Flow Configurations in a Lid Driven Cavity With Throughflow -- A Dragomirescu

U.P.B. Sci. Bull., Series D, Vol. 66, No. 2-4, 2004

FLOW CONFIGURATIONS IN A LID DRIVEN CAVITY WITH THROUGHFLOW

A. DRAGOMIRESCU∗

Această lucrare investighează multiplicitatea configuraţiilor curgerii plane într-o cavitate cu perete mobil parcursă de un debit de fluid, în funcţie de parametrii acestei curgeri. In domeniul de parametri explorat, diagramele de bifurcaţie ce corespund configuraţiilor curgerii posedă o structură interesantă, numărul şi modul de conectare a ramurilor acestor diagrame prezentând modificări substanţiale odată cu parametrii consideraţi. Permiţând o mai bună înţelegere a curgerii, aceste diagrame pot furniza indicii importante privind modul de proiectare şi de operare a dispozitivelor tehnice bazate pe o cavitate cu perete mobil parcursă de un debit de fluid.

In this work the multiplicity of the flow configurations in a lid driven cavity with throughflow with respect to the flow parameters is investigated. In the parameter range explored, the bifurcation diagrams corresponding to flow configurations exhibit an interesting structure, the number and connectivity of the branches showing substantial changes with the flow parameters. With the insight they offer into the flow, these diagrams may give important hints on how to design and operate a technical device that relies essentially on a lid driven cavity with throughflow.

Keywords: lid driven cavity, bifurcation

Introduction

This paper investigates the multiplicity and connectivity structure of the two dimensional flow configurations, i.e. the bifurcation, in a symmetrical lid driven cavity with throughflow. The domain to be considered is illustrated in Fig. 1.a. The fluid enters the two-dimensional cavity through the inflow channel at the bottom. The upper horizontal wall of the cavity consists in a moving lid that forms with the rigid walls of the cavity two horizontal channels through which fluid may leave the cavity to the left and to the right (outflow region). A sketch of the channel with a sudden expansion (Fig. 1.b) shows that if the cavity in Fig. 1.a is long enough, one would expect a decoupling of the flow in the entrance (expansion) region of the cavity from the lid driven outflow. This should bring the

∗ Assist., Dept. of Hydraulics and Hydraulic Machinery, University POLITEHNICA of Bucharest, ROMANIA.

A. Dragomirescu 148

inflow configurations close to those of the sudden expansion channel flow. For short cavities, on the contrary, one would expect two effects of the top wall: a) a stagnation reaction in the flow in the expansion region, possibly having stabilising consequences on the recirculation flows, and b) a driving effect of the moving top wall, which will certainly distort the flow configurations and change them qualitatively.

The flow in the extreme configuration of the lid driven cavity with throughflow, namely in the sudden expansion channel, is certainly a prototype flow, which is fascinating because it shows, from the continuing experimental and numerical effort spent on its investigations, an impressive richness of flow configurations [1]-[3] in spite of its geometric simplicity. In more recent investigations of a channel with a sudden expansion followed by a sudden contraction, the complex interplay of the stagnating flow in the contraction region with the expansion flow has been studied (Mizushima et al. [4]), shedding an additional light on the complexity of the flow structure to be expected in the present problem.

In Section 1 the problem is formulated in terms of geometric configurations, fluid properties, flow parameters, and boundary conditions, summarised in characteristic numbers (aspect ratios and Reynolds numbers). The instrument and the method of analysis are described in Section 2. The instrument is based on the software package FIDAP [5]. The results of the present investigation are presented in Section 3 in terms of bifurcation diagrams and streamline contour plots.

1. Problem formulation

The geometry of the flow domain, which is considered horizontal, is shown in Fig. 2. It is characterised by the width di of the inflow channel, the width

a) b)

Fig.1. The lid driven cavity with throughflow (a) put into context with the channel with a sudden expansion (b).

Flow configurations in a lid driven cavity with throughflow

149

D and the height H of the cavity, and the widths dof and dob of the outflow channels. With a symmetrical entrance expansion and two equal outflow channels, the flow geometry is symmetrical with respect to the middle axis. The fluid, considered to be Newtonian and incompressible, is characterised by its density ρ and its dynamic viscosity µ. The characteristics of the flow regime are the flow rate Qi at the inlet and the velocity Uw of the wall on top of the cavity. The steady state two-dimensional incompressible flow of the Newtonian fluid is governed by the Navier-Stokes equations (momentum equations)

∂∂+∂

∂+∂∂−=

∂∂+∂

∂2

2

2

2

y

u

x

u

x

p

y

uu

x

uu xxx

yx

x µρ , (1)

∂∂+∂

∂+∂∂−=

∂∂+∂

∂2

2

2

2

y

u

x

u

y

p

y

uu

x

uu

yyyy

yx µρ , (2)

and the mass conservation equation

0=∂∂

+∂∂

y

u

x

u yx . (3)

The boundary conditions for the flow are the usual ones of no slip on the fixed and moving walls. Also, the inflow and outflow conditions have to be specified. The velocity profile at the inlet is taken as parabolic with a mean velocity iU equal to two-thirds of the maximal velocity Ui,max. This mean velocity

gives the inflow rate per unit channel width: iii UdQ = (or 3/2 ,maxiii UdQ = ).

Both outlets are considered free of normal stress, which is the so-called “natural

Li

Hd o

f

D

Lob LofUw

y

x

d ob

di

Qi Fig. 2. Geometry, parameters, and coordinate system of the flow domain.

A. Dragomirescu 150

boundary condition” that arises from the application of the finite element method to the flow equations [5]. This condition guarantees that at the outflow boundaries the internal flow in the cavity accommodates to a reference pressure, p = 0 in our case, in as much as the flow in the two channels reaches a state of full development. Hence the lengths Lof and Lob and the heights dof and dob of the “forward” and “backward” outflow channels determine how the inflow rate Qi distributes between these two channels (such as Qof + Qob = Qi). The choices dof = dob (= di) and Lof = Lob (= Lo) in this paper expresses the close relationship for the channel flow with a symmetrical sudden expansion.

To give to this problem a higher degree of generality, the flow equations were solved in their dimensionless form. As reference length the width di of the inflow channel was chosen, while the reference velocity was adopted as the maximum velocity at the inlet, Ui,max. With this, three length ratios and two Reynolds numbers can be introduced as dimensionless parameters for characterising the flow. The length ratios are the expansion ratio l i = D/di, the aspect ratio la = H/D, and the width ratio for the outflow against the inflow

ioo ddl /= The Reynolds numbers are µρµρ 2/3/Re iii,maxi QdU == , which

specifies the inflow, and µρ /Re wow Ud= , which characterises the motion of

the upper wall. Two additional length ratios are introduced by the length Li of the inflow channel and the (common) length Lo of the two outflow channels: ci = Li/di and co = Lo/do. After nondimensionalisation, the geometric parameters of the

chamber become: 1*=id , ilD =* , ai llH =

* , ooobof lddd ===*** , and

oooobof clLLL ===*** (the star denotes a dimensionless variable). The

dimensionless maximum velocity at the inlet is now 1*, =maxiU . Taking

imaxi dU /,µ as reference pressure, the equations governing the flow may be

written in the form:

22 *

*2

*

*2

*

*

*

**

*

**Re

y

u

x

u

x

p

y

uu

x

uu xxx

yx

xi ∂∂+

∂∂+∂

∂−=

∂∂+∂

∂ , (4)

22 *

*2

*

*2

*

*

*

**

*

**Re

y

u

x

u

y

p

y

uu

x

uu yyy

yy

xi ∂∂+

∂∂+∂

∂−=

∂∂+∂

∂, (5)

0*

*

*

*

=∂∂

+∂∂

y

u

x

u yx . (6)

In the above dimensionless equations the fluid density ρ is now replaced by the Reynolds number Rei, while the dynamic viscosity µ by 1.


151

As already specified, the computations were made in terms of the dimensionless parameters. However, because of the relatively large number of them, some of the parameters were kept constant. Thus, the computations were performed for various Reynolds numbers, while l i, la, and lo were kept fixed. This approach seems to be appropriate for the first step of investigation that concentrates on a “neighbourhood” of the sudden expansion flow.

2. Methods and instruments of analysis

The computations of the flow fields were performed with the CFD software FIDAP and its finite element method in the Galerkin formulation. The continuum flow region was divided into 6360 quadrilateral elements forming a mesh with 6156 grid points. Several branches of the solutions of this system were identified by a zero-order, one parameter continuation scheme. One of the continuation parameters was the Reynolds number Rei. Each new solution of the non-linear equation system corresponding to a new Rei was sought by starting a Newton algorithm with a previous solution as initial guess. Furthermore, the influence of the mesh on the bifurcation diagrams was investigated using a second and refined computational mesh consisting of 24260 elements.

Studies of the sudden expansion flow revealed that below a certain value of Rei the flow equations possess a unique solution [3]. Therefore it is convenient to start computations initially from low values of Rei. The detailed path of continuation sequences needed to uncover a picture of the flow multiplicity, especially for the detection of disconnected solution branches, rests on some intuition.

To obtain a compact graphical characterisation of the individual solutions, a suitable functional was chosen. This is the first moment of the axial velocity taken at a given height y = yf in the chamber. It has the expression

∫−

=

2/

2/

*******

*

),(...),(D

Dfyf dxyxuxyf . (7)

The ellipsis stays here for other possible parameters, like Reynolds numbers and/or aspect ratios.

Due to the definition of this functional, its value will be zero for flows which are symmetrical with respect to the symmetry axis of the domain (in this case the function under the integral is antisymmetrical while the integration domain is symmetrical). Positive values of the functional will correspond to situations in which the flow attaches to the “right” wall, while for situations in which the flow attaches to the “left” wall f will take negative values.

A. Dragomirescu 152

3. Results

Computations were performed for a fixed expansion of ratio l i = 6, a fixed aspect ratio la = 2, and a fixed value lo = 1. The inflow and outflow ratios were chosen as ci = 1 and co = 5 respectively. With these, the dimensionless velocity of

the upper wall becomes iwwU Re/Re*= , since do/di = lo = 1. For convenience,

this velocity will be used instead of Rew in the following, so the subscript i in the notation Rei can be dropped (i.e. Re). A remark should be added here: the definition of the dimensionless upper wall velocity implies that a constant value of

*wU means that the real velocity of the upper wall changes proportional to the

inflow velocity. The computations, which were performed for different Reynolds numbers

and different velocities of the upper wall, moved in the two-dimensional

parameter space (Re, *wU ). The path in this parameter space focused on the

influence of the top wall velocity on the flow picture. Results will be presented for

increasing values of this velocity: *wU = 0, 0.1, 0.2, 0.5, 1.0, 2.0, 4.0, 7.0, 7.3, 7.4,

and 8.0. For each velocity and Reynolds number, the functional f was evaluated at

269.2*=fy .

3.1. Top wall at rest – 0* =wU For this situation, the computations revealed at relatively low Reynolds

numbers a symmetrical bifurcation picture (Fig. 3), similar to the flow in a channel with a sudden expansion having the same ratio for D/di. Fig. 3 shows the presence of a bifurcation point at Recr ≈ 37, which agrees well with the results obtained by Alleborn et al. [3].

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 50 100 150 200 250 300

f(R

e)

Re

I I'

II+

II-

Fig. 3. Bifurcation diagram for *wU = 0.


153

The bifurcation picture displays a unique, symmetrical, and stable stationary flow configuration I for Re < Recr and three configurations above Recr: I’ , II + and II -. Fig. 4 shows the streamlines for configurations I’ and II - at Re = 50. Flow I’ is symmetrical with respect to the symmetry axis and has two recirculation vortices. Flow II - is asymmetrically attached to the “left” wall and shows also two recirculation vortices, although not equally sized. Flow II +, which is not shown, is the reflection of II - with respect to the symmetry axis. The theory presented in [6] and the studies on the flow in a sudden expansion allow us only to presume, in the absence of further numerical investigations, that the flow corresponding to branch I’ is unstable while the flows corresponding to branches II + and II - are stable.

The number and structure of the recirculation vortices change when the Reynolds number increases. For Re = 300 (Fig. 5) the symmetrical flow I’ shows at least three vortices on each side of the main stream, the larger ones being

a) b)

Fig. 4. Streamlines for *wU = 0 at Re = 50: a) the symmetrical flow I’

and b) the asymmetrical flow II -.

a) b)

Fig. 5. Streamlines for *wU = 0 at Re = 300: a) the symmetrical flow I’

and b) the asymmetrical flow II -.

A. Dragomirescu 154

situated in the lower half of the cavity. In the asymmetrical flow II - only four vortices appear: one remained at the “left” wall (the other two being destroyed by the asymmetry of the flow), and another three interposed between the main stream and the “right” wall.

Comparing Fig. 4 and 5 one could also see that the increase of the Reynolds number narrows the main flow and enlarges the recirculation zones. Thus the dissipation grows and the quantity of fluid possibly subjected to ageing increases.

3.2. Moving wall – 0* ≠wU

The motion of the upper wall breaks the reflection symmetry of the flow. Because of this motion, the bifurcation diagram changes qualitatively, from a perfect bifurcation in the symmetry case to an imperfect bifurcation. Also, as a consequence of the wall motion, an additional change is to be expected. The two asymmetrical flows will be influenced differently by the wall motion. While the flow attached to the “right” wall of the chamber (II +) is expected to be “stabilised”, the one attached to the “left” wall (II -) will probably be “destabilised” and, for a sufficiently high velocity of the top wall, one would expect that this last flow will be “swept away” from the back wall of the cavity.

Expressed in terms of the bifurcation diagram, it is to be expected that for

0* >wU the symmetrical branch I will be continuated smoothly by the upper

branch II + around Re = Recr. The lower branch, II -, on the other hand, will

probably end at a certain Reynolds number and, the higher the velocity *wU is, the

sooner this end is supposed to appear.

Fig. 6.a and 6.b present the bifurcation diagrams for *wU between 0.1

a) b)

Fig. 6. Bifurcation diagrams for a) *wU = 0.1, 0.2, 0.5, and b) *wU = 1.0, 2.0, 4.0.


155

and 4.0. These diagrams confirm the assumptions made before. Around Recr ≈ 37 branch I turns smoothly into branch II +, which does not change significantly

compared to case *wU = 0. The other two branches, I’ – originating from the

symmetrical branch I’ of the case *wU = 0 – and II - become isolated, being linked

by two turning points T1 and T2 located at Recr,1 and Recr,2. The diagrams in Fig 6.a and 6.b indicate that starting from a Reynolds

number close to 0 and increasing it in small steps, flow configurations corresponding to branches I and II + will be obtained. This means that the main flow will attach to the “right” wall (e.g. Fig. 7.a) and will remain attached to this wall. Flow configurations corresponding to branches I’ (Fig. 7.b) and II - (Fig. 7.c) could be obtained if, for example, significant changes of the Reynolds number from values Re < Recr to values Re > Recr are made.

With increasing *wU , the location of the first turning point, T1, remains

almost unchanged, while the second turning point, T2, moves towards smaller Reynolds numbers and makes the isolas formed by branches I’ and II - to become smaller and smaller. This behaviour could lead to the assumption that for high enough velocities of the upper wall the fluid will attach only to the “right” wall and the isolas in the bifurcation diagram will disappear completely.

According to our computations, the previous assumption seems not to be correct. As Fig. 8 shows, an interesting change in the bifurcation picture occurs at

higher velocities *wU . At a critical velocity *

,crwU , the value of which was found to be

between 7.3 and 7.4, a new bifurcation point seems to exist, connecting the upper

branch, II +, with the lower ones, I’ and II -. For *,

*crww UU > this bifurcation point

disappears again and the connectivity of the branches changes to a hysteresis-like configuration. Branch I is now continued by branch II - that evolves further up to a

a) b) c)

Fig. 7. Streamlines for *wU = 1.0 at Re = 50: a) corresponding to branch II +,

b) corresponding to branch I’ , and c) corresponding to branch II -.

A. Dragomirescu 156

critical Reynolds number Recr,2, for which the turning point T2 appears. At this point, a further increase in the Reynolds number will make the flow to change suddenly its configurations to the ones corresponding to branch II +. On the other side, if at point T2 the Reynolds number is decreased, the flow will evolve according to branch I’ down to

a new turning point '1T located at a Reynolds number ' 1,Recr . At this point, a new

increase of the Reynolds number will have as result flow configurations corresponding to branch II +.

Investigations of the computational mesh influence on the aspect of the

bifurcation diagrams were carried out for upper wall velocities *wU = 0.1, 7.3,

and 7.4. The flow domain was discretised with a finer mesh having 24260 elements. Comparisons between the bifurcation diagrams obtained with this new mesh and those obtained with the initial mesh are presented in Fig. 9. The plots show that the mesh refinement had practically no qualitative and quantitative influence on the bifurcation diagrams in the region of Recr. Only at higher Reynolds numbers the refined mesh produced some small changes of the bifurcation diagrams (the turning points T2 moved slightly to the left, which correspond to a small decrease of Recr,2). This fact leads to the conclusion that the choice of the initial computational mesh was adequate to numerically simulate the flow in the lid driven cavity in the neighbourhood of the (first) critical Reynolds number.

Conclusions

The object of this investigation was a simple flow configuration, a lid driven cavity with throughflow that establishes a link between a well understood

-0.05

-0.04

-0.03

-0.02

-0.01

0

30 35 40 45

Fig. 8. Bifurcation diagram for *wU = 7.0, 7.3, 7.4, and 8.0.


157

category of channel flows and a second flow category, which comes close to certain technical devices in process engineering. The purpose of the investigation was to gain an insight into the number and nature of two-dimensional flow configurations allowed by those flow parameters which in process engineering should be accessible and also sufficient to control the flow completely. The instruments of the investigation were essentially commercial numerical tools for flow computation.

The results obtained show that there is a pronounced qualitative change in the multiplicity of the flow configurations in a lid driven cavity with throughflow as one departs from the relatively simple configurations of a channel with a sudden expansion towards more complex geometric shapes like that of a lid driven cavity. The parameter ranges considered reveal the existence of a rich variety of

flow configurations. At low values of the upper wall velocity *wU , flow

configurations separated from the main branch were found. They form isolas

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 100 200 300 400 500 600

f(R

e)

Re

6360 elements24260 elements

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100 120 140

f(R

e)

Re


a) b)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100 120 140

f(R

e)

Re


c)

Fig. 9. Comparisons between bifurcation diagrams obtained with the initial computational mesh

and with the refined one: a) *wU = 0.1, b) *wU =7.3, c) *

wU = 7.4.

A. Dragomirescu 158

whose size decreases when *wU increases. But instead of disappearing, as one

would expect, these isolas turn into a hysteresis configuration once the upper wall velocities become greater than a critical value found to be between 7.3 and 7.4. Further investigations, either numerical or experimental, are needed to find what branches on the bifurcation diagrams are stable and what branches are instable.

To study the influence of the computational grid on the bifurcation diagrams, some investigations were repeated using a refined mesh. They confirmed the correctness of the initial results obtained at relatively low Reynolds numbers, where the aspect of the bifurcation diagrams remained practically unchanged. Only at high Reynolds numbers more reliable results may demand finer grids.

R E F E R E N C E S

1. W. Cherdron, F. Durst, J. H. Whitelaw, Asymmetric flows and instabilities in symmetric ducts with sudden expansions. Journal of Fluid Mechanics, 84, 1978, 13–31.

2. R. M. Fearn, T. Mullin, K. A. Cliffe, Nonlinear flow phenomena in a symmetric sudden expansion. Journal of Fluid Mechanics, 211, 1990, 595–608.

3. N. Alleborn, K. Nandakumar, H. Raszillier, F. Durst, Further contributions on the two-dimensional flow in a sudden expansion. Journal of Fluid Mechanics, 330, 1997, 169–188.

4. J. Mizushima, Y. Shiotani, Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part. Journal of Fluid Mechanics, 434, 2001, 355–369.

5. FIDAP (Fluid Dynamics Analysis Package), Version 7.52. FDI, Evanston, 1996. 6. G. Ioos, D. D. Joseph, Elementary stability and bifurcation theory. Springer, New York, 1990.

Documents

Flow Configurations in a Lid Driven Cavity With Throughflow -- A Dragomirescu