numbers (Re = 0.05, 500 and 1500). For this benchmark case, the performance of the two most commonly used formulations for viscous term modeling is investigated, as well as the effect of using the XSPH variant. Some points regarding using the symmetric form for pressure gradient modeling are also briefly discussed. Then, the application of SPH is extended to oscillating flows imposed by oscillating body force (Womersley type flow) and oscillating moving boundary (Stokes’ second problem) at different frequencies and amplitudes. There is a very good agreement between SPH results and exact solution even if there is a large phase lag between the oscillating pressure difference and moving boundary and the movement of the SPH particles generated. Finally, a modified formulation for wall shear stress calculations is suggested and verified against exact solutions. In all presented cases, the spatial convergence analysis is performed.     

Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response

We perform a bifurcation analysis of a discrete predator-prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F₁,F₂ and F₃ and collect complete information on this in a single scheme. In the case of F₂ we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM. Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.     

A Five-Dimensional System of Navier-Stokes Equation for a Two-Dimensional Incompressible Fluid on a Torus

A five-mode truncation of Navier-Stokes equation for a two-dimensional incompressible fluid on a torus is studied. Its stationary solutions and stability are presented, the existence of attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through an involved sequence of bifurcations with the changing of Reynolds number, is simulated numerically. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system are revealed.     

Struggles With Developing the Concept of Angle: Comparing Sixth-Grade Students' Discourse to the History of the Angle Concept

This article highlights the similarities between sixth-grade students' developing notions of angle and mathematicians' struggles to define this complex concept. Students learning from a reformed curriculum had the opportunity to voice their ideas concerning the definition of angle. These conversations were compared with past discussions and debates concerning the meaning of angle. The data were classified into three major categories: (a) What exactly is being measured when referring to the size of angles? (b) Can angles contain curves?, and (c) Difficulties with conceiving of 0°, 180°, and 360° angles. The data suggest that using multiple representations such as those examples from history would seem to be more comprehensive and a better preparation as students study the concept of angle.     

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.