1 Sequence of bifurcations of natural convection of air in a laterally heated cube with perfectly insulated horizontal and spanwise boundaries

2025-04-30 0 0 5.12MB 52 页 10玖币

侵权投诉

Sequence of bifurcations of natural convection of air in a laterally heated cube

with perfectly insulated horizontal and spanwise boundaries

Alexander Gelfgat

School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Ramat

Aviv, Tel-Aviv, Israel, 6997801, gelfgat@tau.ac.il

Abstract

A sequence of three steady – oscillatory transitions of buoyancy convection of air in a laterally

heated cube with perfectly thermally insulated horizontal and spanwise boundaries is studied. The

problem is treated by Newton and Arnoldi methods based on Krylov subspace iteration. The finite

volume grid is gradually refined from 1003 to 2563 finite volumes. It is shown that the primary

instability is characterized by two competing eigenmodes, whose temporal development results in

two different oscillatory states that differ by their symmetries. Bifurcations due to both modes are

subcritical. These modes develop into different oscillatory and then stochastic flow states, which,

at larger Grashof number, stabilize and arrive to single stable steady flow. With further increase

of the Grashof number this flow loses it stability again. It is argued that in all the three transitions,

the instabilities onsets, as well as reinstatement of stability, take place owing to an interaction

between a destabilizing centrifugal mechanism and stabilizing effect of thermal stratification.

Key words: natural convection, instability, Krylov methods, SIMPLE iteration

1. Introduction

A primary goal of this study is to examine a chain of three steady – oscillatory bifurcations

that take place in a three-dimensional benchmark configuration of natural convection of air in a

laterally heated cube with perfectly insulated horizontal and spanwise boundaries. The two-

dimensional analog of this problem, i.e., convection in a square laterally heated cavity with

perfectly thermally insulated (adiabatic) horizontal boundaries, is one of the earliest CFD

benchmarks [1] proposed for validation of steady flow calculations. It was then extended for

comparison of calculated critical parameters of the primary bifurcation corresponding to steady to

oscillatory transition. The details and additional references can be found in [2]. Later, the two-

dimensional problem was extended to a three-dimensional one in a cube, assuming the spanwise

boundaries to be also perfectly thermally insulated. The results for 3D steady flows have been

published and cross-verified [3-8], so that the 3D steady flows benchmark is also well established.

However, numerical investigation of the instabilities of 3D steady flows is significantly more

challenging and had been addressed only by straight-forward time integration of the 3D time-

dependent governing equations [9-14].

Recently, a comprehensive linear stability analysis was applied to the primary steady -

oscillatory transition of air convection in a laterally heated cube in two simpler configurations, for

a cube with perfectly thermally conducting horizontal and spanwise boundaries [15], and with

perfectly insulated horizontal and perfectly conducting spanwise boundaries [16]. The second case

appeared to be noticeably more complicated because the critical Grashof number becomes almost

two orders of magnitude larger, and the steady – oscillatory transition is preceded by a symmetry

breaking steady bifurcation. A replacement of the perfectly conducting spanwise boundaries by

the perfectly insulated ones leads to a qualitatively different transition, which exhibits a sequence

of bifurcations described below. This finding is in line with recent computational and experimental

study [17], where the authors concluded that the flow can be stabilized or destabilized by variation

of the heat transfer conditions at the horizontal boundaries.

The time-dependent calculations of [14] showed that with the increase of the Grashof number

, the steady flow becomes oscillatory unstable at  > 4 10, then the stability reinstates at

 > 7 10, and the resulting steady flow becomes oscillatory unstable at  > 2 10.

Following a series of the cited above previous studies, we consider the fixed value of the Prandtl

number,  = 0.71, characteristic for air. The flows with very large [18] and very small [19,20]

Prandtl numbers were also considered for the current configuration, however dependence of the

transition on the Prandtl number is out of scope of the present study.

In this study, we examine the instabilities by applying linear stability analysis, using the

numerical approach and the visualization technique of [15,16]. For calculation of steady flows we

apply the Newton method, whose corrections are calculated by the biconjugate gradient stabilized

(BiCGstab(2)) method combined with generalized minimal residual (GMRES) method. Leading

eigenvalues are computed by the Chebyshev preconditioned Arnold iteration [21]. The

corresponding Krylov vectors, which are divergence free and satisfy all the boundary conditions

are computed by the SIMPLE-like technique proposed in [15,22].

Applying the linear stability analysis to the calculated 3D steady flows, we confirm three

transitions predicted by the time-dependent computations [14]. We discuss which physical

mechanisms destabilize the flow, then stabilize it, and then destabilize it again. Along with that,

we confirm existence of two different most unstable modes of the primary instability, as was

predicted by [11]. We show that these modes differ by broken flow symmetries and become

unstable at close Grashof numbers. Additional time-dependent calculations revealed that beyond

the stability limits, there exist two different oscillatory flow states, which also differ by their

symmetries. It was shown also that the steady – oscillatory transitions caused by any of the two

modes are subcritical. With further increase of the Grashof number, these modes turn into

stochastic regimes with different phase space attractors. Surprisingly, when the Grashof number is

increased to approximately 7 × 10, the two attractor collapse into the same stable focus, thus

producing a single stable steady flow state. This flow state remains stable up to the Grashof number

2.9 × 10, after which the flow becomes oscillatory and then turbulent.

Analyzing the most unstable disturbances and steady flow patterns near the critical points, we

argue that in all three cases the destabilization and stabilization result from an interaction of two

main factors: destabilizing centrifugal instability mechanism and stabilizing thermal stratification.

The centrifugal instability sets in at the cube corners where the flow direction turns from horizontal

to vertical, and vice versa, similarly to what is observed in the 3D lid driven cavity flow (see [22]

and references therein). This mechanism can be enhanced by reverse circulations that appear near

the top and bottom borders, and increase the curvature of streamlines of the main convective

circulation. On the other hand, a strong convective mixing tends to make the isotherms far from

the vertical isothermal boundaries almost horizontal, so that colder and heavier fluid is located

below the warmer and lighter one. This results in a stable stratification, which tends to suppress

all possible instabilities.

2. Formulation of the problem

As in our previous studies [14-16], we consider the natural convection in an incompressible

fluid in a cubic cavity, whose opposite sidewalls are maintained at constant but different

temperatures,  and . The horizontal and spanwise boundaries are perfectly thermally

insulated, as is defined for the AA – AA case in [14]. The Boussinesq approximation is applied.

To render equations dimensionless we choose 

,  

, 

 as scales of the length, time

, the velocity =(,,) and the pressure , respectively, where  is the fluid kinematic

viscosity and  is the density. The temperature is rescaled to a dimensionless function by  

(  ) (  ). Additionally, the dimensionless time, velocity and pressure are

scaled, respectively, by  

,  

, and , where  =(  )

 is the

Grashof number,  is the acceleration due to gravity, and  is the thermal expansion coefficient.

The resulting system of energy, momentum, and continuity equations is defined in a cube 0

,,  1 as:



 +(  )=

 

 (1)



 +(  )= +

 

 + (2)

   = 0 . (3)

where  = 

 is the Prandtl number, and  is the thermal diffusivity. All the boundaries are

assumed to be no-slip. Two vertical boundaries at = 0,1 are kept isothermal, so that

(= 0, ,)= 1, (= 1, ,)= 0. (4)

The absence of the heat flux at the horizontal and spanwise boundaries yields:



 =

 = 0 (5)



 =

 = 0 (6)

For discussion purposes, the areas adjacent to the cube edges (0,y,0), (0,y,1), (1,y,0) and (1,y,1) are

called below the lower left, upper left, lower right, and upper right corners, respectively.

The problem is additionally characterized by three symmetries [11]: (i) reflection symmetry

with respect to the midplane = 0.5, {,,,}(,,)={,,,}(, 1  ,), (ii) 2D

rotational symmetry with respect to rotation through 180° around the line x = z = 0.5,

{,,,}(,,)={,,,}(1 ,, 1  ), and (iii) 3D centro-symmetry

{,,,}(,,)={,,,}(1 , 1  , 1  ), where, =  (1 ). Obviously,

these symmetries are characteristic of steady state flows at relatively small subcritical Grashof

numbers. They can be broken by instability, so that supercritical oscillatory flows can maintain

only one type of symmetry or be fully non-symmetric. Following [16], to examine whether the

steady state solution is symmetric, we use the temperature field and define

 =|(,,) (, 1  ,)| (7)

 =|(,,)+(1 ,, 1  )| (8)

 =|(,,)+(1 , 1  , 1  )| (9)

as measures of the reflection, 2D rotational and 3D centro-symmetries, respectively. For the

symmetric solutions, these values are of the order of 10. Deviation of any of these expressions

from numerical zero indicates symmetry breaking.

3. Numerical and visualization techniques

The numerical approach is the same as in [14-16], where the reader is referred for the details.

Here we only mention that this approach allows for calculation of divergence free Krylov vectors

satisfying all the boundary conditions, which, in its turn, allows for a direct calculation of the

leading eigenvalue [21], without applying the Arnoldi method in shift-and-inverse mode like in

[23].

As mentioned in [16], convergence of the critical parameters for a similar 2D case of

convection in a laterally heated square cavity with insulated horizontal boundaries was established

in [23], the convergence and comparison with independent results for 3D steady states were

reported in [8], and the convergence of slightly supercritical oscillatory states was studied in [14].

Three-dimensional velocity fields are visualized by the method proposed in [24,25] and

applied to similar problems in [15,16], where all the definitions and computational details are

discussed. Here, to clarify the plots below, we only mention that 3D flow is visualized by

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1SequenceofbifurcationsofnaturalconvectionofairinalaterallyheatedcubewithperfectlyinsulatedhorizontalandspanwiseboundariesAlexanderGelfgatSchoolofMechanicalEngineering,FacultyofEngineering,Tel-AvivUniversity,RamatAviv,Tel-Aviv,Israel,6997801,gelfgat@tau.ac.ilAbstractAsequenceofthreesteady–oscillator...

展开>> 收起<<

1 Sequence of bifurcations of natural convection of air in a laterally heated cube with perfectly insulated horizontal and spanwise boundaries.pdf

共52页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 Sequence of bifurcations of natural convection of air in a laterally heated cube with perfectly insulated horizontal and spanwise boundaries

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: