vortices in film flow over strongly undulated bottom profiles at low reynolds numbers

8/11/2019 Vortices in Film Flow Over Strongly Undulated Bottom Profiles at Low Reynolds Numbers

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Vortices in film flow over strongly undulated bottom profilesat low Reynolds numbers

A. Wierschem, M. Scholle, and N. Aksela)

LS Technische Mechanik und Stromungsmechanik, Universitat Bayreuth, D-95440 Bayreuth, Germany

Received 15 August 2002; accepted 24 October 2002; published 8 January 2003

We present an experimental study of gravity driven films flowing down sinusoidal bottom profiles

of high waviness. We find vortices in the valleys of the undulated bottom profile. They are observedat low Reynolds numbers down to the order of 105. The vortices are visualized employing a

particle image velocimeter with fluorescent tracers. It turns out that the vortices are generated

beyond a critical film thickness. Their size tends to a finite value for thick films. The critical film

thickness depends on the waviness of the bottom undulation, the inclination angle, and on the

surface tension but not on the Reynolds number. Increasing the waviness, a second vortex can be

generated. 2003 American Institute of Physics. DOI: 10.1063/1.1533075

I. INTRODUCTION

Gravity-driven film flow is one of the most studied and

best understood systems in hydrodynamics. Although the in-fluence of imperfections in the topography of the bottom

strongly affect the flow of thin films at low Reynolds num-

bers, almost the entire research has been focused on the film

flow down a flat incline. The impact of single small local

imperfections on the flow of thin liquid films down an in-

clined plane has been studied numerically by Pozrikidis and

Thoroddsen.1 Experiments on gravity-driven flow have been

carried out by Decre et al. considering a topography of

steps.2 These studies, however, focus only on the shape of

the free liquid surface induced by steep steps or by small

discrete particles.

The effects of sinusoidal bottom-profile variations on the

film of a liquid flowing down an incline has been studiedanalytically and experimentally only for the case of rather

small waviness. Wang has considered the effect of weakly

undulated bottoms of thick films3 and that of more pro-

nounced waviness on thin films4 in his perturbation analyses.

He suggested that return flow may occur due to surface-

tension effects. However, Wierschem et al. showed that this

is beyond the valid range of the perturbation analyses.5 Fur-

thermore, they have studied the regime of moderate and

small waviness experimentally and could not find any vorti-

ces in this regime. A numerical study of the influence of wall

corrugations on film flow at rather high Reynolds numbers

has been carried out by Bontozoglou6 and by Bontozoglou

and Papapolymerou.7 They found that in a certain range of

Reynolds numbers capillary-gravity waves could be excited

by resonant interaction with the bottom undulation. They

also reported the observation of inertia-driven vortices in the

trough of the bottom undulations and found that these vorti-

ces may be suppressed by the resonant excitation of surface

wavesa fact also observed by Trifonov8 in his calculations

at high Reynolds numbers.

An extensive numerical study of the influence of wavy

bottom profiles on creeping film flow has been carried out byPozrikidis.9 He found good agreement comparing his results

to those obtained analytically by Wang in the range of their

validity. Varying the flow rate, inclination angle, wave am-

plitude, and surface tension, he also studied situations that

are beyond the scope of Wangs analyses. He showed that

surface-tension effects may alter the flow considerably. For

profiles of large waviness, he observed return flow solutions

if the film is thick or at low inclination angles. However,

there are no experimental data on the occurrence of vortices

in film flow over sinusoidal bottom profiles.

Vortices have been found experimentally under creeping

flow conditions by Taneda.10 He visualized the flow in dif-

ferent geometries, such as arcs, steps, and corners and couldrecover most of the theoretical results on vortices in creeping

flows.1113 Different from Tanedas systems, the gravity-

driven film flow over sinusoidal bottoms studied here has a

free surface and thus besides the waviness it has further char-

acteristic length scales that yield critical values for the gen-

eration of vortices. Vortices in free film flow along vertical

corrugated surfaces have been reported by Zhao and Cerro.14

They studied experimentally the free surface shape and visu-

alized the streamlines in the film. For periodic convex half

cycles they observed vortices no matter the thickness of the

film was. For triangle and concave half cycles they stated

that they found flow separation only at the highest Reynolds

numbers and Capillary numbers studied. While the afore-

mentioned vortices appear at the lee side of the corrugations

or in the troughs, Negny et al. observed vortices at the flat-

test part of the bottom undulation at rather high Reynolds

numbers.15

Here, we report on vortices observed in gravity-driven

film flow over sinusoidal bottom profiles. The vortices occur

in the valleys of the undulated bottom profile even under

creeping flow conditions beyond a critical film thickness.

The latter depends on the waviness of the bottom undulation,

aAuthor to whom correspondence should be addressed. Telephone:

49-921-55-72 60; fax: 49-921-55-72 65; electronic mail:

[email protected]

PHYSICS OF FLUIDS VOLUME 15, NUMBER 2 FEBRUARY 2003

4261070-6631/2003/15(2)/426/10/$20.00 2003 American Institute of Physics


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the inclination angle, and is modified by surface-tension ef-

fects. Increasing the waviness, a second vortex can be gen-

erated. In Sec. II we describe the experimental system and

the applied methods. The experimental results on the vortices

are given in Sec. III and are discussed in Sec. IV. Finally, our

conclusions are summarized in Sec. V.

II. EXPERIMENTAL SYSTEM AND SETUPS

A. Experimental system

We have used a channel with sinusoidal bottom profiles

of different waviness. The channel is built from an aluminum

bottom and transparent Plexiglas side-walls. It is divided into

different sections: A flat one and another one with different

bottom profiles. The latter ones are characterized by their

amplitude and wavelength. The parameters of the bottom

waves are given in Table I. The crests of the bottom waves

are all at the same level to minimize surface-tension effects

at the borders of the different sections. The measurements

are performed in the center of each of the sections, which are

large enough to assure that the measurements are not affected

by neighboring regions. The flat section serves to compare

the flow over the wavy profile to that over a flat bottom. The

channel has a width of 1701 mm. The setup with all undu-

lations except that with 1 mm amplitude and 5 mm wave-

length is shown in Fig. 1.

To minimize the Reynolds number, we chose a highly

viscous silicone oil, BC5000cs silicon oil from Basildon

Chemicals. Its viscosity, density, and surface tension were

measured in the temperature interval ranging from 291.15

303.15 K. At 295.15 K, it has a density of 0.972 g/cm3, a

kinematic viscosity of 5780 mm2/s, and a surface tension of

21.4 mN/m. The experimental runs are carried out at ambient

temperatures between 293.15 and 297.15 K resulting in a

deviation of density, viscosity, and surface tension from their

mean values of less than 0.2%, 4%, and 0.5%, respectively.

To study the impact of the Reynolds number we carried out

additional experiments using the silicon oil B1000 from Elbe

Silikone. This offers the advantage to change the Reynolds

number without hardly changing surface tension. Its viscos-

ity, density, and surface tension were measured in the tem-

perature interval ranging from 295.15 K299.15 K. The ex-

periments on the vortices were carried out in a temperature

interval between 297.55 and 297.95 K, where it has a density

of 0.969 g/cm3, a kinematic viscosity of 1213 mm2/s, and a

surface tension of 20.4 mN/m.

B. Experimental methods

The vortices are visualized at the centerline of the chan-

nel with a particle image velocimeter PIV from Dantec

Dynamics using fluorescent tracer particles. It is made up of

a double pulse laser system, light sheet optics, a HiSense

camera with a 2.8/105 mm Nikon objective and a red filter

transmitting wavelengths larger than 550 nm, a PIV 1500

data acquisition unit, and FLOWMANAGERsoftware for evalu-

ation. The laser system is made of two frequency doubled

Nd:YAG laser, working at 532 nm wavelength, with a pulse

energy of 30 mJ each. Part of the system is shown in Fig. 1.

For small time intervals, the PIV system can be used in the

double image mode and for large time intervals it is operated

in the single image mode, acquiring single images at fixed

time steps. The latter is mainly applied in this study, because

of the slow motion of the flow and the spatial resolution

required. The images captured are evaluated with the FLOW-

MANAGER software to determine velocity fields or they are

superposed with the image processing software Optimas

from Media Cybernetics. From a sequence of images, the

maximum brightness at each pixel is determined to obtain

the particle positions during irradiation. If the tracer velocity

is so small, that the tracer moves less than its diameter dur-

ing the time interval between two successive images, this

yields the pathlines. If the tracer velocity is larger, the tracerslook like beads on a string from which the pathlines can be

reconstructed.

Since we are interested in the flow close to the bottom, it

is essential to use fluorescent particles as tracers. The study

has been performed with dry red fluorescent polystyrene mi-

crospheres from Duke Scientific with a mean diameter of 5

m. They have a density of 1.05 g/cm3. Due to the small

density difference of the particles to the silicone oils and due

to the high viscosity of the latter the sedimentation of the

tracers is of the order of 106 mm/s107 mm/s, depending

on the silicon oil used.16 This is still sufficiently smaller than

the smallest velocities detected, which are down to about

TABLE I. Parameters of the bottom waves.

Wavelength mm Amplitude mm Number of valleys

7.5 1 10

7 1 10

14 2 5

18 3 5

5 1 39

20 4 5

20 8 520 9 5

20 10 5

FIG. 1. View of the experimental system and setup. Left from the channel

with different undulated bottom profiles are the PIV camera and the CCD

camera with the microscope objective used to scan the bottom profiles and

the free surface. In the upper right is the neon lamp with the frosted-glass

cover.

427Phys. Fluids, Vol. 15, No. 2, February 2003 Vortices in film flow


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104 mm/s. To prevent the tracer particles from agglomera-

tion, they have been dispersed in small quantities of silicone

oil with a ULTRA TURRAX T50-disperser from IKA-

Werke.

Besides the vortices, we have also measured the positionof the free liquid surface. The surface position of the film

flowing over the flat part of the channel and thus the film

thickness is measured with a micrometer screw. This film

thickness serves as a reference for the flows over the differ-

ent bottom waves. It has the advantage over volume-flux

detection that side-wall effects do not enter in its measure-

ment. When we refer to a film thickness in this article we

always mean the thickness over the corresponding flat in-

cline unless otherwise specified. The Reynolds number, Re,

is also defined for the corresponding film flow over a flat

incline: ReUh/(gh 3 sin)/(22), where U and h are

surface velocity and film thickness over a flat incline, respec-

tively, is the mean inclination angle, the kinematic vis-cosity, and g the acceleration of gravity.

The surface position over the undulated bottom has been

determined from the flow visualization images as shown in

Fig. 2. In this images, the surface appears as a bright line.

Small surface undulation have been measured by scanning

the meniscus from the side with a CV-M300 C/E CCD cam-

era from JAI mounted with a Zoom 70 microscope objective

from Opto Sonderbedarf GmbH. The camera is fixed to an

XYZ-transverse unit and scans the meniscus against the

light. As light source we used a neon lamp with a frosted-

glass cover placed at a distance between 400 to 700 mm

from the channel. The setup is shown in Fig. 1. Further de-

tails on this method can be found in Ref. 5. Finally, the mean

transport velocity at the surface, as averaged over many

waves, has been measured by particle tracking of carbon

powder strewn on the liquid surface with a CCD camera

from above.

III. EXPERIMENTS

We first report our results on single vortices. Unless oth-

erwise stated, they have been carried out with the silicon oil

BC5000cs. After some qualitative features, we cover the ef-

fects of the Reynolds number, film thickness, inclination

angle, and surface tension for a bottom undulation of given

waviness, i.e., the ratio of wave amplitude to wavelength.

Then, we show the influence of the waviness and report on

our observations of a second vortex. After some qualitative

features on the sensitivity of the vortices to perturbations we

finish with the effect of the vortices on the transport flowvelocity.

A series of images of the film flow over a sinusoidal

bottom of 5 mm wavelength and 1 mm amplitude is shown

in Fig. 2. While there is no vortex observable in Fig. 2a, a

slight increase of the film thickness results in a small vortex

in the trough of the bottom undulation as seen in Fig. 2 b.

This vortex is about 100 m thick. Increasing the film thick-

ness further leads to a much larger vortex as shown in Fig.

2c. Remarkably the separatrix seems to be a straight line at

mean inclination angle. A quantitative measure of the vortex

size as a function of the film thickness is given in Fig. 3a.

It shows the distance of the vortex core and of the separatrix

FIG. 2. Film flow as visualized by determining the

maximum brightness at each pixel out of a series of

tracer images. The main flow direction is from the up-

per left to the lower right. The undulated bottom is at

the lower left, the film itself is seen as a flecked area,

and its free surface as a thin bright line. Images of

tracers above this line are due to reflections of the emit-

ted light at the free surface. In image a, no vortex can

be observed. Image b depicts a small vortex of about

100 m thickness in the trough of the undulation and

c shows a large eddy for a thick film. Bottom wave-

length: 5 mm, amplitude: 1 mm, inclination angle: 45.

Film thickness: 1.6 mma, 1.8 mmb, and 12 mmc.

428 Phys. Fluids, Vol. 15, No. 2, February 2003 Wierschem, Scholle, and Aksel


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to the center of the trough. Both distances show the same

qualitative behavior. Up to a critical film thickness, there is

no vortex in the flow. Beyond this critical film thickness the

size of the vortex increases sharply with the film thickness

and tends to an asymptotic value for thick films. Although

the Reynolds number differs strongly for the two silicon oils,

we obtain the same curves for both oils, as shown in Fig.

3a. For the BC5000cs silicon oil, the Reynolds number at

the critical film thickness is 5104 and reaches about 0.2

for the highest value shown in the diagram, while for the

B1000 silicon oil, we arrive at a Reynolds number of 1.1

102 for the critical film thickness and reach for the highest

value shown a Reynolds number of 1.2. Thus, although the

Reynolds number changes about one to two orders in mag-

nitude, we do not see any impact on the vortices.

Figure 3b shows the peakpeak amplitude of the free

upper surface as measured for the BC5000cs silicon oil. Be-yond a peakpeak amplitude relative to the amplitude of the

bottom wave of about 0.3 the data can be fitted properly to

an exponential decay and the surface shape seems to be sinu-

soidal. For thinner films, however, the surface amplitude de-

viates from this fit to smaller values. This is due to restric-

tions resulting from the mean inclination angle and from the

surface tension. Since the bottom contour is not monoto-

nously falling, thin films build puddles at the upward point-

ing side of the undulation, resulting in an almost horizontal

surface in this part of the undulation. The amplitude is fur-

ther decreased by surface tension that flattens sharp corners.

Under these circumstances, the surface shape is not sinu-

soidal anymore. An example observed in our experiments isdepicted in Fig. 11a. For the case shown in Fig. 3, we

remark that the critical film thickness for the vortex genera-

tion is in the regime where the surface amplitude falls off

exponentially.

A typical velocity field of the vortex is shown in Fig.

4a. To focus on the low velocity of the vortex, the overlay-

ing flow field is not resolved. Fig. 4bdepicts the maximum

return velocity in the vortex as a function of the film thick-

ness together with an exponential fit to the data. It is scaled

with a reference velocity URef(g2 sin)/(2), where is

the mean inclination angle, is the wavelength of the bottom

contour, the kinematic viscosity, and g the acceleration of

gravity. The diagram depicts velocities that range from less

than 1 to about 60 m/s.

The dependence of the critical film thickness on the in-

clination angle is shown in Fig. 5 for the case of small

surface-tension effects. For small inclination angles, the criti-

cal film thickness increases strongly with the inclination

angle while it changes rather weakly at angles beyond 20.

The data can be fitted properly with a cotangent function as

shown in the diagram. Surface-tension effects are described

in nondimensional variables by the Bond number

FIG. 3. Vortex size as a function of the film thickness a, and peakpeak amplitude of the free surface for the silicon oil BC5000cs b. Ina, the distance

of the vortex core and of the separatrix to the center of the trough are shown as circles and squares, respectively. Measurements with the oils BC5000cs and

B1000 are depicted as solid and open symbols, respectively. The Reynolds numbers at the onset of the vortices are 5 104 and 1.4102, respectively. The

curves are fits to the experimental data with the Weibull function. The thick curve in b is an exponential fit to the data for thick films of the BC5000cs oil.

Bottom wavelength: 5 mm, amplitude: 1 mm, inclination angle: 45.

FIG. 4. Velocity field of a vortex in the trough of the bottom a and the

maximum return velocity in the vortex as a function of the film thickness

together with an exponential fit to the data b. The velocities are scaled with

a reference velocity that corresponds to the surface velocity of a flat film

with a thickness of the wavelength. Bottom wavelength: 5 mm, amplitude: 1

mm, inclination angle: 9.8. Film thickness in a: 7.13 mm.



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BoBo*sin 2lCa

2

sin 1

where is the mean inclination angle, and is the wave-

length of the bottom contour. The capillary length is defined

as l Ca/(g), with the surface tension , the liquid den-sity , and the acceleration of gravity g. We have seen that

the inclination angle changes the critical film thickness. Thus

to check the influence of surface tension on the vortices

alone for a given waviness without changing the liquid, we

fix the inclination angle and vary the absolute scale of the

bottom undulation. To this end we have studied the case of

small and large Bond numbers at the waviness 0.2 for a

wavelength of 20 and 5 mm at two different inclination

angles. The results are shown in Fig. 6. Comparing the data

for a given inclination angle, Fig. 6a shows that the non-dimensional critical film thickness is larger for higher Bond

number. At an inclination angle of 45 the difference is rather

small; however, at an angle of 9.8 the gap has widened

considerably. On the other hand, comparing the data for dif-

ferent angles, we see that although the Bond number for the

20 mm waves at 9.8 is larger than that for 5 mm waves at

45, its nondimensional critical film thickness is not. With

increasing film thickness the difference in the nondimen-

sional vortex size diminishes so that for thick films it be-

comes independent of the Bond number as well as from the

inclination angle. We further remark that the Reynolds num-

ber is largest for the case of high Bond number. For the

values shown in the graph, the Reynolds numbers range from

about 5102 to about 0.2. The surface amplitude for the

different cases is shown in Fig. 6b. The surface amplitudes

apparently decrease exponentially for thick films. For thin

films, they cannot be described with an exponential law. Es-

pecially the amplitude for the lowest Bond-number case

studied seems to reach a plateau at a peakpeak amplitude ofabout 90 m. Comparing the data for a given inclination

angle, the amplitude for high Bond numbers is higher than

that for the small Bond number for thin films; however, this

is not the case when comparing different angles.

Up to now we have studied the vortices for a fixed wavi-

ness by varying the film thickness, the inclination angle, and

FIG. 7. Vortex with bent separatrix in the trough of a bottom wave as

visualized by adding up series of tracer images. The main flow direction is

from left to right. Bottom wavelength: 20 mm, amplitude: 9 mm, inclination

angle: 45. Film thickness: 5.3 mm.

FIG. 5. Critical film thickness for the generation of a vortex as a function of

the inclination angle. The solid squares mark the minimum film thickness

where the vortex has been observed and the open squares indicate the maxi-

mum film thickness with no vortex observed. The curve is a fit to the data of

the form: HcrH0(cot 0cot ), with Hcr , H0 , and a 0 being the critical

film thickness, and two fit parameters. Bottom wavelength: 20 mm, ampli-

tude: 4 mm.

FIG. 6. Influence of the Bond number on the vortex for given waviness. Distance of the separatrix to the center of the trough as a function of the film thickness

a and peakpeak amplitude of the free surface b. The circles correspond to a wavelength of 5 mm, and the squares to a wavelength of 20 mm. The open

and solid symbols indicate the data for 9.8 and 45 inclination angle, respectively. Waviness: 0.2, Bond numbers: : 3.18; : 0.77; : 0.20; : 0.05.



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the absolute scale of the bottom wave to account for the

effect of surface tension. We now study the vortices for dif-

ferent waviness. First of all, we observe that for stronger

waviness the separatrix is not a straight line but is bent.

Figure 7 shows an example for the waviness 0.45. Second,

we detected the critical film thickness for the vortex as a

function of the waviness. The results are given in Fig. 8

together with a logarithmic fit to the data. At the inclination

angle of 45 considered here, the influence of surface tension

is rather weak as shown in Fig. 6a. We see that the critical

film thickness increases strongly with decreasing waviness.

For the higher wavy bottoms we could not find a film with-

out a vortex at the given inclination angle. Although we de-

creased the film thickness down to less than 0.7 mm, corre-

sponding to about 1/30 of the wavelength.

In the cases of highly undulated bottoms, we found asecond vortex for sufficiently thick films. An example is

shown in Fig. 9. Like for the first vortex, their critical film

thickness decreases with increasing waviness. Figure 10

shows an example of the vortex size and the surface defor-

mation for very high waviness. Concerning the second vor-

tex, we see that it forms again beyond a critical film thick-

ness and grows with the film thickness. The velocities in the

second vortex are much smaller than that of the first vortex.

For a film thickness of about 0.47 times the wavelength, the

ratio of the velocities along the separatrices is about 3000:1

and it reduces to about 1250:1 for a film thickness to wave-

length ratio of 0.62. In the region of thick films, where the

second vortex is observed, the first vortex shows the samequalitative behavior as observed before for smaller waviness

in Fig. 3a, i.e., the growth of the vortex diminishes with the

film thickness. However, for thinner films the diminution of

the vortex lessens and the size seems to tend to a constant

value for zero film thickness. For the thinnest films the po-

sition of the vortex core moves slightly to the steep side of

the bottom contour. The surface amplitude of the thin films

deviates from the exponential behavior and the surface shape

strongly differs from the sinusoidal form. The latter can be

seen in Fig. 11a. The film is so thin that its free surface

invades the wavy region. At the falling edge of the bottom

the film is bent backwards. Here the film is extremely thin

and almost parallel to the bottom. The rising edge restricts

the level of the film, since the films free surface must be

monotonously falling. So in this region, the free surface de-

creases only slightly in flow direction. Thus, the rising edge

enforces a lower limit for the local film thickness over the

trough. Only by increasing the inclination angle this limit

can be lowered. For the waviness of 0.45 studied here we

observed a flow without vortices at an inclination angle of

80 as shown in Fig. 11b.

We found that small vortices over highly undulated bot-toms are sensitive to small perturbations. Perturbing the flow

by slight inclinations along the channel width or by a small

pulsation of the flow may deform the vortex into a spiral. An

example of a single vortex is given in Fig. 12a. Figures

12b and 12c show the second vortex. It can even be dis-

placed to the side as seen from Fig. 12c. We notice that in

this case the first vortex is hardly affected by the perturba-

tion.

The flow separation due to the vortex modifies the

boundary condition of the overlying flow downstream in the

sense that there is no no-slip condition for the downstream

flow along the separatrix. This may modify the overall trans-

port velocity of the flow. To clarify the influence of the vor-

tices on the transport velocity, we measured the mean trans-

port velocity at the films free surface over 1530 waves and

compared them to the theoretical film surface velocity over a

corresponding flat plate at same inclination angle, thus, for

the same volume flux. The result is shown in Fig. 13. We

remark that in this case there exists a vortex even for the

thinnest films. For thin films the transport velocity over the

wavy bottom is considerably smaller than that over a flat

incline. With increasing film thickness the difference tends to

zero. In a certain parameter range, we measured a ratio

slightly larger than one. However, the ratio one is still within

the experimental uncertainty of the data. Finally, for thickfilms, the ratio between the two velocities is about one.

FIG. 9. Film flow with two vortices in the trough of a

bottom wave as visualized by adding up series of tracer

images. The main flow direction is from the upper left

to the lower right. a shows the flow over one bottomwave, and b is an amplification focusing on the sec-

ond vortex. Bottom wavelength: 20 mm, amplitude: 9

mm, inclination angle: 45, film thickness: 11.6 mm.

FIG. 8. Critical film thickness as a function of the waviness. The squares

indicate the critical thickness of the first vortex and the circles that of the

second vortex. Solid and open symbols refer to the thickness where a vortex

is observed or not observed, respectively. The curves are logarithmic fits to

the data Hcr/H0/B ln(A/A0/), with H0, B, and A 0 being fit pa-

rameters. Inclination angle: 45.



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IV. DISCUSSION

A. Discussion of the experimental observations

For strongly undulated bottoms, the vortices are gener-

ated beyond a critical film thickness. We have observed these

vortices for small Reynolds numbers, however, covering sev-

eral orders of magnitude ranging from about 3105 to

about 1.2. In this range, the Reynolds number apparently is

not responsible for the existence of the vortices. As shown in

Fig. 3a, the vortex takes the same size for the same film

thickness irrespective of the Reynolds number, which varies

about one to two orders of magnitude. Furthermore, if the

Reynolds number were important for the generation of the

vortices, the critical film thickness for the generation of a

vortex would be a function of the Reynolds number and

would diminish for increasing Reynolds numbers. However,as shows Fig. 5, the critical film thickness increases with the

inclination angle and so does the Reynolds number. Even

more striking is the fact that the critical film thickness is

higher for larger absolute sizes, as shown in Fig. 6a, and

thus for higher Reynolds number. The Reynolds numbers for

20 mm bottom waves at the critical film thickness is 5

103 and 4.5102 for 9.8 and 45 inclination angle, re-

spectively. This is about two orders of magnitude higher than

the lowest Reynolds numbers at which we have observed

vortices over 5 mm bottom waves. Furthermore, inertia-

driven flow separation usually takes place at the lee side of a

flow. The vortices here, however, are generated in the very

trough of the bottom wave. We suppose that the reason for

the apparent independence of the vortices from the Reynolds

number lies in the fact that the velocities in the bottom

trough are even much smaller than at the free surface and

consequently as shown in Fig. 13 the mean free surface ve-locity is about the same as that of a film flowing over a flat

incline for the same volume flux. The maximum return ve-

FIG. 10. Vortex size as a function of the film thicknessaand peakpeak amplitude of the free surface b. Ina, the distance of the vortex core and of the

separatrix to the center of the trough are shown as open and solid squares, respectively. Squares indicate the first vortex and circles the second one. The thick

curve in b is an exponential fit to the data for thick films. Bottom wavelength: 20 mm, amplitude: 9 mm, inclination angle: 45.

FIG. 11. Film flow, as visualized by adding up series of tracer images, with a vortex in the trough of a bottom wave at 45 inclination angle aand withouta vortex at 80 inclination angle b. The main flow direction is along to mean inclination angle of the bottom contour. The film is bent backwards at the

steeply falling edge. Ina, the free surface is almost flat at the rising edge. bshows the trough region only. The film over the steeply falling edge is hardly

visualized since the free surface is almost parallel to the laser light. Bottom wavelength: 20 mm, amplitude: 9 mm, film thickness: 1.1 mm.



8/11

locity as depicted in Fig. 4 is about three to four orders

smaller than the mean surface velocity.

Instead of the Reynolds number, the critical film thick-

ness depends on the inclination angle and on the surface

tension as shown in Figs. 5 and 6. The dependence on the

inclination angle in Fig. 5 can be fitted properly with a co-

tangent function. This, on the other hand, corresponds to the

ratio of the force acting in the direction perpendicular to the

mean inclination to that acting in flow direction. It also cor-

responds to the ratio of the horizontal to the vertical dis-

tances between the crests of the bottom and thus restricts the

amplitude of the free surface. The flatter the free surface is,

the easier the vortices are generated, i.e., at zero inclination

the free surface has to be flat even for thin films. Surface

tension increases this effect as we have seen in Fig. 6. Low-

ering the Bond number, i.e., increasing the surface tension,

lowers the critical film thickness. Increasing the effect of

surface tension yields a higher local film thickness over the

trough of the bottom wave but also a flatter surface for the

respective critical film thickness. Thus, the curvature of the

surface has an impact on the critical film thickness of the

vortex. This is in accordance with the observation that the

critical film thickness decreases with the bottom waviness, as

shown in Fig. 8.

We speculate that the strong increase in the critical film

thickness for lowering the waviness may yield a minimum

waviness below which no vortices can be generated. Since

this would be the case for infinite film thickness, surface

tension is not supposed to play any role. One the other hand,

the region for which a critical waviness exists is restricted for

high waviness as shows particularly Fig. 10a. Thus, beyonda certain waviness, there exists always a vortex in the trough

of the bottom wave, no matter how low the volume flux may

be. As depicts Fig. 11a this seems to be due to the fact that

the local film thickness reaches a lower limit determined by

the difference in the vertical position of the rising edge and

the trough of the bottom undulation. Like the critical film

thickness for the generation of a vortex, the maximum wavi-

ness for a nonzero critical film thickness also depends on the

inclination angle as shows Fig. 11. The logarithmic fit shown

in Fig. 8 was chosen assuming a minimum waviness for the

possible generation of a vortex and a maximum waviness for

the existence of a critical film thickness. The fact that it fits

well to the data points underlines this conclusion.

Beyond the critical film thickness, the size of the vortex

is very sensitive to the film thickness as shown in Fig. 6a

and then converges apparently to an asymptotic maximum

size as depicted in Fig. 3a. The convergence takes place

when the surface is already almost flat and thus the stream-

lines close to the bottom hardly change anymore with the

film thickness. The asymptotic value is independent of the

Bond number as indicates Fig. 6a, which is clear since the

surface is flat for thick films. But it also seems to be inde-

pendent of the inclination angle. Comparing the results for

the waviness 0.2 in Fig. 6 to those for a waviness of 0.45 in

FIG. 12. Film flow, as visualized by adding up series of

tracer images, with a perturbed vortex in the trough of a

bottom wave. The main flow direction is along to mean

inclination angle of the bottom contour. A single vortex

at an inclination angle of 80 is shown in a;bandc

show a second vortex at 45 inclination angle during

different time intervals. Bottom wavelength: 20 mm,

amplitude: 9 mm, film thickness: 3.67 mm a, and

11.35 mm in b and c.

FIG. 13. Mean transport velocity at the free surface of a film over a wavy

bottom compared to that over a flat incline. The curve is an exponential

decay fit to the data. Bottom wavelength: 5 mm, amplitude: 1 mm, inclina-

tion angle: 9.9.



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Fig. 10 shows that the vortex size seems to be a function of

the waviness.

Although the presence of a vortex changes the boundary

condition for the overlying film flow downstream along the

separatrix from a no-slip condition to a slip condition, it does

not considerably increase the liquid transport as shown in

Fig. 13. For thin films the transport velocity is significantly

lower than that over a flat plane, although there already ex-

ists a vortex in the trough. This is apparently due to an in-

crease of the mean film thickness caused by the bottom un-

dulation as was shown in Ref. 5 for weakly wavy bottoms.

For thick films on the other hand, the bottom undulation is

not supposed to play an important role anymore for the mean

surface velocity. Between these two extremes, we could not

find clear-cut evidence for a possible enhancement within

experimental uncertainty.

In the bottom undulations of highest waviness we ob-

served a second vortex. It apparently shows the same quali-

tative behavior as the first one. As shown in Fig. 8, there

apparently also exists a minimum waviness for its genera-

tion. The critical film thickness also decreases with increas-ing waviness. It also appears from Fig. 10a that its size

tends to an asymptotic value for thick films.

B. Comparison to other studies of vortices at lowReynolds numbers

We have seen that Reynolds number effects are not im-

portant in our study. Therefore, we conclude that the ob-

served vortices are not driven by inertia as those studied in

Refs. 6 8 and start comparing our findings to Pozrikidis

numerical study of Stokes flow over sinusoidal inclined

planes.9 He considered film flow over bottom undulations

with a waviness of 0.01, 0.1, and 0.2 at inclination angles of

9 and 45. Furthermore, he studied the impact of the Bond

number. In our experiments, we covered the waviness rang-

ing from 0.13 to 0.5 at inclination angles ranging from 5 to

80. Both Pozrikidis numerical and our experimental study

coincide in the waviness 0.2 and the inclination angle of 45.

Although he mainly focuses on the free surface profile of the

film while we center our attention on the vortices, we found

good agreement between the two studies whenever compa-

rable. As in our experiments, Pozrikidis observed from his

numerical calculations that the free surface is a nearly sym-

metric sinusoidal wave for thicker films and it becomes al-

most horizontal over increasing slopes of the bottom for thin

films. A comparison of Pozrikidis calculations for the freesurface amplitude at infinite Bond number to our experimen-

tal data with Bo*4.5 shows good agreement and is repro-

duced in Fig. 14. Also the effect of the Bond number seems

to compare well qualitatively. Although the small Bond num-

bers are not the same in both studies, we observe as does

Pozrikidis that the surface amplitude does not tend to the

bottom amplitude for thin films but to much smaller values.

The fact that the Pozrikidis calculations for a waviness

of 0.1 could not reveal any vortex coincides with our obser-

vation of a strong increase of the critical film thickness by

lowering the waviness. For a waviness of 0.2, Pozrikidis did

not observe any vortex for thin films but for thick films.

Although he did not determine the critical film thickness, at

least the points he studied fit to ours, i.e., they show the same

qualitative behavior. Also the wall shear stress from which

he deduced the existence of a vortex is nearly symmetric

coinciding to our symmetric vortices for that parameters.

Pozrikidis stated that vortices occur beyond a critical wavi-

ness and that the critical film thickness vanishes for very

large wave amplitudes, such as we have observed it in our

experiments. Unfortunately, however, he did not embark on

this subject and did not give evidence for it.

Like the systems studied experimentally by Taneda10 we

found that the characteristic velocity of the second vortex is

several orders of magnitude smaller than that of the first one.

In general, one may classify the different Stokes flows with

vortices studied by Taneda10 into two groups: In one class,

vortices occur beyond a certain angle between the flow con-fining boundaries such as Moffatts eddies.13 There is no

length scale entering into this kind of problems and similar-

ity solutions are obtained. In the other group, the vortices are

generated below a certain length scale. Into this second class

fall the vortices between two spheres that are not in contact

of which the flow pattern has been calculated by Davis

et al.11 The film flow studied here belongs to this second

class with the waviness playing the role of a length scale

taking the film thickness fixed. Like, for instance, the case

described by Davis et al.,11 increasing the waviness leads to

a generation of a further vortex. However, different from

those cases studied by Taneda10 we have a free surface flow

and the film thickness enters as a further parameter. Althoughthe experimentally accessible range for the film thickness is

limited, it seems from our experiments that the maximum

number of vortices depends on the waviness and not on the

film thickness.

In their experiments on free film flow along vertical cor-

rugated surfaces, Zhao and Cerro14 have observed vortices

between periodic convex half cycles independent of the film

thickness. It seems to us that these are essentially those de-

scribed theoretically by Moffatt.13 Besides these, they stated

that they found vortices only at the highest Reynolds num-

bers and Capillary numbers for walls shaped as triangles or

as concave half cycles but did not give evidence of these

FIG. 14. Comparison of the free surface amplitude with numerical calcula-

tions for Stokes flow by Pozrikidis. The experimental data are given as solid

squares; the numerical values, taken from Ref. 6, Fig. 5b, are indicated by

open squares. Parameters: waviness: 0.2 mm, inclination angle: 45, Bo *:

4.44 experimental and infinity numerical.



10/11

observations. However, Malamataris and Bontozoglou have

calculated numerically the flow along concave half cycles

and found recirculation zones behind the corner at the lee

side of the half cycles for high flow rate and surface

tension.17 By varying the film thickness, Reynolds number,

and the Capillary number, they showed that the eddy can

disappear. They found as a common feature of these calcu-

lations that in all three cases the local film thickness dimin-

ished. Apart from this, however, we find certain differencesto our study: The vortices that are generated in the concave

half cycles seem to be a flow separation due to a corner flow

while there is nothing like this in our experiments. The vor-

tices in our study are in general symmetric in the trough of

the bottom while those vortices are at the lee side. Finally,

we could show that in our case the Reynolds number is not

important.

Negny et al.15 observed a bulge of the film over the flat-

test part of an undulated substrate and concluded in accor-

dance with numerical calculations18 that the swelling is

caused by an underlying vortex. Different from our vortices,

these ones occur at rather high Reynolds numbers and Negny

et al. suggest that viscous friction prevents the flow from

overcoming the pressure gradient in these regions. From the

fact that these vortices occur in the flattest region of the

corrugations it seems clear that these vortex are different

from ours.

V. CONCLUSIONS

We have presented an experimental study of vortices in

gravity driven films flowing down sinusoidal bottom profiles

of rather high waviness. The vortices were visualized em-

ploying a particle image velocimeter with fluorescent tracers.

They were observed in the troughs of the undulated bottom

profile at low Reynolds numbers down to the order of 10

5.We showed that the Reynolds number is not responsible for

their generation. From the experimental data, it seems that

there exists a minimum waviness below which these vortices

cannot be generated. Beyond this minimum waviness, the

vortices occur beyond a critical film thickness. The minimum

film thickness for their generation increases with the inclina-

tion angle and is lowered by surface tension. The critical film

thickness diminishes with increasing waviness until it

reaches zero. This waviness depends on the inclination angle.

Beyond this waviness, there exists always a vortex irrespec-

tively of the film thickness. Further increasing the waviness

results in the generation of a second vortex beyond a critical

film thickness. This second vortex occurs again in the trough

of the undulation and seems to have the same features as the

first one. The size of the vortices strongly increases beyond

the critical film thickness and tends asymptotically to a finite

value for thick films that is independent of the inclination

angle or surface tension. Finally, we could not observe a

significant increase of the transport velocity due to the vor-

tices.

ACKNOWLEDGMENTS

The authors acknowledge the support of G. Jena, A.Kammerer, A. Dornhofer, F. Meisel, and of our engineer C.

Lepski.

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vortices in film flow over strongly undulated bottom profiles at low reynolds numbers

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