International Journal of Heat and Fluid Flow 32 (2011) 698–707

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate / i jhf f

Wavy wall influence on the hydrodynamic instability of a liquid film flowingalong an inclined plane

Nguyen Chi Cong a, Frédéric Plourde b,⇑a Department of Aeronautical Engineering, HoChiMinh City, University of Technology, Bd C5, 268 Ly Thuong Kiet, District 10, HoChiMinh City, Viet Namb ENSMA, Département FTC – Thermique, UPR 3346, 1, Avenue Clément Ader, BP40109, 86961 Chasseneuil Futuroscope Cedex, France

a r t i c l e i n f o

Article history:Received 24 January 2010Received in revised form 30 January 2011Accepted 14 February 2011Available online 5 March 2011

Keywords:FilmInstabilityEnhancementTransferComplex surface

0142-727X/$ - see front matter � 2011 Elsevier Inc. Adoi:10.1016/j.ijheatfluidflow.2011.02.008

⇑ Corresponding author. Tel.: +33 549498119; fax:E-mail address: plourde@let.ensma.fr (F. Plourde).

a b s t r a c t

The film dynamic of a thin liquid along an inclined and wavy wall was numerically depicted in aweighted-residual integral boundary layer equation. A qualitative and quantitative analysis was initiallycarried out and accurate comparisons were obtained from experimental data on film instability along aflat and inclined as well as a wavy wall. To pinpoint the effect of waviness on film instability, 20 wavywall periods in the computational CFD domain were considered. Several waviness parameters were stud-ied and shown to have taken on a major role in the film instability process. Finally, a wide range of mainwall inclination angles was taken into account, and consequent numerical data permitted identificationof a threshold angle value. For wall angles higher than the threshold angle, the film behaved as though nocorrugations were present. For lower angles, the film was repeatedly altered during the acceleration anddeceleration phases.

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1. Introduction

The dynamics of thin films falling down an inclined plane havebeen studied ever since the pioneering experiments of Kapitza andKaptiza (1949), and this flow is now considered typical of hydrody-namic instability. The large volume of literature is due to the factthat film convection exhibits multiple fundamental physical mech-anisms such as open-flow hydrodynamic instabilities driven by lin-ear and non-linear phases, external noise influence and pulse-to-pulse interaction. Analyses have been conducted either experimen-tally or numerically. In the framework of modeling, only a fewworks have been dedicated to time-dependent Navier–Stokesequations; Malamataris et al. (2002) studied solitary wave interac-tion and found that coalescence occurs only when the waves differappreciably in height. Many major findings are due to the large-scale development of film modeling over recent decades, particu-larly since Benney (1966) used long-wave expansion to obtain asingle equation for g film thickness. To improve modeling predic-tions, integral boundary-layer approximation has combined aself-similar parabolic velocity profile and long waves on the inter-face. Shkadov (1967, 1977) proposed an approach consisting of twoequations, one for g film thickness and the other for q local flowrate. This modeling was found to describe non-linear waves farfrom the critical Reynolds number. Solitary wave solutions havelikewise been accurately depicted by integral boundary layer. For

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+33 549498130.

readers interested in further details, see the reviews of Chang(1994) and Chang et al. (1993, 1996). In addition to modeling,Ruyer-Quil and Manneville (1998, 2000) and Scheid et al. (2006)proposed a weighted-residual approach explaining the accountingfor inertia effects due to changing the evolution of velocity profiles.The models they proposed allow for consideration of velocity pro-files in film thickness as polynomial functions up to degree 6. Mod-el development is linked to the e film parameter, and a two-equation model is consistent with e i.e. at first order (Ruyer-Quiland Manneville, 1998) and at second order (Ruyer-Quil and Man-neville, 2000). Even at first order, a weighted-residual integralboundary layer model accurately predicts instability leading to sol-itary wave train with numerical data, and predictions are fairlyconsistent with the experimental data of Liu and Gollub (1994).One should nonetheless admit that wave amplitudes have not beencorrectly estimated, while the second order model was found to bein excellent agreement with DNS results.

While film falling down an inclined plane has repeatedly beenstudied in basic research, additional complexities such as furtherthermal processes (evaporation and condensation) or corrugatedwalls have many interesting engineering applications. Oron andHeining (2008) emphasized weighted-residual approach modelsdepicting the non-linear dynamics of thin liquid films falling onan undulating vertical wall. To the author’s knowledge, this wasthe first application of weighted-residual models to wavy channels.Oron and Heining (2008) stressed the film unsteadiness whenanalogies between vertical wall with or without waviness wereexhibited. Trifonov (1999) likewise investigated films over periodic

Nomenclature

ak(xs, t) auto-similar coefficients of velocity profileA amplitude of the wavy substratec phase velocityfk�1ð�zÞ polynomial functions for velocity profile~g gravityq(xs, t) local flow rateqN flow rate estimated in the Nusselt assumptionOxGzG global systemOxWzW local orthogonal coordinate

L0 ¼ m2=3f g�1=3 characteristic viscous length

T0 ¼ m1=3f g�2=3 characteristic viscous time

U0 ¼ L0=T0 ¼ ðmf gÞ1=3 characteristic viscous velocityu,w velocity components of local velocity in x and z

directionsuN Nusselt mean velocityKa ¼ r

lf U0Kapitza number

Re ¼ qf uNgN

lfReynolds number

We ¼ rqf gN u2

NWeber number

S temporal mean surface areaxw longitudinal coordinate in the main direction of wavy

wall

Greek symbolsa ¼ SC

Sper� 1

� �kgN

dimensionless surface ratio

b global inclination angle of the wavy walle film parameter i.e. small parameter defined for con-

struction modelk wavy wall characteristic lengthh local inclined angle of solid wall from the gravity ~gqf density of liquidqg gas densityr surface tensionmf kinematic viscosity of liquidmg kinematic viscosity of gasg(xs, t) local height of liquid filmgN Nusselt liquid film height

Operatorð:Þ dimensionless variable normalized by characteristic vis-

cous parameter(.)C parameter evaluated at interface(.)per parameter averaged along the wavy period(.)s parameter evaluated at wall[.] parameter variation across the interface from liquid to

gas

N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707 699

surface and underlined linear and non-linear stability with regardto spatial disturbances of various wave lengths. Häcker and Uecker(2009) recently studied liquid film instability with a second-orderweighted integral boundary equation model and shed light on anew phenomenon i.e. only along an inclined wavy wall does shortwave instability arise; according to the authors, such a phenome-non requires validation from experimental data and/or full Na-vier–Stokes simulations. Like Oron and Heining (2008), Häckerand Uecker (2009) conducted simulations along a single period,but as far as we know, spatial evolution has yet to be studied in de-tail from a numerical standpoint.

With these considerations in mind, a basic set of equations andboundary conditions governing the problem was initially pro-posed. In order to adapt to more complex geometries, the set ofequations was put forward in a coordinate system befitting thewavy wall. Section 3 of our paper is devoted to thin film develop-ment on wavy walls; analysis is derived from direct comparisons offilm development along the wavy inclined wall and the flat in-clined wall respectively and tends to underscore the important roleof waviness in the film development. We have limited our analysisto the overall framework of long-wave assumption i.e. wave heightremains small in comparison with wave length. Several inclinationangles have been studied, while the main film figures (Reynoldsand Weber numbers) were kept constant and equal to those usedin the experiments of Liu and Gollub (1994). In order to shed lighton the role of the wavy geometry and the film dynamics, dedicatednumerical investigations were carried out.

Fig. 1. Sketch of the geometry and the coordinate system.

2. Governing equations and model

Numerical analysis is based on a weighted-residual model facil-itating analysis of instability in thin liquid film flowing on an in-clined and wavy wall. The liquid film is assumed to behave like aNewtonian fluid and the simulation takes into consideration atwo-dimensional curvilinear wall. No heat transfer occurs and nointeraction with the gas phase is considered. Fig. 1 shows the wallgeometry and the main geometrical parameters defining the Oxwzw

global system. The b angle corresponds to the main angle inclina-tion of the wavy wall. A OxSzS local orthogonal coordinate has also

700 N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707

been introduced, the h angle representing the local inclined angleof solid wall with regard to ~g gravity (Fig. 1b).

For spatial discretisation, we have not followed Oron and Hein-ing’s formulation in which the wall is defined as a function of thelongitudinal coordinate. On the contrary, we have defined the wallsurface by a list of points, thereby rendering the model more flex-ible with regard to more complex geometries such as heat exchan-ger geometries. In the following, the wavy wall corresponds to atrigonometric function defined as:

zw ¼ A cosð2pxw=kÞ ð1Þ

Following Ruyer-Quil and Manneville (1998), the set of equationshas been written in a dimensionless form scaled by viscous length,viscous time and viscous velocity. Continuity equation and momen-tum equations were developed in a curvilinear coordinate systemand written as:

Continuity equation

@�u@�xsþ @

�w@�z� �w

dhd�xsþ �z

@�u@�xs

dhd�xs� �z �w

dhd�xs

dhd�xs¼ 0 ð2Þ

Momentum conservation equation in the xs direction

@�u@�tþ �u

@�u@�xsþ �w

@�u@�z� �u �w

dhd�xsþ �u�z

@�u@�xs

dhd�xs� �u �w�z

dhd�xs

dhd�xs

¼ � @�p@�xsþ cos hþ @2�u

@�x2sþ @

2�u@�z2

!� �w

d2hd�x2

s� @

�u@�z

dhd�xs� 2

@ �w@�xs

dhd�xs

� �udhd�xs

� �2

� �z@�p@�xs

dhd�xsþ @

@�xs

@�u@�xs

dhd�xs

� ��zþ �z2 dh

d�xs

� �

� 2 �wd2hd�x2

s

�zþ �z2 dhd�xs

� �� �z

@�u@�z

dhd�xs

dhd�xs

� 2@ �w@�xs

dhd�xs

2�zdhd�xsþ �z2 dh

d�xs

dhd�xs

� �

� �udhd�xs

� �2

2�zdhd�xsþ �z2 dh

d�xs

dhd�xs

� �ð3Þ

Momentum conservation equation in the direction normal to the wall

@ �w@�tþ �u

@ �w@�xsþ �w

@ �w@�zþ �u2 dh

d�xsþ �u�z

@ �w@�xs

dhd�xsþ �u2�z

dhd�xs

dhd�xs

¼ � @�p@�z� sin hþ @2 �w

@�x2sþ @

2 �w@�z2

!þ �u

d2hd�x2

s� @

�w@�z

dhd�xsþ 2

@�u@�xs

dhd�xs

� �wdhd�xs

dhd�xsþ @

@�xs

@ �w@�xs

dhd�xs

� ��zþ �z2 dh

d�xs

� �

þ 2�ud2hd�x2

s

�zþ �z2 dhd�xs

� �� �z

@ �w@�z

dhd�xs

dhd�xs

þ 2@�u@�xs

dhd�xs

2�zdhd�xsþ �z2 dh

d�xs

dhd�xs

� �

� �wdhd�xs

� �2

2�zdhd�xsþ �z2 dh

d�xs

dhd�xs

� �ð4Þ

The boundary conditions (at wall and at liquid interface) were like-wise written in the curvilinear coordinate system:

Non-slip wall

�us ¼ 0�ws ¼ 0

ð5Þ

Balance of stress at the film interface C

�2�l @�u@�xC

� �@�g@�xCþ �l @�u

@�zCþ @ �w@�xC

� �� �1� @�g

@�xC

� �2 !

þ 2�l @ �w@�zC

� �@�g@�xCþ 2�l �w

dhd�xC

� �@�g@�xCþ �l�u

dhd�xC

� �1� @�g

@�xC

� �2 !

¼ 0

ð6Þ

½�p� 1þ @�g@�xC

� �2 !

¼ 2 �l @�u@�xC

� �@�g@�xC

� �2

� �l @�u@�zCþ @ �w@�xC

� �� �@�g@�xC

þ �l @ �w@�zC

� �� �l �w

dhd�xC

� �@�g@�xC

� �2

� �l�udh

d�xC

� �@�g@�xC

!

� Ka@h@�sþ @hC

@�s

� �1þ @�g

@�xC

� �2 !

Kinematic condition at interface C

@�g@�tþ @

@�xs

Z �g

0

�ud�zþ dhd�xs

@

@�xs

Z �g

0

�u�zd�z� dhd�xs

Z �g

0

�w 1þ �zdhd�xs

� �d�z

¼ 0 ð8Þ

Only the main outlines of the weighted-residual method are re-called hereinafter; readers interested in the model formulation areinvited to refer to the pioneering works of Ruyer-Quil and Manne-ville (1998, 2000). From the weighted-residual formulation, a mod-el apt to deal with complex geometries such as wavy walls hasbeen developed. One of the main aims is to develop key variablesin polynomial series of e such as:

�u ¼ �u0 þ e�u1 þ e2�u2 þ � � ��w ¼ �w0 þ e�w1 þ e2 �w2 þ � � ��p ¼ �p0 þ e�p1 þ e2�p2 þ � � �

ð9Þ

Our model extension was limited to the first order in e includingterms at order zero, and all terms involving e at second order orhigher were assumed to be negligible. As a consequence, the equa-tions have been written as:

@�u0

@�xsþ @

�w1

@�z¼ 0 ð10Þ

e@�u0

@�tþ e�u0

@�u0

@�xsþ e�w1

@�u0

@�z¼ �e

@�p0

@�xsþ cos hþ @

2ð�u0 þ e�u1Þ@�z2

� e@�u0

@�zdhd�xs

ð11Þ

in which the gradient of pressure can be written:

@�p0

@�xs¼ @

�pg

@�xsþ @

@�xsð�g sin hÞ � �z cos h

dhd�xs

� Ka ed2hd�x2

sþ e2 @�g

@�xs

dhd�xs

dhd�xsþ 2�g

dhd�xs

d2hd�x2

sþ @

3 �g@�x3

s

! !ð12Þ

In this set of equations, we assumed that the Kapitza number isstrong enough; thus a e first order mode is accurate enough to de-pict liquid film development. According to Oron and Heining(2008), such a model overestimates wave amplitudes and ripplesbut remains accurate in the case of large Kapitza numbers.

The set of boundary conditions may also be expressed at firstorder in e, and Eqs. (5)–(7) become:

�us0 þ e�us1 ¼ 0�ws1 ¼ 0

ð13Þ

N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707 701

where balance of the stress at interface is simplified following:

@ð�uf 0 þ e�uf 1Þ@�z

þ e�uf 0dhd�xs¼

lg

lf

@�ug

@�zð14Þ

The longitudinal velocity profile in the liquid film was assumedto follow the shape proposed hereinafter:

�uð�x;�z;�tÞ ¼ �u0ð�x;�z;�tÞ þ e�u1ð�x;�z;�tÞ

¼ a0ð�x;�tÞð�z=�gÞ þX5

k¼1

akð�x;�tÞfk�1ð�z=�gÞ ð15Þ

in which ak ¼ akðxs; tÞ were assumed to vary in space slowly and

fk�1ð�zÞ ¼ �z�g

� �k� k

kþ1�z�g

� �kþ1were polynomials in ½0; �g� forming differ-

entiable functions �u in time and space.Following the longitudinal fluid velocity profile, the normal

fluid velocity component was determined from the continuityequation. The e profile expansion for all variables was introducedin the boundary layer equations, and all terms at e2 or higher wereneglected. We readily obtained a polynomial of degree 4 and itscoefficients were identified from a polynomial identification yield-ing a 5 equation system of 5 unknowns (a1, . . ., a5). When the ai

coefficients were estimated, the expression of the �qðxs; tÞ local flowrate was straightforward. The system of equations may be ex-pressed as:

@�g@�tþ @�q@�xsþ e�g

@�q0

@�xs

dhd�xs¼ 0

e@�q@�tþ e

177

�q�g@�q@�xs� e

97

�q2

�g2

@�g@�xs

¼ 56

�g cos h� e56

�g sin h@�g@�xsþ 5

6e3Ka�g

@3 �g@�x3

s

� 52

�q�g2 � e

516

�g2 cos hdhd�xs� e

4516

�q�g

dhd�xs� e2 23

16�q0@�q0

@�xs

dhd�xs

þ 56

Ka e2 �gd2hd�x2

sþ e3 �g

@�g@�xs

dhd�xs

dhd�xsþ 2�g

dhd�xs

d2hd�x2

s

! !

ð16Þ

Even though the geometry may be treated in a complex manner,the set of equations obtained above offers similarities to the set ofRuyer-Quil and Manneville (2000); difference arises mainly fromthe way equations are discretized.

From the set of equations above, we may identify the effect le-vel of each term of film dynamic evolution by the order in e of theircoefficients. Mean film velocity and thickness were expressed as�q � U�g and �q0 ¼ U�gþ OðeÞ. The set of Eq. (16) became:

@�g@�tþ @U�g@�xsþ �g

@U�g@�xs

dhd�xs¼ 0

@U�g@�tþ 17

14@UU�g@�xs

� 114

UU@�g@�xs

¼ 56

�g cos h� 56

�g sin h@�g@�xsþ 5

6Ka�g

@3 �g@�x3

s� 5

2U�g

� 516

�g2 cos hdhd�xs� 45

16U

dhd�xs

þ 56

Ka �gd2hd�x2

sþ �g

@�g@�xs

dhd�xs

dhd�xsþ 2�g

dhd�xs

d2hd�x2

s

! !

ð17Þ

The system (17) was solved by finite difference method basedon a spatially uniform staggered grid of characteristic size Dxs.To underscore wave evolution, a second order TVD-Upwindscheme was introduced for the convective terms with slope limit-ers applied on the gradient of variables. The other spatial termswere discretized by means of a second order central scheme. Thetemporal derivation was taken explicitly into account via a first or-

der Euler, a second or even a third order Runge–Kutta scheme.From our own experience, a staggered grid differentiating the filmvelocity nodes from the film thickness nodes enhances stability ofthe scheme. As a result, we have considered the equations with anexplicit scheme, but time increments were of the order of magni-tude of about 10�3 – 10�4 while similar time steps were likewiseprovided from an implicit scheme (Oron and Heining, 2008).

All the simulations presented hereinafter were computed fromthe present model and variable initialisation was carried out withthe Nusselt solution. At inlet, perturbation was applied either at agiven frequency or at a large frequency band such as inlet filmheight and mass flow rate:

�gð0; tÞ ¼ �gNð1þ e sinð2pftÞÞ�qð0; tÞ ¼ �qNð1þ e sinð2pftÞÞ3

ð18Þ

To avoid numerical instability at the outlet, an absorption zonewas created through application of a smooth function (such as ahyperbolic tangent function) at the right-hand side of the set ofEq. (17). The smooth function applied at outlet boundary conditionis defined as:

Cabsðx� x0Þ ¼12

1� e2ðx�x0Þ=dabs � 1e2ðx�x0Þ=dabs þ 1

� �ð19Þ

where Cabs is the absorption coefficient applied to the source termsof (17). The latter is applied on Labs = 6dabs absorbing length at outlet(in the presented numerical data, the absorbing length Labs is equalto 400L0) and x0 corresponds to the mid-location of the absorptionzone considered; along the Labs length, Cabs varies smoothly fromunity to 0. Such a procedure renders possible to consider zero gra-dients on the g(xs, t) film height and its velocity components toclose the set of equations at outlet.

3. Unstable identification

3.1. Flat and inclined wall

One of the first and most important steps in numerical investi-gation is validation. With this in mind, we refer to the experimen-tal configuration studied by Liu and Gollub (1994). Even if thegeometry does not correspond to a wavy wall, their experimentaldata contain exhaustive information on solitary wave evolution.Numerical investigations were carried out for a similar configura-tion i.e. for an b = 6.4� inclined wall and similar Weber and Rey-nolds numbers (Re = 19.33, We = 5.43). Due to a difference indefinition, the non-dimensional figures did not coincide. From anumerical standpoint, a 1% height amplitude noise was introducedat a specified frequency. Fig. 2 presents the spatial wave profile,normalized by Nusselt thickness for four different forced frequen-cies. For fc = 7.0 Hz, the wave profile becomes nearly sinusoidal.When the frequency is smaller, i.e. fc = 4.5 Hz, the wave is no longercharacterized by a sinusoidal shape while interactions with sub-sidiary waves appear. They are amplified with fc = 3.0 Hz wherewave structure is characterized mainly by a primary wave frontwhile secondary waves strongly interact with solitary ones. Thisdistribution is clearly observed for fc = 1.5 Hz. These types ofbehavior fully agree with the experimental observations providedby Liu and Gollub (1994).

At this stage, only a qualitative validation process has beenachieved. In order to quantitatively validate the model, the charac-teristic velocities in the linear and non-linear film instability re-gime were estimated and compared to the measurements. Fig. 3,for a film with Re = 19.33, We = 5.43 at an b = 6.4� inclined wall,shows velocities in the linear and non-linear regime as a functionof forcing frequency, and the numerical data fairly fit the experi-mental data. As stated in the introduction, solitary wave amplitude

xS/ηN

0 200 400 600 800 1000 12000.6

0.8

1

1.2

1.4

1.6 b) fC = 4.5Hz

0 200 400 600 800 1000 12000.6

0.8

1

1.2

1.4

1.6 c) fC = 3.0Hz

0 200 400 600 800 1000 12000.6

0.8

1

1.2

1.4

1.6d) fC = 1.5Hz

xS/ηN

η/η N

η/η N

η/η N

η/η N

0 200 400 600 800 1000 12000.6

0.8

1

1.2

1.4

1.6

C = 7.0Hz

xS/ηN

xS/ηN

xS/ηN

xS/ηN

Fig. 2. Variation of the normalized film height with regard to the downstreamnormalized distance at Re = 19.33 and We = 5.43 along a 6.4� inclined wall andforcing frequencies of (a) 7.0 Hz, (b) 4.5 Hz, (c) 3.0 Hz and (d) 1.5 Hz.

Linear velocity

Nonlinear velocity, Exp. dataLinear velocity, Exp. dataNonlinear velocity

frequency (Hz)

phas

evel

ocity

, c/u

Γ

0 2 4 6 8 10 121.3

1.5

1.7

1.9

2.1

2.3

Fig. 3. Linear and non-linear velocity with regard to the forcing frequencies andcomparison with experimental data extracted from Liu and Gollub (1994) along a6.4� inclined wall and Re = 19.33 and We = 5.43.

(a)

2.4

2.7

3.0

3.3

3.6

3.9

4.2

ηη ηη[m

m]

Exp. data

1st order model

150 200 250 300 350 400 450

xw [mm]

2

3

4

5

6

7

Exp. data

1st order model

(b)

ηη ηη[m

m]

150 200 250 300 350 400 450

xw [mm]

Fig. 4. Film height variation along a period inclined wavy wall with A/k = 5.0% andcomparison with experimental data extracted from Wierschem et al. (2005) at (a)Re = 0.0285 and We = 59.71 and b = 28� and (b) Re = 0.0285 and We = 68.59 andb = 18.05�.

702 N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707

is not thereby adequately predicted and, according to Ruyer-Quilet al. (1998), only higher order models can reasonably be expectedto quantitatively describe wave amplitude. In the framework ofwavy walls, Wierschem et al. (2005) conducted experimentalinvestigations on liquid film along an undulated wavy wall; theymeasured the local film thickness for two main inclination anglesat 28� and 18.05� and the A/k wavy wall aspect ratio was equalto approximately 5%. Wierschem et al.’s configuration was numer-ically simulated; Fig. 4 presents local film thickness along a wavy

wall period for both experimental and numerical data; the numer-ical results fairly fit the experimental data for the two configura-tions studied, even as regards the 18.05� angle for which thenumerical simulations conducted by Wierschem et al. (2005) failto predict film height evolution. To complete the validation pro-cess, we computed the cases studied by Oron and Heining (2008)and results (not shown) were found to be similar to those pub-lished by the two authors. The proposed numerical scheme andmodel may consequently be considered as validated, and we willuse it to investigate the influence of wavy walls on film instabilitydevelopment.

3.2. Wavy wall

In order to depict the effect of wall waviness, the film instabilityconditions experimentally depicted by Liu and Gollub (1994) wereconsidered, and numerical simulations under similar dynamic con-ditions were conducted along several A/k wavy wall factors as highas 4%; the A/k ratio was obtained with a fixed k wavy period equalto 0.213 m (A = 0.0084 m). The main inclination angle was setequal to b = 6.4�, Weber and Reynolds numbers were similar tothose studied in the flat case (Re = 19.33, We = 5.43). A solitarywave development was considered by introducing additional noiseat a frequency equal to 1.5 Hz. So much said, direct comparison be-tween the instantaneous heights for various A/k is not straightfor-ward because the mean flow field evolves as well. For example,Fig. 5 displays numerical steadiness without forced perturbationflow field solution for A/k = 0.5, 1 and 2%. The spatial profiles werenormalized by Nusselt solution to facilitate comparison with thedynamic solution on flat plane presented in Fig. 2. As was to be ex-pected, non-dimensional film velocity and height are no longer

η/ηNu/uN

λ = 0.5%

b) A/λ = 1.0%

c) A/λ = 2.0%

η/η N

,u/u

N

0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

xS/ηN

xS/ηN

xS/ηN

η/η N

,u/u

N

0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

η/η N

,u/u

N

0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

Fig. 5. Film height and velocity film variation along an 6.4� inclined wavy wall forRe = 19.33 and We = 5.43 and (a) A/k = 0.5%, (b) A/k = 1.05% and (c) A/k = 2.0%.

N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707 703

constant and evolve spatially. The higher the A/k ratio, the moreaverage distribution differs from Nusselt distribution. For A/k = 0.5%, a relatively small wavy wall directly interacts with aver-age distribution; in a wavy period, film velocity is first increasedby higher local inclination of the wall; once the film is localizedaway from the trough, local inclination decreases, thereby under-scoring film height in the region. The variations in film heightand film velocity are out of phase, while film characteristics arespatially modulated along the wavy wall. For higher A/k ratios(1% and 2%), the observed modulations are amplified by a strongerlocal variation of the inclination angle. In what way may a solitarywave be affected? With this question in mind, a forced perturba-tion was introduced at a constant frequency fc = 1.5 Hz. Taking intoaccount the findings of Liu and Gollub, forced frequency perturba-tion highlights the development of a solitary wave with subsidiarywaves located upstream from the latter when the film evolvesalong a b = 6.4� inclined and flat wall. In order to compare filmheight and film velocity, we studied height and velocity variationswith regard to their average distribution. Normalized g0 fluctuatingheights are plotted in Fig. 6 for the flat wall and the three A/k wavywalls studied. For A/k = 0.5%, the instantaneous fluctuating filmheight profile wholly fits with the profile obtained in the referencecase. Despite the modulation in film characteristics, there exist no

η'/η

N

Inclined plan - A/λ = 0.5%A/λ = 1.0%A/λ = 2.0%

xS/η0 200 400 600 1000 1200 1400 1600800

-0.5

-0.25

0

0.25

0.5

0.75

Fig. 6. Film height fluctuations with regard to the downstream normalized distanceat Re = 19.33 and We = 5.43 along an inclined wavy channel with (a) flat or A/k = 0.5%, (b) A/k = 1.05% and (c) A/k = 2.0%.

interactions in solitary wave development for A/k = 0.5%. This is nolonger the case for higher A/k ratios. For A/k = 1% the g0 fluctuatingheight is altered compared to the height observed along an in-clined wall; a solitary wave still occurs but maximum peak ampli-tude is reduced and spatially delayed compared to the inclined flatwall. The subsidiary waves driven by surface tension and inertiabalance are significantly altered by the successive increase and de-crease of local gravity and such a trend is amplified with a higherA/k ratio. For A/k = 2%, whole unstable mechanisms are modified,and fluctuations are no longer characterized by emergence of a sol-itary wave. Waviness of the wall would appear strong enough todisturb not only subsidiary waves but the whole film dynamic.The continuous acceleration and deceleration profoundly interactin the force balance – surface tension, inertia and gravity forces –and film height is consequently driven by incoherent fluctuations,at least as regards the characteristic parameter range.

Liquid film instability behavior falling down an inclined wavywall has been studied for a single and relatively small inclination(b = 6.4�) and for a fixed set of film dynamic parameters; to provideexhaustive parametric analysis of all the sensitive variables relatedto film dynamic behavior, extensive parametric analysis is re-quired. Such an analysis is however out of the scope of our objec-tives, since we simply wish to identify the main development of afilm falling down an inclined and wavy wall.

With this in mind, the effect of the b angle was extensivelystudied with regard to a given set of dynamic parameters, whichwere fixed equal to the ones studied by Liu and Gollub (1994)while sensitivity of the wavy factor and the Reynolds number werelikewise studied. The Nusselt parameters are directly dependent onthe inclination angle, while characteristic instability is not fixedwith regard to a forced perturbation. For all the configurationsinvestigated hereafter, a white noise was applied.

Film instability behavior obviously depends on the inclinationangle and dynamic conditions for a given wavy wall. In order tobetter understand film change flowing on a wavy wall at a giveninclination angle, we have followed Valluri et al. (2004), whoquantitatively studied free surface dynamic change by computingthe ratio of the free surface area to the solid surface area. The anon-dimensional surface ratio is then determined by taking intoaccount the relative difference between the free surface area andthe solid surface area.

To underscore the role of waviness geometry in film develop-ment, the a surface ratio was scaled by factor gN/k and computedfor fixed dynamic conditions over a wide range of b inclination an-gles. By calculating the a coefficient with regard to both a wavyand a flat wall, it is easy to perceive the role of waviness in filmdevelopment. Fig. 7a shows the a surface ratio change for A/k = 2and 4%; the figures plotted correspond to time and spatial averageobtained along the nineteenth period; only the last period (thetwentieth) could not be considered for such a balance becausesmooth functions were applied for reasons of numerical stability.Along a flat and inclined wall, film stability changes significantlywith regard to the inclination angle; under the dynamic conditionsstudied, a levels ranged from 0.25 to approximately 0.60; the high-er the b the greater the a surface ratio; the most interesting ap-proach is to follow the a change along a wavy substrate and todirectly compare it with the change occurring along a flat wall.For A/k = 2%, a coefficients are obviously higher than those ob-tained under flat wall conditions while for b P 20� the a surface ra-tio is strictly equal to the ratio reached for an inclined flat wall. Forlow angles (b < 20�), the b inclination angle is a highly significantparameter; a ratio first increases up to approximately 0.50 atb = 10� and is reduced up to a a minimum level close to 0.30; fromthis minimum value, an opposed trend occurs i.e. the a ratio con-stantly increases along with the b increase. It likewise appearsobvious that as soon as the threshold b angle is reached, the

0 10 20 30 40 50 60 70 80 90

ββββ (Deg.)

ββββ (Deg.)0 10 20 30 40 50 60 70 80 90

αα αααα αα

wavy surface - A/L = 4%

wavy surface - A/L = 2%

plane surface

Α/λΑ/λ

(a)

0,00

0,10

0,20

0,30

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0,70

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0,90

wavy surface - A/L = 2%

plane surface

Α/λ

(b)

-0,60

-0,40

-0,20

0,00

0,20

0,40

0,60

0,80

1,00

Fig. 7. Flat and wavy a dimensionless surface ratio with regard to the b inclinationangle on the wavy wall at (a) Re = 19.33 and We = 5.43 and (b) Re = 60 andWe = 5.43.

-0.2000

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

0 4000 8000 12000 16000 20000 24000 28000

αα αα

6.4°

10°

20°

30°

60°

80°

90°

x w

Fig. 8. Change of a dimensionless surface ratio with regard to the downstreamnormalized distance at Re = 19.33 and We = 5.43 along various inclined A/k = 2%wavy channel.

704 N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707

dynamics of a film along a flat or wavy wall cease to be altered. Itwould appear that below a threshold angle, waviness plays a sig-nificant role; under similar dynamic conditions (Re = 19.33,We = 5.43) with a stronger substrate waviness parameter (A/k = 4%), a similar trend is observed. For low b angle, film instabilityis significantly affected, with the a surface ratio attaining a maxi-mum level equal to approximately 0.85 for b = 12�; above this an-gle, similar levels are less frequently reached under flat andinclined geometric conditions. The b threshold angle for A/k = 4%is found to be equal to approximately 30�. Although only two wavysubstrate geometries were carried out, b threshold angle was foundto be dependent to A/k. How may the trend be explained? This kindof influence has been noted under low gravity dependence i.e. for blow angles. A given A/k ratio provides a given Db local angle vari-ation and the latter plays a direct and significant role in alterationof film development. In a gravity regime (i.e. for large b angle), thefilm evolves rapidly towards established unstable behavior be-cause its characteristic data are well above critical regime values.In a hydrostatic regime (i.e. for low b angle), the characteristic filmvelocity is lower and can be significantly affected by Db. Fig. 7aclearly exhibits the a change for A/k = 2% and 4%; the b thresholdis increased with A/k (at least in the A/k range studied). A higherwaviness ratio strengthens film instabilities i.e. providing highera levels.

Even though no extensive parametric analyses were carried out,Fig. 7b shows the a surface ratio change for a stronger film regimei.e. for Re = 60. The Weber number was kept at a constant level andthe A/k equal to 2%. As expected, a stronger regime reduced gravityinfluence and as a consequence, the b threshold angle was equal to10�. The a profile (under wavy and flat geometric conditions) iscompletely altered. For the lowest inclination angle, gravity ac-tively favors film instability and the characteristic a levels are

underscored. Once the dynamic of the film is favored, i.e. byincreasing b angle inclination, surface ratio is reduced and con-verges to levels close to 0.30 for b = 90�.

Though significant in comparison to Nusselt thickness, suchvariations remain low with regard to the solid surface. Could theynonetheless exert drastic influence in the overall scope of filmcharacteristics? Park and Nosoko (2003) and Yoshimura et al.(1996) have studied the impact of film deformation on heat andmass transfer from the surface into the film and found that despitea relatively low surface increase due to unstable film development,the variations induce vortex development, which greatly increasesmass and heat transfer. While the surface is growing larger, flowfilm behavior with the liquid film effectively enhances heat andmass transfer.

With this in mind, we studied a average change along the wavywall and plotted out its average value per period (Fig. 8) under awide range of b inclination angles. The a profile is plotted in con-junction with position of the wavy period in the main direction�xw i.e. the surface of the film is averaged per wavy period. In addi-tion, to better depict the a coefficient influence on film behavior,Fig. 9 plots film distribution for 4 b angles (b = 6.4�, 10�, 30� and60�) and along a given spatial location in order to observe the cor-responding film interface colored by local mean velocity; this helpsto correlate the a value change in accordance with film interfacebehavior. The a profile for b = 6.4� is mainly characterized by twoconstant levels. In the first periods ð�xw < 8000Þ, the a ratio is con-stant and negative i.e. the film interface area is lower than the sur-face wall. In that region, a stationary solution is reached; with alow gravity force due to a small inclination angle, the presence ofa ‘‘bump’’ helps to decelerate the film while increasing its height(Fig. 9a-1). The presence of ‘‘bumps’’ behaves as would a barrageto the flow. The drastic change in the a levels corresponds to thedevelopment of several small capillary ripples along the interface(Fig. 9a-2). The small ripples develop first upstream from the cresti.e. in the accelerating zone while more widespread deformationoccurs in the decelerating zone (Fig. 9a-3). A balance in film defor-mation nonetheless rapidly occurs (in space), and no significantchanges are displayed, whatever �xw. The film is then repeatedly al-tered along the waviness wall (Fig. 9a-3 and a-4). At a more pro-nounced inclination angle (b = 10�), the film develops rapidlytowards unstable behavior through ripples arising quickly alongthe wall. Note that the interface shape displayed for b = 10� for6000 � �xw � 7200 (Fig. 9b-2) is similar to the shape observed forb = 6.4� and that it reaches a maximum value for �xw � 8000. For�xw � 22;000 (Fig. 9b-4), interface structure consists of a singlelarge teardrop hump and several small capillary ripples; the tear-drop hump has a steep front, which forms capillary ripples that

η/η N

(x20

)

4000 4200 4400 4600 4800 5000 5200-50

0

50

100(c3)

η/η N

(x20

)

22000 22200 22400 22600 22800 23000 23200-50

0

50

100(c4)

η/η N

(x20

)

0 200 400 600 800 1000 1200-50

0

50

100(c1)

η/η N

(x20

)

2000 2200 2400 2600 2800 3000 3200-50

0

50

100 (c2)wx

wx

wx

wx

η/η N

(x20

)6000 6200 6400 6600 6800 7000 7200

-50

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50

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η/η N

(x20

)

10000 10200 10400 10600 10800 11000 11200-50

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50

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η/η N

(x20

)

12000 12200 12400 12600 12800 13000 13200-50

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50

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η/η N

(x20

)

22000 22200 22400 22600 22800 23000 23200-50

0

50

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wx

wx

wx

wx

β = 6.4°

β = 30°η/

η N(x

20)

2000 2200 2400 2600 2800 3000 3200-50

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50

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η/η N

(x20

)

6000 6200 6400 6600 6800 7000 7200-50

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50

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η /η N

( x20

)

8000 8200 8400 8600 8800 9000 9200-50

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η/η N

(x20

)

22000 22200 22400 22600 22800 23000 23200-50

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wx

wx

wx

wxη/

η N(x

20)

0 200 400 600 800 1000 1200-50

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50

100(d1)

η/η N

(x20

)

1000 1200 1400 1600 1800 2000 2200-50

0

50

100(d2)

η/η N

(x20

)

2000 2200 2400 2600 2800 3000 3200-50

0

50

100(d3)

η/η

N(x

20)

22000 22200 22400 22600 22800 23000 23200-50

0

50

100(d4)

wx

wx

wx

wx

β = 10°

β = 60°

Fig. 9. Normalized film height at specific spatial location at Re = 19.33 and We = 5.43 for various angles (a) b = 6.4�, (b) b = 10�, (c) b = 30� and (d) b = 60�.

N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707 705

exponentially decrease in amplitude. For b = 30�, which corre-sponds to an inclination angle greater than the threshold angleabove which the film behavior along either a wavy wall or a flatwall remains similar, the a coefficient is greatly increased as soonas the film develops and several humps come into being (Fig. 9c-2).It is obvious that downstream from a crest, i.e. where additionalacceleration is given to the film, a higher teardrop hump developsfrom the coalescence of several smaller ripples (Fig. 9c-3 and c-4).Such behavior explains the significant increase in surface area atthe interface. As soon as the film is fully developed, the a ratio re-mains at a virtually constant level and the interface is mainly char-acterized by a single teardrop hump. A similar trend is displayedfor b = 60� and once the film is established, two main humps con-vect, thereby increasing the a ratio (Fig. 9d-4). In the region of

strong interaction between the film and waviness of the wall(b < 30�) the successive trough and crest modulate film instability,damping the latter upstream from the crest and amplifying humpsdownstream from the crest. As is clearly observed in liquid filmbehavior, the waviness of the wall, successively exhibiting acceler-ation and deceleration to the film, modulates its unstable develop-ment. Wavy wall alternatively takes on the roles of wave filter andwave amplifier. As soon as b is strong enough (i.e. with regard tothe film dynamics), gravity becomes predominant or, at the veryleast, presents an order of magnitude higher than the film modula-tion provided by wall waviness. The increase of b inclination angledoes not play a role in film development along a flat or wavy wall.

In order to shed light on time-dependent evolution of the filmalong either a flat and inclined wall or a wavy wall, Fig. 10 shows

t = t0

t = t0 + Δt

t = t0 + 2Δt

t = t0 + 3Δt

t = t0 + 4Δt

η/η

N(x

20

)

16000 16400 16800 17200wx

-50

0

50

100

η/η

N(x

20

)

16000 16400 16800 17200wx

-50

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50

100

η/η

N(x

20

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16000 16400 16800 17200wx

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t = t0

t = t0 + Δt

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t = t0 + 3Δt

t = t0 + 4Δt

η/η

N(x

20

)

16000 16400 16800 17200wx

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η/η

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20

)

16000 16400 16800 17200wx

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16000 16400 16800 17200wx

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η/η

N(x

20

)16000 16400 16800 17200

wx-50

0

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100

η/η

N(x

20

)

16000 16400 16800 17200wx

-50

0

50

100

Fig. 10. Time dependant spatial film height evolution for a 6.4� inclined flat wall and a 6.4� inclined A/k = 2% wavy wall at Re = 19.33 and We = 5.43 – Dt = 20 between eachplot.

706 N.C. Cong, F. Plourde / International Journal of Heat and Fluid Flow 32 (2011) 698–707

the film height profiles colored by the level of mean velocity withregard to t time for b = 10� under similar film dynamic conditions.Each plot is separated by a Dt = 20. For the flat wall, the timechange exhibits a wave-train propagation for which classical filminstability is repeatedly developed. A different feature is observedalong a wavy wall. The main hump is at its maximum height at thetrough for t = to while two main humps are present with smallercapillary ripples ahead. In the deceleration area ð16;500 <�xw < 17;360Þ the two main humps merge, forming a wider butsmaller hump. At the crest, secondary ripples have been reducedin amplitude and some of them coalesce with the wider hump,but smaller ripples still exist. Then, as clearly identified in theacceleration area ð16; 000 < �xw < 16;800Þ, the secondary ripplesare favored and amplified; film profile instability is strengthenedand such evolution is repeatedly developed along the wavy wall.

4. Conclusion

Using a numerical model based on the weighted-residual inte-gral boundary-layer approach, the dynamics of a thin liquid filmhas been presented along an inclined and wavy wall. Our strategyconsisted in attempting to validate our numerical model on waterfilm flowing on a flat and 6.4� inclined wall, i.e. corresponding tothe exhaustive data provided by Liu and Gollub (1994). Experimen-tal data on film characteristics along an inclined wavy wall allowfor both qualitative and quantitative confirmation of the validationstep, even though the model we proposed was limited to first orderdevelopment of the e film parameter.

Dedicated numerical simulations were carried out in order tounderline the role of waviness in the unstable processes. No sys-

tematic analysis was proposed, but our approach shows that a A/k waviness amplitude exists above which film destabilisation isremarkably pronounced. This observation was made for relativelylow inclination angles. Finally, at a given film dynamic and wavi-ness parameter, influence of the inclined wavy wall was put for-ward as a working hypothesis. A threshold angle was exhibitedbelow which film development is significantly altered and the freeinterface area of the film is significantly enhanced. Several wavygeometries and film dynamics were studied and they all confirmthe same trends. Our analysis is naturally but a first step in liquidfilm analysis, and extension toward a 2D surface model would pro-vide a welcome opportunity to better describe film behavior over acomplex corrugated surface.

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vertically falling film. J. Fluid Mech. 250, 433–480.Chang, H.C., Demekhin, E., Kalaidin, E., 1996. Simulation of noise-driven wave

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