Transition in Convective Flows Heated Internally

2002 ◽  
Vol 124 (4) ◽  
pp. 635-642 ◽  
Author(s):  
Masato Nagata ◽  
Sotos Generalis
Keyword(s):  
Hopf Bifurcation ◽  
Secondary Flow ◽  
Linear Stability ◽  
Three Dimensional ◽  
Vertical Channel ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Grashof Number ◽  
Stability Curve ◽  
Convective Flows

The stability of internally heated convective flows in a vertical channel under the influence of a pressure gradient and in the limit of small Prandtl number is examined numerically. In each of the cases studied the basic flow, which can have two inflection points, loses stability at the critical point identified by the corresponding linear analysis to two-dimensional states in a Hopf bifurcation. These marginal points determine the linear stability curve that identifies the minimum Grashof number (based on the strength of the homogeneous heat source), at which the two-dimensional periodic flow can bifurcate. The range of stability of the finite amplitude secondary flow is determined by its (linear) stability against three-dimensional infinitesimal disturbances. By first examining the behavior of the eigenvalues as functions of the Floquet parameters in the streamwise and spanwise directions we show that the secondary flow loses stability also in a Hopf bifurcation as the Grashof number increases, indicating that the tertiary flow is quasi-periodic. Secondly the Eckhaus marginal stability curve, that bounds the domain of stable transverse vortices towards smaller and larger wavenumbers, but does not cause a transition as the Grashof number increases, is also given for the cases studied in this work.

Three-dimensional stability analysis for a salt-finger convecting layer

2018 ◽  
Vol 841 ◽  
pp. 636-653
Author(s):  
Ting-Yueh Chang ◽  
Falin Chen ◽  
Min-Hsing Chang
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Longitudinal Mode ◽  
Transverse Mode ◽  
Solute Diffusion ◽  
Two Dimensional ◽  
Grashof Number ◽  
Significant Difference ◽  
The Stability ◽  
Mode A

A three-dimensional linear stability analysis is carried out for a convecting layer in which both the temperature and solute distributions are linear in the horizontal direction. The three-dimensional results show that, for $Le=3$ and 100, the most unstable mode occurs invariably as the longitudinal mode, a vortex roll with its axis perpendicular to the longitudinal plane, suggesting that the two-dimensional results are sufficient to illustrate the stability characteristics of the convecting layer. Two-dimensional results show that the stability boundaries of the transverse mode (a vortex roll with its axis perpendicular to the transverse plane) and the longitudinal modes are virtually overlapped in the regime dominated by thermal diffusion and the regime dominated by solute diffusion, while these two modes hold a significant difference in the regime the salt-finger instability prevails. More precisely, the instability area in terms of thermal Grashof number $Gr$ and solute Grashof number $Gs$ is larger for the longitudinal mode than the transverse mode, implying that, under any circumstance, the longitudinal mode is always more unstable than the transverse mode.

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

scholarly journals Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ruiqing Shi ◽  
Junmei Qi ◽  
Sanyi Tang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Three Dimensional ◽  
Distributed Delay ◽  
Dimensional System ◽  
Two Dimensional ◽  
Normal Form Theory ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory

We propose a two-dimensional predatory-prey model with discrete and distributed delay. By the use of a new variable, the original two-dimensional system transforms into an equivalent three-dimensional system. Firstly, we study the existence and local stability of equilibria of the new system. And, by choosing the time delayτas a bifurcation parameter, we show that Hopf bifurcation can occur as the time delayτpasses through some critical values. Secondly, by the use of normal form theory and central manifold argument, we establish the direction and stability of Hopf bifurcation. At last, an example with numerical simulations is provided to verify the theoretical results. In addition, some simple discussion is also presented.

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

The secondary flow and its stability for natural convection in a tall vertical enclosure

1989 ◽  
Vol 200 ◽  
pp. 189-216 ◽  
Author(s):  
Arnon Chait ◽  
Seppo A. Korpela
Keyword(s):  
Three Dimensional ◽  
Pseudospectral Method ◽  
Base Flow ◽  
Temperature Fields ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Grashof Number ◽  
Practical Applications ◽  
Prandtl Numbers ◽  
Dependent Flow

The multicellular flow between two vertical parallel plates is numerically simulated using a time-splitting pseudospectral method. The steady flow of air, and the time-periodic flow of oil (Prandtl numbers of 0.71 and 1000, respectively) are investigated and descriptions of these flows using both physical and spectral approaches are presented. The details of the time dependency of the flow and temperature fields of oil are shown, and the dynamics of the process is discussed. The spectral transfer of energy among the axial modes comprising the flow is explored. The spectra of kinetic energy and thermal variance for air are found to be smooth and viscously dominated. Similar spectra for oil are bumpier, and the dynamics of the time-dependent flow are determined to be confined to the lower end of the spectrum alone.The three-dimensional linear stability of the multicellular flow of air is parametrically studied. The domain of stable two-dimensional cellular motion was found to be constrained by the Eckhaus instability and by two types of monotone instability. The two-dimensional multicellular flow is unstable above a Grashof number of about 8550 (with the critical Grashof number for the base flow being 8037). Therefore the flow of air in a sufficiently tall vertical enclosure should be considered to be three-dimensional for most practical applications.

scholarly journals Large-eddy simulation of three-dimensional dunes in a steady, unidirectional flow. Part 1. Turbulence statistics

2013 ◽  
Vol 721 ◽  
pp. 454-483 ◽  
Author(s):  
Mohammad Omidyeganeh ◽  
Ugo Piomelli
Keyword(s):  
Secondary Flow ◽  
Three Dimensional ◽  
Wall Stress ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Mean Flow ◽  
Form Drag ◽  
Separated Shear Layer ◽  
Large Eddy ◽  
The Mean

AbstractWe performed large-eddy simulations of flow over a series of three-dimensional dunes at laboratory scale (Reynolds number based on the average channel depth and streamwise velocity was 18 900) using the Lagrangian dynamic eddy-viscosity subgrid-scale model. The bedform three-dimensionality was imposed by shifting a standard two-dimensional dune shape in the streamwise direction according to a sine wave. The statistics of the flow are discussed in 10 cases with in-phase and staggered crestlines, different deformation amplitudes and wavelengths. The results are validated qualitatively against experiments. The three-dimensional separation of flow at the crestline alters the distribution of wall pressure, which in turn may cause secondary flow across the stream, which directs low-momentum fluid, near the bed, toward the lobe (the most downstream point on the crestline) and high-momentum fluid, near the top surface, toward the saddle (the most upstream point on the crestline). The mean flow is characterized by a pair of counter-rotating streamwise vortices, with core radius of the order of the flow depth. However, for wavelengths smaller than the flow depth, the secondary flow exists only near the bed and the mean flow away from the bed resembles the two-dimensional case. Staggering the crestlines alters the secondary motion; the fastest flow occurs between the lobe and the saddle planes, and two pairs of streamwise vortices appear (a strong one, centred about the lobe, and a weaker one, coming from the previous dune, centred around the saddle). The distribution of the wall stress and the focal points of separation and attachment on the bed are discussed. The sensitivity of the average reattachment length, depends on the induced secondary flow, the streamwise and spanwise components of the channel resistance (the skin friction and the form drag), and the contribution of the form drag to the total resistance are also studied. Three-dimensionality of the bed increases the drag in the channel; the form drag contributes more than in the two-dimensional case to the resistance, except for the staggered-crest case. Turbulent-kinetic energy is increased in the separated shear layer by the introduction of three-dimensionality, but its value normalized by the plane-averaged wall stress is lower than in the corresponding two-dimensional dunes. The upward flow on the stoss side and higher deceleration of flow on the lee side over the lobe plane lift and broaden the separated shear layer, respectively, affecting the turbulent kinetic energy.

New results in rotating Hagen–Poiseuille flow

2000 ◽  
Vol 417 ◽  
pp. 103-126 ◽  
Author(s):  
D. R. BARNES ◽  
R. R. KERSWELL
Keyword(s):  
Hopf Bifurcation ◽  
Couette Flow ◽  
Poiseuille Flow ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Rotation Rates ◽  
Supercritical Hopf Bifurcation

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

A Calculation Procedure for Three-Dimensional, Time-Dependent, Inviscid, Compressible Flow through Turbomachine Blades of any Geometry

1979 ◽  
Vol 21 (1) ◽  
pp. 39-49 ◽  
Author(s):  
C. Bosman ◽  
J. Highton
Keyword(s):  
Secondary Flow ◽  
Compressible Flow ◽  
Subsonic Flow ◽  
Three Dimensional ◽  
Experimental Results ◽  
Time Dependent ◽  
Two Dimensional ◽  
Through Flow ◽  
Flow Through ◽  
Inviscid Compressible Flow

A method for calculating three-dimensional, time-dependent, inviscid, subsonic flow is presented. Application is made to flow through the rotor of a small radial inflow turbine and comparison with conventional through-flow calculations and experimental results is made. The nature of the strong secondary flow in this rotor indicates the probable inadequacy of the two-dimensional calculations which is confirmed by the comparison.

Stability of a temporally evolving natural convection boundary layer on an isothermal wall

2019 ◽  
Vol 877 ◽  
pp. 1163-1185 ◽  
Author(s):  
Junhao Ke ◽  
N. Williamson ◽  
S. W. Armfield ◽  
G. D. McBain ◽  
S. E. Norris
Keyword(s):  
Boundary Layer ◽  
Natural Convection ◽  
Linear Stability ◽  
Transition Point ◽  
Three Dimensional ◽  
Base Flow ◽  
Convection Boundary Layer ◽  
Two Dimensional ◽  
Integral Boundary ◽  
Convection Boundary

The stability properties of a natural convection boundary layer adjacent to an isothermally heated vertical wall, with Prandtl number 0.71, are numerically investigated in the configuration of a temporally evolving parallel flow. The instantaneous linear stability of the flow is first investigated by solving the eigenvalue problem with a quasi-steady assumption, whereby the unsteady base flow is frozen in time. Temporal responses of the discrete perturbation modes are numerically obtained by solving the two-dimensional linearized disturbance equations using a ‘frozen’ base flow as an initial-value problem at various $Gr_{\unicode[STIX]{x1D6FF}}$, where $Gr_{\unicode[STIX]{x1D6FF}}$ is the Grashof number based on the velocity integral boundary layer thickness $\unicode[STIX]{x1D6FF}$. The resultant amplification rates of the discrete modes are compared with the quasi-steady eigenvalue analysis, and both two-dimensional and three-dimensional direct numerical simulations (DNS) of the temporally evolving flow. The amplification rate predicted by the linear theory compares well with the result of direct numerical simulation up to a transition point. The extent of the linear regime where the perturbations linearly interact with the base flow is thus identified. The value of the transition $Gr_{\unicode[STIX]{x1D6FF}}$, according to the three-dimensional DNS results, is dependent on the initial perturbation amplitude. Beyond the transition point, the DNS results diverge from the linear stability predictions as nonlinear mechanisms become important.

Global stability of multiple solutions in plane sudden-expansion flow

2012 ◽  
Vol 702 ◽  
pp. 378-402 ◽  
Author(s):  
Daniel Lanzerstorfer ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Secondary Flow ◽  
Mirror Symmetry ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Secondary Flows ◽  
Sudden Expansion ◽  
Shear Stresses ◽  
Two Dimensional ◽  
Large Expansion

AbstractThe two-dimensional, incompressible flow in a plane sudden expansion is investigated numerically for a systematic variation of the geometry, covering expansion ratios (steps-to-outlet heights) from $0. 25$ to $0. 95$. By means of a three-dimensional linear stability analysis global temporal modes are scrutinized. In a symmetric expansion the primary bifurcation is stationary and two-dimensional, breaking the mirror symmetry with respect to the mid-plane. The secondary asymmetric flow experiences a secondary instability to different three-dimensional modes, depending on the expansion ratio. For a moderately asymmetric expansion only one of the two secondary flows (the connected branch) is realized at low Reynolds numbers. Since the perturbed secondary flow does not deviate much from the symmetric secondary flow, both secondary stability boundaries are very close to each other. For very small and very large expansion ratios an asymptotic behaviour is found for suitably scaled critical Reynolds numbers and wavenumbers. Representative instabilities are analysed in detail using an a posteriori energy transfer analysis to reveal the physical nature of the instabilities. Depending on the geometry, pure centrifugal and elliptical amplification processes are identified. We also find that the basic flow can become unstable due to the effects of flow deceleration, streamline convergence and high shear stresses, respectively.

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