Steady Symmetric Flow Research Articles

Stability of flows past a pair of circular cylinders in a side-by-side arrangement

The stability and transition of flow past a pair of circular cylinders in a side-by-side arrangement are investigated by numerical simulations and linear stability analyses. Various flow patterns around the cylinders have been reported to appear due to an instability of the steady symmetric flow that is realized at small Reynolds numbers, among which deflected oscillatory flow is particularly noticeable. The physical origin of the flow is explored by bifurcation analyses of the numerical data. We found that the deflected oscillatory flow arises from the steady symmetric flow through sequential instabilities due to stationary and oscillatory unstable modes. Steady asymmetric flow with respect to the streamwise centreline between the two cylinders was also found to be induced by the instability due to a stationary mode in a very narrow range of the gap width between the two cylinders. We classify the instability modes of the steady symmetric flow into four groups in the parameter space of the gap width, and evaluate the critical Reynolds number for each mode of instability.

Coil-stretch transition and the breakdown of computations for viscoelastic fluid flow around a confined cylinder

The breakdown of finite element (FEM) computations for the steady symmetric two-dimensional flow of dilute and ultradilute Oldroyd-B fluids around a cylinder in a channel, at Weissenberg numbers Wi=O(1), is shown to arise due to a coil-stretch transition experienced by polymer molecules in the wake of the cylinder in the vicinity of the location of the stress maximum on the centerline. In dilute Oldroyd-B fluids, due to the modification of the flow caused by the presence of the polymer, the coil-stretch transition leads to the stress maximum diverging toward infinity at a finite value of Wi≈0.7. On the other hand, in ultradilute solutions, the stress maximum approaches infinity only as Wi→∞. In FENE-P fluids, the coil-stretch transition leads to the mean extension of the molecules saturating to a value close to the fully-extended length, with the maximum stress remaining bounded with increasing Wi. An estimation of the number of finite elements required to achieve convergence for ultradilute Oldroyd-B fluids reveals that obtaining solutions at Wi>1 is not feasible.

Effect of power-law index on critical parameters for power-law flow across an unconfined circular cylinder

The conditions for the formation of a wake and for the onset of wake instability for the flow of power-law fluids over an unconfined circular cylinder are investigated numerically by solving the continuity and momentum equations using FLUENT (version 6.2). The effect of power-law index on the critical Reynolds numbers, Strouhal number and drag coefficient has been presented over a wide range of power-law index ( 0.3 ⩽ n ⩽ 1.8 ) thereby establishing the limits of the flow without separation and the steady symmetric flow regimes, respectively. While both the shear-thinning ( n < 1 ) and the shear-thickening ( n > 1 ) seem to lower the value of the critical Reynolds number denoting the onset of wake instability as compared to that for Newtonian fluids, the effect is seen to be more prominent for shear-thickening fluids than that for shear-thinning fluids. The corresponding values of the critical Strouhal number ( St c ) and drag coefficient have also been presented for the critical values of the Reynolds number. Included here are also a series of streamline plots showing the onset of asymmetry and of the time-dependent flow regime.

Symmetry breaking of melt flow typically encountered in a Bridgman configuration heated from below

A computational model is developed to investigate three-dimensional fluid flow instability and transition to unsteadiness. The simulated domain corresponds to a parallelepiped configuration which is a restriction to the fluid phase of commonly used configurations in crystal growth (viz. Bridgman configuration). In the present work, this problem is studied using three-dimensional simulations and focuses on symmetry breaking and transition to unsteadiness occurring for a low Prandtl number fluid ( Pr = 0.01). Three-dimensional simulations presented thus overcome the limitation of the two-dimensional approach. It has been found that the initially steady symmetric flow in the three-dimensional case becomes asymmetric for lower Ra number than for the two-dimensional case. The breaking in symmetry occurs firstly in the transverse plane. For the relatively low Ra number we still have no intensification in the global heat transfer but it appears that the heat transfer increases locally on the bottom and decreases on the vertical active walls. The classical spiral flows typical characterizing the three-dimensional effect (in the third direction) are also identified.

A numerical study on the forced convection heat transfer from an isothermal and isoflux sphere in the steady symmetric flow regime

The effects of Reynolds and Prandtl numbers on the heat transfer characteristics of an unconfined sphere for different thermal boundary conditions (isothermal and isoflux) on the sphere surface have been investigated numerically by using a finite volume method for the range of conditions as 5 ⩽ Re ⩽ 200 and 0.7 ⩽ Pr ⩽ 400 (the maximum value of Peclet number being 2000). Based on the numerical results obtained herein, heat transfer correlations are developed for the constant temperature and the constant heat flux boundary conditions on the solid sphere surface in the steady symmetric flow regime. The variation of local Nusselt number on the sphere surface shows the effect of Prandtl number on heat transfer from a sphere in this flow regime. In addition, this work also demonstrates an approach to solve such flow problems using the Cartesian form of the field equations.

Instability and transition of flow past two tandem circular cylinders

The instability and transition of flow past two circular cylinders arranged in tandem are investigated numerically. A steady symmetric flow is realized at small Reynolds numbers, but the flow becomes unstable above a critical Reynolds number and makes a transition to an oscillatory flow. We obtained the symmetric flow numerically and analyze its stability by applying linear stability theory. The nonlinear oscillatory flow arising from the instability is obtained not only by numerical simulation but also by direct numerical calculation of the equilibrium solution, and the bifurcation diagram for the nonlinear equilibrium solution is depicted. We focused our attention on the effect of the gap spacing between the two cylinders on the stability and transition of the flow. The transition of the flow from a steady state to an oscillatory state is clarified to occur due to a supercritical or subcritical Hopf bifurcation depending upon the gap spacing. We found that there is a certain range of the gap spacing where physical quantities such as the drag and lift coefficients and the Strouhal number show an abrupt change when the gap spacing is continuously changed. We identified the origin of the abrupt change as the existence of multiple stable solutions for the flow.

Linear Stability of Flow past Two Circular Cylinders in a Side-by-Side Arrangement

The linear stability of flow past two circular cylinders in a side-by-side arrangement is investigated theoretically, numerically and experimentally under the assumption of a two-dimensional flow field, in order to explore the origin of in-phase and antiphase oscillatory flows. Steady symmetric flow is realized at a small Reynolds number, but becomes unstable above a critical Reynolds number though the solution corresponding to the flow still satisfies the basic equations irrespective of the magnitude of the Reynolds number. We obtained the solution numerically and investigated its linear stability. We found that there are two kinds of unstable modes, i.e., antisymmetric and symmetric modes, which lead to in-phase and antiphase oscillatory flows, respectively. We determined the critical Reynolds numbers for the two modes and evaluated the critical distance at which the most unstable disturbance changes from the antisymmetric to the symmetric mode, or vice versa.

Bifurcation and stability of forced convection in tightly coiled ducts: stability

Abstract A numerical study is made on the stability of multiple steady flows and heat transfer in tightly coiled ducts by examining their responses to finite random disturbances. It is found that possible physically realizable fully developed flows evolve, as the Dean number increases, from a stable steady symmetric 2-cell flow at lower Dean numbers to a temporal periodic oscillation, a temporal intermittent oscillation, another temporal periodic oscillation, the co-existence of stable steady symmetric 2-cell flow and temporal oscillating flows (either periodic or aperiodic), and the co-existence of three stable steady 2-cell flows (either symmetric or asymmetric) and aperiodic oscillating flows.

Three-dimensional numerical simulations of the transition of flow past a cube

The wake of a cube placed in a uniform flow is numerically studied. The present study concentrates on the first two transitions, one spatial and the other a temporal one. Computations are carried out for a Reynolds number range of 20–300. Starting from a steady symmetric flow, the transition to asymmetric steady flow occurs at a Reynolds number between 216 and 218. The loss of symmetry increases with increasing Reynolds number. The asymmetric steady flow experiences a Hopf bifurcation at a Reynolds number between 265 and 270 and becomes unsteady but maintains the planar symmetry. The transitions of the present study are compared to the transitions of flow past a square cylinder and a sphere. The transition sequence and flow structures match closely to that of a sphere. The drag and side force coefficients are calculated and it is observed that they are in accordance with the spatial structures of the wake. The drag coefficient is found to decrease with Reynolds number in the steady flow regime. Similarly, one component of the side force coefficients increases while the other component remains zero with increasing Reynolds number for the asymmetric flow.

A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder

A finite volume method based on a velocity-only formulation is used to solve the flow field around a confined circular cylinder in a channel in order to investigate lateral wall proximity effects on stability, Strouhal number, hydrodynamic forces and wake structure behind the cylinder for a wide range of blockage ratios (0.1&amp;lt;β⩽0.9) and Reynolds numbers (0&amp;lt;Re⩽280). For blockage ratios less than approximately 0.85 a first critical Reynolds number is identified at which a supercritical Hopf bifurcation of the symmetric solution occurs. For blockage ratios greater than about 0.687 and at Reynolds numbers exceeding the first critical Reynolds number a second curve of neutral stability is seen, representing a pitchfork bifurcation of the steady symmetric solution to one of two possible steady asymmetric solutions. Either side of the neutral stability curve for the pitchfork bifurcation our linear stability analysis and direct numerical simulations demonstrate that although the flow is linearly stable it is unstable to finite two-dimensional perturbations. At blockage ratios larger than about 0.82 the steady asymmetric solutions also become unstable through a Hopf bifurcation. In contrast with the first Hopf bifurcation of the symmetric solution at lower Reynolds numbers numerical calculations of the lift coefficient reveal that the oscillations are no longer symmetric in the rising and falling parts of each cycle. Very strong vortices shed from the cylinder and the wall cause drastic increases in the amplitudes of the lift and drag coefficients. A co-dimension 2 point where pitchfork and Hopf bifurcations occur simultaneously has been located in parameter space. Altogether, four distinct regions in the parameter space (β,Re)∈(0,0.9]×(0,280] have been identified, each corresponding to a different class of flow: (i) Steady symmetric flow, (ii) symmetric vortex shedding, (iii) steady asymmetric flow, and (iv) asymmetric vortex shedding, where a periodic-in-time flow is classed as symmetric or asymmetric depending on whether the time-average over one cycle of the lift coefficient is zero or not. Numerical solutions are computed on meshes having up to 1.8 million degrees of freedom. Extensive comparisons are made with the results available in the literature.

The effect of cylinder rotation and blockage ratio on the onset of periodic flows

The steady two-dimensional laminar flow past a stationary cylinder is well known to lose stability to a periodic flow at a supercritical Hopf bifurcation point as the flow rate is increased. It is less well known that the critical flow rate at which the instability occurs can be increased by rotating the cylinder about is own axis, and that at a fixed flow rate the vortex street can be eliminated entirely by sufficiently large rotation. We confine the flow between two parallel walls and report the effect of cylinder rotation on the Reynolds number and frequency at the Hopf bifurcation point. Two new, and somewhat surprising, results are that for stationary cylinders at large blockage ratios, the steady symmetric flow can restabilize above a critical Reynolds number, and that steady asymmetric flows exist.

Symmetry Breaking of Radially Outgoing Flow between Two Parallel Disks

The instability and transition of radially outgoing flow between two parallel disks with two symmetrically attached inlet pipes through which fluid enters are investigated theoretically and experimentally assuming an incompressible and axisymmetric flow field. The flow is steady and symmetric with respect to the center plane between the two disks at small Reynolds numbers, but becomes asymmetric above a critical Reynolds number. Numerical simulations and the linear stability analysis reveal that steady asymmetric flows result from the instability of the steady symmetric flow. The axisymmetry and the transition were confirmed in our experiments with flow visualization.

Vortex shedding from a row of square bars

Interactions of wakes in a flow past a row of square bars, which is placed across a uniform flow, are investigated by numerical simulations and experiments on the tassumption that the flow is two-dimensional and incompressible. At small Reynolds numbers the flow is steady and symmetric with respect not only to streamwise lines through the center of each square bar but also to streamwise centerlines between adjacent square bars. However, the steady symmetric flow becomes unstable at larger Reynolds numbers and make a transition to a steady asymmetric flow with respect to the centerlines between adjacent square bars in some cases or to an oscillatory flow in other cases. It is found that vortices are shed synchronously from adjacent square bars in the same phase or in anti-phase depending upon the distance between the bars when the flow is oscillatory. The origin of the transition to the steady asymmetric flow is identified as a pitchfork bifurcation, while the oscillatory flows with synchronous shedding of vortices are clarified to originate from a Hopf bifurcation. The critical Reynolds numbers of the transitions are evaluated numerically and the bifurcation diagram of the flow is obtained.

２次元円柱，３次元球及び低アスペクト比円柱周り非対称流れの全体安定性解析

It is well known that the flow around a sphere becomes steady and non-axisymmetric when the freestream Reynolds number (based on the diameter) becomes greater than 210. However such steady asymmetric patterns cannot be found in the case of the two-dimensional flow around a circular cylinder. In this paper, the flow around a three-dimensional low aspect ratio cylinder is calculated, and it is found that the flow around a low aspect ratio cylinder also becomes steady and asymmetric under a certain range of freestream Reynolds numbers. The two-dimensional and three-dimensional incompressible Navier-Stokes equations are numerically solved for flows around these bodies, and the transition mode from steady symmetric flow to asymmetric flow is investigated by global stability analysis. These results show that in the two-dimensional case, the asymmetric non-oscillatory mode decays faster than the oscillatory modes. In the case of the asymmetric non-oscillatory mode of the two-dimensional flow, the region of its influence extends far downstream and reaches the outer boundary of the computational domain, while in the three-dimensional non-oscillatory mode the region of influence is observed only in the vicinity of the body. The similarities of the flow around a sphere and low aspect ratio cylinder are discussed from a viewpoint of the configuration of these transition modes.

Control of circular cylinder wakes using base mass transpiration

The effect of steady base suction and blowing on the stability and dynamics of the cylinder wake is investigated at low Reynolds numbers, using numerical simulation and stability analysis. Simulation results show that, in the supercritical Reynolds number regime (Re&amp;gt;47), slight blowing or high enough suction stabilizes the wake; in the subcritical regime, suction destabilizes the wake and results in vortex shedding, whereas blowing has no detectable effect on the flow stability. At supercritical Reynolds numbers, the transition from unsteady to steady flow at a critical suction flow rate is accompanied by simultaneous symmetry breaking, resulting in strongly asymmetric steady flow. For finite flow domain, the flow undergoes another transition from steady asymmetric to steady symmetric flow at even higher suction flow rates. The dynamics of vortex shedding in the controlled flow can be strongly modified, in comparison to the uncontrolled flow. Global stability analysis confirms the results of numerical simulation, and yields detailed information on the dynamics of linear global modes. Computational domain size can significantly affect the flow transitions, in particular, at high suction flow rates.

Linear stability of steady fluid flow through a constricted channel

Considered in this paper is the linear stability of the steady symmetric flow of a Newtonian fluid through an abrupt planar constriction. Flows with Reynolds numbers up to 2,000 and various ratios of contraction were investigated. In all the cases considered asymmetric disturbances imposed on the steady flow were found to decay in time. The results indicate that the flow is linearly stable even for large Reynolds numbers. This is in agreement with experimental investigations which do not reveal the occurrence of any flow instabilities.

Stability of flow in a channel with a suddenly expanded part

The stability of a two-dimensional flow in a symmetric channel with a suddenly expanded part is investigated numerically and analyzed by using the method of the nonlinear stability theory. From results of the numerical simulation, it is shown that the flow is steady, symmetric and unique at very low Reynolds numbers, while the symmetric flow loses its stability at a critical Reynolds number resulting in an appearance of asymmetric flow. The transition from the steady symmetric flow to the steady asymmetric one is found to occur due to the symmetry breaking pitchfork bifurcation when the aspect ratio, the ratio of the length of the expanded part to its width, is large. It is also found that the bifurcated flow becomes symmetric again when the Reynolds number is increased and the resultant symmetric flow loses its stability becoming periodic in time as the Reynolds number is further increased. On the other hand, when the aspect ratio is small there occurs no pitchfork bifurcation and the direct transition from the steady symmetric flow to a periodic flow occurs due to a Hopf bifurcation. The critical aspect ratio is found to be about 2.3. The critical Reynolds numbers for these bifurcations are evaluated.

MIXED CONVECTIVE HEAT TRANSFER IN A HORIZONTAL CHANNEL HEATED PERIODICALLY FROM BELOW

Two-dimensional Navier-Stokes and energy equations are numerically solved for laminar flow in a horizontal channel with localized heating from below. The relative strength of the forced flow and buoyancy effects are examined for a wide range of Rayleigh Ra, Reynolds Re, and Prandtl Pr numbers. For a fixed geometry and a given Rayleigh number, a complicated solution structure is observed upon increasing the Reynolds number. For Re = 0 (i.e., in the case of pure free convection), a steady symmetric flow pattern is obtained. This pattern becomes asymmetric for Re below a critical value, but the rolls remain attached to the heating elements. Above the critical Re, the rolls are carried downstream with a time-dependent velocity, and the flow becomes periodic in time.

Steady two-dimensional merging flow from two channels into a single channel

Two-dimensional steady symmetric merging flow from two channels into a single one is investigated. The geometry of the configuration has been chosen such that it can be mapped conformally onto a rectangular geometry, thus facilitating the numerical solution procedure for the governing Navier-Stokes equations. Computed velocity profiles and streamline patterns are presented in graphical form. Furthermore, results concerning the inlet length are given.

Heat and Mass Transfer from Elliptical Cylinders in Steady Symmetric Flow

Heat and Mass Transfer from Elliptical Cylinders in Steady Symmetric Flow