SINGLE DROP SPLASH ON THIN FILM: MEASUREMENTS OF CROWN CHARACTERISTICS

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1 ILASS-Eurpe 99 SINGLE DROP SPLASH ON THIN FILM: MEASUREMENTS OF CROWN CHARACTERISTICS A. Cghe, G. Brunell*, G.E. Cssali, M. Mareng Plitecnic di Milan-Milan-Italy * TeMPE-CNR, Milan-Italy Università di Bergam-Dalmine-Italy ABSTRACT The impact f a single drp n a liquid film was studied by phtgraphic techniques. A drp generatr capable t prduce single liquid (water drps f millimetric size (with a gd repeatability was psitined at different distances frm the hrizntal wall nt which drps impacted by gravity. The wall was cvered by a liquid layer whse thickness culd be varied. The fluid mtin was studied by means f a CCD camera (µs expsure time frm different psitins. In particular, the use f a transparent (glass wall allwed t study the impact phenmenn frm belw by a prper ptical set-up. Quantitative analysis f the crwn evlutin was then pssible, crwn diameter (internal and external, crwn elevatin and thickness were measured, with different degree f accuracy, as a functin f impact Weber number and nndimensinal film thickness. Number f jets prtruding frm the upper rim was als recrded as well as their average size and grwth rate. The gathered quantitative data allwed a thrugh analysis f the prcess, and sme interesting features f crwn evlutin are reprted tgether with cmparisn with existing mdels. INTRODUCTION When a drp impact vertically n a liquid film a axial-symmetric liquid sheet (crwn grws up and, depending n impinging drp size and velcity and liquid characteristics (like viscsity, density, surface tensin, liquid jets may prtrude frm the crwn and subsequently break-up t frm secndary drplets. Theretical mdels f crwn evlutins have been prpsed (Engel (967, Maclin & Metaxas (976, Yarin and Weiss (995 and recently many numerical simulatin f the phenmenn have been perfrmed (see fr example: Rieber and Frhn (998, Gueyffier and Zaleski (998. Hwever fewer experimental results are available n the crwn characteristics during evlutin (see fr example Mund et al. (995, Cssali et al. (997, despite f the need f data fr mdels evaluatin. The paper presents experimental results n crwn characteristics (crwn diameter, height thickness, number f jets, fr a limited range f drp impact Weber number and film thickness, fr single water drp impingement. Whenever pssible, cmparisns with available theretical mdels were reprted. EXPERIMENTAL SET-UP The experimental set-up cmprised a needle suspended at a distance frm the impact liquid film ranging between 0.3 and 0.9 m and cnnected t a small water tank; a small pressure pulse prduced by the pening f a slenid valve detached single drps frm the needle. The repeatability f the drp diameter (measured frm enlarged pictures f the falling drps was better than 6%; terminal velcities ranged between 2.3m/s and 4.2 m/s. A thin cylindrical rubber r aluminium ring (diameter >50 mm was stuck n a transparent glass windw (thickness <3mm and the inner regin was filled with water t prduce a thin liquid film nt which drps impinged. T maintain relatively cnstant the film thickness after the impact f several drps, a drain was inserted int the wall. The film thickness was measured by phtgraphic enlargement and cmparisn. By using different rings, film thickness ranging between. and 4.3 mm were btained. Pictures f the phenmenn were btained by a CCD camera (PCO Flashcam 752x286: illuminatin was given by a lamp (flash duratin 0 µs cntrlled thrugh a delay circuit triggered by the bscuratin f a laser beam (imaged nt a phtdide and parallel t the film surface caused by the passage f the falling drp. By varying the delay between the trigger signal and the flash, the entire prcess culd be recrded. The chice f a transparent supprt fr the liquid film allwed the bservatin f the splashing phenmenn frm belw, and thrugh, the liquid film: a mirrr was psitined underneath the glass supprt and the CCD camera was fcused n the film surface thrugh the mirrr, this cnfiguratin allwed t measure the inner and uter crwn radius and t cunt the number f jets prtruding frm the crwn rim. The crwn evlutin was bserved als frm the side t measure the crwn height and the jets grwth.

2 RESULTS AND DISCUSSION The experiments were perfrmed maintaining cnstant the impacting drp diameter (D =3.82mm; the drp generatr was psitined t fur different values btaining fur different Weber number (We=ρD V 2 /σ values; fr each impacting velcity three different nndimensinal film thickness (δ=h/d were set: δ=0.29, 0.67, and.3. Pictures f the event were taken at different times after drp impact (setting t=0 the time at which the drp tuches the film, the uncertainty n timing was evaluated t be abut 0.02ms. Pictures were taken, at equal times after impact, using the tw set-up abve described (frm the side and frm the belw the target, in this way the splash event was bserved at the same time frm tw different pint f view, allwing t measure different crwn characteristics. Crwn diameter The crwn evlutin was analysed by measuring the radial extent f the crwn. Sme uncertainty is hidden int the definitin f the crwn diameter, in fact, as it can be bserved in fig 3, the crwn is nt usually cylindrical, and the diameter f the external surface varies with the distance frm the upper surface f the film. Tw diameters where then defined when analysing the pictures taken frm the side: the upper external diameter (D eu and the lwer external diameter (D el, the first measured at the base f the rim, the secnd at the crwn base. The difficulties in defining exactly the psitin where t measure thse diameters intrduces an uncertainty cnnected t the peratr that manually measures (by means f a prper image analysis sftware the parameters. The errr intrduced by the peratr was estimated by evaluating the discrepancies between parameters measured by tw different peratrs n the same set f pictures and it was fund t be f abut 7%. The upper external diameter was fund t be cnsistently smaller than the lwer external diameter and the differences ranges between 4% and 24% f the average diameter (D em =(D eu +D el /2, depending slightly n drp velcity (the larger the velcity, the larger the difference. The radial evlutin f the crwn was als investigated by examining the pictures btained frm belw the target, (see fig b and again uncertainties rise in defining the crwn diameter. As figure b shws, it is always pssible t bserve a ring having larger luminsity which is evidently cnnected t the liquid crwn. A simple simulatin was perfrmed t explain the particular intensity distributin: a crwn prfile like that reprted in figure X6b was used t evaluate, by a ray tracing methd (taking int accunt Snell and Fresnel laws, the intensity distributin f the light falling n the crwn frm the tp and crssing it. the result, shwn in figure a (and fig. 2b, bears a certain resemblance t the measured light intensity (fig b, the thickness f the inner ring appears t be related t the crwn thickness, and it can be cnsidered, apart f a scaling parameter f the rder f unity, a measure f the crwn thickness. Then, frm the picture taken frm belw, it is pssible t define tw mre crwn diameters (see fig b: the inner diameter (D in and the uter diameter (D u, and figure 4 reprts as an example the relatins amng them and D eu, D el. T cmpare the results cming frm different experimental cnditins, the nndimensinal time τ=v t / D (V = impact velcity, D =impinging drp diameter and the nndimensinal crwn diameters x = D x /D will be used. Fllwing Yarin and Weiss (995 the evlutin f the crwn size is suppsed t fllw the relatin: ( τ n ( x = C τ with n=0.5. Hwever, such a relatin is nt suppsed t hld fr all the crwn evlutin, and the attempts t fit the whle experimental data with equ. were nt successful. In rder t find the cnstants (n, C, τ in the fllwing f τ be the relatin describing the evlutin f the nndimensinal crwn prcedure was used. Let ( x = diameter, then it is always pssible t calculate: ( τ τ f ( τ τf ( ( τ f N τ = and ( ( τ τ N F( τ, τ ( τ τ n τ ( τ τ N( τ n n ( τ τ ( τ τ C =. If equatin hlds, the functin Ν(τ becmes: n+ ( τ τ τ ( τ τ n+ n ( τ τ ( τ τ τ F, = dτ and then t define: n = + = na( τ + B( τ + n+ + where fr τ sufficiently large: A(τ and B(τ 0, s that : N(τ n, C(τ C. The experimental data abut crwn diameters were numerically integrated t evaluate F(τ,τ and then N(τ and C(τ; a typical plt f N(τ and C(τ is presented in fig 5. After a time cmparable t D /V bth C(τ and N(τ attains an almst cnstant value, at least fr a reasnable time lag. Fr all the crwn diameters the expnent n is independent n bth drp velcity and film thickness; the cnstant C is independent n film thickness but shws a dependence n the drp impact Weber number, as reprted in fig 6. τ

3 The value f τ was evaluated by a best fitting prcedure (after n and C were evaluated and its value was ranging between 0 and.5. Crwn height Frm the pictures taken frm the side, it was pssible t fllw the crwn height evlutin and fig 8 reprts the nndimensinal crwn height (η c =H/D versus nndimensinal time fr ne f the investigated experimental cnditins. The crwn height reaches a maximum value (η c,max after a certain time (τ max frm the impact, that was calculated frm the available data by using a prper curve fitting. The nndimensinal crwn height d nt depend nly n the nndimensinal time τ, and the maximum crwn height and the time at which such value is attained depend n the drp impact velcity, whereas the influence f the film thickness appears t be very weak. A mre acceptable hypthesis abut the relatin between η c and τ is: η c =f(τ,we,oh, althugh frm the present data it is pssible t analyse the influence f We (thrugh the impact velcity but nt that f Oh. It is als interesting t ntice that the rati: η c,max /τ max is independent f the drp impact velcity as bth η c,max and τ max seem t depend n the impact velcity in a similar manner and figure 7 shws that a pssible dependence is: η c,max =A We n, τ max =A 2 We n with n= Thse findings are als cnfirmed by the evaluatin f the mean nndimensinal vertical crwn velcity, defined as: H& dηc η = = where the brackets stand fr time average, at the very beginning f crwn elevatin which dτ V results independent f the drp Weber number (impact velcity. Als the crwn falling velcity (i.e. the vertical crwn velcity when the crwn is being reabsrbed by the film is independent n Weber number, hwever, bth velcities are slightly dependent n the film thickness (fig 9. Crwn thickness Frm the gathered data it was als pssible t give an estimatin f the crwn thickness evlutin. As abve explained, and as it can be seen frm fig 3, the crwn is nt a cylindrical wall f unifrm thickness, thus it is nt pssible t define in a unique way the crwn thickness. Hwever, frm the estimated crwn diameters (see fig 4 it is pssible t define a nminal crwn thickness as: s c =(D u -D in /2 (i.e. the half f the difference between the uter diameter and the inner diameter, see als fig b. It is then pssible t fllw the evlutin f the nminal crwn thickness as in fig. 0. The mst imprtant feature is the clear grwth with time: the final value f the nminal crwn thickness is 3 t 5 times the initial ne. The grwth is nt unifrm, but it is nt pssible t state whether the measured fluctuatins are actual thickness variatin r pssibly due t the particular definitin f the nminal crwn thickness. Hwever, there nt appears t exist any crrelatin t the film thickness, and als any influence f the impact velcity cannt be recgnised. Jets number and grwth Frm the images taken frm belw the impactr, it was pssible t evaluate the number f jets prtruding frm the crwn rim with gd accuracy. The results shw (except fr the lwest Weber number hw the number f jets decreases with time in the initial perid (τ<8 and then stabilises n a value f abut 0 (see fig., independently f film thickness and drp impact velcity. This behaviur was already pinted ut by previus researchers (Cssali et al. 997 and, as it was suggested, it is prbably due t the merging f the clsest jets during the crwn evlutin. Frm pictures taken frm the side, it was als pssible t evaluate the jets grwth rate (see fig. 3 and the results are reprted in figure, where L j is the jet elngatin frm the crwn rim and Λ j =L j /D is the nndimensinal value. The nndimensinal grwth rate (dλ j /dτ f the prtruding jets is independent f the film thickness and impact velcity, which means that the grwth rate (dl j /dt depends (linearly n the impact velcity. It shuld be nticed that the grwth rate is a measure f the relative velcity between ejected drplets and crwn rim, and the fact that it is much smaller (dλ j /dτ than nndimensinal crwn radial (ν r and rim vertical (ν v velcity implies that the ejected drp velcities wuld be very clsely represented by the vectr (ν r,ν r. CONCLUSIONS The analysis f the crwn evlutin generated by liquid drps impacting n a thin liquid film revealed the fllwing: x = C τ τ - The crwn hrizntal size ( diameter fllw a lw f the frm: ( n, (at least fr τ 5 with n 0.4 independent n drp velcity and film thickness, whereas C increases with Weber number. - The nndimensinal crwn height reaches a maximum after a nndimensinal time depending, as the nndimensinal crwn height, n the impact Weber number. - The grwing and receding nndimensinal crwn vertical velcities depend slightly n the film thickness but nt n drp Weber number. - The nndimensinal crwn thickness increases with time frm 3 t 5 times the initial value, independent f film thickness and impact velcity. - The number f jets prtruding frm the crwn rim decreases with time reaching, fr τ>8, a srt f asympttic value f abut 0.

ACKNOWLEDGEMENTS The experimental wrk was perfrmed at TEMPE-CNR labratries in Milan; the authrs wuld like t thank Ms. Michaela Olivetti and Ms. Mnica Selenu fr their valuable wrk during experiments.

L., Weiss D.A. (995: Impact f drps n slid surfaces: self-similar capillary waves, and splashing as a new type f kinematic discntinuity, J.Fluid Mech., 283, 4 G.E. Cssali, A. Cghe, M.Mareng.

4 ACKNOWLEDGEMENTS The experimental wrk was perfrmed at TEMPE-CNR labratries in Milan; the authrs wuld like t thank Ms. Michaela Olivetti and Ms. Mnica Selenu fr their valuable wrk during experiments. REFERENCES Mund Chr., Smmerfeld M., Trpea C. (995: Drplet-Wall Cllisins: Experimental Studies f the Defrmatin and Breakup Prcess, Int. J. Multiphase Flw, 2, 5 Yarin A.L., Weiss D.A. (995: Impact f drps n slid surfaces: self-similar capillary waves, and splashing as a new type f kinematic discntinuity, J.Fluid Mech., 283, 4 G.E. Cssali, A. Cghe, M.Mareng. (997, The impact f a single drp n a wetted slid surface ; Experiments in Fluids 22, Rieber M., Frhn A.; (998 Numerical simulatin f splashing drps ILASS-Eurpe Gueyffier D., Zaleski S.; (998; Full Navier Stkes simulatins f drplet impacts n thin liquid films. 3 rd Int. Cnf. n Multiphase Flw ICFM 98, Lyn, France, June 8-2. Macklin W.C., Metaxas G.J. (976: Splashing f drps n liquid layers, J. f Appl. Phys., 47, 3963 Engel O.G. (967: Initial pressure, Initial flw velcity and the time dependence f crater depth in fluid impact, J. f Appl. Phys., 38, 3935 FIGURES (a (b Fig. Simulated a and measured (b light distributin under a liquid crwn.

5 Fig. 2 Simulated linear light intensity distributin and crwn prfile Fig. 3 Crwn seen frm the side. Fig. 4 Crwn diameters time evlutin (an example Fig. 5 Typical plt f N(τ and C(τ. Fig. 6 Cnstant C (equatin ( vs. drp Weber number Fig. 7 Maximum nndimensinal crwn height and crrespnding nndimensinal time vs drp impact velcity.

6 Fig. 8 Nndimensinal crwn height vs nndimensinal time Fig. 9. Vertical crwn velcities at the early ( grwing stage f crwn evlutin. Fig. 0 Nndimensinal nminal crwn thickness vs nndimensinal time Fig. Number f jets vs nndimensinal time Fig. 2 Nndimensinal jet elngatin vs nndimensinal time

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