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1

Flow in a symmetric channel with anexpanded section

By T. Mullin1, S. Shipton1 and S. J. Tavener2

1 Manchester Center for Nonlinear Dynamics, University of Manchester, Oxford Road,Manchester M13 9PL, U.K.

2 Department of Mathematics, Colorado State University, Fort Collins, CO 80623, U.S.A.

Received Oct 31, 2002; revised July 2, 2003; accepted July 2, 2003.

Communicated by S. Kida.

We consider the flow in a symmetric two-dimensional channel with an expanded section.By examining the effect of varying the ratio of the inlet and outlet channel widths, weare able to place the earlier results of Mizushima et al. (1996) within an entire spec-trum of nonlinear phenomena. Laboratory experiments support predicted behaviour intwo regions of parameter space and highlight the importance of imperfections in theexperimental apparatus.

1. Introduction

The two-dimensional flow in a channel with a sudden symmetric expansion has at-tracted considerable interest as it provides an example of symmetry breaking and multi-plicity in an open flow field. Asymmetric flows in an expanding channel were first recordedin experiments by Durst et al. (1974) and Cherdron et al. (1978). In a combined labora-tory and finite-element study, Fearn et al. (1990) established that the asymmetric flowsarise at a pitchfork bifurcation point. They focused on the symmetry breaking in a 1:3expansion and further considered the important effects of physical imperfections on thebifurcation structure. Allenborn et al. (2000), Battaglia et al. (1997) and Drikakis (1997)all investigated the effect of varying the expansion ratio and showed that the criticalReynolds number decreases as the expansion ratio increases. Hawa & Rusak (2000, 2001)proposed a detailed instability mechanism and used both asymptotic and simulationtechniques to investigate the effect of symmetry-breaking perturbations in their model.

Symmetry breaking bifurcations are also found in expanding geometries which arenot abrupt. Shapira et al. (1990) performed a numerical investigation of the flow ina channel where the expanding section had a constant slope other than 90 degrees.They showed that the shallower the slope in the expansion zone, the larger the criticalReynolds number at which the symmetry-breaking bifurcation occurred. Further, Cliffe &Greenfield (1982) showed the existence of a symmetry-breaking bifurcation in a smoothlyexpanding channel.

Recent interest in the suddenly expanding channel has centered on three-dimensionaleffects whose presence has been noted in all experiments, and whose influence and impor-tance increases with Reynolds number. Computational studies have been performed byChiang et al. (1999, 2001) and Schreck & Schafer (2000), and Chiang et al. (2001) providenumerical evidence for breaking of the reflectional symmetry about a plane parallel tothe plane of the expansion. This possibility has yet to be confirmed experimentally.

2 T. Mullin, S. Shipton and S. J. Tavener

inlet

outlet

l

2h 2d 2b

Figure 1. The basic geometry

Mizushima et al. (1996) and Mizushima & Shiotani (2000, 2001) were the first to con-sider an interesting extension to the simple expansion geometry by studying the effect ofa downstream contraction on the bifurcation set. Since expansions in many flow devicesare followed by a contraction, it is of considerable practical benefit to consider the effectof a contraction downstream. Mizushima et al. found that, provided the expanded sectionof the channel is longer than a critical minimum value, the addition of a downstream con-traction channel restabilizes the symmetric state above a critical flow rate. Specifically,the pitchfork bifurcation which gives rise to asymmetric flows is followed by a subcriti-cal bifurcation which restabilises the symmetric state. They also showed the intriguingpossibility of Hopf bifurcations at larger flow rates. Mizushima et al. however, restrictedtheir attention to the case in which the outlet channel is the same width as the inletchannel. We now explore the effects of varying the ratio of the outlet and inlet channelwidths and uncover a systematic set of behaviours which involve multiple bifurcations.These are of interest for further study since they are known to provide organising centresfor global dynamical behaviour including chaos as discussed by Abshagen et al. (2001).

We first consider a perfectly symmetric geometry to establish the bifurcation set. It isknown that physical imperfections have a surprisingly large effect on the disconnection ofthe pitchfork bifurcation in the simple expansion (Fearn et al. (1990)). Here we perform anumerical study of the effects of geometric perturbations on the multiple bifurcation setof the symmetric model, and compare the results of computations with those obtainedin our laboratory experiments. In general, good agreement is obtained.

2. Equations of motion

We consider the steady flow of an incompressible, Newtonian fluid in the domain shownin figure 1. Let ∗ denote dimensional quantities and define

x =x∗

αhand y =

y∗

h, (2.1)

u =u∗

U0

and v =αv∗

U0

, (2.2)

where U0 is the maximum inlet velocity and aspect ratio

α =l

2d. (2.3)

We used this non-dimensionalization of the cross-stream velocity so that the equation

Flow in a symmetric channel with an expanded section 3

of conservation of mass remains independent of the aspect ratio α, i.e.,

∂u∗

∂x∗+

∂v∗

∂y∗=

U0

αh

(

∂u

∂x+

∂v

∂y

)

= 0. (2.4)

The momentum equations are then

Re

α

(

u∂u

∂x+ v

∂u

∂y

)

= −

1

α

∂p

∂x+

(

1

α2

∂2u

∂x2+

∂2u

∂y2

)

, (2.5)

Re

α2

(

u∂v

∂x+ v

∂v

∂y

)

= −

∂p

∂y+

(

1

α3

∂2v

∂x2+

1

α

∂2v

∂y2

)

, (2.6)

where the Reynolds number is Re = U0h/ν. This non-dimensionalization allows theaspect ratio α to be used as a continuously variable parameter in the computation ofsymmetry-breaking bifurcation points or other singularities.

The two remaining geometric parameters are the expansion ratio E and outlet ratioγ, where

E =d

hand γ =

b

h. (2.7)

The expansion ratio was fixed at E = 3 for the entire study.

2.1. Numerical methods

The two-dimensional steady Navier-Stokes equations were discretized using the finite-element method and all computations were performed using the numerical bifurcationcode Entwife (Cliffe (1996)), which has been used to study bifurcation phenomenain a range of fluid flows, for example, in the Taylor-Couette problem. The numericaltechniques implemented within Entwife to compute paths of limit points, symmetry-breaking bifurcation points and Hopf bifurcation points, as well as their convergenceproperties, are discussed in the recent review article by Cliffe et al. (2000). Higher-order singularities, specifically coalescence and quartic bifurcation points were computedusing the techniques described in Cliffe & Spence (1984). Quadrilateral elements withbiquadratic velocity interpolation and discontinuous linear pressure interpolation wereused throughout. Numerical experiments were performed in order to ensure that theresults reported here are independent of the details of the mesh and the lengths of theinlet and outlet channels.

3. Experimental apparatus

The flow rig is shown schematically in figure 2. It is essentially the same as the ap-paratus of the earlier study performed by Fearn et al. (1990) but with new test sectionsadded such that the 1:3 expansion was followed by a contraction of ratio 3:2.

The working fluid was distilled water which was continuously pumped from a lowerholding container to a constant head tank approximately 2m above the channel. A seriesof diffusers, corner vanes, screens, a honeycomb and a machined contraction was usedto provide a steady well-conditioned flow upstream of a long entry section for Reynoldsnumbers up to 400. The entry section was 60 channel heights long, which was sufficientlylong to allow a parabolic flow profile to develop upstream of the expansion for all flowrates reported here.

The height and width of the inlet channel were 4mm and 96mm respectively. Thesidewalls of the inlet channel and expanded section were constructed of ground glassplate. Selected pieces of 10mm thick Perspex were used for the top and bottom plates.


Figure 2. Schematic diagram of the experimental apparatus

Machined spacers were used to bolt the top and bottom plates together with sliconesealant to fill the milled slots which held the side walls. The expanded section had aheight of 12mm and a width of 96mm which previous work of Cherdron et al. (1978)indicates is adequate for nominally two-dimensional flow for the Reynolds number rangestudied here. The outlet section had a height of 8mm and was followed by a diffuser andcorner vanes downstream which further conditioned the flow and reduced any upstreaminfluence effects. Fine control of the flow rate was achieved using a needle bypass valveand the flow rate was measured using a Pelton wheel flow meter.

Mearlmaid AA particles were introduced into the fluid for both the flow visualiza-tion and particle tracking experiments. A projector and mask consisting of two razorblades mounted in a slide case was used to create a thin light sheet and a Cohu 4910monochrome video camera was used to capture images. The particle tracking software,Digimage (Dalziel (1992)) was used to obtain the two-dimensional velocity fields.

4. Results

4.1. Symmetric computations

We first show the results of numerical calculations of the bifurcation set for expansionratio E = 3 and outlet ratio γ = 1.5 in figure 3. The corresponding sequence of qualita-tively different bifurcation diagrams is shown in figure 4. These were calculated at thegiven fixed values of α, and Re was used as the bifurcation parameter. The cross-streamvelocity v at a location αh downstream of the expansion along the centerline of the chan-nel is used as a measure of the flow. The cross-stream velocity along the centerline iszero for symmetric flows and non-zero otherwise. Solid lines indicate solutions that arestable with respect to two-dimensional disturbances and dashed lines indicate unstablesolutions.

At very small aspect ratios, the expanded section has little impact on the flow andthe flow is dominated by the inlet to outlet expansion. In the limiting case as α → 0,


Figure 3. Symmetry-breaking bifurcation points and limit points for E = 3 and γ = 1.5. ARQEis a path of symmetry breaking bifurcations. Q and R are quartic bifurcation points and PQand RS are paths of limit points. C+ and C− are coalescence points.

a single supercritical symmetry-breaking bifurcation occurs, since in this limit a simpleexpanding channel with expansion ratio γ = 1.5 is recovered. The bifurcation diagramshown in figure 4(a) for an aspect ratio of 2.1 is qualitatively similar to this limiting case.The locus of this bifurcation is from A to C− in figure 3. At very large aspect ratios, thewidth of the outlet channel plays a minor role and the behaviour is dominated by theexpanded part. The single supercritical symmetry-breaking bifurcation point that occursin an expanding channel with expansion ratio E = 3 is recovered at large aspect ratiosas shown in figure 4(e). Its path corresponds to the locus connecting C+ to E in figure3. The sequence of bifurcation diagrams 4(b)–(e) show how these two limiting cases areconnected as the aspect ratio is varied.

The point labelled C+ in figure 3 is a coalescence point at which a closed loop ofasymmetric solutions is created as the aspect ratio is increased. A typical bifurcationdiagram highlighting this behaviour is given in figure 4(b) for α = 2.3 where it maybe seen that the initial pitchfork bifurcation is followed by its mirror image as Re isincreased. For Re exceeding the value which corresponds to the subcritical pitchforkbifurcation, the symmetric flow regains stability for a range of Re before losing stabilityto asymmetric flows at a third symmetry-breaking bifurcation point which occurs alongthe curve joining A and C− in figure 3.

The point labelled Q in figure 3 is a quartic bifurcation point at which the secondsymmetry-breaking bifurcation changes from sub- to supercritical as the aspect ratio isincreased, as shown in figure 4(c) corresponding to α = 4. Hence a pair of fold bifur-cations are created at this point and their path is labelled QP in figure 3. The pathsare superposed because perfect symmetry is assumed in these calculations. Yet furtherincrease in α causes the second pitchfork bifurcation to pass through a second quarticbifurcation point labelled R in figure 3. The corresponding bifurcation diagram is givenin figure 4(d) for α = 4.78. At the point R a second pair of folds are created and theirlocus is labelled RS in figure 3. Increase in α causes the second and third pitchforks tomerge at the C− coalescence point labelled C− in figure 3 and the corresponding bifur-cation diagram is given in figure 4(e). The bifurcation diagram given in figure 4(e) willremain qualitatively the same for further increases in α. The solution branches arising


(a) (b)

(c) (d)

(e)

Figure 4. Bifurcation diagrams for E = 3, γ = 1.5 and α = 2.1, 2.3, 4.0, 4.78 and 4.85. Theordinate axis is v along the centerline at a distance αh downstream of the expansion. Solid linesindicate stable solutions, dashed lines are unstable solutions.

at the first pitchfork will be folded, although both folds occur at values of Re which arefar above practically relevant values.

We will now focus on the effect of varying the outlet ratio on the sequence of symmetry-breaking bifurcations. The numerical results are presented in figure 5, where paths ofsymmetry-breaking bifurcation points are plotted for outlet ratios between 1.5 and 2.0.The paths of limit points are not shown for reasons of clarity. For larger values of γ, theC+ and C− merge and the locus of symmetry-breaking bifurcation points is unfolded.

Another way of viewing the emergence of the two coalescence points is given in figure 6where we plot the loci of the coalescence points in the (α, γ) plane for the fixed expansion


Figure 5. Symmetry-breaking bifurcation points for E = 3 and γ = 1.5, 1.6, 1.7, 1.8, 1.9, 2.0from top to bottom. Limit points are not shown for simplicity.

ratio E = 3. This locus partitions the parameter space into regions of qualitativelydistinct behaviour, although the full division would require us to include paths of quarticbifurcation and hysteresis points. We have also marked with a dashed line the outlet ratioγ = 1 which is clearly a critical case as it delineates between an overall expansion andcontraction. Inside the two paths of coalescence points there are two pairs of asymmetricsolution branches, one of which forms a closed, possibly folded, loop. Outside the twopaths of coalescence points there is a single pair of, possibly folded, asymmetric branches.The sequence of bifurcation diagrams shown in figure 4 can be considered as a horizontalslice across this figure. For γ = 1.5, an aspect ratio of 2.1 lies outside the cusped region andthere is a single symmetry-breaking bifurcation point. Aspect ratios of 2.3, 4.0 and 4.78lie inside the cusped region and consequently bifurcation diagrams have three symmetry-breaking bifurcation points. An aspect ratio of 4.85 once again lies outside the cusp-shaped region.

The upper coalescence point moves to large Reynolds number and large aspect ratioas γ approaches one from above. As the outlet ratio increases, the two coalesence pointsget closer and eventually merge at a codimension-two singularity D with normal formx3

− λ3x + αx + βλx = 0. For γ > γD, or alternatively α < αD, the expanded part isinconsequential and the flow behaves as if it were an expanding channel with expansionratio γ. For γ < 1 the first symmetry-breaking can still occur, provided the expandedpart is long enough, i.e. provided α is sufficiently large.

4.2. Experimental results

The results shown in figure 6 and the associated discussion provide motivation for anexperimental investigation. Practical limitations preclude continuous variation of thethree geometric parameters and so experiments were planned for fixed values of theexpansion ratio E = 3, and outlet ratio γ = 2, while two values of α = 2 and 7/3 werechosen. While both choices lie outside the cusp-shaped region, the value of the aspectratio at the codimension-two point αD lies between 2 and 7/3.

An essential consideration for any experimental study of bifurcation events is the roleof imperfections. It was shown by Fearn et al. (1990) that small geometrical imperfec-


Figure 6. Locus of codimension-one points in the (α, γ) plane. The paths of quarticbifurcation points and hysteresis points are not shown.

inlet

outlet

2E2

2γ

∆/22αE

∆

Figure 7. The imperfection introduced into the computational domain as an approximation ofthe physical imperfection that must be present in the experiment.

tions disconnect the pitchfork of a simple expansion by a surprisingly large amount whencompared with internal flows such as those found in the Taylor–Couette problem. There-fore we expect that imperfections will play a significant role in the observed behaviourassociated with the more complex bifurcation studied here.

Any imperfection will break the reflectional symmetry about the midplane and there-fore disconnect the symmetry-breaking bifurcation points. Imperfections in the experi-mental apparatus which break the midplane symmetry may arise in a number of differentways and we chose to assume the imperfection of the type sketched in figure 7, where∆ is the non-dimensional size of the imperfection. Singularity theory suggests that thisis a reasonable assumption to make for steady bifurcation phenomena and, further, weobtained good agreement with experimental measurements for a perturbation which wasin accord with our physical estimates.

Comparisons of experimental streaklines and streamlines computed using both sym-


(a)

(b)

(c)

Figure 8. Streakline and streamline plots for Re = 16 and E = 3, α = 2, γ = 2. (a) Experimentalstreaklines. (b) Streamlines assuming a symmetric domain. (c) Streamlines for a perturbeddomain, ∆ = 0.05.

metric and the perturbed computational domains are shown in 8–11 and 13–18 for α = 2and α = 7/3 respectively. The corresponding bifurcation diagrams are shown in figures12 and 19.

The results displayed in figure 8 for Re = 16 show good agreement between experi-mental streakline plot, and both sets of numerical symmetric and perturbed streamlineresults. Thus the flow is effectively symmetric in practice. However, the situation is verydifferent at Re = 32.2 as may be seen in figure 9. The symmetric flow is stable withrespect to 2D disturbances, but when the reflectional symmetry is broken by inclusion ofthe small imperfection in figure 9(c), an obvious asymmetric flow exists. This is in goodagreement with the experimental streakline plot shown in figure 9(a).


(a)

(b)

(c)

Figure 9. Same as figure 8 except for Re = 32.2.

An asymmetric flow is even more evident at Re = 60.3 as shown in figure 10. How-ever, the comparison between the streaklines (a) and perturbed streamlines (c) is not sofavourable. It should be noted that a flow with the opposite asymmetry was not observed.This point will be commented upon further below. Three-dimensional computations byChiang et al. (1999) at this Re indicate that three-dimensional effects become importantnear the walls, although the flow is nominally two-dimensional across 90% of the channelwidth. We suspect that these may be present in the experimental results presented here,although three–dimensional effects were not obvious in the flow field.

For Reynolds numbers exceeding 60.3, three-dimensional effects became more evidentin the flow, though the flow remained steady, and the comparisons at Re = 94.7 infigure 11 are correspondingly less good. However, details such as the weak ‘Moffatt’ eddy


(a)

(b)

(c)


(Moffatt (1964)) in the bottom right-hand corner of the streakline photograph (a) werepresent in both experiment and asymmetric calculation.

Recall from the discussion of the results presented in figure 5 that for aspect ratioα = 2 there is a single bifurcation point at which the instability is associated with theinlet to outlet expansion ratio. We show a comparison between the experimental andnumerical results for this bifurcation diagram in figure 12. As with the earlier results wepresent numerical results for both the symmetric and imperfect cases. It may be clearlyseen that the perturbation produces strong asymmetry within the expanded section andreasonable agreement is achieved when perturbation ∆ = 0.05 was introduced into thecomputational domain. Most importantly, the perturbed bifurcation diagram clearly in-dicates that a flow in which the symmetry was broken in the opposite sense exists for


(a)

(b)

(c)


Re > 130. This flow is likely to be three dimensional or even time dependent and wasnot observed.

A different range of behaviour was observed for aspect ratio α = 7/3 and this isto be expected from the above discussion. At this increased value of the aspect ratiothere is a single symmetry-breaking bifurcation point, but now, the instability is associ-ated with the expansion from the inlet channel to the expanded section. Therefore thesymmetry-breaking bifurcation occurs at much lower Reynolds number, i.e. well beforethree-dimensional effects become important. Hence, both asymmetric branches shouldbe observed above a Reynolds number of approximately 50.

At Re = 11.8, the results presented in figure 13 provide evidence that the observedflow is essentially symmetric. In figure 14 we show experimental streaklines and com-puted streamlines at Re = 38.1 which is below the symmetry-breaking bifurcation point.


Figure 12. Bifurcation diagram for E = 3, α = 2, γ = 2. The ordinate axis is v along the center-line at a distance αh downstream of the expansion. Solid lines indicate stable solutions, dashedlines are unstable solutions. s: computations performed on a symmetric grid; 0.01: computationsperformed using ∆ = 0.01; 0.05: computations performed using ∆ = 0.05.

The effect of imperfections may clearly be seen with pronounced asymmetry in the ex-perimental streakline plot and asymmetric streamline calculation. The results presentedin figure 15 were obtained at Re = 56.4 which lies above the symmetry-breaking point ofthe symmetric system. The flow fields shown in (a) and (c) correspond to the connectedbranch of the perturbed system and now the stable flow computed on a symmetric do-main (b) is also asymmetric. Interestingly, the degree of asymmetry is similar in all threecases.

A state with the opposite symmetry was also realisable at the nearby Re = 56.1. Forthe imperfection applied here, the Reynolds number exceeds that at the limit point onthe disconnected branch. These flows are shown in figure 16(a) and (c) which lie onthe disconnected branch, while the appropriate asymmetric flow on a symmetric domainis shown in (b). Again, the effect of the domain imperfection does not appear to besignificant in these representations.

Streakline plots are shown in figure 17 for flows along the connected (a) and discon-nected (b) branches at similar flow rates. These flows are at a significant distance in Reabove the symmetry-breaking bifurcation point and the two flows are mirror images ofeach other. This further suggests that the effects of imperfections play a relatively minorrole. The final set of flow visualization and streamline plots are presented in figure 18.Agreement between all three is surprisingly good despite three-dimensional effects whichwere evident in the experiment.

An experimental and numerical bifurcation diagram is presented in figure 19 whereboth connected and disconnected branches may be clearly seen. An imperfection of size0.05 predicts that the branch of disconnected solutions ought to be broken into twoparts, with an isola (see e.g. Golubitsky & Schaeffer (1985), page 133) of disconnectedsolutions at lower Reynolds number. However, no experimental evidence for a gap inthe disconnected solutions was found. Hence, either three–dimensional effects becameimportant or an imperfection of size 0.01 which does not produce this split is moreappropriate.


(a)

(b)

(c)

Figure 13. Streakline and streamline plots for Re = 11.8 and E = 3, α = 7/3, γ = 2. Con-nected branch. (a) Experimental streaklines. (b) Streamlines assuming a symmetric domain. (c)Streamlines for a perturbed domain, ∆ = 0.05.

5. Conclusions

The effect of a downstream contraction on the flow in a channel with an expandedsection can be understood on simple physical grounds. There is a competition betweenthe instability associated with the expansion from the inlet to the expanded section, andthe instability associated with the expansion from the inlet to the outlet. The length ofthe expanded section plays a critical role in determining the outcome of this competition.We provide finite-element computations and experimental evidence which underlies thisargument. It is also interesting to note the very much greater sensitivity to imperfectionsfor the case which the symmetry-breaking bifurcation is associated with the outlet.

6. Acknowledgements

The authors would like to thank P. Reis for his assistance producing the figures.


(a)

(b)

(c)


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Abshagen, J., Pfister, G. & Mullin, T. 2001 A gluing bifurcation in a dynamically compli-cated extended flow. Phys. Rev. Lett. 87, 224501.

Allenborn, N., Nandakumar, K., Raszillier, H. & Durst, F. 2000 Further contributionson the two-dimensional flow in a sudden expansion. J. Phys. D 33, L141–L144.

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Cherdron, W., Durst, F. & Whitelaw, J. H. 1978 Asymmetric flows and instabilities insymmetric ducts with sudden expansions. J. Fluid Mech. 84, 13–31.

Chiang, T. P., Sheu, T. W. H., Wang, R. R. & Sau, A. 2001 Spanwise bifurcation inplane-symmetric sudden-expansion flows. Phys. Rev. E 65 (016306).

Chiang, T. P., Sheu, T. W. H. & Wang, S. K. 1999 Side wall effects on the structure oflaminar flow over a plane-symmetric sudden expansion. Comput. Fluids 29, 467–492.

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Cliffe, K. A. & Greenfield, T. C. 1982 Some comments on laminar flow in symmetrictwo-dimensional channels. Tech. Rep. T.P. 939. UKAEA.

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Taylor problem. In International Series of Numerical Mathematics (ed. H. M. T. Kupper& H. Weber), , vol. 70, pp. 129–144.

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sudden expansion. J. Fluid Mech. 211, 595–608.Golubitsky, M. & Schaeffer, D. G. 1985 Singularities and groups in bifurcation theory .

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sudden expansion. J. Fluid Mech. 436, 283–320.Mizushima, J., Okamoto, H. & Yamaguchi, H. 1996 Stability of flow in a channel with a

suddenly expanded part. Phys. Fluids 8, 2933–2942.


(a)

(b)

(c)

Figure 16. Streakline and streamline plots for Re = 56.1 and E = 3, α = 7/3, γ = 2. Discon-nected branch. (a) Experimental streaklines. (b) Streamlines assuming a symmetric domain. (c)Streamlines for a perturbed domain, ∆ = 0.01.

Mizushima, J. & Shiotani, Y. 2000 Structural in stability of the bifurcation diagram fortwo-dimensional flow in a channel with a sudden expansion. J. Fluid Mech. 420, 131–145.

Mizushima, J. & Shiotani, Y. 2001 Transitions and instabilities of flow in a symmetric channelwith a suddenly expanded and contracted part. J. Fluid Mech. 434, 355–369.

Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.Schreck, E. & Schafer, M. 2000 Numerical study of bifurcation in three-dimensional sudden

channel expansions. Comput. Fluids 29, 583–593.Shapira, M., Degani, D. & Weihs, D. 1990 Stability and existence of multiple solutions for

viscous flow in suddenly enlarged channels. Comput. Fluids. 18, 239–258.


(a)

(b)

(c)

Figure 17. Streakline and streamline plots for E = 3, α = 7/3, γ = 2. (a) Experimentalstreaklines at Re = 107.3. Connected branch. (b) Experimental streaklines at Re = 109.3.Disconnected branch. (c) Streamlines for a perturbed domain. Disconnected branch. ∆ = 0.01.


(a)

(b)

(c)

Figure 18. Streakline and streamline plots for Re = 140 and E = 3, α = 7/3, γ = 2. Con-nected branch. (a) Experimental streaklines. (b) Streamlines assuming a symmetric domain. (c)Streamlines for a perturbed domain, ∆ = 0.05.


Figure 19. Bifurcation diagram for E = 3, α = 7/3, γ = 2. The ordinate axis is v along thecenterline at a distance αh downstream of the expansion. Solid lines indicate stable solutions,dashed lines are unstable solutions. s: computations performed on a symmetric grid; 0.01: com-putations performed using ∆ = 0.01; 0.05: computations performed using ∆ = 0.05.

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