Visualization of Flows in Curved Channels with a Moderate or High Rotation Speed

Flows in channels with streamwise curvature and spanwise rotation are visualized in terms of end-view near the exit of the test sections through injecting smoke into the flows. Two test sections are used, i.e. the rectangular channels with the aspect ratio of and 10, respectively. The work focuses on visualization of Dean and Coriolis vortices under the effects of secondary instabilities and flows in the region with a relatively high rotation speed. The results show that the secondary instabilities cause the Dean and Coriolis vortices oscillating in various forms and the flows at high rotation speeds are controlled by the secondary instabilities rather than the primary instability. In particular, the secondary instabilities lead the flows to be unsteady and turbulent somewhat like the bursting flow in the turbulent boundary layers.

The physical model is illustrated in Fig. (also see Fig. 3).Under the action of the pressure gradient along the channel axis, a viscous fluid is allowed to flow through a channel of rectangular cross section with width hight of (a b).The channel is uniformly curved around the axis o'z' (streamwise curvature).At same time, the channel is rotating about that axis with a constant angular velocity D, (spanwise rotation).The rotation can be positive or negative as shown in Fig.
in terms of angular velocity vector.If positive, the ro- *Corresponding author.Tel." 403-492-3598.Fax:_ 403-492-2200.L. WANG and K. C. CHENG tation direction is in the direction of the relative ve- locity of the fluid inside the channel.When the rota- tion is negative, however, the rotation direction is in the direction opposite to the relative velocity of the fluid.The radial, spanwise and streamwise directions are (X, Y, {)) respectively.
The streamwise curvature and spanwise rotation of the channel introduce centrifugal and Coriolis forces in the momentum equations which describe the rela- tive motion of fluids with respect to the channel.Such body forces can stabilize the channel flow in some flow domain, and destabilize the flow in the other domains.In particular, the stabilizing or desta- bilizing effects due to curvature and rotation may ei- ther enhance or counteract each other depending on the direction of the rotation.This can be intuitively understood through a simple analysis of force direc- tions.In the plane of the cross section, the centrifugal force always acts outwards in the positive X-direc- tion.However, the Coriolis force may act in either positive or negative X-direction depending on the ro- tation direction.If the rotation is positive, it will act along the positive X-direction.When the rotation is negative, however, it will act along the negative X-di- rection.Therefore, we may expect a change in flow pattern as the rotation direction and rate change.For the case of positive rotation with a low rotation rate (Ro o(o-); Ro is the rotation number, Eq.( 2); o- a/Re., the curvature ratio of the channel in Fig. 1), Miyazaki [1971Miyazaki [ , 1973] ] numerically analyzed the fully developed laminar flow in curved, rotating, circular and rectangular channels by a finite-difference method.The secondary flow revealed consists of one pair of counterrotating vortices in a plane perpendicu- lar to the axis of the channel.The interaction of the secondary flow with the pressure-driven main flow shifts the location of the maximum streamwise veloc- ity away from the center of the channel and in the di- rection of the secondary velocity in the central region of the channel.The effects of the rotation direction were not investigated by Miyazaki [1971, 1973].Con- sidering the channel to be a tube of circular cross sec- tions with either positive or negative rotation, the fully developed laminar flow was investigated theoretically by Ito and Motai [1974], and Wang and Cheng  [1996a].The equations of motion were solved by a perturbation method for a small curvature and a low angular rotation rate of the tube.The inward Coriolis forces due to the negative rotation were found to cause the direction of the secondary flow to reverse by over- coming the outward centrifugal forces in the plane of the cross section.The flow reversal occurs by passing through a multi-pair vortex flow region where overall, the effect of the Coriolis force just neutralizes that of the centrifugal force.Such flow reversal and multi- pair vortex flow were also found in rectangular chan- nels by a finite-volume numerical analysis (Wang and   Cheng [1996b], and Wang 1995) and were experimen- tally confirmed by Wang and Cheng [1995].
For the case of moderate rotation in both positive and negative directions (Ro o(1)), the fully devel- oped laminar flow was examined by Daskopoulos and Lenhoff [1990] for circular tubes and by Wang [1995], Wang and Cheng [1996b], and Selmi et al.
[1994] for rectangular channels.The centrifugal and Coriolis forces were found to cause the centrifugal and Coriolis instabilities in the forms of streamwise- oriented roll cells (Dean and Coriolis vortices).The interaction between the two instabilities results in a complex flow structure.In particular, the appearance of the Dean and Coriolis vortices leads to a high in- flectional profile for streamwise velocity.Such a profile may be unstable with respect to streamwise wavy disturbances in accordance with the Rayleigh's in- flection point criterion.
For a channel with an infinite span, Finlay et.al. [1988] and Finlay [1990] examined the secondary stabilities of the Dean vortices and the Coriolis vor- tices with respect to the streamwise wavy distur- bances.They found that the instability leads to the development of undulating vortex flow and twisting vortex flow.As well, the stability of Dean vortices and Coriolis vortices subjected to two-dimensional spanwise periodic disturbances (i.e.Eckhaus stabil- ity) was examined numerically by Guo and Finlay 1991] for infinite span cross section.They found that the Eckhaus instability causes the vortex pairs to split apart or merge together.This motivates the first part of the present study to experimentally confirm these secondary instabilities and to expose the other possi- ble secondary instabilities.
It appears that the literature on high rotation rates (I Ro > 3) is rather limited.For the case of the positive rotation with a high rotation rate, Ludwieg  1951] and Hocking [1967] employed the boundary layer approximation to theoretically examine the fully developed laminar flow in a curved channel with a square and a rectangular cross-section, respec- tively.They found that the secondary flow also con- sists of one pair of counter-rotating vortices in the cross plane.The streamwise velocity profile assumes a Taylor-Proudman configuration in the core region (Greenspan [1968]).While Ludwieg [1951] and Hocking [1967] appear to be the only works on the high rotation case, the validity of the boundary layer approximation has not been established for the flow with secondary flows.As a matter of fact, all the numerical simulations (Wang [1995], Wang and Cheng [1996b], Daskopoulos and Lenhoff [1990], Nandakumar et al. 1991, and Selmi et al. [1994]) failed to find stable flows if the rotation rate is high.This motivates the second part of the present work to experimentally visualize the flows in curved channels rotating at a higher rotation rate.Note that the rotat- ing curved channels are usually operated at a high rotation rate in practical applications, and the results presented in this paper are believed to be not only of theoretical interest but also of practical importance.
In the present work, a smoke visualization appara- tus is designed to visualize the flow in two curved channels rotating spanwise at a moderate or high ro- tation rate.The work mainly contains two parts.The first part is to experimentally visualize flow under the effect of the secondary instabilities of Dean and Coriolis vortices.The second part is to visualize flow at a high rotation rate.In addition, the primary instabil- ity analysis by Wang and Cheng [1995] is extended to the case with a high rotation rate.This leads to a conclusion that the flows at a high rotation rate are controlled by the secondary instability rather than the primary instability.

EXPERIMENTAL APPARATUS AND TECHNIQUE
A schematic diagram of the experimental apparatus is shown in Fig. 2. It consists of a test section, a rotating table with the rotating seal, an air supply system and a smoke generator.
The rotating table is driven by an electric motor with adjustable speed drive and the range of the ro- tating speed is n 0 500 rpm.The rotational speed is measured by using an optical slot switch running on a disc with equally spaced 60 holes near its perimeter.The signal from the switch is fed to a Hewlett Packard HP 5314A Universal counter.With 60 holes in the disc, the frequency in Hz equals the rotational speed in rpm.
A second disc with a single hole provides the sig- nal for firing the General Radio 1540 Strobolume by way of a delay generator/single flash flip-flop which allows visual observation using a slit light source with one flash per rotation and also permits a single, properly timed flash for photographing the whole sec- ondary flow field.
The building compressed air is used as the fluid.The air flow rate is measured by a Meriam flow ele- ment with a calibrated differential pressure trans- ducer.The smoke generated by burning Chinese in- L. WANG and K. C. CHENG cense sticks is injected through a dispersing tube be- fore the test section as shown in Fig. 2. The very tiny smoke particle (10 .210-Mm), subjected to ne- glected gravitational and rotational buoyancy forces relative to the drag forces, marks fluid particle trajec- tories.The smoke patterns are photographed instantly near the exit of the test section (--0.8cmfrom the exit) revealing the flow pattern.This provides an end- view of the secondary flow pattern for an observer looking upstream into the channel cross-section.A Nikon FM2 single lens reflex camera and Kodak T-Max black and white film P3200 are used.
Two test sections, denoted by Ts-A and Ts-B, are shown in Fig. 3.The top view and the exit cross- section of Ts-A are shown in Fig. 3(a).It consists of an entrance spiral square channel with axial length 0.85m and a curved square channel (270 bend with axial length of 1.2m) with a constant radius of curva- ture R, 25.4cm.The curved square channel has cross-section 5.08 5.08cm2, and is made from acrylic sheets.The air flows through a rotating straight tube (with an inside diameter of 4cm) along the axis of rotation, and then enters the spiral square channel before entering the test section.The flow near the exit of the channel is believed to be fully developed because of a ratio of the length over the width of 40.4.
Ts-B, also made from acrylic sheets, consists of an entrance spiral rectangular channel with axial length 1.25m and a curved rectangular channel (270 bend) with a constant radius of curvature R 25.4cm.The top view and the exit cross section for Ts-B are shown in Fig. 3(b).The air flows through a rotating straight tube (with an inside diameter of 4cm) along the axis of rotation and then enters the spiral rectangular channel before flowing into the curved channel.With an entrance spiral channel length of 1.25m and a sub- sequent curved rectangular channel (length 1.2m) with a constant radius of curvature, the flow near the exit of the channel is also believed to be fully devel- oped because the ratio of the length over the width of the channel is 96.5.

RESULTS AND DISCUSSION
The end-view photographs are to be shown for Ts-A and Ts-B with spanwise direction vertical and radial direction horizontal.In each photograph, the convex (inner) wall is on the left and the concave (outer) wall is on the right.For a given test section, the flow is characterized by two parameters, namely, Reynolds where W m, v, 1) are the mean streamwise velocity, the kinematic viscosity of air and the angular rotation speed of the channel.The channel width a is used as the characteristic length in Re and Ro.For a given set of Re and Ro, several photographs, each taken at dif- ferent instant, are usually presented to show the time variation of the flow qualitatively.No attempt was made to specify the exact time intervals.For Ts-B, an alternative characteristic length is the hydraulic diameter dh.The values of Re and Ro with dh as the characteristic length are shown in the paren- theses.

Dean and Coriolis Vortices
An important feature relating to stabilization and de- stabilization is the appearance of the Dean and Cori- olis vortices.It is of special interest because they may cause a significant change to flow structure and the wall heat transfer distribution.In addition, they have an important effect on transition from laminar to tur- bulent flow and relaminarization from turbulent to laminar flow.
The Dean and Coriolis vortices may be present as a result of centrifugal and Coriolis instabilities imposed on the flow by curvature and rotation.Experimen- tally, these vortices are identified by the mushroom- shaped smoke patterns.Such patterns form because of spanwise variations of streamwise velocity due to the different inwash and outwash regions associated with pairs of counter-rotating vortices.The stems of individual mushroom shapes mark the region of flow away from the wall surface between the vortices which form one pair.
The possible various kinds of secondary instabili- ties of the Dean and Coriolis vortices may produce the Dean and Coriolis vortices with different features for different flow regions.The properties of distur- bances triggering initial vortex formation (such as their steadiness, uniformness along the spanwise di- rection etc.) may also lead to various Dean and Co- riolis vortices.Consequently, we may observe differ- ent mushroom-shaped smoke patterns due to single or combined effect of the two causes (secondary insta- bilities and initial disturbances).In a fully developed region which the present work is mainly concerned with, the Dean and Coriolis vortices become fully developed and should be less sensitive to the initial disturbances because once initiated, vortex behaviour and development are strongly controlled by the cen- trifugal and Coriolis instabilities from the curvature and rotation of the channel.The different Dean and Coriolis vortices observed in our experiments are, therefore, believed to be caused by their secondary instabilities.
The Dean and Coriolis vortices observed can be divided into two groups: symmetric and nonsymmet- ric vortices with respect to radial-streamwise plane.The symmetric Dean and Coriolis vortices appear usually at a dynamic parameter close to the critical value for their onset.Such vortices are identified by mushroom stems which form the radial symmetry lines.The mushroom-shaped smoke patterns are the mirror images on each side of the line.In other words, the two vortices in each pair are similar.The nonsymmetric Dean and Coriolis vortices are often observed at higher dynamic parameters.When viewed in cross (radial-spanwise) planes, such vorti- ces are distorted generally by spanwise and/or radial unsteadiness in the flow such that no symmetric line is observed.terns are especially interesting because they show only small unsteadiness.No significant time variation on shape and size can be observed.The narrow mush- room stems indicate narrow outwash regions from the high pressure wall.Also the large mushroom petals imply large, spread-out sidewash and inwash regions.
The fairly uniform size and shape at different span- wise positions indicate that the flow is fully devel- oped since otherwise the difference will result from the sensitivity of developing vortices to initial distur- bances.
A comparison of the smoke patterns in Fig. 4(a) with those in Fig. 4(b) shows that the shape of the symmetric Coriolis vortices changes with the operat- ing parameter.This is consistent with the numerical simulations (Finlay [1990]) which indicate that vor- tex shape changes with spanwise wavenumber, Reynolds number and rotation number.

Radially Oscillating Coriolis Vortex Flow
Two examples of radially oscillating Coriolis vortices are presented in Fig. 5.They occur at Re 600, Ro -1.37 for Ts-A and at Re 193.6(352),Ro -0.663(-1.205)for Ts-B respectively.At this stage of flow, very little spanwise oscillation is present and mushroom patterns seem to move almost exclusively in the radial direction.When these oscillations occur, they are mostly symmetric Coriolis vortices, and the radial unsteadiness decreases in magnitude as the overall average vortex height increases.
The vortex heights at different spanwise locations are fully developed along the streamwise direction and the oscillation is caused by the secondary insta- bility of the Coriolis vortices.
direction provide evidence of fully developed vorti- ces.As well, the oscillation is believed to be caused by the secondary instability.

Spanwise Oscillating Dean and Coriolis Vortex Flow
The mushroom-shaped smoke patterns at Re 500, Ro 0 for Ts A and at Re 581(1056), Ro -1.12(-2.036)for Ts B are shown in Fig. 6.The time variations in shape show that very little radial oscillation is present and mushroom patterns seem to move almost exclusively in the spanwise direction.When these oscillations occur, they are mostly sym- metric.The uniform height and spacing in spanwise 3.1.4Simultaneous Spanwise and Radial   Oscillating Dean Vortex Flow The Mushroom-shaped smoke patterns, oscillating si- multaneously in the spanwise and radial directions may be observed without a twisting motion.One ex- ample is shown in Fig. 7 for Ts-A at Re 550, Ro 0. The vortices in this vortex flow are also nearly symmetric, and generally have larger-scale oscillation and greater unsteadiness than the smoke patterns identified with radial, span or twisting modes.
(a,) Spa,wise oscil[at;iag Dert vor,ices   .t  .1.5Twisting Coriolis Vortex Flow Finlay et al. [1988] and Finlay [1990] simulated curved and rotating channel flows using three-dimen- sional incompressible time-dependent Navier-Stokes equations.They found evidence that the Dean or Coriolis vortices develop two different types of travel- ling waves in the streamwise direction.The undulat- ing Dean or Coriolis vortex flow contains long streamwise wavelengths and very small growth rates which make them difficult to observe experimentally.
The twisting Dean or Coriolis vortex flow contains much shorter streamwise wavelength than the undu- lating Dean and Coriolis vortex flow.This causes them to be observed more readily.
Streamwise fully developed travelling waves gen- erate same events at one time but different stream- wise locations as at one streamwise location but dif- ferent times.With twisting, the vortex centers oscil- late only a little in the spanwise direction.The motion occurs mostly in the radial direction as the vortices in one pair oscillate in shape and strength.
The smoke patterns for twisting Dean and Coriolis vortices are mushroom-shaped with a rocking type of motion.Examples of smoke patterns from twisting Dean vortices are presented in Fig. 8.They were ob- tained at Re 600, Ro 0 for Ts-A and at Re 581(1056), Ro 0(0) for Ts-B.It is observed that the twisting Dean vortices are nonsymmetric.The Reynolds number for the two cases with twisting Dean vortices (Ro 0) are larger than 1.96R G. with Re. as the critical Reynolds number for the onset of the Dean vortices.These are consistent with the numeri- cal simulations given by Finlay et al. [1988].
For Ts-B, it is found that the rocking motion is rarely observed alone, rather it is usually observed in conjunction with at least one other mode of oscilla- tion.In many cases, the rocking vortex pairs are ad- jacent to pairs with a different type of motion.

Small Vortex Pairs
Using a linear stability theory and spectral method, Guo & Finlay [1991] examined the Eckhaus stability of the Dean and Coriolis vortices.They found that the Eckhaus stability boundary is a small closed loop.Outside the boundary, the Eckhaus instability causes the vortex pair to split apart by the formation of a new vortex pair or merge together, and no vortex flow is stable to spanwise disturbances when Re > 1.7RG:.When Re is not too high (Re < 4RG. ),the wavenumber of vortices is selected by the Eckhaus instability.Similar phenomena have also been re- ported by Finlay et al. [1988], Finlay [1990]  and  Bland and Finlay [1991] in their numerical simula- tions of channel flows with curvature or rotation.

Complex Oscillating Models
In some cases, usually with high dynamical parame- ters, the oscillation of the Dean and Coriolis vortices appear to be so chaotic such that the mushroom- shaped patterns are barely recognizable or can not be identified clearly.Photographs of such patterns are  shown in Fig. l0 for Ts-B at Re 193.6(352),Ro 0(0).
3.2 Flows in Curved Channels with a High Rotation Rate A generalized Rayleigh stability criterion was derived by Wang and Cheng [1995] to analyze stable and unstable regions of flows with respect to the primary instability in a rotating curved channel of an infinite span.The result is II-2 The flow is stable in the flow domain with II > O, and unstable in the domain with II < O.This criterion was used to analyze the stable and unstable regions for the case with a small gap (o-< < 1) and a low rotation rate (Ro o()).Here we extend the analy- sis to the case of a small gap with a high rotation rate.
For the small gap (or < < 1) and high rotation rate (Ro o(1)), the generalized Rayleigh's criterion 1I can be approximated as II(x)-2Ro (xW + 2Ro) (5) which vanishes at a radial position x o satisfying dw + 2Ro 0 (6) For the flow in a curved channel with rotation around the axis of curvature, the base flow can be written as (Chandrasekhar 1961 ]) This, with Eq.( 6), yields Xo- The sign of H in different regions is summarized in Table I.The stable and unstable regions are shown in Fig.
(Although Table and Fig. 11 show the re- sults for the whole range of Ro, we should use the results in Wang and Cheng [1995] for the case with a low rotation rate).This shows that the unstable re- gions reduces as Ro increases, and the high rota- tion with Ro > 3 always stabilizes the flow in the whole cross-section.
Examples of smoke patterns from flows in a region with Ro > 3 are illustrated in Fig. 12(a) for T A and in Fig. 12(b) for Ts B. They are obtained at Re-200, Ro-21.42;Re-200 -3.59(-6.53)and Re 452(822), Ro 3.45(6.27)for Ts-B.The patterns are seen to be some- what similar to the bursting flow in the turbulent boundary layer (Kline et al. 1119671), and are far from the one pair of counterrotating vortices predicted by Ludwieg [1951]  and Hocking [1967].Therefore, the boundary layer approximation may not be valid for flow in the curved channels with a high rotation rate.Also the high unsteadiness of flow implies that the flow is controlled by the secondary instability rather than the primary instability.In particular, the second- ary instability can produce the low Reynolds number turbulent flows.An experimental apparatus was designed and con- structed especially to visualize the fully developed flows in two kinds of channels with streamwise cur- vature and spanwise rotation in terms of the end-view secondary flow near the exit of the test sections.The work focuses on the Dean and Coriolis vortices under the effects of the secondary instability, and the flows in the region with Ro > 3.

CONCLUDING REMARKS
Pairs of counter-rotating Dean and Coriolis vorti- ces are observed as the mushroom-shaped smoke pat- terns.The narrow mushroom stems indicate the nar- row outwash flow regions while large mushroom pet- als imply the large, spread-out sidewash and inwash regions.The fairly steady, symmetric Dean and Cori- olis vortices are observed in some parameter regions.Their size and shape change with the operating pa- rameters.
Secondary instabilities of the Dean and Coriolis vortices lead to oscillation in various forms even in the streamwise fully developed flow regions.The ob- served oscillating modes include: (1) one with mostly radial motion, (2) one with mostly spanwise motion, (3) one with significant simultaneous radial and span- wise motion, and (4) one with rocking motion.In addition, the oscillating modes are also present with the motion so complex that it is rather difficult to describe the flow visualization result.The experimental results are to be confirmed both theoretically and numerically in the future.
In the cross planes, the twisting Dean and Coriolis vortices are evidenced by nonsymmetric mushroom- shaped smoke patterns with rocking motion.The re- gion where the twisting Dean and Coriolis vortices are observed is in agreement with that predicted by Finlay et al. [1988] and Finlay [1990].This kind of vortex flow differs from that with simultaneous radial and spanwise motion mainly in three aspects: (1) the twisting vortices are nonsymmetric but the simulta- neous radial and spanwise oscillating vortices are symmetric, (2) with twisting, the spanwise locations of vortex centers oscillate only a little.and (3) the simultaneous radial and spanwise oscillat- ing Dean and Coriolis vortices usually have larger- scale oscillation and greater unsteadiness than the twisting Dean and Coriolis vortices.
The vortex pairs sometime seem to appear and dis- appear as indicated by small vortex pairs observed temporarily in the cross planes for some experimental conditions.This is believed to be caused by the Eck- haus instability discussed by Guo and Finlay [1991].
The flow at large Ro is found to be controlled by the secondary instability rather than the primary in- stability.In particular, the secondary instability leads the flow with large ]Ro to be unsteady and turbulent somewhat like the bursting flow in the turbulent boundary layer.This appears to produce the low Reynolds number turbulent flow and one may conclude that the boundary layer approximation is not valid for this type of flow.

FIGURE
FIGUREFlow in a channel with streamwise curvature and spanwise rotation.
FIGURE 3 Two test sections.

FIGURE 4
FIGURE 4 Symmetric Coriolis Vortices for Ts-B.

FIGURE 7
FIGURE 7 Simultaneous spanwise and radial oscillating Dean vortices at Re 550, Ro 0 for Ts-A.

FIGURE 9
FIGURE 9 Small vortex pairs for Ts-B.

TABLE Sign of
II for the case of small gap and high rotation rate With simulta- neous radial and spanwise oscillating, the spanwise FIGURE 11 Stable and unstable regions for the case of a small gap with a high rotation rate.L. WANG and K. C. CHENG n.. 200,.o2.42 ........... =, .iFlows in region with Ro ]> 3If Ts-B FIGURE 12 Flows in a region with Ro > 3 locations of vortex centers can oscillate significantly,