Direct Numerical Simulation of Non-Newtonian Droplet Deformation

Transcription

1 PAPER ID ICLASS06-85 Direct Numerical Simulation of Non-Newtonian Droplet Deformation Bowen P.J., Phillips T.N. an Zhen Y. Cariff University Abstract Spectral element methos are evelope for solvin two-phase flow problems of relevance to two-phase power eneration systems (e.. IC an as turbines enines), environmental protection (e.. halon-free explosion suppression) an a rane of non-newtonian fluis. In particular, a irect numerical simulation is presente for the benchmark problem of a Newtonian roplet eformin an acceleratin from rest in a uniform as fiel, an here this DNS moel is extene to inclue a rane of inelastic non-newtonian fluis. The overnin equations for incompressible fluis are conveniently written in terms of four imensionless roups for ease of parametric numerical experiments : Reynols number, Weber number an the viscosity an ensity ratios of the two fluis. The four-parameter Cross viscosity moel is introuce to exten the analysis to non- Newtonian fluis. The arbitrary Laranian-Eulerian (ALE) formulation is use to account for the movement of the mesh. Spectral element approximations are use to ensure hih eree of spatial accuracy. The computational omain is ecompose into two reions, one of which remains fixe in time while the other, locate in the vicinity of the roplet, is allowe to eform within the ALE framework. Transfinite mappin techniques are use to map the physical elements onto the computational element, surface tension is treate implicitly an naturally within the weak formulation an the viscosity moel is introuce explicitly within the numerical formulation. Whereas the avantaes an accuracy of the Spectral-element metho is iscusse in etail elsewhere, here sample results emonstrate the potential of the methooloy for free bounary eformation problems by consierin the influence of Weber number, Reynols number, viscosity ratio an an exemplar non-newtonian moel for bloo on the eformation of the free roplet bounary. Bounary oscillations of the extremities of the eforme roplet about its centre are preicte, which stabilise in the upstream an ownstream irection for hih an low Weber number respectively. It is sueste that this may be an inicator of the onset of breakup. Hiher Reynols number appears to ampen oscillatory behaviour, whilst hih viscosity ratio inhibits eformation, an the non-newtonian bloo moel shows unusual eformation topoloies. Keywors: Droplet eformation, spectral-element, arbitrary Laranian-Eulerian metho, non-newtonian, bloo INTRODUCTION Deformation of flui elements (e.. roplets) in a flowstream of a seconary flui, is a precursor to flui breakup, a funamental process in atomisation. These problems occur in a iverse rane of applications both in inustry an nature, from the efficient utilisation of liqui fuel in IC an as turbine enines, to the effective utilisation of water sprays as Halon-replacement suppressants of acciental explosions, to the eformation of rainrops or the atomisation of bloo particles in criminoloy cases. Fiure shows a hih-spee imae of a roplet eformin prior to break-up in an air-stream. Empirical analyses of the roplet breakup process may be foun elsewhere (, ), with most referrin to the critical Weber number approach. Whilst numerical simulation coes have now been evelope to a stae whereby meaninful preictions may be enerate for inustrial two-phase problems (3), there are still consierable limitations ue to the computational requirements of more common numerical techniques such as finite volume methooloies. This provies the motivation to evelop ever more efficient numerical methos to aress complex, computationally expensive problems such as those consiere in this paper. Fiure. Hih-spee Imae of Droplet Deformation Prior to Breakup in an Air-Flow (left-to-riht) The principle challene in the numerical simulation of the interactions between two immiscible fluis is the ability to track the interface between a flui, which is arbitrarilyshape an translatin. There are two stanar approaches that can be employe for trackin the location of the free surface between the two fluis, viz. the Eulerian an Laranian approaches. However, the shortcomins of both approaches prompte the evelopment of what are known

2 as arbitrary Laranian-Eulerian techniques (4, 5, 6, 7). The methos utilise the avantaes of both the Laranian an Eulerian formulations, whilst at the same time avoiin their rawbacks. The ALE approach may be consiere as a computational reference system that can be chosen quite arbitrarily an which is continually chanin in orer to allow for the precise escription of movin interfaces, ensurin the interity of the mesh over time. In previous numerical simulations (8) a boy force is applie to the roplet to prevent it from acceleratin until it reaches its free-stream velocity. This constraint is not impose in other work (9) nor the present stuy, so that as in practical problems, the roplet is free to translate ownstream whilst eformin. Here, sample results for both the Newtonian an non- Newtonian roplet eformation problem are presente. In sections an 3, the Newtonian an non-newtonian overnin equations are presente. Section 4 summarises the numerical schemes evelope, incluin the final alorithm employe for both fluis. Sample results are presente an iscusse in Section 5, before rawin conclusions in Section 6. GOVERNING NEWTONIAN EQUATIONS It may be shown that the imensionless equations of mass an momentum for the two Newtonian fluis may be written u = 0, u = 0 () u Re + u u = p + T t () u ρ Re + u u = p + T t (3) where the extra-stress tensor for each flui is iven by T =, T = η (4) an the imensionless form of the force balance across the interface becomes : Re ( σ σ ) n = κ n We σ i is the Cauchy-stress tensor for flui i, an κ is the local raius of curvature of the interface which can be written (5) κ = + R R (6) The surface tension force at an interface between two fluis epens on the curvature of the interface. In eneral, for an arbitrarily shape surface the principle raii of curvature can be etermine relatively easily. Let the interface be escribe parametrically by the curve x = x(s), y = y(s), where s is measure alon the interface. Then the raii of curvature for the axi-symmetric surface are iven by: x y y x [( x ) + ( y ) ] 3 / R y ( x ) + ( y ) [ ] / R, = x = (7) where the primes enote ifferentiation with respect to s. Hence, the problem is pose in terms of 4 imensionless numbers of practical relevance, facilitatin inepenent parametric stuies of each roupin, which of course is very ifficult to achieve experimentally. These are the familiar (aseous-base) Reynols number, (aseous base) Weber number, ensity ratio an viscosity ratio of the two fluis. 3 NON-NEWTONIAN MODEL The extension of the roplet/flow moel to inclue non-newtonian effects is particularly of interest iven the rane of practical problems involvin fluis which exhibit non-newtonian characteristics, the potential influence of non-newtonian parameters in roplet eformation an the fact that hitherto they have been iven very limite consieration for this problem. However, the inclusion of non-newtonian effects in numerical moellin is known enerally to increase the ifficulty of obtainin convere numerical solutions for complex flow, an so a raual increase in moel complexity has been aopte. Hence, this first evelopment from the Newtonian moel consiers a flui moel without elasticity, namely a shear-rate epenent viscosity moel. Whilst the simplest non-newtonian viscosity moels comprise two-parameter (e.. power law) or threeparameter (e.. Herschel-Buckley, Sisko) moels, here the 4-parameter Cross viscosity moel (0) is utilise in the interest of enerality. The common two an three parameter moels reference are in fact special cases of the Cross moel. The Cross moel has been utilise previously to escribe a broa rane of non-newtonian fluis, incluin polymer solutions an melts, particulate ispersions, yohurt, soy protein suspensions, hair shampoo, bloo, etc.. The Cross viscosity moel as a function of shear-rate has been utilise in the followin mathematical form: η0 η η = η + (6) m + (K & γ ) where the four parameters comprise η 0 - the zero shear-rate viscosity, η, - the hih shear-rate viscosity asymptote, K an m, where K has imensions of time, an m is imensionless. 4 NUMERICAL METHODS As iscusse earlier, a more sophisticate approach to evaluatin the partial time erivative within a movin ri system is the arbitrary Laranian-Eulerian (ALE) approach. In the ALE formulation, the computational mesh evolves in time an can be chosen quite arbitrarily. Given a

3 reference frame with velocity w, then the partial time erivative can be evaluate by : δ = ( w ) (7) t δt where δ/δt is the time erivative with respect to the movin reference frame. In this moel, the reference frame is taken to be the spectral element ri etermine at each time-step. The Navier-Stokes equations, written in component form, are utilise to erive the weak formulation of the problem (). Care must be exercise in efinin suitable test functions in the ALE formulation for the sake of consistency, since the function spaces are efine on the time-epenent omain Ω(t), an the time erivative will involve the spatial frame of reference. The test functions v must be inepenent of time in the reference system. It may be shown () that the weak formulation can be written in the form: δ δt T ( Bui ) = Di p Ai, jui + Fi + Ci (8) D iui =0 (9) where A is the iscrete iffusion matrix, B is the velocity mass matrix, D i is the iscrete raient matrix, F i contains the surface tension contributions an C i contains the convection terms. The noal co-orinates are upate usin δ X i = δ t w i (0) Temporal iscretisation is performe usin a seconorer scheme. The linear terms are iscretise implicitly usin a secon-orer backwar ifferentiation scheme an the nonlinear terms are iscretise explicitly usin a secon-orer Aams-Bashforth metho. The mesh is move usin : n+ n 3 n n ( X X ) = w w i i i i Δt () In orer to set up the corresponin iscrete variational formulation of the Stokes problem, it is necessary to choose conformin iscrete subspaces of the velocity an pressure spaces. In this paper the problem is iscretise usin the spectral element metho. The omain Ω is partitione into K non-overlappin quarilateral spectral elements Ω k, k =,, K. Each spectral element is mappe onto the parent element D = {(ξ, η) : - ξ, η } () usin a transfinite mappin. For elements ajacent to the interface, a hybri isoparametric transfinite mappin is use. Usin appropriate basis functions () the velocity an pressure may be represente in spectral form as u N Nv NQ ( x) = u ( x), p ( x) = p ψ ( x) φ (3) j j N j= k = an the linear system associate with the iscrete form of the system may be shown to be A N u N + B t N p N = f N (4) B N u N - μc N p N = 0 (5) where entries of the matrices A N, B N an C N are specifie elsewhere (). It follows that the followin equation etermines p N (B N A N - B t N + μc N ) p N = B N A N - f N (6) It can be shown that the preconitione conjuate raient (PCG) metho may be use to solve the system, an was use to enerate the results presente in this paper. Finally, the numerical alorithm utilise for this problems takes the form : () Form an initial mesh aroun an within the roplet. Part of the skeleton spectral element mesh (Ω f ) is fixe, while the remainin portion (Ω m ) is allowe to chane in time. () Start with an initial solution at time t = 0 e.. as an roplet at rest or the velocity fiel etermine by solvin an associate problem for a soli sphere at the same Reynols number. (3) Move from time t n to t n+ by performin the followin steps : (a) Solve for the mesh movement by solvin δx δt n+ i + = w n i k k (7) to etermine the position of the new vertices on the skeleton spectral element mesh corresponin to Ω m. (b) In non-newtonian case, evaluate the viscosity function by first evaluatin the rate-of-strain at time t n an then employin within the Cross moel (c) Upate the matrices A i,j n+, B n+, etc.. () Solve for the velocity an pressure at the new time level usin the preconitione conjuate raient metho. (e) Obtain the new mesh velocity an repeat. 5 RESULTS AND DISCUSSION The spectral-element scheme has been shown to be efficient an accurate (). Here, a sample of the results is presente, emonstratin the versatility of the types of numerical experiments that may be unertaken, an the potential usefulness in evelopin a funamental unerstanin for a rane of practical problems.

4 Fiure shows a typical preicte non-imensionalise time sequence, showin the bounary eformation, translation an velocity manitue of a Newtonian roplet. For this benchmark sequence, Re = 000, We = 5, η / η = 6, ρ /ρ = 550. Fiure. Preicte Droplet Deformation, Translation an Velocity Manitue Contours - Benchmark Newtonian Problem. T = 0.03, 0.5, 0., 0.7, 0.33 First, note that the roplet starts acceleratin with the flow ue to the ra force, an also relatively smoothly starts to eform, reucin its waist, whilst elonatin in the vertical irection. Thouh not shown in these selective lobal imaes, the extremities of the eforme roplet in fact reverses its irection of movement relative to the nominal centre of the movin, eformin roplet, i.e. first moves aainst the flow, before sweepin back to alin itself with the flow for the remainer of the time interval. This may be observe more clearly in Fiure 3, where three cases for ifferin Reynols an Weber number are presente. Here it is note that for low Weber number (We = ), the extremities of the roplet first move away from the centre in the upstream irection, before relaxin an seeminly asymptotically approachin a constant eformation in the upstream irection (i.e. x k < 0). This is associate with the increase influence of surface tension at the lower Weber number. By contrast, for hiher Weber number (We = 5), the roplet extremities aain first move in the upstream irection, before reversin an continuin to eform in the ownstream irection with time, ue to less resistance to chane of shape offere by surface tension forces. For hih Weber number an lower Reynols number, the roplet oscillates urin the ominant perio of elonation in the ownstream irection. The oscillations appear to ampen out for hiher Reynols number as the

5 eformation process becomes more stable as the influence of inertia ominates over viscous forces. It is of interest to consier the influence of lare chanes in flui viscosity, with the contemporary interest of utilisation of veetable oil as a fuel for power eneration applications. Fiure 4 shows the ifference in eformation occurrin at rouhly the same instant in time for the same benchmark problem. All parameters apart from viscosity ratio which is increase by an orer of manitue are kept constant. In this case it is clearly visible that minimal bounary eformation has taken place compare with the corresponin time instant for the benchmark case - the final imae in Fiure. thouh it is appreciate that bloo requires more sophisticate visco-elastic moellin for more realistic preictions in complex flows (3). The benchmark case is aain consiere, but with the four aitional parameters require for the Cross moel taken from () for bloo to be :η 0 = 5 mpa.s, η = 5 mpa.s, K = 5.5 s an m = The eree of shear thinnin is represente by the value of m, where m approachin zero tens towars the Newtonian flui, whilst m approachin inicates the most shear-thinnin fluis. Hence, in this moel bloo is consiere to show sinificant shear thinnin characteristics. Atomisation of bloo has been known to be influential in homicie leal cases, where ifferences in bloo spray at the scene require ifferentiatin in terms of the atomisation source (e.. spray ue to impact, involuntary oral spray release ue releasin bloo collectin in boily air passaes, etc..). Sinificant variation in eformation topoloy from the Newtonian case is visible in Fiure 5, as the roplet now extens laterally in the irection of flow - rather than in orthoonal to the flow. These finins require consierable further investiation, which is onoin. Fiure 3. Evolution of the Lateral Displacement of the position of Maximum Axial Deformation relative to the Droplet Oriin for 3 Cases : η / η = 6, ρ /ρ = 550, (i) Full Line We = 5, Re = 000, (ii) Dash Line We =, Re = 000, (iii) Dot-Dash line We = 5, Re = 00 Fiure 5. Droplet Deformation an Contours of Velocity Manitue for a Bloo Droplet represente by the Shearthinin Moel of Cross (0) 6 CONCLUSIONS Fiure 4. Droplet Deformation an Velocity Manitue Contours of a Liqui Droplet in a Flow at 0x the Viscosity Ratio compare with Benchmark Case, all other parameters constant Finally, consieration is iven to the preiction of eformation of the non-newtonian flui. In terms of practical relevance, the Cross moel flui parameters have been chosen to represent a nominal sample of bloo (), () A new DNS of roplet eformation base on Spectral Element spatial iscretisation couple with an arbitrary Laranian Eulerian (ALE) treatment of the free bounary, has facilitate stuy of a Newtonian an non-newtonian flui roplet translatin an eformin in a secon Newtonian flui meium. () The overnin Newtonian two-flui system of equations has been neatly non-imensionalise to express the problem in terms of four nonimensionalise variables, each of practical relevance, an a 4-parameter non-newtonian viscosity moel has been introuce to further exten the rane of fluis for which the moel is applicable. Hence, it is possible to unertake parametric numerical experiments for each of the variables inepenently.

6 (3) Characteristic eometrical features of the eformin roplet vary with Weber number, whilst Reynols number seems to influence the stability of the eformation, ampenin out oscillatory behaviour. (4) An increase in viscosity ratio minimises roplet eformation, whilst introucin the non-newtonian moel is shown to introuce unusual eformation topoloies such as extension in the irection of flow, which require further stuies an analyses. 7 NOMENCLATURE u i, p i velocity fiel, pressure, for flui i Re, We Reynols number, Weber number Ti, σ i extra, Cauchy stress tensor, for flui i i rate of strain tensor, for flui i κ - local raius of curvature of flui interface R, R - the principal raii of curvature at interface η,η 0,η - viscosity function, zero shear-rate viscosity, hih shear-rate viscosity asymtote K, m - parameters of the Cross moel w - reference frame velocity A i,j,b,c i,d i,f i matrices in weak formulation of overnin equations X i - noal co-orinates of mesh D - parent element in tranfinite mappin operation u N, p N - spectral expansion in terms of basis functions φ j, ψ k - lobal basis functions for spectral expansion u j, p k - components of spectral expansion A N,B N,C N - matrices representin the overnin equations t - non-imensionalise time x k - istance between centre of movin roplet an horizontal position of maximum axial eformation 8. Hellenbrock B.T. an Ewars C.F., A two-flui Spectral Element Metho with Application to Drops, Proc. 8 th ICLASS, Paseena, CA, Den Z-T, Jen S-M, Numerical Simulation of Droplet Deformation in Convective Flows, AIAA Journal, 99, 30, No.5, p Cross M.M, J. Colloi Sci., 965, 0, p Bowen P.J., Phillips T.N. an Zhen Y. The Preiction of Droplet Deformation usin a Spectral ALE Metho, submitte to J. Comput. Phys, 006. Barnes H.A., Hutton J.F. an Walters K. An Introuction to Rheoloy, Elsevier, Amsteram, R. G. Owens, A new microstructure-base constitutive moel for human bloo. J. Non-Newtonian Flui Mech. (006), accepte for publication. 8 REFERENCES. Pilch M. an Erman C.A., Use of Breakup Time Data an Velocity History Data to Preict the Maximum Size of Stable Framents for Acceleration-Inuce Breakup of a Liqui Drop, Int. J. Multiphase Flow, 987, 3, No.6, p Shraiber A.A., Povysotsky A.M. an Dubrovsky V.V., Deformation an Breakup of Drops by Aeroynamic Forces, Atomisation an Sprays, 996, 6, p Comer, M.A., Bowen P.J., Sapsfor S.M. an Kwon S.J., A Parametric Sensitivity Stuy of GDI Spray Characteristics usin a 3D Transient Moel, Int. J. Automotive Technoloy, Hirt C.W., Amsen A.A. an Cook J.L., An Arbitrary Laranian-Eulerian Computin Metho for All Flow Spees, J.Comput. Phys., 974, 4, p Noh W.F., A Time-Depenent Two-Space Dimensional Couple Eulerian-Laranian Coe, In. Aler B., Fernback S. an Rotenber M., eitors, Methos in Computational Physics 3, Acaemic Press, New York, Donea J., Giuliani S. an Halleux J.P., An Arbitrary Laranian-Eulerian Finite Element Metho for Transient Flui-Structure Interactions, Comput. Meth. Appl. Mech. Enr, 98, 33, p Belytschko T., Kenney J.M. an Schoeberie D.F. Quasi-Eulerian Finite Element Formulation for Flui Structure Interaction, ASME J.Pressure Vessel Technol., 980, 0, p.6-69