In the present study, we performed experiments on the evolution of gravity-driven nonlinear traveling free surface waves over strongly undulated inclines. We focused on the impact of the excitation frequency and amplitude and the substrates’ shape and periodicity on the traveling wave. Thereby, we revealed phenomena concerning the amplitude evolution of convectively unstable waves. We can classify the wave evolution in three categories: (1) A normal exponential growth of the wave until it reaches a saturation amplitude. (2) An exponential growth of the wave and an abrupt collapse of the wave leading to a wave with a saturated amplitude, which is sensibly smaller than the maximal amplitude. (3) An alternating exponential growth and breaking of the wave. By using Fourier analysis, we investigated the waves in more detail. Furthermore, we report (a) a selection of excitation frequencies for the wave breaking, (b) a major impact of the steady state free surface, especially its mean or global curvature, (c) a bubble formation occurring at the wave breaking, (d) an overlap of the wave front, and (e) a formation of a jet during wave breaking.

1.
I.
Luca
,
K.
Hutter
,
Y. C.
Tai
, and
C. Y.
Kuo
, “
A hierarchy of avalanche models on arbitrary topography
,”
Acta Mech.
205
,
121
149
(
2009
).
2.
K.
Hutter
,
B.
Svendsen
, and
D.
Rickenmann
, “
Debris flow modeling: A review
,”
Continuum Mech. Thermodyn.
8
,
1
35
(
1994
).
3.
R.
Greve
and
H.
Blatter
,
Dynamics of Ice Sheets and Glaciers
(
Springer
,
Berlin, Heidelberg
,
2009
).
4.
A.
Kumar
,
D.
Karig
,
R.
Acharya
,
S.
Neethirajan
,
P. P.
Mukherjee
,
S.
Retterer
, and
M. J.
Doktycz
, “
Microscale confinement features can affect biofilm formation
,”
Microfluid. Nanofluid.
14
,
895
902
(
2013
).
5.
S. F.
Kistler
and
P. M.
Schweizer
,
Liquid Film Coating
(
Chapman and Hall
,
New York
,
1997
).
6.
S. J.
Weinstein
and
K. J.
Ruschak
, “
Coating flows
,”
Annu. Rev. Fluid Mech.
36
,
29
53
(
2004
).
7.
G.
Gugler
,
R.
Beer
, and
M.
Mauron
, “
Operative limits of curtain coating due to edges
,”
Chem. Eng. Process.: Process Intensif.
50
,
462
465
(
2011
).
8.
R. L.
Webb
,
Principles of Enhanced Heat Transfer
(
Wiley
,
New York
,
1994
).
9.
P.
Vlasogiannis
,
G.
Karagiannis
,
P.
Argyropoulos
, and
V.
Bontozoglou
, “
Air-water two-phase flow and heat transfer in a plate heat exchanger
,”
Int. J. Multiphase Flow
28
,
757
772
(
2002
).
10.
P.
Valluri
,
O. K.
Matar
,
G. F.
Hewitt
, and
M. A.
Mendes
, “
Thin film flow over structured packings at moderate Reynolds numbers
,”
Chem. Eng. Sci.
60
,
1965
1975
(
2005
).
11.
J. M.
de Santos
,
T. R.
Melli
, and
L. E.
Scriven
, “
Mechanics of gas-liquid flow in packed-bed contactors
,”
Annu. Rev. Fluid Mech.
23
,
233
260
(
1991
).
12.
P. L.
Kapitza
, “
Wave flow of thin layers of a viscous fluid
,”
Zh. Eksp. Teor. Fiz.
18
,
1
28
(
1948
).
13.
P. L.
Kapitza
and
S. P.
Kapitza
, “
Wave flow of thin layers of a viscous fluid
,”
Zh. Eksp. Teor. Fiz.
19
,
105
120
(
1949
).
14.
T. B.
Benjamin
, “
Wave formation in laminar flow down an inclined plane
,”
J. Fluid Mech.
2
,
554
574
(
1957
).
15.
C. S.
Yih
, “
Stability of liquid flow down an inclined plane
,”
Phys. Fluids
6
,
321
334
(
1963
).
16.
Y. Y.
Trifonov
, “
Viscous liquid film flows over a vertical corrugated surface and the film free surface stability
,”
Russ. J. Eng. Thermophys.
10
(
2
),
129
145
(
2000
).
17.
M.
Vlachogiannis
and
V.
Bontozoglou
, “
Experiments on laminar film flow along a periodic wall
,”
J. Fluid Mech.
457
,
133
156
(
2002
).
18.
A.
Wierschem
and
N.
Aksel
, “
Instability of a liquid film flowing down an inclined wavy plane
,”
Physica D
186
,
221
237
(
2003
).
19.
A.
Wierschem
,
M.
Scholle
, and
N.
Aksel
, “
Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel
,”
Exp. Fluids
33
,
429
442
(
2002
).
20.
A.
Wierschem
,
C.
Lepski
, and
N.
Aksel
, “
Effect of long undulated bottoms on thin gravity-driven films
,”
Acta Mech.
179
,
41
66
(
2005
).
21.
Y. Y.
Trifonov
, “
Stability of a viscous liquid film flowing down a periodic surface
,”
Int. J. Multiphase Flow
33
,
1186
1204
(
2007
).
22.
Y. Y.
Trifonov
, “
Stability and nonlinear wavy regimes in downward film flows on a corrugated surface
,”
J. App. Mech. Tech. Phys.
48
,
91
100
(
2007
).
23.
L. A.
Dávalos-Orozco
, “
Nonlinear instability of a thin film flowing down a smoothly deformed surface
,”
Phys. Fluids
19
,
074103
(
2007
).
24.
L. A.
Dávalos-Orozco
, “
Instabilities of thin films flowing down flat and smoothly deformed walls
,”
Microgravity Sci. Technol.
20
,
225
229
(
2008
).
25.
Y. Y.
Trifonov
, “
Stability and bifurcations of the wavy film flow down a vertical plate: The results of integral approaches and full-scale computations
,”
Fluid Dyn. Res.
44
,
031418
(
2012
).
26.
C.
Heining
and
N.
Aksel
, “
Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability
,”
Phys. Fluids
21
,
083605
(
2009
).
27.
S. J. D.
D’Alessio
,
J. P.
Pascal
, and
H. A.
Jasmine
, “
Instability in gravity-driven flow over uneven surfaces
,”
Phys. Fluids
21
,
062105
(
2009
).
28.
A.
Wierschem
,
M.
Scholle
, and
N.
Aksel
, “
Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers
,”
Phys. Fluids
15
,
426
435
(
2003
).
29.
C.
Heining
and
N.
Aksel
, “
Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline
,”
Int. J. Multiphase Flow
36
,
847
857
(
2010
).
30.
D.
Tseluiko
,
M. G.
Blyth
, and
D. T.
Papageorgiou
, “
Stability of film flow over inclined topography based on a long-wave nonlinear model
,”
J. Fluid Mech.
729
,
638
671
(
2013
).
31.
T.
Pollak
and
N.
Aksel
, “
Crucial flow stabilization and multiple instability branches of gravity-driven films over topography
,”
Phys. Fluids
25
,
024103
(
2013
).
32.
Y. Y.
Trifonov
, “
Stability of a film flowing down an inclined corrugated plate: The direct Navier-Stokes computations and Floquet theory
,”
Phys. Fluids
26
,
114101
(
2014
).
33.
M.
Schörner
,
D.
Reck
, and
N.
Aksel
, “
Stability phenomena far beyond the Nusselt flow—Revealed by experimental asymptotics
,”
Phys. Fluids
28
,
022102
(
2016
).
34.
M.
Schörner
,
D.
Reck
,
N.
Aksel
, and
Y. Y.
Trifonov
, “
Switching between different types of stability isles in films over topographies
,”
Acta Mech.
229
,
423
436
(
2018
).
35.
Z.
Cao
,
M.
Vlachogiannis
, and
V.
Bontozoglou
, “
Experimental evidence for a short-wave global mode in film flow along periodic corrugations
,”
J. Fluid Mech.
718
,
304
320
(
2013
).
36.
M.
Schörner
,
D.
Reck
, and
N.
Aksel
, “
Does the topography’s specific shape matter in general for the stability of film flows?
,”
Phys. Fluids
27
,
042103
(
2015
).
37.
N.
Aksel
and
M.
Schörner
, “
Films over topography: From creeping flow to linear stability, theory, and experiments, a review
,”
Acta Mech.
229
,
1453
1482
(
2018
).
38.
K.
Argyriadi
,
M.
Vlachogiannis
, and
V.
Bontozoglou
, “
Experimental study of inclined film flow along periodic corrugations: The effect of wall steepness
,”
Phys. Fluids
18
,
012102
(
2006
).
39.
M. A.
Mendez
,
B.
Scheid
, and
J.-M.
Buchlin
, “
Low Kapitza falling liquid films
,”
Chem. Eng. Sci.
170
,
122
138
(
2017
).
40.
D.
Reck
and
N.
Aksel
, “
Experimental study on the evolution of traveling waves over an undulated incline
,”
Phys. Fluids
25
,
102101
(
2013
).
41.
Y. Y.
Trifonov
, “
Nonlinear waves on a liquid film falling down an inclined corrugated surface
,”
Phys. Fluids
29
,
054104
(
2017
).
42.
M.
Dauth
,
M.
Schörner
, and
N.
Aksel
, “
What makes the free surface waves over topographies convex or concave? A study with Fourier analysis and particle tracking
,”
Phys. Fluids
29
,
092108
(
2017
).
43.
J. J.
Stoker
,
Water Waves: The Mathematical Theory with Applications
(
John Wiley & Sons, Inc.
,
New York
,
1992
).
44.
B.
Le Méhauté
,
An Introduction to Hydrodynamics and Water Waves
(
Springer Verlag
,
New York
,
1976
).
45.
M.
Schörner
and
N.
Aksel
, “
The stability cycle—A universal pathway for the stability of films over topography
,”
Phys. Fluids
30
,
012105
(
2018
).
46.
J. H.
Spurk
and
N.
Aksel
,
Fluid Mechanics
, 2nd ed. (
Springer
,
Berlin
,
2008
).
47.
A.
Georgantaki
,
J.
Vatteville
,
M.
Vlachogiannis
, and
V.
Bontozoglou
, “
Measurements of liquid film flow as a function of fluid properties and channel width: Evidence for surface-tension-induced long-range transverse coherence
,”
Phys. Rev. E
84
,
026325
(
2011
).
48.
D.
Reck
and
N.
Aksel
, “
Recirculation areas underneath solitary waves on gravity-driven film flows
,”
Phys. Fluids
27
,
112107
(
2015
).
49.
S.
Varchanis
,
Y.
Dimakopoulos
, and
J.
Tsamopoulos
, “
Steady film flow over a substrate with rectangular trenches forming air inclusions
,”
Phys. Rev. Fluids
2
,
124001
(
2017
).

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