Falling liquid films: wave patterns and thermocapillary effects

Transcription

1 Falling liquid films: wave patterns and termocapillary effects Benoit Sceid Cimie-Pysique E.P., Université Libre de Bruxelles C.P. 65/6 Avenue F.D. Roosevelt, Bruxelles - BELGIUM bsceid@ulb.ac.be Abstract Tree-dimensional waves in film flows down a uniformly eated plane are investigated by simulation. For small Reynolds number, regularly spaced rivulets are observed, aligned wit te flow and sustaining two-dimensional waves of larger amplitude and pase speed tan in isotermal conditions. For larger Reynolds number, te picture is similar to te isotermal case and no rivulets appear. Te transition between tese two regimes sows complex cooperative beavior between bot ydrodynamic and termocapillary modes. Key words: Falling film, Hydrodynamic instability, Termocapillary effect Introduction Tree-dimensional (3D) ydrodynamic waves in falling films ave been investigated experimentally by various autors (see for instance [). Experiments by Liu & Gollub [4 and more recently by Park & Nosoko [5 give te clearest picture of te wide penomenology of te interacting 3D waves on isotermal film flows suc as syncronously deformed fronts, subarmonic patterns and orsesoe-like waves. Teoretically, we ave applied a regularization procedure to te weigted integral boundary layer metod providing a two-dimensional modeling tat describe te complex wave patterns observed in experiments up to moderate Reynolds numbers [8. For eated walls, Joo et al. [3 ave modeled rivulet patterns induced by termocapillary effects for small Reynolds number, i.e. using a single evolution equation for te film tickness. However teir simulations experienced finite-time blow-up tat was sown to be intrinsic to te Benney-type equation. Indeed, Ramaswamy et al. [6 ave simulated te full tree-dimensional boundary layer equations and could resolve te wole dynamics of a rivulet formation, from onset to rupture. Neverteless, tey were restricted to a domain size of about one wavelengt in bot streamwise and spanwise directions. We aim in tis paper to propose a two-dimensional model describing te tree-dimensional dynamics of a eated falling film in large-scale domains and to investigate te influence of te Marangoni effect on wave patterns observed in isotermal conditions up to moderate Reynolds number. We will in fact extend to 3D case te model developed by Ruyer-Quil et al. [7 for two-dimensional waves.

2 Matematical formulation of te problem Te flow of a Newtonian liquid down a eated vertical plane is considered. We look for D equations in te streamwise (x) and spanwise (z) coordinates tat mimic te full 3D motion of te fluid. Including te eating of te plane into te regularized tree-field model as detailed in [8 leads to a four-field model independent on te cross-stream coordinate (y), for te film tickness, te streamwise and spanwise flow rates q and p and te temperature field at te interface θ: t = x q z p, (a) [ 9 q δ t q = δ 7 x 7 { [ q 7 xq + δ 8 q z p 7 9 p z q qp z q [ +η 4 q ( x) 9 x q x 3 z p x 43 x p z 73 p xz 6 q xx [ 9 δ t p = δ 7 [ zq z + 3 q ( z ) + 3 p x z 3 q zz xxq + zz q + 7 xzp ( 5 M 4 xθ δ ) 4 q xxθ + 5 } 6 ( xxx + xzz ) ( δ 70 q x + M 5 ) x θ, (b) 56 +η p z 7 p 7 zp 8 7 xp x 4 p ( z) z p z p x q x q z 6 q x p p ( x ) + 3 q x z qp x 5 p 43 z q x 73 q xz 6 p zz 6 6 p xx 5 4 M zθ ( xxz + zzz ), [ ( θ Bθ) 7 δ t θ = 3 + δ Pr 40 ( θ)( xq + z p) 7 + η [( θ 3 )( Pr Bθ ( x ) + ( ) z) +( θ) ( xx + zz ) were te reduced dimensionless numbers are δ = (3Re)/9 Γ /3, B = Bi(3Re) /3, M = + 9 zzp + xx p + 7 xzq (q x θ + p z θ) + x x θ + z z θ (c) + xx θ + zz θ, (d) Ma Γ /3 (3Re) 4/9 and η = (3Re)4/9 Γ /3,

3 based in turn on te usual Reynolds, Biot, Marangoni and Kapitza numbers: Re = g3 N 3ν, αν/3 γ T Bi =, Ma = and Γ = λg/3 ρν 4/3 g /3 σ ρν 4/3 g /3, wit g te gravity acceleration, N te tickness of te uniform film, ν te kinematic viscosity, ρ te density, σ te surface tension, γ = dσ/dt, α te eat transfer coefficient and λ te termal conductivity; Pr = ν/χ is te Prandtl number were χ is te termal diffusivity. Te lengtscales are N and N / η in te wall-normal y and in te in-plane x and z directions, respectively, wile te timescale is ν/g N η. Equation (a) is te mass conservation equation, equations (b,c) express te averaged momentum balances in bot directions x and z, and equation (d) is te energy balance. 3 Computations 3. Small-size domain We first validate our model wit a small-scale simulation investigated by Ramaswamy et al. [6 for small Reynolds numbers, i.e. wen a one-field model can still be valid. Tey demonstrated a mecanism of rivulet formation based solely on instability penomena. Figure sows simulation results, done wit periodic boundary conditions and a simple armonic disturbance of te form (x, z, 0) = + 0. cos(k x x)+ 0. cos(k z z), as initial condition. Te wavenumbers k x and k z ave been cosen to be below tose for te maximal linear growt rate in eac direction, wic are k xmax =0.56 and k z max =0.53. Actually, tese appropriate values allow for interesting secondary flow development. Te initial perturbation creates a troug in te center of te domain (a). Ten, termocapillarity sets in, displacing te fluid from te otter trougs towards te surrounding colder crests. However, te growt rate of te ydrodynamic mode is dominant at initial stage and surface waves develop (b). Te local pase speed being proportional to te square of te local film tickness, te crest travel faster tan te troug (c). In te absence of te mean flow in te spanwise direction, te liquid is displaced laterally due to termocapillarity. Terefore, as time progresses, te tinning of te liquid layer persists and forms a valley surrounded by rivulets aligned wit te flow (d). Tis process is similar to te evolution of a eated tin film on a orizontal substrate. Likewise, it exibits te formation of a secondary rivulet between te main one (e). As found by Boos & Tess [ for orizontal layers, a true cascade of structures takes place in tinner zones (f), precursor to te film rupture. Te last stage before rupture obtained by Ramaswamy et al. wit DNS is at t = 53, wic is in excellent agreement wit Fig.e. However, our simulation runs beyond tis time and exibits finer structures in te film before its rupture (see Fig.g). Te reason wy te full-scale computations wit Navier-Stokes/Fourier equations did not provide suc furter evolution 3

4 (a) t =0 (b) t =5 (c) t = (d) t =90 (e) t = 53 (f) t = 75 Figure : Inception and development of rivulet aligned wit te flow computed wit () arising for Re =/3, Γ = 0, Ma =, Bi = and Pr = 7. Te domain is square of side π/k x wit k x = k z = Te flow direction is from te rigt top towards te left bottom. of te film is likely related to te coice of te number of mes points used in [6. Te autors would ave most probably been able to compute larger time wit refined grid resolution. However, tis would ave been at te expense of computing time, wic demonstrate te great advantage of working wit a model of reduced dimensionality. Indeed, our reduced model involves variables tat are defined only at te film surface and not in te bulk. Actually, te mes points are 3 3 and te time necessary for computing te case of Fig., wit an accuracy of 4, is about one our on a standard desktop computer. Now, let increase te Reynolds number up to Re =, i.e. out of te range of validity of a single evolution equation for te film tickness. Te wavenumbers for te maximal linear growt rate are k xmax 0.6 and k z max 0.34 (approximated values calculated in te long-wave limit), wic yields k x k xmax / wile k z k zmax. Figure sows tat te ydrodynamic mode quickly generates ig amplitude waves (a,b) tat evolves to a solitary-like wave wit preceding capillary ripples (c). However, te termocapillary mode leads te film to rupture before te surface wave saturates. An interesting interaction between te two modes can be pointed out ere: as te termocapillary flow feeds te core of te rivulet, te mean film tickness at te crest increases, and so te local flow rate. Hence, te wave solution 4

5 (a) t =40 (b) t = 400 (c) t = 690 Figure : Same as for Fig. wit Re =. does not saturate and rater follows te cange of te local Reynolds number, by increasing its amplitude and its pase speed. Tis process vanises at t>6 wen te film approaces te wall enoug for te viscous stress to slow down te lateral termocapillary flow. Te ydrodynamic wave and te longitudinal rivulet were found to coexist along a distance tat is about 50 wavelengts ere and tat increases wit Re. 3. Large-size domain Let us turn now to large-scale simulations of a water film at C (Γ = 3375, Pr =7) wit a temperature difference between te vertical wall and te ambient air of 5 C (Ma = 50) and a ig eat transfer coefficient of 00W/cm K(Bi =0.). Te simulations tat follow were started wit a wite noise of amplitude /00 of te flat film tickness. Figure 3 sows again te formation of rivulets due to Marangoni effect: after te development of a parallel wave train (a), drop-like accumulation breaks te D waves into 3D patterns (b,c), precursor of rivulet patterns aligned wit te flow (d,e). As depicted previously for Fig., te liquid ten accumulates into rivulets, wic increases te local Reynolds number and fosters D solitarylike waves of larger amplitude and pase speed tan in isotermal conditions (f). Tis process continues until te rupture of te film, te snapsot of wic is sown in Fig.4a. Similar rivulet pattern, but wit larger wavelengt, is also sown for Re = 5 (b) wile no rivulet develops for still larger Re = (c) were te wave pattern beaves like in isotermal conditions [5. Figure 5 sows finally te pase speed of solitary waves versus Re. For small Reynolds number and small pase speed, i.e. in te drag-gravity regime, inertia effects are small as compared to termocapillary effects wile it is te contrary for large Reynolds number and large pase speed, i.e. in te drag-inertia regime, were inertia is dominant. Te resulting patterns are drastically different (a,c). Te transition between tese two regimes for 4 <Re<6 sows complex cooperative beavior between bot ydrodynamic and termocapillary modes, mostly wen teir 5

6 (a) 0 (b) 500 (c) 700 (d) 900 (e) 0 (f) 500 Figure 3: Water film free surface at different time for Re = for Ma = 50, Bi = 0., Pr = 7 and Γ = Te domain is a square of side π/k x were k x = k z =0.05. Brigt (resp. dark) zones correspond to elevations (resp. depressions). amplitudes are comparable, as illustrated for Re = 5 in Fig.4b. 4 Discussion Te dynamical coupled equations for te film tickness, te surface temperature θ and te streamwise and spanwise flow rates q and p given in () as been sown to be robust and accurate in describing te competition between ydrodynamic waves and termocapillary effect responsible for te rivulet formation in eated falling liquid films. For constant temperature difference across te film, it as been found tat in te drag-gravity regime, quasi-regularly spaced rivulets arise and grow up until rupture (a). Meanwile, te rivulet confines te flow in suc a way tat waves beaves like two-dimensional solitary waves, but of iger flow rate because of te local increase of te Reynolds number. On te contrary, no qualitative influence of te Marangoni effect as been observed in te drag-inertia regime, at least during te 6

7 (a) Re = (b) Re =5 (c) Re = Figure 4: Wave patterns for Ma = 50, Bi =0., Pr = 7 and Γ = (b) (c) c.5 (a) Re Figure 5: Pase speed c of solitary waves versus Re for Ma = 50 (solid line) and Ma = 0 (dased line). Te letters refer to Fig.4. time of te simulations, sowing tat inertia fully dominates te dynamics of te film (c). Finally te transition between tese two regimes is also instructive in terms of rupture wic occurs at only one place and not between all rivulets at te same time. Tis is due to te dispersion of te rivulet size and amplitude, allowing also to observe different kind of waves: D wave trains, D solitary waves and 3D wave structure (b). Having at one s disposal reliable low-dimension model, one is now able to undergo systematic numerical analyzes at low cost (in terms of CPU time) and for large domains. Tis will allow to explore in te wole parameter space te complex 3D interactions between ydrodynamic and termocapillary effects. Notice owever tat, to our knowledge, no experimental data of film flowing down uniformly eated substrates are yet available for comparison purpose. 7

8 Acknowledgment Te autor would like to tank Serafim Kalliadasis and Cristian Ruyer-Quil for fruitful discussions. References [ S.V. Alekseenko, V.E. Nakoryakov, and B.G. Pokusaev. Wave flow in liquid films. Begell House (New York), tird edition, 994. [ W. Boos and A. Tess. Cascade of structures in long-wavelengt marangoni instability. Pys. Fluids, :633, 999. [3 S.W. Joo and S.H. Davis. Instabilities of tree-dimensional teory viscous falling films. J. Fluid Mec., 4:59, 99. [4 J. Liu, J.B. Scneider, and J.P. Gollub. Tree-dimensional instabilities of film flows. Pys. Fluids, 7:55 67, 995. [5 C. D. Park and T. Nosoko. Tree-dimensional wave dynamics on a falling film and associated mass transfer. AICE J., 49():75 77, 03. [6 B. Ramaswamy, S. Krisnamoorty, and S.W. Joo. Tree-dimensional simulation of instabilities and rivulet formation in eated falling films. J. Comp. Pys., 3:70 88, 997. [7 C. Ruyer-Quil, B. Sceid, S. Kalliadasis, M.G. Velarde, and R.K. Zeytounian. Termocapillary long waves in a liquid film flow. part. low dimensional formulation. J. Fluid Mec., 538:99, 05. [8 B. Sceid, C. Ruyer-Quil, and P. Manneville. Wave patterns in film flows: Modelling and tree-dimensional waves. J. Fluid Mec., to appear, 06. 8